Born-Infeld inspired modifications of gravity
Jose Beltran Jimenez, Lavinia Heisenberg, Gonzalo J. Olmo, Diego Rubiera-Garcia
aa r X i v : . [ g r- q c ] M a y Born-Infeld inspired modifications of gravity
Jose Beltr´an Jim´enez
Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, Marseille, France.
Lavinia Heisenberg
Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich, Switzerland.
Gonzalo J. Olmo
Depto. de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Burjassot-46100, Valencia, Spain.Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-900 Jo˜ao Pessoa, Para´ıba, Brazil.
Diego Rubiera-Garcia
Instituto de Astrof´ısica e Ciencias do Espa¸co, Universidade de Lisboa,Faculdade de Ciencias, Campo Grande, PT1749-016 Lisboa, Portugal.
Abstract
General Relativity has shown an outstanding observational success in the scales where ithas been directly tested. However, modifications have been intensively explored in theregimes where it seems either incomplete or signals its own limit of validity. In particular,the breakdown of unitarity near the Planck scale strongly suggests that General Relativityneeds to be modified at high energies and quantum gravity effects are expected to beimportant. This is related to the existence of spacetime singularities when the solutionsof General Relativity are extrapolated to regimes where curvatures are large. In thissense, Born-Infeld inspired modifications of gravity have shown an extraordinary abilityto regularise the gravitational dynamics, leading to non-singular cosmologies and regularblack hole spacetimes in a very robust manner and without resorting to quantum gravityeffects. This has boosted the interest in these theories in applications to stellar structure,compact objects, inflationary scenarios, cosmological singularities, and black hole andwormhole physics, among others. We review the motivations, various formulations, andmain results achieved within these theories, including their observational viability, andprovide an overview of current open problems and future research opportunities.
Keywords:
Born-Infeld gravity, Astrophysics, Black Holes, Cosmology, Early universe,Compact objects, Singularities
Email addresses: [email protected] (Jose Beltr´an Jim´enez), [email protected] (Lavinia Heisenberg), [email protected] (Gonzalo J. Olmo), [email protected] (Diego Rubiera-Garcia)
Preprint submitted to Physics Reports May 18, 2017 ontents1 Preamble 5 f -mode and I -Love- Q . . . . . . . . . . . . . . 773.3.6 Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813.4 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832 Black Holes 84 f ( R ) gravity . . . . . . . . . . . . . . . . . . . . . . . . 1755.5.2 Ricci scalar in the determinant . . . . . . . . . . . . . . . . . . . . . 1795.6 Other extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1805.6.1 Gravity coupled to Born-Infeld . . . . . . . . . . . . . . . . . . . . . 1805.6.2 Teleparallel inspired Born-Infeld . . . . . . . . . . . . . . . . . . . . 1835.6.3 Born-Infeld in Weitzenb¨ock space-time . . . . . . . . . . . . . . . . . 1855.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1883 Concluding remarks, open questions and prospects 190Acknowledgments 196References 196 . Preamble General Relativity (GR) is nowadays firmly established as the standard theory to de-scribe the gravitational interaction with the same mathematical framework and physicalprinciples as those used by Einstein more than one hundred years ago. After all thistime, it still stands out as the most successful theory able to explain all the gravitationalphenomena in a wide range of scales. Direct tests comprise from sub-milimeter to SolarSystem scales, where the Parameterised Post-Newtonian formalism has allowed to con-strain deviations from GR in the weak field limit at the level of ∼ − [364]. Moreover,the amazing direct observation of gravitational waves by the LIGO collaboration is alsocompatible with the prediction of GR for the merging of two black holes, where strongfield effects are relevant [2, 1]. On the other hand, we have witnessed how the accuratemeasurements of the CMB anisotropies and galaxy surveys have established ΛCDM asthe standard model of cosmology, which is based on a homogeneous and isotropic Uni-verse governed by GR as the theoretical framework for gravity. This picture requires anunobserved cold dark matter source plus a tiny cosmological constant to account for thecurrent accelerated expansion of the Universe. Furthermore, the ΛCDM model needs tobe supplemented with the inflationary paradigm so that the primordial perturbations aregenerated during a short period of accelerated expansion at very early times. For a re-view on the current status of the ΛCDM model, its challenges and possible alternatives,see Bull et al. [89]. Further observational tests, for instance via the Euclid satellite [16],will hopefully shed light on all the additional elements above and their contributions tofundamental physics.Despite its observational success, there are strong arguments supporting and/or mo-tivating to seek for theories beyond GR. These arguments are of two kinds. On thetheoretical side, GR itself predicts the unavoidable existence of spacetime singularities,i.e., events where our ability to make predictions comes to an end [326]. Such singularitiesare unavoidably developed during the gravitational collapse of a fuel-exhausted star toform a black hole [223], as well as during the cosmological evolution in the early Universe.In this sense, the requirement that “nothing should cease to exist suddenly” and that“nothing should emerge out of nowhere” should be seen as basic consistency conditionsfor any physical theory, including GR. The existence of singularities in GR unavoidablyleads to the breakdown of these conditions, and gives clear indications that we have pushedthe theory beyond its regime of validity. According to the standard lore [90], GR is a goodeffective field theory up to a scale somewhere near the Planck mass and, therefore, thosesingular behaviours are regarded as manifestations that the higher order operators shouldbe included. For this reason, quantum gravity is usually expected to regularise such sin-gularities, although it is possible that high energy modifications of GR might allow toclassically regularise some of those singularities before reaching the cut-off of the theorywithout invoking any quantum gravity effects.On the phenomenological side, the unprecedented experimental precision reached byobservational cosmology requires the aforementioned ad hoc extra ingredients in order toaccount for the observations. While the cosmological constant is fundamental part of the5heory and its difficulty resides in its aesthetic value that poses naturalness problem, darkmatter and inflation require the introduction of new physics and, as a consequence, a largedegeneracy among all the proposed models. This degeneracy is more prominent owing tothe lack of experimental signatures from laboratory experiments and particle accelerators,despite the existence of different ongoing galactic [342, 5], cosmic rays [23], CMB [7],collider [100] and underground laboratory [9] searches.In view of the above situation, one may wonder if the difficulties and lack of naturalnessfaced in GR indicates that a new framework to describe gravity is needed, which wouldyield different astrophysical and cosmological observational signatures from the ΛCDMmodel [224]. From a conservative perspective, one may stick to the point of view thatgravitation is a manifestation of the curvature of spacetime, but one that is not suffi-ciently well described by GR. As a matter of fact, the common factor to all the issuesdiscussed above is the extrapolation of GR to regions where it has not been directly welltested and this may introduce significant bias in the interpretation of astrophysical andcosmological observations. The consideration of additional curvature contributions to theEinstein-Hilbert action, usually under the form of curvature invariants, has been used inthe literature as a way to enlarge the phenomenology of gravity. This typically involves anumber of problems such as higher-order field equations, which usually entail the presenceof ghost-like instabilities [341, 340, 266, 107], or the difficulty to make these models com-patible with solar system tests due to the existence of new degrees of freedom [270, 108] .The arbitrariness in the choice of curvature invariants also implies a strong lack of natu-ralness in these models. The main references regarding such models and their applicationsare provided by de Felice and Tsujikawa [130], Capozziello and de Laurentis [95], andNojiri and Odintsov [261] (see also Faraoni and Sotiriou [338]).The difficulties with ghost-like instabilities in higher curvature modifications of gravitycan be avoided by formulating those theories in the so-called Palatini or metric-affineformalism [272]. Though this approach is sometimes viewed as a shortcut to obtain thefield equations of GR (and rightly so for some specific Lagrangians), it actually representsan inequivalent formulation of gravity in which metric and affine structures are regarded asindependent geometrical entities. The fact that, when formulated `a la Palatini [163], metricand connection are compatible in the case of GR has spread the view that such conditionshould always hold regardless of the form of the gravity Lagrangian. However, this is nottrue in general. In the metric-affine approach, the specific relation between metric andconnection is determined by the field equations, not imposed a priori by mathematicalconventions. In fact, whether the affine connection is determined by the metric degreesof freedom or not is as fundamental a question as the number of spacetime dimensions orthe existence of supersymmetry.The metric-affine or Palatini approach, therefore, avoids the problems with ghoststhat affect extensions of GR in the usual metric formulation. In vacuum configurations,the field equations of these theories boil down to Einstein’s equations with an effectivecosmological constant [164] which, apparently, supports their compatibility with orbital Those models avoiding these shortcomings and, at the same time, being able to provide a consistentcosmological expansion which is coherent with the GR limit are usually termed as viable , see e.g. [15, 121,131]. ? ], with the result that the divergence of the self-energy of a point-like chargeis regularised. This type of high-energy modification is analogous to the transformationthat leads from a free particle in Newtonian mechanics to a free relativistic particle,whose maximum speed is bounded by the speed of light. The same Lagrangian structuredescribes the electromagnetic fields of p -branes in string theories [180, 86, 94]. It is naturalto wonder whether such an approach, now fully defined in terms of geometrical objects,could play a similar role in order to avoid divergences and spacetime singularities in thehigh-energy/curvature regime and, accordingly, different proposals have been consideredin the literature. Indeed, a major reason for the investigation of such models is the factthat, using standard matter sources satisfying the energy conditions, they naturally leadto non-singular cosmologies, inflationary scenarios without the need for scalar fields, andblack hole spacetimes without singularities, among other appealing results. Moreover, thephysics of these gravity theories has been studied in numerous astrophysical, black holeand cosmological scenarios where high-energy physics is relevant.In this work we shall refer to this kind of models, which are close to the original spiritof Born-Infeld electrodynamics, as Born-Infeld inspired modifications of gravity . They aredefined by the following basic principles: • Square-root form : Some geometric object(s) appears under a square-root with a de-terminantal structure in the action which defines the gravitational theory, alongsidewith some new mass/length scale. • Consistency : No obvious pathologies are present, among which the absence of ghost-like instabilities is of utmost importance. In turn, this almost unavoidably enforcesthe use of a metric-affine formulation. • High-energy modification : The modifications of GR mostly occur in the ultravioletregime, i.e., in regions of large mass/curvature or short scales. This implies that GRis recovered in the low-energy limit.Nonetheless, as there are available proposals in the literature for these theories thatrun away from one (or both) of the two last requirements, for completeness of this workwe shall also discuss such proposals. A more precise description and classification of suchtheories will be presented in section 2, alongside a criticism of each of them.This review is intended to fill a gap in the recent literature of Born-Infeld inspiredmodifications of gravity by providing a comprehensible account of the many different sce-narios on which these classes of theories have been considered, including the astrophysicsand internal structure of compact objects, solar physics constraints, modifications on blackhole structure, non-singular black holes and wormholes, early universe and bouncing so-lutions, inflation, and dark energy, among others. Its aim is to summarise, classify and7nify the different theoretical approaches, to clarify the assumptions on which the differentapproaches to build the theory are formulated, to discuss the numerous theoretical andphenomenological results, to highlight the experimental constraints these theories are sub-jected to, to clarify some existing misunderstandings, and to provide an overview of thefuture research opportunities. It is designed to be useful both for pure theorists and forastrophysicists/cosmologists working on alternatives to the ΛCDM (plus inflation) model.For a review on modified gravity in cosmology mainly focused on infrarred modifica-tions of gravity in connection with late-time solutions (but with little contact with Born-Infeld-inspired theories or the Palatini formalism), see instead Clifton et al. [120]. Foradditional astrophysical and cosmological observational constraints over different modifiedtheories of gravity deviating from GR predictions, see Berti et al. [67].
The main content of this review is split in four sections, according to the context onwhich Born-Infeld-inspired theories of gravity have been investigated.In section 2 we will briefly review the original Born-Infeld electrodynamics theory fromwhich the motivation for analogue constructions within gravity emerges. After explainingthe early attempts that resulted in pathological theories, we will introduce what representsthe most extensively studied theory of gravity with the Born-Infeld structure. The slightlydifferent formulations of such a theory will be discussed as well as the main equations.Along the way, we will spend some time discussing the two frames existing in these theoriesand clarify the physical meaning of the different geometrical objects arising in them. Wewill end this section with a survey on the different Born-Infeld inspired theories of gravityexisting in the literature and we will provide a general mathematical framework for thesetheories. The general developments introduced in this section will serve as starting pointsfor the practical applications discussed in the subsequent sections.In section 3 some attempts to place observational constraints on the Born-Infeld theoryusing stellar models are reviewed. We will make special emphasis on the central role playedby the energy density in the modified dynamics of this theory, which affects in a nontrivialway the mass-radius relation and maximum mass limit of compact objects, the energytransport mechanisms and oscillation frequencies of stars, the intensity of neutrino fluxesfrom the Sun, . . . providing numerous tests to confront the theory with observations. Theneed for a careful description of the outermost layers of compact objects is also discussedin detail, considering for this purpose some relevant examples in which the peculiaritiesof metric-affine theories demand additional modeling beyond the canonical approaches ofGR.In section 4 we will review the counterparts of the Schwarzschild and Reissner-Nordstr¨omblack hole solutions of GR, where a coupling to a Maxwell field is considered. We willspend some time explaining the procedure for derivation of the corresponding solutions,so as to highlight some important subtleties. Then we will explain the main differences ofsuch solutions as compared to the GR ones, in particular, regarding the modifications onthe horizon structure, which bear some resemblance to that of black holes supported byBorn-Infeld electrodynamics in GR. On the other hand, we will study how these black holesmay affect the description of strong gravitational lensing as well as the physics regard-8ng mass inflation. An important issue will be the existence of non-singular geometriesin these theories, whose nature and properties is tested using different well-establishedcriteria. We also review some wormhole solutions constructed out of anisotropic fluids.Finally, different extensions to higher and lower dimensions, as well as to magneticallycharged solutions will be discussed.The section 5 will be devoted to the effects of Born-Infeld inspired theories of gravityin cosmological scenarios. We will discuss the existence of homogeneous and isotropicsolutions free from Big Bang singularities with standard matter sources as well as cou-plings of these theories to other types of fields. Anisotropic models and inhomogeneousperturbations will also be discussed. Since the Born-Infeld inspired theories are designedto modify gravity in the high curvatures regime, their natural domain of applicability isthe early universe. However, there have also been studies where Born-Infeld theories areconsidered for late time cosmology and we will revisit them.We will end in section 6 by giving a summary of all the material presented in the coreof this review. We will discuss the most outstanding achievements and will make specialemphasis on the open questions that remain as well as the prospects for future researchwithin the field.
In this section we will review some basic ingredients of differential geometry that wewill use throughout the different parts of this review. We will assume that the reader isfamiliarized with the concepts presented here and the main purpose of this section will be tofix the notation and the conventions for the different choices of signs and numerical factorsin the definitions of relevant geometrical objects. It does not intend to be an exhaustiveand rigorous exposition, but rather it should be regarded as a brief compendium of usefulconcepts and formulae. For a more detailed treatment we urge the reader to consult her/hisfavourite book on differential geometry or General Relativity or, in the lack thereof, seee.g. [325, 251, 359]. One reference particularly useful and with numerous applications ingravitation and gauge theories is [148].
Connection, curvature and torsion conventions
The theories that will be considered throughout the present review will be formulatedeither in (pseudo-)Riemannian or non-Riemannian geometries. In order to construct thenecessary geometrical framework, we first introduce a 4-dimensional manifold M that willeventually constitute our spacetime. In that spacetime we introduce a general connectionΓ that defines the covariant derivative of a 1-form A µ as ∇ µ A ν = ∂ µ A ν − Γ λµν A λ . (1.1)This definition results in the following covariant derivative for a vector field A µ : ∇ µ A ν = ∂ µ A ν + Γ νµλ A λ . (1.2)These expressions can then be easily generalised to arbitrary tensors T µ ··· µ p ν ··· ν q so that ∇ α T µ ··· µ p ν ··· ν q = ∂ α T µ ··· µ p ν ··· ν q − Γ λαν T µ ··· µ p λν ··· ν q − · · · − Γ λαν q T µ ··· µ p ν ··· ν q − λ + Γ µ αλ T λµ ··· µ p ν ··· ν q + · · · + Γ µ p αλ T µ ··· µ p − λν ··· ν q . (1.3)9n addition to objects with tensorial transformation properties under changes of co-ordinates, we will also find objects with other transformation properties throughout thisreview. In particular, we will encounter vector densities, which pick up some power ofthe Jacobian under a change of coordinates. If A µ is a vector density of weight w , ittransforms as ˜ A µ = (cid:18) det ∂ ˜ x α ∂x β (cid:19) w ∂ ˜ x µ ∂x ν A ν . (1.4)This modified transformation property makes necessary to add a piece to the definition ofthe covariant derivative to maintain its tensorial character, that reads ∇ µ A ν = ∂ µ A ν + Γ νµλ A λ + w Γ λµλ A ν . (1.5)Again, this formula can be generalized for an arbitrary tensorial density T µ ··· µ p ν ··· ν q byadding a term w Γ λαλ T µ ··· µ p ν ··· ν q in (1.3).After introducing the connection, we can start computing geometrical objects fromthe commutator of covariant derivatives acting on different tensorial fields. The firstcommutator we can compute is that of two covariant derivatives acting on a scalar field,which reads (cid:2) ∇ µ , ∇ ν (cid:3) φ = −T λµν ∂ λ φ (1.6)with T λµν ≡ Γ λµν − Γ λνµ (1.7)the torsion tensor. Let us notice that it has tensorial transformation properties because itcan be seen as the difference of two connections. The next geometrical important objectis obtained by computing the commutator of two covariant derivatives acting on a vectorfield, which can be written as (cid:2) ∇ µ , ∇ ν (cid:3) A α = R αβµν A β − T λµν ∇ λ A α (1.8)where we have introduced the curvature Riemann tensor, defined as R αβµν ≡ ∂ µ Γ ανβ − ∂ ν Γ αµβ + Γ αµλ Γ λνβ − Γ ανλ Γ λµβ (1.9)Out of this general Riemann tensor, we can build two independent traces, namely the Riccitensor defined as usual R µν = R αµαν and the homothetic tensor given by Q µν = R ααµν .While the Ricci tensor does not have any symmetry (even for a torsion-free connection),the homothetic tensor is antisymmetric. A quantity that we will need to compute fieldequations is the variation of the Ricci tensor under an infinitesimal displacement of theconnection Γ → ¯Γ + δ Γ, which reads δ R µν = ¯ ∇ λ δ Γ λνµ − ¯ ∇ ν δ Γ λλµ + ¯ T λρν δ Γ ρλµ (1.10)where the bars denote quantities corresponding to the background connection ¯Γ. Thisrelation reduces to the usual Ricci identity for torsion-free connections. This is true for true tensorial densities. For pseudo-tensorial densities the transformation also picksup a sign for parity odd transformations. etric convention After setting-up the notation and convention for the objects directly related to theconnection, we will turn to the conventions for the metric tensor g µν . This object isassumed to be non-degenerate and its inverse is denoted with upper indices g µν so that g µα g αν = δ µν and so on. Furthermore, this object is used to raise and lower indicesof arbitrary tensors (i.e. it establishes an isomorphism between the tangent and the co-tangent spaces). We will use the mostly plus signature for the metric so that the Minkowskimetric is η µν = diag( − , + , + , +). The covariant derivative of the metric defines the non-metricity tensor Q αµν as ∇ α g µν = Q αµν . (1.11)Notice that the non-metricity is symmetric in the last two indices. This expression can besolved in the usual way to write the connection asΓ αµν = 12 g αλ (cid:16) ∂ ν g µλ + ∂ µ g λν − ∂ λ g µν (cid:17) + L αµν ( Q ) + K αµν ( T ) (1.12)where the first term is the standard Levi-Civita piece, the second term depends on thenon-metricity and the last term (usually called contorsion) is determined by the torsion.If the non-metricity vanishes and the connection is symmetric (i.e. vanishing torsion), theconnection reduces to the Levi-Civita connection given by the Christoffel symbols. Witha metric at hand, there is yet a third rank-2 tensor we can construct from the Riemanntensor of the full connection, known as co-Ricci tensor and defined as P αµ ≡ g βν R αβµν .Of course, for the Levi-Civita connection all three objects coincide up to a sign so theonly independent trace of the Riemann is the Ricci tensor R µν . Throughout this reviewwe will denote with calligraphic letters R µν , ... the objects corresponding to an arbitraryconnection, while the curvature objects associated to the Levi-Civita connection will bedenoted with normal characters R µν , ... The determinant of the metric det g µν ≡ g is a tensorial density of weight − √− g is a tensorial density of weight − ∇ µ √− g = ∂ µ √− g − Γ λµλ √− g. (1.13)We can thus use √− g to tensorialize tensorial densities. For instance, if A µ is a tensorialdensity of weight w , then A µ ≡ ( √− g ) w A µ has weight zero. Another important use of thisobject is to construct invariant volume elements. Since d V generates a Jacobian undera change of coordinates, we can compensate for that by adding a factor of √− g so that √− g d V will be invariant. Let us notice that this is a choice and actually we could use ϕ d V with ϕ being whatever scalar density of weight −
1. For instance, p det a µν with a µν being an arbitrary rank 2 tensor will do the job.The totally antisymmetric tensor is defined as ε µνρσ = √− g (cid:2) µνρσ (cid:3) (1.14)with (cid:2) µνρσ (cid:3) the totally antisymmetric Levi-Civita symbol with [0123] = 1. The con-travariant version of it is ε µ µ µ µ = g µ ν g µ ν g µ ν g µ ν ε ν ν ν ν = − √− g (cid:2) µ µ µ µ (cid:3) . (1.15)11he Levi-Civita tensor allows to introduce the Hodge dual that establishes an isomor-phism between p -forms and ( D − p )-forms. If F µ ··· µ p is a p -form, its dual is defined as˜ F µ ··· µ D − p = 1 p ! ǫ µ ··· µ D − p ν ··· ν p F ν ··· ν p . (1.16)As a specific example that we will use throughout the review, the dual of a 2-form F µν infour dimensions is given by ˜ F µν = 12 ǫ µναβ F αβ . (1.17)For an antisymmetric rank 2 tensor we can introduce the so-called electric E µ and magnetic B µ components relative to an observer with 4-velocity u µ as E µ = F µν u ν and B µ = ˜ F µν u ν . (1.18)For an observer with u µ = (1 ,~
0) these definitions reduce to the usual expressions F i = E i and F ij = ǫ ijk B k . Tetrads formulation
An alternative language to describe the geometrical framework of gravity theories isprovided by the formalism of frames. We start by introducing a set of vectors defined onthe tangent space e a = e aµ ∂ µ with a Lorentz index a so that they satisfy the followingorthonormality condition e aµ e bν g µν = η ab . (1.19)with respect to the Minkowski metric η ab . These objects receive several aliases in the liter-ature: tetrads , vierbein or frames . The corresponding dual objects e a = e aµ d x µ belongingto the cotangent space are defined in the usual way e aµ e bµ = δ ab . This relation in turnsalso implies e aµ e aν = δ νµ . They are sometimes interpreted as the square root of the metricbecause g µν can be expressed as e aµ e bν η ab = g µν . (1.20)The vierbein can be used to transform tangent space indices into spacetime indices forarbitrary tensors. All the geometrical objects introduced above thus have their corre-sponding object in the tetrads formulation. If we introduce the so-called spin connectiongiven by the set of 1-forms ω aµ b , the associated curvature 2-form is given by R ab = d ω ab + ω am ∧ ω mb (1.21)where d is the exterior derivative and ∧ stands for the exterior product. The existence ofthe tetrad allows to define the torsion 2-form as T a = d e a + ω ab ∧ e b . (1.22) Let us remember that a form is nothing but a completely antisymmetric tensor. T a + ω ab ∧ T b = R ab ∧ e b (1.23)that relates all the relevant objects, namely, the tetrads, the spin connection, the torsionand the curvature. Taking a second exterior derivative of this expression will yield theusual Bianchi identities, which we do not need to display here. Instead, let us focus ontwo special connections that will be of relevance for this review. The first one is defined bythe condition of being torsion-free, so it is defined by d e a + ω ab ∧ e b = R ab ∧ e b = 0 and itis the relevant one for the usual formulation of General Relativity. The second connectionis curvature-free so we have d ω ab + ω am ∧ ω mb = d T a + ω ab ∧ T b = 0 and defines theso-called Weitzenb¨ock space. This is the natural place for the Teleparallel formulation ofGR. Energy conditions
A perfect fluid can be defined as one in which the energy-momentum tensor is locallyseen as isotropic and it is fully determined by its density ρ and its pressure p . Accordingto this definition, the energy-momentum tensor of a perfect fluid as seen by an observerwith 4-velocity u µ ( u = −
1) is given by. T µν = ( ρ + p ) u µ u ν + pg µν (1.24)where it is immediate to see that ρ = T µν u µ u ν and p = ( g µν + u µ u ν ) T µν . For a comovingobserver with u µ ∝ ∂ t we have that T = − ρ and T ij = pδ ij .For a general energy-momentum tensor, there is a set of conditions known as energyconditions that play an important role in theories of gravity in relation with singularitytheorems, instabilities, superluminal propagation or entropy bounds. In the following welist them for future reference: • Weak Energy Condition (WEC). This condition states that T µν v µ v ν ≥ v µ ( v < ρ ≥ ρ + p ≥ • Dominant Energy Condition (DEC). This condition is satisfied if T µν w µ w ν ≥ w µ ( w ≤
0) and − T µν w ν is a future-oriented causal vector. Fora perfect fluid, this condition translates into ρ ≥ | p | . • Strong Energy Condition (SEC). The SEC is satisfied if T µν v µ v ν ≥ − T for everytime-like vector v µ ( v < ρ + p ≥ ρ + 3 p ≥ • Null Energy Condition (NEC). The NEC is satisfied if for any null vector n µ ( n =0) the condition T µν n µ n ν ≥ ρ + p ≥ atrix notation Given a rank-2 tensor, we will often use a hat to denote the corresponding matrix.Thus, the metric tensor g µν will also appear as ˆ g and its inverse g µν will be denoted byˆ g − and similarly for other objects. The determinant of a matrix ˆ M will be explicitlyspelled out as det( ˆ M ) or will be alternatively denoted as | ˆ M | where no confusion withabsolute value should occur. In the special case of a metric g µν , we will alternatively usethe broadly used notation g for its determinant. Analogously, for the trace of a matrixwe will use either the explicit notation Tr( ˆ M ) or the more compact notation [ ˆ M ] where,again, the context should clarify when the square brackets stand for the trace or simplyplay the role of actual brackets.A recurrent matrix formula that we will use throughout this review is the expansionvalid for an arbitrary n × n matrix ˆ M given bydet (cid:16) + ˆ M (cid:17) = n X i =0 e i ( ˆ M ) (1.25)where is the n × n identity and e i the elementary symmetric polynomials which, for thecase of interest here of n = 4, read: e ( ˆ M ) = 1 ,e ( ˆ M ) = [ ˆ M ] ,e ( ˆ M ) = 12! (cid:16) [ ˆ M ] − [ ˆ M ] (cid:17) ,e ( ˆ M ) = 13! (cid:16) [ ˆ M ] −
3[ ˆ M ][ ˆ M ] + 2[ ˆ M ] (cid:17) ,e ( ˆ M ) = 14! (cid:16) [ ˆ M ] −
6[ ˆ M ] [ ˆ M ] + 8[ ˆ M ][ ˆ M ] + 3[ ˆ M ] −
6[ ˆ M ] (cid:17) . (1.26)It is useful to notice that the last elementary symmetric polynomial coincides with thedeterminant of ˆ M . Moreover, if ˆ M is antisymmetric its trace is identically zero and, thus, e and e vanish. Units and constants
Unless otherwise stated, we will use units with ~ = c = 1. We will mostly use thereduced Planck mass, related to Newton’s constant as M − = 8 πG N . We will also makeuse of the Einstein’s constant κ = 8 πG N . 14 . Born-Infeld theories The class of theories that generally go under the name of Born-Infeld all share the samebasic feature of being defined in terms of some square root structure aimed at regularisingthe presence of divergences. The inception of these theories originated from the pioneeringworks by Born and Infeld in the 1930’s [72, 73, 74, 75] where they assumed a principleof finiteness, according to which physical quantities are always bounded and can neverbecome infinite. The self-energy of the electron, or a general point-like charged particle,is infinite in the classical Maxwell’s theory so they searched for a non-linear modificationcapable of regularising this divergence as to comply with the principle of finiteness, i.e.,a non-linear theory where point-like charges had finite self-energy . Motivated by theexistence of an upper bound for the velocities of particles in relativistic mechanics, in thesummer of 1933 Born proposed to introduce the same square root structure for electro-magnetism in order to have an upper bound for the electric fields [72, 73]. A few monthslater Infeld joined Born and together worked on a better version of this construction be-cause they wanted a theoretically better motivated argument for such a theory and, then,they argued that the square root structure should come in from symmetry arguments.In analogy with mechanics where going from Newtonian to relativistic mechanics meansupgrading Galilean transformations to the fully relativistic Lorentz group, Born and Infeldassumed that the Lorentz symmetry of Maxwell’s theory should be enlarged in the newtheory. They considered the new symmetry to be the full group of coordinate transfor-mations which, after imposing the recovery of Maxwell’s theory in the appropriate limit,led to the non-linear theory now known as Born-Infeld electromagnetism, expressed as thesquare root of a certain determinant [74, 75]. It is no surprise that the use of symmetries asa guiding principle gave rise to a remarkable theory of non-linear electromagnetism which,not only classically regularises the self-energy of point like charges, but it also shares someinteresting features with Maxwell’s theory and found a natural arena in the realm of othertheoretically appealing theories, like e.g. string theory [307, 308, 374].Given the success of Born-Infeld theory to classically regularise divergences in elec-tromagnetism, it is perhaps surprising that the same ideas were applied to resolve thesingularities of General Relativity (GR) only in the late 1990’s . The first attempt in thisdirection came about in a work by Deser and Gibbons [140], where they finally took overthe idea and tried to apply it to the case of gravity. However, as usual with gravity, thingscan very quickly go wrong when one tries to modify the Einstein-Hilbert action. Themost straightforward application of the Born-Infeld philosophy by introducing a squareroot structure of a determinant involving the Ricci tensor gives rise to the presence of We should perhaps remark here that, at the time when Born and Infeld developed their theory forelectromagnetism, the full machinery of quantum electrodynamics and the renormalization techniques werenot available. Today we know that quantum electrodynamics is a renormalizable quantum field theorywhere physical quantities are finite and, in particular, the charge of a particle acquires radiative correctionsat high energies owed to virtual processes. A possible reason for this was the relative lack of interest in these topics until the seminal works byHawking and Penrose [299, 300, 197] concerning the singularity theorems in GR. On the other hand, theextraordinary success of quantum field theory perhaps motivated to invoke quantum gravity effects as themost likely mechanism that should regularise gravity in the high curvatures regime. f ( R ) theories that contain one extra degree of freedom with respect to GR and,thus, it would deviate from the original Born-Infeld spirit where the theory is modified insome high energy regime by changing the structure of the theory in that regime insteadof adding additional modes.Some years later, Vollick re-considered Born-Infeld type of actions for gravity froma different perspective [356]. Similarly to Deser and Gibbons, Vollick also resorted toa straightforward translation of the Born-Infed action to the case of gravity. However,instead of adopting the metric formalism, he considered the action within a metric-affineapproach so that the connection is left arbitrary and promoted to an independent field.Within that formalism, the problem of the ghosts encountered in the metric formalism areavoided and, thus, no additional terms to remove undesired interactions are needed. Thisapproach can actually be seen as a combination of the Born-Infeld ideas together with theoriginal purely affine theory of gravity proposed by Eddington. Later on, Ba˜nados andFerreira took on Vollick’s approach with a slight modification of the original action, thatnow goes under the name of Eddington-inspired Born-Infeld gravity (EiBI), and showed theexistence of non-singular cosmological and black hole solutions. This particular realisationof Born-Infeld gravity theories has since then received a considerable attention and hasbeen extensively explored in different contexts with promising results.The proposal by Vollick and its relative by Ba˜nados and Ferreira finally succeeded toimplement the ideas of Born-Infeld electrodynamics to the case of gravity. However, it isfair to say that this initial proposal merely consisted in obtaining a gravitational action`a la Born-Infeld, but it lacked any underlying guiding principle, based on symmetries likein Born-Infeld electrodynamics or any other equally valid motivation. In fact, it is verysimple to come up with more general actions that could also be catalogued as Born-Infeldtheories and could be considered on the same footing as EiBI. It does not come as asurprise then that very soon, modifications, extensions or alternative implementations ofthe Born-Infeld ideas to gravity appeared in the literature.In this section we will review in detail the developments discussed above that led tothe formulation of Born-Infeld gravity theories. We will start by reviewing Born-Infeldelectrodynamics as a good starting point to motivate the search for analogous theorieswithin gravitational contexts. We will show how the first attempts formulated in themetric formalism did not succeed due to the presence of ghosts. After that, we will turnto the formulation of Born-Infeld actions for gravity within a metric-affine approach andexplain how the ghost issue is avoided. The general properties of these theories will bediscussed in detail and, in particular, we will explain the existence of two frames. We willend this section by performing a classification of the different Born-Infeld inspired theories16f gravities considered in the literature so far and briefly discuss them. The underlying idea used by Born and Infeld to develop a modification of the Maxwellaction as a potential mechanism to regularise some divergences associated to point-likecharges was motivated by the appearance of an upper bound for the speed of particles whenupgrading Newtonian mechanisms to relativistic mechanics. In that case, the NewtonianLagrangian for a massive particle of mass m is simply L = m v , where v is its velocityand can take any value. When including the principles of relativistic mechanics, theLagrangian for the massive particle becomes L = − m c p − ( v/c ) , where the speedof light c makes its appearance as an upper bound for the velocities due to the squareroot. Taking inspiration from this, Born came up with the idea of modifying Maxwell’sLagrangian in such a way that the divergences of the Coulomb potential are automaticallyregularised due to the existence of a natural upper bound in the theory. In [72, 73],he followed the most straightforward application of this idea and proposed the followingreplacement of Maxwell’s Lagrangian: L = − F µν F µν → L = b "r − b F µν F µν − , (2.1)with b representing the desired upper limit of possible electric fields. Although this simplereplacement could do the job of regularising the infinities associated to point-like charges,it is not completely satisfactory from a theoretical point of view since there is no guidingprinciple for it other than the principle of finiteness. That is the reason that motivatedBorn, this time in collaboration with Infeld, to look for a more theoretically appealingmodification of Maxwell electromagnetism. They noted that, when going from classicalmechanics to relativistic mechanics, the symmetry group is enlarged from the Galileoto the Lorentz group and it is precisely this group structure that nicely introduces thedesired square root. Born and Infeld embraced this line of reasoning and looked fora non-linear theory of electromagnetism enlarging the group of special relativity as therelevant one. The idea is then that, very much like Newtonian mechanics is the limitof special relativity for small velocities and the Lorentz group reduces to the Galileantransformations, Maxwell electromagnetism should be the limit of some theory with alarger group of symmetries which, in some suitable limit, should reduce to the usualrelativistic Lorentz transformations. Motivated by recent developments in gravity wherethe relevant group was shown by Einstein to be general coordinate transformations, theyopted by enlarging the symmetry group of electromagnetism from the Lorentz group to thefull group of general coordinate transformations . Then, to have general covariance, the Incidentally, they were aware and noticed similarities with earlier attempts by Einstein, Weyl andEddington, among others, in the same direction as a way to unify gravity and electromagnetism in ageometrical theory. However, Born and Infeld motivation was completely different and, as themselvesclaimed, their approach had nothing to do with those theories, except for some formal analogies, speciallywith Eddington’s developments in [147]. Remarkably, Eddington’s theory eventually served as guidanceto develop gravity theories `a la Born-Infeld, as we will see in the section 2.4. S = R d x p det a µν , with a µν some rank-2 covariant tensorwhose symmetric part can be identified with the metric tensor and its antisymmetric partis identified with the electromagnetic field strength F µν . After imposing that Maxwell’stheory should be recovered for small electromagnetic fields and neglecting some boundaryterms, they arrived at the celebrated Born-Infeld action S BI = − b "Z d x r − det (cid:16) η µν + 1 b F µν (cid:17) − . (2.2)This action has the properties they were after, namely, it introduces the square rootstructure by means of enlarging the symmetry group of Maxwell’s theory. The constant b is the only free parameter of the theory and it precisely gives the maximum allowed valuefor electric fields. Born and Infeld assumed the value of b to be such that the correctionsarise at the electron radius, although that value is now experimentally ruled out (see [171]for a recent review on experimental bounds for non-linear electromagnetism). In order tosee the appearance of a maximum value for the electric field, let us notice that the actioncan be written in several useful ways by expanding the determinant in (2.2) to obtain S BI = − b Z d x "r b F µν F µν − b ( F µν ˜ F µν ) − (2.3)= − b Z d x s − ~E − ~B b − ( ~E · ~B ) b − , with ˜ F µν ≡ ε µναβ F αβ the dual of the field strength, ~E and ~B the corresponding electricand magnetic parts and we have used the matrix identitydet (cid:18) δ µν + 1 b F µν (cid:19) = 1 + 12 b F µν F µν − b (cid:16) F µν ˜ F µν (cid:17) . (2.4)Notice that this implies a Z symmetry F µν → − F µν owed to the property of the deter-minant det( + ˆ M ) = det( − ˆ M ) for an arbitrary matrix ˆ M . From the above expressionit is straightforward to see that Maxwell’s electromagnetism is recovered for electromag-netic fields much smaller than b and that, for configurations without magnetic field, wealso recover the first Lagrangian (2.1) considered by Born. Furthermore, written in thisway, we can easily understand why the self-energy of point-like charged particles is regu-larised. Since a particle at rest (or in its own rest frame) only generates electric field, theLagrangian reduces to L BI = − b s − ~E b (2.5)and we clearly see that the electric field is bounded by b . Given the gauge character ofthe theory, we still have the constraint equation generating the gauge symmetry (or theequivalent of Gauss’ law) given by ~ ∇ · ~ Π = ρ with ~ Π = ∂ L BI ∂ ~E = ~E q − ~E b (2.6)18nd ρ is the density of electric charge. As usual, for a point-like particle of charge Q wecan integrate the equation over a sphere to obtain I ~ ∇ · ~ Π d x = Q ⇒ | ~ Π | = Q πr , (2.7)where Q is the total charge enclosed by the sphere Q = H ρ d x . By inverting the relation(2.6) between ~ Π and ~E we can obtain the solution for the electric field generated by theparticle ~E = ~ Π q ~ Π b . (2.8)As promised, for | ~ Π | ≪ b we have | ~ Π | ≃ | ~E | ∝ /r which is the usual result in Maxwell’selectromagnetism, while in the opposite regime with ~ Π ≫ b the electric field saturates tothe value | ~E | = b . This saturation is in turn the responsible for the regularization of theself-energy of the particle, that is given by U = Z d x H = b Z d x s ~ Π b − = 4 πb Z ∞ r d r s (cid:18) Q πbr (cid:19) − , (2.9)where we have used the expression for the Hamiltonian density H = ~ Π · ~E − L andthe corresponding solution (2.7). The integral diverges in the case of Maxwell electro-magnetism due to the unbounded contributions from the small scales where one has H Maxwell ∼ ~E ∝ r − . In the Born-Infeld case however, the small scales region is modifiedand we have H BI ∼ ~ Π ∝ r − which makes the integral convergent (see Fig. 2.1). Theintegral can be exactly computed in terms of the gamma function Γ( x ) and the total finiteresult is U = Γ (1 / π p bQ . (2.10)Let us return to the solution for the electric field given in (2.11) and express it directlyin terms of the generating charge as | ~E | = 1 r (cid:16) Q πbr (cid:17) Q πr . (2.11) For the more careful reader, let us clarify that the Hamiltonian density including the interactionbetween the electric potential and the charge is H = ~ Π · ˙ ~A − L BI + A ρ . However, we can use the definitionof the electric field ~E = ˙ ~A − ~ ∇ A to express the Hamiltonian density, up to total derivatives giving rise toboundary terms, as H = ~ Π · ~E − L BI + A ( ρ − ~ ∇ · ~ Π). The term depending on A will then be responsiblefor the gauge constraint giving Gauss’ law that vanishes on-shell, so that it will not modify the self-energyof the particle. For the amusement of the reader familiarised with screening mechanisms in modified gravity, let usnotice that this solution realises a screening mechanism for the electromagnetic interaction resembling theso called K-mouflage or Kinetic screening of scalar fields. x E (cid:144) b MaxwellBorn - Infeld x x H (cid:144) b MaxwellBorn - Infeld
Figure 1: In this plot we show the regularisation occurring in Born-Infeld electromagnetism (solid lines)as compared to the case of Maxwell’s theory (dotted lines). In the left panel we show the profile (as afunction of x ≡ p πb/Qr ) for the electric field generated by a point-like charge. We can clearly see thechange from the usual 1 /r behaviour at large distances to the saturation for the electric field due to theBorn-Infeld corrections on small scales. In the right panel we show how this modified behaviour at smallscales also regularises the energy density of the particle. This expression allows for an alternative interpretation of Born-Infeld electromagnetism.Instead of having modified Maxwell equations in the sector of the electromagnetic field,we can equivalently interpret Born-Infeld electromagnetism as a modification in the sourceterm, i.e., the way in which charges generate electric fields is modified on small scales. Inother words, we can interpret it as an effective scale-dependent charge, showing a certainformal resemblance with the renormalisation of the charge when radiative correctionsare included in standard QED, but here from a purely classical standpoint without anyquantum effect. This re-interpretation of Born-Infeld electromagnetism will be useful forthe case of gravity where the Born-Infeld inspired modified gravity theories will also admitan interpretation as a modification of the way in which matter gravitates at high energies.We will conclude by stressing that the resulting theory turned out to have a seriesof remarkable features that make the Born-Infeld action be very special among all pos-sible non-linear extensions of electrodynamics. Such properties are related to its specialstructure, giving additional motivation and support to the idea of implementing the prin-ciple of finiteness by enlarging the symmetry group of Maxwell theory. This is nothingbut another example of the power of using symmetries as guiding principles to formulatephysical theories. In order to avoid further delays in entering into the main topic of thisreview, namely Born-Infeld inspired theories of gravity, we will abstain our desire of goingthrough all the fascinating features of Born-Infeld electromagnetism and we will contentourselves with briefly enumerating some of its more remarkable properties. For more de-tailed information we refer to [303, 181, 228] or standard textbooks on string theory wherethe Born-Infeld Lagrangian naturally appears, as e.g. [307, 308, 374]: • The Born-Infeld action arises in string theory from T -duality invariance when de-scribing an open string in an electromagnetic field, i.e., the Born-Infeld action is theappropriate one to couple strings to electromagnetic (or more general gauge) fields.20 Born-Infeld electromagnetism shares with its Maxwellian relative (and other non-linear theories of electromagnetism) the so-called electric-magnetic self-duality [69,182]. This is a highly non-trivial invariance of the theory corresponding to a dualtransformation of the electric and magnetic fields. See [27] for a review on manyinteresting aspects of duality rotations and theories with duality symmetry. • Despite the highly non-linear character of the Born-Infeld action, the correspondingequations of motion give rise to causal propagation and avoid the presence of shockwaves and birrefringence phenomena. • The equations of Born-Infeld electromagnetism admit solitonic solutions with finiteenergy, known as BIons [94, 180].As we can see, the Born-Infeld theory for electromagnetism not only conforms to thetask it was devised for, namely the regularisation of divergences associated to point-likecharges, but it is kind enough as to also provide a number of additional gifts that were notrequired a priori . In the remaining of this section we will overview the attempts to applysimilar ideas to the case of gravity. In general, we could say that, by the time of writing,there is not a gravitational analogue of Born-Infeld electromagnetism exhibiting all thesuccesses and remarkable properties discussed above, but the search for it has neverthelessyielded very interesting gravitational theories `a la Born-Infeld, both from a theoretical anda phenomenological points of view. We will start our tour however by reviewing the firstattempts in this direction that led to pathological theories.
The original idea by Born and Infeld to regularise divergences in electromagnetismwas taken over by Deser and Gibbons [140] as a potential mechanism to regularise thesingularities that commonly appear in General Relativity, like e.g. the divergences at thecenter of black holes or the original Big Bang singularity. Following the same scheme, theyconsidered an action for the gravitational interaction including the same determinantal andsquare root structures that appear in Born-Infeld electromagnetism. A straightforwardtranslation of the Born-Infeld action for electromagnetism to the case of gravity would bethe naive replacement of field strength F µν by the Ricci tensor R µν so that the first naivetentative action for a gravitational version of Born-Infeld electromagnetism would be S = Z d x r − det (cid:16) ag µν + bR µν (cid:17) , (2.12)where a and b some parameters, g µν the spacetime metric and R µν the Ricci tensor of thecorresponding Levi-Civita connection. However, this naive procedure leads to a theoryplagued by ghost-like instabilities. The reason is clear from the well-known fact thatan arbitrary action containing a non-linear function of the Ricci tensor will give rise tohigher order gravitational field equations and, thus, it will be prone to the Ostrogradskiinstability [366]. In order to avoid the presence of ghosts in the theory, Deser and Gibbonsconsidered instead the action S DG = Z d x r − det (cid:16) ag µν + bR µν + cX µν (cid:17) , (2.13)21here the fudge tensor X µν must be tuned in order to get rid of the ghost. The form X µν can be obtained perturbatively to remove the ghost at a given order and its effects arethen pushed to higher orders. We can use the identity r det (cid:16) + ˆ M (cid:17) = r M ] + 12 (cid:16) [ ˆ M ] − [ ˆ M ] (cid:17) + O ( ˆ M )= 1 + 12 [ ˆ M ] + 18 [ ˆ M ] −
14 [ ˆ M ] + O ( ˆ M ) , (2.14)valid for an arbitrary matrix ˆ M , to expand the action in powers of the curvature as S DG = Z d x q − det( ag µν ) (cid:16) bR + cX (cid:17) a + (cid:16) bR + cX (cid:17) a − (cid:16) bR µν + cX µν (cid:17) a + · · · (2.15)where R = g µν R µν is the Ricci scalar and X = X αα . In this expression we can seethat, omitting X µν for a moment, we have a cosmological constant at zeroth order, whileat first order we recover the usual Einstein-Hilbert term. At higher orders however theappearance of the quadratic terms R µν R µν will lead to higher order equations of motion,thus rendering the theory unstable due to the presence of ghosts. Since we know that, atquadratic order, only the Gauss-Bonnet prevents the appearance of such ghosts, we mustuse the leading order contribution from X µν in order to remove the undesired terms. Wecan then assume an expansion in curvatures starting at quadratic order for the fudgetensor of the form X µν = X (2) µν + · · · and choose X (2) µν to satisfy cX (2) µµ + b a (cid:16) R − R µν R µν (cid:17) = α (cid:16) R µνρσ R µνρσ − R µν R µν + R (cid:17) , (2.16)with α some constant. The above choice thus only leaves the Gauss-Bonnet contributionat second order. By iterating this procedure one could remove the ghosts at any desiredorder. However, we already see at quadratic order that only the trace of X µν is determinedand, therefore, a large variety of fudge tensors can do the job (see [188, 187] for explicitconstructions). In fact, except for some singular actions, one can presumably write almostany gravitational action in the form of (2.13) by means of an appropriate choice of X µν .We can exemplify this by taking the Born-Infeld gravity theory developed by Nieto in[260]. Motivated by the MacDowell-Mansouri formalism, Nieto considered a spacetimemanifold endowed with a connection giving rise to a total curvature R aµ that can be splitas R aµ = R aµ + λe aµ , where R aµ is the usual curvature of the Levi-Civita connection, e aµ is the vielbein field and λ a constant parameter. For this connection, he then considers aLagrangian in D dimensions given by L = det R aµ . (2.17) We could also add lower order terms for the fudge tensor as, e.g. X (1) µν = Rg µν , but that will notintroduce the discussion other than adding some more terms in the equations.
22f we use the previous splitting, we can write the Lagrangian as L = λ D e det (cid:18) δ µν + 1 λ R µν (cid:19) = λ D e D X n =0 L ( n ) ( R ) , (2.18)where e = det e aµ and we have used the matrix identity det( + ˆ M ) = P Dn =0 e n ( ˆ M ),with e n ( ˆ M ) denoting the n -th elementary symmetric polynomial of the matrix ˆ M (see(1.25)). In the present case, the matrix is the Ricci tensor and its elementary symmetricpolynomials are precisely the Lovelock invariants, that we denote by L ( n ) ( R ), so that theconsidered action is nothing but a combination of all the Lovelock terms and, thus, thetheory is ghost-free. One can then rewrite this Lagrangian in the Deser-Gibbons form bysimply defining a matrix ˆ G given by ˆ G = − ˆ R so that the Lagrangian can be alternativelywritten as L ∝ s − det (cid:18) g µν + 2 λ R µν + 1 λ R µα R µα (cid:19) , (2.19)where we have used the commutativity of the determinant and the square root (wheneverit exists). This is the form found by Nieto and which he then related to Born-Infeld gravity.However, as we have seen, it is nothing but Lovelock gravity written in an obscure way.Furthermore, no additional work is necessary to know that the theory does not containany ghosts. This example perfectly illustrates the necessity of a better defined strategy toconstruct theories of gravity `a la Born-Infeld in order not to be deluded with well-knownhealthy theories in mysterious disguises. In the procedure presented in the precedent section, we have been careful to imposethat only the Lovelock invariants should remain at a given order in the expansion. This is acrucial requirement for the consistency of the theory, as the presence of ghosts invalidatesany background classical solution. The approach followed by Deser and Gibbons canbe seen as a way to make sense of the theory by pushing the scale at which the ghostbecomes relevant at higher scales, but the lack of any other guiding principle obstructsthe construction of an appealing and well-defined full theory.One might however take a less demanding approach and impose instead a weakercondition without compromising the stability of the theory due to the presence of ghosts,but at the expense of partially abandoning the original Born-Infeld spirit. For instance,instead of using the fudge tensor to only leave Lovelock invariants at each order, onecould allow for some arbitrary functions of them. Thus, at quadratic order we could haveallowed for terms involving some linear combination of the squares of the Ricci scalar andthe Gauss-Bonnet term. This would find motivation in the fact that arbitrary functionsof these two scalar quantities are known to be particular cases where the Ostrogradskiinstability is bypassed. In the end, this would be nothing but a complicated way ofrewriting the class of theories described by an arbitrary function f ( R, G ), with R and G the Ricci scalar and the Gauss-Bonnet term respectively. Although perfectly legitimate,these theories introduce additional scalar degrees of freedom and, thus, they can hardly be23onsidered as genuine Born-Infeld modifications of gravity. Of course, this does not meanthat those alternatives are uninteresting, but rather they should be regarded as belongingto another class of theories.In case one is interested in obtaining gravitational theories with an upper bound forthe curvature, then one can simply write a specific model of an f ( R ) theory where thefunction f presents a branch cut at some high but finite curvature R . The square rootfunction typical from Born-Infeld would achieve this, but other functions involving e.g.logarithms could serve as well. Feigenbaum et al. [158] explored this route in two dimen-sions where the curvature is fully determined by the Ricci scalar and they studied someblack holes solutions. In a subsequent work [157], Feigenbaum extended the analysis tofour dimensions where he considered an action of the following type: L = R + β q − k R µνρσ R µνρσ − k R µν R µν − k R , (2.20)with k i and β some constants. Again he studied black hole solutions that we will brieflyreview in section 4.1. However, the problem of ghosts arising from the explicit dependenceon the full Riemann and the Ricci tensors is not discussed. In fact, from the own equationsof motion given in [157], one can see that they are fourth order and, thus, it would beexpected to have ghosts. This pathology renders the black hole solutions of limited physicalinterest, as the perturbations around them are likely to be unstable. The same problemapplies to the theories considered by Comelli and Dolgov in [123] constructed in terms ofthe Lagrangian L = det q A ( R ) g µν + B ( R ) R µν , (2.21)with A and B some given functions of the Ricci scalar. This Lagrangian combines theDeser and Gibbons proposal with f ( R )-type of theories, but without taming the pres-ence of ghosts so that the obtained cosmological solutions are again of limited realisticapplicability.A more interesting proposal that is also closer to the Born-Infeld spirit was given byWohlfarth in [365]. The theory is based on a symmetric object defined as R AB ≡ R [ a a ][ b b ] , (2.22)where the indices A ≡ [ a a ], B ≡ [ b b ] should be regarded as ordered pairs of indices.He then introduces the new metric and Kronecker delta g AB ≡ g a b g a b − g a b g a b (2.23) δ AB ≡ δ a b δ a b − δ a b δ a b (2.24)that are then used in the usual way to manipulate capital indices. Moreover, one has theidentity det g AB = (det g ab ) d − valid in d dimensions. The proposed Lagrangian withinthis formalism is L = √− g h det (cid:16) δ AB + λR AB (cid:17)i ζ , (2.25)with λ some constant and ζ a parameter with the only restriction to be a fractionalnumber in order to allow for a regularization of curvature divergences. This represents24n extension of Deser and Gibbons construction since more general curvature invariantsappear in the Lagrangian. However, it shares the same problematic of containing ghosts(typically appearing at the scale determined by λ ) which is then resolved in a similarfashion, i.e., the Lagrangian is corrected as L = √− g h det (cid:16) δ AB + λM AB + λ N AB (cid:17)i ζ , (2.26)where M AB and N AB are general expressions containing linear and quadratic curvatureterms, respectively. The relative parameters among all the terms must be tuned to removethe ghosts at quadratic order, although one would expect to find again the ghost at higherorders. Thus, similarly to the Deser and Gibbons construction, additional requirementsare necessary to find a satisfactory Born-Infeld theory of gravity within this formalism.Another approach that has been taken in the literature consists in choosing the fudgetensor X µν such that some specific gravity theories are recovered in the low curvaturesregime. In [185], the authors followed this path to construct a Born-Infeld extension ofthe so-called New Massive Gravity theory [66], whose action is given by S NMG = 1 κ Z d x √− g (cid:20) − R + 1 m (cid:16) R µν R µν − R (cid:17)(cid:21) (2.27)and describes a massive graviton in 3 dimensions . One can then see that this action isrecovered at quadratic order from (2.13) in 3 dimensions by choosing X µν proportional to Rg µν and appropriately tuning the parameters (with the possible addition of a cosmologicalconstant). Interestingly, the resulting action that they consider recovers at cubic order theextension of New Massive Gravity found in [332] by imposing the existence of a c − theorem.The same authors pursued a similar approach in [186] to construct theories that recoverHorava’s gravity [210, 211] in 3 dimensions at quadratic order. In the previous sections we have seen that a straightforward implementation of theBorn-Infeld idea to the case of gravity is not an obvious task. It is not difficult to convinceoneself that the main difficulty is the avoidance of ghosts and this is hardwired in theuse of the metric formalism in the action. One can however seek for Born-Infeld inspiredmodifications of gravity within the realm of affine theories of gravity where the connec-tion is regarded as an independent object. Within this framework, it is very natural toremember the purely affine theory of gravity introduced by Eddington and described bythe following action [147]: S E = Z d x q | det R ( µν ) (Γ) | , (2.28) Since the graviton propagator trivialises in 3 dimensions, the problem of the potential ghosts discussedabove are less virulent. Deser and Gibbons already made reference to this approach in [140], but they did not consider it anyfurther in favour of a metric formalism. R ( µν ) (Γ) is the symmetric part of the Ricci tensor of an arbitrary connection Γ αµν .In vacuum, this theory is equivalent to GR . This is easy to understand, since this theorycan be seen as GR after integrating out the metric tensor. If we start with GR in thepresence of a cosmological constant and in the Palatini formalism, we have S = 12 M Z d x √− g (cid:16) R (Γ) + 2Λ (cid:17) (2.29)that gives the Einstein equations R ( µν ) − R g µν = Λ g µν . (2.30)We can now take the trace to obtain R = − R ( µν ) = − Λ g µν . This relation can be used in the action to remove the dependence on themetric tensor and we then recover the Eddington action. This procedure of integratingout the metric tensor is also valid when including matter fields as long as they coupleminimally, i.e., the metric tensor will only enter algebraically. In that case, the resultingaction will be more involved, but it allows to write a fully affine theory of gravity, as itwas Eddington’s original idea.An important consequence of using the connection as a fundamental geometrical objectin Eddington’s theory is the avoidance of introducing ghosts associated to higher orderequations of motion for the metric tensor. This is not a specific feature of Eddington’stheory, but it is a general result for theories formulated `a la Palatini. In view of theseresults, Eddington’s action seems to be a better suited starting point to implement theBorn-Infeld construction for theories of gravity. This approach was taken by Vollick [356],who considered the action S EBI = M M Z d x "s − det (cid:16) g µν + 1 M R µν (Γ) (cid:17) − p − det g µν , (2.31)where M BI is a mass scale determining when high curvature corrections are important. Thesecond term is introduced to remove a cosmological constant, thus allowing for Minkowskisolutions in vacuum. The above action for a theory of gravity combines the ideas ofEddington’s theory with the Born-Infeld construction, resulting in a theory of gravity for-mulated in a metric-affine approach and incorporating the square root and determinantalstructures characteristic of Born-Infeld electrodynamics. The recovery of the GR equations in vacuum is not specific of Eddington’s theory and, in fact, it is ageneral result for any theory of gravity. The generality of this result actually boils down to the covarianceof the field equations which imposes that, in vacuum, the Ricci tensor must be proportional to the metric.In theories of gravity with additional degrees of freedom, the extra fields should be regarded as matter fieldsand the recovery of GR in vacuum also applies. Another complementary way of understanding this generalresult is provided by the fact that GR is the only Lorentz invariant and unitary theory for a self-interactingmassless spin-2 field in 4 dimensions, usually called graviton. Thus, if by gravity we understand a theoryfor such a particle, we will inevitably find GR in vacuum. Differences can however show up when includingmatter fields, as we will discuss later. Here we use the dimension 1 parameter M BI as the Born-Infeld scale instead of the constant b used in[356]. The relation between both is b = M − . M . When taking that limit, theleading order correction is S EBI ( |R µν | ≪ M ) ≃ M Z d x √− gg µν R µν (Γ) (2.32)thus reproducing the Einstein-Hilbert action in the first order formalism, which is knownto coincide with GR on-shell and provided the matter fields couple minimally (see forinstance [201, 288]). Let us pause here for a moment and seize the opportunity to clarifysome subtleties concerning this point which are well-known in the community but arestill source of a little confusion in some works (see for instance the discussion at thisrespect in section 2.3.1 of [120]). When considering the Einstein-Hilbert action in thePalatini formalism in the presence of minimally coupled fields, the field equations of theconnection can be recast as a metric compatibility condition for the metric tensor and,thus, a solution of the equations is the Levi-Civita connection of the spacetime metric. Animportant point to note however is that such a solution represents a solution, but the mostgeneral solution for the connection field equations involves an arbitrary 1-form, which canbe taken to be the trace of the non-metricity or the trace of the torsion tensor. Thisis of course nothing but a reflection of the fact that the metric compatibility conditionobtained from the connection field equations does not fully determine the connection andthe Levi-Civita connection is only obtained after imposing a symmetric condition. It issometimes stated that such a condition must be supplemented for the Einstein-Hilbertaction to give GR in the Palatini formalism. However, one must also notice that theEinstein-Hilbert action has a projective invariance Γ λµν → Γ λµν + ξ µ δ λν which also involvesan arbitrary 1-form ξ µ , and this is precisely the undetermined mode obtained when solvingthe connection equation. The gauge character of the projective invariance is discussed ingreat detail in [225, 128].In the case of the action (2.31), the projective invariance is only obtained as a lowcurvature accidental symmetry, but it is generally broken by higher order interactions,unless the initial theory is defined only in terms of the symmetric part of the Ricci tensor,in which case the projective invariance is a symmetry of the full theory. Considering onlythe symmetric part of the Ricci tensor is a widely adopted (and very convenient) optionin the literature and, in addition, it would be closer to Eddington’s original theory. This The equivalence between the metric and the Palatini approaches has also been considered for moregeneral actions in, e.g. [156, 76, 127]. A particularly interesting result is that the equivalence of bothformulations extends to the whole series of Lovelock invariants, among which the Einstein-Hilbert actionrepresents nothing but the lowest order term. See for instance [288] for details. We will also show more details on how this is achieved in section2.7.1 within the context of more general theories. In section 2.5 we will show that this symmetry is shared by all theories defined in terms of the symmetricpart of the Ricci tensor and we will compute the associated conserved current.
