Accelerated expansion as manifestation of gravity: when Dark Energy belongs to the left
UUniversidade Federal do Esp´ırito Santo
Programa de P´os-Gradua¸c˜ao em Astrof´ısica, Cosmologia e Gravita¸c˜ao
Accelerated expansion as manifestation of gravity: when Dark Energy belongs to the left
Leonardo Giani
Thesis submitted as part of the requirements for the degree ofDoctor of Philosophy in Astronomy & PhysicsSupervisor:
Prof. Oliver Piattella
Universidade Federal do Esp´ırito Santo, Vit´oria, Brasil
Co-supervisor:
Prof. Luca Amendola
Institut f¨ur Theoretische Physik der Universit¨at Heidelberg, Germany a r X i v : . [ g r- q c ] F e b niversidade Federal do Esp´ırito SantoCentro de Ciˆencias Exatas Programa de P´os-Gradua¸c˜ao em Astrof´ısica, Cosmologia e Gravita¸c˜ao
Accelerated expansion as manifestation of gravity: when Dark Energy belongs to theleft
Leonardo Giani
A presente tese “
Accelerated expansion as manifestation of gravity: whenDark Energy belongs to the left ” foi submetida por
Leonardo Giani ao PPG-Cosmo, tendo sido apresentada e aprovada no ano de 2020 como parte dosrequisitos para a obten¸c˜ao do t´ıtulo de Doutor em Astronomia e F´ısica.Comiss˜ao avaliadora:Prof. Dr. Oliver Fabio Piattella (UFES), orientador,Prof. Dr. Luca Amendola (ITP Heidelberg, Alemanha), co-orientador,Prof. Dr. Alexandr Kamenchtchik (UNIBO, Italy), examinador externo,Prof. Dr. Jorge Zanelli (CECs, Chile), examinador externo,Prof. Dr. Saulo Carneiro de Souza Silva (UFBA), examinador externo,Prof. Dr. Hermano Velten (UFOP), examinador externo,Prof. Dr. J´ulio C´esar Fabris (UFES), examinador interno. bstract
In order to explain the Late-times accelerated expansion of the Universe we must ap-peal to some form of Dark Energy. In the standard model of cosmology, the latter isinterpreted as a Cosmological Constant Λ. However, for a number of reasons, a Cos-mological Constant is not completely satisfactory. In this thesis we study Dark Energymodels of geometrical nature, and thus a manifestation of the underlying gravitationaltheory. In the first part of the thesis we will review the ΛCDM model and give a briefclassification of the landscape of alternative Dark Energy candidates based on the Love-lock theorem. The second part of the thesis is instead devoted to the presentation ofour main results on the topic of Dark Energy. To begin with, we will report our studiesabout nonlocal modifications of gravity involving the differential operator (cid:3) − R , withemphasis on a specific model and on the common behavior shared by this and similartheories in the late stages of the evolution of the Universe. Then we introduce a novelclass of modified gravity theories based on the anticurvature tensor A µν (the inverse ofthe Ricci tensor), and assess their capability as source of Dark Energy. Finally, we willdiscuss a type of drift effects which we predicted in the contest of Strong GravitationalLensing, which could be employed both to study the effective equation of state of theUniverse and to constrain violations of the Equivalence Principle. esumo Para explicar a expans˜ao acelerada do Universo tardio ´e necess´ario recorrer a algumaforma de Energia Escura. No modelo padr˜ao da cosmologia, ela ´e interpretada comouma constante cosmol´ogica Λ. Por´em, devido a v´arias raz˜oes, a constante cosmol´ogican˜ao ´e totalmente satisfat´oria. Nesta tese n´os estudamos modelos alternativos onde aEnergia escura ´e de natureza geom´etrica, e ent˜ao ´e uma manifesta¸c˜ao da teoria degravita¸c˜ao subjacente. Na primeira parte da tese revisaremos o modelo Λ e vamosdar uma breve classifica¸c˜ao do panorama dos modelos alternativos de energia escurabaseada no teorema de Lovelock. A segunda parte da tese, por outro lado, ´e dedicadaa apresenta¸c˜ao dos nossos resultados principais no t´opico da Energia Escura. Paracome¸car, n´os iremos relatar nossos estudos sobre modifica¸c˜oes n˜ao locais da gravita¸c˜aoque envolvem o operador diferencial (cid:3) − R , com ˆenfase no modelo VAAS e no compor-tamento similar compartilhado por esta classe de teorias nos est´agios finais da evolu¸c˜aodo Universo. Em seguida, n´os apresentamos uma nova classe de teorias de gravita¸c˜aomodificada que ´e baseada no tensor de anticurvatura A µν , inverso do tensor de Ricci, eavaliamos o potencial dessas teorias como fonte de Energia Escura. Para concluirmos,discutiremos um tipo de efeitos de deriva que previmos no contexto de lentes gravita-cionais fortes que poderiam ser empregados tanto para estudar a equa¸c˜ao de estadoefetiva do Universo quanto para restringir viola¸c˜oes do princ´ıpio de equivalˆencia. cknowledgments This work would not have been possible without the support and the help of a huge number ofcolleagues and professors who I had the pleasure and honour to met at the Federal Universityof Espirito Santo, and at the Institute for Theoretical Physics of Heidelberg.In particular, I owe my deepest gratitude to Prof. Oliver Fabio Piattella, who did not justcompletely fulfilled his duties as a supervisor, but has also been the strongest influence andinspiration in my scientific training. It is possible that I have disappointed him as a student,but I hope I did not as a friend. I want also to thank Prof. Luca Amendola for makingpossible my visiting stages in Heidelberg, and for letting me feel welcome and comfortablein its group. Finally, I need to thank Doctors Tays Miranda and Emmanuel Frion, and M.ScGiorgio Laverda, who actively contributed to part of the results I am going to present inthis thesis, and even more importantly have been close to me when keeping going was noteasy.The results presented in this thesis have been possible thanks to the financial support fromthe Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Fi-nance Code 001, and from the DAAD Co-financed Short-Term Research Grant Brazil, 2019(57479964). ontents
1. Introduction 12. Overview Of the Λ CDM model 3 H and σ . . . . . . . . . . . . . . . . . . . . . 152.3.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3. Dark Energy Bestiarium 18 able of Contents RT model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.6. Giving up locality and Lorentz invariance . . . . . . . . . . . . . . . . . . 243.1.6.1. Unimodular gravity . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.6.2. Hoˇrava-Lifshitz gravity . . . . . . . . . . . . . . . . . . . . . . . 253.2. Ethology of Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1. The η and Y parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.2. Linear theory of structure formation . . . . . . . . . . . . . . . . . . . . . 263.2.3. Equation of state of Dark Energy . . . . . . . . . . . . . . . . . . . . . . . 273.2.4. Variation of the electromagnetic coupling α EM . . . . . . . . . . . . . . . 27
4. Nonlocal gravity 29 RR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3.3. The VAAS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5. Personal Contribution:Nonlocal gravity 37
6. Personal Contribution:Ricci Inverse Gravity 48 R + αA − . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3.1. Evolution with Dust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3.2. Evolution with Dust and a cosmological constant . . . . . . . . . . . . . . 51 able of Contents
7. Personal Contribution:Strong Lensing for testing Gravity and Cosmology 55
8. Final Considerations 64A. Critical points in VAAS gravity 67B. Qualitative dynamic of nonlocal models 71C. Field equations in Inverse Ricci gravity 76D. Estimating Time delay Uncertainties with PyCS3 78References 83
HAPTER 1.
Introduction
The layman always means, when he says”reality” that he is speaking of somethingself-evidently known; whereas to me itseems that the most important andexceedingly difficult task of our time is towork on the construction of a new idea ofreality.
Wolfgang Pauli
This thesis is presented in candidacy for the degree of doctor of philosophy, and its main goalis to report and collect the scientific results we were able to achieve in the last four years towardsthe understanding of the fundamental nature of Dark Energy. The latter, whatever it is, becamea fundamental ingredient in our description of the Universe after the discovery, in the late 90’s, ofits accelerating expansion [1, 2]. In particular, our research focuses on those proposals in whichDark Energy is the manifestation of a different theory of gravitation, i.e. on those geometricaltheories in which no new degrees of freedom are introduced to drive the present acceleratedexpansion.However, it is a legitimate question to ask why we should consider such things when the currentstandard model of cosmology, the ΛCDM, has proven to be in excellent agreement with a numberof different observations. From our point of view, there is indeed no completely satisfactory answerto this question since the cosmological constant is the simplest and yet effective candidate ofDark Energy we can think of. It does not introduce any new physics nor changes significantly thebehavior of gravity at small scales (where General Relativity has been tested with astonishingprecision). On the other hand, there are a number of less satisfactory answers which motivatethe quest for a different description of Dark Energy.First, General Relativity is a century old and some physicists start to get bored, or at leastfrustrated, of it. There is no commonly accepted framework in which its quantization can beachieved, and it is extremely difficult to explore its properties in strong gravity regimes. Forthese reasons Cosmology, in particular at early and late times, is a fertile ground both for testingand speculating on the nature of gravitational interaction.A slightly more satisfactory reason is that in the last decade we witnessed the appearance of agrowing tension between the result of measurements from the local Universe and at early times.Part of the scientific community believe that these tensions are due to systematic, but a lot ofpeople think that they are actually indications of new physics. At the end of the day, whetherthere are good reasons for studying Dark Energy or not, it seems to us that the following quoteby Weinberg remarkably describes the situation: ”It seems that scientists are often attracted to ntroduction Section 1.0 beautiful theories in the way that insects are attracted to flowers — not by logical deduction, butby something like a sense of smell” .Motivated by the above considerations, we decided to dedicate chapters 2 and 3 of this thesisto a review of the ΛCDM model and of the landscape of Dark Energy candidates. The topicspresented there are fairly standard and already covered in many textbooks, and the expert readermay freely decide to skip them. On the other hand, in our treatment (by no means complete) wetried to privilege our personal perspectives on the topics, which ultimately provide the motivationfor the work done in the subsequent chapters. The third chapter is also introductory in nature,and aims to review the main motivations which led to consider nonlocal modifications of gravityas Dark Energy candidate, as well as some technicalities typical of this approach. The remainingchapters of this thesis contain instead a summary of our results. In chapter 5 we show someinteresting cosmological features of the nonlocal model of gravity proposed in Ref. [3], whichwe studied in Ref. [4]. Here we also try to explain, based on [5], the apparently coincidentalcommon behavior shared by different nonlocal models in the late stages of the evolution of theUniverse. In chapter 6 we introduce a novel class of modified gravity models we recently proposedin Ref. [6]. In chapter 7 we will discuss how to extract cosmological information from a noveltype of observables proposed by us in Ref. [7] in the context of Strong Gravitational Lensing,and how they can be used to test Dark Energy and the Equivalence Principle [8]. HAPTER 2.
Overview Of the Λ CDM model
Indeed, it has been said that democracy isthe worst form of government except allthose other forms that have been triedfrom time to time.
Winston Churchill
The discovery of the accelerated expansion of the Universe [1, 2] is undoubtedly one of thecornerstone of modern cosmology. After roughly two decades, it is commonly accepted thatthe best description of our universe on cosmological scales relies on the ΛCDM model. In thischapter we will give a brief introduction of the model, highlighting its agreement with the mainobservational evidences and describing the theoretical framework at his foundation. Finally, wewill conclude the chapter mentioning some open problems of the standard model. Most of thematerial presented here is covered (surely better) in many standard textbooks, and was largelyinfluenced by Refs. [9, 10], which I recommend for a detailed treatment.
One of the cornerstone that led Einstein to the formulation of General relativity is the Equiv-alence Principle. Historically, it is formulated with the statement that the gravitational andinertial mass are equivalent , and it is also a pillar of Newtonian theory of gravity. Roughly 300hundreds years after its verification by Galileo, Einstein realized that one of the consequences ofthe principle is that no static homogeneous external gravitational field could be detected fromphysics experiments performed by free-falling observers located in a sufficiently small spacetimeregion.In the context of General Relativity, a useful statement of the Equivalence Principle is thefollowing [11] : at every spacetime point in an arbitrary gravitational field it is possible to choosea locally inertial coordinate system such that, within a sufficiently small region of the point inquestion, the laws of nature take the same form as in unaccelerated Cartesian coordinate systemin the absence of gravitation . Usually one refers to the above statement as the strong EquivalencePrinciple , to distinguish it from the aforementioned equivalence between inertial and gravitationalmass, which is instead labelled as weak Equivalence Principle .Experimental tests of the Equivalence Principle, either in its weak of strong version, are of cru-cial importance for our fundamental understanding of gravity. Indeed, many alternative theoriesof gravitation result in some kind of violation of the Equivalence Principle, so that the precision verview Of the Λ CDM model Section 2.1 within which we can trust its validity can be used to rule out a certain class of models.
The main goal of a cosmological model is to describe the dynamical evolution of the Universein agreement with data. In order to relate the dynamics of the Universe to its components, atheory of gravitation is required. The ΛCDM model assumes that the appropriate descriptionof the gravitational interaction on cosmological scales is given by the Einstein Field Equations(EFE). We prefer to speak of Einstein Field Equations instead of General Relativity because, asshowed in Ref. [12], it is possible to obtain the same equations from other geometrical theoriesthat differs from GR at fundamental level.The Einstein Field Equations are: R µν − g µν R + Λ g µν = 8 πG N T µν , (2.1)where R µν is the Ricci tensor, R its trace, g µν the spacetime metric, G N the Newton’s constant,Λ the Cosmological Constant (CC) and T µν the energy momentum tensor. The Ricci tensor andscalars are construed from the Riemann tensor R µνλσ , also called curvature tensor, which satisfiesthe Bianchi identities: R µνλη + R µηνλ + R µλην = 0 , (2.2) R µνλη ; σ + R µνσλ ; η + R µνησ ; λ = 0 , (2.3)where we use the notation A µ ; ν to represent the covariant derivative of A µ with respect to x ν .It is possible to show [13, 11] using the Bianchi identities that the left hand side of Eqs. (2.1) isdivergenceless, enforcing the validity of the continuity equation T µν ; µ = 0 in curved spacetime. In 1922 the Russian mathematician Alexander Friedmann obtained an analytical solution ofEqs. (2.1) under the assumption that the spacetime is homogeneous and isotropic [14]. A similarresult was obtained independently by the Belgian astronomer George Lemaˆıtre in 1927 [15], andlater on by the American mathematician Howard Robertson [16] and the British mathematicianArthur Walker [17]. The resulting spacetime is described in terms of a metric usually denoted
FLRW , named after them. The FLRW line element can be written: ds = − dt + a ( t ) (cid:18) dr − Kr + r d Ω (cid:19) , (2.4)where Ω is the solid angle, the function a ( t ) is the scale factor and the constant K is relatedto the curvature of the spatial slices. A negative, vanishing or positive value of K correspondsrespectively to Hyperbolic, Euclidean or Spherical spatial geometry.The assumptions that at background level the Universe is homogeneous and isotropic areusually referred to as the Cosmological Principle . They are also at the core of Newtonian gravityand Galilean relativity, where they are stated as the existence of a universal time and the lackof any preferred direction in space. These definitions on the other hand are not completelysatisfactory in the contest of General Relativity, where differential geometry concepts are requiredto unambiguously define them. A very rigorous definition by Wald is the following, see Ref. [18]:
Homogeneity:
A space-time is said to be homogeneous if ∃ a family of 1-parameter ofspacelike hypersurfaces Σ t foliation such that ∀ t and p, q ∈ Σ t ∃ an isometry I : I ( p ) → q . verview Of the Λ CDM model Section 2.1
Isotropy: a space is spatially isotropic if ∃ at each point a congruence of timelike curveswith tangents u α such that ∀ p ∈ the congruence, given two s α , s α spacelike vectors orthogonalto u α ∃ I of g µν : s α → s α leaving p and u α fixed . A slightly less technical definition of the Cosmological Principle could be instead found in Wein-berg’s book [11]:
A globally hyperbolic spacetime is homogeneous and isotropic if : i) Hypersurfaces with cosmic standard time are maximally symmetric subspaces of thewhole space-time. ii) g µν , T µν and all the other cosmic tensor are form invariant with respect to the isometriesof these subspaces.We recall that a manifold is globally hyperbolic if it possesses a Cauchy surface, i.e. there existsa surface which every causal curve on the manifold crosses exactly once. Roughly speaking thismeans that a surface exists from which, once specified the initial conditions, it is possible to trackpast and future evolution of the causal curves through the field equations. A space of dimension D is maximally symmetric if it admits D ( D + 1) / isometry is a bijective function in a metric space that preserve distances.We hope the reader could forgive the latter brief technical digression on the cosmological princi-ple, but once a proper definition was given we are now able to highlight some of its consequences.First we notice that no concept from General Relativity was used to define the cosmologicalprinciple. Indeed, it is an assumption (which is well motivated from the observational point ofview, as we will discuss later) independent of the specific metric theory of gravitation we areconsidering. We also note that the existence of a preferred foliation of the spacetime in terms ofa time parameter implies the existence of a privileged class of observers, i.e. free falling observers,whose clocks measure the cosmic time. This could be misleading from the perspective of GeneralRelativity, because of general covariance and of the Equivalence Principle. The main point isthat the goal of cosmology is to describe our Universe, which is just a particular realization, orsolution, of the Einstein Field Equations (or any alternative metric theory of gravitation), andalthough the EFE are generally covariant, a particular solution of them does not have to be. Computing Eqs. (2.1) for the FLRW metric (2.4) we obtain the Friedmann equations: H + Ka − Λ3 = 8 πG N T , (2.5) g ij (cid:18) H + 2 ¨ aa + Ka − Λ (cid:19) = − πG N T ij , (2.6)where we have defined the Hubble function H ( t ) = ˙ a/a . Note that the high symmetry of thecosmological principle restricts the allowable choices of T µν . Since the FLRW metric depends onlyon time, the same must hold for the components of the energy momentum tensor. Furthermore,due to spatial isotropy, the 0 i components must vanish. Finally, since the left hand side ofEq. (2.6) is proportional to g ij the same must be true for T ij .Usually in cosmological applications we consider perfect fluids, which satisfy the above listedproperties and can be written in general as: T µν = ( ρ ( t ) + P ( t )) u µ u ν + P ( t ) g µν , (2.7) verview Of the Λ CDM model Section 2.1 where we have defined the rest energy density of the fluid ρ and the pressure P . We have alsointroduced the 4-velocity u µ , which is normalized by definition so that g µν u µ u ν = −
1. Thus,in the comoving frame, where the fluid is at rest, we have u i = 0 and u = √ | g | . Note thatEqs. (2.5),(2.6) are not completely independent; indeed, since in General Relativity the zerocomponent of the EFE is a constraint equation that contains only first derivatives, Eq. (2.5)contains only the first time derivative of the scale factor. It is possible to obtain Eq. (2.6)combining the derivative of Eq. (2.5) with the continuity equation of the fluid:˙ ρ ( t ) + 3 H ( ρ + P ) = 0 . (2.8)If we consider barotropic fluids it is possible to relate the pressure and the density through theEquation of State (EoS): P ( t ) = wρ ( t ) (2.9)where w is the EoS parameter. It is possible to obtain analytical solutions of Eqs. (2.5),(2.6) for barotropic fluids by solving thecontinuity equation Eq. (2.8). Indeed, we have:˙ ρρ = − aa (1 + w ) , (2.10)from which: ρ = a − w ) . (2.11)For most cosmological applications the parameter space for the equation of state parameter w is very simple; one usually consider pressureless non-relativistic matter with w = 0, also dubbed dust , and relativistic matter with w = 1 /
3, denoted radiation , which includes for example photonsand neutrinos. Let us consider a model of flat Universe K = 0 filled with a perfect fluid definedby Eq. (2.11). In this case we can rewrite Eq. (2.5) in terms of the scale factor only:3 ˙ a a = 8 πG N a − w ) , (2.12)which solved with respect to the scale factor gives: a ( t ) = (cid:104)(cid:112) πG N (1 + w ) t (cid:105) w ) . (2.13) Let us now focus on the evolution of the Universe for particular solutions of Eq. (2.13) relevantfor cosmological purposes. If only a species is present, it is straightforward to integrate the latterequation and obtain analytical solutions for the Hubble function. In Table 2.1 is reported thebehavior of the scale factor, the density and the equation of state parameter for a flat Universedominated by matter, radiation and Cosmological Constant. It is important to realize that in anexpanding Universe, since a ( t ) is a growing function, the density function of matter and radiationis decreasing. Thus, if the Universe contains only these species, they will eventually dilute andthe Hubble function approaches H ( t ) →
0. In these models, the Universe approaches a stableMinkowski attractor in the future. On the other hand, if a Cosmological Constant is present, inthe future the Universe reaches a stable de Sitter attractor and the scale factor starts to growexponentially. verview Of the Λ CDM model Section 2.1
Radiation Dust CC a ( t ) t / t / e Ht ρ ( a ) a − a − Λ / w / − a ( t ), the density ρ ( a ) and the EoS parameter w for a flat FLRW Universe during radiation, matter and Cosmological Constant(CC) domination. When Einstein was considering cosmological applications of its theory he had in mind a staticUniverse with ˙ a = ¨ a = 0. From Eq. (2.5), since both Λ and ρ are positive, we must impose K = 1, i.e. a closed Universe. Moreover, the acceleration equation implies: ρ + 3 P = 0 , (2.14)and since ρ is positive definite, the only possibility is that there must be something with negativepressure that compensates. This was the main motivation that brought Einstein to introduce aCosmological Constant Λ into its equations. Indeed, since ρ Λ = − P Λ we have:2 ρ Λ = ρ m , (2.15)which is a critical point of the dynamical equations. On the other hand such a point is unstable,and depending on the sign of a small perturbation the Universe evolves into a de Sitter or aMinkowski critical point. The ΛCDM model describe a FLRW flat Universe filled with a mixture of dust, in the form ofCold Dark Matter and baryons, radiation and a Cosmological Constant. As we will show later,there are observational evidences that justify the hypothesis of spatial flatness. The Hubblefunction is given by the first Friedmann equation: H ( z ) = (cid:113) Ω K (1 + z ) + Ω m (1 + z ) + Ω rad (1 + z ) + Ω , (2.16)where the Ω i are the present day densities of the species i , and where Ω m = Ω CDM + Ω baryons is the total matter. In Fig. 2.1 the Hubble function for the ΛCDM model is plotted as a functionof the redshift z = 1 /a − m ≈ .
3, Ω r ≈ − and Ω Λ ≈ .
7. If we define the normalizedenergy density of a species x as Ω x = 8 πG N ρ x Ω x / H , it is possible to rewrite the Friedmannequation in the form: 1 = Ω m + Ω r + Ω Λ . (2.17)In Fig. 2.2 are plotted the normalized energy densities in terms of the e-fold time parameter N = log a . It is straightforward to realize that the Universe evolution could be divided intodifferent epochs, during which one species is dominant with respect to the others and determinatethe rate of expansion. Since radiation dilutes the fastest it will be dominant at earlier times,followed then by matter and finally by the Cosmological Constant. In Fig. 2.3 the logarithmictime derivative of the Hubble function is given in terms of N . For N ≤ −
15, in the radiationdominated epoch, H ∼ √ ρ rad , so that H (cid:48) /H ≈ −
2. Then radiation dilutes, and around N ≈ − verview Of the Λ CDM model Section 2.2 H ( z ) Λ CDM
Figure 2.1.: The Hubble diagram as function of redshift for the ΛCDM modelits density equals the matter one. When matter starts to dominate the Hubble factor behave as √ ρ m , so that H (cid:48) /H ≈ − /
2. Finally, around today, matter dilutes and its density equals the oneof the Cosmological Constant, so that H (cid:48) /H tends asymptotically to → Λ CDM model
Observational dataset
The most important cosmological probes that support the ΛCDM model are the Cosmic Mi-crowawe Background (CMB), type Ia Supernovae observations, and Baryonic Acustic Oscillations(BAO). Combined, they favor a model of flat Universe | Ω k | ≤ .
003 where the total matter den-sity today Ω m = Ω DM +Ω b is of order Ω m ≈ . ≈ .