27s the option adopted by Ba˜nados and Ferreira in [45] , where they considered the action S EiBI = M M Z d x "s − det (cid:16) g µν + 1 M R ( µν ) (Γ) (cid:17) − λ p − det g µν (2.33)that has now become the standard version of the so-called Eddington-inspired-Born-Infeldgravity (EiBI). In this version, it is customary to let a cosmological constant term beencoded in the parameter λ as Λ = ( λ − M . An important notational convention thatmight lead to some misinterpretations but is very common in the community is to use R µν to denote the symmetric part of the Ricci tensor without the explicit symmetrisation. Toavoid any confusion, we will always make explicit the corresponding symmetrisation. In the literature there is a number of subtle points in the derivation of the field equa-tions that are sometimes overlooked or omitted, so we will provide a detailed derivationhere. The main differences that one can encounter eventually boil down to whether onlythe symmetric part or the full Ricci tensor is considered and if the connection is assumedto be symmetric a priori or not. The former condition is related to the presence of aprojective invariance, while the latter has to do with the presence of torsion. In manypractical applications, these differences do not make a huge impact in the results, but oneshould nevertheless be careful to obtain the correct field equations. Let us then considerthe action S = M M Z d x "s − det (cid:16) g µν + 1 M R µν (Γ) (cid:17) − λ p − det g µν + S M [Ψ , g µν , Γ](2.34)where no assumptions are made a priori on the connection and the full Ricci tensor R µν with both its symmetric and its antisymmetric parts. Let us stress again that Vollick [356]used the full Ricci tensor but constrained the connection to be symmetric, while Ba˜nadosand Ferreira left the connection fully undetermined but considered only the symmetricpart of the Ricci. We have also added general matter fields Ψ that can, in principle,couple to both the metric and the connection. Then, we will detail where the differencesarise when making the different assumptions. For later convenience and to comply withstandard notation in the literature, let us introduce the notation q µν ≡ g µν + 1 M R µν . (2.35) Here we prefer to restore all the dimensionful constants as opposed to [45], where the authors set8 πG = 1. Furthermore, we correct a typo in form of a factor of 2 appearing there, which has propagatedin the literature. δ S = M M Z d x (cid:20) √− q (cid:0) ˆ q − (cid:1) νµ (cid:18) δg µν + 1 M δ R µν (cid:19) − λ √− gg µν δg µν (cid:21) + δ S M [Ψ , g µν , Γ](2.36)where q = det ˆ q and we have used the formula δ q − det ˆ M = 12 q − det ˆ M Tr h ˆ M − δ ˆ M i (2.37)valid for an arbitrary matrix ˆ M . The field equations for the metric tensor are thenimmediately seen to be √− q (cid:0) ˆ q − (cid:1) ( µν ) = √− g (cid:18) λg µν − M M T µν (cid:19) (2.38)with the energy-momentum tensor of the matter fields defined as T µν ≡ √− g δ S M δg µν (cid:12)(cid:12)(cid:12) Γ . (2.39)Notice that this energy-momentum tensor is defined at constant connection . For minimally-coupled bosonic fields this is not relevant and the energy-momentum tensor will have thestandard form. However, when considering fermionic and non-minimally coupled bosonicfields, the expression for the energy-momentum tensor will be in general different fromthe one that would be obtained in a purely metric formalism. It is important to note thesymmetrisation of the object ˆ q − in the field equations as a consequence of the symmetryof the metric tensor. Had we considered only the symmetric part of the Ricci tensor inthe starting action, this symmetrisation would be innocuous. Furthermore, as said before,in most practical applications in cosmological contexts or spherically symmetric solutions,the matrix ˆ q is symmetric and then one could omit the symmetrisation, but in the generalcase it is important to properly include it. We will come back to this point in section 2.7.1for more general Lagrangians.The derivation of the connection field equations requires a bit more of work. In orderto obtain them, we need the variation of the Ricci tensor: δ R µν = ∇ λ δ Γ λνµ − ∇ ν δ Γ λλµ + T λρν δ Γ ρλµ , (2.40)where T λρν = Γ λρν − Γ λνρ is the torsion tensor. Equipped with this relation, we can now pro-ceed to compute the variation with respect to the connection. Leaving aside the variation In the literature of Born-Infeld theories it is customary to denote the inverse of the matrix q µν simplyas q µν , in accordance with the usual convention of denoting the inverse of a metric with upper indices.Since we will have two metrics, we prefer to explicitly keep the inverse for the moment in order to avoidany confusion to the unfamiliar reader in these first steps into the formalism of Born-Infeld theories, since q µν could very well be confused with g µα g µβ q αβ . We will eventually drop the explicit mention for theinverse of ˆ q to alleviate the notation and whenever there is no risk of confusion.
29f the matter sector for a moment, we have δ S Γ = M Z d x √− q (cid:0) ˆ q − (cid:1) νµ δ R µν = M Z d x √− q (cid:0) ˆ q − (cid:1) νµ (cid:16) ∇ λ δ Γ λνµ − ∇ ν δ Γ λλµ + T λρν δ Γ ρλµ (cid:17) = − M Z d x n ∇ λ h √− q (cid:0) ˆ q − (cid:1) νµ i δ Γ λνµ − ∇ ν h √− q (cid:0) ˆ q − (cid:1) νµ i δ Γ λλµ − √− q (cid:0) ˆ q − (cid:1) νµ T λρν δ Γ ρλµ o + M Z d x n ∇ λ h √− q (cid:0) ˆ q − (cid:1) νµ δ Γ λνµ i − ∇ ν h √− q (cid:0) ˆ q − (cid:1) νµ δ Γ λλµ io . (2.41)Let us take a moment here to elaborate on the terms in the last line. Usually in (pseudo-)Riemannian geometries without torsion, these terms correspond to total derivatives thatcan be simply dropped and do not contribute to the equations of motion. However, thedivergence of a vector density A µ of weight w = − ∇ µ A µ = ∂ µ A µ + T λλµ A µ . (2.42)Since √− q is indeed a scalar density of weight w = −
1, we then see that the usualboundary terms generated when integrating by parts, actually contribute non-trivially tothe field equations whenever torsion is present. Let us stress that the crucial element hereis the torsion, i.e., even if there is non-metricity, the boundary terms would not contributeto the equations in the absence of torsion. This is in fact one of the important differencesarising from considering a torsion-free connection from the beginning. After taking intoaccount these considerations in the variation (2.41) we obtain δ Γ S = − M Z d x n ∇ λ h √− q (cid:0) ˆ q − (cid:1) νµ i δ Γ λνµ − ∇ ν h √− q (cid:0) ˆ q − (cid:1) νµ i δ Γ λλµ − √− q (cid:0) ˆ q − (cid:1) νµ T λρν δ Γ ρλµ o + M Z d x n √− q (cid:0) ˆ q − (cid:1) νµ δ Γ βνµ − √− q (cid:0) ˆ q − (cid:1) βµ δ Γ λλµ o T ααβ . (2.43)After an appropriate re-shuffling of the indices, the connection field equations can finallybe expressed as ∇ λ h √− q (cid:0) ˆ q − (cid:1) µν i − δ µλ ∇ ρ h √− q (cid:0) ˆ q − (cid:1) ρν i = ∆ µνλ + √− q h T µλα (ˆ q − (cid:1) αν + T ααλ (ˆ q − (cid:1) µν − δ µλ T ααβ (ˆ q − (cid:1) βν i (2.44)where, for completeness, we have added the hypermomentum of the matter fields∆ µνλ ≡ M δ S m δ Γ λµν (cid:12)(cid:12)(cid:12) g µν . (2.45)Analogously to the energy-momentum tensor, the hypermomentum must be computed atconstant metric. In most of the cases, we deal with minimally coupled bosonic fields inwhich case we have ∆ µνλ = 0. However, the standard way of coupling fermionic fields to30ravity is by resorting to the vierbeins formalism that allows to generalise the definition ofthe gamma matrices to curved spacetime. In that formalism, the fermions couple directlyto the spin connection and, thus, contributions to the hypermomentum typically arise. Wewill leave this case aside and will assume vanishing hypermomentum. For this simplifiedcase, we have the full set of equations for the Born-Infeld gravity that we display groupedtogether here for future reference √− q (cid:0) ˆ q − (cid:1) ( µν ) = √− g (cid:18) λg µν − M M T µν (cid:19) , (2.46) ∇ λ h √− q (cid:0) ˆ q − (cid:1) µν i − δ µλ ∇ ρ h √− q (cid:0) ˆ q − (cid:1) ρν i = √− q h T µλα (ˆ q − (cid:1) αν + T ααλ (ˆ q − (cid:1) µν − δ µλ T ααβ (ˆ q − (cid:1) βν i (2.47)where q µν ≡ g µν + 1 M R µν . (2.48)These will be the fundamental set of equations that need to be solved in Born-Infeldgravity. In most practical situations, the equations are greatly simplified and the generalcase is rarely required. Thus, instead of tackling the full set of equations directly, letus first first consider a simplified case where most of the results will be sufficient for theastrophysical and cosmological applications discussed in the subsequent sections. We will start by considering the simplest possible case with vanishing torsion a pos-teriori and where the action is constructed out of the symmetric part of the Ricci tensorsolely, and we will postpone the general case for later. The busy reader rushing to explorethe different applications of the theory will be able to grasp the essential details in thissection, since this simplified scenario is the most extensively considered case in sphericallysymmetric and cosmological solutions. The thorough reader will hopefully be satisfied withthe more detailed discussion provided in section 2.7.1 for more general theories (where infact we will see that getting rid of the torsion does not represent any limitation for a classof theories among which we find EiBI). Let us notice that the assumption on R µν refersto the own definition of the theory while the torsion-free condition restricts the consideredclass of solutions within the theory.The fact that we only consider the symmetric part of the Ricci tensor in the action hastwo important consequences. On one hand, the object q µν will inherit the symmetry of theRicci tensor (along with that of the metric g µν ). On the other hand, we are enlarging thesymmetries of the theory by introducing a projective invariance and, thus, this conditioncan be naturally introduced by imposing such a symmetry in the gravitational sector. Theprojective invariance corresponds to a shift in the connection of the formΓ λµν → Γ λµν + ξ µ δ λν (2.49)for an arbitrary 1-form ξ µ . That this is in fact a symmetry of the theory containing only thesymmetric part of the Ricci tensor can be easily seen from (2.40) by taking δ ξ Γ λµν = ξ µ δ λν
31o obtain that, under a projective transformation, the full Ricci tensor transforms as δ ξ R µν = ∇ µ ξ ν − ∇ ν ξ µ + T λµν ξ λ . (2.50)We can clearly see from here that the variation of the Ricci tensor under a projective trans-formation of the connection is antisymmetric and, thus, its symmetric part is invariant δ ξ R ( µν ) = 0. A consequence of this symmetry is that one of the traces of the connec-tion field equations vanishes identically, i.e., the constraint associated to the projectivesymmetry is δ λν δ S δ Γ λµν = 0 . (2.51)Let us stress here that the projective symmetry will not be broken by the presence ofminimally coupled fields. Bosonic fields with minimal couplings will only couple to themetric, so the projective invariance is obvious. On the other hand, minimally coupledfermions do couple to the connection, but such a coupling still respects the projectivesymmetry (see for instance [202]). Finally, it is also interesting to note that the projectiveinvariance is so-called because it is in fact a symmetry of the geodesics equations, sinceits effect can be re-absorbed into a re-definition of the affine parameter. For minimallycoupled fields this is irrelevant because they are only sensitive to the Levi-Civita part ofthe full connection.The field equations under the conditions at hand now reduce to √− q (cid:0) ˆ q − (cid:1) µν = √− g (cid:18) λg µν − M M T µν (cid:19) (2.52) ∇ λ h √− q (cid:0) ˆ q − (cid:1) µν i − δ µλ ∇ ρ h √− q (cid:0) ˆ q − (cid:1) ρν i = 0 (2.53)where we have set T λµν = 0 and dropped the explicit symmetrization for (cid:0) ˆ q − (cid:1) µν since it isautomatically symmetric. We can check that the trace of the connection equations withrespect to λ and ν vanishes identically, as a consequence of the projective symmetry, whilethe trace with respect to λ and µ gives ∇ λ h √− q (cid:0) ˆ q − (cid:1) λν i = 0 . (2.54)This constraint can then be plugged back into the connection equations to finally obtain ∇ λ h √− q (cid:0) ˆ q − (cid:1) µν i = 0 . (2.55)Since the action only depends on the symmetric part of the Ricci, the object q µν is sym-metric and the above equation tells us that the connection must be compatible with the auxiliary metric q µν , i.e., the connection is given by the Levi-Civita connection of the met-ric q µν . It is important to notice that the metric compatibility condition only determinesthe symmetric part of the connection and, in general, leaves a vector component of the an-tisymmetric part undetermined. However, the assumption of a symmetric condition fixes32his undetermined part. At this point, one could fairly object that we have not solved theconnection yet, as the auxiliary metric q µν is defined in terms of the Ricci tensor, whichdepends on the connection itself. The resolution to this comes about by going back tothe metric field equations (2.52). From there, we can see that the auxiliary metric canbe fully expressed in terms of the spacetime metric g µν and the matter content throughits energy-momentum tensor T µν , so that the solution for the connection has actuallybeen achieved. An important feature of this procedure that should not go unnoticed isthat the connection has been obtained by solving algebraic equations and, therefore, nodegrees of freedom are actually associated to it. In other words, there are no additionalboundary conditions that we need to provide to solve for the connection, which meansthat it is nothing but an auxiliary field. This is the reason why the Born-Infeld theorymodifies gravity without introducing new degrees of freedom. We will come back to thispoint later for more general cases.Now that we have the solution for the connection, we can proceed to complete theresolution of the problem. This is not a very difficult task, since the field equationsdetermining the auxiliary metric (that then gives the connection) are simply R µν (ˆ q ) = M (cid:16) q µν − g µν (cid:17) (2.56)where we need to remember that ˆ q = ˆ q (ˆ g, Ψ) is obtained from the metric field equations.However, instead of using these equations directly in this form, it is convenient to workthem out a little bit to recast them into a more suitable form for direct applications. Letus begin by introducing some additional notation that is commonly used in the literatureand which will allow to make contact with more general theories. We will denote by ˆΩthe deformation matrix relating the auxiliary and the spacetime metrics as q µν = g µα Ω αν (2.57)or, in matrix notation, ˆ q = ˆ g ˆΩ. In the present case, this matrix is simply ˆΩ = + M ˆ g − ˆ R ,obtained from the definition of ˆ q . However, the advantage of introducing this notation isthat we can very easily solve the metric field equations (2.52) for ˆΩ. When plugging (2.57)into (2.52), we obtain the relationˆΩ − = 1 p det ˆΩ (cid:18) λ − M M ˆ T ˆ g (cid:19) . (2.58)Now, we can multiply (2.56) by ˆ q − and use (2.57) to obtainˆ q − ˆ R ( q ) = 1 M p det ˆΩ h M M (cid:16)p det ˆΩ − λ (cid:17) + ˆ T ˆ g i . (2.59)This will be the starting point for many of the discussions in the subsequent sectionsdevoted to astrophysical, black holes and cosmological applications. Let us stress that the Again, remember that we are considering minimally coupled fields, so the energy-momentum tensordoes not depend on the connection, but only on the matter fields and, perhaps, the spacetime metric g µν . T ˆ g ), i.e., in terms of the metric andthe matter content through the combination ˆ T ˆ g . For some important material contents,this combination does not depend on the metric but only on the energy density ρ andthe pressure p . This is the case for instance for a perfect fluid or an electromagneticfield (see sections 4.3, 4.4 and 5.2). In that case, solving (2.58) will yield ˆΩ = ˆΩ( ρ, p ).After obtaining these expressions, one can then complete the resolution of the problem bysolving the differential equations (2.59).To end this section, let us notice that the equations (2.59) admit yet another formu-lation in terms of the Born-Infeld gravitational Lagrangian defined by means of S BI = R d x √− g L BI . If we restore the components notation, we have R µν ( q ) = 1 M p det ˆΩ (cid:16) L BI δ µν + T µν (cid:17) , (2.60)where we have used the metric q µν to raise the first index of the Ricci tensor, i.e., R µν ( q ) ≡ q µα R αν . The interest of writing the equations in this form is twofold. Firstly, as we willsee in section 2.7.1, this form of the field equations is valid not only for the Born-Infeldgravity theory considered here, but also for a large variety of theories formulated in thePalatini formalism. Thus, given a certain specific theory, we can immediately obtain thecorresponding field equations by using (2.60) directly. Secondly, this will be the startingpoint for many of the developments for practical applications that will be discussed in thesubsequent sections of this review.Another important feature of (2.60) is that we can directly compare it with the usualEinstein equations of GR written as R µν = 1 M (cid:18) T µν − T δ µν (cid:19) . (2.61)We can then see that the equations for Born-Infeld gravity written as (2.60) can be in-terpreted as the usual Einstein equations for the auxiliary metric but with a modifiedsource term, i.e., matter fields gravitate in a non-standard way. This closely resembles theanalogous interpretation for Born-Infeld electromagnetism given from (2.11). A perhapsmore apparent way of showing this is by re-writing (2.60) in a more familiar form. If wetake the trace, we obtain the relation R ( q ) = 1 M p det ˆΩ (cid:16) L BI + T (cid:17) (2.62)so the Einstein tensor associated to the auxiliary metric G µν ( q ) = R µν ( q ) − R ( q ) δ µν canbe expressed as G µν ( q ) = 1 M p det ˆΩ (cid:20) T µν − (cid:16) T + 2 L BI (cid:17) δ µν (cid:21) . (2.63)34hese equations show even more clearly how the Born-Infeld theory can be seen as usualgravity for the auxiliary metric with a modified source term (let us remember once againthat Ω is algebraically related to the matter content through (2.58)). Furthermore, fromhere we can also easily understand a very distinguishing property of the theory. If wenow use the relation between the two metrics ˆ q = ˆ g ˆΩ to expand the Einstein tensorin (2.63) in terms of ˆ g -related objects we can immediately see that, since the Einsteintensor contains up to second derivatives of the metric, we will end up with up to secondderivatives of the deformation matrix ˆΩ. This deformation matrix depends on the energymomentum tensor through (2.58)) so that the evolution equations for the spacetime metric g µν will contain derivatives of the energy-momentum tensor components . This is a verydistinctive feature of these theories that gives rise to new effects and, among others, adependence of the gravitational potential on the local density and not only on an integrateddensity as in the usual case (see (3.6)). In fact, this effect has been claimed to lead to veryserious drawbacks. We will give a careful discussion about this issue in section 3.1. Finally,this feature will also be the responsible for a dependence of the background cosmologyevolution on the sound speed and not only on the equations of state parameter as inordinary gravitational theories. We will see in section 2.7 that these properties are in factshared by a large class of theories. This non-standard interplay between the gravitationalsector and the matter fields has been noticed and extensively used in the literature. Seefor instance [138] for a devoted discussion on this point and [297] where it is shown thatgravity theories with generic auxiliary fields exhibit these properties.In the next subsection we will re-obtain this result in a slightly different and comple-mentary way that will allow to clarify the role played by both metrics. Already here wecan sense that the auxiliary metric carries physical relevance and it is not simply a math-ematical object. We will postpone a thorough discussion about this point for the nextsubsection. Let us notice now that, very much like for the electromagnetic case, whenwe take curvatures much smaller than M (or, equivalently, densities much smaller than M M ), the deformation matrix is approximately the identity ˆΩ = + O ( R/M ) so that q µν and g µν coincide up to corrections O ( R/M ). In that case, we also have L BI ≃ M R and (2.63) reduces to the usual Einstein equations, confirming that the modifications onlyappear when the curvatures become order one as compared to the Born-Infeld scale M BI .Equivalently, the Born-Infeld modifications will appear when | T µν | ∼ M M .To end this section, we will give some good news that will appease the less thoroughreader. Despite having neglected the torsion, all the results obtained here are completelyvalid for the general case with torsion provided the projective symmetry is imposed. Wewill show this explicitly in section 2.7.1 One could object that second derivatives of T µν will give rise to higher than second order derivativesof the matter fields because the energy-momentum tensor typically contains first derivatives and, thus, thesystem might be prone to the very same Ostrogradski instabilities we claimed to be avoided. However, oneshould keep in mind that the matter fields will have their own second order field equations so they will inany case propagate the correct number of degrees of freedom. .6. The two frames of Born-Infeld gravity and the physical relevance of the auxiliarymetric We have seen that Born-Infeld gravity naturally leads to the appearance of two metrictensors, namely the spacetime metric g µν and the auxiliary metric q µν . The former playsthe role of the metric to which matter fields couple, while the latter has been introducedas an auxiliary object to solve the equations so that the connection is the one compatiblewith it. So far, we have not provided this object with any physical meaning and it simplyappeared as a mathematical trick to facilitate the resolution of the field equations or,equivalently, it appears as a result of integrating out the connection. The aim of thissection will be to clarify the role of this object and unveil its physical significance.The bi-metric character of the theory can be better appreciated by rewriting the EiBIaction in the equivalent form S EiBI = 12 M M Z d x √− q (cid:20)(cid:0) ˆ q − (cid:1) µν (cid:18) g µν + 1 M R ( µν ) (Γ) (cid:19) − (cid:21) + S M [Ψ , g µν ] (2.64)where we have introduced an auxiliary field that we have suspiciously called q µν . To seethat this is in fact equivalent to the EiBI action we can compute the field equations forthis auxiliary field − (cid:20)(cid:0) ˆ q − (cid:1) µν (cid:18) g µν + 1 M R ( µν ) (cid:19) − (cid:21) q αβ + g αβ + 1 M R ( αβ ) = 0 . (2.65)If we contract this equation with (cid:0) ˆ q − (cid:1) αβ we obtain the relation (cid:0) ˆ q − (cid:1) µν (cid:18) g µν + 1 M R ( µν ) (cid:19) = 4 (2.66)which can be plugged into the field equation to obtain the solution q µν = g µν + M R µν ,that justifies our original name for this auxiliary field, since it turns out to be nothingbut the auxiliary metric defined above. If we insert the solution into (2.64) we see thatwe recover the original determinantal form of the EiBI action after integrating out theauxiliary field q µν , proving the equivalence of both representations. The bimetric repre-sentation however provides a more orderly arrangement of the two metrics that allowsto unveil their role in the theory. The role of the metric tensor is already clear fromthe beginning as the metric seen by matter fields and, therefore, determining their causalstructure. In particular, point-like particles will follow the geodesics of the Levi-Civitaconnection corresponding to g µν . There is nothing special here as this is a consequenceof considering minimally coupled fields, the only difference with respect to the usual casebeing that the solution for the metric tensor will be different. In order to properly identifythe physical role of the auxiliary metric, let us notice two important features in (2.64).The first one is that the spacetime metric g µν only enters the action algebraically, i.e., Here we consider the λ − term as part of the matter sector, where it will contribute as a cosmologicalconstant. g µν is an auxiliary field that can be integratedout. In fact, its equation of motion is given by √− q (cid:0) ˆ q − (cid:1) µν = − M M √− gT µν (2.67)which allows to obtain g µν algebraically in terms of the matter fields and the auxiliarymetric q µν . For some types of matter fields this step might not be possible and, thus, thefollowing discussion would not apply. Barring these singular cases, we can integrate outthe spacetime metric and we will end up with an action of the form S EiBI = 12 M Z d x √− q (cid:0) ˆ q − (cid:1) µν R ( µν ) (Γ) + ˜ S M [Ψ , q µν ] (2.68)where ˜ S M represents the new form of the matter sector after replacing the solution for g µν obtained from (2.67). We thus arrive at an equivalent action with the Einstein-Hilbertterm in the Palatini formalism to describe the dynamics of the auxiliary metric q µν , butnow the coupling of this auxiliary metric to the matter fields will have a complicatedform. As discussed above, the Einstein-Hilbert sector will state that the connection Γmust correspond to the Levi-Civita connection of the auxiliary metric q µν , which is againthe result obtained when working with the determinantal form of the action. This versionof the action reveals a more profound role for the auxiliary metric since now we can seethat the gravitational waves can be straightforwardly interpreted as the tensor part ofthe perturbations of the auxiliary metric. To further clarify this point, let us assumethat we have a background configuration for both metrics given by ¯ g µν and ¯ q µν . Inthis background geometry, the matter fields will propagate in the metric ¯ g µν that willdetermine the corresponding causal structure. In particular, the light cone for photonswill be determined by this metric. Furthermore, massive objects will be coupled in thestandard way to the gravitational potentials and will follow the geodesics of ¯ g µν . On theother hand, gravitational waves will propagate on the background metric ¯ q µν and it is thisauxiliary metric that determines the causal structure for them so that gravitons will followthe geodesics of the auxiliary metric ¯ q µν . Since matter fields couple in a non-standard wayto this metric, the interaction of the gravitational potentials encoded in the perturbationsof q µν with the matter fields will differ from the usual case. We can then summarise thisdiscussion by saying that the spacetime metric determines the propagation of matter fieldsand the auxiliary metric determines the propagation of gravitons.The result obtained here and that boils down to the equivalent action (2.68) for EiBIgravity is equivalent to the finding presented at the end of 2.5.1 where the field equationswere eventually written as (2.63) in the form of Einstein equations for the metric q µν with a modified source term. This is exactly what the action (2.68) is telling us, sincethe corresponding field equations will consist of the Einstein tensor obtained from varyingthe gravitational sector which will then be sourced by the energy-momentum tensor ofthe matter sector as computed with respect to the metric q µν . In other words, the field37quations are G µν = 1 M ˜ T µν (2.69)where ˜ T µν is the effective energy-momentum tensor defined as˜ T µν ≡ √− q δ ˜ S M δq µν (cid:12)(cid:12)(cid:12) Γ . (2.70)We thus recover the field equations for Born-Infeld gravity written in an Einsteinian formas in (2.63) where we need to identify the non-standard source term in the right handside with the effective energy-momentum tensor ˜ T µν , which is non-trivially related to T µν . It is important to notice that both energy-momentum tensors will satisfy theircorresponding conservation equations, namely: ˜ ∇ µ ˜ T µν = ∇ µ T µν = 0 with ˜ ∇ and ∇ thecovariant derivatives associated to q µν and g µν respectively. The result found here will helpexplaining why singular solutions like the Big Bang and/or black holes can be regularisedwithout violating the null energy condition, because the object that will need to violate an effective null energy condition is not the standard energy-momentum tensor of the matterfields (see also sections 4.3, 4.4 and 5.2).A certain familiarity with modified gravity allows to appreciate a close analogy betweenthe above discussion and the existence of two frames in scalar-tensor theories. In theJordan frame matter fields are minimally coupled to the metric, but gravity is describedby a scalar-tensor theory. In the Einstein frame however gravity is described by theEinstein-Hilbert term, but matter fields couple to a conformal metric whose conformalfactor depends on the scalar field. In the case of Born-Infeld, the situation is alike, butwith the crucial difference that there are no additional propagating degrees of freedom. Inthe original description of the theory, that we will call the Born-Infeld frame , matter fieldscouple in the standard way to the metric but the gravitational action is non-standard.In this frame, we have that gravity reacts differently to the presence of matter when thedensities are very high and particles follow the geodesics of the metric just as in standardgravity. In the alternative description exposed in this section, that we will call
Einsteinframe for obvious reasons, gravity has the standard Einstein-Hilbert action, but now thecouplings of the matter fields to gravity are not the usual ones, i.e., we cannot simplyfollow the usual minimal coupling rule from flat spacetime and replace the Minkowskimetric by the curved one appearing in the Einstein-Hilbert term.The existence of the Einstein frame also helps understanding the Born-Infeld inspiredgravity theories from a particle physics perspective. The common wisdom says that GRis the only consistent theory for a massless spin 2 field in 4 dimensions and this isusually used to state that modifications of gravity either introduce additional degrees of Of course, we need to integrate out the connection. Since for the Einstein-Hilbert term at hand weknow that the connection is given by the Levit-Civita connection of the metric q µν , we omit this step hereand assume that this operation has already been carried out. By consistent one usually means unitary and Lorentz invariant, although locality is a frequent implicitcondition. See for instance [71] for consistent theories including non-localities. See also [207, 208] for otherconstructions based on higher but finite derivatives. M Pl and theBorn-Infeld scale M BI . These two scales are assumed to satisfy M BI ≪ M Pl and thishierarchy introduces yet another relevant scale in the problem given by their geometricalmean ¯ M BI = √ M Pl M BI . The introduced hierarchy has the purpose of having Born-Infeldcorrections before hitting the quantum gravity regime, that takes place at some scalenear M Pl , so that we can have a range of scales between ¯ M BI and M Pl where gravitybehaves differently but the quantum gravity effects can still be safely neglected. From theaction in the Einstein frame expressed as 2.68, we see that the Born-Infeld scale M BI canbe completely moved to the matter sector and, in combination with M Pl through ¯ M BI ,it controls the scale at which the generated non-linear interactions of the matter fieldsbecome relevant . Interestingly, even fields that do not interact directly in the Born-Infeld frame will couple in the Einstein frame and the coupling will again be controlledby ¯ M BI . The fact that all fields will be generically coupled in the Einstein frame and thecoupling constant ¯ M BI is universal can be nicely interpreted as a consequence of dealingwith a gravitational theory, i.e., as a sort of additional Born-Infeld equivalence principle .In other words, the Born-Infeld inspired theories have the usual equivalence principle,according to which all matter fields couple to gravity with a universal coupling constant M Pl (fully valid on scales below ¯ M BI ), and what we have called Born-Infeld equivalenceprinciple, according to which all the generated couplings in the matter sector come inwith another universal coupling constant ¯ M BI . Since we have not observed any anomalousinteractions beyond those of the standard model at LHC, we can straightforwardly imposethe very conservative constraint ¯ M BI &
10 TeV, which translates into M BI & − eV sothat the Born-Infeld corrections can only have effects in regions of spacetime where thecurvature is larger than 10 − eV .The couplings generated in the matter sector bring about one important point thatis usually overlooked in the literature and has not been properly addressed yet, namelywhether, or under which conditions, the quantum corrections can remain under control in Since the source of gravity in most situations is the energy density ρ , the transition between the usualGR and the Born-Infeld regimes in the gravitational sector is expected to occur when ρ ∼ ¯ M , as we willconfirm in the numerous applications studied in the subsequent sections. However, we should point outthat this is only true in the simplest scenarios, but, in general, the Born-Infeld corrections will becomerelevant whenever some interactions reach ¯ M BI . To give an example, one could imagine a situation wherethe densities are small as compared to ¯ M , but some anisotropic stresses or heat fluxes are of order 1 ascompared to the scale ¯ M BI . a priori because the couplings generated in thematter sector controlled by ¯ M BI will usually contain non-renormalisable operators and, infact, one would naively expect ¯ M BI to play the role of a strong coupling scale and, thus,the effects at that scale will require non-perturbative analysis for large background fieldconfigurations. This however does not necessarily mean that the Born-Infeld regime willinevitably face strong coupling problems. We will give here a taste on a possible situationwhere the Born-Infeld regime can be safe, but a more careful analysis should definitely beperformed. If we consider a massless scalar field in the Born-Infeld frame, in the Einsteinframe we will have a K -essence type of theory where the interactions will be controlledby ¯ M BI . The strong coupling scale in these theories around a trivial background is ¯ M BI and one can apply the standard perturbative analysis because the background value of thefield is smaller than ¯ M BI . The worry comes when the background field takes values near¯ M BI and non-perturbative effects would be expected to become relevant. However, aroundthese non-trivial backgrounds the vacuum value of the scalar field re-dresses the strongcoupling scale so that it can be pushed to values higher than ¯ M BI . This mechanism is atwork for instance in theories featuring a K-mouflage/Kinetic or Vainshtein screening (seefor instance [135, 133, 91, 136, 204, 137, 85]). In these situations the non-linear classicalsolutions can be trusted in the Born-Infeld regime. In this case the scalar will also coupleto other matter fields through ¯ M BI , but again the coupling scale will be re-dressed by thebackground value of the scalar field, so that these interactions can also remain small. Aswe have emphasised, this is only a potential resolution of the strong coupling problemsthat one would expect in these theories, but one should carefully check whether this is theactual situation.Let us end this section by noting that the discussion presented here is not particular ofthe Born-Infeld gravity, but it is a feature of a general class of gravity theories formulated`a la Palatini. We will show this explicitly in section 2.7.1 The Born-Infeld theory of gravity discussed in the previous sections are naturally for-mulated on a spacetime manifold endowed with a general affine connection. Thus, giventhe richness offered by this geometrical framework, it is of no surprise that the Eddington-Born-Infeld theory described so far has found extensions in different directions. Unlikethe case of Born-Infeld electrodynamics, the Eddington-Born-Infeld theory has not beensingled out by resorting to symmetries principles or any other guiding criteria, but ratherit originates from a straightforward transcription of the Born-Infeld Lagrangian for elec-tromagnetism to gravity and taking inspiration from the Eddington affine theory. Thus,Born-Infeld inspired gravity is more prone to modifications and extensions than its electro-magnetic relative. However, before proceeding to review the existing models and in viewof the zoology of Born-Infeld inspired gravity theories found in the literature, we find itconvenient to introduce some taxonomic system. We will classify the theories according totheir proximity to the original Born-Infeld spirit, consisting in modifying the high curva-ture regime of gravity without introducing additional fields or pathologies. Furthermore,we will take the EiBI theory as the baseline because it is the most extensively studiedmodel. After these considerations, we have decided to make the following classification:40
Class 0 . We start our classification with a class comprising all those early attemptsof building gravity theories `a la Born-Infeld which did not succeed due to the pres-ence of pathologies. Subsequent proposals sharing these pathologies will also beconsidered to belong to this class. • Class I . Here we will include the EiBI theory and the modifications that are theclosest to the Born-Infeld spirit and do not introduce additional ingredients, be itnew degrees of freedom or additional geometrical objects. • Class II . A next step with respect to the Class I is to allow for more generalgeometrical objects, but respecting the Born-Infeld philosophy, i.e., only the highcurvature regime is modified and no additional degrees of freedom are present. • Class III . Under this category we will classify those models where the Born-Infeldstructure remains but additional degrees of freedom are included. • Class IV . Finally, in this class we will include theories that, although resemble Born-Infeld theories in some aspect, they could be very well classified within a differentclass of theories.The above classification does not intend to be exhaustive nor having sharp edges. Forinstance, sometimes the presence of additional degrees of freedom might depend on somesubtle assumptions on the theory or its solutions so that the same theory can have slightlydifferent versions belonging to different classes. In those cases, we have opted by classifyingit according to the most extensively used version in practical applications.A substantial part of the formal developments and equations for many of the Born-Infeld inspired theories share numerous similarities among them and with the theory dis-cussed so far. For that reason, prior to discussing specific theories we will present a generalframework applicable to most of them.
In this section we will discuss some features that are common to a large class of theories,that include many of the proposed extensions and which are shared with EiBI gravity. Letus consider a general theory of the form S = 12 M M Z d x √− gF (cid:0) g µν , R µν (Γ) (cid:1) (2.71)where F is a function of the inverse of the metric and the Ricci tensor. Notice that we haveincluded a factor √− g in the measure so that F behaves as a true scalar. For simplicity, wewill assume that the function will only depend on the combination P µν = g µα R αν /M ,where we have introduced the scale M BI for dimensional reasons. This is also the usual casein the literature so it will suffice for us . An important consequence of the function being ascalar is that F ( ˆ A − ˆ P ˆ A ) = F ( ˆ P ) for any non-degenerate transformation ˆ A . Furthermore, A treatment of more general theories can be found for instance in [60]. P can be expressed as traces of powers of [ ˆ P n ]. Byusing the Cayley-Hamilton theorem, we can express any power higher than 4 in terms oflower powers so that the action could in principle be written as F ( X , X , X , X ) with X n = [ ˆ P n ]. This is useful to show some general properties of this general class of theories.We will not make extensive use of the advantages introduced by writing the action in thisform and we will instead consider the action written as S = 12 M M Z d x √− gF (cid:0) ˆ P (cid:1) . (2.72)In order to recover GR in the limit | P µν | ≪ ∂F∂P µν (cid:12)(cid:12)(cid:12) ˆ P =0 = δ µν . (2.73)Notice that this is not really a constraint and any analytic function will satisfy it up to aconstant factor that can be absorbed into M . The Einstein-Hilbert action is recoveredfor F ( ˆ P ) = P αα , in which case the above relation is exactly fulfilled for all values of ˆ P andnot only at ˆ P = 0. To be completely precise we should say that the above condition willguarantee the existence of one branch of solutions that will be continuously connected withGR at low curvatures. Nevertheless, the non-linearity of the equations can, in general,present several branches and some of them will give a different behaviour for the lowcurvatures regime. We will encounter specific examples where this situation occurs whenstudying explicit solutions.For the general action considered, we can obtain the corresponding field equations bytaking with respect to both the metric and the connection, yielding δ S = 12 M M Z d x √− g (cid:20) − F g µν δg µν + 1 M ∂F∂P µα (cid:16) δg µν R να + g µν δR να (cid:17)(cid:21) . (2.74)For the subsequent developments, it is convenient to write the above variation in matrixnotation δ S = 12 M M Z d x √− g Tr (cid:20) − F ˆ gδ ˆ g − + 1 M (cid:18) ∂F∂ ˆ P ˆ R T δ ˆ g − + ˆ g − ∂F∂ ˆ P δ ˆ R T (cid:19)(cid:21) (2.75)Now it will be useful to introduce some definitions before proceeding any further. First,let us define √− qq αν ≡ √− gg νµ ∂F∂P µα (2.76)or in matrix notation √− q ˆ q − ≡ √− g (cid:18) ˆ g − ∂F∂ ˆ P (cid:19) T . (2.77)This definition is not an innocent choice and we will see later that q µν will actually playthe role of the auxiliary metric determining the connection, as in the EiBI case. We cantake determinants in both sides of (2.77) to obtain the relation gq = 1det ˆ F ˆ P (2.78)42here we have introduced the notation ˆ F ˆ P ≡ ∂F/∂ ˆ P . Then, we can re-write the definition(2.77) as ˆ q − = 1 q det ˆ F ˆ P (cid:18) ˆ g − ∂F∂ ˆ P (cid:19) T , (2.79)or, if we invert both sides, we finally obtain an expression for q µν as follows:ˆ q = q det ˆ F ˆ P "(cid:18) ∂F∂ ˆ P (cid:19) − ˆ g T . (2.80)For the Einstein-Hilbert term, the derivative of F gives the identity and ˆ q exactly coincideswith the spacetime metric, as expected. If we consider f ( R ) types of theories for which F f ( R ) = F ( P αα ), the derivative gives ∂F f ( R ) ∂ ˆ P = F ′ f ( R ) (2.81)so we have that q µν = F ′ f ( R ) g µν , recovering the known result that in these theories thetwo metrics are conformally related. Finally, in the case of the EiBI action (2.34) we have F EBI = 2 q det (cid:0) + ˆ P (cid:1) so its derivative is (cid:18) ∂F EBI ∂ ˆ P (cid:19) T = q det (cid:0) + ˆ P (cid:1)(cid:16) + ˆ P (cid:17) − . (2.82)When inserting this expression into the definition (2.77) we obtain √− q ˆ q − = √− g r det (cid:16) + ˆ P (cid:17)(cid:16) + ˆ P (cid:17) − ˆ g − = s − det (cid:18) ˆ g + 1 M ˆ R (cid:19) (cid:18) ˆ g + 1 M ˆ R (cid:19) − (2.83)and we recover that q µν = g µν + M R µν as it should. After this little satisfaction, wecan continue with the computation of the field equations. Another useful relation for thevariation of the action that we can obtain from the definition of q µν is the following: √− gM ∂F∂ ˆ P ˆ R T = √− q (cid:16) ˆ g ˆ P ˆ q − ˆ g (cid:17) T . (2.84)With the new jargon, we can re-write the variation (2.75) as δ S = 12 Z d x Tr h(cid:16) √− g L G ˆ g − δ ˆ g − M M √− q (cid:0) ˆ q − (cid:1) T ˆ P T (cid:17) δ ˆ g + M √− q ˆ q − δ ˆ R i (2.85)where we have used the ciclic property of the trace and the identity ˆ gδ ˆ g − = − δ ˆ g ˆ g − .Furthermore we have re-introduced the Lagrangian L G = M M F . From the last termwe can already sense that q µν will be related to the metric generating the connection, sincethat piece resembles the variation one would obtain from the Einstein-Hilbert action in43he Palatini formalism with a metric q µν . A word of caution is necessary though, since q µν does not need to be symmetric at this point. Again, if we assume a projective symmetryso only the symmetric Ricci enters, only the symmetric part of q µν will contribute and,thus, it will exactly be the auxiliary metric. Prior to the discussion of the connection fieldequations, let us first write the metric field equations:12 M M √− q (cid:20) ˆ P ˆ q − + (cid:16) ˆ P ˆ q − (cid:17) T (cid:21) − √− g L G ˆ g − = √− g ˆ T (2.86)where the symmetrization follows from the symmetry of g µν and we have also added theenergy-momentum tensor of the matter sector. For the sake of completeness, we will alsogive the expression of this equation in components M M √− qq α ( µ P ν ) α − √− g L G g µν = √− gT µν . (2.87)As one of our favourite exercises, let us check that we recover the expected results whenthe above equation is particularised to known cases. For the Einstein-Hilbert action,we have already seen that q µν = g µν and it is immediate to see that (2.86) reduces to R ( µν ) − R g µν = M T µν . For the Born-Infeld inspired theory with F EBI = 2 q det (cid:0) + ˆ P (cid:1) we have also shown that ˆ q reduces to the expected result. In that case, it is easy to see fromˆ g − ˆ q = + ˆ P that ˆ P ˆ q − = ˆ g − − ˆ q − . If we insert this relation into (2.86) and use that √− g L G = M M √− q , we can see that the equations reduce to − M M √− qq ( µν ) = √− gT µν , in agreement with (2.38) (taking λ = 0).Let us pause a bit before moving on to the connection field equations to discuss thestructure of the metric field equations. In general, the symmetry of the metric resultsin a set of ten independent equations. The general treatment of theories `a la Palatinirequires the use of these equations to solve for the Ricci tensor (or connection-dependentobjects for more general theories) in terms of the metric and the matter fields. Thisstep is algebraic and it is crucial for the subsequent resolution of the connection as theLevi-Civita connection of some auxiliary metric. However, while the Ricci tensor has ingeneral sixteen components, the metric field equations are limited to ten and, therefore,the full Ricci cannot be obtained from them. This means that the method to solve theconnection as the Levi-Civita connection will fail. This motivates considering theorieswith the projective symmetry for simplicity reasons.Let us know turn to the computation of the connection field equations. By lookingat the last piece of (2.85) we can see that it reads exactly the same as the correspondingvariation for the Born-Infeld case in (2.41). Hence, the derivation will follow analogouslyand we can simply use the equations already obtained in (2.44) ∇ λ h √− q (cid:0) ˆ q − (cid:1) µν i − δ µλ ∇ ρ h √− q (cid:0) ˆ q − (cid:1) ρν i = ∆ µνλ + √− q h T µλα (ˆ q − (cid:1) αν + T ααλ (ˆ q − (cid:1) µν − δ µλ T ααβ (ˆ q − (cid:1) βν i (2.88)where we only need to remember that now ˆ q is defined in (2.76) and we have added the hypermomentum of the matter fields defined in (2.45). If only the symmetric part of the44icci tensor enters in the action, so that we have a projective symmetry Γ αµν → Γ αµν + ξ µ δ αν ,we can see from (2.74) that only the symmetric part of q µν will contribute to the connectionfield equations. In that case we can easily see that the trace with respect to ρ and ν vanishes identically, as a consequence of the projective invariance. Sometimes, this isregarded as a flaw of these theories because, in case the symmetry is not present in thematter sector, there is no reason to expect to have ∆ µρρ = 0 and this would be the sourceof an inconsistency in the equations. However, there is an obvious way to evade thisapparent problem by assuming that matter fields do not couple to the connection directlyso that we actually have that the full hypermomentum vanishes. Again, this is the casefor minimally coupled bosonic fields, but complications might arise due to fermions. Inany case, even if we need to have ∆ µρρ = 0, this should be regarded as a constraint inthe matter sector and there is no reason a priori to assume that solutions satisfying thatconstraint cannot be found . For simplicity and to comply with most of the literature wewill take ∆ µνρ = 0 in the following.On the other hand, if there is no projective symmetry in the action, the object q µν will not have, in general, any defined symmetry. In that case, the equations have a formalresemblance with non-symmetric gravity theories [255, 129, 350, 58] so one could try toapply the same techniques to solve the equations. However, the similarities are purelyformal and, in fact, there are profound conceptual differences between the non-symmetricgravity theories and the ones under study here, mainly the absence of an actual non-symmetric metric.We will manipulate the equations to recast them in more useful forms. We can firsttake the trace of the equations with respect to µ and λ to obtain that ∇ λ (cid:16) √− qq λν (cid:17) = 12 T λλα √− qq αν (2.89)If we plug this relation back into the equations we obtain ∇ λ (cid:16) √− qq µν (cid:17) + √− q (cid:18) δ µλ T ααβ q βν − T µλα q αν − T ααλ q µν (cid:19) = 0 . (2.90)Now, it is convenient to introduce the shifted connection˜Γ αµν = Γ αµν − T λµλ δ αν (2.91)that satisfies ˜Γ ααµ = ˜Γ αµα and it is invariant under a projective transformation of the originalconnection Γ αµν , i.e., we have that ˜Γ αµν → ˜Γ αµν when Γ αµν → Γ αµν + ξ µ δ αν for an arbitrary ξ µ . This will play a crucial role in the following because it means that the connection ˜Γ In order to illustrate this point, let us remember the case of a Proca field coupled to conservedcurrents whose equations read ∂ ν F µν + m A µ = J µ . The gauge invariance of the charged sector impliesthe conservation of the current ∂ µ J µ = 0, while the mass term for the vector field breaks the gaugeinvariance in the vector field sector. However, this does not introduce any inconsistency in the equationsas, by taking their divergence one obtains the constraint ∂ µ A µ = 0 which, not only it does not represent aninconsistency, but it plays in fact a crucial role to remove additional polarizations for the massive vector. √− q ∂ λ (cid:16) √− qq µν (cid:17) + ˜Γ µαλ q αν + ˜Γ νλα q µα − ˜Γ αλα q µν = 0 . (2.92)If we take the two possible traces of these equations and subtract them we find ∂ λ (cid:16) √− qq [ µλ ] (cid:17) = 0 (2.93)and, thus, the antisymmetric part of q µν satisfies a Maxwell-like equation. Another usefulrelation is obtained by multiplying (2.92) by √− qq µν to obtain ∂ λ log √− q = ˜Γ ααλ , (2.94)which can then be used in (2.92) to finally write the equations as ∂ λ q µν + ˜Γ µρλ q ρν + ˜Γ νλρ q µρ = 0 (2.95)or, if we multiply by q αµ q νβ , in the equivalent way ∂ λ q αβ − ˜Γ µβλ q αµ − ˜Γ µλα q µβ = 0 . (2.96)These equations will determine the connection ˜Γ in terms of q µν and, thus, the originalconnection Γ up to the aforementioned projective mode. We can do a bit better byfollowing the usual procedure to compute the connection in terms of the metric, i.e., wesubtract appropriate permutations of indices from (2.96) to write it in the following form: q ( µλ ) ˜Γ µαβ = 12 (cid:16) ∂ α q βλ + ∂ β q λα − ∂ λ q αβ (cid:17) + q [ αµ ] ˜Γ µβλ + q [ µβ ] ˜Γ µλα . (2.97)This expression is crucial to understand many features of the theories under considerationthat will in turn determine many of their properties. Let us stress that we have notconsidered any simplifying assumption, so our result is completely general. This pays ourdebt to the meticulous reader, who was promised a more thorough analysis in section 2.5.1.The first thing to notice is that, for a symmetric q µν , the solution for the connection ˜Γ isnothing but the usual Levi-Civita connection of q µν . Of course, the matrix q µν as definedin (2.76) depends on the Ricci and, thus, on the curvature. As usual, the resolution tothis is that q µν can be algebraically solved from the metric field equations (2.87). Theconnection ˜Γ is thus solved as the Christoffel symbols of q µν and this is how ˆ q earns itsdenomination of auxiliary metric in the general case. All this reasoning is however basedon the assumption that q µν is symmetric, but this is not an outrageous wish to ask and,in fact, it will be nicely granted by the projective invariance. To see this, we can expressthe definition of q µν in terms of derivatives with respect to the Ricci tensor as follows √− qq αν ≡ √− gg νµ ∂F∂P µα = √− gg νµ ∂F∂ R ρσ ∂ R ρσ ∂P µα = M √− g ∂F∂ R να (2.98)46here we have used the definition of ˆ P to compute its derivative with respect to ˆ R .From here, we see that the matrix q µν will inherit the symmetries of the Ricci tensor.In particular, if only the symmetric part of the Ricci enters the action, then q µν willautomatically be symmetric and its Levi-Civita connection will be the solution for ˜Γ.Equivalently, if only the symmetric part of the Ricci appears in the action, only thesymmetric part of q µν will contribute to the connection field equations. On the otherhand, it is also very easy to see that, in that case, the metric field equations permit toobtain q µν (the number of equations will coincide with the number of components of ˆ q )and, thus, the usual procedure giving the connection as the Levi-Civita of the auxiliarymetric q µν is fully consistent.This is an appropriate place to make some remarks on these results. The first one isthat the connection has only been obtained up to a projective mode. However, this doesnot represent a flaw and, in fact, rather the opposite for theories based on a symmetricRicci. For those theories, there is a projective gauge symmetry that will necessary beresponsible for the presence of undetermined modes in the solutions. In other words, theapparent undetermined projective mode will be innocuous and can be removed by a simplegauge fixing. This also applies to the case of the Einstein-Hilbert action and it is preciselythe discussion we exposed below (2.32). Hence, for theories with the projective symmetry,the whole resolution of the field equations is consistent and, at least formally, achievable.So far we have discussed the case when q µν is symmetric by definition. Things can bequite different when this condition is abandoned. In that case, we find problems in the twosets of equations, namely the metric and the connection equations. For the metric equa-tions, we find the trouble already discussed above that, while the metric field equationsprovide ten independent equations, q µν will in general have sixteen independent compo-nents and, therefore, it cannot be fully expressed in terms of the spacetime metric and thematter content. Concerning the connection field equations and its polished expression in(2.97), simply obtaining ˜Γ in terms of the non-symmetric q µν is an arduous task. In fact, intheories with non-symmetric metrics, the solution is usually obtained only perturbativelywith respect to the antisymmetric part of the metric [255, 129, 350, 58]. This gives furthermotivation to consider only theories with the projective symmetry, but theories without itwill definitely present a much richer structure. In particular, they will likely contain addi-tional degrees of freedom, among which there could be propagating torsion. Additionally,the results obtained here give support to the simplifying assumption of vanishing torsionupon which the results of section 2.5.1 were obtained.For the projectively invariant theories, we can also make contact with the previousformalism developed in the case of EiBI theories and the definition of the deformationmatrix relating the spacetime and the auxiliary metrics. If we remember the relationbetween both metrics defined in (2.57) as ˆ q = ˆ g ˆΩ we see that we can re-write (2.79) in asimilar form by defining ˆΩ − = 1 q det ˆ F ˆ P (cid:18) ∂F∂ ˆ P (cid:19) T . (2.99)As one would require, the condition (2.73) imposed to recover GR in the low curvaturesregime implies that ˆΩ ≃ in that limit, so that both metrics coincide when | ˆ P | ≪