7, with the radiation energy density of order Ω r ≈ − [19, 20, 21]. In a FLRW background it is possible to compute the age of the Universe t by integrating theHubble function: t = (cid:90) ∞ dz (1 + z ) H ( z ) . (2.18)It is straightforward to realize that the main contribution to the above integral comes from recenttimes, i.e. small redshift z . In this regime we can neglect the contribution of radiation, whosedensity is of order 10 − . Using the first Friedmann equation to eliminate Ω m = 1 − Ω wecompute the value of the integral in Eq. (2.18) as function of the Cosmological Constant valueΩ . In Fig. 2.4 the result of the integral assuming flatness is reported in units of H − , and onecan appreciate that for small values of the Cosmological Constant, i.e. without Dark Energy, theUniverse is younger, while in the opposite limit Ω m → verview Of the Λ CDM model Section 2.2 - - - - Ω Ω m Ω r Ω Λ Figure 2.2.: The evolution of the energy densities of the species in the ΛCDM modelglobular clusters, i.e. clusters of 10 − stars with a high density of around 10 stars for ly − that share the same age and the same chemical composition, usually found in the galactic halo.These stars are remnants of galaxy formation, and are among the oldest objects observed in thesky, see Ref. [22]. It is possible to infer their age by using spectroscopic mesurements, i.e. bystudying their abundance of heavy elements. It is believed that in the early Universe there weremostly Hydrogen and Helium produced during the Big Bang Nucleosynthesis, as a result thestars that were produced at the time should lack heavy elements. The oldest globular clustersfound are dated around 11 Gyr, which correspond roughly to t H ∼ . Incorporating Λ in the total density T in Eq. (2.5) we can write: H = 8 πGρ tot − K a , (2.19)and we can define the critical density: ρ crit = 3 H πG , (2.20)which is the value of ρ such that K = 0. Observations indicate that the today total density ρ tot is of order: ρ tot = 3 H πG ∼ ρ crit ∼ − g/cm . (2.21)This means that the Universe is spatially flat and that its average density is of around 10 protonsper cubic meter. On the other hand the existence of baryonic compact objects, like us, indicates verview Of the Λ CDM model Section 2.2 - - - - - - - - H ' / H Λ CDM
Figure 2.3.: The evolution of ξ = H (cid:48) /H in the ΛCDM modelthat the Universe contains highly nonlinear regions which are uniformly distributed accordingto the cosmological principle. Baryon’s perturbations at recombination, around z = 1100, wereproportional to the CMB fluctuations which are of order 10 − . By solving the perturbations equa-tions for ρ b at linear order during the matter dominated epoch we know that matter overdensitiesgrow linearly with the scale factor. This in turn imply that today, z ∼
0, these fluctuations shouldbe of order 10 − and thus still linear. We can conclude that if only baryonic matter is present,its perturbations from the recombination would not have been in time for growing non-linearlyand form compact objects. On the other hand, if another matter species decoupled from photonsprior to recombination soon enough, it would be able to catalyze baryonic structure formation.Thus Dark Matter is a crucial ingredient for structure formation. Spiral galaxies, like the one in which we locate ourselves, are common objects in the Universe.The distribution of luminous matter is peaked in the center and, using Newtonian argumentsand assuming spherical symmetry, one expects that the centrifugal force is compensated by thegravitational attraction: v r = GM ( r ) r . (2.22)Since the mass contained within a radius r is proportional to the volume r , the velocity of thestars in the galaxy drops down as we move to higher r as v ∝ r . On the other hand observationsare not compatible with the above simple profile, see for example [23], and show instead that thevelocity of stars in the outer arms of spiral galaxies approaches a constant value. This problemis known as the flatness of velocity curve of stars , and can be explained by assuming a differentdistribution of matter from the visible one, thus invoking the presence of a “dark” matter species. During the 1998 Riess et al. [1] and Perlmutter et al. [2] realized through type Ia Supernovaeobservations that the rate of expansion of the Universe is accelerating. Supernovae are extremelybright stellar explosions which occur in the last stages of a massive star evolution or during the verview Of the Λ CDM model Section 2.2 Ω Λ H t Figure 2.4.: The age of the Universe in unit of H − for a flat FLRW Universe filled with matterand Cosmological Constant depending on the value of Ω nuclear fusion of a white dwarf. The brightness of these astronomical transient events is compa-rable with the one of an entire galaxy, and last for several weeks or months. The classification ofSupernovae is made via spectroscopic measurements and depends on which absorption lines arepresent. If there is no Hydrogen line in the spectrum they are classified as type I supernovae, andtype II otherwise. If they contain an absorption line of singly ionized silicon they are classifiedas Ia, whereas they are classified Ib if they contain Helium. Finally if they lack both Heliumand Silicon they are classified Ic. Type Ia supernovae are of the utmost importance in cosmologybecause their absolute luminosity is roughly constant at the peak of brightness. They are formedin stellar binary systems containing a white dwarf that increases its mass by absorbing gasesfrom the companion, eventually causing it to exceed the Chandrasekhar limit and triggering theexplosion. For their properties the type Ia Supernovae are called standard candles, and observingthem at various redshifts it is possible to reconstruct the cosmological evolution. Indeed, it iswell-known that the apparent magnitudes of two sources m i are related with their apparent fluxes F i : m − m = −
52 log (cid:18) F F (cid:19) . (2.23)From the apparent flux of a source and its absolute luminosity L s it is possible to define theluminosity distance d L : d L = L s π F , (2.24)finally, the apparent and the absolute magnitude of a source m and M are related as: m − M = 5 log (cid:18) d L pc (cid:19) , (2.25)i.e. the absolute magnitude of a source is defined as the magnitude that it would have if observedat a distance of 10 pc. For type Ia supernovae M is roughly constant and equal to ∼ −
19, so verview Of the Λ CDM model Section 2.3 that the apparent magnitudes m and m of two of them can be related to their distances: m − m = 5 log (cid:18) d L d L (cid:19) . (2.26)The theoretical prediction for the luminosity distance in a FLRW Universe is: d L ( z ) = c (1 + z ) H (cid:113) Ω K sinh (cid:18)(cid:113) Ω K (cid:90) ∞ d ¯ z H (0) H (¯ z ) (cid:19) , (2.27)which for a flat Universe K = 0 and small values of z can be expanded at second order as: d L ( z ) ∼ cH (cid:20) z + (cid:18) − H (cid:48) (0)2 H (cid:19) z + .... (cid:21) , (2.28)and if radiation is negligible Ω r ∼ d L ( z ) = cH (cid:20) z + 14 (cid:0) − ω DE Ω DE (cid:1) z (cid:21) . (2.29)From Eq. (2.29) it is straightforward to realize that the presence of DE, remembering that w DE <
0, pushes d L to higher values with respect to the case without it. In the Big Bang paradigm the early Universe was filled with a dense plasma of baryons electronsand photons. Baryons and electrons, which have opposite charge, interact via Coulomb forces.Photons instead bounce between electrons via Thompson scattering. As the Universe expandsits temperature drops, and electrons and baryons merge to produce hydrogen atoms in the so-called epoch of recombination , around z ∼ ∼ . ∼ − K through satellite experiments like Planck[19]. The temperature-temperature (TT) power spectrum of the CMB is usually studied inspherical harmonics and its modes are labeled with the harmonic number (cid:96) . In Fig. 2.5 is reportedthe CMB angular power spectrum for TT anisotropies, together with the theoretical predictionfor the ΛCDM model. The position and the amplitude of the peaks of Fig. 2.5 strongly constrainthe energy densities of the species in the ΛCDM model today. For example, the amount of totalmatter and the ratio of the densities can be extrapolated measuring position and amplitudes ofthe first three peaks. For a detailed description of the impact on the TT CMB power spectrumof the cosmological parameters see for example Refs. [28, 29]. Λ CDM model
Even being the most accepted paradigm to describe the cosmological evolution of the Universe,the ΛCDM suffers because of some theoretical and observational issues which lack a satisfactory verview Of the Λ CDM model Section 2.3
Figure 2.5.: The TT angular Power spectrum of the CMB as a function of the angular scale (cid:96)
The Cosmological Constant Λ is the simplest natural candidate for Dark Energy. On the otherhand, the phenomenological value required by the observations to produce the accelerated ex-pansion is quite challenging to predict from the theoretical point of view. From a quantum fieldtheory (QFT) perspective the behavior of the Cosmological Constant is at a phenomenologicallevel equivalent to the expected behavior of vacuum quantum fluctuations. Unfortunately, thevacuum fluctuations of the fields described in the standard model of particle physics would resultin a value for the Cosmological Constant which span from 123 to 55 orders of magnitude higherdepending on the scenario considered. The above incompatibility is usually referred to as the
Cosmological Constant Problem , see for example Refs. [30, 31, 32] for a detailed account of theproblem. Another problem associated with the Cosmological Constant is the so called
Coinci-dence Problem , see for example Refs. [9, 33]. The coincidence relies on the fact that the presentday energy density of DE and DM are roughly of the same order. Such an occurrence, if notexplained dynamically, would require an extremely severe fine-tuning in the initial condition ofthe Universe. Indeed, since the Cosmological Constant density is, of course, constant and theDM density dilutes as ρ m ∼ a − , in the early stage of the evolution of the Universe, say thePlanck scale for which a ∼ − , the ratio ρ Λ /ρ m would be of order ∼ − . This means thatthe initial condition for the Universe should be set with the astonishing precision of 96 digits; aone digit difference would result today in a factor 10 difference on the respective energy densities, verview Of the Λ CDM model Section 2.3 well outside the parameter space allowed by observations.
It turns out from numerical simulations of structure formation that on small scales, around ∼ kpc , and for mass scales smaller than M ≤ M (cid:12) , CDM is not completely satisfactory. Fora review on the topic, see for example Ref. [34]. One of the problems that arise in this frameworkis known as the cusp/core problem. Numerical simulations of the ΛCDM shows that DM halosshould present a steep growth of the density profile at small radius, of order ρ Halo ( r ) ∼ r γ ,with 0 . ≤ γ ≤ .
4. On the other hand several observations of small galaxies with well measuredrotation curves prefer 0 ≤ γ ≤ .
5, showing that pure ΛCDM simulations are too cuspy comparedto the observations.Another issue with CDM which appeared in the late 90’ is the
Missing satellites problem .Simulations show that DM clumps should exist in a broad range of masses and should resultsin thousands of satellite objects with mass M ≤ M (cid:12) trapped in those clumps. On the otherhand, at the time, only a bunch of these satellites were observed. The problem persisted forroughly two decades, but it seems that nowadays, with the improvement of the observations andof the numerical simulations, the missing satellites problem had been turned inside out. Indeed,in the last years astronomers found thousands of low mass objects, which could be too manycompared to those predicted by the simulations, see for example Ref. [35].Finally, a third known issue of the CDM paradigm is the Too big to fail problem. Fromobservations, it seems that galaxies fail to form in the most massive subhalos, while at the sametime satellites of lower mass form in less dense subhalos. This appears to be a contradiction,since most massive satellites should be too big to fail to form in most dense halos while smallersatellites do so in the lighter ones.