1. An47mportant derived relation is that, in four dimensions, we have det ˆΩ = det ˆ F ˆ P . Let usalso notice that the Lorentzian signature for the auxiliary metric will be guaranteed aslong as the derivative ˆ F ˆ P is positive definite or, equivalently, if the deformation matrixˆΩ is positive definitive. In Born-Infeld inspired theories of gravity, this is usually relatedto the existence of the square root of a matrix characteristic of those theories, which isthen imposed as a condition on physical solutions. In the general case, we will need toimpose the deformation matrix be positive definite for physical solutions. This will in turnguarantee that q det ˆ F ˆ P is a real quantity. We can now follow the same procedure as wedid with the EiBI theory and obtain an algebraic equation for the deformation matrix byintroducing ˆ q − = ˆΩ − ˆ g − into (2.86) and multiplying by ˆ g to obtainˆΩ − ˆ P = 1 M M p det ˆΩ (cid:16) L G + ˆ T ˆ g (cid:17) , (2.100)where we have used that ˆ P and ˆΩ commute and the property g − ( ˆΩ − ˆ P ) T ˆ g = ˆΩ − ˆ P .This is an algebraic equation for the deformation matrix provided ˆ P can be expressed interms of ˆΩ by inverting (2.99). Now, if we use that ˆ q − ˆ R = M ˆ q − ˆ g ˆ P = M ˆΩ − ˆ P wefinally obtain the differential equations satisfied by the auxiliary metric R µν ( q ) = 1 M p det ˆΩ (cid:16) L G δ µν + T µν (cid:17) (2.101)in complete analogy with the equations (2.60) obtained for the EiBI case. This provesour claim that those equations are valid for general theories. Furthermore, the sameconclusions drawn there are automatically valid for this more general case. In particular,in the low curvatures regime the deformation matrix is the identity and the Lagrangian is L G ≃ M R by construction and, thus, we recover the usual Einstein equations.To end this section, let us extend the discussion on the existence of two frames shownfor the Born-Infeld case to the more general theories considered here. In view of thediscussions so far about the structure of the theories, it should be clear by now thatassuming a projective symmetry would be a wise decision on the grounds of simplicity.Very much like we did for EiBI, let us go to a bi-metric representation of the theory byintroducing an auxiliary field Σ µν as follows: S = 12 M M Z d x √− g (cid:20) F ( g µν , Σ µν ) + ∂F∂ Σ µν (cid:18) M R ( µν ) − Σ µν (cid:19)(cid:21) + S M [Ψ , g µν ](2.102)We can see that Σ µν can be integrated out by solving its own equation of motion andwe recover the original action. We have considered the case with projective invariance to Both properties can be easily proven by assuming that F is an analytic function so that ˆ F ˆ P and, as aconsequence, ˆΩ are analytic matrix functions of ˆ P . If we have an arbitrary analytic function ˆ F of ˆ P we canexpand it as ˆ F = P n c n ˆ P n from where it is trivial to see that it commutes with ˆ P . Furthermore, we can alsoshow that ˆ g ˆ F ( ˆ P )ˆ g − = ˆ F (ˆ g ˆ P ˆ g − ) = ˆ F ( ˆ P T ) = ˆ F T ( ˆ P ), where we have used that ˆ g ˆ P ˆ g − = M − ˆ R ˆ g − = ˆ P T which is valid whenever the Ricci tensor ˆ R is symmetric or, as in our case, when only its symmetric partis considered. From this relation we can obtained the desired property by simply taking ˆ F = ˆΩ − ˆ P . µν is symmetric. Now we can introducea field re-definition as √− qq µν = √− g ∂F∂ Σ µν (2.103)that can be used to obtain Σ µν = Σ µν (ˆ g, ˆ q ) so that the action can be expressed as S = 12 M Z d x h √− qq µν R µν + √− gM U (ˆ g, ˆ q ) i + S M [Ψ , g µν ] (2.104)with U (ˆ g, ˆ q ) = F − ∂F∂ Σ µν Σ µν . (2.105)In this action, the spacetime metric g µν appears as an auxiliary field (provided the matterfields are minimally coupled) so it can be integrated out. Its equation is simply ∂ U ∂g µν − U g µν = 1 M M T µν (2.106)which allows to solve algebraically for g µν in terms of q µν and the matter fields, similarlyto the case of Born-Infeld. Thus, the original action can be alternatively expressed as S = 12 M Z d x √− qq µν R µν (Γ) + ˜ S M [Ψ , q µν ] (2.107)and we see again that the theory is equivalent to GR but with modified couplings tothe matter fields. Hence, the same discussion presented in section 2.6 applies to themore general class of theories considered here. The Born-Infeld frame introduced in thatsection naturally extends to a more general affine frame within the framework of thegeneral class of theories discussed here. This naturally motivates an extension of the Born-Infeld equivalence principle to a more general affine equivalence principle with the sametheoretical and phenomenological consequences, in particular the constraint M BI & − eV obtained by imposing the absence of anomalous interactions at LHC also applies here.Notice however that some exceptions exist where this argument fails, since we are assumingthat (2.105) can be inverted to express Σ µν in terms of g µν and q µν and similarly for (2.106)that allows to integrate out g µν . One important example of theories where this argumentis not applicable is the case of f ( R ) theories. In that case, only the trace of Σ enters(2.103) so it is not possible to invert it and obtain Σ µν = Σ µν (ˆ g, ˆ q ). As it is well-known,in that case it is a better idea to add a scalar field in the Legendre transformation insteadof Σ µν . As we showed above, for these theories q µν and g µν are conformally related.After the general considerations discussed in this section, let us turn to consideringspecific examples of extensions of Born-Infeld gravity corresponding to the classificationintroduced above. We will classify under this category those theories aiming at modifying GR in the highcurvature regime with a Born-Infeld type of modification, but which fail in fulfilling some49rucial consistency requirement, like the presence of unavoidable ghost-like instabilities.Into this class will go the first attempts towards Born-Infeld gravity explained in thesections 2.2 and 2.3 that were based on the metric formalism. As extensively discussedthere, the higher order field equations for the metric arising in those theories compromisetheir stability due to the presence of ghosts.As another example of Born-Infeld inspired gravity theories that would belong to thisclass we can mention theories consisting of a Born-Infeld sector formulated in the affineapproach (similar to the EiBI Lagrangian) supplemented with another sector formulatedin the metric formalism. This type of action was already considered by Ba˜nados in [44] andsome phenomenological consequences were explored in [319, 47, 46]. In view of the analysisperformed in section 2.7.1, it is clear that these theories are generally plagued by ghost-likeinstabilities, similarly to the original attempts made in the pure metric formalism. Theproblem with these theories is precisely the presence of the metric sector. We can repeatthe same construction leading to (2.104), but now with the additional sector formulatedin the metric formalism. If we take such a sector to consist of an Einstein-Hilbert termfor the metric g µν , as it was the case considered in [44], we end up with the equivalentaction S = 12 M Z d x h √− gg µν R µν ( g )+ √− qq µν R µν + √− gM U (ˆ g, ˆ q ) i + S M [Ψ , g µν ] , (2.108)so we have a bi-metric theory where both metrics are coupled through U (ˆ g, ˆ q ). If there wasno metric sector explicitly making g µν a propagating field, the spacetime metric could beintegrated out and we would be left with only one propagating metric, as we obtained in(2.107). However, having an independent Einstein-Hilbert term for the spacetime metricmakes it a propagating field and, thus, the action (2.108) shows that we can no longerintegrate the metric g µν out. The result of this is that we have a bi-metric theory wherethe two metrics are dynamical and interact through the potential U (ˆ g, ˆ q ). Unless theinteractions encoded in that potential belong to the class of ghost-free bi-gravity type[134, 196], the theory will contain the so-called Boulware-Deser ghost [82] and, therefore,the theory will be unstable. In general, the absence of ghosts will then be guaranteed ifthe following condition holds U (ˆ g, ˆ q ) = F − ∂F∂ Σ µν Σ µν = X n =0 β n e n (cid:0)p ˆ g − ˆ q (cid:1) , (2.109)where the terms in the last sum are the massive gravity and bi-gravity potentials writtenin terms of the the elementary symmetric polynomials e n defined in (1.26). One can easilycheck that the EiBI action does not fulfill this condition and, thus, the theory will containthe undesired ghostly mode. A construction with auxiliary fields that somehow connectthe EiBI Lagrangian with bi-gravity theories as different branches of the same underlyingtheory was presented in [324], but our discussion here differs from the one given there.We should notice that this is in fact a general result for theories mixing sectors formu-lated in the metric and in the affine formalism that go under the name of hybrid theories The same will however apply if we consider more general metric sectors like, e.g., f ( R ) terms. We identify this class as the one containing the most extensively studied case in theliterature, i.e., the EiBI reviewed in the precedent subsections, as well as its extensions.The most immediate class of extensions of the EiBI gravity is to consider some sort offunctional extension. As we have extensively seen above, the fundamental object in EiBIgravity is the determinant det (cid:20) g µν + 1 M R µν (Γ) (cid:21) (2.110)in terms of which the action is written. One of the reasons to introduce the determinant isto guarantee the diffeomorphisms invariance of the volume element because the determi-nant of a rank-2 covariant tensor transforms as a scalar density of weight w = −
2. Thus,in order to introduce functional extensions of the EiBI theory, it is more convenient torewrite the action in the following form: S EiBI = M M Z d x √− g r det (cid:16) + ˆ P (cid:17) (2.111)with P µν ≡ M g µα R αν . (2.112)From here one can straightforwardly perform functional extensions in different directionsthat are discussed in the following sections. There can be slightly different versions of thetheory depending on whether the connection is assumed symmetric a priori or if only thesymmetric part of the Ricci tensor is considered, as we have seen above, and this couldalso be the origin of differences in the formulation of the theories. • Arbitrary function of the determinant
It is a common practice in modified gravity to generalise theories by introducing ar-bitrary functions of the defining quantities as it is done for instance in f ( R ) or f ( R, G )theories where arbitrary functions of the Ricci scalar and/or the Gauss-Bonnet term areconsidered. Thus, probably the first extension one could think of for the EiBI theory is51aking an arbitrary function of the defining determinant. This was done in [268] wherethe following extension of Born-Infeld was considered S = M M Z d x √− gf (cid:0) X (cid:1) (2.113)with X = det( + ˆ P ). The EiBI theory is recovered for f ( X ) = X / . As usual, thefunction f should be chosen so that we recover GR at small curvatures. The action inthat limit can be obtained by expanding the function around X = 1 so we have S ≃ M M Z d x √− gf ′ (1) (cid:16) X − (cid:17) , (2.114)which only differs from the original EiBI theory by the factor f ′ (1) so we need to impose f ′ (1) = 1 to have the correct limit at low curvatures. The generality introduced byconsidering an arbitrary function of the determinant can be handled in the usual way byintroducing a Legendre transformation with an auxiliary field φ as S = M M Z d x √− g h f ( φ ) + f φ (cid:16) X − φ (cid:17)i , (2.115)followed by a field redefinition ϕ = f φ so that the action can be written in the equivalentway S = M M Z d x √− g h ϕ X − V ( ϕ ) i . (2.116)The field equation of the scalar field imposes the constraint X = V ,ϕ (2.117)which can be eventually incorporated in the final form of the equations. The procedurepresented above for general theories can be straightforwardly applied to this case and onefinds that the auxiliary metric reads q µν = ϕ √X (cid:18) g µν + 1 M R ( µν ) (cid:19) (2.118)while the deformation matrix ˆΩ relating the two metrics is given byˆΩ = ϕ √X (cid:16) + ˆ P (cid:17) . (2.119)After some more manipulations along the lines of the general case depicted in the previoussection, one can finally write the equations as R µν ( q ) = 12 M ϕ X / (cid:16) M M f ( X ) δ µν + T µν (cid:17) , (2.120) We adapt the notation of that reference to be consistent with the notation of this review, so we reserveˆ q and ˆΩ for the auxiliary metric and the deformation matrix respectively. • Extension to all the elementary symmetric polynomials
A slightly different way of writing the EiBI action leads to another class of extensions.By commuting the square root and the determinant in (2.111), we can alternatively writethe action as S EiBI = M M Z d x √− g det ˆ M (2.121)where the matrix ˆ M has been defined as ˆ M ≡ p ˆΩ = p + ˆ P . Now, since the determinantof a matrix is nothing but the invariant elementary symmetric polynomial of highestdegree, the EiBI action rewritten as (2.121) calls for a natural extension including the fullseries of elementary symmetric polynomials of the fundamental matrix ˆ M . This is thepath taken in [59] that led to the family of Born-Infeld inspired theories described by thefollowing actions S GBI = M M Z d x √− g X n =0 β n e n ( ˆ M ) . (2.122)with β n some dimensionless constants and e n ( ˆ M ) the elementary symmetric polynomialsdefined as e ( ˆ M ) = 1 ,e ( ˆ M ) = [ ˆ M ] ,e ( ˆ M ) = 12! (cid:16) [ ˆ M ] − [ ˆ M ] (cid:17) ,e ( ˆ M ) = 13! (cid:16) [ ˆ M ] −
3[ ˆ M ][ ˆ M ] + 2[ ˆ M ] (cid:17) ,e ( ˆ M ) = 14! (cid:16) [ ˆ M ] −
6[ ˆ M ] [ ˆ M ] + 8[ ˆ M ][ ˆ M ] + 3[ ˆ M ] −
6[ ˆ M ] (cid:17) . (2.123)As commented above, the fourth symmetric polynomial coincides with the determinanti.e. e ( ˆ M ) = det ˆ M so that the β term contributes the usual EiBI Lagrangian. The lowcurvature limit | g µα R αν | ≪ M gives S ≃ M M Z d x √− g h(cid:16) β + 4 β + 6 β + 4 β + β (cid:17) + 12 M (cid:16) β + 3 β + 3 β + β (cid:17) g µν R µν (Γ) i (2.124)which coincides with the Einstein-Hilbert action in the Palatini formalism supplementedwith a cosmological constant term (which can be cancelled by tuning the parameter β ),provided we impose β + 3 β + 3 β + β = 1. The projective symmetry always appearshere as an accidental symmetry of the low curvature action and it will only be a symmetryof the full theory if the elementary symmetric polynomials are constructed in terms of thesymmetric Ricci tensor. Let us now consider the high curvature limit where | g µα R αν | ≫ . This means that ˆ M ≃ p ˆ P and, therefore, the action in this regime turns intoa combination of the elementary symmetric polynomials of p ˆ P . In the presence of allthe polynomials, this regime will be dominated by the fourth one and we will recover anEddington-like action S ≃ β M M Z d x q det R µν (Γ) . (2.125)In the general case, the Born-Infeld regime will be determined by the highest degreepolynomial present in the action. The case of e admits an amusing interpretation sinceits Born-Infeld regime gives S = ˜ m Z d x √− g (cid:18)h ˆ g − ˆ R i − hq ˆ g − ˆ R i (cid:19) (2.126)with ˜ m some scale. This theory could even be treated in the metric formalism. Nowif we interpret the operation of tracing as a type of averaging, the above action canbe interpreted as being the variance of q ˆ g − ˆ R . Despite its amusing interpretation, itsphysical viability is dubious since it likely gives rise to observational conflicts and a lackof hyperbolicity in the field equations might. However, these issues should be exploredbefore reaching a definite conclusion.Again, we can apply the machinery developed above for the general case to this par-ticular family of theories with the identification F ( ˆ P ) = 2 X n =0 β n e n ( ˆ M ) (2.127)where ˆ M = p + ˆ P . The derivative of this function can be computed as ∂F∂ ˆ P = 2 X n =1 β n X k =1 ∂e n ∂ [ ˆ M k ] ∂ [ ˆ M k ] ∂ ˆ M ∂ ˆ M∂ ˆ P (2.128)where we have made extensive used of the chain rule and dropped the term with n = 0because that is just a cosmological constant term. Now, we will introduce the notation E kn = ∂e n /∂ [ M k ], whose explicit form is given by E kn = e e − e e − e e e − e e − e , (2.129)and use that ∂ [ ˆ M k ] /∂ ˆ M = k (cid:0) ˆ M k − (cid:1) T and ∂ ˆ M /∂ ˆ P = (cid:0) ˆ M − (cid:1) T to finally obtain theauxiliary metric as given in (2.77) for the present case: √− q ˆ q − = √− g (cid:18) ˆ g − ∂F∂ ˆ P (cid:19) T = √− g X n =1 β n X k =1 E kn ˆ M k − ! ˆ g − . (2.130)54his is precisely the result found in [59], where the sums in the brackets corresponds tothe matrix ˆ W defined in that reference. Since the sum over k runs from 1 to 4, the righthand side of (2.130) will contain powers of ˆ M from − √− q ˆ q − = √− g (cid:16) f ˆ M − + f + f ˆ M + f ˆ M (cid:17) ˆ g − (2.131)with f = β e + β e + β e + β e (2.132) f = − ( β e + β e + β e ) (2.133) f = β e + β e (2.134) f = − β e . (2.135)From these expressions one can now straightforward adapt the general formalism for thisfamily of theories and obtain all the relevant equations, which, of course, coincide withthose in [59]. As a particularly simple case, we can take a theory containing only e sothat the action reads S Min = M M Z d x √− g Tr (cid:20)q + M − ˆ g − ˆ R − (cid:21) . (2.136)This is the model that was studied in more detail in [59] and subsequently used in [61]to develop an inflationary scenario. For that case, we have that f = f = f = 0 and f = β is a constant, which is set to 1 in order to recover GR at low curvatures. In thisvery simple case, we can easily compute the deformation matrix from (2.99), which yieldsˆΩ = 1 p det ˆ M ˆ M . (2.137)If we use this relation together with ˆ P = ˆ M − obtained from the definition of ˆ M , theequation for the deformation matrix given in (2.100) can be written in terms of ˆ M asˆ M − − ˆ M − h Tr (cid:16) ˆ M − (cid:17)i = 1 M M ˆ T ˆ g (2.138)which exactly coincides with the equation found in [59]. This equation will give the matrixˆ M in terms of the matter sector and then one can follow the common procedure to solvethe equations. This will be explicitly done in section 5.3, where the cosmology of thismodel will be studied. There is a second class of extensions of the EiBI theories that makes use of additionalgeometrical objects. Let us remind that the original EiBI theory only utilizes the Riccitensor and the metric and its natural arena is a non-Riemannian geometry. The metricaffine formulation of the theory implies the presence of a completely independent connec-tion and its associated curvature encoded in the Riemann tensor R αβµν . For this general55iemann, we can take three independent traces, namely: the Ricci R βν = R αβαν , thehomothetic tensor Q µν = R ααµν and the co-Ricci P αµ = g βν R αβµν . The traces of thesethree objects are all the same and give the Ricci scalar R = g µν R µν . We can see thatthe EiBI theory only makes use of the Ricci tensor, but a much larger variety is possiblethanks to the rich geometrical structure at our disposal. For instance, the determinantalform of EiBI can be extended to include an arbitrary combination of the three differenttraces of the Riemann tensor so we could consider actions of the type S = M M Z d x s − det (cid:20) a g µν + 1 M (cid:16) a R µν + a Q µν + a P µν (cid:17)(cid:21) (2.139)where a i can be arbitrary scalar functions of curvature invariants. In the simplest casewe could take a i = a i ( R ), but other scalars like Gauss-Bonnet combinations or R µν R µν could also be envisaged. Obviously, here we encounter once again a similar obstacle as inthe Deser and Gibbons construction discussed in section 2.2 (although this time avoidingthe ghost problem), namely the lack of a guiding principle. Thus, very much like in thatcase, one can foreshow that any gravitational theory (except for some singular cases) canbe recast in the above form by appropriately tuning the free functions a i . The EiBI theorycorresponds to possibly the simplest among the possible theories described by the action(2.139). Let us stress that we always remain within the Born-Infeld spirit, so we leaveout here well-known theories, like those written in terms of Lovelock invariants, rewrittenin a way that resemble the characteristic square root structure of Born-Infeld theories.Let us notice that we can consider even more general actions by including generaliseddeterminants for the Riemann tensor itself.Faced with the obstruction of lacking some motivation to select extensions of EiBIwithin the Class II, people have resorted to the always welcomed principle of simplicity.In this case, it means that extensions along the lines depicted here have predominantlyresorted to adding new terms only containing the Ricci scalar. This has been consideredin two fashions, either by writing new R -dependent terms outside the EiBI action, as inthe Born-Infeld plus f ( R ) models introduced in [245], or by modifying the determinantalstructure with only R -dependent terms, as was considered in the appendix of that samework [245] and in [106] for a specific case.Another possibility that differs more profoundly from the one sketched so far is toconsider other geometrical frameworks. At this respect, an interesting class of theoriesformulated on a Weitzenb¨ock space was introduced in [166, 167]. It is well-known that GRadmits a formulation in a Weitzenb¨ock space, where the connection is constrained to havevanishing curvature and all the gravitational effects are encoded in the torsion tensor T λµν .This construction goes under the name of Teleparallel Equivalent of General Relativity(TEGR) and it has been used as the starting point of some modifications of gravity, amongthem some Born-Infeld inspired gravity theories that are of interest for us (see section 4.6.4for applications of this theory on black holes). An extensive and comprehensive review onTEGR can be found in [12]. Even though TEGR is GR in disguise, this mask shows aninteresting face for GR as a gauge theory of the inhomogeneous part of the Poincar´e groupwhere the vierbeins are precisely the gauge fields of translations. Thus, TEGR provides a56ery appealing starting point for Born-Infeld modifications of gravity that deserves to beexplored.After discussing some of the different approaches that can be taken to obtain Born-Infeld inspired theories of gravity within the Class II, let us briefly review some specificexamples. • Born-Infeld plus f ( R )A simple extension within this class is to combine the Born-Infeld action with the well-known f ( R ) theories in the metric-affine formalism, as considered in [245, 244, 246, 151].The resulting action adapted to our notation is given by S = 12 M M Z d x " s − det (cid:18) g µν + αM R ( µν ) (cid:19) + √− gf ( R ) = 12 M M Z d x √− g " r det (cid:16) + α ˆ P (cid:17) + f ([ ˆ P ]) (2.140)with α some dimensionless constant. Since the small curvature limit of the EiBI sector inthe above action already gives the Einstein-Hilbert term, we need to impose α + f [ P ] (0) = 0to recover GR at low curvatures. The above action is simply a combination of the EiBI andthe f ( R ) and, as such, the corresponding solutions are expected to interpolate betweenthese two cases. The general formulae obtained in 2.7.1 can be straightforwardly appliedto this case. For instance, the definition of the auxiliary metric given in (2.76) yields √− q ˆ q − = s − det (cid:18) ˆ g + αM ˆ R (cid:19) (cid:18) ˆ g + αM ˆ R (cid:19) − + √− gf [ P ] ˆ g − , (2.141)which coincides with the result found in the literature.A second possibility to extend EiBI gravity by including the Ricci scalar is to includeit in the determinantal structure. This was considered in an appendix in [245] where theauthors considered an action of the form S = M M Z d x s − det (cid:20)(cid:16) f ( R ) (cid:17) g µν + αM R ( µν ) (cid:21) (2.142)as another example of the addition of an f ( R ) piece to the EiBI action. In this case,recovering GR at low curvatures requires to have 4 M f R (0) + α = 0. In [106] this pathwas considered in more detail and the authors explored the cosmology of the followingspecific case: S = M M Z d x s − det (cid:20)(cid:16) βM R (cid:17) g µν + αM R ( µν ) (cid:21) (2.143)with α and β some constants satisfying α + 4 β = 0 in order to recover GR in the limit ofsmall curvatures. 57 Born-Infeld actions in Weitzenb¨ock spaces.
Another example of extensions that we classify within the Class II, but which take adifferent direction, are those based on the teleparallel equivalent of GR. In this descriptionof GR, one makes a fundamental use of the vierbein language and Weitzenb¨ock spaces,characterised by having a curvature-free connection so that the torsion is the only relevantobject. In terms of the vierbein e aµ and its inverse e µa , the connection is given by Γ λµν = e λa ∂ ν e aµ so that the torsion tensor reads T λµν = e λa ( ∂ ν e aµ − ∂ ν e aµ ). From the torsion wecan built a useful quantity called the super-potential and that is given by S ρµν = 14 (cid:16) T νµρ + T νµρ − T ρµν (cid:17) + 12 (cid:16) δ µρ T αν α − δ νρ T αµα (cid:17) . (2.144)With this object, we can construct the Weitzenb¨ock invariant defined as T = S ρµν T ρµν . (2.145)Then, the so-called Teleparallel Equivalent of General Relativity is described by the action S TEGR = 12 M Z d xe T (2.146)where e = det e aµ . That this action is equivalent to GR can be seen from the fact that T for the Weitzenb¨ock connection differs from the Ricci scalar of GR by a total divergence,so that both theories give rise to the same equations of motion. The TEGR, however,serves as an alternative starting point to develop modifications of gravity `a la Born-Infeld.This was pursued in [166, 167, 170], where the authors considered a general expression ofthe form S BITG = M M Z d x "s(cid:12)(cid:12)(cid:12)(cid:12) g µν + 1 M (cid:16) α S µλρ T νλρ + α S λµρ T λνρ + α g µν T (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) − λ √− g , (2.147)with α i some parameters that must satisfy α + α + 4 α = 1 in order to recover (2.146)in the limit of T ≪ M . Similarly to the case of EiBI gravity, the parameter λ controlsthe presence of a cosmological constant. Unlike the proposals discussed so far, this class oftheories must be formulated with the vierbein being the fundamental fields and the generalframework presented in section 2.7.1 cannot be applied to this case. The Born-Infeldextensions along these lines are substantially less explored than those based on the EiBIformulation. However, the Born-Infeld theories based on the TEGR also show interestingfeatures and, furthermore, could be seen to be closer to Born-Infeld electromagnetism,since TEGR can be seen as a gauge theory where the vierbeins play the role of gaugefields associated to the translations group. One might be concerned however with thefact that, similarly to what happens in the models belonging to the class 0 formulatedin the metric formalism, generic theories described by the action (2.147) will introduceinstabilities. In particular, the loss of local Lorentz symmetry when going from TEGRto the action (2.147) will likely introduce additional degrees of freedom. At the time ofwriting this review, it lacks a full analysis of the fields content and their stability aroundrelevant backgrounds of those theories. 58 .7.5. Class III In this category we will include theories based on the Born-Infed structure but whichmake use of additional fields. The most natural example of these theories would be tocombine EiBI gravity with its electromagnetic predecessor or with a Dirac-Born-Infeldscalar field φ , resulting in actions of the form S = M M Z d x s − det (cid:20) g µν + 1 M (cid:16) b R µν + b F µν + b ∂ µ φ∂ µ φ (cid:17)(cid:21) . (2.148)This type of actions are perhaps the most natural combination of Born-Infeld actions forgravity, electromagnetism and/or scalar fields. Already Vollick considered a combinationof this type in [357]. A different approach is to simply add the corresponding Lagrangiansand consider actions of the form S = M M Z d x "s − det (cid:18) g µν + 1 M R µν (cid:19) + c s − det (cid:18) g µν + 1 M F µν (cid:19) + c s − det (cid:18) g µν + 1 M ∂ µ φ∂ µ φ (cid:19) . (2.149)This was considered for instance in [218, 219]. More general actions that belong to this classcan be obtained from the EiBI action formulated in higher dimensions after a dimensionalreduction, for instance by compactifying one extra dimension as done in [159]. Besides the extensions or variations around the EiBI theory discussed so far, thereare other alternatives that make use of some of the ideas characteristic of Born-Infeldtheories, but they could be classified as belonging to other classes of theories. Withinthis category we could mention some of the early attempts to build a Born-Infeld inspiredgravity theory in the metric formalism already discussed in section 2.3. Among them, wecould cite here specific f ( R ) models with a square root or some other bounded function(see also [235, 236]). Although the square root structure introduces some resemblancewith Born-Infeld theories, those models could be classified as belonging to the f ( R ) classof theories. The same would apply to theories involving not only the Ricci scalar, but alsothe higher order Lovelock invariants, in particular, the Gauss-Bonnet term G which is theonly relevant one in four dimensions besides the Ricci scalar. It is possible to constructtheories of the type f ( R, G ) that incorporate some square root structure, as it is consideredin [122]. These theories would then belong to the general class of f ( R, G ) theories. Thesame would apply to theories based on the teleparallel equivalent of GR as for instancein [165] that can be classified as belonging to the f ( T ) extensions of teleparallel theories[92]. We will also include in this class theories making use of the determinantal structure Actually, in [122], the author considers a family of theories that would generically belong to the Class0, but a particular model is eventually selected that would belong to the Class IV and is the one we referto here.
In this section we have provided the reader with a general framework for the study ofBorn-Infeld theories, as well as an overview of the different classes of these theories existingin the literature. We have started by briefly reviewing Born-Infeld electromagnetismand surveyed the attempts to adapt the same ideas to the case of gravity as potentialmechanisms to regularise the divergences appearing in GR. The first attempts formulatedin a metric formalism faced serious shortcomings due to the presence of ghosts. In order tobypass these pathologies, one can introduce higher order corrections to remove the ghostsat a given order, but the large freedom existing in the choice of the counter-terms rendersthe procedure unappealing. It is fair to say that, to date, there is no compelling theoryfree from ghosts that comply with the Born-Infeld philosophy in the metric formalism.A step forward was given when considering Born-Infeld types of actions in the affineapproach. In that case, the ghost is not present from the beginning and the theory canreally be regarded as a proper Born-Infeld theory of gravity, meaning that it modifies thegravitational interaction at high curvatures where a natural bound appears. We wouldlike to remark once again that other attempts that resemble Born-Infeld theories actuallycontain additional degrees of freedom so that they deviate from what we consider shouldbe the spirit of Born-Infeld theories. At this respect, we have introduced a classificationof the different Born-Infeld inspired theories of gravity attending to their closeness to theBorn-Infeld realm.Since the most extensively explored theory within the framework of Born-Infeld ex-tensions of gravity is the EiBI model, we have devoted a substantial effort to showing indetail its main properties, although we have later shown that the same features are sharedby a much larger class of theories. We have provided a detailed derivation of the fieldequations and highlighted the importance of the projective symmetry in the constructionof the theories. In particular, we have seen that theories with that invariance can befully solved in terms of an auxiliary metric and the torsion only enters as an irrelevantprojective mode (under some assumptions on the matter sector). Even though this aux-iliary metric makes its first appearance as an object allowing to solve the connection, wehave seen that it carries physical relevance. This was apparent when we discussed the twoframes existing in these theories. From there, we clearly saw that, while the spacetimemetric determines the causal structure of matter fields, the auxiliary metric determinesthe causal structure of the gravitational waves. This in turn implies that while photonstravel along null geodesics of the spacetime metric, gravitons move along null geodesics ofthe auxiliary metric and, thus, even if both particles are massless, their motion will differin regions where the curvature is large as compared to the Born-Infeld scale.An issue that remains within the affine formulation of Born-Infeld gravity is the lackof clear guiding principles to select a unique family of theories. Born and Infeld followeda symmetry principle that allowed them to single out their non-linear electrodynamics,which was later shown to have a number of remarkable features and it was even related to60tring theory. The same is currently lacking for Born-Infeld inspired theories of gravity.In fact, modifications and extensions of Born-Infeld gravity have flourished in severaldirections. By studying some of the proposed extensions, one can convince oneself thatsome families of theories seem to lead to much simpler equations than others. While thissimplicity principle can be useful to explore the physical consequences of these families oftheories, a more profound and appealing principle would be desired.61 . Astrophysics
A generic feature of extended theories of gravity in which the connection is regardedas independent of the metric (Palatini approach) is the emergence of a dependence of themetric on the local stress-energy densities. This property was soon noticed in the case of f ( R ) theories [272] and its extensions with Ricci-squared R µν R µν terms [283, 275], and isalso present in Born-Infield inspired theories of gravity and its known generalisations, seesection 2.5. This local dependence on the matter fields may at first appear as somethingexotic but is such a basic and fundamental issue in metric-affine theories of gravity thatit must be properly understood in order to handle these theories correctly and properlydefine strategies to test their viability.In this section we will explore situations of astrophysical interest in which the depen-dence of the metric on the local densities of energy and momentum manifests itself veryclearly. In fact, numerous observables of stellar objects are very sensitive to the physicalprocesses taking place in their interiors, whose properties strongly depend on the localdensity. This is the case, for instance, of the mass-radius relation, the mechanisms ofenergy transport, the seismic properties of stars, the type and intensity of neutrino fluxes,the speed of sound profile of acoustic waves in the sun, the potential existence of phasetransitions in terms of ordered (crystalline) and superfluid phases inside neutron stars, thedeconfinement of quarks or the mechanisms of generation of very large magnetic fields.For some reviews on these topics see e.g. [203, 184]. This dependence on the local densitycan thus be used to efficiently test some aspects of this type of modified theories of gravitybut it may also lead to unexpected subtleties. In particular, we will see that the fluid ap-proximation and some models regularly used in the context of GR must be handled withcare or conveniently adapted in order to avoid fictitious forces induced by the averagingprocedure employed in the transition from the microscopic description to the continuouslimit. This will be particularly relevant in the discussion of the outer boundaries of somestelar models both in the relativistic and in the non-relativistic limit.We will begin this section by considering the weak-field, slow-motion limit of theEddington-inspired Born-Infield (EiBI) theory first introduced in section 2.4 , and itsimplications for non-relativistic stars. This will allow us to visualise in a very simpleway where the subtleties of the fluid approximation may arise, which will help us betterunderstand the peculiarities of these theories and identify situations in which an improveddescription of the matter sector may be necessary in order to construct realistic models.We will then move to consider relativistic stars, their structure, and their observationalproperties. A word on the notation of this section
For operational convenience and to make contact with existing literature, both in thissection and in the black holes section 4, we shall redefine part of the notation employedin section 2.4 and redefine Born-Infield mass as M BI = 1 /ǫ and reintroduce Einstein’sconstant in the action via M Pl = 1 / (8 πG ) = 1 /κ . This way, by dimensional consistency For the purpose of this section, we shall consider just the case of both symmetric connection and Riccitensor for this theory, a case discussed in detail in section 2.5.1. has dimensions of length squared, while the Einstein-Hilbert action of GR reads S GR = κ R d x √− gR . To better visualise the local dependence of the spacetime metric on the stress-energydensities, it is useful to study the weak field, non-relativistic limit of EiBI theory given byEq.(2.33). For this theory one finds that, to leading order in the EiBI parameter ǫ , theright-hand side of the Ricci tensor on the field equations (2.59) takes the form R µν ( q ) ≈ κ (cid:18) T µν − g µν T (cid:19) + ǫκ (cid:18) S µν − S g µν (cid:19) , (3.1)where S µν = T αµ T αν − T T µν , while T and S are the trace of T µν and S µν , respectively.This equation indicates that the deviation of the auxiliary metric q µν from the Minkowskimetric will be determined by the total amount of energy-momentum appearing on theright-hand side of this equation. For weak sources, therefore, q µν will be given by theMinkowski metric plus corrections which depend on integrals of the elements on the right-hand side. Now, since g µν is related to q µν via the deformation matrix ˆΩ as defined inEq.(2.57), which in the low density limit is given byΩ µν ≈ δ µν + ǫκ (cid:18) T µν − T δ µν (cid:19) (3.2)the relation between the perturbations in q µν ≈ η µν + t µν and g µν ≈ η µν + h µν turns outto be t µν = h µν + ǫκ (cid:18) T µν − η µν T (cid:19) . (3.3)The left-hand side of (3.1), once the standard gauge choice ∂ λ ( h λµ − h δ λµ ) = 0 is made, leadsto R µν ( η + t ) ≈ − (cid:3) t µν , where (cid:3) is the flat d’Alembertian. For weak sources, therefore,the above equations lead to − (cid:3) t µν = κ (cid:18) T µν − η µν T (cid:19) , (3.4)where only the leading order contributions on the right-hand side have been kept. Forthe (weak field and slow-motion) Newtonian limit we just focus on the t -componentassuming, as usual, a pressureless fluid with T µν ≈ ρu µ u ν , where ρ is the energy densityof the fluid. Defining t = − φ N and h = − φ N , such that ¯ φ N = φ N − ǫκ ρ , the aboveequation in the non-relativistic limit can be written as ∇ φ N = κ ρ + ǫκ ∇ ρ , (3.5)which admits a general solution of the form φ N ( t, ~x ) = κ π Z d ~x ′ ρ ( t, ~x ′ ) | ~x − ~x ′ | + ǫκ ρ ( t, ~x ) . (3.6)63he first term in (3.5) represents the standard Newtonian source, while the second onecorresponds to a new source of gravity that involves derivatives of the matter density.Whenever those gradients become important, significant deviations from Newtonian grav-ity will arise. To estimate the scale at which such deviations occur and the kind of effectsone may find, it is illuminating to take the Fourier transform of (3.5) [29], which leads to k ˜ φ N ( ~k ) = κ (cid:18) ǫk − (cid:19) ˜ ρ ( ~k ) , (3.7)where ˜ φ N ( ~k ) and ˜ ρ ( ~k ) are the momentum space counterparts of φ N and ρ . It is clearfrom this expression that in the GR limit, ǫ →
0, the right-hand side of (3.7) is alwaysnegative. For any finite (but positive) ǫ , however, one finds a scale k J = p /ǫ beyondwhich the right-hand side of (3.7) flips its sign, thus leading to repulsive rather thanattractive gravity. This allows us to interpret the effective Jeans length λ J = 2 π/k J asthe critical scale below which the collapse of pressureless dust is not possible due to thedominance of repulsive interactions. One obvious consequence of this is that for ǫ < ǫ >
0. Another important consequence that can be derived from the ǫ < ∼ − − − munless 8 πGǫ < − m s − kg − [29]. Our current understanding of nuclear and particlephysics, therefore, requires ǫ < × m or, equivalently, √ ǫ .
800 m.
The application of the modified Poisson equation (3.5) to the study of non-relativisticself-gravitating fluids was first carried out in [294] (see also [295] for more details and someclarifications). For fluids in hydrostatic equilibrium, one must supplement the modifiedPoisson equation (3.5) with the fluid conservation equation ∇ µ T µν = 0 in the appropriatelimit. For spherically symmetric systems, the conservation equation boils down to dp/dr = − ( ρ + p )Γ rtr (where p is the pressure of the fluid), and for weak sources Γ rtr ≈ − ∂ r h tt leads to p r = − κ π m ( r ) ρr − ǫκ ρρ r , (3.8)where p r ≡ dp/dr , ρ r ≡ dρ/dr , m ( r ) = 4 π R r dxρ ( x ) x , and an equation of state p = p ( ρ )must be specified.An immediate solution of this equation corresponds to the case in which p ( r ) = 0. Un-like Newtonian gravity, where pressureless solutions cannot be in hydrostatic equilibrium,the above equation yields a nontrivial solution when ǫ >
0. This case simply requiressolving the equation m ( r ) = − πǫr ρ r . Applying on this equation a radial derivative, itcan be cast as the Lane-Emden equation of a polytrope with index n = 1 (recall that poly-tropes have equation of state p ( ρ ) = Kρ n , where K and n are real positive constants,64nd n is the so-called polytropic index). If the regularity condition ρ (0) = ρ c is imposedat the centre to get rid of the 1 /r term, the solution to this equation takes the form ρ ( r ) = ρ c sin( k J r ) k J r . (3.9)As is standard in the study of polytropes, the authors in [294, 295] restricted the rangeof validity of this solution to the interval r ∈ [0 , π/k J ] to avoid the presence of a negativeenergy density beyond the first zero at r = π/k J (see Fig.2). Though this restriction isnatural and harmless in the usual Newtonian theory, the fact is that it forces a discontinuityin ρ r at r = π/k J , thus causing a divergence on the right-hand side of (3.5). Figure 2: Density profile (3.9) of the pressureless configurations. Note that the density is not positivedefinite beyond k J r > π . The example above illustrates an important property of this type of theories of gravity,namely, that the matter fields must satisfy certain differentiability conditions that are notnecessary in the context of GR. The matter/energy profiles must be continuous and dif-ferentiable up to some degree. This requirement may certainly be inconvenient , becauseit forces us to pay more attention to the modeling of our energy sources in certain ap-plications, but is not a fundamental problem because matter and radiation are ultimatelydescribed in terms of quantum fields, which are sufficiently smooth to comply with thedifferentiability requirements of these theories. Therefore, the solution (3.9) admits twopossible interpretations: 1) that we are dealing with an unconventional fluid or 2) that animproved description of the matter fields (with a different fluid or even beyond the fluidapproximation) is necessary near the surface at k = π/k J to avoid undesired or fictitiousunphysical effects. • Regarding option 1), note that in the transition from the (relativistic) weak-fieldapproximation (3.4) to (3.5) we (implicitly) assumed that the stress-energy tensor T µν of the matter fields could be averaged to yield that of a perfect fluid withoutcausing any harm to the theory. In this process, the fluid we had in mind wassome distribution of localised particles (or wavepackets) such that when averaged65ver a certain scale should yield a continuum distribution characterisable by the T µν of a perfect fluid. In particular we expected a positive definite density function ρ ( x ), which turns out to be in conflict with our solution (3.9) beyond r = π/k J .Our microscopic interpretation of the fluid, therefore, does not fit well with thepredictions of this theory, which indicates that we are dealing with an unconventionalmatter source. Note in this sense that the authors in [294, 295] argued in favor ofthis solution representing some kind of dark matter, which might give plausibilityto this result. The effects on the galactic metric of a dark matter density profile ofthis kind has been studied in detail in [193]. • Regarding option 2), if the fluid is interpreted as made out of standard particles, animproved microscopic description of those particles should be considered near theouter boundary (where the density is close to zero) to get a smooth transition to theexterior region in the neighborhood of r = π/k J . Thus, a refinement of the physicsnear the surface is necessary to build a complete solution. As mentioned above, thismight be inconvenient but is not a fundamental problem. In fact, as shown in [29],different averaging procedures in the transit to the continuum fluid approximationmay lead to different (fictitious) acceleration fields associated to the specific weightfunctions employed in the averaging. The emergence of negative densities in theouter regions of these solutions can thus be interpreted as a manifestation of ficti-tious effects which should be regarded as unphysical and avoidable by an improveddescription.The view that one should go beyond the fluid approximation or consider a suitabletransition thick shell in the description of the surface region is further reinforced by theanalysis presented in [295] regarding the process of dust collapse. Starting with genericstatic profiles, it was found that the fate of the system is to reach a universal configurationwhich oscillates around the pressureless solution (3.9) with a period that coincides withthe fundamental mode of proper oscillations of the pressureless case. This means thatthe configurations provided by (3.9) are not a fine-tuned solution of an exotic matter fieldbut, rather, they are a universal, regular final state for the collapse of reasonable mattersources. The role of the EiBI dynamics is, clearly, to stabilise the object against collapseby generating repulsive gravitational forces at short scales. The problems arising on thesurface can be regarded as artifacts of the particular fluid approximation considered. Similar problems affecting the exterior boundary of some polytropic stellar models werealso found in [296]. Due to the divergence of derivatives of the energy density with respectto the pressure as p → ρ r at the surface). This occurs, in particular, for polytropic indices γ = 1 + n > / γ = 5 /
3) or thecase γ = 10 / ? ]. Since themicroscopic constituents are individually well behaved, the curvature divergences on thesurface of polytropes are likely to be a manifestation of fictitious accelerations inducedby the continuum approximation [29]. Secondly, a careful analysis of the validity of thepolytropic equation of state near regions of divergent curvature was carried out in [230].The idea was not to estimate the corrections due to finite temperature or electromagneticrepulsion between charged particles, as is necessary in realistic models to properly accountfor the opacities in stellar atmospheres, but to explore how the microscopic definition ofpressure could be affected near curvature divergences. By analysing the geodesic devia-tion equation (4.132), the frequency of the interactions between a particle and a nearby(fictitious) wall was found to increase with respect to the corresponding statistical esti-mate in flat space-time. This represents an additional pressure which changes the effectiveequation of state for the case 3 / < γ < < γ <
3, which is also problematic, it is found that the fluid is repelled fromthe surface. It is then argued that such fluids would not be appropriate to describe thesurface and that some other type of matter should be necessary. The conclusion is thatthe fluid reacts as the curvature grows on the surface and that an improved description ofthe matter there is necessary.We thus see that the fluid approximation and/or the modeling of certain objects in theEiBI theory of gravity may require some refinements to avoid unphysical effects that ariseat the outer boundaries of some solutions. This occurs when the derivatives of the matterdensity diverge too rapidly as the pressure goes to zero or when the matter profile andits derivatives are abruptly set to zero at some point in order to match with the external(idealised) Schwarzschild solution. By smoothing the behavior of the matter profiles,these problems can, in principle, be overcome. Though this is certainly an inconvenience,it is not that far from what realistic models require. In fact, in order to qualitativelyand quantitatively understand numerous observational features of neutron stars, such astheir electromagnetic spectra, envelope composition, X-ray bursts, surface temperatureprofiles, etc, it is not only necessary, it is essential, to carefully describe the microphysicsof the outer layers. Some of these layers are very thin as compared to the radius of thestar, with a height of ∼ . −
10 cm and density, ρ ∼ − − g/cm [209] in the Note that a particle and a wall are necessary to define the pressure microscopically. g/cm on the outer 10 cm of the envelope.The composition of this region is dominated by a gas of (partially) ionized atoms andelectrons plus radiation, with the electron equation of state transitioning from an idealto a degenerate gas as one goes deeper into the star [105, 291], which has a crucial effecton the efficiency of the different energy transport mechanisms and, thus, dramaticallyaffects the observable features of the star [310]. We thus see that in these layers, finitetemperature, radiation fields, chemical composition, electromagnetic repulsion, magneticfields, etc, induce significant deviations from the basic polytropic equations of state [231],which are nonetheless very useful to estimate the gross properties of these objects. Thoughmodels with this level of refinement have not been yet constructed in the EiBI gravityscenario, as we will see below, the evidence so far indicates that there is no fundamentalreason to believe they are not possible. From the discussion in the previous section, it is now clear that the external boundaryof stars should be modeled in such a way that the matter and pressure as well as its firstand second-order derivatives should smoothly vanish to guarantee a correct matching withthe exterior empty solution. This refinement should be done if one is really interested inunderstanding observational features of the models such as electromagnetic spectra, butcan be overlooked in situations in which only structural aspects are important. In thissense, the standard approach in which the stellar surface is identified as the region wherethe pressure is sufficiently low can be retained as valid, as long as one accepts that a thintransition shell should be added to correctly complete the model.Having understood the peculiarities that the matter profiles should satisfy on the outerboundaries of stars, we now focus on the information that stellar models can provide to testthe viability of EiBI gravity and constrain its parameters. The results of [295] establisha limitation for the existence of polytropic solutions with regular boundary condition atthe centre, ρ ≈ ρ c + ρ r , which requires ǫκ > − K (cid:18) n (cid:19) ρ − n c . (3.10)The reason for this bound in static configurations is related to the monotonicity of ρ ( r ),which requires ρ <
0. An expansion of (3.8) around the centre puts forward that if thebound (3.10) is not satisfied, then ρ >
0. Going beyond polytropic models, a non-rotating,zero temperature white dwarf model with parametric equation of state p ( x ) = πm e c µ e m P h h x (2 x − p x + 3 sinh − x i , (3.11) ρ ( x ) = 8 πm e c µ e m P h x , (3.12)was studied for different values of ǫ [295]. It was found that for ǫ >
0, the mass of theseobjects is not limited by the Chandrasekhar bound M ≈ . M ⊙ . It turns out that themass can be arbitrarily large while the radius tends to a minimum value which scales as68 ǫ . In the relativistic version of these objects, however, an upper bound for the mass doesappear, though it can be much larger than in GR (see Fig.3). - - (cid:144) R Ÿ M (cid:144) M Ÿ Κ Ρ WD = - Κ Ρ WD = - Κ Ρ WD = - Κ Ρ WD = - Κ Ρ WD = - Figure 3: Zero-temperature relativistic white dwarfs in EiBI theory (in this plot, ǫ → κ ) in units of thetypical density ρ WD = 10 kg/m . The horizonal line denotes the Chandrasekhar limit, M = 1 . M ⊙ . Notethat an upper limit on the mass of these stars arises but can be much larger than in GR. Figure takenfrom Ref.[295]. A closer confrontation with observations is certainly possible by considering the effectsof the modified Poisson equation on the properties of the Sun [103]. Since the hydrostaticequilibrium and energy transport ultimately depend on this equation, any correction wouldhave an impact on the thermal balance and temperature profile inside the star, which canleave observable traces. In fact, neutrino fluxes are very sensitive to the temperatureprofile inside the Sun [36, 349]. An increase or decrease of the innermost conditions dueto a modified Poisson equation will necessarily leave a trace on the amounts of emittedneutrinos, which are relatively well understood observationally. Something similar happenswith helioseismic data, which provide very accurate information on the solar acousticmodes, the sound speed profile, and the depth of the convective envelope, see e.g. [116]for a review. In order to extract the necessary information to use solar neutrinos andhelioseismic data to test EiBI gravity, the hydrostatic equilibrium equation (3.8) and thecontinuity equation, dm/dr = 4 πr ρ ( r ), must be supplemented with the conservation ofthermal energy equation dLdm = q ( r ) − r dsdt , (3.13)where q ( r ) represents the rate of heating from nuclear reactions and s is the entropyper unit mass [119], plus the corresponding equation for the convective energy transport,69hich takes the form dTdm = − (cid:20) Gm ( r )4 πr + ǫκ ρ dρdm (cid:21) Tp τ . (3.14)Here τ ≡ d log T /d log p is the temperature gradient, which for adiabatic changes becomes τ = (Γ − / Γ , where Γ is one of the adiabatic exponents [362]. In the radiative zone,the transport energy equation is unmodified dTdm = − λ R σT L π r , (3.15)where λ R is the Rosseland mean opacity and σ the Boltzmann constant.Implementing the above equations in CESAM [290], a numerical code for stellar struc-ture and evolution, the authors of [103] constructed a number of models able to fit the solarproperties with an accuracy of 10 − in the interval − . GR ⊙ < ǫκ < . GR ⊙ . Forsmaller values of ǫκ , no equilibrium stars were found, in agreement with the constraint(3.10) for polytropic models. For ǫκ > . GR ⊙ , solutions do exist but are unable tomatch simultaneously the observed values for age, radius, mass, luminosity and metallicityof the Sun.Qualitatively, models with ǫ > ǫ = 0), whereas for ǫ < B with an intensity that scales as φ B ∝ T c . This flux iscurrently measured with high precision by neutrino telescopes: (5 . ± . × cm − s − . From the numerics one observes a decay in the neutrino flux for ǫ > ǫ <
0, such that with the current data it is possible to conclude that ǫκ . − . GR ⊙ is ruled out.The precision with which acoustic modes are currently measured by helioseismic mis-sions allows to probe the solar interior revealing the sound speed and density profiles downto 10% of the solar radius [146]. The separation between the frequencies of modes withdifferent degree l and radial order n , δν n,l = ν n,l − ν n − ,l +2 , is a quantity very sensi-tive to the temperature gradient. The case of l = 0 is particularly important because itcorresponds to waves that traveled through the entire solar radius, carrying valuable infor-mation about the density profile. The small uncertainties associated with these quantitiesallow to rule out the regions ǫκ > . GR ⊙ and ǫκ < − . GR ⊙ . On the other hand,the agreement between the sound speed profile and the solar standard model reaches anaccuracy better than 1% in most of the solar interior, with larger uncertainties right belowthe convective envelope. Comparison with this model and the data, one can safely ruleout the region ǫκ > . GR ⊙ . Constraints on the depth of the convective envelopeand the helium surface abundance, which also follow from helioseismic data, imply that − . GR ⊙ < ǫκ < . GR ⊙ and ǫκ > − . GR ⊙ , respectively. These examplesclearly illustrate the capabilities of solar observations to constrain modifications of gravitywith new couplings in the matter sector. 70 .3. Relativistic stars White dwarfs and neutron stars are by far the natural scenarios where the highestpressures can be achieved, which offers an excellent opportunity to test our theories aboutnuclear matter and also the modified dynamics of alternative gravity theories. It is wellknown that the masses and radii of neutron stars depend critically on the equation of stateof dense matter [183, 238]. For a given equation of state, a mass-radius relation and amaximum mass can be derived. The so-called stiffness of the equation of state depends onhow many bosons are present. Since bosons do not contribute to the Fermi pressure, theytend to soften the equation of state, which leads to low maximum neutron star masses( ∼ . M ⊙ ). GR sets a maximum mass ∼ . M ⊙ , and it is expected that the maximumachievable mass in nature is of order ∼ . M ⊙ , but this depends on the stiffness of theequation of state [226] and is thus open to observational scrutiny. The density-dependentmodifications induced by the EiBI dynamics can be seen as the effect of a modified source[138] and, for this reason, must also leave an imprint in the mass/radius relation of thesecompact objects. In this section we consider the efforts carried out so far to understandthe impact of the EiBI modified dynamics on the structure and stability of neutron starsas well as some strategies to distinguish its predictions from those of GR. In the EiBI gravity scenario, the equations of stellar structure in the full relativisticcase have been studied in numerous works [295, 329, 192, 333], and for a line element ofthe form d s = − e φ ( r ) dt + e λ ( r ) dr + f ( r ) d Ω , (3.16)can be written as dmdr = r ǫ (cid:18) − ab + ab (cid:19) (3.17) dpdr = − (cid:2) mr + r ǫ (cid:0) ab + ab − (cid:1)(cid:3)(cid:2) − mr (cid:3) h ρ + p + ǫ (cid:16) b + a c s (cid:17)i (3.18)with f ( r ) = r ab (3.19) a ≡ p ǫκ ρ (3.20) b ≡ p − ǫκ p (3.21) c s ≡ dpdρ . (3.22)Given a barotropic equation of state p = p ( ρ ) and appropriate boundary conditions,concrete models can be studied. The boundary conditions at the centre typically involvethe specification of the central density ρ c (or the central pressure p c ), and the condition m (0) = 0. For rough structural considerations, the radius of the star is defined by the71ondition p ( R ) ≤ p , where p is ideally zero but in practice is represented by a smallpositive number. At that point one assumes that the Schwarzschild solution should takeover, provided a sufficiently smooth transition profile is used. This last step is usuallyassumed to hold and can be omitted (more details on this will come later).From the definitions of the functions a and b above, the constraints ǫκ p c < , for ǫ > | ǫ | κ ρ c < , for ǫ < ρ c in neutron stars is of the order ρ c ∼ kg/m and p c ∼ N/m , we get that | ǫκ | . kg − s − . Numericallyone verifies that compact objects only exist if p ( r ) ≈ p c + p r near the centre has p < ǫ >
0. These objects have amaximum compactness of
GM/R ∼ . ǫ . The fact that the current cosmological model requires a significant componentof cold dark matter particles with p = 0 makes these solutions particularly interesting, sincethey indicate that such particles could aggregate in structures with the typical compactnessand mass of most neutron stars.Models in EiBI gravity with realistic equations of state based on nuclear physics havebeen studied in detail in [329] (see also [333] and [295]). These equations of state are usuallypresented in tabulated form and require numerical interpolation for their implementationin the codes. Though this is not a problem in the case of GR, the interpolation method mayintroduce numerical noise and artificial effects which should be avoided. In [329] this wassolved by using smooth analytic functions to model the tabulated equations of state (seeFig.4), while in [295] a piecewise polytropic interpolation was implemented. As a generalfeature, it is observed that the mass of the solutions increases with the central densityuntil a certain maximum value. This maximum mass is larger than the GR prediction if ǫ > ǫ <
0. The maximum appears at a lower central density than in GRif ǫ >
0. As pointed out in [295], the larger mass predicted by models with ǫ > M ≈ . M ⊙ was a critical test (see also [312] for updated dataon massive pulsars). Different examples were studied in [192], including a model with acausal stiff fluid for which the speed of sound equals the speed of light, a radiation-typeequation of state, a polytrope of index n = 3 (relativistic neutrons), and the quark matterequation of state. An exact (and exotic) analytical solution of the relativistic equationswas also found there. More recently, the influence of hyperons in the equation of statehas been explicitly considered in [312] to illustrate that the “hyperon problem” found inneutron star models within GR may be avoided in the EiBI theory. Their conclusions arein agreement with the previous literature on this topic. Slowly rotating relativistic starswere also considered in [295] using the perturbative approach introduced by Hartle [195].72 log ( ε / g cm -3 ) l og ( P / dyn c m - ) - . l og ( ε / g c m - ) BBB2FPS
Figure 4: Analytic fit of the BBB2 and FPS models, taken from Ref.[329]. The crosses and pluses representthe data points in the EOS tables and the lines are the analytic fit functions.