By looking at the angular spectrum of CMB alone it seems that a model of Universe with aslightly negative curvature is preferred with respect to a flat one. The situation changes if wealso include priors from CMB lensing and BAO, which carry information about H and Ω m . Theconstraints on the curvature density from Planck 2018 [19] are reported in Fig. 2.6If we assume that BAO, CMB lensing and CMB polarization data should not be combined,there is indeed space left for curvature being non-vanishing from Planck 2018 data, see alsoRef. [36]. The impact of such a point of view on the cosmological standard model was consideredby the authors of Ref. [37], which claims that our current understanding of the Universe couldbe biased and that would imply a possible crisis for cosmology. As discussed by the authors, thetendency towards a closed Universe could just be a signal of systematic, but is stronger in Planck2018 than in Planck 2015 [27], and could indicate a strong disagreement between CMB powerspectrum and BAO measurements. However, it must be noted that there is a strong degeneracyin the CMB power spectrum between the curvature Ω K and the lensing amplitude A lens . If theUniverse is closed, data favor a higher amount of Dark Matter, which in turn enhance the lensingeffect allowing for a better fit to the data at lower multipole. Whether there is a conspiracy fora flat Universe or not, the results of [37] show the kind of dangers hidden behind the corner inthe era of high precision cosmology. verview Of the Λ CDM model Section 2.3
Figure 2.6.: Constraints on a non-flat Universe from CMB angular power spectrum (dashed line),CMB + lensing (solid green), and CMB + lensing + BAO (purple region). Picturetaken from [19] H and σ The history of cosmology is strongly entangled with the history of one of its parameters, thevalue of the Hubble factor today. Indeed, while the first measurement of H by Hubble buriedthe philosophical preconceptions about a static Universe, its measure today possibly uncoversand targets the Achilles heel of the ΛCDM model.We can classify brutally most of the sources of cosmological information in two groups, i.e.measurements of early and Late-times Universe. With Late-times Universe sources we refer tothose measurements performed at low redshift, like for example type Ia supernovae or stronglensing time delays. With early time Universe measurements we refer mostly to CMB and BAOobservations. State of the art experiments seem to indicate that measurements of H from lateand early Universe are in disagreement and this tension is estimated to be significantly above4 σ [38, 39, 40]. In Fig. 2.7, taken from [39], measurements of H from different experiments andtheir combined results are reported, and the tensions quantified.Another cosmological parameter suffers from the same kind of issue, the σ parameter. It refersto fluctuations of the matter density on scales of 8 h − M pc : σ = (cid:90) ∞ dkk (cid:20) j (8 k )8 k (cid:21) ∆ ( k ) , (2.30)where j = sin ( x ) /x − cos ( x ) is proportional the first order spherical Bessel function and ∆ ( k )is the dimensionless matter power spectrum ∆ = k P m ( k ) / π . It seems to be difficult to solveboth the σ and the H tensions within the same framework. One of the reasons is that a highervalue of H could be obtained with new physics that reduces the size of the sound horizon r s with early time modifications, whilst in order to tackle the σ tension one needs to suppressthe linear power spectrum of matter at Late-times or decrease Ω m . These modifications usuallypoint towards opposite directions, making it difficult to relieve both tensions within the same verview Of the Λ CDM model Section 2.3
Figure 2.7.: The tension on the value of H arising from late and early time measurements. Figuretaken from [39].framework. Several modifications of gravity were proposed in order to alleviate the tensions, seefor example Refs. [41] and [42] where interactions between DM and neutrinos and dynamicalDE were considered, or Ref. [43] where a scalar quintessence field couples to DM, reducing itsamount and clustering at Late-times thus alleviating the σ tension. A promising new approachto the problem emerged in recent years, which tackles the tensions by employing the machineryof DE models proposed to explain the accelerated expansion of the Universe together with aCosmological Constant Λ. This was done for example in [44] and [45] for the Brans-Dicke model,fitting the data better than ΛCDM. An Occam’s razor logic makes the ΛCDM model the most successful description of the Universeas we know it. At the cost of six parameters we are able to explain a plethorae of observationscoming from a very broad landscape of physic ranging from astrophysical to cosmological scales.However, out of these 6 parameters, two are so obscure that we need to label them as dark ,and the situation is even worse when we realize that the darkness fills roughly the 95% of theUniverse. Moreover, beyond the challenging nature of DM and DE, in the past few years stateof the art observations disclose a Pandora’s box of inconsistencies of the ΛCDM model whichcosmologists are now forced to deal with, above all the cosmological tensions on H and σ . verview Of the Λ CDM model Section 2.3
With our current understanding of the Universe under siege, it is of the utmost importance tolook with fresh eyes and open mind to alternatives of the ΛCDM model. An army of scientistsgrouped in surveys is currently working on new experiments and exploring the consequences ofdifferent models. Maybe in 20 years from now they will still rely on a Cosmological Constant andCold Dark Matter to describe the evolution of the Universe, or maybe they will have the luckof witnessing the appearance of new physics. Whether this is the case or not, these are excitingtimes to live for cosmologists. HAPTER 3.
Dark Energy Bestiarium
The miracle of physics that I’m talkingabout here is something that was actuallyknown since the time of Einstein’s generalrelativity; that gravity is not alwaysattractive
Alan Guth
The goal of this chapter is to give an overview of the possible Dark Energy models beyondthe standard cosmological model, i.e. we would like to present a DE
Bestiarium . Keeping theanalogy with biology, in the first part of the chapter we will attempt to classify the models basedon their taxonomy , i.e. on how they look from the mathematical point of view. In the secondpart we will try instead to address their ethology , i.e. the way in which different DE modelsaffect observable quantities. General relativity has been proven to be, at least at some scales, our best description of thegravitational interaction. The Einstein Field Equations have the nice properties of being localand covariant differential equations of the metric and its first and second derivatives, and linearin the latter. One could ask whether there are alternatives to the EFE in order to describe theinterplay between matter and geometry of the space-time. Of the utmost importance in thisdirection is Lovelock’s theorem, see Refs. [47, 48], which can be enunciated as follows [49]:
Theorem 1
The only possible second-order Euler-Lagrange expression obtainable in a four di-mensional space from a scalar density of the form L = L ( g µν ) is: E µν = α √− g (cid:20) R µν − Rg µν (cid:21) + √− gg µν Λ , (3.1) This choice of terminology is inspired by chapter 9 of Profumo’s book [46]:
Bestiarium: A Short, BiasedCompendium of Notable Dark Matter Particle Candidates and Models In biology, taxonomy, from Ancient Greek taxis, meaning ’arrangement’, and -nomia, meaning ’method’, is thescience of naming, defining (circumscribing) and classifying groups of biological organisms on the basis of sharedcharacteristics The term ethology derives from the Greek words ethos, meaning ”character” and -logia, meaning ”the studyof”. In Biology refers to the scientific and objective study of animal behaviour ark Energy Bestiarium Section 3.1 where α and Λ are constant. Note that the above theorem is a statement about the form of the field equations resulting froma scalar Lagrangian, not a statement about the form of the Lagrangian itself, which then couldbe different from the standard Einstein Hilbert action.To rephrase it in other words, Lovelock’s theorem states that if we want a geometrical theory ofgravity in 4 dimensions arising from a scalar Lagrangian of the metric, the only possibility are theEinstein field equations plus a cosmological constant. The importance of the above result is thatit clearly indicates which kind assumptions we have to relax in order to obtain a gravitationaltheory different from general relativity plus a cosmological constant. Indeed the only options leftare: • Increasing the number of degrees of freedom, i.e. considering other fields together with themetric tensor g µν . • Include higher order derivatives in the field equations • Consider spaces with dimension N (cid:54) = 4. • Giving up Lagrangian formulations • Abandoning locality and Lorentz invariance of the field equations.
The most general Lagrangian containing a tensor field and a scalar field which gives secondorder equation of motion was discovered by Horndenski in 1974 in Ref. [50]. Later on, it wasrediscovered in the context of the so called generalized Galileon theories, see for example Ref. [51],and the equivalence between the two theories was shown in Ref. [52]. The general form of theHorndeski Lagrangian can be written : L ( g µν , φ ) = G ( φ, X ) + G ( φ, X ) (cid:3) φ + G ( φ, X ) R + G ,X ( φ, X ) ( (cid:3) φ − φ µν φ µν ) + G ( φ, X ) φ µν R µν − G ,X (cid:104) ( (cid:3) φ ) − (cid:3) φφ µν φ µν + 2 φ µν φ νλ φ µλ (cid:105) , (3.2)where X = ∇ µ φ ∇ µ φ/ G i,X = ∂ X G i and φ µν = ∇ µ ∇ ν φ . By taking appropriately the freefunctions G i of the above Lagrangian one is able to reproduce any second order scalar tensortheory as a specific case. Choosing G = M Pl and G i = 0 for i (cid:54) = 4 reproduces the Einstein Hilbertaction. The function G can account for any free Lagrangian of the scalar field, for examplequintessence. Note that in Eq. (3.2) the functions G and G must have an X dependence,otherwise they can be absorbed into G and G up to a total derivative. Note also that generalLagrangians of the Ricci scalar, i.e. f ( R ) theories, belong to the Horndeski family since they canbe cast in a scalar tensor form by defining φ = df /dR and performing a Legendre transformationof the action functional. The same apply for other geometrical theories which result in secondorder equation of motion; for example also a non minimally coupled Gauss-Bonnet is containedin the Hordenski Lagrangian [52]. ark Energy Bestiarium Section 3.1 Another option is to increase the number of degrees of freedom by means of a vector field.The main advantage of this approach over a multi-scalar field theory is that it generally resultsin a richer dynamics. This family of theories is called generalized Proca theories, and wererecently proposed in Ref. [53]. Previous attempts of introducing vector fields in a gravitationalcontext were made already in the 2000’s, with the goal of modelling anisotropic Dark Energy, seeRefs. [54, 55]. The Lagrangian of the generalized Proca theories is: L GP = − F µν F µν + G ( X ) + G ( X ) ∇ µ A µ + G ( X ) R + G ,X (cid:104) ( ∇ µ A µ ) + c ∇ ρ A σ ∇ ρ A σ + (1 − c ) ∇ ρ A σ ∇ σ A ρ (cid:105) + G ( X ) G µν ∇ µ A ν − G ,X (cid:104) ( ∇ µ A µ ) − d ∇ µ A µ ∇ ρ A σ ∇ ρ A σ − − d ) ∇ µ A µ ∇ ρ A σ ∇ σ A ρ +(2 − d ) ∇ ρ A σ ∇ γ A ρ ∇ σ A γ + 3 d ∇ ρ A σ ∇ γ A ρ ∇ γ A σ ] , (3.3)where F µν = ∇ µ A ν − ∇ ν A µ is the Maxwell tensor and X = A µ A µ /
2. The G i ’s are free functionsof the Proca field and G µν is the Einstein tensor. Note that in the above Lagrangian the usual U (1) symmetry of the vector field is broken, i.e. A µ is not Abelian. The above property becomesuseful and interesting for cosmological implications, indeed it allows for an isotropic backgroundevolution, see for example Ref. [56]. It is interesting to note that in this class of theories, on FLRWbackground, the vector field equation allow for constant solutions which are of de Sitter type,thus being potentially capable of describe the Late-times accelerated expansion of the Universeas well as an inflationary epoch. It is possible to construct a consistent covariant theory which combines together scalar, vectorand tensor interations, known as SVT theories, see Ref. [57]. The resulting theory is richer thana theory built just from the Horndeski and generalized Proca theories. Indeed, it allows for thevector and the scalar fields to interact in non-trivial way. Thus, beyond the standard scalars ofHorndeski and Proca theories, the free functions appearing in the SVT Lagrangian depend alsoon the scalars: X = − A µ ∇ µ φ , Y = ∇ µ φ ∇ ν φF µα F να , Y = ∇ µ φA ν F µα F να . (3.4)In the full SVT Lagrangian we also have terms containing the double dual Riemann tensor: L µναβ = (cid:15) µνρσ (cid:15) αβγδ R ρσγδ , (3.5)where the (cid:15) ’s are the Levi Civita symbols in 4 dimensions.The general form of the Lagrangian is given in Ref. [57] and the background and perturbedequations are developed in Ref. [58]. They depend in general on whether the U (1) invariance ofthe Proca field is broken or not. It turns out that the resulting theories are useful in Dark Energyapplications and bouncing scenarios, allowing for an accelerated epoch of expansion while beingcapable of producing transient contracting phases, which can avoid the appearance of cosmologicalsingularities. ark Energy Bestiarium Section 3.1 It is also possible to increase the number of degrees of freedom by introducing a new metrictensor field f µν . This class of theory is usually known as bimetric gravity and it was formulatedas an attempt to generalize massive gravity models, see Refs. [59, 60, 61]. The action functionalin the original formulation takes the form: S = M g (cid:90) √− gd xR g + M f (cid:90) (cid:112) − f d xR f + M eff m (cid:90) √− gd x L int , (3.6)where M i and R i are the Planck masses and Ricci scalars relative to the metric f and g . M eff = (cid:16) M − g + M − f (cid:17) − is the effective Planck mass and m a mass term associated to the massivegraviton of the metric f . The interaction Lagrangian is given by: L int = 12 (cid:0) X − X µν X µν (cid:1) + c (cid:15) µνρσ (cid:15) αβγσ X µα X νβ X ργ + c (cid:15) µνρσ (cid:15) αβγδ X µα X νβ X ργ X σδ , (3.7)where X µν = δ µν + γ µν , with γ µν defined by the relation γ µσ γ σν = g µσ f σν . The above form of the inter-action Lagrangian has been proposed by the authors to avoid the appearance of Boulware-Deserghost in both the g and f metrics. Bimetric models offer a rich and interesting phenomenologyfor cosmological implications and have been studied extensively in the literature, see for exampleRefs. [62, 63] It is well known that General Relativity is not renormalizable from the point of view of QFT dueto the fact that the Newton constant has dimensions of an inverse squared mass G N ∝ m − . Oneof the main historical reasons to consider higher order derivative theories is that they modifythe propagator improving its UV behavior. Let us consider for example the introduction of a R µν R µν term; in this case the propagator can be symbolically written as:1 k + 1 k G N k k + 1 k G N k k G N k k + ... = 1 k − G N k . (3.8)The propagator of Eq. (3.8) is dominated at high Energy by the k − term and its UV behavioris improved. On the other hand we can rewrite it as:1 k − G N k = 1 k − k − /G N , (3.9)from which we can see that it decomposes in the standard graviton mode k − together with the − / (cid:0) k − G − N (cid:1) mode, which has negative sign and thus corresponds to a ghost.The appearance of ghost modes is a recurring theme in higher order derivative theories and isstrongly related to Ostrogradsky instability, see Refs. [64, 65]. To briefly illustrate how Ostro-gradsky instability works let us consider a non degenerate Lagrangian containing second orderderivatives L ( x, ˙ x, ¨ x ). The associated Euler Lagrange equations are: ∂ L ∂x − ddt ∂ L ∂ ˙ x + d dt ∂ L ∂ ¨ x = 0 , (3.10)and the non degeneracy conditions implies ∂ L ∂ ¨ x (cid:54) = 0. Ostrogradsky showed that if we choose thefollowing 4 canonical coordinates: Q = x , Q = ˙ x , P = ∂ L ∂ ˙ x − ddt ∂ L ∂ ¨ x , P = ∂ L ∂ ¨ x , (3.11) ark Energy Bestiarium Section 3.1 it is possible to perform a Legendre transformation and obtain the Hamiltonian: H = P Q + P A ( Q , Q , P ) − L ( Q , Q , P ) , (3.12)where the function A ( Q , Q , P ) is obtained inverting Eqs. (3.11) for ¨ x . The Hamiltonian (3.12)is linear in the canonical momentum P and thus is unbounded from below, i.e. the system isunstable. Note that the only assumption made here is invertibility of the Lagrangian with respectto ¨ x , so in order to have well-behaved higher order derivative theories we should consider onlydegenerate Lagrangians. An interesting example of a degenerate theory is given by the beyondHorndeski Lagrangian, also known as Degenerate Higher Order Scalar Tensor (DHOST) theories,see Refs. [66, 67] for a detailed discussion. A similar construction can be done also for vectortensor theories, and we end up with the beyond generalized Proca theories, see Ref. [68]. A way out from Lovelock’s theorem that was explored by Lovelock himself is to consider ageometrical description of the gravitational interaction in more than four dimensions. On theother hand, we have no observational evidence of the presence of such extra dimensions, thushigher dimensional theories need a mechanism that hides or compactifies these dimensions atscales which do not emerge in standard experiments. Higher dimensional theories have attracteda lot of interest in the past decades, with the most popular being probably string theory. To givea flavour of the potential of higher dimensional theories, we will discuss briefly here the Kaluza-Klein model, which inspired and motivated subsequent works on compactifications of higherdimensions and unifications of the fundamental interactions. We will also introduce the Lanczos-Lovelock gravity, which is essentially a generalization of General Relativity to an arbitrary numberof dimensions.