A detailed analysis of the stability under radial perturbations of relativistic stars wascarried out in [329] and [334], both focusing on the fluid modes and neglecting the space-time modes. In [329] a fixed, static physical background was assumed but it was notedthat the auxiliary metric could develop a non-zero contribution in the t − r sector due tothe perturbations in the fluid. The approach of [334] is different, as the author adoptsa crude Cowling approximation forcing both the physical metric perturbation, δg µν , andthe auxiliary metric perturbation, δq µν , to vanish.Denoting by ξ the radial Lagrangian displacement and ˙ ξ its time derivative, the four-velocity of the fluid is given by u µ = ( − e φ/ , e − φ/ ˙ ξ, , T µν given by T rt = − (¯ ρ + ¯ p ) ˙ ξ , with the bar denotingbackground values. Assuming a time dependence e iωt for all the perturbed quantities, therelevant eigenvalue equation for the radial oscillation modes can be written as χ ′′ = − W χ − W χ ′ , (3.25)where χ ≡ r Q (¯ ρ + ¯ p ) ξ , and the functions W , W , and Q depend on the backgroundfields and the frequency ω (see Appendix A of [329] for the explicit expressions of thesefunctions). The analogous equation for linear radial perturbations in the non-relativisticlimit was studied in [295] in the context of polytropic fluids finding a more tractableexpression: − ρ h γ ¯ pr ( r ξ ) ′ i ′ + 4¯ ρr ξ ¯ p ′ + ǫκ (cid:20) ξ ¯ ρ ′ r − ξ ′ ¯ ρ ′ − (cid:16) ¯ ρr ( r ξ ) ′ (cid:17) ′ (cid:21) = ω ξ , (3.26)where γ defines the adiabatic index of the perturbations. In both cases, the resultingeigenvalue equation must be solved subject to standard regularity condition at the centre,73 M ( M s un ) πκε = 0 (GR)8 πκε = 0.18 πκε = -0.1 (a) APR ε c ( 10 g cm -3 ) -2000200400600 ω ( s - ) M ( M s un ) πκε = 0 (GR)8 πκε = 0.18 πκε = -0.1 (b) BBB2 ε c ( 10 g cm -3 ) -2000200400600 ω ( s - ) Figure 5: Gravitational mass M and fundamental mode frequency square ω plotted against the centraldensity ρ c (denoted as ε c in the plots) for two EOS models: (a) APR and (b) BBB2. Three different valuesof the Born-Infield parameter ǫ , denoted as κ in the plots (not to be confused with the Einstein constant,as follows from the notation employed in this section) are considered. The circle on each M -density curvecorresponds to the maximum-mass configuration. Plots taken from [329]. ξ (0) = 0 = ξ ′ (0), being ξ ( R ) finite at the surface. An instability corresponds to aneigenmode with ω < ω remains positive up to the value of the central density at whichthe stellar mass reaches its maximum (see Fig.5). This critical density sets the onset ofa dynamical instability against radial perturbations. At lower densities, the solutions arelinearly stable. This behavior is qualitatively identical to that already observed in GR. Thenumerical results in Fig.5 show that in the EiBI gravity theory the mass of the solutionsmay attain a local minimum at larger central densities. The location of this extremumcoincides with a zero in the frequency square of the first radial harmonic. While thefrequency of the first and higher harmonics depend on the specific value of ǫ chosen, for agiven mass, the fundamental mode is quite insensitive to ǫ .For non-relativistic stars, in the presureless case one finds that, for a given ǫ , there isone fundamental mode, which is numerically determined as ω = αρ / c , where α ≈ . ǫ [295]. These solutions do not have unstable modes, confirmingthat they are linearly stable. For polytropic models, the stability is improved as comparedto the case of GR. In GR, models with adiabatic index γ = 4 / n , whereas in the Born-Infeld theory these models are always stableif ǫ > ǫ <
0. 74 .3.3. Observational discriminations from GR
The exceptional conditions of matter inside neutron stars, with densities that may beseveral times above nuclear densities, turn these objects into natural laboratories fromwhich to extract valuable information on the properties and behavior of nuclear matter.The study of their structure, rotation, and magnetic fields can thus offer a powerful win-dow on the microscopic properties of matter, complementing in this way the knowledgeobtained from laboratory experiments. In the context of GR, this could help constrainthe form of the equation of state of nuclear matter, which has led to the study of empiri-cal relations between the structural properties of neutron stars (mass, radius, moment ofinertia, ...) and their equations of state. Alternatively, the potential existence of relationsweakly dependent on the equation of state could also be used to constrain the gravitationaldynamics.The possibility of discriminating between the predictions of GR and those of EiBIgravity using a special kind of neutron stars was raised in [333]. As mentioned above, themass-radius relation of neutron stars is intimately linked to the details of the equations ofstate and the internal gravitational dynamics. Extracting useful information about thesetwo inputs of the theory requires not only measuring the radii of neutron stars for a broadrange of masses but also breaking the potential degeneracies that may arise between thematter and the gravity sectors. In this respect, a normal neutron star with M ∼ . M ⊙ ismore sensitive to the high density properties of the equation of state than a low mass starwith M ∼ . M ⊙ , and the uncertainties in the corresponding equations of state of eachmodel are very different. The importance of the analysis of [333] lies on the observationthat the radii of neutron stars with M ∼ . M ⊙ are strongly correlated with the neutronskin thickness of P b nuclei in a way which is independent of the particular equationof state [101]. This suggests that laboratory measurements of the neutron skin thicknessof
P b combined with the precise observational determination of the radii of neutronstars with 0 . M ⊙ could provide a direct estimate of ǫ .Quantitatively, [333] finds that the radius of 0 . M ⊙ neutron stars, R (measured inkm), for different equations of state admits a linear parametrisation of the form (see Fig.6) R = c + c ∆ R , (3.27)where ∆ R (measured in fm) represents the neutron skin thickness of P b . It shouldbe noted that the value of ∆ R depends on the particular equation of state consideredbut is independent of ǫ . The constants c and c depend on the value of ǫκ ρ , where κ ρ / π = 1 . × − km − follows from taking ρ = 2 . × g/cm (nuclear saturationdensity). The dependence of these parameters on ǫκ ρ is explored in the range − . ≤ ǫκ ρ ≤ .
04, finding that In the determination of the relation between pressure and energy density in nuclear matter equationsof state, the so-called symmetry energy quantifies the change in nuclear energy associated with modifyingthe neutron-proton asymmetry. Accurate determination of the thickness of the neutron skin of neutron richheavy nuclei would provide crucial experimental constraints on the symmetry energy and, as a consequence,on the structural properties of neutron stars. .1 0.12 0.14 0.16 0.18 0.20 0.2210111213141516 ! R (fm) R ( k m ) OI (230, 73.4)OI (230, 42.6)OI (280, 97.5)OI (280, 54.9)ShenMiyatsuFPSSLy4BSk19BSk20BSk21
Figure 6: Radii of neutron stars with 0 . M ⊙ , R in Eq.(3.27), as a function of the neutron skin thicknessof Pb for ǫκ ρ = − . ,
0, and 0 .
02, using various EOS as shown in [333]. The curves are insensitiveto the EOS but do depend on the Born-Infield parameter ǫ . c km = 8 .
21 + 60 . × ( ǫκ ρ ) (3.28) c km/fm = 31 . − . × ( ǫκ ρ ) . (3.29)Combining these relations with Eq.(3.27), one finds that ǫκ ρ = R − . − . R . − . R , (3.30)where, recall, R is measured in km while ∆ R in fm. Using this expression, with ob-servational values of R and ∆ R one can, in principle, determine the value of ǫ . In thisregard, assuming that R and ∆ R had ±
10% variances, the uncertainties in the determi-nation of ǫκ ρ would reach ± .
04 for R = 12 km and ± .
06 for R = 14 km even inthe case of GR. According to the best current data, the situation is even worse because∆ R = 0 . +0 . − . fm does not allow to constrain ǫ even if R were exactly known. Giventhat the measurement of R is expected to be more difficult than that of ∆ R , because0 . M ⊙ is exceptionally small for a neutron star, the use of this approach to constrain thetheory is really challenging. Nonetheless, there is still hope that the observation of neutronstars with ∼ . M ⊙ , for which this qualitative analysis is still valid, could be used in thefuture (the lowest mass of neutron stars observed so far is (0 . ± . M ⊙ [316]). Let us now focus our attention on phenomena taking place in the high density family ofneutron stars. The potential existence of phase transitions in the nuclear matter of massiveneutron stars could have more dramatic effects in Born-Infield theories of gravity than inGR due to the role that matter gradients play in this theory. The relativistic hydrostaticequilibrium equation (3.18) and the study of stellar pulsations put forward the appearancein the equations of terms associated with the sound speed and its first derivative, which76re dependent on the first and second-order derivatives of the pressure with respect to thematter density. This, in part, motivated the use of interpolating functions to approximatethe tabulated equations of state in order to avoid artificial numerical effects. However,should first-order (or second-order) phase transitions take place in the interior of neutronstars, discontinuities in the matter density (sound speed) would occur. The potentialeffects of first-order phase transitions have been investigated in [328].The first thing to note in the case of first-order phase transitions, is that in the limit inwhich c s →
0, Eq.(3.18) behaves as dp/dr ∝ − c s /ǫ thus implying that dρ/dr = c − s dp/dr isfinite even when dp/dr vanishes. If ǫ >
0, one finds that in the region of constant pressure dρ/dr is continuous, constant, and negative, generating in that way a discontinuity in thefunction p = p ( ρ ). This region of constant pressure is self-supported due to the repulsivegravity generated by the strong density gradient, in much the same way as pressurelesssolutions are stabilized. The continuity of ρ ( r ) in this scenario contrasts with the case ofGR, where a discontinuity in ρ is unavoidable. If ǫ <
0, both dp/dr and dρ/dr are positiveat the c s ≈ dpdr ∝ − (cid:20) ρ + p + κ ǫ π (cid:18) b + 1 a c s (cid:19)(cid:21) − . (3.31)Given that the ρ + p term is positive and that the other one is negative and grows as c s →
0, a divergence in dp/dr is unavoidable, which indicates the impossibility of havingequilibrium static solutions when ǫ < ǫ > c s , which is discontinuous at the phase transition. The physical implicationsof this divergence have not been studied in detail (see the section 4.5.2 on black holes forclosely related discussions), though as suggested in [230] they could induce a backreactionable to avoid them. In fact, as acknowledged in [328], there is no evidence whatsoever thatcompact stars in nature exhibit phase transitions in their interiors. As a final comment, wejust note that curvature divergences of this type are common in many physical problemsinvolving thin shells, in which a certain thick boundary is idealised in the form of ahypersurface that separates two regions [257, 141, 173]. The delta-like divergences areexpected to disappear on physical grounds once small perturbations are allowed in thedensity/pressure profiles. f -mode and I -Love- Q . Aside from the mass-radius relation in the low range band of neutron stars, other em-pirical relations connecting parameters of neutron stars have been proposed. In particular,a correlation between the scaled moment of inertia
I/M R and the compactness M/R hasbeen observed [57, 239]. Also the frequency and damping rate of the quadrupolar f − mode,associated to internal fluid oscillations, can be related to global properties such as M and I in a way that depends very weakly on the equation of state [18, 19, 63, 64, 348, 240].Also, the values of M, R , and the moment of inertia I can be accurately inferred fromthe f − mode gravitational wave signals [240]. More recently, a universal relation involving77 M/J and M/R , where Q is the spin-induced quadrupole moment and J the angularmomentum, has been found. Other universal relations between I, Q and the tidal androtational Love numbers λ tid and λ rot have also been discovered [370, 369]. These so-called I -Love- Q relations are relevant for the understanding of gravitational wave signalsin neutron stars binary mergers.The f − mode universality relations and the I -Love- Q relations of [370, 369] have beeninvestigated in the context of Born-Infield gravity in [330]. For this purpose, the authorswrote the field equations in GR-like form following the approach of [138] and computedthe oscillation frequencies also in that representation. The consistency of this approachto the problem of stellar perturbations with that provided in [329] was also confirmed,within numerical accuracy, putting forward the usefulness of this representation of thefield equations.Let us first focus on the properties of the f -mode. Neutron star oscillations are dampedby the emission of gravitational waves, which implies the existence of a complex part inthe oscillation eigenfrequencies (quasi-normal modes), ω = ω r + iω i , with ω i representingthe damping rate of the oscillation mode. Within the frame of GR, it turns out that ω r and ω i of the f -modes (fluid oscillations) can be related to global parameters of the staraccording to M ω r = − . . η + 0 . η (3.32) I M ω i = 0 . − . η , (3.33)where the factor η ≡ p M /I is dimensionless (in the appropriate units). These relationsare more insensitive to the equation of state than previous relations where the radius R was chosen as a parameter. The motivation for this choice comes from the fact that R issensitive to the low-density part of the equation of state, while the moment of inertia I measures the mass distribution globally, which is more closely related to the f -mode oscilla-tions of the star. The approach of [330] consisted on writing the Born-Infield field equationsin GR-like form [138], solving the stellar structure equations for several nuclear matterequations of state, and then computing perturbations around the different backgroundsobtained to identify the f -mode frequency using well-established methods developed inGR [233]. Considering the cases ǫκ ρ = − . , . , .
1, with ρ ≡ g/cm , for which M can change up to 30%, it was found that the relations M ω r ( η ) and I ω i /M ( η ) areessentially independent of the chosen equation of state and ǫ , being in excellent agreementwith Eqs. (3.32) and (3.33), respectively.The moment of inertia of a star is defined by I ≡ J/ Ω, where J and Ω are theangular momentum and the angular velocity of the star, respectively. For a given J , I determines how fast a star can spin and, for this reason, it is expected to be correlatedwith the spin-induced quadrupole moment Q of the star. Interestingly, in [370, 369] itwas found a relation between I and Q which is independent of the equation of state.Related to this, the (traceless) quadrupole moment induced on a neutron star by a nearbycompanion is determined by Q ij = − λ tid E ij , where E ij is the tidal tensor and λ tid isthe so-called Love tidal number. Though, in principle, there is no reason to expect an78quation-of-state-independent relation between the variables ¯ I ≡ I/M , ¯ Q ≡ − Q/ ( M χ ),and ¯ λ tid ≡ λ tid /M , where χ ≡ J/M , it turns out that they are related by an expressionof the form ln y i = a i + b i ln x i + c i (ln x i ) + d i (ln x i ) + e i (ln x i ) , (3.34)where the pairs ( x i , y i ) represent (¯ λ tid , ¯ I ), (¯ λ tid , ¯ Q ), and ( ¯ Q, ¯ I ) (see Figs.7) and the co-efficients a i , b i , c i , d i , and e i are constant. The numerical analysis puts forward that the I -Love- Q relations for the EiBI theory of gravity are the same as the GR ones for therange of parameters explored, | ǫκ ρ | ≤ . l n I APR (0.1)APR (0)APR (-0.1)FPS (0.1)FPS (0)FPS (-0.1)SLy4 (0.1)SLy4 (0)SLy4 (-0.1)WS (0.1)WS (0)WS (-0.1)Fit ln Q -4 -3 -2 -1 | I F it - I | / I F it l n I APR (0.1)APR (0)APR (-0.1)FPS (0.1)FPS (0)FPS (-0.1)SLy4 (0.1)SLy4 (0)SLy4 (-0.1)WS (0.1)WS (0)WS (-0.1)Fit ln λ tid -4 -3 -2 -1 | I F it - I | / I F it l n Q APR (0.1)APR (0)APR (-0.1)FPS (0.1)FPS (0)FPS (-0.1)SLy4 (0.1)SLy4 (0)SLy4 (-0.1)WS (0.1)WS (0)WS (-0.1)Fit ln λ tid -5 -4 -3 -2 -1 | Q F it - Q | / Q F it Figure 7: Universality relations between the moment of inertia, the spin-induced quadrupole moment, andthe tidal Love number as shown in [330]. The curves are insensitive to both the EOS and the Born-Infieldparameter ǫ . For the sake of completeness, we briefly comment now on the approach of [138] usedto study the above universality relations. Following our notation and manipulations, thefield equations of the theory can be written in Einstein-like form as (recall Eq.(2.63)) G µν ( q ) = κ | ˆΩ | / (cid:20) T µν − (cid:18) L G + T (cid:19) δ µν (cid:21) , (3.35)79here L G represents the gravity Lagrangian, being L G = ( | ˆΩ | / − λ ) / ( κ ǫ ) in the Born-Infield theory, and | ˆΩ | is the determinant of the deformation matrix that relates q µν and g µν , as defined by Eq.(2.57). In the case of GR one gets L G = R/ (2 κ ) = − T / T µν term alone on the right-hand side of the equations. Note that while the Einsteintensor on the left-hand side is defined in terms of the auxiliary metric q αβ , the matterterms on the right-hand side (including L G and | ˆΩ | / ) depend on the physical metric g µν .Since the (algebraic) relation between these two metrics depends on T µν , which can havesome dependence on g µν (typically through kinetic terms), the resulting field equationsmay become highly nonlinear in the matter variables. The case of a perfect fluid, with T µν = ( ρ + p ) u µ u ν + pδ µν , is particularly simple because the metric dependence of T µν only appears through the covariant vector u ν ≡ g να u α . For this matter source Eq.(3.35)takes the explicit form G µν ( q ) = κ " ( ρ + p ) | ˆΩ | / u µ u ν + 1 | ˆΩ | / (cid:18) ρ − p L G (cid:19) δ µν , (3.36)with | ˆΩ | / = ( λ + ǫκ ρ ) / ( λ − ǫκ p ) / . This representation suggests a redefinition ofvariables such that the right-hand side of (3.36) can be interpreted as an effective perfectfluid coupled to the geometry defined by q µν . The proposal of [138] thus follows naturally,defining p q = 1 | ˆΩ | / (cid:18) ρ − p L G (cid:19) (3.37) ρ q + p q = ( ρ + p ) | ˆΩ | / (3.38) v µ v ν = u µ u ν , (3.39)where v µ is normalised using the auxiliary metric, q µν v µ v ν = −
1, and v ν ≡ q να v α . UsingEq.(3.39) and the fact that in this model q αβ = Ω g αβ + Ω u α u β , withΩ = p ( λ + ǫκ ρ )( λ − ǫκ p ) (3.40)Ω = ǫκ ( ρ + p ) s λ − ǫκ pλ + ǫκ ρ , (3.41)the relation between v µ and u ν can be readily established. In fact, contracting (3.39) with v ν , one finds v µ = − ( v · u ) u µ , where v · u ≡ v µ u ν g µν is the usual scalar product betweenvectors. If the contraction is done with u ν , one finds instead v ν = − u ν / ( v · u ). Using thedefinition of q µν above to compute u α v α = ( v · u )(Ω − Ω ) and the relation v ν = − u ν / ( v · u )to get the alternative expression u α v α = 1 / ( v · u ), one finds that ( v · u ) = 1 / (Ω − Ω ),which completely specifies the relation between v µ , v ν and u µ , u ν . These new variableshave mapped the EiBI gravity theory into the usual Einstein equations, which can nowbe manipulated and solved using standard methods. The spacetime metric follows fromthe relation g µν = ( q µν − Ω u µ u ν ) / Ω . This approach should, in principle, be applicableto other matter sources as well. 80 .3.6. Magnetic fields Magnetic fields are thought to play an important role in supernova explosions [368],gamma-ray bursts [302], soft gamma repeaters and quasi-periodic oscillations, anomalousX-ray pulsars [345, 234], etc, and are also fundamental to understand basic observationalfeatures of neutron stars. In particular, it is well known that the spectrum of radiationemergent from a neutron star atmosphere can significantly differ from a blackbody spec-trum, and its angular distribution be far from isotropic due to the presence of magneticfields [372]. In this sense, it is important to note that the radiation properties of neutronstars are strongly conditioned by their superficial layers [289], which can be in a gaseousstate (atmosphere) or condensed state (liquid or solid) depending on surface temperature,magnetic field, and chemical composition. A condensed surface, for instance, may arise atlow temperatures and very strong magnetic fields ( T . K and B = 10 G or T . Kand B = 10 G). On the other hand, the strong gravitational field on the surface layers,which is usually regarded as constant and of order g ∼ − cm/s , rapidly sinks theheaviest elements, leaving the lightest available ones at the surface, which will then be re-sponsible for the radiative properties of the atmosphere, critically affecting its spectrum.A thin layer of Hydrogen of just 10 − M ⊙ , for instance, is sufficient to condition the wholespectrum. This is so because magnetic fields are able to shift the ionization energy of Hy-drogen up to 160 eV if B = 10 G (or 310 eV if B = 10 G). The intensity of magneticfields on the neutron star outer layers is thus essential to understand the features of theirradiation spectra, polarization, and thermal conductivity. The presence of magnetic fieldsabove B ∼ − G, therefore, may affect the opacity of the outer layers resultingin a nonuniform surface temperature distribution, which may lead to pulsations of thethermal radiation due to rotation. At lower intensities, however, its impact on the opacityis negligible and can be safely neglected.The effects of the Born-Infield gravitational dynamics on the magnetic fields of neutronstars have been investigated in [335] focusing on the axisymmetric dipole configurations,which are expected to dominate in old neutron stars, and assuming spherically symmetricconfigurations. This assumption implies that the magnetic energy in the star is muchsmaller than the gravitational binding energy, which allows to neglect any deformationinduced by the magnetic pressure. The stellar structure is thus determined by the fluid,while the magnetic field is just computed on top of the resulting geometry. The equationsgoverning the magnetic field follow from Maxwell’s equations F [ µν ; α ] = 0 (3.42) ∇ µ F µν = 4 πJ µ , (3.43)while the coupling between the fluid and the magnetic field result from the conservationequation ∇ µ T µν = 0, which in the ideal magneto-hydrodynamic approximation takes theform ( ρ + p ) u ν ∇ ν u µ + ( δ νµ + u ν u µ ) ∂ ν p = F µν J µ . (3.44)With the appropriate gauge condition, A µ can be written as A µ = (0 , A r , , A ϕ ), and81xpanding A ϕ as A ϕ = a l ( r ) sin θ∂ θ P l (cos θ ), where P l (cos θ ) is the Legendre polynomialof order l , the equation describing the dipole magnetic field ( l = 1) becomes a ′′ + ( φ ′ − λ ′ )2 a ′ + (cid:18) ζ e − φ − f (cid:19) e λ a = − πe λ j , (3.45)where the line element (3.16) has been used, prime denotes radial derivative, j ≡ c f ( r )( ρ + p ), with c a constant, and the constant ζ is related to A r = ζe ( λ − φ ) / a l P l . The compo-nents of the magnetic field, B µ = ǫ µναβ u ν F αβ / B r = 2 a e λ/ f cos θ (3.46) B θ = − a ′ e − λ/ sin θ (3.47) B ϕ = − ζa e − φ/ sin θ , (3.48)from which it is apparent that ζ controls the strength of the toroidal magnetic field. As-suming that the exterior geometry is described by the Schwarzschild solution, the externalpoloidal magnetic field ( ζ = 0) is determined by a ( ex )1 = − µ b r M (cid:20) ln (cid:18) − Mr (cid:19) + 2 Mr + 2 M r (cid:21) , (3.49)where µ b is the magnetic dipole moment at infinity. This solution sets the external bound-ary condition for a and a ′ . From (3.45), one finds that at the centre a ( r ) ≈ α r + O ( r ),with α a constant. The constants α and c (which appear in j ) should be chosen so asto guarantee the continuity of a and a ′ at the surface. The magnetic field strength canthus be written as B ≡ ( B µ B ν g µν ) / = f − [4 a cos θ + a ′ f e − λ sin θ + ζ a f e − φ sin θ ] / . (3.50)At the stellar centre, one finds that B = 2 α p ǫκ ρ c p − ǫκ p c .The analysis of [335] considered stellar models with M = 1 . M ⊙ , a range of parameters | ǫκ ρ s | < .
05, with ρ s = 2 . × g/cm representing the nuclear saturation density, andtwo different realistic equations of state for nuclear matter, FPS [243] and SLy4 [143]. Thischoice was necessary in order to compare the effects of the modified dynamics with thoseof different equations of state in different regions of the star. The magnetic distributionsobserved in the pure poloidal case, ζ = 0, are qualitatively the same as in GR, withdeviations smaller than 10% in some regions and reaching departures of less than 0 .
5% inthe crust. The mixed case, ζ = 0, manifests some peculiar features depending on the valueof ζ , but roughly are also very similar to those of GR. Thus, the differences with respectto GR in the internal regions are comparable to the uncertainties due to the equation ofstate. The magnetic fields on the crust, however, depend very weakly on the couplingconstant ǫ , while properties of this region such as its thickness are very sensitive to theequation of state. It was suggested in [335] that this could be used to extract informationon the equation of state by exploring physical processes associated to the crust, such as82tellar oscillations. However, given that stellar oscillations are also very insensitive to theBorn-Infield parameter due to the universality relations discussed in [330], it seems thatthe magnetic field is a poor probe for this type of theories. The use of astrophysical objects to constrain the magnitude of the non-linearity pa-rameter in the EiBI theory of gravity has shown that with current data reasonable boundscan be placed on the theory. However, several important degeneracies arise which make itdifficult to distinguish the theory from GR or discriminate its effects from those comingfrom the matter sector. The exploration of other Born-Infeld inspired theories in thesescenarios could help better understand whether these degeneracies are proper of the EiBIor are common to a larger family of gravity theories. For all such theories, a realisticand satisfactory modeling of the transient from the top layers of the star to the external(idealized) vacuum solution is still missing. 83 . Black Holes
According to General Relativity (GR), a fuel-exhausted star with a mass exceedingthe refined Tolman-Oppenheimer-Volkoff limit, which may raise up to ∼ . M ⊙ , depend-ing on the equation of state for dense matter (see section 3.3) may end up its lifetimecollapsing to form a region of spacetime causally disconnected from asymptotic observers,and which is called a black hole [331]. The three-dimensional null hypersurface markingthe boundary of this region, which acts as a one-way membrane, is the event horizon .According to the unicity theorems formulated by Israel [215, 214], Carter [102] and Hawk-ing [198] and others [318] (together with the no-hair conjecture, see [253]), starting fromany initial (non-necessarily symmetric) configuration the final state of the gravitationalcollapse corresponds to a stationary and axisymmetric object solely described in termsof three parameters: mass, charge and angular momentum, leading to the Kerr-Newmanfamily of solutions [227, 258] (see [222] for a review on gravitational collapse). With therecent detection of gravitational waves ascribed to black hole merger processes by theLIGO collaboration [2] , which is added to the classical observations from compact X-raysources (with Cygnus X-1 as the first historical and most influential example [286]), theastrophysics of compact objects has entered into a golden era, where GR can be testedwith an unprecedent precision in new regimes [3].Black holes have been and still are a very active area of research as they pose a numberof challenges to our comprehension of gravitational interaction. These problems are ofdifferent kinds. First, it has been convincingly established in the literature that, if oneassumes the validity of the Einstein’s equations all the way down to the innermost regionof a black hole, a spacetime singularity unavoidably develops [299]. Moreover, this result isnot due to an artifact of an excessively simplified modelling, but instead grounded on somephysically reasonable restrictions [326] (spacetime singularities and non-singular blackholes in the context of Born-Infeld inspired modifications of gravity will be extensivelydiscussed in section 4.5). To avoid the breakdown of predictability and determinism,Penrose introduced the cosmic censorship conjecture [300], by which it is assumed thatan event horizon covering the singularity is always developed during the gravitationalcollapse process, and thus a naked singularity cannot be seen from external observers.Second, there is a tension between the classical description of gravitational phenomenaprovided by GR and the fundamental tenet of quantum mechanics, namely, unitarity, asgiven by the apparent disappearance of information inside a black hole, known as theblack hole information loss problem [200, 248]. On the other hand, the very connectionbetween Hawking’s radiation [199] and standard thermodynamic systems still calls for anunderstanding in terms of hypothetical black hole microscopic degrees of freedom, andthe controversy about the potential existence of firewalls at the event horizon still goes on[14]. Finally there are apparent counterexamples of solutions with hair when adding thenew ingredient of superradiance [205] (see [206] for a recent review), with related intensivesearches for observational discriminations from the Kerr solution [98]. Indeed, the existence of gravitational waves was already indirectly hinted by the observations of theHulse-Taylor binary pulsar [213] and others.
84s black holes allow to test the strong field limit of GR, determining the deviationsof black hole solutions from the Kerr one of GR and comparing them with astrophysicalobservations has become a major test on the viability of any modification of gravity .Their study could shed new light on the understanding of all the open questions discussedabove. In the context of Born-Infeld inspired theories of gravity we have already seenthat the vacuum solutions, in Palatini approach, yield the same dynamics of GR with acosmological constant term. Thus, the class of static, spherically symmetric vacuum blackhole solutions of such theories is represented by the Schwarzschild one, characterized bymass M . In order to excite the dynamics contained in the new couplings of this theoryone needs to couple it to some matter source. The available literature so far amounts totwo such sources, namely, electrovacuum fields and anisotropic fluids. In this sectionwe shall review in detail the corresponding deviations from the GR solutions and theircontributions to fundamental and observational issues of black hole physics. Along the years, a number of Born-Infeld inspired actions have been considered in theliterature regarding the search for spherically symmetric solutions. A quick review onsome of the first proposals will prepare us to deal with the Eddington-inpired Born-Infeldgravity introduced by Ba˜nados and Ferreira [45], for which most of the research on blackhole physics in the literature has been carried out. Note that in the original proposal ofDeser and Gibbons [140] the field equations of Born-Infeld gravity were derived using apurely metric variation, which results in fourth order equations of motion and presence ofghosts (see section 2.2), rendering the problem of finding exact solutions to such equationsalmost intractable. Nonetheless, Feigenbaum [157] considered the metric formulation ofthe four-dimensional, Class-0 action (recall the classification of theories of section 2.7): S = Z d x √− g (cid:20) R + β (cid:18)q − k R − k R µν R µν − k R αβµν R αβµν − (cid:19)(cid:21) (4.1)where β, k , k , k are some constants. Despite the unavoidable trouble with ghosts, Feigen-baum investigated spherically symmetric solutions in the approximation R µν ≃ k = k = 0, which imposes the constraint of R αβµν R αβµν ≤ k upon the Kretschman scalar as long as β = 0. Given the limited physicalinterest of this scenario due to the ghost problem, let us just mention that Feigenbaumobtained analytical solutions (perturbatively to lowest order in ǫ ) under the form See Berti et.al. [67] for an overview on experimental constraints on the many gravitational modifica-tions of GR proposed in the literature. It should be stressed that, though in astrophysically realistic situations the amount of net electriccharge is negligible, its consideration for black holes may yield relevant lessons regarding gravitationalphysics beyond GR, in particular, on the spacetime singularities issue. s = − f ( r ) dt + dr h ( r ) + r d Ω (4.2) f ( r ) = r − Mr (cid:20) − k M βr (cid:18) r − Mr − M (cid:19) + O (cid:0) k β M (cid:1)(cid:21) (4.3) h ( r ) = 1 − k M βr (cid:18) r − Mr − M (cid:19) + O (cid:0) k β M (cid:1) . (4.4)In this expression M is the total mass of spacetime, as seen from a far away observer.In the limit of negligible β these solutions reduce to the Schwarzschild black hole of GR.When increasing the constant β the event horizon disappears (the transition value followsfrom a non-trivial relation between k , β and M that can only be numerically determined)and a kind of “bare mass” objects free of curvature divergences arise. We will see laterthat the existence of solutions without curvature divergences turns out to be a feature ofother Born-Infeld inspired theories of gravity as well.The explicit addition of matter, and the corresponding search for spherically symmetricblack hole solutions, was first explored with some detail by Vollick. In [357] he consideredthe following action: S = 1 κβ Z d x (cid:18)q | g µν + β R µν + κβM µν | − q | g µν | (cid:19) (4.5)where β is a constant (whose interpretation shall be clear later), M µν contains the mattercontribution and the connection is taken to be symmetric.When M µν = 0, the purely metric variation of this action (Class-0) has been consideredby Feigenbaum, Freund and Pigli [158], and Feigenbaum [157]. Working in the Palatiniapproach (Class-III theories), Vollick finds electrostatic, spherically symmetric solutions.In this case, one takes the matter contribution M µν = αF µν , where α is a constant and F µν = ∂ µ A ν − ∂ ν A µ is the field strength tensor of the vector potential A µ . In order toobtain a system of equations that can be solved exactly Vollick assumes sufficiently weakfields and compute the field equations up to quadratic terms in the fields as G µν ( g ) = β h g µν R − RR µν − g µν R αβ R αβ + 8 R µα R αν i − α κ β (cid:20) F µα F να − g µν F αβ F αβ (cid:21) . (4.6)Since the last term in brackets in this expression corresponds to the energy-momentumtensor of a Maxwell field, in order to obtain Einstein’s equations to lowest order in β one must take α = 1 / ( κβ ), which implies the positivity of β . Now let us consider(electrostatic) spherically symmetric solutions using the gauge A µ = ( φ ( r ) , , , F tr ( r ) = 0.After a bit of algebra, this restriction allows to cast the field equations (4.6) as G µν ( g ) = κ F µα F να q − F b − b g µν " − r − F b (4.7)86here by convenience we have introduced a new constant as b = 1 / ( κβ ) and F = F αβ F αβ denotes the electromagnetic field invariant. Note that the contribution on the right-hand-side of these equations is formally similar to that of the energy-momentum tensor ofBorn-Infeld theory of electrodynamics [75] with a reversed sign in front of second termand inside the square root. The Einstein equations (4.7) have to be compatible with theequations for the matter, which follow from variation of the action (4.5) with respect to A µ as ∇ µ h | ˆ q | (cid:0) ˆ q − (cid:1) [ µν ] i = 0, where the object q µν ≡ g µν + β R µν + √ κβF µν . To quadraticorder, and in the notation above, these equations become ∇ µ F µν q − F b = 0 (4.8)which is nothing but the field equations of Born-Infeld electrodynamics with a reversedsign inside the square root. In Vollick’s solutions, the mass function is given by (byconvenience, we shall absorb here the factor 4 π from the integration of the electromagneticfield equations as Q → πQ ) dm ( r ) dr = b hp r − Q /b − r i (4.9)again with the reversed sign inside the square-root.The main novelty of Vollick’s reversed sign solution is that it is only defined beyond aradius r = r c , where r c = p | Q | /b . The replacement of the point-like singularity of GR bya finite-size structure is a feature that will re-appear later when discussing electromagneticgeons in section 4.4. In the present case, at the radius r = r c one finds that the curvaturescalar behaves as R = − b " − r − r c r p r − r c (4.10)and thus there is a curvature singularity, displaced here from r = 0 to a finite radius. Tofind the horizons of these solutions one considers the zeros of the metric component g tt ,which can be found by solving the equation h ( r ) = r − M +2 b ( | Q | /b ) / R ∞ r/r c h u − √ u − i du =0. A careful analysis of this equation reveals the presence of charged black holes with eithertwo horizons, a single (degenerate) one or none (and a time-like singularity at r = r c ), orblack holes with a single horizon and a time-like, space-like or null singularity, dependingon the parameters of the solutions.Black holes with a cosmological constant λ can also be implemented within this frame-work via the Class-III action [358] S = 1 κβ Z d x (cid:18)q | g µν + β R µν + p κβF µν + βλg µν | − q | g µν | (cid:19) . (4.11)Now Vollick consider both electrostatic, E ( r ) ≡ F tr , and magnetostic fields, B ( r ) ≡ F θϕ ,via the two field invariants, F = F µν F µν and G = F µν ˜ F µν . In analogy with the solutions87bove, now the Lagrangian, L = L ( F, G ), corresponding to the energy-momentum on theright-hand-side of the gravitational field equations, is obtained as L BI = b − r − F b − G b ! (4.12)with the same redefinitions as in the asymptotically flat case above. One can still assumea spherically symmetric line element given by Eq.(4.16) and follow a similar procedure tosolve the field equations, which yields the expression for the mass function dm ( r ) dr = Λ2 r + b (1 + ˜ λ ) − (cid:20)q r − ˜ Z /b − r (cid:21) (4.13)where Λ = λ (cid:16) βλ βλ (cid:17) plays the role of the cosmological constant term, while we havedefined ˜ λ = λ/ ( κb ), ˜ Z = ˜ Q + ˜ p , with ˜ Q = (1 + ˜ λ ) Q and ˜ p = (1 + ˜ λ ) p being there-scaled electric and magnetic charges, respectively. From the computation of the Ricciscalar constructed out of the spacetime metric R = − − b (1 + ˜ λ ) − " − r − r c / r p r − r c (4.14)it follows that a curvature singularity is still present at the finite radius r = r c = q | ˜ Z | /b .To close this part, let us briefly mention that spherically symmetric solutions were inves-tigated in the context of f ( R ) models with a square root (Class-IV), see [235], but onlymundane (Anti-)de Sitter solutions were found. There is a remarkable parallelism between the modifications on the structure of hori-zons for some of the solutions above and those of Born-Infeld electrodynamics coupledto GR. As this parallelism will re-appear later in the literature, it is instructive to con-sider the spherically symmetric solutions of Born-Infeld electrodynamics. In this sense,the framework of Einstein’s gravity coupled to non-linear electrostatic fields has been de-veloped to a great detail in the literature, particularly for Born-Infeld electrodynamics[321, 139, 132, 87, 162]. The action is written as S = Z d x √− g (cid:20) R κ − L ( F ) (cid:21) (4.15)where the case of Born-Infeld electrodynamics is given by Eq.(4.12) with G = 0. Due tothe symmetry of the energy-momentum tensor for electrostatic solutions, T tt = T rr , onecan write a line elementd s = − (cid:18) − m ( r ) r (cid:19) dt + (cid:18) − m ( r ) r (cid:19) − dr + r d Ω (4.16)88here d Ω = dθ + sin ( θ ) dϕ is the solid angle element, and the mass function m ( r ) isdetermined through the resolution of the Einstein’s equations as dm ( r ) dr = r T tt ( r ) = b hp r + Q /b − r i (4.17)(compare this equation with (4.9)). This can be explicitly integrated (with the constraintof recovering Schwarzschild black hole as r → ∞ ) as m ( r ) = M − πb r (cid:20) r − p r + Q /b + 2 Q b r F (cid:18) , , , − Q b r (cid:19)(cid:21) (4.18)where M is the Schwarzschild mass. Due to the finiteness of the self-energy associatedto a point-like charge in Born-Infeld electrodynamics, see Eq.(2.10), the behaviour of themetric component g tt at r = 0, with the expressions (4.16) and (4.18), becomes there: g tt = g − rr ≃ r → − − πbQ − M − ˜ U ) r + O ( r ) ! (4.19)where ˜ U = 4 π / U , with U defined in Eq.(2.10) and the factor 4 π / comes from theredefinition Q → πQ above. The zeros of g tt in (4.19) set the location of the horizons. Inthe (asymptotically flat) Reissner-Nordstr¨om solution of GR, such horizons are obtainedas r ± = M ± p M − Q , where the signs ± refer to the outer (event) and inner (Cauchy)horizons, respectively. For these horizons to exist, the inequality M > Q has to befulfilled (when this bound is saturated, M = Q , one has an extreme black hole with adegenerated horizon), otherwise one ends up into a naked singularity. In the Born-Infeldelectrodynamics case, due to the finite character of ˜ U , it turns out that the behaviourof the metric at the center determines the existence of three classes of configurationsdepending on the hierarchy between M and U . In this sense, if M < ˜ U the solutionsresemble the Reissner-Nordstr¨om configurations of GR, in that two, a single (degenerate)horizon or none can be found, while for M > ˜ U a single horizon is always found, withsimilar features to those of the Schwarzschild black hole of GR. Finally, when M = ˜ U themetric at the center is finite and equal to − (1 − πbQ ), which consequently yields eithera single horizon or none. This description is depicted in Fig.10 for a particular choice of b = Q = 1 /
2. In all these cases a curvature divergence is always present at r = 0 and thisway Born-Infeld electrodynamics fails to solve the singularity problem within GR. We willsee that quite a similar structure of horizons arises when considering Eddington-inspiredBorn-Infeld gravity in section 4.4, while the issue with singularities will be reviewed insection 4.5. Note in passing by that finiteness of the self-energy can be achieved in other non-linear theories ofelectrodynamics, which indeed share most of the features regarding the structure of horizons and behaviourof curvature scalars when coupled to GR [142]. igure 8: Born-Infeld black holes in GR for b = Q = 1 / U ≃ . M = ˜ U − . M ≃ ˜ U − . M = ˜ U − . M = ˜ U ), and black holes with a single horizon( M = ˜ U + 1). Note the transition between Reissner-Nordstr¨om-like configurations ( M < ˜ U , blue curves) toSchwarzschild-like black holes ( M > ˜ U , red) via the critical case M = ˜ U (brown). Solutions with M = ˜ U and no horizons are also possible. All solutions are asymptotically flat (horizontal dashed green line). We now turn our attention upon to the most widely employed proposal in the literaturefor Born-Infeld inspired modifications of gravity, and where the influential electrovacuumblack hole solutions of Ba˜nados and Ferreira were found [45]. This proposal is definedvia the action (2.33), and nowadays is usually known as Eddington-inspired Born-Infeldgravity (EiBI), which is a Class-I action (see section 2.7 for details on this classification).By convenience, let us write this action in the notation employed in this section as S EiBI = 1 κ ǫ Z d x "r − det (cid:16) g µν + ǫ R ( µν ) (Γ) (cid:17) − λ p − det g µν + S M ( g µν , ψ m ) (4.20)where ψ m denote the matter fields. A few remarks are in order: for the purpose of thissection we shall assume hereafter that the (symmetric) connection Γ is not coupled to thematter sector in the action (4.20), in agreement with Einstein’s equivalent principle, thatdictates that free-falling particles should follow geodesics of the background geometry g µν (see 4.2.2 for specific details). On the other hand, in vacuum, S M = 0, the equation ofmotion for g µν implies g µν = ǫλ − R ( µν ) so that an effective cosmological constant termemerges as Λ = λ − ǫ (thus asymptotically flat solutions correspond to λ = 1). This isconsistent with the non-relativistic limit described in section 3.1, where post-newtoniancorrections only emerge under variations on the energy density of the matter fields.90 .2.1. Geometry and properties Let us now study spherically symmetric configurations in EiBI gravity sourced byelectrovacuum (Maxwell) fields, as given by the action S M = − π Z d x √− gF µν F µν . (4.21)The energy-momentum tensor for this source is written as T µν = 14 π (cid:18) F µσ F σν − g µν F σρ F σρ (cid:19) . (4.22)Ba˜nados and Ferreira considered the spherically symmetric line element for the metric g µν as d s g = − ψ ( r ) f ( r ) dt + dr f ( r ) + r d Ω (4.23)and solved the EiBI equations for an asymptotically flat geometry, λ = 1. It is instructiveto consider in detail the obtention of the field equations, which is not provided in [45],but derived in detail by Wei et. al. in Ref. [360] for arbitrary λ . This will be useful tounderstand the different results obtained in similar but slightly different scenarios in EiBIgravity. In many of such scenarios it is much simpler to solve the field equations for theauxiliary metric q µν , and then transform the solution back to the spacetime metric g µν using Eq. (2.57). In the present case one proposes a line element for q µν as ds q = − G ( r ) F ( r ) dt + 1 F ( r ) dr + H ( r ) d Ω . (4.24)The five metric functions { ψ ( r ) , f ( r ) , G ( r ) , F ( r ) , H ( r ) } are to be determined via the fieldequations (2.60) and the transformations (2.57). The gravitational field equations forma compatible set with the electromagnetic ones, ∂ r ( ψ − r E ) = 0, which gives the result E ( r ) = Qr ψ ( r ), where Q arises as an integration constant associated to the electric charge.Note that there is one redundant equation between the former and the latter, and conse-quently there are several ways to proceed. Wei et al. [360] choose to replace the expressionfor the electromagnetic field into the energy-momentum tensor (4.22), and insert the resultinto the field equations for the auxiliary metric (2.60), which yields4 G ′ G H ′ H + 2 F ′ F H ′ H + 3 G ′ G F ′ F + 2 G ′′ G + F ′′ F = 1 ǫF (cid:18) λ − ǫQ r − (cid:19) , (4.25)4 H ′′ H + 2 F ′ F H ′ H + 3 G ′ G F ′ F + 2 G ′′ G + F ′′ F = 2 ǫF (cid:18) λ − ǫQ r − (cid:19) , (4.26) − H F + F ′ F H ′ H + G ′ G H ′ H + H ′ H + H ′′ H = 1 ǫF (cid:18) λ + ǫQ r − (cid:19) , (4.27)where primes stand for derivatives with respect to the radial coordinate r . On the otherhand, the transformations (2.57) lead to the relations between the metric functions in each91ine element G = ψ (cid:18) λ − ǫQ r (cid:19) ; F = f (cid:18) λ − ǫQ r (cid:19) − ; H = r r λ + ǫQ r . (4.28)Now one just needs to solve the field equations (4.25), (4.26) and (4.27) with the relations(4.28), imposing the asymptotic GR limit: ψ ( r → ∞ ) → f ( r → ∞ ) → − Mr + Q r − Λ r M , charge Q ,and cosmological constant Λ. Now a bit of algebra yields the following expressions for themetric components and the electromagnetic field in the EiBI case ψ ( r ) = r p r + ( ǫ/λ ) Q (4.30) f ( r ) = r p ǫQ + λr λr − ǫQ (cid:20) (cid:0) r − Q − ( λ − r /ǫ (cid:1) p ǫQ + λr r + 13 s Q π √ ǫλ Γ (1 / s iQ √ ǫλ F i arcsinh s iQ r λǫ r , − − √ λM (cid:21) (4.31) E ( r ) = Q p r + ( ǫ/λ ) Q (4.32)where F (Φ , m ) = R Φ0 (1 − m sin θ ) − / dθ (with − π/ < Φ < π/
2) is the elliptic integralof first kind. These explicit expressions were given in Ref.[360], and refine that of f ( r )appearing under the form of an integral in Ba˜nados and Ferreira paper [45], besides cor-recting a factor 2 under the square root of the function ψ ( r ) of the latter. Regarding thehorizon structure, one finds the remarkable result that, for any value of the EiBI parame-ter ǫ , its mere presence induces a change in the causal structure of these black holes (seeFig.9), moving from the two-horizons description of the Reissner-Nordstr¨om solution ofGR to a configuration with a single horizon (resembling the Schwarzschild solution of GR)or none, depending on the combination of parameters [360].Exploring further EiBI black holes, the expression for the electric field (4.32) bearsa remarkable similarity with that obtained in Born-Infeld electromagnetism [75]. In thepresent case, despite the finiteness of the electric field everywhere, the metric functions aresingular at the finite radius r = r c , where r c = √ ǫQ , which may be hidden or not behindan event horizon. In Ref.[360] Wei et al. compute, for asymptotically flat solutions, λ = 1,the following curvature scalars: 92 igure 9: Behaviour of the function f ( r ) in Eq.(4.31) for EiBI gravity with ǫ > ǫ = Q = 1 and vary the mass. Left plot: a Reissner-Nordstr¨om black hole with two horizons may transformeither into a naked singularity ( M = 1 .
07) or in a Schwarzschild-like black hole with a single horizon( M = 1 .
15) in EiBI gravity. Right plot: a naked singularity in Reissner-Nordstr¨om black hole ( M = 0 . r > r c with r c = √ ǫQ . All solutions are asymptotically flat (horizontal dashed green line). R ( g ) ≡ g µν R ( µν ) ( g ) ∝ r − r c ) (4.33) R ( g, q ) ≡ q µν g µν R ( µν ) ( q ) = g µν ( q µν − g µν ) /ǫ = 8 ǫ (4.34) R ( q ) ≡ q µν R ( µν ) ( q ) = 8( r + r c / √ r − r c / √ r + r c )( r − r c ) . (4.35)It is thus immediately seen that the curvature scalar constructed either out of the metric g µν or of q µν blows up as the surface of radius r = r c is approached. However, nointerpretation on the nature of such a surface is given, and the presence of divergences oncurvature scalars could be interpreted as signal of the breakdown of the geometry and thusof the presence of a physical singularity. To overcome this point, Ba˜nados and Ferreiraargue that the geometry (4.23) describes just the exterior of a charged object, so a realisticmodel should consider the process of gravitational collapse to explore such a question indetail. Nonetheless, we shall see later when discussing non-singular solutions in section(4.5) that EiBI gravity hides some surprises regarding the singularity issue. The new non-trivial gravitational dynamics introduced by EiBI gravity, that modifiesthe shape of the geometry, necessarily has its impact upon the geodesic behaviour of bothnull (associated to light rays) and timelike (associated to massive particles) geodesics.As already mentioned, the fact that in EiBI action (2.33) the connection does not coupledirectly to the matter sector, implies that Einstein’s equivalence principle holds (see section2.6). This way, the equations of motion for a geodesic curve γ µ = x µ ( u ), where u is some93ffine parameter, can be derived from the action S = Z du L = 12 Z du r g µν dx µ du dx ν du (4.36)from which the geodesic equation, in a coordinate system, follows as [359, 104] d x µ du + Γ µαβ ( g ) dx α du dx β du = 0 , (4.37)where Γ µαβ ( g ) is the affine connection constructed as the Christoffel symbols of the space-time metric g µν . Eq.(4.37) represents a set of second-order differential equations thatprovide a unique solution once initial conditions, { x µ (0) , dx µ /du | } , are given. Now, re-placing the line element (4.23) for g µν into the Lagrangian density of Eq.(4.36) one getsthe result L = − ψ ( r ) f ( r ) ˙ t + f ( r ) − ˙ r + r ( ˙ θ + sin ( θ ) ˙ ϕ ) (4.38)where dots denote derivatives with respect to the affine parameter u . From the Hamilto-nian description of the system it follows that there are two conserved quantities, namely, E = ψ ( r ) f ( r ) ˙ t and L = r sin( θ ) ˙ ϕ . For timelike observers these quantities can be inter-preted as the energy per unit mass and angular momentum per unit mass, respectively,while for null geodesics we can identify b = L/M as an apparent impact parameter fromasymptotic infinity. In addition, due to spherical symmetry one can assume the motion tobe confined to a plane, that can be chosen to be θ = π/ ψ (cid:18) drdu (cid:19) = E − V . (4.39)This is just the equation of motion of a one-dimensional particle moving in an effectivepotential of the form V ( r ) = s f ψ (cid:18) k + L r (cid:19) , (4.40)where ψ and f are defined in Eqs.(4.30) and (4.31), respectively, while the causal vector u µ = dx µ /du satisfies u µ u µ = − k , with k = 0(+1) for null (time-like) geodesics. Now, ifone considers the circular motion of a test massive particle ( k = +1) around an electricallycharged EiBI black hole, this implies the constraint dr/du = 0 which, via Eq.(4.39), yields E = V ( r ). This orbit is realised, indeed, at the minimum of the effective potential V ( r ).In [336] Sotami and Miyamoto perform a numerical analysis of such a motion, using fixedvalues of Q/M and ǫ/M and varying the ratio L/M , depicted in Fig.10 (left). The main In general, imposing a symmetry and obtaining the equations of motion do not commute. The con-ditions under which these two operations do commute are established by the Palais criticality theorem[293].
L/M decreases, the maximum of the effective potential decreasesas well, while its minimum gets closer to the centre of the EiBI black hole, in such a waythat there is a minimum bound for
L/M (depending on ǫ ), below which no minimum ofthe potential occurs. This bound determines the innermost stable circular orbit (ISCO),which is the minimum radius below which no stable circular orbit of a test massive particlecan exist around an EiBI black hole. These results are qualitatively similar to those ofGR, though the specific quantitative details depend on the particular value of the EiBIconstant ǫ . xx r / M V ( r ) l / M = 3.3 l / M = 3.5 l / M = 3.7 l / M = 4.0 Q/M = 0.5 ! / M = 6 r / M V ( r ) Mb c M Q / M = 0.5 ! / M = 6 x r = r Figure 10: Effective potential V ( r ) for time-like geodesics, k = +1 (left) and null geodesics, k = 0 (right)in Eq.(4.40) for the choice Q/M = 0 . ǫ/M = 6 (in the notation of this plot, ǫ → κ ). Left figure:four values of the impact parameter b ≡ L/M = 3 . , . , . , L → l ) havebeen depicted. On each of such curves the open circle corresponds to the radius of the innermost circularorbit (ISCO). Right figure: three values of the impact parameter b = 2 M, b c , M . On the b = 7 M curvethe photon is scattered by the black hole at r = r . Figures taken from Ref.[336] and [337], respectively. Two works [360, 337] have been carried out in the literature to determine the effect ofthe parameter ǫ of charged EiBI black holes regarding the lensing in a strong gravitationalfield. Gravitational lensing is indeed a powerful test to determine the nature of a compactobject, which may allow to find deviations from GR predictions in the strong field regime[311]. For a massless particle, k = 0, Eq.(4.39) can be conveniently rewritten as ψ (cid:18) drdu (cid:19) = 1 − b f ψ r (4.41)where we have redefined u → u/E . To characterise the orbits of photons in the effec-tive potential (4.40) one first establishes the existence of the photon sphere, namely, theinnermost region for a photon in orbit around a black hole, which for static, sphericallysymmetric spacetimes coincides with the unstable circular orbit (UCO) radius. Accordingto the analysis carried out by Virbhadra and Ellis [351, 118, 352], for a line element of theform (4.23) this radius is simply defined by the equation ( ψf ) ′ r = 2 ψ f . Explicitly, forthe EiBI black hole metric defined by the functions (4.30) and (4.31), the UCO radius, r UCO , corresponds to the solution of the equation (in units 2 M = 1, which is equivalent95o making dimensionless the black hole parameters as r → r/ (2 M ), Q → Q/ (2 M ) and κ → κ/ (2 M ) ):8 ǫ / Q r (cid:0) r − Q (cid:1) p ǫQ + r = (cid:0) ǫ Q + 2 ǫQ r + 3 r (cid:1) (4.42) × (cid:20) − √ iQ / rF ( i arcsinh( s i √ κQ r ) , − − Q / r Γ (cid:0) (cid:1) √ π + √ ǫ (cid:16) r − p ǫQ + r (cid:17) (cid:21) , which is consistent with the fact that, when the charge Q = 0, the above equation yieldsthe result r UCO = 3 / r UCO = 3 M ), whichcorresponds to that of the Schwarzschild black hole. For non-vanishing Q , however, findinganalytic solutions to (4.42) is highly non-trivial. One may note instead that the integrationof the photon sphere equation above, and comparison with the effective potential (4.41),tells us that the UCO radius r UCO corresponds to the solution of the equation dV /dr = 0with d V /dr <
0. This way photons will be swallowed by the black hole if V ( r UCO ) < r = r if V ( r UCO ) >
1, and move indefinitely around it inabsence of perturbation if V ( r UCO ) = 1. This can be translated into the condition b ⋚ b c ,where b c is a critical number that depends non-trivially on the black hole parameters andthe EiBI constant. As a comparison, in the Reissner-Nordstr¨om case of GR, ǫ = 0, one has r UCO = 3 M (1 + p − Q / (9 M )) / b c = r UCO / [( r UCO − r + )( r UCO − r − )]. In Fig.10(right) the effective potential for the choice ǫ/M = 0 . r . The dependence of the UCO radius r UCO at fixed charge with the EiBI constant can also be studied numerically, with the resultthat it monotonically decreases with increasing ǫ [337, 360], meaning that it is harder tocapture a photon by the EiBI black hole than in the Reissner-Nordstr¨om black hole ofGR.Let us now consider the scattering process of a photon by the electrically charged EiBIblack hole, which can only take place for b > b c . First, from Eqs.(4.39) and (4.40) weobtain the equation dϕdr = bψr p r − f ψ b . (4.43)We assume a photon that travels from infinity, is scattered at r = r and ϕ = 0 (see Fig.10,right), and returns to infinity. By construction, this turn-around point satisfies dr/dϕ = 0,which implies b = r / ( f ( r ) ψ ( r )), where the subindex 0 means that functions are beingevaluated at r . This way, the integration of (4.43) yields the result φ ( r ) − φ ( r ) = Z rr bψr p r − b ψ f dr . (4.44)With this expression, the deflection angle α ( r ) of the photon, which is defined as ∆( ϕ )( r ) =2 φ ( ∞ ) − π [353], can be written for the EiBI metric as∆( ϕ )( r ) = 2 b Z ∞ r ψr p r − b ψ f dr − π . (4.45)96espite the presence of a pole in the integrand of (4.45) at r = r , this can be isolated andproperly handled using the variable z = 1 − r /r , which finally yields a finite result (see[337] for details). This way, for the EiBI black hole the deflection angle can be numericallycomputed and compared to the GR solution, and the result is plotted in Fig.11. Thereit is seen that the deflection angle increases as r decreases. As the ratio r /M decreasesthe deflection angle increases until it reaches the value 2 π corresponding to the pointwhere the massless particle completes a loop around the black hole before reaching theasymptotic observer. By decreasing further the ratio r /M one gets subsequent values2 πn ( n an integer number) of the deflection angle, which means that the photon performs n loops around the black hole before escaping from it. Indeed, should r be able to reachthe UCO radius r UCO , then the deflection angle would diverge, meaning that the photonwould turn indefinitely around the EiBI black hole, again, in absence of any perturbation.These light rays passing close to the UCO radius give rise to multiple images on bothsides of the optical axis, called relativistic images . The position of such images in this casedepends strongly on the value of the EiBI parameter ǫ/M , i.e., on the gravitational theory.Thus, this strong gravitational lensing represents a promising scenario to experimentallytest EiBI gravity in the strong field limit.As already mentioned, when r = r UCO the integrand in (4.45) diverges, and it has tobe handled with care via the new variable z = 1 − r /r . In both Refs.[360, 337] this allowsto perform the integration of (4.45) around the region r ≃ r UCO , with the (finite) result ! ! ! ! r / M ! " Q / M =0.5 GR " x " x +5 x +10 ! ! ! ! r / M ! " Q / M =1.0 GR " x " x " x " " Figure 11: Deflection angle ∆ ϕ in Eq.4.45 for fixed charge Q/M = 0 . Q/M = 1 . r /M for different values of ǫ/M (dashed lines) as compared to theGR case (solid), corresponding to ǫ = 0. Figures taken from Ref.[337]. ∆ ϕ ( b ) = − a log (cid:18) bb c − (cid:19) + a + O ( b − b c ) / (4.46)(alternatively one can write this expression in terms of r UCO , as it is done by Wei et al.[360]) where the strong deflection coefficients a and a depend on the EiBI parameter ǫ ina non-trivial way (see [337] for details). For fixed charge, it turns out that increasing (andpositive) ǫ implies an increasing of the deflection angle as compared to the Schwarzschildblack hole (see Fig.5 of Ref.[360]), that could be used to obtain information on ǫ usingstrong gravitational lensing. 97ext, to investigate the position and magnification of the relativistic images in stronggravitational lensing, one considers the lens geometry, where it is assumed that the blackhole lens is located between the source and the observer, which are both required to befar away from the black hole so that the gravitational fields there are weak enough to bedescribed by a flat metric. Under such constraints the form of the lens equation was foundby Virbhadra and Ellis [351] astan ω = tan θ − D LS D OS [tan(∆( ϕ ) − θ ) + tan(Θ)] (4.47)where ω and Θ correspond to the lens/source and the lens/observer angular separationbetween, respectively, while D LS and D OS stand from the distance between lens andsource, and observer and source, respectively. In the strong deflection limit source, lens andobserver can be assumed to be highly aligned, i.e., ω ≪ ≪ ϕ n − Θ) ≪ ϕ n ≡ ∆ − πn is the deflection angle when all the loops of photons around theEiBI black hole are removed [83]), and using also that in the lens geometry b ≃ D OL Θone gets the deflection angle∆ ϕ (Θ) = − a log (cid:18) D OL Θ b c − (cid:19) + a . (4.48)The relativistic images correspond to ∆ ϕ (Θ) = 2 πn , which yieldsΘ n = b c D OL (cid:20) (cid:18) a − nπa (cid:19)(cid:21) (4.49)where Θ n is the angle of the n th relativistic image. Due to the exponential contribution thefirst relativistic image, Θ , is the brightest one, while the other are greatly demagnified. InFig.12 the position of such an image is depicted as a function of the EiBI parameter ǫ/M for several values of the electric charge (set of curves) with assumed values of D OL = 8 . M = 4 . × M ⊙ , corresponding to the supermassive black hole at the centre ofthe Milky Way [176]. From this figure it is clear that the deviation from the GR predictionincreases with stronger EiBI coupling ǫ/M (in the range ∼ −
5% for
Q/M = 0 . | ǫ/M | = 10), which is consistent with the fact that the location of the scattering radius r decreases as the EiBI parameter increases.There are, in addition, other quantities that can be constructed to be compared withastronomical observations. In order to take the simplest situation for observation, onecan assume that the first relativistic image Θ can be resolved from the others, that arecollectively packed at Θ ∞ [83]. This way, one finds three observables: the position of therelativistic images except the first one, Θ ∞ , and the two quantities s ≡ Θ − Θ ∞ = Θ ∞ exp (cid:18) a − πa (cid:19) (4.50) R = exp (2 π/a ) (4.51)corresponding to the angular separation between the first image and all the others, andto the ratio between the flux of the first image and all the others, respectively. The latter98
10 –8 –6 –4 –2 0 2 4 6 8 102021222324252627 ! / M ! [ ! a r c s ec ] x GR xx x x x x x xx x Figure 12: First relativistic image in Eq.(4.49) as function of the EiBI parameter ǫ/M (in this figure, ǫ → κ ) for different values of the electric charge (set of curves), as compared to GR (solid thick line).Figure taken from Ref.[337]. defines a more convenient observable, R m = 2 . R , which is the relative magnificationof the images. This way, given an EiBI parameter ǫ one can numerically compute the strongdeflection coefficients a and a and thus the three observables above. By comparing themwith astronomical observations one can test the nature of black holes via gravitationallensing and, in particular, put experimental constraints on the value of the EiBI constant ǫ . This has been explored, for ǫ >
0, by Wei et al.[360] by assuming that the EiBI blackhole describes the supermassive black hole at the centre of our galaxy, and compare itto the description provided by Schwarzschild black hole [351]. In table 1 of that paper,an explicit computation of these three observables for different values of ǫ has been done.The main result is that these observables fulfill the inequalities Θ EiBI ∞ < Θ RN ∞ < Θ Sch ∞ , R EiBI < R RN < R Sch and s Sch < s RN < s EiBI (for ǫ < s do notnecessarily hold for all values of ǫ [337]). For instance, the difference in the observableΘ ∞ between the charged EiBI black hole and the Schwarzschild black hole is of order ∼ µ arcsecs, which seems to be far from the reach of current astronomical instruments[84]. On the other hand, the relative magnification R m may significantly deviate fromthe GR prediction, for instance, with the choice Q/M = 0 . . − . ǫ/M = ∓
10. This way,strong gravitational lensing can complement other techniques for testing deviations fromthe Kerr solution such as the measurement of the iron Kα line observed in the X-rayfluorescence spectrum produced by the illumination of a cold accretion disk by a hotcorona of (stellar-mass or supermassive) black hole candidates [220, 221, 39]. The innermost structure of black holes in the presence of accretion has been studied fordecades, with the striking result first found by Israel and Poisson [305, 306], and furtherextended by Ori [284] and others, that over the inner (Cauchy) horizon of a rotating blackhole there occurs an exponential growth of the local Misner-Sharp mass, which in turnsinduces un unbounded growth of the curvature, a phenomenon known as mass inflation (see [189] for a review on the topic). It is triggered by the relativistic counter-streamingeffects between ingoing and outgoing streams, which occurs not only in the context of GR,99ut also in black hole solutions of other theories of gravity. In the case of EiBI gravity, thisquestion has been investigated by Avelino [31] using electric charge instead of rotation inorder to simplify the problem. The reason for this choice lies on the fact that the interiorstructure of a charged black hole closely resembles that of rotating black hole, where thenegative pressure generated by the electric field yields a gravitational repulsion analog tothat produced by the centrifugal force in a rotating black hole.Since the inner structure of charged EiBI black holes can be drastically affected bythe accretion of mass, one has to employ some simplifying assumptions in order to obtainanalytic solutions. In particular, the homogeneous approximation assumes the ingoing andoutgoing streams to be equal. This implies that all relevant quantities can be written as afunction of a radial (timelike) coordinate, which has been shown to be useful for studyingsome of the most important aspects of mass inflation [190, 34, 33]. This allows to writetwo spherically symmetric line elements as ds q = A ( r ) dt + B ( r ) dr + H ( r ) d Ω (4.52) ds g = g tt dt + g rr dr + r d Ω (4.53)where A ( r ), B ( r ), H ( r ), g tt ( r ) and g rr ( r ) are functions of the radial coordinate r alone.The total energy-momentum tensor is split into two pieces T µν = e T µν + f T µν (4.54)where e T µν and f T µν are the electromagnetic and fluid contributions, respectively. Thecomponents of such an energy-momentum tensor can be written as T rr = − ρ = − ρ e − ρ f ; T tt = p k = − ρ e + w k ρ f ; T θθ = T φφ = p ⊥ = ρ e + w ⊥ ρ f (4.55)where ρ e = Q πr is the electromagnetic energy density, ρ f the fluid energy density and thefactors { w k , w ⊥ } are the fluid equations of state for the radial and tangential pressures, re-spectively. Since the electromagnetic and fluid contributions are assumed to be conservedindependently, the conservation equation of the energy-momentum tensor of the lattercan be explicitly integrated as ρ f,f = ρ f,i (cid:16) g tt,i g tt (cid:17) (1+ w k ) / (cid:0) r i r (cid:1) w ⊥ ) , where the subscripts { i, f } mean that physical quantities are evaluated at some initial and final radius, respec-tively. The above setup describes a charged EiBI black hole that accretes mass, the latterbeing described by a fluid, from an initial state which is the Reissner-Nordstr¨om solutionof GR. Now, using Eqs.(2.57) the following relations between the metric functions in theline elements (4.52) and (4.53) are obtained A = g tt (1 + ¯ ǫρ ) / (1 − ¯ ǫp ⊥ )(1 − ¯ ǫp k ) / , (4.56) B = g rr (1 − ¯ ǫp k ) / (1 − ¯ ǫp ⊥ )(1 + ¯ ǫρ ) / , (4.57) H = r (1 + ¯ ǫρ ) / (1 + ¯ ǫp k ) / , (4.58)100here ¯ ǫ ≡ πǫ . When the fluid energy density vanishes, ρ f = 0, the solution reduces tothe Reissner-Nordstr¨om one of GR.To obtain analytical solutions one must introduce additional constraints. In particular,Avelino [31] studies the mass inflation regime in which w k ∼ | ¯ ǫ | ρ ≪
1, whichsimplifies the relations between metrics as A = g tt , B = g rr and H = r . In addition, it isassumed that mass inflation takes place near the inner horizon, r ∼ r − , and since duringthis regime the energy density becomes much larger than that of the electromagnetic field(so ρ ∼ ρ f ) one can approximate H ∼ r (1 + ¯ ǫρ ) / (1 − ¯ ǫρ ) / ∼ r − [1 − (¯ ǫρ/ ]. Underthese conditions, the tt and rr components of the field equations read − H ′ H B ′ B − BH − (cid:18) H ′ H (cid:19) + 2 H ′′ H = 8 πBT tt , (4.59) − H ′ H A ′ A + BH − (cid:18) H ′ H (cid:19) = 8 πBT rr . (4.60)Mass inflation takes place for r − ¯ ǫ ρ | ρ ′ | ≪
1, where one obtains the additional simplifica-tions H ′ ∼ A ∼ g tt , and B ∼ g rr . This way, Eqs.(4.59) and (4.60) become approximately g ′ rr g rr ∼ − πr − ρg rr ; g ′ tt g tt ∼ − πr − ρg rr . (4.61)Combining the last two equations and integrating the results one gets g rr g tt | [MI] ∼ constantwhere MI stands for quantities evaluated during mass inflation. This equation means that g rr g tt | [start] ∼ g rr g tt | [end] and we recall that g tt [start] ∼ − g − rr [start] (for the Reissner-Nordstr¨omsolution of GR). Now, since mass inflation starts when the energy density of the fluid be-gins to dominate over the electromagnetic contribution, for the sake of finding analyticalsolutions one can assume ρ f = αρ e where α is some constant of order unity. The combi-nation of the above equations implies that the ratio between metric components duringmass inflation satisfies g rr g tt (cid:12)(cid:12)(cid:12)(cid:12) [MI] ∼ − α Q π ρ f,i g tt,i r − w ⊥ ) − r w ⊥ ) i . (4.62)Finally, assuming that mass inflation ends at r − ¯ ǫ ρ | ρ ′ | = β , where β is another constantof order unity, one gets the maximum energy density attained at the end of mass inflation: ρ [end] ∼ β / π / α g / tt,i r − w ⊥ − r w ⊥ i Q ρ / f,i | ǫ | , (4.63)which implies the presence of a threshold of energy density for mass inflation not to betriggered in EiBI gravity, i.e., ρ / f,i | ǫ | < α π / β / Q g / tt,i r − w ⊥ − r w ⊥ i . (4.64)101his threshold depends on the solution’s mass and charge, the accretion rate, and theEiBI parameter ǫ . Note that mass inflation can always occur if the accretion rate is largeenough, independently of the value of ǫ . To see the effect of this threshold in the behaviourof the local mass inside a sphere of radius r in the innermost region of these solutions, oneconsiders the Misner-Sharp mass (MS), defined as M MS = r (cid:16) Q r − g rr (cid:17) [252], whosemaximum is attained at the end of mass inflation, g rr [end] . The calculation of this mass inthe present case yields the result M MS[end] ∼ − r − g rr [end] ∼ π / β / α g / tt,i r − w ⊥ − r w ⊥ i Q ρ / f,i | ǫ | . (4.65)These analytical calculations complement and are in agreement with the numerical analysispresented also by Avelino in [30]. As depicted in Fig.13, for small values of ρ f,i the slopeof the contours indicates that the Misner-Sharp mass is a function of ρ f,i /ǫ / and thatno significant mass inflation occurs below a threshold on the fluid energy density, whichis fully consistent with the analytic result obtained in Eq.(4.65). The conclusion of thisanalysis is that, under the restricted conditions considered in these works, in EiBI gravitythere is a minimum accretion rate below which no mass inflation occurs, no matter howclose the theory is to GR (which is obtained in the limit ǫ → Figure 13: The maximum value of Misner-Sharp mass M MS as a function of the logarithm of the finaldensity ρ f,i and the EiBI parameter ǫ , assuming ω ⊥ = 1 (left), resulting from a numerical simulation(taking r i ∼ . r − and a Reissner-Nordstr¨om solution of GR as initial conditions) of the field equationsto obtain the metric component g rr [end] . Figure taken from Ref.[30]. Lorentzian wormholes are geometric structures representing a shortcut or tunnel be-tween two asymptotically flat regions of spacetime. Such a geometry, for a static sphericallysymmetric and traversable (i.e. without horizons) solutions can be written as [354] ds = − e r ) dt + 11 − b ( r ) r dr + r d Ω (4.66)102here the (gravitational) redshift function Φ( r ) and the (wormhole) shape function b ( r )characterise the geometry. In order to describe a wormhole, two charts for the two asymp-totically flat regions are needed, r ∈ [ r , + ∞ ), where r is the radius of the minimum areasurface at which the two regions are joined. This defines the throat of the wormhole, forwhich b ( r ) = r is fulfilled. In addition, from embedding calculations of the wormholegeometry, it follows that for the throat to be a minimum the flare-out condition b ( r ) − b ′ ( r ) rb ( r ) > , (4.67)must be satisfied there by any wormhole geometry [256]. In GR, the flare-out conditionat the wormhole throat (4.67) implies the violation of the null convergence condition viaRaychaudhuri equation which, for a congruence of light rays with vanishing shear androtation, is given by (for further details see [359], chapter 9) d ˆ θdu + 12 ˆ θ + R αβ ˆ u α ˆ u β = 0 , where ˆ u µ is the four-velocity of a light ray and ˆ θ the expansion of the congruence. In turn,via the Einstein equations, the Raychaudhuri equation entails the violation of the nullenergy condition [354], implying that in the context of GR wormholes are unavoidablesustained by exotic matter. However, such a restriction does not necessarily apply toextensions of GR and thus one could, in principle, obtain wormhole geometries withoutviolations of the energy conditions. To investigate this issue in the context of EiBI gravityit is useful to write the field equations as G µν = R µν − δ µν R = κ S µν (4.68)where R µν ≡ R µν ( q ) and R = R µµ , and the effective energy-momentum tensor S µν isgiven by S µν = τ T µν − (cid:18) − τκ ǫ + τ T (cid:19) δ µν (4.69)with τ ≡ p g/q = | δ µν − κ ǫT µν | − / and T = g µν T µν is the trace of the energy-momentumtensor. This representation of the field equations makes clear that the effective energy-momentum tensor S µν , assumed to be exotic, could be able to sustain wormhole geometrieswithout violations of the null energy condition on the physical energy-momentum tensor T µν .In this section we shall consider the construction of such wormhole geometries in EiBIgravity. Consider a static spherically symmetric geometry, described by the line elementsof the physical and auxiliary metrics asd s g = − e ν ( r ) dt + e ξ ( r ) dr + f ( r ) d Ω (4.70)d s q = − e β ( r ) dt + e α ( r ) dr + r d Ω (4.71)103here { ν ( r ) , ξ ( r ) , f ( r ) , β ( r ) , α ( r ) } are some functions of the radial coordinate r . Observethat the gauge freedom has been imposed in this setup upon the line element for q µν in order to obtain two free functions there, which contrast with the Ba˜nados-Ferreirageometry, where this restriction is made instead upon g µν (see Eqs.(4.23) and (4.24) insection 4.2.1). As a matter source, let us consider an anisotropic fluid given by the energy-momentum tensor T µν = ( ρ + p t ) u µ u ν + p t g µν + ( p r − p t ) χ µ χ ν (4.72)where u µ is the four velocity in the metric g µν , normalized as u µ u ν g µν = − χ µ is theunit vector in the radial direction, i.e. χ µ = e ξ/ δ µr , while { ρ ( r ) , p t ( r ) , p r ( r ) } are theenergy density, tangential pressure (measured in the direction of χ µ ) and radial pressure(measured in the orthogonal direction to χ µ ) of the fluid, respectively. With the lineelement (4.71), and assuming asymptotic flatness, λ = 1, the gravitational field equationsfor the auxiliary metric q µν read [194]1 r − e − α r + α ′ e − α r = 12 ǫ (cid:18) ahc − hac − ah + 2 (cid:19) , (4.73) − r + e − α r + β ′ e − α r = 12 ǫ (cid:18) ahc − hac + 2 ah − (cid:19) (4.74) e − α r (cid:2) β ′′ r − (cid:0) α ′ − β ′ (cid:1) (cid:0) β ′ r (cid:1)(cid:3) = 2 ǫ (cid:18) ahc + hac − (cid:19) , (4.75)with the functions a = p κ ǫρ , h = p − κ ǫp r , and c = p − κ ǫp t , respectively,while we have τ = ( ahc ) − / . Like in the Ba˜nados-Ferreira solutions, two of the metricfunctions can be removed using the relations (2.57), which imply e β = hc a e ν , e α = ac h e ξ ,and f = r ah . In addition, from the assumption of minimal coupling of the matter to thespacetime metric, the energy-momentum tensor of the fluid satisfies the conservation equa-tion ∇ µ T µν = 0, computed with the covariant derivative constructed with the spacetimemetric g µν . This equation reads explicitly dνdr = 4 r p t − p r ρ + p r − p r + ρ dp r dr = 4 r h − c a − h + 4 da − h dhdr . (4.76)Now, the flare-out condition (4.67), which can be written in this case as ξ ′ e − ξ <
0, togetherwith the field equations (4.73) and (4.74), and the relations above between q µν and g µν ,imply that, for the energy conditions to be satisfied in these geometries, the inequality κ ǫ ( ρ + p r ) < ǫh r ( c ) ′ c (cid:18) − br (cid:19) (4.77)(where we have redefined e − ξ ( r ) = 1 − b ( r ) /r to convert (4.70) into the standard form of thewormhole geometry (4.66)) must be satisfied. Evaluation of this condition at the throat b ( r ) = r , implies that if the factor ( c ) ′ /c is finite, then (4.77) is violated, which meansthat exotic matter is needed in order to thread these geometries, like in GR. However, if104 c ) ′ /c diverges or, alternatively, ( c ) ′ /c e − ξ → K (with K some constant) as r → r ,then the condition (4.77) is satisfied if 0 < ρ + p r < K .It is important to note that the set of field equations and relations provided so far donot constitute a closed system, since there are more independent functions than equations.Thus some restrictions have to be made. Harko et al [194] provide a particular wormholegeometry in this framework by introducing the equation of state p r ( r ) = ρ ( r )1 + κ ǫρ ( r ) , (4.78)which is equivalent to choosing the restriction a ( r ) h ( r ) = 1 on the matter components,and in turn implies f ( r ) = r via the transformations between the metric q µν and g µν above. To close the system of solutions one introduces the additional constraint β = 0,and upon solving of the field equations one obtains the result ds = − dt + (cid:18) ǫr /r − r /r (cid:19) dr + r d Ω (4.79)where ǫ > r , so that r < r < + ∞ . Alternatively onecan describe both sides of the wormhole using the radial coordinate l defined as r = l + r ,so now −∞ < l < + ∞ and the throat is located at l = 0. The wormhole geometry (4.79)reduces, in the GR limit ǫ →
0, to the Ellis and Bronnikov (EB) wormhole sustainedby an exotic (phantom) scalar field [88]. In the present case, the energy density, ρ ( r ) = κ ǫ (cid:16) ǫr /r − (cid:17) is negative throughout all space, so the NEC is violated everywhere nomatter the value of the EiBI constant ǫ , which is an outrageous result. On other hand, theflare-out condition at the throat, ξ ′ e − ξ = − r ǫ + r <
0, is satisfied. It should be stressedthat if ǫ < r > | ǫ | , which suggestsa lower bound of r = p | ǫ | for the wormhole throat in this case.On each side of the wormhole throat l = 0 the masses seen by an observer can becomputed as [355]: M ± = ± π R ±∞ ρr drdl dl , which in the case under consideration yieldsthe result [344]: M ± = ± r ± κ r ∓ κ r + . . . . Thus, despite the fact that on each sideof the throat an observer orbiting the wormhole would measure a mass M ± , the totalmass M = M + + M − adds exactly to zero, which is a manifestation of the mass-without-mass mechanism proposed by Wheeler [363] long ago (see section 4.4 for a more completediscussion of this issue).Regarding the effects on physical observers crossing the wormhole throat, Tamang etal. [344] analyse the effect of ǫ on the tidal forces experienced by a free falling observer byconsidering the relative tidal acceleration, ∆ a j , between two nearby parts of the observerfalling into the wormhole. In an orthonormal basis { e ˆ0 , e ˆ1 , e ˆ2 , e ˆ3 } of the observer radiallymoving towards the wormhole, this acceleration is given by [256]∆ a j = −R ˆ0ˆ j ˆ0ˆ p ξ p (4.80)where ξ p is the deviation vector between these two parts and R ˆ i ˆ j ˆ k ˆ l are the components ofthe Riemann tensor. For the wormhole geometry (4.79) one has [344] R ˆ0ˆ j ˆ0ˆ p = γ/ (2 ǫ + r )105where γ = (1 − v /c ) − / and v = ± p | g rr /g tt | dr/dt ), which is finite for any non-vanishing ǫ , and thus the presence of a throat at r may avoid the infinitely large tidalforces found in the EB black hole.Shaikh [327] also uses an anisotropic fluid (4.72) to investigate wormhole structureswithin EiBI gravity, taking the equations of state p r = − ρ and p t = αρ (where 0 ≤ α ≤ s g = − ψ ( r ) f ( r ) dt + dr f ( r ) + r ( dθ + sin θdφ ) (4.81)d s q = − G ( r ) F ( r ) dt + dr F ( r ) + H ( r )( dθ + sin θdφ ) . (4.82)Integration of the conservation equation ∇ µ T µν = 0 yields the result ρ = C r α +1) , where C is a constant (of dimension 2(1 − α )) whose explicit form will be determined fromthe asymptotic behaviour of the metric. Following the same strategy as in the previousspherically symmetric spacetimes considered in this section, the field equations (with λ =1) provide the relations between the metric functions in the line elements (4.81) and (4.82)as f ( r ) = F ( r )(1 − ¯ ǫαρ ) , ; ψ ( r ) = G ( r )(1 − ¯ ǫαρ ) − ; H ( r ) = r p ǫρ , (4.83)where ¯ ǫ ≡ κ ǫ . With these relations, Eq.(2.57) can be explicitly written for this case, givinga set of three independent differential equations. Together with the fluid conservationequation, one can obtain the following solutions for the components of the line element(for ǫ <
0) as [327] ψ ( r ) = " r α +1)0 r α +1) − (4.84) f ( r ) = 1 − r α +1)0 r α +1) α r α +1)0 r α +1) − r α +1)0 | ǫ | r α − Mr r − r α +1)0 r α +1) − α + 1) r α +1)0 | ǫ | r r − r α +1)0 r α +1) I ( r ) , (4.85)where r α +1)0 ≡ | ¯ ǫ | C , the constant − M arises from imposing the asymptotically flatbehavior of the metric component f ( r ), and the function I ( r ) satisfies dIdr = 1 r α r − r α +1)0 r α +1) . (4.86)The interpretation of the radius r is that of the minimum value of the radial coordinate,at which ψ ( r ) → ∞ . The reason to choose ǫ < θ = π/
2. For the line element (4.81) one has ˆ u t = 1 / ( ψ f ) and ˆ u r = ± /ψ and thus the different contributions to (4.68) readˆ θ = ± r r ǫC r α +1) ; d ˆ θdλ = − r (cid:20) α + 2)¯ ǫC r α +1) (cid:21) ; R ( αβ ) ˆ u α ˆ u β = 2( α + 1)¯ ǫC r α +2) , (4.87)with ± for ingoing (outgoing) rays. From these expressions it is clear that, since oneneeds to have C > ≤ α ≤ ǫ <
0. On the other hand, comparing the line element (4.81) with the canonical form of awormhole geometry (4.66) it follows that e = ψ f and (1 − b/r ) = f , which translatesthe flare-out condition (4.67) into f ′ − f ) >
0. Regarding the regularity of the spacetimeone can compute curvature scalars, with the result that they generically diverge, exceptwhen the mass is tuned to the value M = − ( α + 1) r α +1)0 | ǫ | I ( r ) = ( α + 1) r α − | ǫ | F (cid:20) , α − α + 2 , α + 12 α + 2 ; 1 (cid:21) (4.88)(where the second equality is valid provided that α = 1 /
2) for which all curvature scalarsare finite. If this constraint on the mass is assumed, then the parameter x = r / | ǫ | separates those states without a horizon, x <
1, corresponding to traversable wormholes,from black holes with horizons, x >
1, and for which the curvature divergence is avoided.However, the mass M in Eq.(4.88) can only be positive if α > /
2. In Fig.14 the metriccomponents for the case α = 3 / x = x = x = - r Ψ H r L f H r L x = x = x = - r f H r L Figure 14: The metric functions g tt = ψ ( r ) f ( r ) and g − rr = f ( r ) for the case α = 3 / ǫ = −
4. In these plots, x = r / | ǫ | , in such a way that x = 1 sets the appearance/dissappeareanceof a horizon. When x >
1, no horizon is found and the minimum value of the radial coordinate correspondsto r , where the wormhole throat is located. Figure taken from Ref.[327]. Following the analysis of Tamang et al. [344], Shaikh also discusses the tidal forcesupon an observer travelling through the wormhole [327]. Using the tidal acceleration107quation (4.80) one can compute the components of such equation in the present case as(the subindex 0 denotes evaluation of the quantities at the wormhole throat)∆ a ˆ1 (cid:12)(cid:12) r = − αc r (cid:18) − x (cid:19) ξ ˆ1 ; ∆ a ˆ2 (cid:12)(cid:12) r = 1 r (1 − x ) γ v ξ ˆ2 ; ∆ a ˆ3 (cid:12)(cid:12) r = 1 r (1 − x ) γ v ξ ˆ3 (4.89)with the definition x = r /ǫ (in terms of this variable, the flare-out condition reads f ′ − f ) | r = (1 − x ) /r >
0, which is satisfied if x <
1, implying that the wormholethroat r < | ǫ | / ). Restricting the acceleration felt by a traveller of typical size ξ ∼ m tobe below a certain value g , i.e, ∆ a ˆ1 ′ < g , one obtains that the minimum wormhole throatradius is r ≥ αc g (cid:18) − x (cid:19) . (4.90)As r is directly related to EiBI constant, this is translated into a maximum bound for ǫ . Now, from solar physics (see section 3.2.1) one has the constraint [103] | ǫ | /κ . . × m . Take now for instance a model with α & / x .
1, which implies a boundon the acceleration | ∆ a ˆ1 ′ | min ≃ . × sec − , or roughly 17 times Earth’s gravity g E . However, such a wormhole would have a typical minimum size r = p c / (3 g ) = p c / (51 g E ) ≃ . × m, which is roughly 2 . ǫ , which in turn implies stronger accelerationsat the throat. Note that the angular components of the tidal acceleration in Eq.(4.89)impose limits upon the radial velocity v at the wormhole throat r . Let us emphasizethat the general solution for ǫ > ǫ < α = 1, for which the structure of the energy-momentum tensor(4.72) coincides with that of a standard (Maxwell) electromagnetic field, have been derivedand studied in detail, which we review thoroughly next in sections 4.4 and 4.5.It should be pointed out that there are several difficulties on the consistence andviability of this kind of approaches to construct wormhole geometries supported by exoticfields in the context of EiBI gravity, such as the potential instabilities at the quantumlevel [343], which would require to perform stability analysis in the context of this theory,something not available in the literature yet. The solutions we are going to discuss now correspond to EiBI gravity (2.33) coupledto the Maxwell Lagrangian (4.21). However, they will differ from the Ba˜nados-Ferreirasolutions [45] in that i) only the case of ǫ < ∇ µ F µν = 0, for a generic static, sphericallysymmetric line element of the formd s = g tt dt + g xx dx + r ( x ) d Ω (4.91)and an electrostatic field, yield the only non-vanishing component of the field strengthtensor F tx = Qr ( x ) √− g tt g xx . Though this component depends explicitly on the metriccomponents g tt and g xx , the energy-momentum tensor (4.22) does not, which implies T µν = X π (cid:18) − ˆ I × ˆ0 × ˆ0 × ˆ I × (cid:19) ⇒ | ˆΩ | / ( ˆΩ − ) µν = (cid:18) ( λ + ˜ X ) ˆ I × ˆ0 × ˆ0 × ( λ − ˜ X ) ˆ I × (cid:19) (4.92)(where we have combined Eqs.(2.52) and (2.57) for the second equality) and hats denotematrices. Here we have introduced by convenience a new length scale as ǫ → − l ǫ (to dealwith ǫ < X = − l ǫ κ π X . Given the structure ofthe right-hand-side of (4.92), one can introduce the ansatzˆΩ = (cid:18) Ω + ˆ I ˆ0ˆ0 Ω − ˆ I (cid:19) ⇒ Ω − = ( λ + X ) ; Ω + = ( λ − X ) (4.93)for the ˆΩ matrix, where the explicit expressions of Ω − and Ω + follow from solving Eq.(4.92).This way, the gravitational field equations, with the assumption of vanishing torsion andsymmetric Ricci tensor [276] become R µν ( q ) = − l ǫ (Ω − − − ˆ I ˆ0ˆ0 (Ω + − + ˆ I ! , (4.94)where R µν ( q ) ≡ q αµ R ( αν ) . At this point it should be noted that the length-squared scale l ǫ characterises the high-curvature corrections, as follows from the expansion of the EiBIaction in series of l ǫ ≪ S = 12 κ Z d x √− g (cid:20) R −
2Λ + l ǫ (cid:18) − R R ( µν ) R ( µν ) (cid:19)(cid:21) + O ( l ǫ ) + S M ( g µν , ψ m ) (4.95)where Λ = − λ l ǫ plays the role of the effective cosmological constant. Remarkably, thefield equations for the action (4.95) up to order l ǫ , turn out to be exactly the sameas those of EiBI gravity in Eq.(4.94), as can be explicitly verified from Ref.[275]. Theunderlying reason for this result lies on the algebraic properties of the EiBI action andgoes as follows: given the linear relation between T µν and | ˆΩ | / , the diagonal characterof T µν will make the matrix P αν ≡ g αµ R ( µν ) (Γ) to be diagonal as well. Now, if P has twodouble eigenvalues, like happens in this case, ˆ P = diag( p , p , p , p ), then the fourth-orderpolynomial | ˆΩ | / ( ˆΩ − ) µν | = | ˆ I + ǫ ˆ P | turns into the second-order polynomial appearingin Eq.(4.95) when the square root is evaluated. Moreover, this quadratic polynomialexactly coincides with the series expansion of the EiBI action. As a result, all the higher-order corrections beyond l ǫ order cancel out, which means that the electrostatic spherically109ymmetric solutions of EiBI gravity exactly coincides with those obtained for the quadraticLagrangian at order l ǫ appearing in Eq.(4.95). Indeed, electrovacuum solutions in thecontext of such a quadratic gravity models (in Palatini approach) were previously foundin Refs.[273, 274, 275].Now, to solve the field equations (4.94) we introduce the static, spherically symmetricline element for the geometry q µν asd s q = − A ( x ) e ψ ( x ) dt + 1 A ( x ) dx + ˜ r ( x ) d Ω . (4.96)The three functions in this line element can be reduced to a single one by noting thatthe combination R tt − R xx = 0 of the field equations, which follows from the symmetry T tt = T xx of the energy-momentum tensor (4.92), implies that ˜ r xx = ψ x ˜ r x . This allowsto redefine the metric function and the radial coordinate as A → A/ ˜ r x and ˜ r x dx → dx ,respectively, so the line element can be written in the Schwarzschild-like formd s q = − B ( x ) dt + 1 B ( x ) dx + x d Ω . (4.97)This leaves a single independent function to be determined from the R θθ component of thefield equations, which after introducing a standard mass ansatz, B ( x ) = 1 − M ( x ) /x , reads − l ǫ M x = x (Ω − − / Ω − . Resolving this equation requires comparison with the spacetimemetric Eq.(4.91) using the transformations (2.57), which can be split into two 2 × q ab = g ab Ω + and q mn = g mn Ω − . The latter implies the relation of coordinates in thetwo line elements x = r Ω − → dxdr = ± Ω + Ω / − (4.98)which will play an important role later. This way, the equation to be solved reads − l ǫ M r = r (Ω − − + / Ω / − , whose integration can be formally written as M ( z ) = M (1 + δ G ( z )), where M is an integration constant associated to the Schwarzschildmass, G ( z ) contains the electromagnetic contribution, and δ isolates all the relevant con-stants out of this integration. A full solution for the spacetime line element (4.91), in theasymptotically flat case, λ = 1, can now be found explicitly asd s g = − A ( x ) dt + dx A ( x )Ω + r ( x ) d Ω (4.99)with the expressions 110 ( x ) = 1Ω + " − r S r ( x ) (1 + δ G ( r ( x )))Ω / − (4.100) δ = 12 r S s r Q l ǫ (4.101)Ω ± = 1 ± r c r ( x ) (4.102) r ( x ) = x + p x + 4 r c r c = p r Q l ǫ , with r Q = κ Q / (4 π ) a length scale associated to the electric chargewhich, together with the Schwarzschild radius, r S = 2 M , and the EiBI length scale, l ǫ , characterises the solution via the constant δ in Eq.(4.101). Note that the relation(4.103) follows from explicitly solving Eq.(4.98). The function G ( z ) in Eq.(4.100), withthe dimensionless variable z = r/r c , follows directly from the field equations as dG/dz = − Ω + / ( z Ω / − ), and can be explicitly written as G ( z ) = − δ c + 12 p z − (cid:18) F (cid:20) , , , − z (cid:21) + F (cid:20) , , , − z (cid:21)(cid:19) , (4.104)where F [ a, b, c ; t ] is a hypergeometric function and δ c ≈ . z ≫
1. In this limit one has G ( z ) ≃− /z , Ω − ≃ z ( x ) ≃ x ), and the metric function reduces to A ( x ) ≈ − r S r + r Q r , (4.105)which is the standard Reissner-Nordstr¨om solution of GR. This is confirmed by consideringthe behaviour of the curvature scalars for z ≫ R ( g ) ≈ − r c r + O (cid:18) r c r (cid:19) ; Q ( g ) ≈ r Q r (cid:18) − l ǫ r + . . . (cid:19) ; K ( g ) ≈ K GR + 144 r S r c r + . . . (4.106)where R ( g ) = g µν R ( µν ) , Q ( g ) = R ( µν ) R ( µν ) and K ( g ) = R αβµν R αβµν are the curvaturescalar, the Ricci-squared and the Kretchsman, respectively. These expressions smoothlyconverge to their GR counterparts with higher-order corrections in l ǫ . It should be noted that the line element (4.99) can be written in a standard Schwarzschild-like form by absorbing the Ω + factor via a redefinition of the radial coordinate as d ˜ x = dx / Ω . This must be done with care since the radial coordinate x ∈ ] − ∞ , + ∞ [, while r ≥ r c , as depicted in Fig.15, where one observes that the area of the two-spheres111 = 4 πr ( x ) reaches a minimum of size S c = 4 πr c at x = 0, where it bounces offand re-expands again. As already discussed in section 4.3, the presence of a minimumvalue for the radial coordinate allows to infer the presence of a wormhole structure (herethe flare-out condition (4.67) is automatically satisfied). Indeed, from the relation (4.98)between coordinates, it follows that it is ill-defined at r = r c , because dr/dx = 0 at thispoint, and thus the use of r as a radial coordinate is limited to those regions where r ( x )is a monotonic function. Thus, in agreement with wormhole physics lore, one needs twocharts of r to cover the entire manifold, but a single chart in terms of x . Figure 15: Representation of the radial function r ( x ) in Eq.(4.103) as a function of the coordinate x andmeasured in units of r c . In this plot the wormhole throat is located at x = 0 ( r = r c ). The dotted linerepresents | x | (the two asymptotic spaces). The line element (4.96) can be alternatively written in Eddington-Filkenstein coordi-nates using a local redefinition of the time coordinate, dt = dv − dx/ ( A Ω + ), which bringsthe metric into the form ds = − A ( x ) dv + 2Ω + dvdx + r ( x ) d Ω , (4.107)For null and time-like radial geodesics, ds ≤
0, which implies − Adv + σ + dvdx ≤ A <
0, which means that all such geodesics move in the decreasingdirection of x as the advanced time coordinate v moves forward. Now, since the radialfunction r ( x ) has a minimum at x = 0, the relation (4.98) becomes dx/dr = Ω + / Ω / − inthe region x > dx/dr = − Ω + / Ω / − for x <
0. This way, ingoing geodesics, whichalways move in the direction of decreasing x , propagate in the direction of decreasingarea of the radial function r ( x ) if x >
0, but in the growing direction if x <
0, i.e., theyapproach the wormhole throat if x >
0, but move away from it if x < x = 0.As already stated, the geometry described by (4.99) reduces to the Reissner-Nordstr¨omof the Einstein-Maxwell field equations, Eq.(4.105), for z ≫
1, but important departuresare found as r ≈ r c . From the expansion of the G ( z ) function (4.104) in that region, G ( z ) ≈ − /δ c + 2( z − / − (11 / z − / + . . . , one finds that the expansion of the112etric function A ( z ( x )) there yields the result A ( x ) ≈ N q N c ( δ − δ c ) δ δ c r r c r − r c + N c − N q N c + O (cid:0) √ r − r c (cid:1) , (4.108)where N q = Q/e ( e is the electron charge) is the number of charges and N c = (2 /α em ) / ≃ .
55 (with α em the fine structure constant). From the expression (4.108) it is clear thatthere exists a transition on the behaviour of A ( x ) for δ = δ c , yielding three differentsituations. This is consistent with the analysis of the horizons of these configurations,as given by the zeroes of A ( x ). A detailed analysis of such zeroes reveals the followingstructure for the horizons [282]: • If δ < δ c a single horizon is located on each side of the wormhole throat r = r c ,resembling the structure of the Schwarzschild spacetime. • If δ > δ c there may be two, one (degenerate) or none horizons, depending on thenumber of charges N q . These are Reissner-Nordstr¨om-like configurations. • If δ = δ c a single horizon is found for N q > N c and none otherwise. The spacetimemetric g µν is finite at the throat r = r c , and the geometry there is Minkowski-like.It is worth pointing out the resemblance of the three classes (in terms of horizons) ofconfigurations above with those solutions resulting from the coupling of Born-Infeld elec-trodynamics to GR [321, 139, 132, 162, 142], described in section 4.1.1. Nonetheless, thepresence of a finite-size wormhole throat introduces new features as compared to that case.In this sense, as the region z → r c R ( g ) = − δ (cid:18) − δ c δ (cid:19) (cid:20) z − / + a √ z − (cid:21) + (cid:18) − δ c δ (cid:19) + O ( z −
1) (4.109) r c Q ( g ) = (cid:18) − δ c δ (cid:19) (cid:20) δ − δ δ ( z − / + a √ z − (cid:21) + (cid:18)
10 + 86 δ δ − δ δ (cid:19) + O ( z −
1) (4.110) r c K ( g ) = (cid:18) − δ c δ (cid:19) (cid:20) δ − δ )3 δ ( z − / + a √ z − (cid:21) + (cid:18)
16 + 88 δ δ − δ δ (cid:19) + O ( z −
1) (4.111)where { a , a , a } are some constants. These expansion reveal the divergence of all cur-vature scalars at the wormhole throat, z = 1, but for the particular choice δ = δ c theyall become finite. The latter condition sets a particular charge-to-mass ratio, but says113othing on the particular amounts of them. Remember that, from the discussion on thestructure of horizons above, the case δ = δ c corresponds to the transition between theReissner-Nordstr¨om-like and Schwarzschild-like case, where the geometry at z = 1 becomesMinkowskian. As already said, this is somewhat analogous to the case of Born-Infeld elec-trodynamics coupled to GR, though in such a case curvature divergences at r = 0 arealways present. It should be noted that in the EiBI case, the presence of curvature di-vergences or not has no influence on the existence of a wormhole structure, so one couldwonder about the physical meaning of such divergences (see section 4.5.2).In the context of GR one could wonder what is the location of the sources generatingthe mass M and charge Q of the geometry of the Reissner-Nordstr¨om solution. It turnsout that it is not possible to have a well defined point-like source generating both massand charge of the Reissner-Nordstr¨om geometry and, at the same time, being a solution ofthe Einstein equations everywhere [287]. It is equally natural to wonder about the natureof both mass and charge that generate the wormhole geometries discussed here and insection 4.3. As first shown by Misner and Wheeler [253], the non-trivial topology of thewormhole allows to define by itself a charge without the need of considering sources forthe electric field, an effect coined charge-without-charge in that paper. Indeed, an electricflux flowing through a 2-dimensional, spherical S surface enclosing one of the sides of thewormhole mouths defines a charge asΦ = 14 π Z S ∗ F = ± Q (4.112)where ∗ F is the two-form dual to Faradays’s tensor and the two signs ± come from thedifferent orientation of the normal on each side of the wormhole throat. Note that thisresult holds true regardless of the particular details of the configurations as long as awormhole throat exists and the topology does not change. In particular, it is not affectedby the presence of curvature singularities. For the solutions considered in this section, thedensity of lines of flux crossing the spherical wormhole throat can be computed asΦ4 πr c = Qr c = s c ( ~ G ) (4.113)which is independent of the particular amount of charge and mass, i.e, independent on thepresence or not of curvature divergences.In a similar fashion one could wonder about the origin of the mass generating thegeometry (4.99). Following also the mass-without-mass mechanism introduced by Misnerand Wheeler [253], and in analogy with the energy of an electric field in a Minkowskispacetime, S M = R dt × E e one can estimate the total mass of the spacetime by evaluatingthe gravitational + matter action for these configurations, i.e., S = R dt × ( E G + E e ), whichcan be performed in terms of the variable dx = dz / Ω − , with the result [282] S = 2 M c δ δ c Z dt (4.114)where the factor 2 comes from the need of integrating on both sides of the wormholethroat. Like the electric flux above, this result is finite and independent of the existence114r not of curvature divergences. The explicit implementation of both charge-without-charge and mass-without-mass mechanisms make these objects be explicit realizationsof Wheeler’s geon [363], understood as self-gravitating electromagnetic entities withoutsources. It remains to be seen whether the case with ǫ > δ = δ c seems to have no influence on thephysical properties of the solutions such as total charge, mass and density of lines of electricfield, which are as well defined as in the case δ = δ c , where no curvature divergences arise.This is somewhat similar to the thin-shell approach to construct wormhole solutions, bywhich two spacetimes are joined together at a given hypersurface, where the throat islocated [257, 141, 173]. The resulting manifold is geodesically complete by construction,but curvature divergences arise at the wormhole throat [304], which is interpreted as asurface layer with an energy-momentum tensor on it. To get an intuitive idea of thesimilarities and differences between the smooth, δ = δ c , and divergent, δ = δ c solutionsdescribed in this section, one can construct Euclidean embeddings of the spatial equatorial, θ = π/
2, and t =constant section of the line element (4.99), expressed in terms of thecoordinates dx = Ω dr / Ω − , which reads dl = Aσ − dr + r dϕ , to embed it into athree-dimensional space with cylindrical symmetry as dl = dξ + dr + r dϕ , (4.115)where the function ξ must be chosen so as to match the equatorial t =constant line elementabove. Around the wormhole throat r c one can make use of the expansions of the metricfunctions there, together with Ω − ≃ r − r c ) /r c , which yields dl = ( N c − N q )8 N c r c ( r − r c ) dr + r dϕ if δ = δ cN c N q δ δ c ( δ − δ c ) q r c r − r c dr + r dϕ if δ = δ c (4.116)(4.117)so that one has ξ ( r ) = ± ( N c − N q )4 N c √ r c √ r − r c if δ = δ c ± N c N q δ δ c ( δ − δ c ) r c (cid:16) r − r c r c (cid:17) / if δ > δ c (4.118)In Fig.16 we have depicted these Euclidean embeddings for δ = δ c (top figures) and δ > δ c (bottom figures), in those cases where no horizons are present (recall the discussionof section 4.4.1). In both cases, the presence of a wormhole structure is manifest. The115 igure 16: Left plot: Euclidean embedding of the equatorial θ = π/ t =constant section of the caseswith curvature divergences for electromagnetic geons. The vertical axis represents the function ξ ( r ). Figureextracted from Ref.[279]. Right plot: Euclidean embedding of the wormhole described by Eqs.(4.84) and(4.85) with α = 3 / ǫ = 4 and x = 1 /
2. Figure extracted from [327]. two-dimensional curvature, however, as given by the expression of the Kretchsman: K D = N c − N q ) N c r c r if δ = δ cN q N c ( δ − δ c ) δ δ c r c ( r − r c ) r if δ > δ c (4.119)is finite for the former, but divergent for the latter. This highlights the fact that twosimilar wormhole structures can show very different properties regarding the behaviour ofcurvature invariants. This can also be observed in the case of wormholes supported byanisotropic fluids found in [327] and discussed in section 4.3, whose Euclidean embeddingfor the model with α = 3 / The coupling of EiBI gravity to Born-Infeld electrodynamics (4.12) has been consideredby Jana and Kar in [218]. The strategy followed to obtain electrovacuum solutions to thefield equations is pretty much the same than the one employed for the Maxwell field above,and therefore we shall omit the details. The resulting line element, with the redefinition b = α/ (4 ǫ ), becomes 116 s = − U ( x ) e ψ ( x ) dt + V ( x ) U ( x ) e − ψ ( x ) dr + r (cid:0) dθ + sin θdφ (cid:1) (4.120) U ( x ) = 2 − α − α ) − α − α ) 1 q ǫQ (1 − α ) αx V ( x ) = 2 − α − α ) − α − α ) r ǫQ (1 − α ) αx , where the metric function ψ ( x ) splits into two subcases and takes the form ∞ > α > e ψ ( x ) = 1 − Mx + αx ǫ ( α − "r − ǫQ ( α − αx − (4.121)+ α / (4 Q ) / ǫ / ( α − / x F arcsin (cid:0) ǫQ ( α − (cid:1) / α / x ! , − ! −∞ < α < e ψ ( x ) = 1 − Mx − αx ǫ (1 − α ) "r ǫQ (1 − α ) αx − (4.122)+ 4 Q x F (cid:18) ,
12 ; 54 ; − ǫQ (1 − α ) αx (cid:19) , and the radial coordinates are related as r = V ( x ) x . (4.123)From the analysis of these solutions it follows that for α < r = r in Eq.(4.123), where r = (cid:16) ( α − ǫQ (1 − /α )) / α − (cid:17) / , and thusin this case the charge is distributed over a 2-sphere of radius r . This is in agreementwith the analysis carried out in [277] where wormhole solutions with geonic propertieswere obtained in an extension of GR including quadratic corrections in the curvature andcoupled to Born-Infeld electrodynamics. Now, for α > r c = ( κQ α/ (1 − α )) / , while in the range 0 < α ≤
2, no minimum isfound and a point-like charge arises (note that α = 0 corresponds to the GR case). Thisimplies that, in particular, for α >
0, depending on the interplay between EiBI gravity andBorn-Infeld electrodynamics wormhole solutions might be found, but this is not explicitlyinvestigated by Jana and Kar. Nonetheless, they investigate in detail the case α = 1.By variation of the EiBI constant ǫ , solutions with one or two horizons may be found inthat case (which is thus similar to what is found in Born-Infeld electrodynamics in GR,see section 4.1.1, and to geonic solutions supported by a Maxwell field, see section 4.4.1).Curvature divergences are always present either at the location of the throat r = r c (whena wormhole is present) or at r = 0, although the energy density of the electromagneticfield remains finite everywhere. 117o investigate the geodesic behaviour in these geometries, one first notices that in non-linear theories of electrodynamics photons do not propagate along null geodesics of thebackground metric, but instead on null geodesics of an effective geometry [265] given by g µνeff = (cid:0) b F (cid:1) g µν + b F µσ F σν , where F µν is the background electromagnetic field. Forany value of α = 1, the effective metric for the photon propagating in the EiBI backgroundreads [218]d s eff = − U ( x ) e ψ ( x ) dt + U ( x ) e − ψ ( x ) dx + (cid:18) αV ( x ) x + 4 κQ αV ( x ) x (cid:19) ( dθ + sin θdφ )(4.124)from which the expression for the deflection angle (see section 4.2.3 for the basic definitions)of the photon moving on this effective metric is obtained as∆( ϕ ) = 2 Z ∞ x tp U ( x ) r ( x ) " U ( x tp ) e ψ ( x tp ) r ( x tp ) − U ( x ) e ψ ( x ) r ( x ) − / dx − π (4.125)with the expressions appearing in the line element in Eq.(4.120) and with the definitionsof Eqs.(4.121) and (4.122). Numerical integration of (4.125) yields the plots of Figs.17,where the light deflection angle ∆Φ is depicted against the turning point radius r tp (forwhich dr/dφ r th = 0) for different values of α , both positive and negative, and comparedto Maxwell case. This way, like in the case of EiBI gravity coupled to a Maxwell field (seesection 4.2.3), gravitational lensing could be used in order to put experimental constraintson the size of ǫ and on possible nonlinear corrections to Maxwell theory. Maxwell Field Α = 8.0 Α = 1.8 Α = 1.0 Α = 0.51.5 2.0 2.5 3.0 3.5 4.0 4.5 5.002468101214 r tp D Φ Maxwell Field Α = − 8.0 Α = − 1.8 Α = − 0.51.5 2.0 2.5 3.0 3.5 4.0 4.5 5.002468101214 r tp D Φ Figure 17: Deflection angle ∆Φ for EiBI gravity coupled to Born-Infeld electrodynamics as a function ofthe turning radial point r tp (defined as dr/dφ r th = 0), for different values of b = α/ (4 ǫ ), both positive(left plot) and negative (right plot). The dashed curve represents the coupling of EiBI gravity to Maxwellelectrodynamics. Figures taken from [218]. .5. Non-singular solutions Let us now consider an aspect of utmost importance regarding the internal structure ofblack holes resulting from gravitational collapse, namely, the presence of a singularity attheir center. This is an unavoidable consequence of the singularity theorems provided thati) a trapped surface exists, ii) the null congruence condition holds and iii) global hiper-bolicity is fulfilled [299, 300, 197, 177] (see [326, 125] for more pedagogical discussions ofthis issue). These theorems are formally based on the notion of geodesic (in)completeness,namely, on the impossibility of extending null and time-like geodesics to arbitrarily largevalues of their affine parameters. As null geodesics are associated to the transmission ofinformation and time-like geodesics to the free-falling paths of physical observers, geodesic(in)completeness has become the most widely accepted criterium to detect the presence ofspacetime singularities . However, as geodesics are geometrical structures that representidealized point-like observers without internal structure, it is unclear what a quantum the-ory of gravity should say about them. Indeed, from an intuitive point of view, since gravityis a matter of curvature, the blow up of curvature scalars could be seen as an indicationof the presence of large tidal forces that would potentially rip apart a physical (extended)observer, which has shaped numerous approaches to get rid of spacetime singularitiesthrough bounded curvature scalars [35, 21, 20, 259, 254, 241, 65]. Indeed, the standardlore of the field states that, as the curvature grows to reach Planckian values, an improvedtheory of gravity properly incorporating quantum effects should avoid the formation ofsingularities during the last stages of the gravitational collapse [212, 41, 373, 320, 49, 247].In this section we will discuss the regular/singular character of the geonic configurationsdiscussed in section 4.4 making use of these concepts. Our aim in this section is to determine whether the spacetimes considered in section 4.4are geodesically complete, i.e., whether any time-like or null geodesic can be extended be-yond the wormhole throat, since the latter can be reached in finite affine time. We will usethe notations and conventions described in section 4.2.2. We shall focus on asymptoticallyflat spacetimes, λ = 1. In terms of the line element (4.99) the two conserved quantities ofmotion read E = A dtdu and L = r dϕdu or, alternatively, in terms of Eddington-Filkensteincoordinates (4.107), E = A dvdu − + dxdu . This way, the line element (4.99) can be used towrite the modulus of the tangent vector u µ = dx µ /du , which satifies u µ u µ = − k , with k = 0(1) for null (time-like) geodesics, as − k = − A (cid:18) dtdu (cid:19) + 1 A Ω (cid:18) dxdu (cid:19) + r ( x ) (cid:18) dϕdu (cid:19) . (4.126)In terms of the conserved quantities above, Eq.(4.126) reads1Ω (cid:18) dxdu (cid:19) = E − V ( x ) ; V ( x ) = A (cid:18) k + L r ( x ) (cid:19) , (4.127) Note, however, that a given spacetime can be geodesically complete and still be pathological since itcan contain finite paths for observers with bounded acceleration, see Geroch [177]. V ( x ). For radial ( L = 0) null ( k = 0) geodesics, Eq.(4.127) simplifies to1Ω − (cid:18) drdu (cid:19) = E , (4.128)which admits an analytical integration of the form ± E · u ( x ) = F [ − , , ; r c r ] r if x ≥ x − F [ − , , ; r c r ] r if x ≤ , (4.129)where x = F [ − , , ; 1] = √ π Γ[3 / / ≈ . ± corresponds to in-going/outgoing geodesics. For x → ∞ , series expansion of the solution (4.129) yields Eu ( x ) ≈ r ≈ x and the GR behaviour is naturally recovered. In the GR case one has( dr/du ) = E everywhere, whose integration is r ( u ) = ± Eu . Since in that case thefunction r ( u ) is strictly positive, then the affine parameter u ( x ) is only defined on thepositive/negative (ongoing/ingoing) axis and thus geodesics cannot be extended beyond x = 0, hence such spacetime is geodesically incomplete. In the present case, however,the presence of a wormhole throat introduces significant deviations from the GR solu-tion, and from (4.129) one finds that at r = r c ( x = 0) the affine parameter behaves as Eu ( x ) ≈ ± x + √ r − r c ≈ x ± x/