The first higher dimensional extension of General Relativity was suggested by Kaluza in Ref. [69],in a similar framework as in a previous attempt by Nordstrom [70]. In this model it is possibleto obtain both Maxwell and Einstein equations in four dimensions from a geometrical vacuumtheory with a fifth dimension. The five dimensional metric, ˜ g ab , becomes here a function of thestandard four dimensional metric g µν plus a vector field A µ and a scalar field φ , and could bewritten as: ˜ g ab = (cid:18) g µν + φ A µ A ν φ A µ φ A ν φ (cid:19) . (3.13)Kaluza imposed on the metric ˜ g ab the so-called cylinder condition, i.e. that it does not depend onthe fifth coordinate ∂ ˜ g ab /∂x = 0. The five dimensional vacuum Einstein field equations reduceto: R µν − g µν R = k φ T EMµν − φ [ ∇ µ ( ∂ ν φ ) − g µν (cid:3) φ ] , (3.14)while the Maxwell field equations are: ∇ µ F µν = − ∂ µ φφ F µν . (3.15)The so called Kaluza’s miracle is that the standard four dimensional Einstein and Maxwell fieldequations, with the electromagnetic term appearing in the former as a source term, are recovered ark Energy Bestiarium Section 3.1 in the limit φ = 1. However, the latter condition is not consistent with the Klein Gordon equationfor the scalar field: (cid:3) φ = k φ F µν F µν . (3.16)A lot of criticism was made to Kaluza’s proposal due to the cylindrical condition, i.e. theintroduction of a fifth dimension that plays no role in the dynamics. To overcome this issue,Klein suggested in Ref. [71] a mechanism of compactification of the fifth dimension, demandingthat it has the topology of a circle S of very small radius r . Thus, the whole spacetime hastopology R × S and physical fields must depend on the fifth dimension only periodically. The topological space’s shape of an object is identified by a constant number ξ , called Eulernumber, or Euler characteristic, regardless of the way in which the space is bent. The Eulercharacteristic in 2 n dimensions can be written as the integral of the Euler density R n whichreads: R n = (2 n !)2 n δ µ [ α δ ν β δ µ α δ ν β ...δ µ n α n δ ν n β n ] n (cid:89) r =1 R α r β r µ r ν r , (3.17)where the square bracket indicate antisymmetrization. The Lovelock Lagrangian is the sum ofthe Euler densities: L = √− g n (cid:88) t =0 α t R t , (3.18)and yields to conserved second order Euler Lagrange equations of motion, see for exampleRefs. [47, 72] for a detailed derivation. Expanding the above Lagrangian up to second orderwe obtain: L = √− g (cid:2) α + α R + α (cid:0) R + R µνρσ R µνρσ − R µν R µν (cid:1) + O ( R ) (cid:3) , (3.19)which shows that at zero and first order the Lovelock Lagrangian reproduces the standard EinsteinHilbert action plus a cosmological constant, while from the second order term inside the bracketwe appreciate that it contains the Gauss-Bonnet gravity term. Note that in four dimensions thesecond and higher order terms become trivial and we are left with standard GR. Several proposals of modified gravity are based on ad hoc modifications of the EFE which arenot derivable from an action functional, often with interesting cosmological applications. Toillustrate the potential of this kind of modifications we will briefly present two theories belongingto this class, the Rastall gravity and the RT nonlocal model. Following the idea that the stress energy tensor T µν could be not conserved in curved spacetime,Rastall proposed in Ref. [73] the following modification of the Einstein field equations: R µν − g µν R = 8 πG (cid:18) T µν − γ − g µν T (cid:19) , (3.20) ark Energy Bestiarium Section 3.1 with a non conserved continuity equation for T µν : T µν ; µ = γ − T ν . (3.21)It has been shown that Rastall gravity is very interesting from the cosmological point of view,being able to reproduce ΛCDM at the background level, see for example Ref. [74], while beingdifferent at perturbative level and resulting in a type of Dark Energy capable of clustering.It is a matter of debate if it is possible or not to derive the Rastall equations from a Lagrangiandensity. In the 80’, in Ref. [75], the Rastall field equation where obtained by a variational principleof a Lagrangian density, but the latter was not a scalar Lagrangian and thus the derivation isnot completely satisfactory. Some more recent attempts were made in Refs. [76, 77], where thefield equations were obtained as a particular case of an f ( R, T ) theory of the type f ( R, T ) = f ( R ) + f ( T ), or from a matter Lagrangian non minimally coupled to gravity. However, somecriticism emerged since for f ( R, T ) theories of this type has been claimed that the f ( T ) typeterm should be included in the matter Lagrangian and not in the gravitational part, see forexample Ref. [78]. It was also suggested in Ref. [79] that Rastall gravity is equivalent to generalrelativity and Rastall’s stress–energy tensor corresponds to an artificially isolated part of thephysical conserved one. This point of view, however, was criticized in Ref. [80] and the debate isstill open. RT model The RT model was proposed in Ref. [81] and consist of a nonlocal modification of the EFEinvolving the inverse d’Alembertian of the Ricci scalar (cid:3) − R : R µν − g µν R − m (cid:0) g µν (cid:3) − R (cid:1) T = 8 πG N T µν , (3.22)where the superscript T denotes the extraction of the transverse part, which is in itself alreadya nonlocal operation.We will discuss in detail nonlocal modifications of gravity in chapter 4, for the moment wewill just mention that the within the RT model one is able to reproduce a viable cosmologicalhistory both at background and perturbative level. Contrary to other nonlocal models withsimilar features, the RT model is also compatible with experiments at Solar System scales, inparticular Lunar Laser Ranging constraints [82], making it very appealing despite the lack of aLagrangian formulation. Another class of theories that escapes Lovelock’s theorem is based on modifications of gravitywhich include nonlocal terms or which broke explicitly Lorentz invariance. We will discuss indetail the former in Sec. 4, while we present here as prototypical examples of the latter class oftheories the Unimodular and the Hoˇrava-Lifshitz gravities.