2, with the sign + ( − ) corresponding to the region with x > < u ( x ) can be smoothly extendedbeyond x = 0 and thus radial null geodesics are complete regardless of the value of δ . Thisis a relevant result since in the cases δ = δ c curvature divergences arise at the wormholethroat, but they do not have any impact on the behaviour of the affine parameter, whichis the same in all cases, being free of curvature divergences or not (see section 4.5.2 for adiscussion on the impact of such divergences). Figure 18: Affine parameter u ( x ) (in this plot u → λ ) as a function of the radial coordinate x for nullradial geodesics in Eq.(4.129), compared to the GR behaviour (dashed green curve). In this plot E = 1,and the horizon axis is measured in units of r c . Figure taken from [279]. For null geodesics with L = 0 and time-like geodesics, the effective potential in (4.127)120an be approximated near the wormhole throat x = 0 as [280] V ( x ) ≈ − a | x | − a ; a = (cid:18) k + L r c (cid:19) ( δ c − δ )2 δ c δ ; b = (cid:18) k + L r c (cid:19) ( δ − δ )2 δ . (4.130)From this expression it is clear that if δ > δ c , corresponding to Reissner-Nordstr¨om-likeconfigurations (see section 4.4.1), an infinite potential barrier prevents any such geodesicsto reach the wormhole throat x = 0, which is the same behaviour found in the GR case.But if δ < δ c , corresponding to Schwarzschild-like solutions, these geodesics see an infiniteattractive potential as x → r = 0 as λ ( r ) ≈ ± r ( r/r S ) / and, likewisein the case of radial null geodesics above, the fact that r > x = 0, with no possibility of further extension, and thereforegeodesics in this case are incomplete. In the geonic wormhole case, however, the geodesicequation (4.127) can be integrated as [279] dudx ≈ ± (cid:12)(cid:12)(cid:12) xa (cid:12)(cid:12)(cid:12) → u ( x ) ≈ ± x (cid:12)(cid:12)(cid:12) xa (cid:12)(cid:12)(cid:12) , (4.131)and again, the fact that x ∈ ] − ∞ , + ∞ [ allows to smoothly extend the affine parameter u ( x ) across x = 0 to the whole real axis, which contrasts with the geodesics ending at x = 0 of the GR case. Finally, if δ = δ c (finite curvature cases), the leading order termof the expansion of the effective potential in Eq.(4.130) vanishes, and the new expressionbecomes V ( x ) L = (1 − N q N c ) + O ( x ) (remember that in this case an event horizon is presentif N q > N c ) for both null geodesics with L = 0 and time-like geodesics, which means thatthe potential is regular at x = 0. This way, all geodesics with energy E greater than themaximum of the potential V max will be able to go through the wormhole, while boundedorbits can exist for 0 < V max < E . The comparison of the behaviour of the effectivepotential for the three classes of configurations and different values of the number ofcharges is depicted in Fig.19, corresponding to time-like geodesics with L = 0.From the description above, it follows that the presence of a spherically symmetricwormhole structure replacing the point-like singularity of GR allows for geodesically com-plete spacetimes, which is in agreement with the standard lore of wormhole physics [354].Nonetheless, the physical meaning of curvature divergences at the wormhole throat re-quires a separate analysis. We have already discussed in section 4.3, following Shaikh [327], that for wormholesolutions supported by anisotropic fluids, tidal forces at the wormhole throat can be finite.In this section we shall follow a different approach to determine the impact of curvaturedivergences on physical (extended) observers and review the results of [281]. This approachis based on the concept of strong singularities, originally introduced by Ellis and Schmidt[152]. Such singularities are identified by the property that all objects approaching themare crushed to zero volume, no matter what their internal constituents or forces holdingthem might be. This is opposed to weak singularities, for which a body could retain its121 igure 19: Effective potential V ( x ) for time-like geodesics with L = 2. Plots A, B and C depict the Reissner-Nordstr¨om-like ( δ > δ c ), Minkowski-like ( δ = δ c ) and Schwarzschild-like ( δ < δ c ) cases, respectively, forthree curves corresponding to N q = 1 , N c , N c (solid, dashed, dotted, respectively). Plots D, E and F depictthree values of charge N q = 1 , N c , N c , respectively, for three curves corresponding to δ = δ c , . δ c , . δ c , (solid green, dashed red, dotted blue, respectively). Figure taken from [279]. identity while crossing the divergent region. Built on the precise mathematical frameworkintroduced by Tipler [346, 347], Clarke and Krolak [117] and others [263, 285], the idea is toidealize a physical (extended) observer as a set of points following their own geodesic path(i.e. a congruence), and to determine the relative separation between nearby geodesics asthe divergent region is crossed. The congruence is characterised as x µ = x µ ( u, ξ ), where u corresponds to the affine parameter along a given geodesic and ξ labels the differentgeodesics on such a congruence. The separation between nearby geodesics (for fixed u ) ismeasured by the Jacobi field Z µ ≡ ∂x µ /∂ξ , which satisfies the geodesic deviation equation D Z α du + R αβµν u β Z µ u ν = 0 . (4.132)Given the second-order character of this equation, it follows that there are six independentJacobi fields along a given geodesic, which are obtained as Z a ( u ) = A ab ( u ) Z b ( u i ), where Z b ( u i ) corresponds to the value of the Jacobi fields at some initial instant u i and A ab ( u ) isa 3 × u = u i ). If all Jacobi fields vanish at u = u i , one can insteadwrite Z a ( u ) = A ab ( u ) DZ b du (cid:12)(cid:12)(cid:12) u = u i , where A a is a 3 × u = u i . Thisway, three linearly independent solutions of (4.132) allow to define a volume element: V ( u ) = det | A ( u ) | V ( u i ) (4.133)(or as det |A ( u ) | if Z a ( u i ) = 0). Thus a strong singularity is met if lim u → V ( u ) = 0 [117],where the singularity is approached if u → { Z (1) = B ( u )( u x /A, Au t , , , Z (2) = (0 , , P ( u ) , , Z (3) =(0 , , , Q ( u ) / sin( θ )) } , which are orthogonal to the time-like radial geodesic vector u µ =( u t , u x , , u t ≡ dt/du = E/A and u x ≡ dx/du . Thegeodesic deviation equation (4.132) allows to obtain the functions B ( u ), P ( u ) and Q ( u )via the equations P ( u ) = P + C R dur ( u ) , Q ( u ) = Q + C ′ R dur ( u ) and B uu + A yy B ( u ) = 0.Close to the wormhole throat, the behaviour of these functions can be computed and theresult compared to their GR counterparts as B GR ( u ) ≈ C | u | / − | u | / | u i | / ! → B EiBI ( u ) ≈ C ′ | u | / − | u | / | u i | / ! (4.134) P GR ( u ) ≈ C (cid:18) | u i | / − | u | / (cid:19) → P EiBI ( u ) ≈ C ′ ( u − u i ) (4.135) Q GR ( u ) ≈ C (cid:18) | u i | / − | u | / (cid:19) → Q EiBI ( u ) ≈ C ′ ( u − u i ) , (4.136)where { C , C ′ , C , C ′ , C , C ′ } are arbitrary constants. Now, from [263] the resulting vol-ume from these spacetimes can be written as V ( u ) = | B ( u ) P ( u ) Q ( u ) | r ( u ) , (4.137)Now, since in GR one has r GR ≈ (9 r S / / u / and in EiBI geons r ( u ) ≈ r c + x /
2, thenone finds that the volume in the former is V GR ≈ u / , while in the latter V EiBI ≈ /u / .Thus, in the GR case the volume vanishes as u → u →
0, a scenario that hasbeen independently discussed by Nolan [264] and Ori [285].To investigate in more detail the effect of such a divergent volume on physical observerslet us rewrite the line element (4.99) in free-falling coordinates as [280, 281] ds g = − du + ( u y ) dξ + r ( u, ξ ) d Ω , (4.138)where ξ measures the radial separation between nearby geodesics and u y ≡ dy/du , where dy = dx/ (1 + r c /r ( x )). For the Scharwarzschild-like configurations, δ < δ c , which is theonly case in which time-like observers can go through the wormhole (recall the discussionof section 4.5.1), the vector ( u y ) can be approximated near the wormhole throat as( u y ) ≃ a/ | x | ≃ ( a | u − Eξ | ) − . This turns (4.138) into ds g ≈ − du + (cid:18) a | u − Eξ | (cid:19) − / dξ . (4.139)This expression states that, as the wormhole throat is approached, the distance betweentwo infinitesimal nearby geodesics diverges, dl P hys = (cid:0) a | u − Eξ | (cid:1) − / dξ . However, for123nite comoving separation between nearby geodesics, l ξ ≡ ξ − ξ , the physical separation l P hys ≡ R | u y | dξ can be computed as l P hys ≈ (cid:16) a (cid:17) / E (cid:12)(cid:12)(cid:12) | u − Eξ | / − | u − Eξ | / (cid:12)(cid:12)(cid:12) , (4.140)which is finite. Due to the divergent volume carried by a physical observer, the meaning ofthis result is that, as the wormhole throat is approached, infinitesimally nearby geodesicsare infinitely stretched in the radial direction, followed by an identical contraction asthe wormhole is left behind, in a sort of spaghettisation process. The danger lies on thepossibility that the constituents that make up and keep cohesioned the body could losecausal contact due to the spatial stretching affecting their infinitesimal elements, whichwould result in the unavoidable destruction of the body. To check this one can considerthe propagation of radial null rays, ds = 0, in the background (4.139), so the photonpath satisfies dξdu = ± (cid:12)(cid:12)(cid:12)(cid:12) a ( u − Eξ ) (cid:12)(cid:12)(cid:12)(cid:12) / . (4.141)Using a numerical integration, in Fig.(16) two main results are observed: i) a fiducialobserver at ξ = 0 never loses causal contact with its nearby geodesics (left figure) andii) the proper time taken in a round trip by a light ray from ξ = 0 to a nearby geodesicis always finite and casual as the wormhole throat is crossed (right figure), with just anadditional delay in the travelling time. Thus, in these geometries, physical observers nearthe wormhole throat can remain in causal contact despite the spaghettisation processexperienced as u → Eξ , and can apparently cross this region with curvature divergences,without experiencing absolutely destructive deformations. - - Ξ- - Λ - - Λ DΛ Figure 20: Left plot: trajectories of light rays (in this plot u → λ ) emitted by a time-like observer from ξ = 0 at different times before reaching the wormhole throat (oblique line u − Eξ = 0) in a Schwarzschild-like configuration, δ < δ c (in this plot, E = 1 , a = 3). Right plot: proper time ∆ u taken in a round tripby a geodesic at ξ = 0 to another separated by a distance ξ = { . r c , . r c , . r c } , as a function ofthe proper time u at which the light ray was sent. At u = 0 the divergent region is reached, but the (finite)travelling time tends to zero as the comoving distance tends to zero too. Figures taken from Ref.[281]. .5.3. Tests with scalar waves As a third test to determine whether the presence of curvature divergences endangersthe well posedness of the physical laws in these geonic geometries one can study thepropagation of scalar waves near the wormhole throat, following the description of [280].This analysis considers the case of Reissner-Nordstr¨om-like configurations, δ > δ c , wherethe presence of a time-like Killing vector allows for a separation of variables. The fieldequation for a massive scalar field, ( (cid:3) − m ) φ = 0, can be decomposed in modes of theform φ ω,lm = e − iωt Y lm ( θ, ϕ ) f ω,l ( x ) /r ( x ), where Y lm ( θ, ϕ ) are spherical harmonics and thefunctions f ω,l ( x ) /r ( x ) are governed, in the radial coordinate dy/dx = 1 /A (1 + r c /r ), bythe Schr¨odinger-like equation − f yy + V eff f = ω f ; V eff = r yy r + A ( r ) (cid:18) m + l ( l + 1) r (cid:19) , (4.142)where the effective potential V eff converges to the GR result for r ≫ r c , but behaves nearthe wormhole throat r = r c as V eff ≈ k | y | / ; k ≡ s ( δ − δ c ) N q δ δ c N c ( N c [ m r c + 1 + l ( l + 1)] − N q ) N c (8 r c ) / . (4.143)While low-energy modes cannot overcome the potential barrier and are almost entirelyreflected, much like in the GR case, high-energy models may overcome such a barrier andend hitting the wormhole throat. Considering an incoming wave packed travelling fromnull infinity (when no horizon is present, or from the event horizon otherwise), the waveequation (4.142) reduces to f y ′ y ′ + (cid:18) α ± √ y ′ (cid:19) y = 0 , (4.144)where the parameter α = | k | − ω encodes all the relevant information for this problem.The sign ± determines an infinite well or potential, the former leading to a transmissioncoefficient that tends to one as α grows, while the latter has a typical sigmoid profileof barrier experiments, where a threshold around α = α th ∼ . ω there is another threshold, l = l max , such that the cross section, σ , can be roughlyestimated by considering that the transmission factor is one for l > l max (almost entiretransmission) and zero for l < l max (almost entire reflection) as σ = πω l max X l =0 (2 l + 1)1 = πω (1 + l max ) (4.145)which is depicted in the right panel of Fig.21, where for ω → ∞ one has σ ∝ ω − / .As a summary of this section, the well posedness of the wave scattering problem, to-gether with the geodesic completeness for null and time-like geodesics and all spectrumand mass and charge, and the fact that the constituents making up physical (extended)125 +++++++++++++++++++++++++++++++++++++++ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ Α T Ω Σ Figure 21: Left plot: transmission coefficient for the potential well −| y ′ | − / (blue dots) and the barrier+ | y ′ | − / (red crosses). Right plot: transmission cross section in Eq.(4.145) as a function of ω for δ = 1and N q = 10 (configurations without horizons), from numerical calculation of V eff (dots) in Eq.(4.142)and compared to the approximation σ ∝ ω − / (continuous line). Figures extracted from Ref.[280]. observers can remain in causal contact as the wormhole throat is crossed, imply the exis-tence of classical non-singular black hole geometries in EiBI gravity. It should be pointedout that similar electromagnetic solutions as those analyzed here and in section 4.4 canbe found in functional extensions of the form f ( X ) = X n , where X ≡ det(ˆ g − ˆ q ) and theparameter n labels different models ( n = 1 / / < n ≤ / n > / The setup derived and discussed in Sec.(4.4) can be extended to their higher-dimensional,
D >
4, counterparts. Most of the corresponding expressions are easily obtained followinga similar approach, see Bazeia et al [55]. The field equations in the q µν geometry read now R µν ( q ) = ǫ | ˆΥ | D − [ L BI δ µν + T µν ] ; L BI = | ˆΥ | D − − λǫκ , (4.146)with the definition ˆΥ ≡ | ˆΩ | / (Ω − ) µν = λδ µν − ǫκ T µν (4.147)(remember that ˆΩ − ≡ ˆ q − ˆ g , while the definition (4.147) implies | ˆΩ | / = | ˆΥ | D − ). Therelation between the auxiliary q µν and the physical metric g µν is now given by q µν = | ˆΥ | D − (Υ − ) µα g αν ; q µν = 1 | ˆΥ | D − g µα Υ αν . (4.148)It is easy to see that the system of equations (4.146), with the definitions (4.147) andthe transformation (4.148), satisfies the same second-order field equations and ghost-free126haracter of their four-dimensional partners, besides the recovery of the D -dimensionalMinkowski spacetime in vacuum.Electrovacuum solutions of the field equations (4.146) are easily derived following thesame steps as in section 4.4, using now the set of transformations (4.147). One thus settwo static, spherically symmetric line elements of the formd s g = g tt dt + g xx dx + r ( x ) d Ω D − (4.149)d s q = − A ( x ) e ψ ( x ) dt + 1 A ( x ) dx + x d Ω D − , (4.150)(where d Ω D − is the angular sector in the maximally symmetric subspace) for the metric g µν and q µν , respectively, so that the electromagnetic field satisfies F tx = Qr ( x ) D − √− g tt g xx .The ansatz for ˆΩ compatible with the symmetry of the electromagnetic field becomesˆΩ = Ω + ˆ I × ˆ0 ( D − × ˆ0 × ( D − Ω − ˆ I ( D − × ( D − ! ⇒ Ω − = ( λ + ˜ X ) D − ; Ω + = ( λ − ˜ X )( λ + ˜ X ) D − D − (4.151)where we have used Eq.(4.147). Like in the four dimensional case, the combination R tt − R xx = 0 allows to rewrite the line element for q µν in Eq.(4.149) in standard Schwarzschild-like form, while the introduction of a mass ansatz, A = 1 − M ( x )( D − x D − , allows to solve thefield equations for M ( x ), and transforming that solution back to g µν using that { q ab = g ab Ω + ; q mn = g mn Ω − } , where ( a, b ) contains the 2 × m, n ) the maximallysymmetric sector, one obtains the final solution for g µν as ds g = − A Ω + dt + (cid:18) Ω + A (cid:19) (cid:18) dx Ω + (cid:19) + z ( x ) d Ω (4.152) A ( z ) = 1 − δ G ( z ) δ Ω D − − z D − ; G z = − z D − Ω − − / − ! (cid:18) λ + 1 z D − (cid:19) (4.153) δ ≡ ( D − r D − c M l ǫ ; δ ≡ ( D − r D − c M , (4.154)with the definition z ≡ r/r c , where r D − c ≡ l ǫ r D − Q with ǫ ≡ − l ǫ and r D − Q ≡ κ Q / (4 π ), while M is Schwarzschild mass. Again, to detect the presence of wormholestructures, we just need to inspect the relation between radial coordinates in the two lineelements (4.149), obtained as x = r Ω − ⇒ (cid:18) | x | r c (cid:19) ( D − = 1 z D − (cid:16) z D − − (cid:17) , (4.155)127hich is just a standard quadratic equation for z d − , which can consequently be solved as r d − = | x | D − + q | x | D − + 4 r D − c x comes from the fact that a square root has been extracted toobtain (4.156). The behaviour of the radial function r ( x ) is depicted in Fig.22 (left),where we observe the typical bouncing behaviour of a wormhole for any dimension D ,with the throat located at x = 0 ( z = 1). As follows from the analysis of Bazeia et al [55],expansions of the metric functions and the curvature scalars at the throat reveal that, asopposed to the four dimensional case, they always diverge there (in four dimensions, inthe case δ = δ c they become finite, see section 4.4.1). However, a similar analysis of thegeodesic structure near the wormhole throat as in section 4.5.1 reveals the completenessof null and time-like geodesics for all the spectrum of mass and charge of the solutions.The case of radial null geodesics is depicted in Fig.22 (right), where we observe that theycan be naturally extended beyond the wormhole throat x = 0. However, the impact ofsuch curvature divergences on physical observers crossing the wormhole throat has notbeen analyzed in the literature yet. The geonic properties of such solutions have been alsoanalyzed in [55], with similar qualitative results as those found in section 4.4.1. - - - x r H x L Figure 22: Left plot: representation of r ( x ) in Eq.(4.156) for D = 4 (solid), D = 6 (dashed) and D = 10(dotted), with both axes measured in units of r c . The wormhole throat is located at x = 0. Right plot:representation of the affine (null radial geodesics) parameter Eu ( x ) (in this plot u → τ ) as a function of theradial coordinate x in D = 4 (solid), D = 5 (dashed) and D = 10 (dotted). Figures taken from Ref.[55]. The original idea of extra dimensions was implemented by Kaluza and Klein by as-suming that the four dimensional energy-momentum tensor of the electromagnetic field isoriginated from a part of a five dimensional metric tensor. This idea was reemployed inthe EiBI scenario in Ref.[159], where they assume a five-dimensional metric given byˆ g AB = (cid:18) g µν + αA µ A ν αA µ αA µ (cid:19) (4.157)128here α is a parameter, latin indexes run from A = 0 , . . . , µ = 0 , . . . , R , to be given by 2 π ˜ R/ ˆ G = 1 /G = 1 (whereˆ G and G are the five and four dimensional Newton’s constant, respectively), and takingby convenience α = 4, one obtains that the five dimensional EiBI action reduces to [159] S = 18 π ˆ G ǫ Z d x "r(cid:12)(cid:12)(cid:12) (ˆ g AB + ǫ ˆ R ( AB ) ) (cid:12)(cid:12)(cid:12) − λ p | ˆ g | ⇒ πG ǫ Z d x hp ǫF (4.158) × vuut(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g µν + ǫ ( R ( µν ) + 2 F µβ F βν ) + ( ∇ δ F δµ ∇ β F βν ) ∞ X n =0 ( − n +1 ǫ n +2 F n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − λ p | g | where it is clearly seen that it transforms into a four dimensional, Class-III gravitationalaction containing a number of curvature-matter couplings, and where F µν arises as the fieldstrength tensor associated to the vector potential A µ . The field equations correspondingto the action (4.159) are highly involved, even to lowest order in ǫ (see Eqs.(4.4)-(4.7) ofRef.[159]). Nonetheless, in the spherically symmetric case, (electrostatic) solutions to firstorder in ǫ can be obtained under the form ds g = − f ( r ) dt + f ( r ) − dr + r d Ω (4.159) f ( r ) = (cid:18) − Mr − Λ r Q r (cid:19) + ǫ (cid:18) Q r − Λ Q r (cid:19) + O ( ǫ ) (4.160)which corresponds to a modification of the Reissner-Nordstr¨om-Anti-de Sitter solution ofGR ( ǫ → R = 4Λ + 6 ǫQ /r + O ( ǫ ), yields a curvature singularity at r = 0, but no further properties of these solutions(such as horizons, geodesic structure, etc) are investigated in that work. Braneworld scenarios represent an interesting development of the Kaluza-Klein idea,boosted by the proposals introduced by Randall-Sundrum [313, 314] and Arkani-Hamed-Dimopoulos-Dvali [24, 22] models. They assume that the four-dimensional world to whichstandard model particles are attached (the brane), is embedded in a higher-dimensionalspacetime (the bulk) with a warped geometry, in such a way that gravitons can propagatealong the extra dimension (see e.g. [315] for a review). Though in the original proposalsthe brane is infinitely thin, in this section we shall consider instead a thick brane, namely,a five-dimensional bulk with a scalar field propagating in the extra dimension, and whoseenergy density is assumed to be localized around the point (say) y = 0 of the extra dimen-sion. The analysis of this scenario can be carried out to a large degree of generality, by129onsidering a Born-Infeld inspired modification of gravity (in Palatini formalism) definedby an arbitrary function F of the object P µν ≡ g µλ R ( λν ) and coupled to a scalar field as S = 12 κ Z d D x √− gF ( ˆ P ) + Z d D x √− g L ( X, φ ) (4.161)where D = d + 1 is the number of spacetime dimensions. The Lagrangian density L ( X, φ )contains, in general, a non-canonical contribution from the scalar field kinetic term X ≡ g αβ ∂ α φ∂ β φ (see [25] for the inception of these theories in Cosmology). The field equationsfor this system are derived in the usual way, i.e., by performing independent variations ofthe action (4.161) with respect to the metric and the connection, which can be handledalso by introducing a new metric q µν as [53] q µν = 1 | ˆ F ˆ P | D − g µλ ( F ˆ P ) λν ; q µν = | ˆ F ˆ P | D − ( F − P ) µλ g λν (4.162)where ( F ˆ P ) λν ≡ ∂F∂P λν and | ˆ F ˆ P | represents its determinant. The resulting field equationsare quite similar to those of EiBI gravity given by (4.146), which are written here byconvenience as R να ( q ) = κ | ˆ F ˆ P | D − (cid:16) L G δ ν α + T ( φ ) να (cid:17) , (4.163)where L G corresponds to the particular Lagrangian density considered. To implement thethick brane scenario one sets the line element for the physical metric g µν :d s g = a ( y ) η ab dx a dx b + dy (4.164)where a ( y ) is the warp factor, which is assumed to depend only on the extra dimension y ,and η ab is the metric on a d -dimensional spacetime brane of constant curvature K . Thecorresponding field equations (4.163) in this case can be conveniently written with thehelp of a similar ansatz for the auxiliary metric q µν :d s q = ˜ a (˜ y ) η ab dx a dx b + d ˜ y . (4.165)Using (once more) the relation (2.57) it follows that in this caseΩ µλ ≡ | ˆ F ˆ P | D − ( F − P ) µλ = (cid:18) Ω + I d × d ˆ0ˆ0 Ω − (cid:19) , (4.166)from where one obtains that ˜ a (˜ y ) = Ω + a ( y ) and d ˜ y = Ω − dy . The gravitational fieldequations follow now immediately as (see Bazeia et al [53] for details) d ( d − K − H ] = κ | Ω | / h ( d − L G + d · T + − T − i (4.167)( d − K + H ˜ y ] = κ | Ω | / ( T + − T − ) (4.168)130here H ≡ ˜ a ˜ y / ˜ a , while T + = −L ( φ, X ) / T − = L X φ y + T + are the components ofthe scalar energy-momentum tensor, and Ω ± are model-dependent functions of X and φ .For the sake of the search for solutions below, L G in this equation represents EiBI gravityLagrangian (2.33).For the case of standard canonical kinetic term with a potential, L = X − V ( φ ), specificsolutions were obtained by Liu et al. [242], using a fully equivalent approach to the onedepicted above though written directly in terms of the functions { a, ˜ a } , see Eqs.(17a), (17b)and (17c) of that paper. Specifically, they look for a kink solution interpolating betweendifferent vacua at asymptotic infinity y = ±∞ . This can be achieved by introducingan additional constraint φ ′ ( y ) = Ka ( y ), where K is a constant conveniently defined as K = ± (cid:0) (cid:1) √ ǫκ . This way, the scalar field equation4 a ′ a φ ′ + φ ′′ = ∂V ( φ ) ∂φ (4.169)can be integrated with the result V ( y ) = K a ( y ) + V , where V is an integration constantthat can be interpreted as the scalar field vacuum energy, fixed here as V ( φ ) = − λ/ ( κǫ ).Inserting this result into the gravitational field equations (4.167) and (4.168) one finds ananalytic solution for the warp factor and scalar field profile in closed form as a ( y ) = sech (cid:18) √ ǫ y (cid:19) , (4.170) φ ( y ) = ± / × / κ (cid:20) i E (cid:18) iy √ ǫ , (cid:19) + sech (cid:18) y √ ǫ (cid:19) × sinh (cid:18) y √ ǫ (cid:19)(cid:21) , (4.171)where E is an elliptic integral of second kind. This way the potential can be expressed as V ( y ) = √ ǫκ sech ( y √ ǫ ) − λǫκ , which allows to compute the energy density associated tothese solutions as ρ ( y ) = 7 √ ǫκ sech (cid:18) y √ ǫ (cid:19) . (4.172)These scalar field and energy density profiles naturally implement the defining propertiesof a kink, namely, its interpolating character between different vacua at y = ±∞ , aswell as the localized nature of the energy density around the center of the kink, y =0. Regarding the curvature of the solutions, a simple calculation yields the result R = g MN R ( MN ) = ǫ h − ( √ ǫ ) y i , which asymptotically approaches the value R →− /ǫ <
0, corresponding to an Anti-de Sitter space.An important aspect of configurations on the brane is to determine its stability againsttensorial perturbations there. As shown by Bazeia et al. [53] this can be also done infull generality for a theory F ( ˆ P ) . The idea is to write two perturbed line elements inGaussian normal coordinates as A general treatment of tensorial perturbations in EiBI gravity can be found in [371]. s g = a ( y ) ( η ab + h ab ) dx a dx b + dy (4.173) ds q = ˜ a (˜ y ) ( η ab + h ab ) dx a dx b + d ˜ y (4.174)where the scalar and vector modes are decoupled by imposing the conditions δg ab = a ( y ) h ab and δg ay = 0 = δg yy . From (4.173) and (4.174), the perturbation of the fieldequations (4.163) in the q µν geometry reads simply δ R µν ( q ) = 0 → δ R ( µν ) ( q ) = R µβ t βν ,where t ab = ˜ a h ab is the only non-vanishing component of t βν . Now, using standardcovariant perturbation methods, and after some algebra, the tensorial modes, assumedto be written as h ab = X ( z ) ǫ ab ( t, ~x ) (where we have introduced a new coordinate z as dz = d ˜ y / ˜ a ), satisfy two sets of equations, namely ( η ) (cid:3) ǫ ab − Kǫ ab − p ǫ ab = 0 (4.175) − Y zz + V eff ( z ) Y = p Y (4.176)where p is a constant. (4.176) is a Klein-Gordon-type equation for the massless, p = 0,and massive, p = 0, gravitons, while in the Schr¨odinger-like equation (4.176) for theKaluza-Klein modes we have redefined X = ˜ a − ( d − Y , and the effective potential V eff ( z )is given by V eff = ( d − H z + ( d − H , (4.177)where H ≡ ˜ a z / ˜ a . The operator on the left-hand side of Eq.(4.176) can be factorized as (cid:18) ddz − ( d − H (cid:19) (cid:18) ddz − ( d − H (cid:19) (4.178)which is a non-negative operator, guaranteeing in this way that p >
0, which impliesthe tachyonic-free and stable character of this class of theories of gravity under tensorperturbations. For the particular case of EiBI gravity, Liu et al. [242] computed the zeromode, p = 0, as Ψ ( z ) = N a / ( z ), where the normalization condition R Ψ ( z ) dz = 1 fixesthe constant N ≈ . / √ ǫ . This gravity zero mode is localized at the center of the kink, y = 0, while vanishes at y = ±∞ . The effective potential (4.177) has a (asymptoticallyvanishing) volcano-like profile with a well at the center of the kink, with the result thata continuous set of massive Kaluza-Klein modes (not localized on the brane) arises for p > φ ′ ( y ) = Ka ( y ) n , so that Liu et al. case [242]corresponds just to n = 1. The corresponding solutions for the warp factor and the scalarfield can also be obtained in closed analytical form as a ( y ) = sech n ( ky ) ; φ ( y ) = 2 Kk h i E( iky/ , / ( ky ) sinh( ky ) i (4.179)132here the constants K = ± (1+4 n/ / ( n +1) p nǫκ and k = n √ ǫ (4 n +3) , while the energy densitycan be computed simply as ρ = n +1 n K sech ( ky ). These configurations show similarfeatures as those of n = 1 below, namely, interpolation of the kink between two asymptoticvacua at y = ±∞ and localized character around y = 0 with a maximum of the energydensity there. The impact of increasing the value of n is just to decrease the width of thekink and to lower the maximum of the energy density. One could go on further in thestandard strategy in the field, by investigating additional models which allow to modifythe physical properties of the kink at will, but we shall stop here. Let us simply emphasizethat the zero mode for any n not localized in the brane, while more complex models like φ ′ ( y ) = K a ( y ) (1 − K a ( y ) ) allow to find quasi-localized states on the brane for massiveKK gravity modes. Electrovacuum solutions of EiBI gravity in D = 3 dimensions requires a separateanalysis from that of section 4.6.1, due to the peculiarities of the integration of the metricon such a case. In this sense, the expressions (4.146), (4.147), (4.148), (4.149), (4.151) arestill valid, but the integration of the metric (with a cosmological constant term, λ = 1)yields now the result ds g = − A ( r )Ω + dt + 1 A ( r ) dx Ω / ! + r ( x ) dθ (4.180) A ( x ) = − λ M − λ − s | ǫ | r − Q (cid:18) sr c r + 1 λ ln (cid:20) r + sr c /λr (cid:21)(cid:19) , (4.181)where s in ǫ = s | ǫ | is the sign of ǫ , and r is an integration constant. The line element(4.180) represents a natural generalization of Ba˜nados, Teitelboim and Zanelli (BTZ)solution [48], which is recovered both in the limit ǫ →
0, and asymptotically, r ≫
1. TheBTZ solution raised a great deal of interest due to the fact that the states with M = − r ( x ) in Eq.(4.180) can be explicitly written as | r ( x ) | = | x | ± p | x | − sλr c λ , (4.182)which attains a minimum at r = r c /λ / both for s = ±
1. When s = − s = +1 a similar construction as the Einstein-Rosen bridge [149]can be obtained. In both cases, null and time-like geodesics can be indefinitely extendeddespite the presence of curvature divergences at the wormhole throat. However, this isdone via two different mechanisms: when s = − (see Fig.23, left), while when s = +1, the wormhole is reached on A similar result has also been found in other theories of gravity formulated in the Palatini approach,like f ( R ) [278, 40, 56].
133 finite affine time but, like their four and higher dimensional counterparts (see sections4.5.1 and 4.6.1, respectively), it can be extended beyond this point to arbitrarily largevalues of the affine parameter (see Fig.23, right). This way, all the electrically chargedsolutions of EiBI gravity with a wormhole structure are geodesically complete in D ≥ Figure 23: Left plot: affine parameter u ( x ) (in this plot u → σ ) for ingoing and outgoing radial nullgeodesics in the case s = −
1, as compared to the GR case (dashed lines). In this case, the wormhole lieson the future (or past) boundary of the spacetime. Right plot: Affine parameter u ( r ( y )) (where y is a newsuitable radial coordinate) for radial null geodesics in the case s = +1, where the wormhole is reached infinite time but can be indefinitely extended. Figures extracted from Ref.[54]. Three dimensional, asymptotically flat, circularly symmetric charged solutions withinthe context of Born-Infeld inspired gravity formulated in Weitzenb¨ock spacetime (Class-II)have been found by Ferraro and Fiorini [166]. This is a formulation of classical gravity interms of a spacetime possessing absolute paralelism (or teleparallel gravity, see Ref.[202]).The action considered in this work is defined as S BIT = 12 κ ǫ ( A + B ) Z d x (cid:20)q | g µν + 2 ǫF µν | − λ q | g µν | (cid:21) , (4.183)where F µν = AS µλρ T λρν + BS λµρ T λρν (with A and B some constants) is quadratic in theWeitzenb¨ock torsion T ρ µν = e ρa ( ∂ µ e aν − ∂ ν e aµ ) build out of the set of 3-forms { e a ( x ) } , withthe definitions (2.144) and (2.145). In the limit ǫ → R , can be written as R = S µνρ T ρ µν (+ total derivative terms).Upon resolution of the corresponding field equations for this theory one obtains theline element [166] ds = (cid:18) J r + M (cid:19) dt − (cid:18) Y ( r ) J / (4 r ) + M (cid:19) dr − r (cid:18) − J r dt + dθ (cid:19) (4.184)described by a mass M and an angular momentum J , while the function Y is determinedvia the cubic equation Y − Y = ǫJ / (4 r ) = ∆, and out of the three solutions of this134ystem, imposing recovery of the GR limit, Y = 1 for ∆ →
0, one gets the result3 Y = 1 + (cid:18) − − p
3∆ (27∆ − (cid:19) − / + (cid:18) − − p
3∆ (27∆ − (cid:19) / . (4.185)This geometry can be written, using a suitable change of coordinates given by { t, r } →{ T = M t + J θ/ (2 M ) , ρ = M − ( J / M r ) / } , as ds = dT − Y ( ρ ) dρ − M ρ dθ (4.186)so that the TEGR limit, ǫ →
0, is naturally recovered. To further understand the geometry(4.186) one can consider the behaviour of the curvature scalars (in the case ǫ < R = 2 Y ( ρ ) ′ ρY ( ρ ) = 2 Y ( r ) ′ rY ( r ) ; R ( µν ) R ( µν ) = 12 R ; R αβγδ R βγδα = R . (4.187)Ferraro and Fiorini analyse the structure of this spacetime in the two regions of interest.At asymptotic infinity, ρ → ∞ , where Y →
1, all these scalars vanish, and the geometry(4.186) describes a BTZ-type spacetime with a conical singularity. On the central region, r →
0, the scalars vanish as well. In particular, the curvature scalar behaves as R ∼− (cid:16) √ r | ǫ | J (cid:17) / . The physical interpretation of this geometry is that of a spacetime with adeficit angle ranging between 2 π (1 − M ) at spatial infinity and 2 π at r = 0, correspondingto the circle of minimum radius ρ = J/ (2 M ) that can be attained in this geometry.Nonetheless, as radial null geodesics satisfy dT = Y dρ , this means that T diverges as alight ray approaches the minimal circle of radius ρ , so they take an infinite affine time toreach it and the same applies for time-like geodesics. Therefore, this approach succeeds inremoving the conical singularity of GR (and, as the same time, it removes the possibilityof existence of closed time-like curves) in much the same way than electrically chargedblack holes in the s = +1 case of EiBI discussed above in this section, i.e., by setting thelocation of the wormhole throat at the future (or past) boundary of spacetime.Further analysis in three-dimensional scenarios, involving a Born-Infeld extension ofNew Massive Gravity [185] with a Chern-Simons term (Class-III), and defined by theaction S = 2 m κ Z d x (cid:20)q − det( g µν − m − G µν + aF µν ) − (cid:18) m (cid:19) q − det( g µν ) (cid:21) + µ Z d xε µνρ A µ ∂ ν A ρ (4.188)where m is a mass scale, Λ represents a cosmological constant and a , µ are some constants,has been considered in [13]. In that work only Anti-de Sitter spacetimes are studied,while black holes were investigated instead in [179] and subsequently in [178] where, byexpanding the New Massive Gravity action (4.188) to four and six derivative terms, theauthors develop a method to find evidence of uncharged and charged black holes, butlittle is said about the deviations of such solutions with respect to the structure of the GRcounterparts relevant for this review. 135 .7. Magnetically charged solutions with cylindrical symmetry Cylindrically symmetric solutions have only been considered in EiBI theories in thecontext of magnetically charged configurations (i.e. Melvin-type [250]) by Bambi et al[42]. The two line elements compatible with such a symmetry can be conveniently writtenas d s g = f ( ρ )( − dt + dz ) + g ( ρ ) dρ + h ( ρ ) ρ dϕ (4.189)d s q = ˜ f ( ρ )( − dt + dz ) + ˜ g ( ρ ) dρ + ˜ h ( ρ ) ρ dφ . (4.190)From the line element (4.189) the only non-vanishing component of Maxwell field equa-tions, ∇ µ F µν = 0, reads F ρϕ = β/ ( ρf √ gh ), where β is an integration constants related tothe intensity of the magnetic field. The energy-momentum tensor (4.22) for these solutionsallows to find the matrix Ω in Eq.(4.147) as T µν = X π diag(1 , , − , − ⇒ ˆΩ = (cid:18) Ω + ˆ I ˆ0ˆ0 Ω − ˆ I (cid:19) ; Ω ± = 1 ± f c f , (4.191)where X = − β /f , f c = l ǫ /l β and l β = 4 π/ ( κ β ). With this matrix at hand, by thetransformation (2.57) one finds the relations { ˜ f = Ω + f, ˜ g = Ω − g, ˜ h = Ω − h } between themetric functions in Eqs.(4.189) and (4.190). The first of these relations can be writtenas f = ˜ f + √ ˜ f − f c and implies that ˜ f ≥ f c , the equality corresponding to f = f c andΩ − = 0. Now, since EiBI Lagrangian density reads now L G = √ det ˆΩ − − κ l ǫ = β f c πf , the fieldequations for q µν become R µν ( q ) = − ǫ β f + ˆ I ˆ0ˆ0 − − ˆ I ! . (4.192)Computing the components of the Ricci tensor corresponding to the line element (4.190),and by taking appropriate combinations of the field equations (4.192) one obtains twoindependent equations ˜ h ρ ˜ h + 2 ρ + 2 ˜ f ρ ˜ f ! = 2 ˜ f ρρ ˜ f ρ ; ˜ f ρρ −
34 ˜ f ρ ˜ f = κ β π ˜ ff (4.193)The first equation (4.193) can be directly integrated as ˜ hρ = α ( ˜ f ρ / ˜ f ) , where α is anintegration constant. To solve the second one in (4.193), in [42] the definitions ˜ f =2 f c φ ( x ), ρ = πf c κ β x are introduced together with the new function Ω = φ x (so that d Ω /dφ = 2 φ xx ), in terms of which one finds the solutionΩ = Cφ + 4 φ (cid:16) φ − p φ − (cid:17) − φ F (cid:18) ,
12 ; 54 ; 1 φ (cid:19) (4.194)where tuning the integration constant C = − / √ π Γ (cid:0) (cid:1) / Γ (cid:0) (cid:1) ≈ . φ ≈
1. Unfortunately, it is not possible to136ntegrate Ω to obtain φ ( x ) in analytic form, though one can resort to analytical expansionsin the relevant regions. For φ ( x ) ≫ φ ( x ) = 4(1+( Cx ) / /C ,which is nothing but the Melvin solution of GR [250], such that the line element reads ds f c ≈ (cid:18) C (cid:19) (cid:18) C x (cid:19) (cid:2) − dt + dz + dρ (cid:3) + (cid:18) C (cid:19) ρ (cid:16) C x (cid:17) dϕ , (4.195)which, via a constant rescaling of ( t, z, ρ ) → ( λt, λz, λρ ) with λ = 64 f c /C , becomes theGR solution. In the other limit, φ ( x ) →
1, the corresponding field equation φ x ≈ φ − dx /x = dy , yields the line element ds f c ≈ (cid:20) − dt + dz + ρ dy (cid:21) + α f c ρ y dϕ , (4.196)up to first order in y . This is just another Minkowski spacetime near the axis as followsfrom the definitions r = ρ y/ α ≡ f c ρ (the constant factor f c can be reabsorbedvia another global rescaling of units). This kind of Melvin-type spacetimes are of greatinterest in the context of the generation of pairs of entangled black holes in high-intensitymagnetic fields via instantons [175, 174, 144, 153]. Indeed, very recently O (4) instantonshave been studied in the context of the EiBI theory [26], with the result that both thephysical metric and curvature scalars are finite. However, curvature divergences arise onthe auxiliary metric, which in turn may induce the formation of singularities, as discussedin detail in section 2.6, and be problematic at the quantum level. In view of this, it wouldbe convenient to investigate further and clarify the physical role played by the auxiliarymetric. In this section we have reviewed the developments on black hole physics in Born-Infeld inspired modifications of gravity described in section 2. Due to the fact that theSchwarzschild black hole is a vacuum solution of such theories, the literature on the topichas searched for scenarios going beyond it. In this sense, though astrophysically realisticblack holes are not expected to have a significant amount of charge, the investigation ofcharged black holes is relevant in order to find theoretical insights on the modifications totheir innermost structure, as well as observational deviations from the predictions of theKerr black hole. In the influential paper of Ba˜nados and Ferreira [45], where a couplingto Maxwell field was considered, a static, spherically symmetric geometry is obtained (for ǫ > ǫ < .For ǫ >
0, EiBI gravity black holes still require further analysis regarding its innermoststructure and the possibility of finding a wormhole core there, and the physics of massinflation requires further refinement beyond the approximations employed in the analysisof [31, 30]. On the other hand, though the physics at the photon sphere has been exploredand understood to some detail [360, 337], much research is still needed in order to obtainobservational signatures for gravitational waves out of the merging of two such black holes,as well as the potential existence of gravitational echoes in this context [99, 4, 97, 50].For ǫ < , as well as to extend thethermodynamic laws studied in other Palatini theories of gravity such as f ( R ) [38] to theBorn-Infeld scenario. To conclude, though many appealing results have been found in thecontext of Born-Infeld inspired modifications of gravity, there is plenty of room for furtherresearch in many different directions. In this sense we point out that the applicability of the Janis-Newman method (which allows to obtaina rotating solution from a seed static metric, see Erbin for a review [154]) in the context of Born-Infeldinspired modifications of gravity is still to be understood. Indeed, very recently it was found evidence on the existence of wormhole configurations above a certainmass threshold when a free static and spherically symmetric scalar field is let to gravitate under the Born-Infeld dynamics [8]. . Cosmology
The high precision of the cosmological observations made cosmology an ideal placeto test fundamental theories of gravity [301, 317, 339, 6, 150]. On the assumption ofGeneral Relativity being the underlying theory of gravitational interactions together withthe homogeneity and isotropy, cosmologists were able to construct the standard model ofBig Bang cosmology. Even if this model is simple and stood up to intense scrutiny, it stilllacks a fully satisfactory theoretical foundation. One of the challenges is the cosmologicalconstant problem, posing a naturalness problem due to the giant mismatch between itsobserved value and the radiative contributions from known massive particles to the vacuumenergy [361, 292, 249]. On the other hand, the observation of the accelerated expansionof the universe introduced the necessity of dark energy independently of the cosmologicalconstant problem [25, 298, 124, 120, 16, 89, 224, 17]. Furthermore, another problem thatone has to face within the realm of General Relativity is the necessity of yet an additionaldark component, dubbed dark matter, in order to correctly account for the formationof large scale structures, the anisotropies of the CMB, weak lensing measurements orobservations of rotation curves of galaxies. Albeit great efforts [68, 342, 70], the truenature of dark matter still remains unknown.The aforementioned challenges concern the late time evolution of the universe andthus, they motivated the consideration of infrared modifications of gravity. Remarkably,the tremendous progress made in observational cosmology also enabled us to probe theunderlying physics of the early universe, which in fact shares a similar burden. In order toexplain the observations the standard cosmological model is supplemented with the infla-tionary paradigm requiring an initial phase of accelerated expansion of the universe, thatis commonly ascribed to yet another ingredient: the inflaton. It is believed that the pri-mordial quantum fluctuations during inflation eventually become the seeds in the densityfield responsible for the cosmic large-scale structure via gravitational instability. Infla-tion is the most prominent model for a successful implementation of an extremely rapidexponential expansion, in which the perturbations of the inflationary field successivelytranslate into the fluctuations of the gravitational potential. Since gravity is coupled toall other fields, these fluctuations are then imprinted onto all existing cosmic fluids. Thesedensity fluctuations leave imprints in the cosmic microwave background as temperatureanisotropies and also in the matter distribution, that then can be probed by gravitationallensing and formation of galaxies. The inflationary scenario is realised in many differentmodels based on different fields, and observations seem to favour models with a nearlyscale invariant red power spectrum, a small value for the scalar to tensor ratio and a smallnon-Gaussianity. While the late time cosmology triggered searches for infrared modifica-tions of gravity, the need for a primordial inflationary phase motivates modifications ofgravity in the opposite regime. Furthermore, within the standard picture one is also proneto encounter a primordial classical singularity which calls for new physics beyond Gen-eral Relativity at these scales. Moreover, the breakdown of unitarity at the Planck scalerequires modifications of gravity in the ultraviolet regime to describe gravitational effectsbeyond M Pl . These additional challenges motivate to modify gravity at high energies.It was precisely the cosmological Bing Bang singularities one of the motivations behindthe inception of Born-Infeld inspired gravity theories in cosmology. The original construc-139ion by Deser and Gibbons [140] formulated in the metric language was an early attemptin this direction. Unfortunately, this approach leads to the presence of ghostly degreesof freedom due to the presence of higher order field equations (see section 2.2 for moredetails), so any regular cosmological solution will not be reliable. In spite of the mentionedghost instabilities of the metric formulation, a first quest of the cosmological implicationsof similar theories was pursued in [123]. There, although different realisations of (quasi)de Sitter solutions were shown to exist for appropriate choices of the parameters, due tothe unavoidable ghost nature of the higher order derivative interactions, these solutionsare unviable. More promising cosmological solutions without pathologies were found byconsidering Born-Infeld inspired gravity theories `a la Palatini, where the connection is leftarbitrary. In fact, Ba˜nados and Ferreira showed the existence of non-singular solutions in[45] in the EiBI model, which have since then been extensively studied, and also foundin other Born-Infeld theories of gravity. Although the avoidance of the singularities wasthe initial motivation, they provide very rich cosmological phenomenology, for instancethese theories can support quasi de Sitter solutions with more standard forms of matter,like dust or radiation, as a consequence of modifying the high curvature regime of gravity.This behaviour permits to develop inflationary scenarios different from the more tradi-tional models based on some scalar (or more general) degree of freedom. As we will see,in most of the modifications `a la Born-Infeld, the different cosmological evolution can betraced to a highly non-trivial dependence of the Hubble expansion rate on the density andpressure of the matter fields in a modified Friedman equation. In other words, the effectsof the modifications in the gravity sector translates into a non-linear contribution fromthe matter fields density to the expansion rate. A remarkable property of these theoriesis that, while in most modified gravity theories the background expansion is determinedby the equation of state parameter, Born-Infeld theories introduce a dependence on thesound speed already at the background level and not only for the perturbations. Thisis the cosmological analogue of the modified Poisson equation (3.5) with gradients of thedensity sourcing the equation for the gravitational potential, with its general case beingdiscussed in section 2.5.1.The goal of this section will be to review all these cosmological applications and showthe novel and interesting phenomenology derived from Born-Infeld inspired gravity the-ories. However, before starting with that, let us take a moment to fix the notation thatwe will use throughout this section. Cosmological observations seem to indicate a ho-mogenous and isotropic universe. Compatible with these symmetries, we will assume themetric tensor to be of the Friedman-Lemaitre-Robertson-Walker (FLRW) form, so the lineelement will read d s = − N ( t ) d t + a ( t )d ~x , (5.1)where N represents the lapse, a the scale factor and t the cosmic time. Sometimes, itwill be useful to work in conformal time η defined as a d η = d t . We will also extensivelyrefer to the Hubble function H = ˙ a/a or, in conformal time, H = a ′ /a , where a dot anda prime denote derivatives with respect to cosmic and conformal time, respectively. Itwill be sometimes convenient to keep the lapse explicitly because of the presence of twometrics in the Born-Infeld theories, as we extensively discussed in section 2.140 .1. Eddington-inspired Born-Infeld gravity We will start our survey on the cosmological applications of Born-Infeld inspired gravitytheories by considering the most extensively studied case of EiBI, whose action we rewritehere for convenience as S BI = M M Z d x (s − det (cid:18) g µν + 1 M R ( µν ) (Γ) (cid:19) − λ √− g ) + S matter , (5.2)where S matter stands for the action of the standard matter fields, that we assume minimallycoupled to the metric g µν . As shown in section 2.5.1, varying the above action with respectto the metric yields the modified field equations s det (cid:16) g µν + 1 M R ( µν ) (Γ) (cid:17) √− g "(cid:18) ˆ g + 1 M ˆ R (cid:19) − µν − λg µν = − M M T µν . (5.3)Similarly, we can vary the action with respect to the independent connection. Since theconnection Γ does not carry any dynamics, its algebraic equation can be used to solve it interms of g µν and R ( µν ) . The resulting solution is such that the connection can be writtenas the corresponding Christoffel symbols of the effective metric q µν = g µν + 1 M R ( µν ) . (5.4)On the other hand, we can use the metric field equations to express R ( µν ) in terms of g µν and the matter fields. This amounts to writing the equations as in General Relativity butwith a modified non-linear matter coupling. See section 2.5.1 for more details on that. Thisfeature becomes more apparent when we write the resulting modified Friedman equationof a homogeneous and isotropic background (5.1). Compatible with the symmetries of thebackground metric, we assume the following Ansatz for the stress energy tensor T µν =diag( − ρ ( t ) , p ( t ) , p ( t ) , p ( t )), where ρ and p represents the energy density and pressure ofthe matter fields, respectively. The Friedman equation modifies into the general form (seesection 5.2 for more details) H = f ( ρ, p, c s ) , (5.5)with a non-trivial function f , that depends non-linearly on ρ , p and c s . In the case of theEiBI model, one can compute this function exactly [45]. In terms of the auxiliary metricwe have q = − s (1 − ¯ p T ) (1 + ¯ ρ T ) and q ij = a p (1 + ¯ ρ T )(1 − ¯ p T ) δ ij , (5.6)where ¯ ρ T ≡ ρ T M M and ¯ p T ≡ p T M M with the total energy density ρ T = ρ +( λ − M M and total pressure p T = ρ − ( λ − M M and the lapse set to N = 1. In terms of thesequantities, the function f ( ρ, p ) corresponds to [45] f ( ρ, p ) = 13 GF , (5.7)141ith the short-cut notations standing for F = 1 − ρ T + ¯ p T )(1 − w − ¯ ρ T − ¯ p T )4(1 + ¯ ρ T )(1 − ¯ p T ) G = M (cid:18) − q − − ¯ p T )(1 + ¯ ρ T ) (cid:19) , (5.8)and the equation of state parameter w = p/ρ . Note, that the dependence on c s dropsin f because we have so far w = const. In general, the dependence ˙ w will appear aswell, as we will see in section 5.1.2 and also in section 5.2 for the more general case. Wecan study this background equation for two different epochs. At late times for a dustfilled universe ( w = 0) together with a cosmological constant, one recovers the standardFriedman equation in General Relativity3 H ∼ ρ + Λ + (cid:20) ρ Λ − ( ρ + Λ) (cid:21) κ Λ + O ( κ Λ) with Λ = ( λ − /κ , (5.9)where κ = M and M = 1 with the notation used in [45]. On the other hand, at earlytimes, when the universe is dominated by radiation, we have w = 1 / H = 1 κ (cid:18) ¯ ρ − √ p (1 + ¯ ρ )(3 − ¯ ρ ) (cid:19) (1 + ¯ ρ )(3 − ¯ ρ ) (3 + ¯ ρ ) . (5.10)As one can see from the above expression, for ¯ ρ = 3 (with κ >
0) one obtains H = 0. Thesame is true for ¯ ρ = − κ < a B = 10 − κ / a , with a representingthe scale factor today. Depending on the sign of κ , the scale factor can evolve in twodifferent ways. If κ < H ∝ a − a B ∝ | t − t B | , which corresponds to auniverse undergoing a bounce. On the other hand, if κ > H ∝ ( a − a B ) , sothat ln( a/a B −
1) = p / (3 κ )( t − t B ). In this scenario there is no bounce, and the universeloiters for a long time. These two behaviours can be visualised nicely by plotting thescale factor normalised by the scale a B as a function of time, which can be seen in figure25. A more detailed analysis of these cosmological solutions was further investigated in[323, 111, 79, 80, 78].For positive values of κ , the primordial nucleosynthesis constraints were used in [28]in order to impose stringent restrictions on the allowed region in the parameter space.The agreement between the observed light element abundances and the predictions of theprimordial nucleosynthesis is only ensured if the dynamics of the universe deviates fromGeneral Relativity only at a few percentage level at the initial epoch of nucleosynthesis.This, on the other hand, imposes the stringent constraint on the energy density at thestart of nucleosynthesis to be of the order ρ nuc ∼ H / (8 πG ) < /κ , which translatesinto κ < × m kg − s − . 142 igure 24: This figure is taken from [45] and illustrates the dependence of the Hubble rate in terms of theenergy density for a radiation dominated universe in the EiBI model. In [45] the notation ρ B stands forthe maximum energy density where H = 0. Furthermore, κ = M and M = 1 in terms of our notation.Figure 25: This figure is taken from [45] and shows the evolution of the scale factor in the EiBI modelin the presence of a radiation fluid. The scale factor is normalised by the minimum length scale a B . For κ <
0, the universe undergoes a bounce (in the upper panel), whereas for κ > a B for t → −∞ (in the lower panel). p = 0 and a field with p = ρ/
2. They show that also in three dimensions the branchof solutions with M − > M − <
0, they also find non-singular solutions in thesame spirit as the four dimensional Born-Infeld gravity model.