The ideas behind Unimodular Gravity (UG) are almost as old as GR itself, and were consideredby Einstein already in Refs. [83, 84]. From the mathematical point of view, UG is equivalent tostandard GR with the following gauge choice, called Unimodular condition: √− g = (cid:15) , (3.23) ark Energy Bestiarium Section 3.1 where (cid:15) is a fixed scalar density which provides a fixed volume elements. Thus, UG is essentiallyGR with less symmetry, being invariant only with respect to the restricted group of diffeomor-phisms respecting the Unimodular condition. The interesting property of UG is that, at classicallevel, its field equations coincide with the traceless EFE. Then, taking into account them to-gether with the Bianchi identities, one obtain the standard EFE with a cosmological constantappearing as an integration constant. Quantum corrections to the energy-momentum tensor ofmatter which are of the form Cg µν , where C is a constant over spacetime, do not contribute tothe traceless EFE. In particular, vacuum fluctuations in the trace of the energy-momentum ten-sor of matter do not affect the metric. With the latter interpretation the cosmological constantdoes not couple to gravity, and consequently UG solves the cosmological constant problem, seeRef. [30]. Several generalizations of UG have been proposed, see for example Ref. [85], wherethe right hand side of Eq. (3.23) is equal to the divergence of a vector density field, or Ref. [86]where an ADM decomposition of the spacetime is assumed with the requirement that the lapseis a function of the determinant of the spatial metric N = N ( γ ) only. Of course UG and itsgeneralizations differ from GR at a quantum level, and their quantization is an active field ofresearch, see for example Refs. [87, 88, 89, 90, 91]. The model was suggested in Ref. [92] as a viable candidate of quantum gravity, and is inspiredby physics of condensed matter systems. Its characteristic feature is that space and time aretreated at fundamental level on a different ground, in such a way that they scale anisotropicallyin the UV limit. The degree of anisotropy between space and time is measured by the anisotropicparameter z , and the resulting theory is power counting renormalizable for certain values of z .The starting point of the construction is that the line element has the following ADM shape: ds = − N dt + g ij (cid:0) dx i + N i dt (cid:1) (cid:0) dx j + N j dt (cid:1) , (3.24)in which N is the shift, N i the lapse and g ij is the spatial metric. In GR we have the gaugefreedom of representing the line element in this way foliating the space-time in terms of spacelikesurfaces Σ t , whilst in Hoˇrava gravity the above decomposition is not just a choice of coordinatesbut rather the fundamental structure of the spacetime. The kinetic term of the action is givenby: S kin = 1 g K (cid:90) d xd t N √ g (cid:2) K ij K ij − λK (cid:3) , (3.25)where the main difference with respect to standard GR is in the constant parameter λ , whichmust be unity if we demand Lorentz invariance. The potential part of the action, due to theanisotropic scaling, allow for the presence of higher order derivative terms of the spatial Riccitensor P ab , defined in terms of the spatial metric g ab . To achieve power counting renormalizabilityin 3+1 dimension we need z = 3, which implies that we can have term up to cubic order in the3D Ricci tensor and its spatial derivatives. The specific form of the potential depends on theformulation of Hoˇrava gravity we are considering. In the original formulation of Ref. [92] it isgiven by: V HL = − g K ω C ij C ij + g k µ ω (cid:15) ijk P il ∇ j P lk − g K µ P ij P ij + g K µ − λ ) (cid:18) − λ P + Λ P − (cid:19) , (3.26) ark Energy Bestiarium Section 3.2 where C ij is the Cotton tensor and g k and ω are coupling parameters. At long distances thispotential is dominated by the last two terms, the cosmological constant and the spatial curva-ture, and the theory flows in the infrared to z = 1 so that Lorentz invariance is accidentallyrestored. There is a very interesting phenomenology arising from Hoˇrava gravity in cosmologicalapplications. It has been shown that certain choices of the potential are able to mimic DM, seeRef. [93]. It is also possible to seed cosmological perturbation without inflation, see Ref. [94],realizing bouncing scenarios, see Ref. [95], and model Dark Energy, see Refs. [96, 97, 98]. As we saw in the previous section, there is a theoretically broad landscape of Dark Energycandidates. Thus, it is of the utmost importance to have a framework in which to study theimpact of each particular theory on cosmological or astronomical observables. The standardapproach consist of studying the specific form that a bunch of observed parameters takes in amodified gravity model and compare it with experimental data. η and Y parameters A given theory of DE which allows for a background compatible with the accelerated expansionof the Universe will have, at perturbative level, some impact on smaller scales. For cosmologicalimplications one is usually interested only in scalar perturbations, and it is convenient to workin the Newtonian gauge: ds = − (1 + 2Ψ) dt + a ( t ) (1 + 2Φ) δ ij dx i dx j , (3.27)where Ψ and Φ are two scalar functions, i.e. the two gravitational potentials. One should studythe perturbed modified EFE for this metric and compare with the ones of standard GR. Formany purposes it is useful to work in the quasi static approximation (QSA), i.e. within theassumptions that spatial derivatives dominate over time ones. This approximation is valid onlyon scales well inside the Hubble Horizon, k/aH (cid:29)
1, see Ref. [99] for a detailed discussion aboutthe scope of validity of the QSA. From the modified EFE we obtain the two generalized Poissonequations in Fourier space for the potentials Ψ and Φ in the case of pressureless matter: k Φ = Y ( k, t ) η ( k, t ) ρ m ( t ) δ m ( k, t ) , (3.28) k Ψ = Y ( k, t ) ρ m ( t ) δ m ( k, t ) , (3.29)where we have defined the anisotropic stress parameter η = − Φ / Ψ and the Y parameter, whichdescribes an effective gravitational coupling G eff and measures deviations from the Newtonconstant G N for matter. Both these parameters can be constrained by observations; for example η has been constrained to be η ≤ − on solar system scales by the Cassini spacecraft, seeRefs. [100, 101, 102]. It is also possible to constrain Y and its time derivative at various scales,see for example Refs. [103, 104, 105, 8]. As we saw before, several Dark Energy models can be expressed in the form of a scalar-tensortheory, i.e. they belong to the general class of Horndeski theories. Through an effective fieldtheory approach (EFT) for the Horndeski theories, it was shown in [106] that the cosmologicalinformation about linear perturbation theory can be encoded in four parameters: ark Energy Bestiarium Section 3.2 • α K is a parameter related to the kinetic energy of the scalar field and to it contribute allthe G i functions of the Hordenski Lagrangian (3.2). It is also called Kinecity • α B is a parameter related to the clustering properties of DE. It is also called Braiding andcomes from the mixing of the kinetic terms of both the metric and the scalar field. To itcontribute the functions G , G and G . • α M encodes the effects of a varying effective Planck mass M and generates anisotropicstress. To it contribute G and G • α T is related to the velocity of propagation of tensor modes. It leads to the emergenceof anisotropic stress by modifying the Newtonian potential Ψ even in absence of scalarperturbations. To it contribute both G and G .A similar EFT approach could be applied also to Generalized Proca Theories, see Eq. (3.3),and SVT theories.As an example of the capability of the method, let us consider the measurements of α T madepossible from the event GW170817 and its electromagnetic counterpart. Since no significant de-viation on the velocity of the gravitational waves was detected with respect to the value predictedby GR, the observation suggests α T = 0. This in turns implies G = const and G = 0, thusruling out roughly half of the Hordenski theories, see Refs. [107, 108, 109, 110, 111]. The sameapplies for the G and G function of generalized Proca and SVT theories. There is however stilla caveat in the above argument, which relies on the fact that the event GW170817 comes froma fairly close distance, and thus we only got information about the value of α T from Late-timesobservations. It was showed in Ref. [112] that it is possible to have a class of theories, whichexhibits scaling behavior, capable of reaching dynamically an attractor solution compatible with α T = 0. We do know that the equation of state parameter w DE of Dark Energy in the case of a cosmologicalconstant behave as the one of vacuum energy, and has the value w Λ = −
1. Current observationsare compatible with this value, but do not exclude a wider parameter space with enough accuracy.It must be stressed however that a measure of w DE alone cannot tell us too much about thefundamental nature of Dark Energy, see for example Refs. [113, 114]. On the other hand, aprecise measurement of w DE could be used to rule out particular Dark Energy candidates. Forexample, a measurement of w (cid:54) = − DE models predict a value for w DE > −
1, thus a measurement in thisdirection would not be particularly enlightening about the nature of DE. On the other hand, astatistically significant measure of w DE < − phantom , it is indeedassociated to the fact that either gravity is not minimally coupled to matter, or that DE is not aperfect fluid which can interact with other species. Indeed, it is a well known fact that a perfectfluid or a minimally coupled scalar field in a phantom regime would carry ghost or gradientinstabilities, see Ref. [115]. α EM As we mentioned before for the case of the Newton constant G N , some models of Dark Energyresult in a violation of the equivalence principle. The specific case of a possible variation of the ark Energy Bestiarium Section 3.2 fine structure constant was discussed by Bekenstein in the pioneering work [116]. Bekenstein,at the time, concluded that tests of the equivalence principle rule out spacetime variability of α EM at any level. In the last decades, however, huge improvements have been made from theexperimental point of view, leading to tight constraint on the variation of α EM , see for exampleRefs. [101, 117, 118, 119, 120], and claims of statistical evidences of α EM variations, see forexample Refs. [121, 122]. For the above reasons, the subject is nowadays very popular anda violation of the equivalence principle could potentially confirm or rule out several alternativetheories of gravity [123]. One could in general distinguish between variations on the value of α EM on large or local scales. On local scales the variation of α EM is related to the local gravitationalfield, see for example Refs. [124, 125, 126]. On cosmological scales these could be motivated by amodification of the gravitational theory due to the dynamical behavior of the DE field and wereextensively studied in the literature, see for example Refs. [127, 128, 129, 130, 131, 132, 133, 134,135]. It is important to note that the connection between DE and the variation of α EM is of greatimportance from the observational point of view, since it is indeed possible to relate constraintson ∆ α to constraints on DE parameters, see for example Refs. [136, 137, 138]. HAPTER 4.
Nonlocal gravity
I am a Quantum Engineer, but onSundays I Have Principles.
John Stewart Bell
Most of our research in the last years focused on nonlocal modifications of gravity as DEcandidates. We mention this class of theories in the previous chapter, since giving up from localityis a possible escape route from Lovelock’s Theorem. Amongst the possible choices of nonlocalmodifications, particularly interesting for DE applications are those which introduce the inversed’Alembert operator acting on the Ricci scalar (cid:3) − R . In this chapter we will briefly review thisparticular branch of modified gravity models, with emphasis on their fundamental motivationsand their general features. In particular, we will introduce the Deser Woodard (DW), the RR and the VAAS nonlocal models. The first two are amongst the most popular and analyzedmodels belonging to this class, while the latter has been proposed recently in Ref. [3]. Nonlocalities emerged from quantum mechanics already in the early stage of its formulation,see for example Ref. [139], and Ref. [140] for a nice historical review. Phenomena observedexperimentally like the quantum entanglement and the Ahronov-Bohm effect show indeed thatan effective description of quantum mechanics, or rather of reality, must be nonlocal. As itis widely known, it is difficult to construct a consistent theory of quantum gravity startingfrom GR, which in order to be renormalizable requires the introduction of infinite counterterms.Thus, just like the Fermi description of the weak interaction, one could think that GR is just aneffective geometrical description of a most deep underlying theory, and it makes sense to lookfor phenomenological modifications of the EFE that arise from quantum effects. The idea thatthe latter could be used to explain the nature of DE or other open problems of the ΛCDM isparticularly intriguing, and the