In the previous subsection, we have seen that the original EiBI theory yields interest-ing homogeneous and isotropic solutions, where the cosmological singularities might beavoided by a bounce. We have seen that a bouncing solution with H = 0 at a B is achiev-able in the presence of a radiation fluid with w = 1 /
3. We have also seen the presence ofloitering solutions, where the scale factor approaches a B for t → −∞ . As next, we shallsee whether the perturbations on top of these possible cosmological solutions are stablein order for them to be viable. This was investigated in detail in [155, 237]. We shallsummarise their results here. For this purpose let us start with the tensor perturbationsand describe the tensor modes of the spacetime and auxiliary metric in conformal timed η = d t/a in the following form g = − a , g ij = a ( δ ij + h ij ) and q = − ˜ N , q ij = ˜ A ( δ ij + f ij ) . (5.11)The tensor perturbations h ij and f ij are transverse and traceless, respectively. Since weare interested in the dynamics of the perturbations in the early universe epoch, we willagain assume a relativistic perfect fluid for the matter fields and hence the backgroundevolution will be as in section 5.1. First of all, using the field equations˜ N ˜ A a ˜ A f ij + λa h ij = h ij ˜ N ˜ A a ˜ A + λa ! (5.12)we immediately observe that the two perturbations are identical even if the backgroundscale factors were different, namely h ij = f ij . (5.13)This is a remarkable property of the EiBI model. In fact, only in the presence of anisotropicstresses, the two tensor perturbations will be different from each other. This proportion-ality of the tensor perturbations turns out to be a generic feature of Born-Infeld inspiredgravity theories beyond the standard formulation. We will see that for a general functionof the metric and the Ricci tensor in section 5.2. See also [60] for more details. The tensorperturbations of the dynamical metric follow the evolution equation [155] h ′′ ij + A ′ ˜ A − ˜ N ′ ˜ N ! h ′ ij + ˜ N ˜ A ! k h ij = 0 , (5.14)where we made use of the background equations of motion. In the regime of low energydensities, one recovers the standard evolution equation of the tensor modes as in General144elativity. On the other hand, in the Born-Infeld regime at high energy densities, themodifications in the evolution equation due to the scale factor of the auxiliary metricbecome appreciable. For the stability of the tensor perturbations, it will be crucial thatboth scale factors are well-behaved. It is not enough to impose this condition solely onthe background variables of the spacetime metric. Similarly, one has to guarantee thatthe auxiliary metric does not vanish. In fact, as we have seen in the previous section, theevolution of the scale factor for κ > a/a B −
1) = p / (3 κ )( t − t B ), hence thelapse and the scale factor of the auxiliary metric evolve as˜ A = 2 / a r exp (cid:16)p / (3 κ )( t − t B ) (cid:17) , ˜ N = 1 √ A a . (5.15)As it becomes clear from these expressions, the scale factor of the auxiliary metric becomessingular for t → −∞ . This non-singular behaviour has a crucial impact on the tensorperturbations, since their evolution equation scales with the quantities of the auxiliarymetric. In the far asymptotic past, the pre-factors of the last two terms in equation (5.14)are suppressed and the evolution equation simply becomes h ′′ ij ∼
0. The solution for themetric perturbations is hence of the form h ij ≈ Aη + B . This represents an unstablegrowth and therefore, the loitering solution in the case κ > κ <
0, we have seenin previous section that a bouncing solution is obtained since H ∼ a − a B ∼ | t − t B | . Interms of the conformal time, the scale factor evolves as a = a B (cid:2) ( βη ) (cid:3) , (5.16)with β ≡ a B p / (3 | κ | ). This, on the other hand, means that the lapse and the scale factorof the auxiliary metric evolve this time as˜ N = a / | tan( βη ) | , ˜ A = a / | tan( βη ) | . (5.17)We can Taylor expand these expressions around the bounce η = 0. By doing so, theevolution equation of the tensor perturbations close to the bounce becomes h ′′ ij + 2 η h ′ ij + k β η h ij = 0 . (5.18)The solution scales this time as h ij ≈ η n with n = − ± p − (4 k / (3 β )), representingan unstable growth. Hence, the bouncing solution suffers also from an instability in thesame way as the loitering solution, even though in the latter case it was much milder.145hus, in the EiBI model in the presence of a radiation fluid the interesting loitering andbouncing solutions suffer from tensor instabilities. This unsatisfactory result might changeif one considers a more general fluid with varying equation of state parameter or if oneextends the EiBI model to a more general Born-Infeld inspired gravity model. In the previous subsections we have seen that the EiBI theory admits interesting bounc-ing and loitering solutions for early universe cosmology in the presence of a radiation fluid.However, as we have seen, these solutions are plagued by tensor instabilities if the matterfield is assumed to be a perfect fluid with the equation of state parameter w = 1 /
3. Itis possible to find more general solutions if we abandon this restriction and this mightalleviate the found tensor instabilities. In fact, this was precisely considered in [32]. Itcould be that additional dynamical fields are present in the early universe, giving rise tomatter fields with ˙ w = 0. In this case, the modified Friedman equation (5.5) generalizesto [32] H = (cid:18) a √ g + ˙ wg g (cid:19) , (5.19)with the functions g i given by the energy density and pressure of the matter fields g = 2 M − (cid:18) ρM (cid:19) (cid:18) − ρwM (cid:19) (cid:20) − M (1 + 3 w ) + 2 D (cid:21) ,g = 4 + 2 M ρ (cid:20) − w (cid:18) − M ρ (cid:19) + 3 w (cid:18) M ρ (cid:19)(cid:21) ,g = − ρ (cid:18) M ρ (cid:19) , (5.20)with D = q (1 + ρ/M )(1 − p/M ) and the choice of units M Pl = 1 and | κ | = 1 used in[32] (remember that κ = M − in our units). In that work it is shown that the possibilitywith time varying equation of state parameter can ameliorate the tensor instabilities foundfor w = const. As an example, a scalar field with a general kinetic and potential termis considered. In the presence of this scalar field, with the Lagrangian L ( X, φ ) where X = − ∂ µ φ∂ µ φ , the equation of state parameter is given by w = L X L ,X − L , (5.21)with the pressure p = L and energy density ρ = L ,X − L accordingly. It turns out, thatfor κ = −
1, the instability of the tensor perturbations cannot be avoided. This is thereason why the authors in [32] consider the case κ = 1. Since ˙ w = 0, one achieves abouncing solution with H = 0 and ˙ H = 0, which differs from the case studied in [155],where ρ → w − as η → −∞ . Remember that the authors in [32] use the units | κ | = 1.For an initial density of ρ i = 10 − and w i = 0, this behaviour is illustrated in figure 26 fora scalar field with L = X − m φ . The tensor perturbations on top of this background146 igure 26: This figure is taken from [32] and shows the evolution of the scale factor in the presence of ascalar field with varying equation of state parameter in the EiBI model. The universe undergoes a bounceat t = 0. The initial values are chosen to be ρ i = 10 − and w i = 0 and the mass of the scalar field isassumed to be m = 100. Note, that they use the unfortunate choice of units M Pl = 1 and | κ | = 1. are given by h ′′ ij + g h ′ ij + g k h ij = 0 , (5.22)with the two functions g = 2 H + κ ˙ ρ κρ ,g = 1 − κρw κρ . (5.23)For the case κ = 1, the pre-factor in the friction term near the bounce vanishes g ∼ We have seen above that the reported tensor instabilities of the interesting cosmo-logical solutions might be avoided by considering matter fields with varying equation ofstate parameter. As a specific model, one can consider the presence of a scalar field asmatter field. This was for instance done in the works [155, 237, 112, 371]. In this way,the underlying physics of the early universe will be determined by both the Born-Infeldmodification and the presence of the scalar field. As a simple realisation one can considera scalar field with a quadratic potential. In standard General Relativity an inflationaryscenario with sufficiently long duration based on such a simple scalar field might requirevery large field values. This is due to the fact that the time derivative of the scalar fieldincreases rapidly as going back in time with the scalar field itself climbing up the potentialgiving rise to an increasing energy density until the Planck scale is reached quickly. The147ope to use this same scalar field in the Born-Infeld inspired gravity theory is to alleviatethis requirement. The crucial point with this respect is that the pressure in EiBI gravityis bounded from above due to the square root structure. Hence, there is an upper boundfor the value of the field velocity as it was shown in [112]. This guarantees a real value forthe Hubble parameter. Due to this upper limit, one does not run into the same problemas in the standard inflationary model. Let us consider the following action [237] S BI = M M Z d x (s − det (cid:18) g µν + 1 M R ( µν ) (Γ) (cid:19) − λ √− g ) + Z d x √− g (cid:18) − g µν ∂ µ ϕ∂ ν ϕ − m ϕ (cid:19) . (5.24)In this model, the curvature scale remains finite thanks to the square root structure of EiBIgravity and the early universe undergoes a pre-inflationary accelerated expansion in orderthen to end in an ordinary chaotic inflationary epoch. Since the scalar Lagrangian is thesame as in General Relativity, it follows the same evolution equation. For a homogeneousand isotropic background metric and correspondingly only time dependent scalar field,the equation of the scalar field is simply ¨ ϕ + 3 H ˙ ϕ + m ϕ = 0. The maximum value forthe field velocity is achieved when ˙ ϕ = m ϕ + 2 λM . So we can define this moment ofmaximum velocity by ˙ ϕ = q m ϕ + 2 λM with the Hubble parameter taking the form H = − m ϕ/ q m ϕ + 2 λM at this point. These equations can be integrated to havethe evolution of the scalar field and the scale factor giving rise to solutions that respect themaximal pressure condition. By doing so, the explicit analytic solutions with this boundare given by ϕ = q λM m sinh [ m ( t − t )] and a = a (2 λM ) / cosh − / [ m ( t − t )] . (5.25)These solutions describe a universe that expands until the bouncing stage is achieved at t = t and then starts contracting whereas the scalar field tracks the symmetric potential.At early times t → −∞ , there is no singularity and the universe expands exponentiallywith a ∼ a (2 /λM ) / e m ( t − t ) and ϕ ∼ − q λM / (2 m ) e m ( t − t ) . During this period,the Hubble parameter is nearly constant and purely determined by the scalar field’s mass H ≈ m/ m ϕ ≫ λM , i.e. the potential of the scalar field is largerthan λM . Thus, the upper limit in the pressure guarantees that the curvature scaleremains finite.In figure 27 an example of the phase map is plotted for ϕ and ˙ ϕ , where the Hubblefunction is denoted by the colour. We borrowed this figure from [112], where one cannicely see the evolution of the Hubble parameter and the scalar field and the realisationof the different phases. One can see that the universe starts off close to the region of theupper bound of the pressure or field velocity respectively and decreases as time passes.Sufficiently away from this region, the universe undergoes the first slow-roll where the field148 igure 27: This figure illustrates an example of the phase map taken from [112] with the parameters chosenas m = 1 / M − = 1 / λ = 1, where the authors use the units M Pl = 1. In the left panel one cansee the behaviour of the Hubble parameter denoted by different colours. The red region corresponds to H >
1, whereas the blue colour shows the regions with small Hubble parameter. The region encoded inwhite is the physically forbidden region. In the right panel the behaviour of ˙ ϕ is represented. The blueregion corresponds to the field space where the upper bound limit is violated. The different trajectoriescorrespond to different initial conditions for the scalar field. The grey region is the high-curvature regimeand the solid and dashed curves represent trajectories that start from the left top and right bottom,respectively. r can be suppressed significantlyin difference to the standard chaotic inflation in General Relativity. For the analysis ofthe scalar perturbations of the model, let us adapt to the useful approach of consideringparallel variables for the g metric and the auxiliary metric, as we did above for the tensorperturbations. Let us consider the following scalar perturbations [237]d s q = ˜ a (cid:26) − (1 + 2 φ q ) Z d η + 2 B ,i √Z d η d x i + [(1 − ψ ) δ ij + 2 E ,ij ] d x i d x j (cid:27) d s g = a (cid:8) − (1 + 2 φ g )d η + 2 B ,i d η d x i + ((1 − ψ ) δ ij + 2 E ,ij ) d x i d x j (cid:9) . (5.26)Similarly, we shall perturb the scalar field as ϕ = ϕ + δϕ . Note that the auxiliarymetric carries the additional background quantity Z = ρ /M − p /M and the overall scalefactor ˜ a = (1 + ρ /M ) / (1 − p /M ) / a with ρ = ϕ ′ / (2 a ) + m ϕ / p = ϕ ′ / (2 a ) − m ϕ /
2. One can use the gauge freedom in order to eliminate some of theperturbations. One could for instance choose ψ = 0 and E = 0. However, not all of theremaining quantities are dynamical. In fact, except for the scalar field, all the remainingperturbations of the metrics can be integrated out using their algebraic equations. Thisis to be expected, since the q µν metric is related algebraically with the g µν metric andthe scalar perturbations in the space-time metric are not dynamical (see sections 2.5.1150nd 2.6 for more details). After introducing the new perturbation variable χ = W δϕ where W stands for W = (cid:16) Z − Z +3 Z ( Z +1)(3 Z− (cid:17) / a and performing the time transformationd τ = W /f d η , the equation of the dynamical scalar field perturbation takes the simpleform at the attractor stage ¨ χ + (cid:18) k − τ − τ ) (cid:19) χ ≈ , (5.27)where τ denotes the end of inflation and f = Z − Z +3( Z +1)(3 Z− a here. In the works [113,115, 109] it was shown that the perturbations start from an initial point where themaximal pressure condition holds and evolve towards an intermediate stage, where theWKB approximation can be applied to then end at the attractor stage. Finally, thesolutions of these three stages are matched together. Furthermore, they compute the co-moving curvature perturbation R ψ + Hδϕ/ϕ and from that the scalar power spectrum P R = k |R| / (2 π ). Last but not least, from this they were able to evaluate the spectralindex n R − P R / d log k . They observe that the spectral index is of second orderin the slow-roll approximation and a suppression of the tensor-to-scalar ratio. The exactform of the scalar power spectrum and the spectral index can be extracted from [109, 114]and we refer the reader to these works for more details.We have seen that in the presence of a scalar field one can realise different epochsin the early universe. One can have a preinflationary scenario followed by a standardchaotic inflationary expansion. Due to the squared root structure of the gravitationalinteractions, there is an upper limit for the pressure and hence the field velocity. So farwe have considered the case where the scalar field is minimally coupled to the gravity andhas standard kinetic and mass terms. In the following subsection we will pay attention tothe case where the scalar sector obeys the Born-Infeld structure as well. In the following we would like to discuss the EiBI gravity theory in the presence ofa scalar Born-Infeld matter field. The Born-Infeld structure in both the gravity andmatter sector with their corresponding scales might have interesting implications. Thisidea was pursued by S. Jana and S. Kar in [219], where they provide interesting analyticalcosmological solutions for a particular choice of the time derivative of the Born-Infeldscalar. For a positive constant M − >
0, they were able to realise solutions with twoseparate de Sitter expansions with an intermediate sandwiched phase of deceleration. Theaction of this model is given by [219] S BI = M M Z d x (s − det (cid:18) g µν + 1 M R ( µν ) (Γ) (cid:19) − λ √− g ) + α T Z d x √− gV ( φ ) q α − T g µν ∂ µ φ∂ ν φ , (5.28)with the scales M BI and α T representing the Born-Infeld scales in the gravity and mattersector, respectively, and V ( φ ) denoting a potential for the scalar field. The scalar sector151s a Dirac-Born-infeld like action. The equation of motion of the scalar field yields ∂ ν V √− gg µν ∂ µ φ q α − T g µν ∂ µ φ∂ ν φ = α T V ′ √− g q α − T g µν ∂ µ φ∂ ν φ . (5.29)Similarly, the corresponding stress energy tensor reads T µν = V ( φ ) ( g µα g νβ − g µν g αβ ) ∂ α φ∂ β φ − g µν α − T q α − T g µν ∂ µ φ∂ ν φ . (5.30)We will be interested in the possible cosmological solutions that one can construct inthis particular model. For this purpose, we will again consider a FLRW metric for thebackground metric g µν with lapse N and scale factor a . For this specific simple background,the scalar field equation becomes¨ φα T N − ˙ φ + 3 ˙ φHα T N + V ′ V − ˙ φ ˙ N N ( α T N − ˙ φ ) = 0 . (5.31)The equation of motion of the scalar field can also be written as ˙ ρ φ /ρ φ = − H ˙ φ / ( N α T ),where the corresponding energy density of the field is given as ρ φ = α T V q − ˙ φ N − α − T . (5.32)Similarly, we can compute the pressure of the scalar field, which for the considered DBIaction yields p φ = − α T V q − ˙ φ N − α − T , which we can use to define the correspondingequation of state parameter of the scalar field. In contrast to the standard single fieldinflation where one assumes a specific form of the potential, here one can choose a specificform for ˙ φ . In [219] the following solution for ˙ φ is contructed:˙ φ = N α T C a n , (5.33)with positive constant variables C and n . For n = 3 for instance one can obtain a constantnegative pressure p φ = − α T C with the integration constant C . Using the metric fieldequations √− qq µν − λ √− gg µν = √− gT µν M M , (5.34)we can relate the lapse ˜ N and shift ˜ a of the auxiliary metric ˆ q with the scale factor of theˆ g metric and the energy density of the scalar field. By doing so, one obtains a = ˜ a q ˜ N /N ˜ α T ,ρ φ = M M (cid:18) ˜ α T N ˜ N − (cid:19) , (5.35)152 igure 28: This figure is extracted from [219] and illustrates the deceleration parameter (which in theirnotation is denoted by q ) as a function of X in the case M − >
0. One can see the transition of thedeceleration parameter from negative to positive values and then afterwards to become negative again.The outcome for different values of the constant C is shown by the different curves. with the new constant parameter ˜ α T = 1 + α T C / ( M M ). Similarly, the equations ofmotion for the connection yield the following relations˙˜ a ˜ a = M N + N − N C N ! , dd t (cid:18) ˙˜ a ˜ a (cid:19) − ˙˜ a ˙˜ N a ˜ N = M − N + ˜ N C N ! . (5.36)In [219] this system of equations is analysed for a particular solution of the scale factor ofthe auxiliary metric, namely, ˜ a = ˜ a e ¯ H b t with two constants ˜ a and ¯ H b . Introducing thequantity Y φ = C (1 − H b ˜ N − M − ), the last equation of (5.36) translates into˙ Y φ Y φ − q Y φ + 3 = − H b . (5.37)In terms of this new variable, the deceleration parameter d can be expressed as d = − aa ′′ a ′ = − q Y φ + 3 − Y φ q Y φ + 3 − Y φ Y φ q Y φ + 3 + q Y φ + 3 + Y φ C C − Y φ ) , (5.38)where prime denotes the derivative with respect to cosmological time τ = R √ N d t . Oneimmediate observation is that for a → ∞ ( Y φ →
1) one has d = − a → Usually, q is used for the deceleration parameter in the literature but we use d here in order to avoidconfusion with the auxiliary metric. igure 29: This figure is borrowed from [219], where one can see the evolution of the scale factor in thedifferent regimes. In [219] the following values have been chosen for this plot: M Pl = 1, M − = 0 . α T = 5, C = 0 .
001 and finally C = 1 . a = 0 .
06 at τ = 0 .
06. In the left panel,one can see the evolution of the scale factor during the loitering, deceleration and acceleration phases. Theright panel is just a zoom of the first two phases. ( Y φ → −∞ ) one also has d = −
1. Hence, one obtains two de Sitter phases, one at earlyand one at late time universe. In figure 28 taken from [219] one can see the dependence ofthe deceleration parameter from X for different values of C . Furthermore, one can choosethe value of the constant C such that one can realise an initial loitering phase with anacceleration and subsequently a decelerated and again an accelerated expansion phaseafterwards. This can be seen in figure 29, where the scale factor is plotted as a functionof cosmological time. During the loitering phase, the scale factor grows approximatelyas a loit ∼ a e √ M BI τ/ √ . During the period of inflation the universe grows by 60 e-folds in 10 − seconds. This on the other hand puts the bound M − . . × − m .The second phase of accelerated expansion at late times has the scale factor growing as a DE ∼ e √ C / α T τ/M Pl . As it becomes clear from the expressions of the scale factor inthese two regimes, the evolution at early times depends on the Born-Infeld scale M BI ofthe gravity sector whereas at late times on the scalar Born-Infeld parameter α T . Theintermediate phase depends on both as α T M − . Similarly, one can study the cosmologicalsolutions of the background equations in the other case when M − <
0. Of course, dueto the Born-Infeld term in the gravity sector, one can construct bouncing solutions aswe have seen before. The novelty of the scalar Born-Infeld term results in an additionalaccelerating phase at late time universe. Nevertheless, this solution with M − < M BI , where one caneither realise loitering, accelerating or bouncing solutions, whereas the physics at latetimes is governed by the scalar Born-Infeld scale α T , which gives rise to an acceleratedexpansion. 154 .1.5. Anisotropic cosmological solutions So far we have studied the cosmological applications of the EiBI gravity theory forhomogeneous and isotropic backgrounds. We have seen the appearance of different inter-esting cosmological solutions for early universe, including loitering, quasi de Sitter andbouncing solutions and discuss their stability. Another interesting question along thisline is the evolution of cosmological backgrounds with anisotropies. Some of the anoma-lies observed in the cosmic microwave background might be due to the presence of smallanisotropies. In this context, Bianchi type models could be a natural and simple extensionof the standard FLRW with small anisotropies, which could be for instance at the originof the power suppression at large scales of the cosmic microwave background. This wasexactly pursued in [229, 191]. Let us consider the following Bianchi type I background forthe dynamical metric g µν d s g = g µν d x µ d x ν = − d t + g ( t )d x + g ( t )d y + g ( t )d z , (5.39)where g i denote the scale factors in the x , y and z direction, respectively. By definingthe quantities A = 1 + ρ/ ( M M ) and B i = 1 − p i / ( M M ) we can construct theauxiliary metric with a similar Ansatz with the three different scale factors. It takes thefollowing form d s q = q µν d x µ d x ν = − d t + q ( t )d x + q ( t )d y + q ( t )d z , (5.40)with q i standing for the quantities q i = g i A/B i . In terms of these variables, the fieldequations of the Bianchi Type I geometry of the EiBI theory can be calculated easily.Plugging the two Ansaetze into the covariant field equations yields1 − AB B B = M (¨ q q q + q ¨ q q + q q ¨ q ) q q q , (5.41)1 − B AB B = M ( ˙ q ˙ q q + ˙ q q ˙ q + ¨ q q q ) q q q , (5.42)1 − B AB B = M ( ˙ q ˙ q q + q ˙ q ˙ q + q ¨ q q ) q q q , (5.43)1 − B AB B = M ( ˙ q q ˙ q + q ˙ q ˙ q + q q ¨ q ) q q q . (5.44)First, we can consider the simple case with isotropic pressure where p i = p and therefore B i = B . For clarity of the notation, we can further introduce the Hubble functions in thedifferent spatial directions as H i = ˙ q i /q i and ∆ H i = H − H i where H = 1 / P i =1 H i isthe mean Hubble expansion rate. We can define the degree of anisotropy as the shear σ = 13 X i =1 (cid:18) ∆ H i H (cid:19) . (5.45)In the field equations the multiplication of the scale factors appears very often. For thisreason, we can define a new variable here as Q = q q q and express the field equations in155erms of H and Q , which read3 ˙ H + X i =1 H i = M − (cid:18) − AB (cid:19) , (5.46)1 Q dd t ( QH i ) = M − (cid:18) − AB (cid:19) . (5.47)After simple manipulations of the field equations, they can be combined into dd t [ Q ( H i − H )] =0, which can be simply integrated to give H i = H + C i /Q with integration constants C i . Further integration gives for the scale factors q i = q i Q / exp (cid:16) C i R (cid:16) Q d t d Q (cid:17) d Q (cid:17) .For consistency, the integration constants have to satisfy C + C + C = 0. Fur-thermore, the product of the scale factors follows the second order differential equation¨ Q = 3 M − (1 − / ( AB )) Q , which can be also integrated easily. From these solutions, wecan also determine the quantities of the ˆ g metric. For instance, the Hubble functions ofthe ˆ g metric in the different directions can be obtained from H gi = H i + ˙ B/B − ˙ A/A andsimilarly the mean Hubble parameter as well. The ordinary matter fields couple to thestandard ˆ g metric, therefore their conservation equation is dictated by the mean Hubbleparameter of the ˆ g metric. Thus, they follow as˙ ρ + 3 H + ˙ BB − ˙ AA ! ( ρ + p ) = 0 . (5.48)In terms of the energy density and pressure of the matter fluid, we can also compute thedegree of anisotropy in the ˆ g sector, which takes the following from σ g = 3 C ( ρ + p ) Q ˙ ρ , (5.49)with C = C + C + C . In [191] the quantity Q is used as a parameter in order to obtain thegeneral solution in a parametric form. Furthermore, they provide the general solutions forthe Hubble functions and the anisotropy parameter in the ˆ g sector for three different fluidtypes: for stiff, radiation and dust fluid. Let us for example consider the dust componentwith p = 0 and hence B = 0. After making the following change of variables θ = tM BI and ρ = rM , the factor Q = q q q becomes Q = M − ρ / ( r (1 + r ) / ) with the initialdensity ρ . From the differential equation ¨ Q = 3 M − (1 − / ( AB )) Q one obtains in thiscase2( r + 1)(5 r + 2) rr ′′ − (7 r (5 r + 4) + 9) r ′ − √ r + 1 r ( r − (1 + r ) / + 1) = 0 (5.50)where prime denotes here the derivative with respect to θ . The volume element in the ˆ g metric sector, so in other words G = g g g , is given by G = M − ρ /r . The evolution of therescaled energy density r and the volume element is plotted in figure 30 for different choicesof the initial energy density, which we took from [191]. In the presence of a dust fluid, themean anisotropy parameter is given by ( M − ρ ) / (3 K ) σ g = r (1 + r ) /r ′ and is plottedin figure 31. As it can be clearly seen in this figure, the mean anisotropy parameter evolves156 igure 30: This figure represents the evolution of the rescaled energy density r = ρM − and the rescaledvolume element G M − ρ (which is v g in the notation used in [191] and furthermore M Pl = 1) as a functionof the rescaled time θ = tM BI for three different initial conditions: r (0) = 0 . r (0) = 0 . r = 0 . r (0) = 0 .
75 (dashed line) in a universe filled with dust.Figures were taken from [191].Figure 31: In this figure taken from [191] the evolution of the physical Hubble function in a dust filleduniverse in a Bianchi type I space-time together with the mean anisotropy parameter σ g (in terms of thenotation used in [191] this corresponds to a ( g ) p ) are shown as a function of the rescaled time θ = tM BI . to zero after some time elapses and the universe ends up in an isotropic phase in a Bianchitype I space-time in the EiBI gravity theory. This seems to be complementary to thestandard picture in General Relativity where shear decays in the presence of a cosmologicalconstant [51]. Within the framework of EiBI gravity this property is maintained in thepresence of a dust component as well. However, this property does not seem to be general.For instance, instead of a dust fluid, if one considers a radiation fluid with w = 1 / g metric sector takes ratherthe form σ g = r (1 + r ) / ((1 − r ) r ′ ). Its evolution together with the evolution of theHubble function can be extracted from figure 32, which we borrowed from [191] as well.As one can see in figure 32, the mean anisotropy parameter increases with time. Startingfrom an initial state with a vanishing anisotropy, the degree of the anisotropy increasesuntil it reaches a maximum constant value.157 igure 32: This figure borrowed from [191] illustrates the evolution of the Hubble function and the meananisotropy parameter σ g (which corresponds to a ( g ) p in the notation used in [191]) as a function of therescaled time θ = tM BI in a universe filled with stiff fluid with different initial values. As stressed several times throughout this review, the general motivation for Born-Infeld inspired theories of gravity is to modify the gravitational interactions in the highcurvatures regime. This means that deviations with respect to GR will typically arisewhen the curvatures become of the order of the Born-Infeld scale M . Since the source ofgravity is weighted by the Planck scale, an equivalent formulation of this statement is thatone only expects deviations from standard gravity when the densities are of the order of ρ ∼ M M . For this reason, the natural place where these theories manifest themselvesin cosmological scenarios is the early universe, being ideal candidates for inflationarymodels or bouncing solutions as we have reviewed above. Applications of these theoriesfor models of the late-time universe are instead dissonant as a consequence of their veryown defining properties. Since the Born-Infeld effects will become negligible wheneverthe cosmological energy density drops below M M , from that moment on we will havethe usual cosmological evolution with GR governing the gravitational interaction. If wewant to have non-negligible effects on cosmological scales today (or somewhere betweendecoupling and today) that would mean that the whole cosmological evolution would havetaken place in the Born-Infeld regime. For this reason, late-time cosmology constitutesan inefficient way to constrain EiBI theories and dark matter and/or dark energy modelsbased on this type of theories are likely to fail in their goal. Models for the dark componentsof the universe find a better suited arena within the framework of infrared modificationsof gravity so that they become relevant in the late-time evolution of the universe. If wewant to be on the safe side, we can impose the Born-Infeld corrections be important onlybefore the onset of Big Bang Nucleosynthesis (BBN). Since BBN takes place when thetemperature of the universe is roughly 1 −
10 MeV, we obtain the conservative constraint More precisely, one expects modifications whenever any component of the energy-momentum tensorbecomes comparable to M M . In the most standard cosmological backgrounds the energy density is therelevant quantity. but in more general scenarios other components of the energy-momentum tensor couldplay an important role as well. This is the case for instance of anisotropic cosmological solutions whereanisotropic stresses can be present. BI & H BBN ≃ T /M Pl ≃ − eV. Notice that this bound is less stringent that the onediscussed in section 2.6 where we obtained M BI & − eV from the absence of anomalousinteractions in collider experiments.Despite the general arguments given in the precedent paragraph, there are some worksin the literature attempting to explain the dark matter problem in galaxies as an effect ofmodifying gravity as in EiBI theory [322]. In [193], the authors study spherical dark matterhaloes and conclude that the value of the Born-Infeld parameter that allows to realisticallyreproduce the dark matter haloes is M − ≃ cm which translates into M BI ≃ − eV.Even if this value allows to reproduce the haloes, we must remember that the Born-Infeldcoupling is universal and this value is in contradiction with the constraints discussed in theprevious paragraph so that it is excluded. Analogous studies like e.g. those in [309, 216]find similar results and are, thus, subjected to the same limitations. Similarly, the boundsobtained on M BI from other cosmological and astrophysical probes explain the resultsfound in [145] where the authors compute the matter power spectrum for the EiBI theory.They find that the deviations of the power spectrum with respect to that of ΛCDM iscompletely negligible for realistic values of M BI .Let us also notice that, still within the class of Born-Infeld inspired gravity theories, inorder to avoid the aforementioned triviality for late-time cosmological applications, therehas been some attempts in the literature to include additional corrections to the EiBIaction that could give some effects at late times. We should note however, that this goesagainst the Born-Infeld spirit and it is very likely that the EiBI sector will not play anyrole and the whole effect will come from the new terms. As an example of this approach,some works introduced an Einstein-Hilbert term supplementing the Born-Infeld sector,but this class of modifications seriously compromise the stability of the theory. In fact,such variations of the Born-Infeld actions belong to the Class 0 described in the section2.7.1 and which are precisely characterised by the presence of pathologies. Thus, even ifone can achieve non-negligible effects in the late-time cosmology, this would come at theexpense of possibly losing the ghost freedom of the theory. Explicit examples of this type ofmodifications will be summarised in section 5.6, but it is worthwhile to stress here that thisroad of tracing late-time cosmological solutions are doomed to fail due to the mentionedinstabilities. As discussed in 2.7.1, if one really wants to add an additional Einstein-Hilbertterm in the Lagrangian, then the Born-Infeld interactions have to be modified so that theghost-free massive (bi-) gravity potential interactions in its formulation in terms of theauxiliary metric are recovered.From the above discussion it is clear that Born-Infeld inspired theories of gravitycannot play a relevant role for the cosmological evolution from roughly BBN (where weneed to have standard gravity) until today. There is however a more natural place tostudy potential effects of EiBI in the late-time cosmology residing within the frameworkof future cosmological singularities. The properties of dark energy are crucial for the futureevolution of the universe and its eventual fate. At this respect, some models predict theexistence of future singularities that can be broadly classified according to the divergenceof some cosmological quantity (see for instance [52, 262, 126, 93, 160, 81, 62, 11] forsome related literature). It is indeed common to perform such a classification attendingto which derivative of the scale factor diverges first [161]. In some scenarios with future159ingularities, the Hubble expansion rate or its derivative show divergences so that the Born-Infeld corrections will eventually be relevant again and one could wonder if such futuresingularities could be tamed. This was studied in [79, 78] and it was found that generallyfuture cosmological singularities can remain, although in some cases the divergences canbe somewhat smoothed. In [77] the authors argued that the classical big rip singularitymight be avoided by applying the quantisation based on the Wheeler-DeWitt equation tothe EiBI model. After reviewing the cosmological applications of the EiBI model, we will discuss thecosmological studies performed for other Born-Infeld inspired theories of gravity. Mostof them share the same underlying features and mechanisms, although leading to differ-ent cosmologies depending on the specific model under consideration. Thus, instead ofstudying the individual extensions one by one, we will develop here a general frameworkto study cosmological solutions within these theories. In fact, without increasing the levelof difficulty, we can consider the general class of theories already analysed in 2.7.1 andharvest the results of that section to obtain the relevant equations to study the cosmologyof these theories. To avoid the reader to thumb through the review, we will rewrite themain equations here for convenience. The starting action is given by S = 12 M M Z d x √− gF (cid:0) ˆ P (cid:1) , (5.51)with P µν = M − g µα R αν (Γ). The analysis of the general field equations, even in thepresence of torsion, was discussed in great detail in section 2.7.1. For our purposes here, theimportant equations will be those relating the auxiliary metric q µν , which determines theconnection as its Christoffel symbols, with the spacetime metric and the matter content.The two metrics are related by ˆ q = ˆ g ˆΩ, where ˆΩ is the deformation matrix defined asˆΩ − = 1 q det ˆ F ˆ P (cid:18) ∂F∂ ˆ P (cid:19) T . (5.52)Notice that this definition relates ˆΩ and ˆ P so that all the equations below will admitequivalent formulations in terms of ˆΩ or ˆ P alone. By using the definition of the deformationmatrix, the metric field equations can be expressed asˆΩ − ˆ P = 1 M M p det ˆΩ (cid:16) L G + ˆ T ˆ g (cid:17) (5.53) We are not assuming any projective symmetry a priori on the Ricci tensor so, in principle, we couldconsider both the symmetric and the antisymmetric parts of the Ricci tensor. The background cosmologicalevolution where all the relevant objects are assumed to be diagonal will be, in general, oblivious to thepresence of the projective symmetry. However, it is crucial when studying the perturbations. For ananalysis of the cosmological scenarios in an extended class of theories see [60]. L G = M M F is the Lagrangian. These equations give the deformation matrixˆΩ (or the fundamental object ˆ P ) in terms of the matter content and the spacetime metric.The resolution of the problem will be completed with the differential equations for theauxiliary metric (see 2.7.1) R µν ( q ) = 1 M p det ˆΩ (cid:16) L G δ µν + T µν (cid:17) . (5.54)After briefly reviewing the relevant equations, we can proceed to the study of cosmologicalscenarios. As usual, we will consider a homogeneous and isotropic background metricdescribed by the FLRW line elementd s g = − N ( t )d t + a ( t )d ~x (5.55)and a perfect fluid with isotropic pressures as matter sector T µν = (cid:18) − ρ pδ ij (cid:19) . (5.56)As an additional condition, we will assume that all relevant quantities inherit this form sothat we will have ˆΩ = (cid:18) Ω
00 Ω δ ij (cid:19) and ˆ P = (cid:18) P P δ ij (cid:19) . (5.57)As we have explained several times above, the recovery of GR at low curvatures imposesˆΩ ≃ for P , P ≪
1. The form of the deformation matrix ensures that the auxiliarymetric will also have a FLRW line elementd s q = − ˜ N ( t )d t + ˜ a ( t )d ~x , (5.58)with ˜ N = N Ω and ˜ a = a Ω . We keep the explicit dependence on the lapse function N ( t ) for later convenience. Once we have specified the assumptions for our homogeneousand isotropic Ans¨atze, we can now proceed to write the background metric field equations(5.53), which read P Ω = 1 M M p Ω Ω (cid:16) L G + T (cid:17) ,P Ω = 1 M M p Ω Ω (cid:16) L G + 13 T ii (cid:17) . (5.59)As anticipated above, for a given function F ( ˆ P ), these equations will allow to obtainthe components of ˆΩ (or those of ˆ P ) in terms of the energy density ρ = − T and thepressure p = T ii of the matter fields. An important point to keep in mind is thatthese equations are non-linear so that, in general, we will find several branches. Out ofthose branches, the condition (2.73) on the function F will guarantee the existence of oneparticular branch that will be continuously connected with GR at low curvatures. This161ill be the interesting branch for most applications, although other branches might alsooffer interesting cosmological scenarios.As we have seen in the previous section for the EiBI model, the crucial step to studythe cosmological evolution is to extract the dependence of the Hubble expansion rate interms of the energy density and pressure of the matter fields. For this purpose, we willmake use of the Einstein tensor of the auxiliary metric and express its 00 component intwo different ways. We will start from the definition of the Einstein tensor of q µν given byˆ G ( q ) = ˆ R −
12 ˆ q Tr (cid:16) ˆ q − ˆ R (cid:17) . (5.60)First, we will compute its 00 component in terms of the auxiliary metric, which will simplygive the corresponding Hubble expansion rate: G ( q ) = 3 ˜ H = 3 (cid:18) d ln ˜ a d t (cid:19) = 3 (cid:20) H + 12 d ln Ω d t (cid:21) . (5.61)Since Ω = Ω ( ρ, p ) as obtained from (5.59), we can express the time derivative of Ω in terms of derivatives with respect to ρ and p . Thus, if we use that matter fields areassumed to be minimally coupled so that they satisfy the usual conservation equation˙ ρ + 3 H ( ρ + p ) = 0 , (5.62)we can finally arrive at G ( q ) = 3 H (cid:20) − (cid:0) ρ + p (cid:1)(cid:16) ∂ ρ ln Ω + c s ∂ p ln Ω (cid:17)(cid:21) (5.63)where we have introduced the sound speed c s ≡ ˙ p/ ˙ ρ . If we further assume a barotropicequation of state p = p ( ρ ) the sound speed can also be written as c s = d p/ d ρ . Thiscompletes the first part of our computation of the Hubble expansion rate. The secondpart consists in writing the Einstein tensor of the auxiliary metric by using the definitionˆ P = M − ˆ g − ˆ R and the relation between the two metrics through the deformation matrixˆ q = ˆ g ˆΩ so that we obtainˆ G ( q ) = ˆ R −
12 ˆ q Tr (cid:16) ˆ q − ˆ R (cid:17) = M ˆ g (cid:20) ˆ P −
12 ˆΩ Tr (cid:16) ˆΩ − ˆ P (cid:1)(cid:21) . (5.64)We can again extract the expression for G , this time in terms of the components of ˆ P and ˆΩ, as follows: G ( q ) = 12 M g (cid:18) P − Ω P (cid:19) . (5.65)If now we equal the right hand sides of (5.63) and (5.65) and solve for H we finally obtain3 H M N = 3Ω P − P Ω h − ( ρ + p ) (cid:16) ∂ ρ ln Ω + c s ∂ p ln Ω (cid:17)i , (5.66)162here we have used that g = − N . This is the master equation providing the modifiedFriedman equation for the theories under consideration, i.e., it gives the dependence ofthe Hubble function in terms of the matter field variables. Let us remember that thecomponents of ˆ P and ˆΩ are functions of ρ and p as obtained from the resolution of(5.59) and, hence, the right hand side of the above equation only depends on the mattersector variables. A very distinctive feature of these theories is the appearance of c s inthis modified Friedman equation. This means that, unlike the case of GR and manyother modified gravity theories, the sound speed not only affects the evolution of theperturbations, but it also affects the background evolution. In particular, this includesone additional parameter for the homogeneous cosmologies of these theories. While in themost extensively studied modified gravity theories the equation of state fully determinesthe background evolution, in the theories under consideration here (among which manyBorn-Infeld inspired theories are included) there is a further dependence encoded in c s .Moreover, some matter sources can actually have a non-constant sound speed (like in thecase of several interacting fluids) and it could even depend on H ( t ) so that (5.66) willbe an implicit equation for the Hubble expansion rate. We have already encountered aparticular case of this result in the EiBI theory rephrased in terms of a time-dependentequation of state parameter and we saw that the background cosmology depends not onlyon w ( t ) but also on ˙ w .There is a number of interesting general features that can be directly inferred from(5.66). The first thing to notice is that now it is very easy to understand the mechanismby which these theories can give rise to bouncing solutions without violating the NEC.For that, let us rewrite the modified Friedman equation in the more familiar form H = 8 πG eff ( ρ, p, c s )3 ρ (5.67)that is closer to its usual form and we have encoded all the modified effects into theeffective Newton’s constant8 πG eff ( ρ, p, c s ) = M P − P Ω h − ( ρ + p ) (cid:16) ∂ ρ ln Ω + c s ∂ p ln Ω (cid:17)i (5.68)where we have momentarily set the lapse to N = 1. If we take the time derivative of (5.67)and use the conservation equation (5.62) we find˙ H = − πG eff (cid:16) ρ + p (cid:17) + 4 π ˙ G eff H ρ. (5.69)In GR with minimally coupled fields, the existence of bouncing solutions (omitting thepossible presence of spatial curvature for the sake of simplicity) characterised by an evo-lution where ˙ H is initially negative (contracting phase) and becomes positive (expandingphase) is subjected to a regime where the NEC is violated in the initial regime and it holdsin the final stage. For the theories under consideration here , the presence of the time This discussion is not specific of these theories, but it also applies to other modified gravity theories or G eff (cid:0) ρ b , p b , c s,b (cid:1) = 0 (5.70)where the subscript b stands for their values at the bounce. By looking at (5.68) we cansee that the bounce can generally happen in two ways, namely: • i ) The numerator vanishes so that 3Ω P − P Ω = 0. • ii ) The denominator diverges so that Ω h − ( ρ + p ) (cid:16) ∂ ρ ln Ω + c s ∂ p ln Ω (cid:17)i →∞ .Let us stress that these two possibilities are the most straightforward (and perhaps smooth)ways to realise the bouncing solution, but they are not exhaustive. For instance, one couldenvisage situations where both the numerator and the denominator diverge (or vanish)while the quotient is a well-behaved function with some roots at ρ = 0. Leaving thispossibility aside, the bounce realised by means of ii ) will generally rely on the existence ofa divergence either in Ω (or one of its derivatives) or in c s . Since both of this quantitieshave a physical relevance, Ω relates the two metrics and c s typically gives the adiabaticsound speed, a divergence in them can potentially give rise to divergent physical effects.On the other hand, the bounce characterised by i ) takes place when 3Ω P − P Ω = 0.From (5.59), we can obtain that3Ω P − P Ω = 1 M M r Ω Ω (cid:16) L G + T (cid:17) (5.71)with T = T µµ the trace of the energy momentum tensor. Thus, a bouncing solution whereboth metrics are regular (i.e., finite and non-vanishing Ω and Ω ) will be characterisedby the equation 2 L G + T = 0. Interestingly, for a radiation dominated universe theenergy-momentum tensor is traceless and the condition reduces to F ( ˆ P ) = 0.In a similar way as we studied the tensor perturbations for the EiBI model in theprecedent sections, we can extend the analysis to the general class of theories consideredhere. We will closely follow the analysis in [60] where tensor perturbations are analysein detail for an even larger class of theories formulated in the affine formalism. We willrecognise that most of the properties we discussed for the EiBI theory are actually genericfeatures for the theories described by (5.51). Let us then consider tensor perturbations ontop of the homogeneous and isotropic background defined as δg µν = (cid:18) a h ij (cid:19) δq µν = (cid:18) a Ω f ij (cid:19) and δT µν = (cid:18) ij (cid:19) (5.72) theories involving non-minimally coupled fields. In general, the argument presented here will be valid forall theories giving rise to a non-constant effective Newton’s constant for the cosmological evolution. Onceagain, the distinctive features of the theories considered here arise from the dependence on c that is notpresent in other classes of modified gravity theories, and this is what can introduce novel features. ij representing the anisotropic stress . An important property that considerablysimplifies the computations with tensor perturbations is that they only live in the spatial 3-dimensional space and all the background quantities are diagonal in that box. This meansthat, at first order in tensor perturbations, any possible pair of matrices appearing inthe equations will commute. Furthermore, the tensor perturbation of any scalar quantityvanishes identically, for instance we will have δ det ˆΩ = 0 and so on. The equation (5.53)at first order in tensor perturbations reads δ ˆΩ − ˆ P + ˆΩ − δ ˆ P = 1 M M p Ω Ω ˆΠ . (5.73)Again, ˆΩ and ˆ P are related by means of the definition of ˆΩ so the above equation can beseen as an equation for δ ˆΩ, whose solution will have the general form δ Ω ij = ω ( ρ, p ) Π ij (5.74)with ω ( ρ, P ) some function obtained from solving (5.73) which only depends on back-ground quantities. This expression for the perturbation of ˆΩ allows to express the pertur-bation of the auxiliary metric as δ ˆ q = δ ˆ g ˆΩ + ˆ g δ ˆΩ ⇒ δq ij = Ω δg ij + a δ Ω ij = Ω δg ij + a ω Π ij (5.75)where the spatial indices have been lowered with the Kronecker delta. In terms of h ij and f ij , we then have f ij = h ij + ω Ω Π ij . (5.76)This result generalises the one already found for the EiBI theory in section 5.1.1. Animportant property is that, in the absence of any anisotropic stresses, the tensor per-turbations of the two metrics are identical and we can simply talk about metric tensorperturbations without referring to any specific metric. In other words, there is only oneclass of gravitational waves.This roots in the conformal relation for the two backgroundmetrics in the spatial 3-hypersurfaces. The field equations for these gravitational wavescan be easily computed from (5.54), whose tensor perturbation yields δR ij ( q ) = 1 M p Ω Ω Π ij (5.77)As it becomes clear, this evolution equation for gravitational waves is exactly the same asthe one found in GR barring the replacement M → M q Ω Ω . (5.78)In terms of the metric perturbations the equation (5.77) can be equivalently written inthe familiar form ¨ f ij + H ( t ) − ˙˜ N ( t )˜ N ( t ) ! ˙ f ij − ˜ N ( t ) ˜ a ( t ) ∇ f ij = 16 πG gw Π ij , (5.79) We are dropping here the perfect fluid assumption in the perturbed sector for generality. G gw = G N / p Ω Ω . In the regime of small energy densities, Ω ≃ ≃ G gw ≃ G N . Once the solution for f ij is computed from the above equation, the evolution ofthe perturbation h ij is determined by the relation (5.76). It should not come as a surpriseby now that the gravitational waves f ij satisfy the usual equation for cosmological tensorperturbations but with respect to the auxiliary background metric and a modified couplingto the source. This is simply the cosmological application of the discussion presented in2.7.1 where it was shown that, in the Einstein frame, the auxiliary metric acquires thestandard Einstein-Hilbert kinetic term, but it is coupled in a non-standard way to thematter fields. Since we are working at first order in tensor perturbations, the modifiedcoupling to matter fields was expected to appear as a modified Newton’s constant. From(5.79) we can also understand the rising of tensor perturbations discussed in section 5.1.1for the bouncing and loitering solutions of EiBI as a consequence of having a non-regularauxiliary metric. We will now consider the Class-I theory introduced in [59] that we already discussed insection 2.7.3. This family of theories consists in extending the EiBI theory to include all theelementary symmetric polynomials and is described by the actions (2.122). The cosmologyof the general case including all the elementary polynomials has not been performed yetin the literature. The fourth polynomial coincides with EiBI so that its cosmology is theone extensively discussed above. The other polynomial whose cosmology has also beeninvestigated is the first one, that was called Minimal model. The corresponding action isgiven by S min = M M Z d x √− g Tr (cid:20)q + M − ˆ g − ˆ R − (cid:21) , (5.80)where the constants have been chosen as to match GR without a cosmological constantin the low curvature regime. A possible cosmological constant term will be consideredas part of the matter sector. As shown in the corresponding part of section 2.7.3, it isconvenient to introduce the fundamental matrix of the model given byˆ M ≡ q + M − ˆ g − ˆ R . (5.81)This fundamental matrix must be positive definite on physically acceptable solutions and isrelated to the deformation matrix by ˆΩ = ˆ M / p det ˆ M , as shown in equation (2.137), thusguaranteeing that both the auxiliary and the spacetime metrics have the same signature.Furthermore, ˆ M satisfies the equation (2.138), that we write here againˆ M − − ˆ M − h Tr (cid:16) ˆ M − (cid:17)i = 1 M M ˆ T ˆ g . (5.82)Before studying the more general cosmological solutions within this model, we shall firstconsider Einstein space solutions characterised by R ( µν ) (Γ) = R E g µν , with R E some con-stant curvature. In terms of the fundamental matrix, this means M µν = m δ µν where166 = q R E M − and the field equations (5.82) simplify to (cid:18) − m − m (cid:19) g µν = 1 M M T µν . (5.83)The conservation of the energy-momentum tensor implies that m = const. Since wehave that R E = ( m − M , the curvature R E must indeed be a constant and cannotbe promoted to some arbitrary function. In other words, only a fluid corresponding to acosmological constant can support Einstein space solutions, as one would have expected.For T µν = − ρ Λ g µν , the above equations give4 − m − m + ¯ ρ Λ = 0 , (5.84)where ¯ ρ Λ = ρ Λ / ( M M ). The solution of this equation is m = 4 + ¯ ρ Λ ± p ρ Λ (8 + ¯ ρ Λ )6 . (5.85)For these Einstein space solutions the deformation matrix is simply ˆΩ = m − so thatboth metrics are conformally related as q µν = m − g µν . In the absence of the cosmologicalconstant ρ Λ = 0, i.e., the vacuum solutions, the two branches give m = 1 and m = 1 / R ( µν ) = 0 and represents thebranch continuously connected with GR, whereas the latter gives R ( µν ) = ( − M / g µν and represents a de Sitter (anti-de Sitter) space for negative (positive) M without theneed for a cosmological constant. The two branches of solutions for m in terms of ρ Λ areillustrated in figure 33. We can clearly see how the physical condition m > ρ Λ ≥ √ − M µν = diag[ M ( t ) , M ( t ) , M ( t ) , M ( t )] andthe equations (2.138) read 1 M + 3 M = 4 + ¯ ρ,M + 2 M + 1 M = 4 − ¯ p, (5.86)with the dimensionless density and pressure¯ ρ ≡ ρM M , ¯ p ≡ pM M . (5.87)These equations are of course (5.53) adapted to the present case. We can now obtainfrom them the quantities M and M algebraically in terms of ¯ ρ and ¯ p , i.e., we will have M (¯ ρ, ¯ p ) and M (¯ ρ, ¯ p ). If we solve for M from the first equation and plug it in the secondone we obtain(4 + ¯ ρ ) M + (cid:20) ¯ P (4 + ¯ ρ ) + 23 (1 + ¯ ρ ) − (cid:21) M − (cid:20) ¯ P + 43 (1 + ¯ ρ ) (cid:21) M + 23 = 0 . (5.88)167 - - - - Ρ L m - - - - Ρ L R E (cid:144) M B I Figure 33: In this plot (adapted from [59]) we show the two branches of solutions for m as a functionof ¯ ρ Λ given in (5.85) (left panel) and the characteristic scale R E normalized to M of the correspondingEinstein space (right panel). The blue-solid solution is continuously connected with GR in vacuum, whilethe red-dashed solution represents the branch giving rise to dS/AdS in vacuum. Interestingly, the dS/AdSbranch is almost insensitive to the presence of ¯ ρ Λ . In this branch, the value of R E quickly saturatesto − M and remains constant irrespectively of the cosmological constant. Finally, we can see how thepositivity of m selects the physical solutions and imposes some bounds on the values of the ¯ ρ Λ that canbe accommodated. It is possible to solve this equation analytically, but the explicit expression is not veryilluminating, so we will omit it here (the interested reader can find it in [61]) and insteadwe will plot the solutions in figure 34 for some interesting matter sources. As usual, onefinds several branches of solutions, 3 in this case owed to above equation being cubic. Outof those 3, one is always unphysical because either M or M is negative and we do notconsider it. The remaining two branches satisfy the positivity requirement, but only oneis continuously connected with GR in the low densities regime (see figure 34). Even forthese physical branches, the positivity of M and M impose constraints on the allowedvalues of ¯ ρ and ¯ p as shown in figure 35. From that figure we can see that the Born-Infeld corrections impose the bounds p . M M and ρ & − M M . In particular,these constraints make the allowed region for a radiation fluid be compact, i.e., there isa maximum allowed value for its energy density. As can be easily understood from theleft panel in figure 35, this will be the case for fluids with constant and strictly positiveequation of state parameter. However, for dust or a cosmological constant, the energydensity can grow arbitrarily large.The definition of the fundamental matrix ˆ M also allows to obtain the correspondingcurvature as R (Γ) = g µν R ( µν ) (Γ) = M (cid:0) M + 3 M − (cid:1) . (5.89)Plugging in the physical branches of solutions for M and M we can then obtain thedependence of the curvature on the density. In the low energy density limit the curvature168ecomes R I = ρ − pM + O (¯ ρ ) (5.90) R II = − M − M ( ρ − p ) + O (¯ ρ ) (5.91)where the Branch I refers to the solution that connects with GR at low energy densitieswhereas Branch II stands for the branch that connects with the dS/AdS solutions invacuum. In figure 34 we show the full solutions. Three different types of fluids areconsidered: dust with p = 0, radiation with p = ρ/ w = − .