Leitmotiv of many nonlocal theories of gravity. The standardrecipe is to postulate an ansatz for the functional form of the nonlocal modification motivatedby fundamental physics. Then one studies the phenomenology of the modification at backgroundand perturbative level and test it against observations. The RR model takes its name from the structure of its Lagrangian term R (cid:3) R . This model takes its name from the initials of authors Vardanyan, Akrami, Amendola and Silvestri. onlocal gravity Section 4.1 Nonlocal effects could naturally arise when we move from the classical action functional of a givenfield theory to its quantum effective action. We will briefly review here the construction of thequantum effective action and its generalization to curved background following Ref. [141].To begin with, consider a scalar field ϕ with classical action S [ ϕ ] in a flat space of dimension D . Once we introduce an auxiliary, classical source J ( x ) we can define the generating functionalof the connected Green’s function W [ J ]: e iW [ J ] = (cid:90) Dϕe iS [ ϕ ]+ i (cid:82) Jϕ , (4.1)where the path integral measure Dϕ denotes integration over all the possible configurations ofthe field ϕ and (cid:82) J ϕ is a shortcut for the integral (cid:82) d D xJ ( x ) ϕ ( x ), so that the spatial dependenceof the source has been integrated out. Functional variation of W [ J ] with respect to the sourcegives the vacuum expectation value of the field ϕ in presence of a source, i.e.: δW [ J ] δJ ( x ) = (cid:104) | ϕ ( x ) | (cid:105) J ≡ φ [ J ] , (4.2)where we have defined the scalar field φ to indicate the vacuum expectation value of ϕ as functionof the source J . The quantum effective action Γ[ φ ] is a function of the vacuum expectation valueand is defined as the Legendre transform:Γ[ φ ] ≡ W [ J ] − (cid:90) φJ [ φ ] , (4.3)where J [ φ ] is obtained by inverting Eq. (4.2). By varying the quantum effective action Γ weobtain: δ Γ[ φ ] δφ ( x ) = − J ( x ) , (4.4)where the implicit spatial dependence of φ [ J ( x )] has been exploited. As we can see, since on theright hand side of Eq. (4.4) we have the source J ( x ), the variation of the quantum effective actiongives directly the equation of motion for the vacuum expectation value of the field. A useful pathintegral representation of the quantum effective action is: e i Γ[ φ ] = e iW [ J ] − i (cid:82) φJ = (cid:90) Dϕe iS [ ϕ + φ ] − i (cid:82) δ Γ[ φ ] δφ ϕ , (4.5)which explicitly shows that the quantum fluctuations of the field ϕ have been integrated outin the quantum effective action Γ[ φ ], which is instead a functional of the vacuum expectationvalue and the source only. It is straightforward to generalize the above construction on curvedbackground described by a metric g µν using a semi-classical approach, i.e. treating the metricat a classical level while the other fields as quantum objects. The representation of quantumeffective action then becomes: e i Γ[ g µν ,φ ] = e iS EH (cid:90) Dϕe iS m [ g µν,φϕ ] − i (cid:82) δ Γ [ gµν,φ ] δφ ϕ , (4.6)from whose variation we obtain the semi-classical EFE G µν = (cid:104) | T µν | (cid:105) . onlocal gravity Section 4.2 To understand which kind of nonlocal modifications could appear in the quantum effective actionlet us consider the case of Quantum Electrodynamics (QED). The quantum effective action takesthe form, see for example Ref. [142]: Γ QED = − (cid:90) d x (cid:20) F µν e ( (cid:3) ) F µν + O (cid:0) F (cid:1)(cid:21) . (4.7)where e ( (cid:3) ) is called form factor . When the electron mass is small with respect to the relevantenergy scale we have: 1 e ( (cid:3) ) (cid:39) e ( µ ) − π ) log (cid:18) − (cid:3) µ (cid:19) , (4.8)where the µ is the renormalization scale and e ( µ ) the renormalized charge. Nonlocality emergesbecause of the logarithm of the d’Alembert operator (cid:3) . It is defined by its integral representation:log (cid:18) − (cid:3) µ (cid:19) = (cid:90) ∞ dm (cid:20) m + µ − m − (cid:3) (cid:21) , (4.9)where nonlocality emerges due to the appearance of the (cid:3) − operator. In the above examplewe have explicitly shown how from the classical Lagrangian of QED we could obtain nonlocalcontributions due to the running of the coupling constant. The main character of the nonlocal class of theories we are considering here is the inverse of thed’Alembert operator acting on the Ricci scalar (cid:3) − R . In order to perform calculations involvingthis quantity an extremely useful trick was developed in Ref. [143], which makes it is possibleto cast these models in the form of a scalar tensor theory. The starting point is to define anauxiliary scalar field U = − (cid:3) − R whose Klein Gordon equation immediately follows: (cid:3) U = − R . (4.10)Thus a general Lagrangian density containing an arbitrary function f ( (cid:3) − R ) could be rewrittenas: L NL = f ( U ) + λ ( (cid:3) U + R ) , (4.11)where we introduced the Lagrange multiplier λ . If negative powers of the d’Alembert operatorsappear in the original action, the above procedure can be iterated and the theory is mapped ina multi-scalar tensor theory. For example, if the Lagrangian contains a term (cid:3) − R , as in themodel proposed in Ref. [144], it can be localized by introducing 4 coupled auxiliary fields withtheir respective Lagrange multipliers: (cid:3) U = − R , (cid:3) S = − U , (cid:3) Q = − , (cid:3) L = − Q . (4.12)It is important to properly carry on the procedure of localization without introducing modifi-cations of the original theory. The equivalence between the two formulations was debated after When we integrate out the quantum fluctuations of the electron and restrict ourselves to terms involving thephoton field only for simplicity onlocal gravity Section 4.3 the papers [145, 146] and [147], which were analyzing structure formation in the DW model andobtained initially different results. It turns out that the analysis of Ref. [145] was not correct,but the arising discussion about the localization procedure helped to outline its possible stabilityissues and the appearance of ghosts [148, 149] The introduction of auxiliary fields naturally rises the question of whether there are or not newdegrees of freedom generated by the nonlocal operator (cid:3) − R . The question is subtle and somecare must be taken in the procedure of localization. Moreover, by looking at Eq. (4.10), it isstraightforward to realize that the kinetic energy of the auxiliary field would be of negative sign,i.e a ghost.To properly understand this point, let us consider the Lagrangian of a massive Proca field: L P roca = − F µν F µν − m A µ A µ , (4.13)which describe a massive boson, with the U (1) gauge invariance broken by the mass term. It hasbeen shown in Ref. [150] that the above Lagrangian is equivalent to the following nonlocal butgauge invariant one: L = − F µν (cid:18) − m (cid:3) (cid:19) F µν . (4.14)If we now proceed with the localization procedure and define an auxiliary field U µν = (cid:3) − F µν ,the latter would introduce new degrees of freedom, which are surely not present in the La-grangian (4.13). In order to avoid the appearance of such spurious degrees of freedom, in thelocalization procedure we have to select only a particular family of solutions of Eq. (4.10). Thegeneral solution for the field U would be the sum of the homogeneous and the particular one: U = U hom + U par . In order to avoid any further propagating degrees of freedom, during the lo-calization procedure we should specify that U is not the most general solution defined by (cid:3) − R ,but only a particular solution selected by fixing its boundary conditions. Following this recipe weavoid the appearance of ghosts associated to U after quantization. Of course the above proceduredoes not prevent the theory from developing instabilities at classical level, which however do notnecessarily imply a pathological behavior of the theory. In particular, if such instabilities emergeon cosmological scales and at Late-times, they could be able to drive the present acceleratedexpansion of the Universe. In this section we will briefly review a bunch of different nonlocal models proposed in the lastyears which are able to provide a viable cosmological history at background level, and are thuspotentially very interesting Dark Energy candidates.
One of the most popular nonlocal gravitational theory was proposed in Ref. [151] and it is knownas Deser Woodard (DW) model. The fundamental idea is to incorporate nonlocal effects without As long as we impose that the inverse d’Alembertian is defined in terms of the retarded Green’s function onlyto obtain casual solutions. onlocal gravity Section 4.3 postulating a priori any specific form of the nonlocal modification. This is achieved by introducinga free function of the inverse d’Alembertian of the Ricci scalar, called distortion function, andreconstruct it in such a way that it produce a background cosmological history identical to theone of the ΛCDM without cosmological constant. Then one can study the perturbative regimeand test its compatibility with observations. For a review on the main features of the model weaddress the reader to Ref. [152]; the issue of ghosts is studied in Ref. [149]. A detailed study of itsdynamics is performed in Ref. [153], while its Newtonian limit is studied in Ref. [154]. The effectsof such kind of modification for structure formation are studied in Refs. [146, 145, 147], whileconstraints from observational datasets are found in Ref. [155]. Finally, an improved version ofthe model has been recently proposed in Ref. [156].The Lagrangian of the model is: L = L EH + Rf (cid:18) (cid:3) R (cid:19) , (4.15)and it could be mapped in a localized scalar tensor theory by introducing the auxiliary fields, seeRef. [143]: (cid:3) U = − R , (4.16) (cid:3) V = ¯ f ( U ) R , (4.17)where ¯ f indicates the derivative of the distortion function f with respect to U . The EFE onFLRW flat background using the e-fold number N as time parameter are:(1 + f − V ) = − U (cid:48) V (cid:48) − f (cid:48) + V (cid:48) + Ω R + Ω M h , (4.18)(2 ξ + 3) (1 + f − V ) = V (cid:48)(cid:48) − f (cid:48)(cid:48) + (cid:0) V (cid:48) − f (cid:48) (cid:1) (2 + ξ ) + U (cid:48) V (cid:48) − Ω R h , (4.19)and the KG equations for the auxiliary fields are: U (cid:48)(cid:48) + (3 + ξ ) U (cid:48) = 6 (2 + ξ ) , (4.20) V (cid:48)(cid:48) + (3 + ξ ) V (cid:48) = − ξ ) ¯ f . (4.21)To solve the above system of equations without introducing ghost we need to fix the initialconditions for the auxiliary fields. If we impose the latter in such a way that they are compatiblewith a radiation-dominated epoch, h i ∼ Ω Ri and ξ i ∼ −
2, Eqs. (4.18) and (4.19) provide the twofollowing constraints: f i − V i = − U (cid:48) i V (cid:48) i − f (cid:48) i + V (cid:48) i , (4.22) − f i + V i = − V (cid:48) i − f (cid:48)(cid:48) i + U (cid:48) i V (cid:48) i , (4.23)where in Eq. (4.23) we used Eq. (4.21) evaluated at ξ i = − U i is unconstrained since it appears on the field equations onlythrough the function f ( U i ); to compute the time derivative of the latter we use the chain rule f (cid:48) = ¯ f U (cid:48) , so that: f (cid:48)(cid:48) = ¯¯ f U (cid:48) − ¯ f U (cid:48)(cid:48) . (4.24) We are using a different definition for the field U with respect to the one of Ref. [143]. The original ones areobtained by making the substitutions U → − U , ¯ f → − ¯ f . onlocal gravity Section 4.3 Evaluating Eq. (4.24) at N = N i we get: f (cid:48)(cid:48) i = ¯¯ f i U (cid:48) i − ¯ f U (cid:48) i , (4.25)where we have used Eq.(4.20) with ξ i = − f ( U ) = 0 . (cid:2) tanh (cid:0) . Z + 0 . Z + 0 . Z (cid:1) − (cid:3) , (4.26)where Z = − U + 16 .
7. Note that the above distortion function satisfies the condition f ( U i ) (cid:39) U i = 0. RR model Another very popular nonlocal theory is the RR model, proposed by Maggiore and Mancarella inRef. [158]. The model attempts to ascribe the accelerated expansion of the Universe to nonlocalmodifications of the quantum effective action caused by the appearance of a new mass scale m dynamically generated in the infrared. For a complete review on the model we address thereader to Ref. [141]; the cosmological perturbation theory and the impact on structure formationare studied in Ref. [159]. A dynamical system analysis of the model was performed numericallyin Ref. [160], while in Ref. [161] the model is tested against observation and compared using abayesian approach with the ΛCDM.In this theory one adds to the usual Einstein-Hilbert Lagrangian a nonlocal modification ofthe form: L = L EH − m R (cid:3) R , (4.27)and it is possible to localize the theory by introducing the auxiliary fields: (cid:3) U = − R , (4.28) (cid:3) S = − U . (4.29)Defining the dimensionless quantity V = H S and varying the action we obtain the followingbackground EFE, written in terms of the e-fold number N , see Ref. [160]: h = Ω M e − N + Ω R e − N + γ U γ (cid:0) − V − V (cid:48) + U (cid:48) V (cid:48) (cid:1) , (4.30) ξ = − M − R h + 3 γ (cid:0) Uh + U (cid:48) V (cid:48) − V (cid:48) (cid:1) − γV ) , (4.31)where we have defined γ ≡ m / H . The KG equations of the auxiliary fields are instead: V (cid:48)(cid:48) + V (cid:48) (3 + ξ ) = Uh , (4.32) U (cid:48)(cid:48) + U (cid:48) (3 + ξ ) = 6 (2 + ξ ) . (4.33)In order to avoid the introduction of new degrees of freedom we have to fix properly the initialconditions. Compatibility with radiation domination h i ∼ Ω Ri and ξ i ∼ − onlocal gravity Section 4.3 and (4.31) at some initial time N = N i become: U i h i = − V i − V (cid:48) i + 12 U (cid:48) i V (cid:48) i , (4.34) V i = U i h i + 14 U (cid:48) i V (cid:48) i − V (cid:48) i , (4.35)providing two constraints for the four initial conditions required on V i , V (cid:48) i , U i , U (cid:48) i . This model was proposed in Ref. [3], where the possibility of a nonlocal interaction term in abimetric theory of gravity was investigated for the first time. The action is the following: S = M P l (cid:90) d x √− gR + M f (cid:90) d x (cid:112) − f R f − M P l (cid:90) d x √− gα (cid:18) R f (cid:3) R + R (cid:3) R f (cid:19) + S m [ g, Ψ] , (4.36)where Ψ is a shortcut notation for all the matter fields, including CDM, and f µν is the auxiliarymetric which does not couple to matter. It turns out, from computing the Bianchi constraints,that R f must be constant and thus the action become: S = M P l (cid:90) d x √− g (cid:18) m (cid:3) (cid:19) R + S m [ g, Ψ] , (4.37)and the arising field equations can be cast as follows:(1 − αV ) G µν + m (1 − U/ g µν + 2 α ∇ µ ∇ ν V + α ∇ ρ V ∇ ρ U g µν − α ∇ µ U ∇ ν V = M Pl T µν , (4.38)where the auxiliary metric f enters only through m ≡ − αR f and all the other geometricalquantities are computed from the metric g µν . The two auxiliary fields U and V were introducedin order to localize the theory and satisfy the following KG equations: (cid:3) U = R , (cid:3) V = − α m . (4.39)Later on, in Ref. [144], an equivalent formulation of the theory was obtained motivated bynonperturbative lattice quantum gravity. The background cosmology was numerically studied inRef. [3], where the compatibility with the standard cosmological history of the ΛCDM is showed.The EFE and the Klein gordon equations in a flat FLRW background, written in terms of thee-fold number N , are: 3 ˜ V + m U H + 3 ˜ V (cid:48) + U (cid:48) ˜ V (cid:48) = ρM Pl H , (4.40) − ˜ V (3 + 2 ξ ) + m H (1 − U/
2) + ˜ V (cid:48) + U (cid:48) ˜ V (cid:48) = M Pl H P , (4.41) U (cid:48)(cid:48) + (3 + ξ ) U (cid:48) + 6 (2 + ξ ) = 0 , (4.42)˜ V (cid:48)(cid:48) + (3 + ξ ) ˜ V (cid:48) = − m H , (4.43) onlocal gravity Section 4.3 where we have defined ˜ V ≡ − αV . Imposing initial conditions compatible with a radiation-dominated era, i.e. h i ∼ Ω Ri and ξ i ∼ −
2, the EFE for a flat FLRW at some initial time N = N i read: ˜ V i + γU i h + ˜ V (cid:48) i + 16 U (cid:48) i ˜ V (cid:48) i = 1 , (4.44)˜ V i − γh i (cid:18) − U i (cid:19) + ˜ V (cid:48) i + 12 U (cid:48) i ˜ V (cid:48) i = 1 . (4.45)The latter equations provide two constrains among the four initial conditions on the auxiliaryfields U i , U (cid:48) i , ˜ V i , ˜ V (cid:48) i . HAPTER 5.