8. As we commented above, a radiation fluid naturally shows an upperbound on its possible energy density, which further translates into an upper bound for thescalar curvature ∼ M . This actually happens for equation of state parameters 0 < w < ρ is no longer compact and, in fact, ¯ ρ is not bounded from above. However, it is interestingto notice that, even if ¯ ρ can grow arbitrarily large, the scalar curvature saturates beyond¯ ρ ≃ M , thus avoiding curvature singularities. Finally,fluids with equation of state close to that of a cosmological constant do not have a compactallowed region and the curvature divergences at high energy densities are even more severethan those in GR in the branch I. While GR gives R ∝ ρ , in the Minimal model we have R I ∝ ρ . Finally, for the cases with a non-compact allowed region, the branch II showsthe same behaviour found for the Einstein space solutions above, i.e., the curvature isinsensitive to the value of ¯ ρ .Once we have the fundamental matrix in terms of ρ and p we can easily compute theauxiliary metric by using that ˆΩ = ˆ M / p det ˆ M so that˜ N ( t ) = N ( t ) q M M − , ˜ a ( t ) = a ( t ) √ M M . (5.92)Then, we can use the general expression for the Hubble expansion rate (5.66) adapted tothe Minimal model to obtain H = M M + (¯ p + ¯ ρ ) M − n M (¯ ρ +¯ p )4[(4+¯ ρ ) M − [1 + (4 + ¯ ρ ) ( ∂ ¯ ρ ln M + c ∂ ¯ p ln M )] o , (5.93)where we have used the equations (5.86) to express M in terms of M , which can then besolved for from (5.88). This expression for the Hubble expansion rate shows once againthe distinctive property of depending on ρ , p and c s . Let us notice that the allowed regiondiscussed above arising from the positivity of the fundamental matrix ˆ M constrained thepossible values for ρ and p . Here we have one additional constraint for the cosmologicalsolutions given by the condition H ≥
0. This condition will in fact be more restrictivesince it will depend on c s . In other words, if we consider a barotropic fluid with p = p ( ρ ) (not necessarily linear) so that c s = d p/ d ρ , the constraint H ≥ Ρ M , M - - - - - Ρ R (cid:144) M B I - Ρ M , M - - Ρ R (cid:144) M B I - Ρ M , M - - Ρ R (cid:144) M B I Figure 34: Figure adapted from [59]. In the left panels we show the solutions for M (green) and M (blue) for the Branch I (solid) and Branch II (dashed). In the right panels the corresponding solutions forthe scalar curvature R are illustrated. Three types of fluids are considered from top to bottom: radiation( p = ρ/ p = 0) and a fluid with p = − . ρ . The dotted-purple lines represent the correspondingsolutions in GR. We can see that the solutions for radiation are bounded for both ρ and R , for dust thedensity can grow to infinity but the curvature is bounded by ∼ M and, finally, for the fluid with w = − . parameter space ( ρ, p, d p/ d ρ ), while having a positive definite ˆ M only gives constraints onits subspace ( ρ, p ).In the right panel of figure 35 it is shown the Hubble expansion rate as a functionof the density for radiation, dust and a cosmological constant. These fluids are impor-tant representatives of the following typical behaviour depending on the equation of state170 - - - Ρ (cid:144)H M BI M Pl L p (cid:144) H M B I M P l L w = (cid:144) w = w =- Physical region Ρ (cid:144)H M BI M p L H (cid:144) M B I w = (cid:144) w = w =- Figure 35: Figures adapted from [61]. Left panel: physical region determined by imposing the positivityof the fundamental matrix ˆ M . It is also shown some important equation of state parameters to illustratethe bounds on ρ and p imposed by the Born-Infeld corrections. Right panel: Evolution of the Hubblefunction in terms of the energy density for different equation of state parameters. At low energy densitiesthe standard evolution of GR is recovered (depicted by the dotted line), whereas at high energy densitiesthe translated modifications in the matter source from Born-Infeld become dominant. A crucial propertyof this model is that the Hubble function becomes constant in the Born-Infeld regime for a dust componentwith w = 0 giving rise to a de Sitter phase. parameter w = p/ρ : • Fluids with the equation of state parameter in the range 0 < w < ρ . M M p . We had already observed thistype of behaviour for the standard EiBI theory in the previous section. Thus, it isquite typical to find an upper bound for the allowed energy densities in theories `a laBorn-Infeld. • For fluids with − / < w ≤ ρ . Inter-estingly, the Hubble function can become constant at high energy densities. • Finally, for fluids with − ≤ w < − / H ∝ ρ which is even worse than in GR in terms of singularities at high energy densities.For this type of fluids the realisation of the Born-Infeld mechanism fails.One distinctive and crucial feature of this minimal Born-Infeld extension is the satu-ration of the Hubble function to a constant value at high energy densities appearing for − / < w ≤
0, which could offer an interesting alternative to realise a de Sitter inflation-ary epoch in the presence of a dust fluid. This idea was developed in [61], where, in orderto achieve an inflationary scenario eventually evolving to a radiation dominated phase, itwas considered a cascade of decaying dust fluids at the end of which there is a radiation171omponent. This system is thus described by the following system of equations:˙ ρ i + 3 Hρ i = Γ i − ρ i − − Γ i ρ i i = 1 , ..., n (5.94)˙ ρ r + 4 Hρ r = Γ n ρ n (5.95)where Γ = ρ = 0, ρ r is the energy density of radiation representing the final stateand Γ i is the decay rate of the i th particle. In order to ensure the stability of the dustcomponents during inflation we need to impose Γ i < H dS with H dS the (nearly constant)Hubble expansion rate during the inflationary phase and that will be H dS ≃ M BI . Theidea is then that the quasi de Sitter phase is supported by the dust components as longas ρ dust ≫ M M . Since ρ dust ∝ a − , the energy density of the dust components willeventually drop below M M and the Born-Infeld regime will be abandoned. This willdetermine the end of the inflationary regime and the beginning of the reheating phase.In this phase, the Hubble expansion rate will evolve as H ≃ ρ dust / (3 M ) so that thedifferent decay rates will become larger than the expansion rate and, therefore, the dustwill start decaying. At the same time, the radiation component will be populated and,after all the dust components have decayed, we will be left with a radiation dominateduniverse.This inflationary model has some interesting features that we will summarise herewithout entering into too many details and refer to [61] for a more rigorous treatment. Thefirst important property is that there is a maximum value for the allowed energy densityin the inflationary regime. This can be traced back to the very presence of the cascadethat will lead to a non-trivial and time-dependent sound speed c s ( t ) and a non-vanishingpressure. As we discussed above, this will give rise to a bounded range of values for theenergy density in the physical space. This bound on the energy density will depend on thedecay rates and, in turn, it will lead to a maximum number of e-folds for the inflationaryphase. Thus, imposing that the inflationary phase lasts for at least 60 e-folds will givebounds on the parameters of the model, namely M BI and Γ i . Another constraint can beobtained from the fact that the reheating phase should end before the onset of BBN. Sincethe end of the reheating period is determined by the last decaying dust component, this willgive a direct constraint on the smallest decay rate. These constraints are summarised infigure 36. Finally, a remarkable property of this inflationary scenario is that the first slowparameter ǫ ≡ − d log H/ d log a is negative so that we actually have a super-inflationaryphase. Nevertheless, one cannot directly infer anything on the perturbations from heresince the gravity sector is highly modified with respect to the standard inflationary models.This can be illustrated by looking at the tensor perturbations. Since they propagate onthe auxiliary metric we need to compute the effective expansion seen by them. It turnsout that they see an effective equation of state w = 1 and, thus, they are oblivious to theinflationary background and no primordial gravitational waves are generated within thismodel. Therefore, this inflationary model would come with the distinctive feature of theabsence of primordial gravitational waves so that the detection of B -modes in the CMBgenerated by primordial gravitational waves would immediately rule it out.172 - - - - n M B I (cid:144) M P l - - - - - -
10 0 10 - - - - - - H G i (cid:144) M Pl L L og H M B I (cid:144) M P l L Excludedby BBN Excluded frominflation stability n = = = = Figure 36: Figure adapted from [61]. The left panel shows the bounds on M BI to have at least 60 e-folds ofinflation as a function of the number of dust species n and assuming the decay rates so that reheating endsright before BBN. The black line denotes the Planck scale and the dotted line gives the lower bound on M BI so that the dust components are stable during inflation. It becomes clear that for n = 1 the allowedregion is above Planck scale and hence no realistic inflationary scenario can be constructed and for morethan 20 species BBN constraints and stability during inflation cannot be realized at the same time. Inthe right panel the bounds on (Γ i , M BI ) are shown together by assuming that all the decay rates are ofthe same order. The bound on Γ i coming from the condition that reheating should end before BBN isencoded by the orange region. The blue region represents the stable region of dust components duringinflation. Furthermore, the green curves indicate the bounds for M BI in order to have 60 e-folds of inflationfor different number of components. .4. Functional extensions of Born-Infeld gravity In the previous section we have studied the cosmological implications of the minimalextension of Born-Infeld inspired gravity theory, which was based on the trace of thesquare root structure rather than the determinant. We have seen how one can constructinteresting quasi de Sitter solutions with a dust component in this model. In this section,we shall draw our attention to another interesting extension of the original EiBI gravityand discuss its potential impact to the early universe cosmology. Instead of a square rootone could consider an arbitrary function of the determinant. This modification would stillshare the same properties as in EiBI gravity in the sense that General Relativity wouldbe recovered in low energy density regimes with the modifications becoming appreciableonly at very high energy density regime. Exactly this idea was pursued in detail in [268]and we shall summarise the main results of this study here. For this purpose, let us adaptto the notation used in [268] with Ω αβ = g αµ q µβ where again q µν = g µν + M − R ( µν ) (Γ).In terms of Ω the Born-Infeld inspired gravity theory can be simply expressed as S = M M Z d x √− g (cid:18)q | ˆΩ | − λ (cid:19) + S matter . (5.96)A natural extension arises by promoting the square root to an arbitrary function as it wasproposed in [268]. In this case, the action generalises to S = M M Z d x √− g (cid:16) f ( | ˆΩ | ) − λ (cid:17) + S matter . (5.97)The gravitational Lagrangian density, L G , in (5.97), can be conveniently expressed in termsof an auxiliary scalar field A via the general function f ( A ) with the Lagrange multiplier( | ˆΩ | − A ) f A ) as L G = M M (Φ | ˆΩ | − V (Φ) − λ ) . (5.98)where Φ = d f / d A and V (Φ) = Af A − f ( A ). Written in this language, the connection fieldequations are simply given by ∇ µ (cid:16) Φ | ˆΩ | / √− qq µν (cid:17) = 0. One can use the same trick asin the previous sections to express the Riemann tensor in terms of the quantities of thematter field. By doing so, one obtains [268] R αβ = L G δ αβ + T αβ M Φ | ˆΩ | / . (5.99)For the cosmological application, we will consider a perfect fluid as a representative of thematter fields. Using exactly the same procedure as above, we can compute the evolution ofthe Hubble function in terms of the matter field quantities. The resulting Hubble functionin this particular extension takes the following form H = M / (2 H ∆)) " ( ρ + 3 p ) / ( M M ) + 2(Φ | ˆΩ | − V − λ )Φ | ˆΩ | + V + λ + ρ/ ( M M ) (5.100)where the short-cut notation is introduced ∆ = 2Φ | ˆΩ | / / (Φ | ˆΩ | + V + λ − p/ ( M M )).In [268] a family of power law functions f ( | ˆΩ | ) = | ˆΩ | n was investigated in detail. For174 = (cid:144) n =
12 n = n = (cid:144) = (cid:144) = (cid:144) n = = ΕΚ Ρ Ε H Ω = (cid:144) n = (cid:144) = (cid:144) = = =
10 n = ΕΚ Ρ Ε H Figure 37: These figures are taken from [268] where the evolution of the Hubble function is plotted interms of the energy density of the fluid ρ/ ( M M ) for different values of the index n . The notation usedthere corresponds to ǫ → M − and κ → M − in our notation. In the left panel we see the evolution fora radiation fluid with w = 1 /
3. The solutions represented by the dashed lines correspond to M M BI < M >
0) represent unstable solutions with H = 0 and H ,ρ = 0 at high energy densities. Onecan further see that these non-singular solutions have very similar behaviour for small deviations in n . Inthe right panel the evolution is shown for a fluid with equation of state parameter w = − /
5. In this caseone striking observation is that the solid solutions start resembling bouncing solutions for sufficiently largevalues for the index n . values of the parameter close to n = 1 /
2, of course the features are very close to theoriginal Born-Infeld gravity. In general, one has again two types of branches of solutions,the branch with M − > M − <
0. It turns out that the firsttype of solutions are more sensitive to the changes in the index n . The second type ofsolutions representing a bounce are more robust. In figure 37 extracted from [268] we cansee the evolution of the Hubble function for different values of the index n for a fluid with apositive equation of state parameter in the left panel and with a negative equation of stateparameter in the right panel, respectively. The bouncing solutions are depicted by thedashed lines and the solid lines represent the unstable solutions with H = 0 and H ,ρ = 0for sufficiently high energy densities. The qualitative behaviour of these two branches ofsolutions remains the same for small deviations in the index parameter. On the otherhand, for fluids with negative equation of state parameter the solid line solutions starthitting the horizontal line converting more and more into a rather bouncing solutions forlarge values of n . There exist extensions of the original EiBI gravity theory that relies on the presenceof an additional Ricci scalar, which we will review in this section. These modificationsare constructed either by including a Ricci scalar R (Γ) directly into the determinantalstructure, or by including an additional function as a separate sector into the theory(belonging to the Class II theories in section 2.7.4). f ( R ) gravity One of the early predominant extensions of the EiBI gravity theory is the Born-Infeld- f ( R ) gravity, where the original EiBI gravity theory is combined with an additional func-175ion, that depends on the Ricci scalar [245]. In order to avoid any ghost instabilities, thetheory is constructed in the Palatini formalism in both sectors. The Riemann tensor andthe Ricci scalar both depend only on the connection and not on the metric. This ideaof combining Born-Infeld and f ( R ) gravity was further investigated in [246, 244, 151].Due to the presence of the Ricci scalar, the model exhibits more freedom for simultaneousapplications to early and late time universe cosmology. In [245] it was shown, that for f ( R ) = a R , the model does not alter much the physical properties of bouncing solutionsfound in the original EiBI model, but it does have crucial impact on the loitering solutions.The Lagrangian proposed in [245] has the following explicit structure: S BI = M M Z d x (s − det (cid:18) g µν + 1 M R ( µν ) (Γ) (cid:19) − λ √− g ) + αM Z d x √− gf ( R ) + S matter , (5.101)where the first term is the standard EiBI Lagrangian and the second term is the novelty inform on an additional function of R = g µν R ( µν ) (Γ). The matter fields in S matter couple ina standard manner to the metric. The variation of this action with respect to the metricyields the modified metric field equations with respect to the standard EiBI model √− q √− g q µν − (cid:20) g µν (cid:18) λ − α M f (cid:19) + αf R M g µβ g νγ R ( βγ ) (cid:21) = 0 , (5.102)where again we can define the ˆ q metric as q µν = g µν + M R ( µν ) (Γ) and f R is the derivativewith respect to R . Similarly, the variation with respect to the connection can be writtenas ∇ σ (cid:0) √− qq µν + αf R √− gg µν (cid:1) = 0 . (5.103)The connection equation can be written in the for us more useful form ∇ σ (cid:0) √− ˜ q ˜ q µν (cid:1) = 0where ˜ q plays now the role of the auxiliary metric and is defined as ˆ˜ q = | ˆΣ | / ˆΣ − ˆ g andits inverse as ˆ˜ q − = | ˆΣ | − / ˆ g − ˆΣ with Σ representing Σ µν = | ˆΩ | / ( ˆΩ − ) µν + αf R δ µν with the standard notation ˆΩ = ˆ g − ˆ q and ˆ M = p ˆΩ in the previous sections. For thecosmological application of the model, we are interested in homogeneous and isotropicbackgrounds. We consider again the metric to be FLRW with N ( t ) = 1 and similarlywe make an homogeneous and isotropic Ansatz for ˆ q or ˆΣ directly ˆΣ = diag( σ , σ δ ij ).This on the other hand determines the form of the metric ˆ˜ q to be ˜ q = − p − σ /σ and˜ q ij = √ σ σ a δ ij . For the matter fields, we again assume a perfect fluid with ˆ T = ( ρ, pδ ji ).We can use the same procedure as in standard EiBI gravity model in order to obtain theevolution equation of the Hubble function in terms of the energy density and pressure ofthe matter fields (see section 5.2 for the general cosmological framework). For that wecan use the field equations and the definition of the Einstein tensor and equal them. Bydoing so, one obtains [245] H = M σ − σ − | ˆΩ | / ( σ w − σ w )2 σ (cid:16) − w ) ρ ∆ ρ (cid:17) , (5.104)176here ∆ = √ σ σ and ∆ ρ = ∂ ∆ /∂ρ . In this way, we again have a parametric representa-tion of the Hubble function of the energy density and pressure of the matter fluid. For aparticular choice of the function f ( R ) and the equation of state parameter of the matterfluid w , one can estimate the evolution of the Hubble function and examine whether dif-ferent bouncing and loitering solutions exist in this extension of the EiBI model. In [245]a simple example was studied assuming a quadratic dependence in the form f ( R ) = a R ,since this allows to compute ˆΩ and H analytically in terms of the variables of the matterfield. For this simple model, the presence of bouncing solutions is assured for M − < H = 0 at | ρ/ ( M M ) | = 1 independently of the sign of the equation ofstate parameter. On the other hand, for M − >
0, the Hubble function strongly dependson the sign of w and shows a divergent behaviour for w ≤
0. The parameter a in thefunction f ( R ) does not effect significantly the type of bouncing solutions for M − < a = H a =
12 , Ω= L H a = Ω= LH a = Ω= L H a =
12 ,
Ω=- LH a = Ω= L H a = Ω=- L -Ε Κ Ρ -Ε H @ Ρ D Figure 38: This figure from [245] represents the evolution of the dimensionless Hubble function − H /M in terms of the dimensionless energy density − ρ/ ( M M ) for both the original EiBI theory (solid blue)and the modification with the function of the form f ( R ) = a R , with the value a = 1 / a = 1 (dashed red), in the presence of a matter fluid with two different equations of state ( w = − / , , and 1 /
3) respectively. The presence of bouncing solution does not alter with the difference in a and henceis a robust property of the M − < /κ → M and1 /ǫ → M . from f ( R ) becomes apparent in the other branch of solutions when M − > w = 1 / Ε Κ Ρ Ε H @ Ρ D Ω=- (cid:144) Ε Κ Ρ- Ε H @ Ρ D Ω= (cid:144)
202 4 6 8 10
Ε Κ Ρ Ε H @ Ρ D Ω= (cid:144)
10 0.5 1.0 1.5 2.0 2.5 3.0
Ε Κ Ρ Ε H @ Ρ D Ω= (cid:144) Figure 39: In this figure we show the evolution of the dimensionless Hubble function H /M as a functionof the dimensionless energy density ρ/ ( M M ) for three different cases: the original EiBI theory a =0 (solid blue) and the Born-Infeld- f ( R ) theory with f ( R ) = aR , with two different values a = 1 / a = 1 (dashed red), and different equations of state ( w = − / , / , / , and1 /
3) respectively. One immediate observation is that the zero of ǫH in the case ω = 1 / a . Furthermore, when the equation of state saturates to ω →
0, theHubble function H /M might become again zero for sufficient high densities. However, the correspondingderivative of the function ˙ H/M would vanish, thus representing rather a minimum of H /M . Thisdoes not correspond to a bounce but rather signals an instability representing a state of minimum volume. due to the presence of a in the function f ( R ). On the contrary, for different equation ofstate parameters one encounters novel loitering solutions, which were not present in theEiBI theory. These properties are shown in figure 39 taken from [245]. One additionalinteresting property is observable for the case w = 1 /
10. After reaching a local maximum, H evolves towards a non-zero minimum to then diverge at a finite value of large energydensities. The non-zero value of the minimum depends on the parameter a . Since H doesnot reach the solution H = 0 in this case, one does not have a bounce. Nevertheless, theycould offer an interesting alternative for a quasi de Sitter inflation due to the long plateaubetween the local minimum and maximum. In this way, one could achieve an inflationaryscenario in the presence of radiation. In the minimal extension of the EiBI theory in sec-tion 5.3 we saw that one could realise a quasi de Sitter evolution in the presence of dustwith w = 0. In this modification with f ( R ) this is achievable with radiation.It is worth mentioning that this same model of Born-Infeld- f ( R ) theories was alsoused in [151] in order to construct singular inflationary cosmologies. For this purpose,they borrow ideas from singular f ( R ) inflation [267]. A requirement is that the scalefactor evolves in the following form a ( t ) = e − ( c ( − t + t s ) c ) / (1+ c ) , (5.105)with the constant variables c , c and t s . The Hubble parameter in this case is H = c ( − t + t s ) c . With this Ansatz of the scale factor, one can establish the required relation178etween the dynamical and auxiliary metric. For large values of the constant c > Other modifications based on the Ricci scalar have been constructed in the literature,where the Ricci scalar enters directly the determinantal structure of Born-Infeld [106]. Theinclusion of this pure trace term in the determinant might offer interesting and promisingcosmological implications. The proposed model has the following action [106] S BI = M M Z d x (s − det (cid:18) g µν + 1 M ( α R ( µν ) (Γ) + βg µν R (Γ)) (cid:19) − λ √− g ) + S matter . (5.106)In order to recover General Relativity in the low energy density limit, the parameters ofthe theory have to satisfy α + 4 β = 1. Furthermore, in the corresponding limits, onerecovers Palatini R theories or the original EiBI theory. In the following we will followthe notation of [106], where M Pl = 1. The variation of the action yields the modified fieldequations √− q √− g (cid:20)(cid:18) β R M (cid:19) q µν − βM q αβ g αβ g µρ g νσ R ( ρσ ) (cid:21) − λg µν = − T µν M , (5.107)where q µν = g µν + M ( α R ( µν ) (Γ) + βg µν R (Γ)) in this particular modification of the EiBItheory. The variation with respect to the connection, on the other hand, results in ∇ ν h √− q ( αq µν + βq αβ g αβ g µν ) i = 0 . (5.108)As in the previous sections, we can manipulate the equations on top of a homogeneousand isotropic background such that the Hubble expansion rate can be expressed in termsof the energy density and pressure of the matter fluid. For the homogeneous and isotropicevolution we can make a diagonal Ansatz for ˆΩ = ˆ g − q as Ω = Ω and Ω ji = Ω δ ji .In terms of a dimensionless parameter x these components can be also written as Ω = x | ˆΩ | / and Ω = | ˆΩ | / /x . After the adequate manipulations, the dependence of theHubble expansion rate in terms of the energy density can be expressed as follows [106] H = 2 M α + | ˆΩ | (4 βz − x ) + 3 σ σ ( | ˆΩ | ( x − βzx ) − αx )3 α (cid:16) − q d˜ q d ρ ρ (1 + w ) (cid:17) , (5.109)with the short-cut notations σ = α + β (1 + 3 x ), σ = α + β ( x − + 3) and ˜ q = √ σ σ Ω .Furthermore, the variable z satisfies x + 3 x − = 4 z . After having brought the expressionof the Hubble function H in the desired form, we can study its evolution for differentequation of state parameter of the matter fluid as we did in the previous sections. This willenable us to directly compare the type of bouncing, loitering and quasi de-Sitter solutions179 - - ΚΡ- - Κ H Figure 40: This figure is taken from [106] and shows the dependence of the Hubble expansion function interms of the energy density for a radiation fluid with w = 1 / β > β = 0 (solid blue), β = 10 − (solid red) and for β = 10 − (dashed blue) areplotted respectively. In the notation used in [106] κ = M − . within this class of modifications with respect to the standard EiBI gravity theory. In [106]this analysis was performed for radiation with w = 1 / β . It wasobserved, that on the contrary to the previous modification in form of an additional f ( R ),the inclusion of the Ricci scalar into the determinant alters the robustness of the bouncingsolutions for M − <
0. These solutions seem to be very sensitive to the presence of theparameter β for even very small values. This behaviour can be seen in figure 40. TheHubble function scales as H ∼ ρ at large energy densities in this case. Another differencein the model rises for the loitering solutions of the standard EiBI model when M − > H ∼ ρ − ρ max , where ρ max represents the maximum energy density at the bounce. The evolution of the Hubblefunction in the case 0 < β ≤ / β > / β getting closer to 1 /
4, the cosmological singular solutions resemblemore those obtained in R + R theories with H ∼ ρ/ M − <
0. For the oppositecase with M − >
0, for instance for β = 1 /
10 and β = 3 /
25, the loitering solutions ofthe standard EiBI theory become again a bounce in the past with the Hubble functionsaturating to H ∼ ρ − ρ max , which has a quasi-sudden singularity in the past. For othervalues of β , for example β = 1 / β = 21 / H ∼ ( ρ − ρ max ) − corresponding to a big freezesingularity in the past. Born-Infeld inspired gravity theories were mainly applied to early universe cosmology,since the effects of the modifications become appreciable at high energies. We have seenthat interesting alternatives to the standard inflationary paradigm can be constructedwithin this framework and promising roads to avoid cosmological singularities can besuccessfully realized. Since the modifications `a la Born-Infeld are dominant at early times,for a possible application to dark energy and dark matter a change of the framework is180
ΚΡ- - Κ H - - - ΚΡ- - Κ H Figure 41: This figure is taken from [106] for the evolution of the Hubble function in the case 0 < β ≤ / β > / κ = M − . needed. This was pursued by Ba˜nados in the work [44], where the standard Einstein-Hilbert Lagrangian of GR is coupled to a “Born-Infeld” field in the hope to reproduceinteresting phenomenology for late-time universe. Even if it was proposed as a modificationof the original EiBI gravity theory, we would like to emphasise once again that these modelsdo not comply the original Born-Infeld spirit of not modifying the field content. The actionconsidered in [44] can be expressed as S = M Z d x ( √− gR + 2 M α s − det (cid:18) g µν − M R ( µν ) (Γ) (cid:19)) + L matter , (5.110)where α is a dimensionless parameter and R is the Ricci scalar associated to the metric g and R ( µν ) is the Ricci curvature of the independent connection Γ. This model constitutesGeneral Relativity with the Einstein-Hilbert term coupled to the Born-Infeld connectionΓ. In fact the model can be analogously written as a bimetric theory, where the potentialinteractions of the two metrics ˆ g and ˆ q do not satisfy the potential structure of massivegravity. Therefore, the theory probably might contain dangerous ghostly degrees of free-dom. Independently of these ghost issues, the model was studied in [44], where it wasfound that the model admits de Sitter solutions at late times. In fact, it is argued that theparameters can be chosen such that the Born-Infeld field contributes ∼
73% of the totalenergy density in form of vacuum energy and 23% in form of dark matter with the equa-tion of state parameter varying between w = − w = 0 respectively. The constructedcosmological solution is such that the scale factor evolves as a ∼ e Ht at late times and181 ∼ t / at early times. The field equations of the model are given by G µν = − M p q/gg µρ q ρβ g βν + T µν M , R µν = M ( g µν + αq µν ) , (5.111)where q µν is the metric associated to the connection Γ. As it can be seen from the fieldequations, the structure of the interactions between ˆ q and ˆ g in − M p q/gg µρ q ρβ g βν doesnot correspond to the ghost-free massive gravity interactions, signalling the presence ofghostly degrees of freedom. For Einstein space solutions R µν = Λ g µν , the two metrics haveto be proportional to each other q µν = C g µν , where the constant C is determined by thefield equations to be C = 1 / (1 − α ). The modification of the Einstein equations is encodedin the term − M p q/gg µρ q ρβ g βν in equation (5.111) and acts as a cosmological constantfor the Einstein space Ansatz, where its corresponding value can be expressed as Λ = C M = M / (1 − α ). As mentioned above, even if these interactions allow for a constantcontribution in form of a cosmological constant, they correspond to ghostly interactions,which will render the cosmological solutions unviable. For general cosmological solutionsbeyond Einstein space solutions, the following homogeneous and isotropic Ansatz for thetwo metrics were considered in [44]:d s g = (cid:0) − N ( t ) d t + a ( t ) d ~x (cid:1) , d s q = (cid:16) − ˜ N ( t ) d t + ˜ a ( t ) d ~x (cid:17) . (5.112)The background field equations (5.111) for these metrics become H = M H ˜ a ˜ N a + ρρ c , ˜ a = 3 ˜ N ˜ aa H ,H q = ˜ N M H (cid:18) −
12 ˜ N + α a ˜ a (cid:19) , (5.113)where N = 1 and ρ c = 3 H / M with H = ˙ a/a , H q = ˙˜ a/ ˜ a and H denoting the Hubbleparameter today. The Born-Infeld field contributes to the field equations in form of a fluidwith the following effective energy density and pressure: ρ BI = M M a ˜ N a and p BI = − M M N ˜ aa . (5.114)We can now study the behaviour of the equations at late and early times. For large valuesof the scale factor we can neglect the contribution of the ordinary matter fields. In thiscase, the scale factors evolve as a = a e t/c and ˜ N = 1 √ − α , ˜ a = a √ − α e t/c , (5.115)182 igure 42: This figure is borrowed from [44], where one can see the evolution of the scale factor in theBorn-Infeld gravity model in the presence of the standard Einstein-Hilbert action versus the standardFriedman universe in the left panel. Both evolutions are almost indistinguishable if one chooses α = 0 . w BI isillustrated. with the constant variable c = p − α ) M BI H . This corresponds to the de Sittersolution with Ω Λ = 1 /c . For small values of the scale factor, on the other hand, thesolutions can be approximated as a = a t / (1 + O ( t / )) and ˜ N = ˜ N (1 + O ( t )) , ˜ a = ˜ N (1 + O ( t )) . (5.116)Hence, the scale factor evolves between a ∼ t / at early times and a ∼ e Ht at late times.In [44] Ba˜nados provides also the numerical solutions to confirm the approximate solutionsof these two regimes. In figure 42, we see the numerical solution for the scale factor. Inorder to achieve the standard evolution as in ΛCDM model, the parameter α should bevery close to 1, whereas the exact value α = 1 is singular.As next, we can compute the effective equation of state parameter of the Born-Infeldfield. In terms of its energy density and pressure, it can be simply expressed as w BI = p BI ρ BI = − a ˜ N ˜ a ! . (5.117)As it can be seen in the right panel of figure 42, at early times the pressure is p BI = 0behaving as matter and at late times p BI = − ρ BI behaving as dark energy.As mentioned above, even if this model provides an interesting phenomenology fordark energy and dark matter, the ghostly interactions between the ˆ g and ˆ q metrics castserious doubts on the physical viability of these cosmological solutions. A Born-Infeld approach to the Teleparallel equivalent of General Relativity was alsopursued in the literature in the hope that Born-Infeld teleparallelism might cure the cos-183ological singularities. For this purpose, Fiorini and Ferraro have considered an extensionof a teleparallel model `a la “Born-Infeld” in [165, 169] with the following action: S = M M Z d xe (s S · T − M − ) , (5.118)where e aµ represents the four one-forms, T µνρ the torsion and S the super-potential S ρµν = −
14 ( T µνρ − T νµρ − T ρµν ) + 12 δ µρ T σν σ − δ νρ T σµσ . (5.119)As before, this model can be barely categorised as a Born-Infeld inspired gravity theoryaccording to our criterium, but rather it should be better considered as belonging to theclass of f ( T ) theories (Class-IV). For a cosmological background e aµ = diag(1 , a, a, a ), thefield equations read 1 − M q − M − H M − ρM M , (5.120)(1 − M ) (cid:16) H M + H dM − M (cid:17)(cid:16) − M − H M (cid:17) / + 1 = pM M , (5.121)where d is here the deceleration parameter d = − a ¨ a ˙ a . A key feature of these modified fieldequations is that the scale factor approaches a ∼ exp s M (1 − M )12 t (5.122)at early times a →
0. This, on the other hand, means that the Hubble parameter reachesa maximum value H max = q M − , regularizing the divergences of standard GeneralRelativity. Let us consider the above action (5.118) without the cosmological constantΛ = 0. We can combine the two field equations (5.120) into a single equation [165]1 + d = 32 1 + w (cid:16) ρM M (cid:17) (cid:16) ρ M M (cid:17) . (5.123)A negative deceleration parameter can be achieved with a sufficiently large energy densitywithout the need of a fluid with negative pressure as in GR. An accelerated expansionnaturally arise if the energy density satisfies the condition2 ρM M > − √
13 + 12 w . (5.124)184 igure 43: In this figure we illustrate the evolution of the scale factor of the modified teleparallel model inthe presence of a radiation fluid w = 1 / α = H max /H , which we extracted from[165]. In the case, where the energy density saturates to ρ → ∞ , one has d → − a w ) ρ = const = a w )0 ρ , the first fieldequation can be rewritten as (cid:18) ˙ aa (cid:19) + M a a vuut β X i Ω i (cid:18) aa (cid:19) − w i ) − = 0 , (5.125)with the subscript “0” denoting the values today and β = (1 − H /M ) − / −
1. Thesecond term in equation (5.125) is always negative and approaches zero if w > − / a → ∞ . In the contrary case, if w < − / a → w > −
1, one has a ∼ e √ M / t and hence H max = q M . The evolution ofthe scale factor is depicted in figure 43. Even if one can achieve interesting phenomenologyfor the early universe cosmology, this model barely represents a modification `a la Born-Infeld but should be rather considered as a f ( T ) model. A model more close to the spiritof Born-Infeld will be discussed in the following. Still within the same framework of the previous section, there has been also the attemptin the literature to consider more general setups in the Weitzenb¨ock space-time and studythe consequences for early universe cosmology [167]. The main ingredients are againthe super-potential S defined in equation (5.119) and torsion T µνρ . Let us consider the185ollowing (Class-III) action S = M M Z d x (cid:26)q det η ab e aµ e bν + 2 M − F µν − λ det( e aµ ) (cid:27) , (5.126)where the tensor F µν has the following general form F µν = α S µρσ T νρσ + α S ρµσ T ρνσ + α η ab e aµ e bν T . (5.127)For the interesting cosmological applications we shall consider a homogeneous and isotropicbackground for the vielbein e a = diag(1 + a ( t ) , a ( t ) , a ( t )) and a barotropic matter fieldwith p = wρ . By varying the action with respect to the vielbein, one obtains the followingmodified Friedman equation √ A H √ − A H (1 + 2 A H − A A H ) − ρM M , (5.128)with the short-cut notations A = 6( α + 2 α ) M − and A = 2(2 α + α + 6 α ) M − . (5.129)Since the matter fluids are assumed to couple minimally to e , they follow the standardconservation equation ˙ ρ +3( ρ + p ) H = 0, which imposes the evolution of the energy densityin the form ρ = ρ (cid:0) a a (cid:1) w ) with the subscript “0” denoting again the present day value.In order to obtain General Relativity in the low energy density regime, we have to fulfilthe condition α + α + 4 α = 1. For simplicity, let us first concentrate on the case with A = 0, in other words, 2 α + α + 6 α = 0. This leaves A = 12 M − . In this particularcase, the Friedman equation simplifies to1 q − H M − − ρM M . (5.130)This specific case with A = 0 recovers the modified Friedman equation of the previoussubsection that we had categorised as f ( T ) theories. Therefore, in this case one obtainsthe same non-singular cosmological solutions for radiation and dust matter as the onesreported in the previous subsection 5.6.2. For a more general case of the background withspatial curvature, the Ansatz for the vielbein is a little bit more involved e = d t, e = a ( t )˜ e , e = a ( t )˜ e and e = a ( t )˜ e , (5.131)where the ˜ e i components are given by˜ e = K ( − K cos θ d ψ + sin( Kψ ) sin θ cos( Kψ )d θ − sin ( Kψ ) sin θ d φ ) , ˜ e = K ( K sin θ cos φ d ψ − sin ( Kψ )(sin φ − cot( Kψ ) cos θ cos φ )d θ − sin ( Kψ ) sin θ (cot( Kψ ) sin φ + cos θ cos φ )d φ ) , ˜ e = K ( − K sin θ sin φ d ψ − sin ( Kψ )(cos φ + cot( Kψ ) cos θ sin φ )d θ − sin ( Kψ ) sin θ (cot( Kψ ) cos φ − cos θ cos φ )d φ ) , (5.132)186ith K denoting the spatial curvature. In this case, the modified Friedman equation(5.130) becomes (1 ± M − a − ) / q − H M − − ρM M , (5.133)where ± represents the closed ( K = 1) and open ( K = −
1) universe, respectively. Inthe high energy density regime, the solution to the modified Friedman equation gives thefollowing evolution for the scale factor in the presence of a radiation fluid a ( t ) ≈ exp( q M / t ) as a/a → . (5.134)This is again the same type of non-singular solution as we found in the previous subsection,which cures the initial singularity and seems to be insensitive to the presence of spatialcurvature. Another interesting case arises when one chooses the parameters as α = 0 and α − α = 0. For this particular case, the Friedman equation modifies to1 ± M − a − q ± M − a − − H M − − ρM M . (5.135)We can again abstract the evolution of the scale factor for the high energy density regime.However, now the solution depends highly on the sign of curvature. For the closed universescenario with the + sign, the scale factor evolves as a ( t ) ≈ t in the high energy regime,which therefore corresponds to a singular solution with the singularity appearing at t = 0.Maybe a more interesting scenario appears for the case of open universe, where the scalefactor evolves as a ( t ) ≈ a min + M BI t with a min = M − , constituting a bounce at t = 0.The accelerated expansion period takes over when the energy density has its maximumvalue ρ max ∼ a − = M and the volume its minimum value a = M − .This model with the three components in (5.127) was further investigated in [168],where the realisation of a primordial brusque bounce was studied in detail. The authorinvestigates the unexplored case with A = A and finds yet other type of interestingcosmological solutions. We shall summarise his findings for this case in the following. Inthe case with A = A , the modified Friedman equation becomes6 H (cid:18) − H M (cid:19) = ρM , (5.136)where α = α and α was reabsorbed into M BI . We can solve this equation for H , whichresults in H = M (1 ± √ − ρ ) /
9, where ¯ ρ stands for the dimensionless energy density¯ ρ = ρ/ ( M M ). The conservation equation for the energy density has the standardform ˙¯ ρ = − w ) H ¯ ρ . Due to the different signs in the expression for H , we havedisconnected two different branches. The branch with the positive sign corresponds toa solution completely disconnected from the GR limit and therefore we can discard thisbranch. Concerning the negative branch, because of the presence of M BI one will havedifferent solutions depending on the sign of this parameter. The type of solutions with M < M > H = M / ρ max = 3 M M . The conservation equation together with the equation (5.136) canbe solved exactly. These exact solutions can be found in [168]. We shall only reporton the behaviour of these exact solutions in the interesting limits. For a universe filledwith radiation ( w = 1 /
3) and for the branch with M >
0, the scale factor evolvesapproximately as (cid:18) a ( t ) a (cid:19) = ρ M M (1 ± M BI t ) + O ( M t ) . (5.137)Accordingly, the dynamics of Hubble function are recast by H ( t ) = ∓ M BI / (1 ± M BI t ) + O ( M t ). From these expressions one immediately observes that there is a minimumvalue for the scale factor at t = 0 (cid:18) a min a (cid:19) = ρ M M , (5.138)with H ( t = 0) = M . This represents a brusque bounce. Even if the Ricci scalar suffersfrom indefiniteness at this point, the solution does not represent a singularity. The authorshows explicitly that the geodesics are well behaved at the bounce in the sense that apoint particle traveling along causal geodesics does not experience any singular behaviour.Furthermore, the author extends this analysis to a finite size extended object and confirmsthe same behaviour. As we have seen in detail in this section, Eddington-inspired Born-Infeld gravity and itsknown extensions have received much attention in the literature. Since the modifications`a la Born-Infeld become appreciable at large energy densities or in high curvature regimes,the direct cosmological applications can be only for early universe physics. The main goalof most of the studies was to construct cosmological solutions curing the standard BigBang singularities. The inflationary scenario with a standard single field suffers from cos-mological singularities. The idea behind using the Eddington-inspired Born-Infeld theoryor its extensions was to deliver an alternative scenario for early universe, for instance inform of a bounce or loitering solutions. We have also seen in this section that interestingbouncing and loitering solutions can be constructed within these theories, that are basedon a radiation or dust, depending on the explicit model. In the standard inflationary sce-nario the matter fields couple minimally to the gravity sector. As we have seen in variousoccasions in this section, Born-Infeld type theories can be seen as nothing else but Gen-eral Relativity with a non-trivial and non-linear matter coupling. Specially, concerningthe cosmological solutions, the essence of the modifications can be encapsulated in theFriedman equation as H = f ( ρ, p ) with a non-linear function. The resulting cosmologicalevolution correspond to either quasi de-Sitter or bouncing or loitering solutions. We haveseen that in the simplest realisation of the bouncing and loitering solutions in the EiBImodel, the tensor perturbations were becoming unstable in the presence of matter fields188ith constant equation of state parameter. This of course renders these simplest reali-sations unviable. More general scenarios with non-constant equation of state parametershowever can alleviate these issues. In this respect, we have seen concrete examples ofan additional scalar field as matter field with varying equation of state parameter, whichavoids the tensor instabilities in the bouncing and loitering solutions.After reviewing the works of the standard EiBI model, we then systematically sum-marised similar cosmological studies of other extensions and modifications of Born-Infeldinspired gravity theories. Since most of these extensions were sharing the same mech-anisms and features, before studying the individual cases, we have first developed thegeneral framework of cosmological solutions for a general class of theories constructed outof the Ricci tensor and the inverse metric. Within this general framework, we have derivedthe master equation that determines the Hubble function in terms of the matter field vari-ables and discussed the general mechanism that provides bouncing solutions. As next, forconcrete models we considered the family of Born-Infeld inspired gravity theories basedon all of the elementary polynomials and discussed in detail the cosmological solutions ofthe first polynomial as an example. We considered again fluids with different equation ofstate parameters and saw that interesting quasi de Sitter solutions can be constructed ina universe with dust component. We summarised briefly the arising inflationary scenariowith a cascade of dust components in the early universe. Another interesting extensionof the original Born-Infeld gravity is the functional extension in the sense that the squareroot of the EiBI model is replaced by an arbitrary function of the determinant. The re-sulting evolution of the Hubble function is such that the bouncing solutions are robust tothe functional enlargement, whereas the loitering solutions do undergo notable changes.We have also discussed extensions of the original theory by means of an additional Ricciscalar, appearing either directly in the determinantal structure or as an additional separatefunction and explored the features of new cosmological solutions beyond the ones presentin the EiBI model. Finally, we have also reported on other extensions along the line ofteleparallel formulations of Born-Infeld theories and discussed the presence of interestingbrusque bouncing solutions. 189 . Concluding remarks, open questions and prospects This review has been devoted to provide a comprehensive survey on theories of modifiedgravity that take inspiration from the Born and Infeld approach to nonlinear electrody-namics. The underlying logic is that a modification of the high curvature/density regimeof the gravitational interaction could effectively introduce upper bounds that cannot besurpassed. As we have seen in this review, the richness of the theory transcends the merebound of certain invariants, leading to physically sound results even in the presence ofcurvature divergences in black hole scenarios.We started our journey on Born-Infeld gravity from the most reasonable place, namely,by reviewing the original construction of Born and Infeld for electromagnetism and thedifferent routes leading to a transcription of its remarkable properties into gravity, speciallyits determinantal structure. The unsuccessful first attempt of the work of Deser andGibbons to construct gravity theories `a la Born-Infeld was rooted in the use of the metricformalism, which inevitably leads to the appearance of ghosts. So far the only ghostlesstheories in the metric approach are the so-called f ( R ) theories, but these can hardlybe considered proper Born-Infeld gravity theories. The scrutiny presented in section 2suggests that any theory of gravity realizing the Born-Infeld construction and formulatedin the metric formalism will either be pathological or reduce to other known theories ofgravity. A challenging problem is to find counter-examples to this general no-go result. Thestory becomes more interesting when resorting to a metric-affine framework, as Vollick did.When the connection is regarded as an independent field, the aforementioned pathologiesarising in the metric formalism are avoided. A further refinement introduced by Ba˜nadosand Ferreira, making the matter-gravity coupling more standard, resulted in the mostextensively explored Born-Infeld inspired gravity theory so far, dubbed EiBI. In this theory,the connection is generated by an auxiliary metric q µν that is non-trivially related to themetric g µν . Although q µν made its debut as an auxiliary object helping to solve the fieldequations, it soon showed its real significance and allowed to establish the existence oftwo frames for these theories, similar to what happens in scalar-tensor theories. In theoriginal Born-Infeld frame, matter fields are minimally coupled to the spacetime metric g µν , which satisfies second-order dynamical equations. In the Einstein frame, the metric q µν behaves as in standard gravity, with an Einstein-Hilbert term governing its dynamics,but it couples in a non-standard way to the matter fields. Elucidating the existence ofthese two frames allowed to discern that, while matter fields follow geodesics of g µν , thegeodesic motion of gravitons is determined by q µν .Most of the existing developments in the literature make two important assumptions(though not always explicitly said) that we also adopted here. The first one is related to theclass of theories considered where only the symmetric part of the Ricci tensor is included.This condition is useful to simplify the field equations and express the solutions for theconnection solely in terms of the auxiliary metric. Although it could seem to be rather adhoc, imposing a projective symmetry naturally results in this type of theories. However,it remains to be explored more general frameworks without the projective invariance andclarify to what extent such a symmetry should be considered as a fundamental ingredient.The second condition that is usually made has to do with the class of solutions that areconsidered, where the torsion is set to zero. Very little can be found at this respect in the190iterature of Born-Infeld inspired gravity theories and it is not rare to find works wherethis issue is simply omitted. Certainly, in most applications assuming vanishing torsion isa consistent Ansatz, but studying the stability of such solutions should also consistentlyincorporate fluctuations of the torsion. As with the projective invariance, the actual roleplayed by the torsion is to be clarified within the context of these theories. In fact, it wouldnot be too surprising to find links between the projective symmetry and the absence (orirrelevance) of torsion in the solutions. Let us remember that for the Einstein-Hilbertaction in the Palatini formalism, the full solution for the connection only contains thetrace of the torsion and it precisely enters as a projective mode, thus being pure gauge.Our first contact with Born-Infeld inspired theories of gravity concluded with a glanceat the different approaches adopted in the literature to incorporate the Born-Infeld ideasinto gravity and a classification of the existing proposals. We first presented a generalformalism showing that most of the features are actually shared by a wide variety oftheories. We then decided to perform a classification based on the proximity to the orig-inal Born-Infeld construction, and taking the most studied case of EiBI as baseline. Wecould appreciate the richness of the field where the imagination of the community led tonumerous searches along several directions. This was in high contrast with the case ofBorn-Infeld electrodynamics where the Lagrangian does not admit immediate alterations.This is so because such a Lagrangian was singled out by precise conditions, among whicha symmetry guiding principle was paramount. In the case of gravity, an analogous guid-ing principle permitting to isolate some unique Lagrangian has not been found yet. Theprojective invariance could be invoked, but we have seen its incapability to sufficientlyreduce the number of possible actions. Finding a better suited principle would consid-erably reduce the different possibilities and would give improved guidance to pursue theexploration of Born-Infeld gravity in closer relation with its electromagnetic ancestor, thatturned out to exhibit a number of remarkable features. Until then, a prolific family ofdifferent Born-Infeld gravity theories is expected to remain. So far, most of them are basedon the EiBI model and extensions along different paths. An interesting alternative wasintroduced taking TEGR as starting point. This allowed to study a different branch oftheories which are formulated in a Weitzenb¨ock space. Since TEGR gives an alternativedescription of GR as a gauge theory of the translational group, this route could lead toBorn-Infeld theories of gravity closer in their structure to the original construction forelectromagnetism. This gauge character could be appropriately exploited and it couldserve as the desired symmetry principle so it deserves a further exploration.Once the general landscape of Born-Infeld inspired theories was overviewed, we movedon to the different territories where these theories find applications. The first pertinentplace to test the modifications introduced by Born-Infeld inspired theories of gravity wasinhabited by the illustrious family of astrophysical objects. Since the Born-Infeld correc-tions are designed to only appear at high curvatures or densities, compact objects exhibitexcellent prospects to put these theories on trial. The first explicit applications, however,already showed some subtleties in the weak-field limit associated with the energy-densitydependence of the modified dynamics proper of metric-affine theories. In diluted systems,the fluid approximation may lead to unphysical effects depending on the weight functionsused in the transition from the discrete to the continuum description. Potential patholo-191ies associated to this can be found in Newtonian pressureless fluids and in compact starmodels based on polytropic equations of state, for instance. As discussed in detail forwhite dwarfs and neutron stars, polytropic equations of state are very useful to modeltheir structural properties, but the transition to the external (idealized) Schwarzschild so-lution must be improved in order to construct realistic models able to account for certainobservational features (like electromagnetic spectra and radiation fluxes), which at thesame time may avoid artificial effects associated with the peculiarities of certain equationsof state and/or the fluid approximation. After clarifying the importance of correctly mod-eling the outermost regions of stars, a number of results related with the structural anddynamical properties of compact objects and the Sun were reported. We can highlight theability of solar observations to constrain the EiBI model via neutrino emission and seismicwaves, the possibility of accommodating higher masses with soft equations of state in neu-tron stars, the stability of these objects against radial perturbations, and the possibility ofusing the low-mass spectrum of neutron stars to discriminate EiBI from GR. On the otherhand, it really came as a surprise the existence of universality relations between quantitiesconstructed using the moment of inertia, the quadrupolar moment, and the Love numbers( I -Love- Q relations), which turn out to coincide with those of GR. Dipolar magnetic fieldsalso converge to the GR prediction at the crust and surface of neutron stars. These resultsimply a degeneracy which poses obstacles to the observational discrimination between GRand the EIBI theory.After spending some time with the best known members of the family of compactobjects, we continued our trip and arrived at the place where some of their more exoticacquaintances dwell, namely, black holes and their closest relatives. Obviously, as theblack hole terrain has occupied the efforts and imagination of countless theoreticians andastrophysicists alike for decades, is not surprising that a few years of research in thecontext of Born-Infeld inspired theories of gravity has only allowed to touch a few of therelevant physical aspects regarding these objects. In this sense, our trip quickly went oversome dubious proposals for these theories, either because they are formulated in metricapproach (and thus plagued by ghostly-like instabilities) or because matter is included inan unconventional form. Nonetheless, this allowed us to naturally introduce the well knownblack hole solutions for Born-Infeld electromagnetism within GR, of which the familiarReissner-N¨ordstrom solution is a particular (limiting) case. We thus introduced some of thetrademark features of such black holes that bear a close resemblance with those obtainedin Born-Infeld inspired theories of gravity, such as the appearance of different numberand types of horizons, depending on characteristic parameters of the matter and gravitymodels. This way we naturally entered into the terrain of black hole solutions within EiBIgravity, where most research in this context has been carried out in the literature. First wereviewed and enlarged the description provided in the paper by Ba˜nados and Ferreira andother works in the field, were we paid special attention to the deviations regarding geodesicmotion, strong gravitational lensing and mass inflation. But we also described a differentfamily of solutions, whose study revealed the presence of all types of exotic objects, likegeons or wormholes. Geons are self-sustained electromagnetic objects without charges.On the other hand, wormholes represent the promised behaviour of Born-Infeld theoriesso that the center of black holes in GR is replaced by a regular object of finite size. We192aw that the construction of wormhole solutions without any pathologies (i.e. violation ofenergy conditions) is a hard task, but EiBI gravity managed to surpass our expectationsand provided, in analytical form, both wormholes and geons.Our analysis of the geodesic structure over the innermost region of these objects re-vealed that, although these places look inhospitable at first, they actually are less perilousthan expected and, in fact, geodesics can smoothly pass through, while the impact of cur-vature divergences on physical observers did not seem to pose any absolutely destructivethreat. These results were confirmed by the well posedness of the problem of scattering ofscalar waves off the wormhole. Further developments on this field involve higher and lowerdimensional models, though with much less impressive results. It should be pointed outthat the counterparts of the rotating Kerr solution of GR (and Kerr-Newman when chargeis included) in Born-Infeld inspired theories of gravity are not available in the literatureand, without such solutions, realistic black hole scenarios for astrophysical purposes cannotbe put to a test. This is a very relevant point, since the present (and future) observationsfrom gravitational wave astronomy offer a great opportunity for testing deviations withrespect to GR solutions. We cannot but to encourage researchers working in the contextof these theories to look for such rotating black hole solutions.The last stage of our pilgrimage throughout the Born-Infeld land took us to a com-pletely different scenery shaped by cosmological applications. It should not come as asurprise at this point that the natural home for such applications is the early universebecause Born-Infeld theories are designed to affect the regime of high densities. In thatcontext, it has been extensively shown that both EiBI and other Born-Inspired theoriescan provide singular-free solutions of two types, namely: bouncing solutions, where theuniverse transits from a contracting phase to an expanding one without hitting a singu-larity, and loitering solutions, where the universe asymptotically approaches a Minkowskiuniverse as the energy density goes to infinity. Both of these solutions were shown topresent some tensor instabilities for the original EiBI theory and in the simplest caseof one single perfect fluid, although it was later shown that such instabilities could beavoided in more contrived scenarios. These solutions have recurrently been found in otherformulations of Born-Infeld inspired gravity. Furthermore, other singular-free solutionshave also been found like, e.g. the brusque bounce solution where the Hubble expansionrate is not defined at the bounce, but all relevant geometrical quantities are smooth. Inmost treatments of these solutions, the analysis is limited to studying the isotropic back-ground evolution and, at most, the tensor perturbations. In some works, homogeneousand anisotropic solutions have also been studied, what is closely related to the analysisof tensor perturbations. However, the full viability of the bouncing solutions can only beclaimed once all potential sources of instabilities have been shown to be under control.This is a paramount issue that needs to be properly addressed.Providing singularity-free cosmological evolutions was precisely the job the Born-Infeldtheories were designed for. However, it did not take long to find other jobs for which thesetheories could serve just as well. In fact, since they are constructed to modify the regime ofhigh densities for gravity, they are also compelling frameworks to have models of inflation.This is achieved for theories that exhibit a nearly constant Hubble expansion rate in theBorn-Infeld regime. Such a behavior has been found in several of the proposed models,193ome even for a radiation dominated universe. In particular, a specific model of inflationwas developed in detail where inflation is supported by a dust component that decays intoradiation, giving a model of the early universe similar to the usual inflationary scenarioswith a re-heating phase, but from a completely different perspective. This in turn led todifferent predictions, in particular, a super-inflationary phase is achieved where primordialgravitational waves are not produced. One important feature of inflationary models basedon these theories is that we have a naturally graceful exit of inflation. This is due tothe fact that the density will typically decrease during the inflationary phase so it willeventually become smaller than the transition scale given by M M and the GR regimewill be restored, thus matching the standard cosmological evolutions. In general, andas with most of the cosmological analysis within Born-Infeld gravity theories, a propertreatment of the scalar and vector perturbations is still to be performed. This is crucialfor the viability of these inflationary scenarios since it is of paramount importance toshow that a red and nearly scale invariant spectrum of primordial scalar perturbations isgenerated, in order to be compatible with CMB measurements. However, given the highlynon-standard gravitational sector of theses theories, a general and rigorous analysis of thesubject will likely be an arduous task. At this respect, simplified models and, perhaps,the use of the Einstein frame could permit advancing in this direction.Since the expansion of the universe makes the total energy density be diluted during thestandard radiation and matter dominated epochs, the Born-Infeld corrections are expectedto be negligible for the late-time evolution of the universe. In fact, a safe assumption isto impose that the transition to the GR regime is achieved before BBN. An importantconsequence of this is that Born-Infeld theories are not fruitful frameworks for dark matterand/or dark energy models and the use of cosmological observables to constrain them bystudying their late-time evolution is futile. These theories can only affect the late-timecosmological observables by modifying the initial conditions in the early universe, perhapsset during a Born-Infeld inflationary scenario. There is however a family of cosmologieswhere Born-Infeld theories can become relevant at late times, namely models with futuresingularities. If the dark energy component happens to have some exotic features likea phantom behaviour, the asymptotic evolution of the universe in GR will end up withsome type of singularity. In the presence of Born-Infeld gravity theories, these models willlead to scenarios where the Born-Infeld regime is reached again when the growing densitytrespasses the transition density M M . Some works have studied the effects of theBorn-Infeld corrections on these future singularities and found that, in general, there isnot a universal regularisation of such singularities. This might not be too surprising sincethe existence of future singularities in GR is tightly linked to exotic properties of darkenergy and, thus, the very cosmological model containing those singularities could alreadypresent pathologies. At this respect, we find fair to say that Born-Infeld inspired theoriesare entailed to regularise the Big Bang singularity with standard forms of matter, thatis radiation and/or matter. The failure in regularising more general types of singularitiesshould not be regarded as a flaw, but rather their eventual success in this task would bean additional gift granted by these theories.As we have extensively discussed, the most outstanding feature and the raison d’ˆetreof Born-Infeld inspired theories of gravity is the possibility of regularising the singularities194f GR without resorting to quantum gravity effects that should appear at the Planck scale M Pl . At this respect, we should say that the actual problem in GR is not the existence ofsingularities per se, but rather that the classical solutions near those divergences go beyondthe regime of validity of GR as an effective field theory (presumably near the Planck scale)and, thus, we cannot trust those solutions anymore. The main idea behind Born-Infeldinspired theories is to introduce a new scale M BI at which the gravitational interactionis modified so that curvature divergences are classically regularised before reaching thequantum regime. However, a proper treatment of the validity of Born-Infeld theoriesas actual effective field theories is still missing. In particular, an issue that should beclarified is the existence of some regime above M BI , that one could naively identify withthe strong coupling scale of the theory, where quantum corrections remain under controland, thus, the resulting classical solutions without singularities can be trusted. As withother open questions, perhaps the best starting point to address this issue would be theEinstein frame where all the effects are moved to the matter sector. It is not hopelessto expect a nice quantum behaviour, at least for some matter fields. For instance, ifwe start from a massless scalar field in the Born-Infeld frame, in the Einstein frame wewould have a K − essence type of theory whose Lagrangian would be of the form K ( X ), forwhich the quantum stability of the classical action has already been discussed in detail in[137, 85]. A crucial point to notice here is that for the singular free-solutions provided byBorn-Infeld theories, NEC violations are not required and, therefore, the usual argumentsfor the instability and breaking of unitarity of these solutions do not directly apply. Ingeneral, the question would be as to what extent the Born-Infeld scale determines thestrong coupling scale or the cutoff of the theory and the radiative stability for knowntypes of matter.The voyage undertaken throughout this review has permitted us to encounter an inter-esting family of gravitational theories that revealed fascinating novel effects in astrophysics,black hole physics and cosmology. They offer excellent opportunities for the explorationof the gravitational interaction and the open questions exposed above should serve, evenif not exhaustively, as a guidance for future research within the field. We hope the ac-companying traveller profited and enjoyed reading this work as much as we did in itselaboration. 195 cknowledgments We are grateful to many colleagues for useful discussions and comments regarding themany topics and results reported in this work. In particular, we thank Pedro Avelino, Ale-jandro C´ardenas-Avenda˜no, Joaqu´ın D´ıaz-Alonso, Rafael Ferraro, Franco Fiorini, SoumyaJana, Sayan Kar, Tomi S. Koivisto, Yu-Xiao Liu, Francisco S. N. Lobo, ChristopheRingeval, Diego S´aez-G´omez, Bayram Tekin, Ke Yang, and Shao-Wen Wei. J.B.J. ac-knowledges the financial support of A*MIDEX project (n ANR-11-IDEX-0001-02) fundedby the Investissements d’Avenir French Government program, managed by the FrenchNational Research Agency (ANR), MINECO (Spain) projects FIS2014-52837-P, FIS2016-78859-P (AEI/FEDER) and Consolider-Ingenio MULTIDARK CSD2009-00064. L.H.is funded by Dr. Max R¨ossler, the Walter Haefner Foundation and the ETH ZurichFoundation. G. J. O. is supported by a Ramon y Cajal contract, the grant FIS2014-57387-C3-1-P from MINECO and the European Regional Development Fund, and thei-COOPB20105 grant of the Spanish Research Council (CSIC). D. R.-G. is funded bythe Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT, Portugal) postdoctoral fellowshipNo. SFRH/BPD/102958/2014 and the FCT research grant UID/FIS/04434/2013. Thiswork is also supported by the Consolider Program CPANPHY-1205388, the Severo Ochoagrant SEV-2014-0398 (Spain), and the CNPq (Brazilian agency) project No. 301137/2014-5. J.B.J. thanks the Institute of Astrophysics and Space Sciences (IA) at Lisbon Univer-sity (Portugal) for financial support under the “IA Visitor Program” to visit the IA. L.H.thanks the STSM fellowship of CANTATA for financial support to visit the IA. BothJ.B.J and L.H. acknowledge the “Unveiling the dynamics of the Universe” group of IAfor their hospitality during the elaboration of this work. This article is based upon workfrom COST Action CA15117, supported by COST (European Cooperation in Science andTechnology).
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