Personal Contribution:Nonlocal gravity
The worthwhile problems are the ones youcan really solve or help solve, the onesyou can really contribute something to.No problem is too small or too trivial ifwe can really do something about it.
Richard Feynman
In this chapter I will present the results of my research about nonlocal models of gravity. Inparticular, the first part of the chapter is devoted to the results of Ref. [4] on VAAS gravity.The second part of the chapter addresses instead the study of the general features of the Late-times asymptotic equation of state for a general class of nonlocal models, following the results ofRef. [5].
In Ref. [3] the cosmological behavior of VAAS gravity was analyzed numerically and found tobe compatible with the one of ΛCDM, but an analytical understanding of its dynamics wasnot addressed. In Ref. [4] we tried to fill this gap using a dynamical system analysis approach,revealing a number of interesting features. In particular, we addressed the existence of criticalpoints and their stability, and studied in a qualitative but analytical way the Late-times dynamicsof the model. We also studied the impact on small scales of the nonlocal modification, i.e. wehave studied its Newtonian limit on solar system scales and within the quasi static approximation(QSA), showing explicitly the existence of static solutions and the changes in the perturbationsequation for the density contrast. ersonal Contribution:Nonlocal gravity Section 5.1 Defining X ≡ ˙ U and Y ≡ ˙ V it is possible to write down Eqs. (4.38) and (4.39) for a flat FLRWbackground in the form of a closed dynamical system:˙ H = 11 − αV (cid:20) ρ − P M P l + m − U ) + 2 αHY (cid:21) − H , (5.1a)˙ ρ = − H ( ρ + P ) , (5.1b)˙ X = − HX − H − − αV (cid:20) ρ − P M P l + m − U ) + 2 αHY (cid:21) , (5.1c)˙ Y = m α − HY , (5.1d)˙ U = X , (5.1e)˙ V = Y . (5.1f)We also have to keep into account the following constraint coming from the first Friedmannequation: (1 − αV )3 H + m U − HαY − αXY = ρM P l . (5.2)We define as critical point of the dynamical system a point in the phase space for which the righthand side of equations (5.1) vanish. The details of the analysis of the above system are reportedin Appendix A, from which we can draw the following conclusions: • For m = 0 and U = const , the only critical point at finite distance represents Minkowskispace. • For m = 0, U = − Ht and H constant, the only critical point at finite distance representsa de Sitter space. • For m (cid:54) = 0 there are no critical points at finite distance; • At infinite distance we found an unstable hyperplane of critical points of Minkowski type.The above results are particularly interesting because m is the free parameter of the theorythat set the strenght of the nonlocal modification. The fact that for m (cid:54) = 0 we do not havestable critical points reflects and confirms the classical instability typical of these models we weretalking about at the end of subsection 4.2.2. By looking at Eqs. (4.40) and (4.41) we see that there is a complicated interplay between theauxiliary fields
U, V and the Hubble function H , which makes not trivial the qualitative under-standing of the dynamical behavior of the system. To get some insight, let us begin with theKlein Gordon Eqs. (4.39) written in terms of X ≡ ˙ U and Y ≡ ˙ V : X (cid:48) + (3 + ξ ) X + 6 (2 + ξ ) = 0 , (5.3) Y (cid:48) + (3 + ξ ) Y = − m H . (5.4) ersonal Contribution:Nonlocal gravity Section 5.1 The latter have the formal solutions: X ( N ) = C e − F ( N ) − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) [2 + ξ ( ¯ N )] , (5.5) Y ( N ) = C e − F ( N ) − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) m H ( ¯ N ) , (5.6)with C and C integration constants and: F ( N ) ≡ (cid:90) NN i d ¯ N [3 + ξ ( ¯ N )] . (5.7)As we discussed in 4.2.2, it is important to fix the initial conditions in order to avoid the appear-ance of spurious propagating degrees of freedom. From the formal solutions of X and Y we seethat: C = X ( N i ) , C = Y ( N i ) , (5.8)and choosing vanishing initial conditions for the fields, which are compatible with the constraintsof Eqs. (4.44) and (4.45), we obtain: X ( N ) = − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) [2 + ξ ( ¯ N )] , (5.9) Y ( N ) = − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) m H ( ¯ N ) . (5.10)It is straightforward to realize from the above equations that if ξ > −
2, which is compatible withthe standard cosmological evolution, then both X and Y are always negative. It is also easy toprove the following inequality: X + 6 > . (5.11)It is interesting to study the behavior of U and V during the different phases of the cosmologicalevolution. The explicit calculations are reported in Appendix B, the results of which are: • During the radiation domination (RD), ξ = −
2, we obtain: U RD = 0 , (5.12)˜ V RD = − m H Ω r e N − m H Ω r e N i − N + m H Ω r e N i + 1 . (5.13) • During the matter dominated epoch (MD), ξ = − /
2, we have: U MD = C − N , (5.14)˜ V MD ∼ − m H Ω m e N . (5.15)Let us now try to understand how these solutions evolve at Late-times when matter and radiationare diluted enough. Following the reasoning of Appendix B, we can conclude that at Late-times(LT) the field V evolve as: ˜ V LT ∼ − m U LT H . (5.16) The details of the calculation are reported in Appendix B ersonal Contribution:Nonlocal gravity Section 5.1 From the latter equation we can estimate then the behavior of ξ : ξ ∼ − − V LT m U LT H → , (5.17)so that at Late-times ξ vanishes approaching the value it would have in the ΛCDM scenario,independently of the value of m . The latter occurrence is not a coincidence, and we will discussit in detail in section 5.2. Using the above asymptotic expressions for ξ we found: U LT = C − N , (5.18)from which we finally obtain the evolution of the Hubble factor at Late-times: ξ = − C − N = − U LT , (5.19) H = C | C − N | / = 3 | U LT | / . (5.20)Fig. 5.1 shows the agreement between our analytical approximation for the evolution of theHubble factor at Late-times and the numerical solution, as well as the agreement between thelatter and the ΛCDM model in the past. h VAAS Λ CDMAnalytic
Figure 5.1.: Behavior of the VAAS model and the analytical approximation of Eq.(5.20) comparedto ΛCDM for m = 0 . H .Summarizing, our qualitative analysis shows that if choosing vanishing initial condition deepinto the radiation dominated epoch results in the following cosmological evolution: • During the RD epoch the field U is constant and vanishing, while V ∼ e N . • During the MD epoch the field U becomes linear in N and starts to grow, while V ∼ e N . • At Late-times we found that V ∝ U/H , which in turns implies that ξ →
0. As a result, U goes linearly as U ∼ − N and H = 3 | U | / , so that ξ = H (cid:48) /H ∝ U − approaches 0 as ∼ N − . ersonal Contribution:Nonlocal gravity Section 5.1 In order to understand the effects of the VAAS nonlocal term at small scales we will considerfirst order perturbations of the FLRW metric in the Newtonian gauge, i.e.: ds = − dt (1 + 2Ψ( x,t )) + a ( t ) (1 + 2Φ( x,t )) δ ij dx i dx j , (5.21)where Ψ and Φ are the gravitational potentials. We also need to consider perturbations of theauxiliary fields: U ( x , t ) = U ( t ) + δU ( x , t ) , V ( x , t ) = V ( t ) + δV ( x , t ) , (5.22)where U and V are the background solutions of the Klein Gordon equations (4.42)(4.43). Thefirst order perturbed Friedmann equation in Fourier space is then:(1 − αV ) (cid:18) Hφ , − H ψ + 2 k φa (cid:19) − αδV H − m (cid:18) − U (cid:19) ψ + m δU
2+ 2 α (cid:18) δV , − ψ , V , − i k i ψa V ,i (cid:19) − α (cid:16) V , δU , + V k, δU ,k + δV , U , + δV k, U ,k (cid:17) − αψ (cid:16) V , U , + V k, U ,k (cid:17) − α ( δU , V , + U , δV , ) = δρM P l , (5.23)while the acceleration equation is:(1 − αV ) 23 a k ( φ + ψ ) + (cid:18) k i k j k − δ ij (cid:19) (cid:20) − αδV G ij + m (cid:18) − U (cid:19) aδ ij φ − m δU δ ij a + 2 α (cid:8) δV ,ij − δ ij a , aδV , − δ ij a [ H + 2 H ( φ − ψ ) + φ , ] V , − iφ ( δ ki k j + δ kj k i − δ ij k k ) V ,k (cid:9) + αaδ ij (cid:16) V , δU , + V k, δU ,k + δV , U , + δV k, U ,k (cid:17) + 2 φaδ ij α (cid:16) V , U , + V k, U ,k (cid:17) − α ( δU ,i V ,j + U ,i δV ,j ) (cid:21) = (cid:18) k i k j k − δ ij (cid:19) δT ij M P l . (5.24)Finally, the Klein Gordon equations for the auxiliary fields are: δU µ,µ + HδU , + ψ , U , + ik i ψU i, + φ , U , + iφk j U j, = − ψ (cid:16) H + a , a (cid:17) − a ∇ ψ + 6 φ , − H ( ψ , − φ , ) − a ∇ φ ; (5.25) δV µ,µ + HδV , + ψ , V , + ik i ψV i, + φ , V , + iφk j V j, = 0 . (5.26) For analyzing physics at solar system scales we will adopt an approach similar to the one ofRef. [154] and make the following approximations: • We will ignore the cosmological expansion, so we set the scale factor a ≈ H ≈ • We look for static spherically symmetric solutions of the gravitational potential. ersonal Contribution:Nonlocal gravity Section 5.1 • We set matter perturbations to 0.Note that since in VAAS gravity there is no Minkowski solution at background level for m (cid:54) = 0,we have to consider m as a perturbative quantity in the following calculations. The perturbationsequations under these approximations become: ∇ Φ = 0 , (5.27) ∇ (Φ + Ψ) + α ∇ δV = 0 , (5.28) (cid:18) ∂ i ∂ j − δ ij ∇ (cid:19) (Φ + Ψ − αδV ) = 0 , (5.29)2 α ∇ δV = − m , (5.30) ∇ δU = − ∇ (2Φ + Ψ) . (5.31)Solving the above equations for the gravitational potentials we can draw the following conclusions: • The Φ potential is a solution of the standard Poisson equation and thus has the same formas in GR: Φ =
GM/r . • The Ψ potential is instead sourced by the m term and its solution is Ψ = − GM/r + m r /
12. The term ∝ m r is particularly interesting because it closely resembles the onethat would appear in GR for the Schwarzschild-deSitter solution, i.e. the nonlocal termbehave similarly to a cosmological constant. • We can write down the post Newtonian parameter γ = − Φ / Ψ = m r / GM , which canbe used to constraint the value of m . It turns out that the current constraint on γ ≤ − are satisfied for the value m ∼ H required for a well-behaved cosmological evolution. In order to address the impact of the nonlocal modifications on the process of structure formationat small scales we will work within the QSA. So we consider scales for which k (cid:29) H , in such away that for a given perturbation we can neglect in the first order perturbation equations thoseterms containing time derivatives, or those proportional to H , with respect to terms proportionalto k . Combining the Euler and the continuity equation for dust we obtain the equation for thedensity contrast δ M : ¨ δ M + 2 H ˙ δ M + k a Ψ = 0 . (5.32)The equations for δU and δV are instead: δU = − , k a δV = m α Ψ . (5.33)Finally, the EFE are: 2 (1 − αV ) k Φ a + m δU − m Ψ = δρ M M Pl , (5.34) − − αV ) k a (Ψ + Φ) − m δU − α k a δV = 0 . (5.35)Combining the latter equations and using Eq. (5.33) to eliminate δU and δV we obtain: − (cid:20) − αV ) k a − m (cid:21) Φ = (cid:20) − αV ) k a − m (cid:21) Ψ , (5.36) ersonal Contribution:Nonlocal gravity Section 5.1 from which we can read off the slip parameter η as: η ≡ − ΦΨ = 2 (1 − αV ) k − m a − αV ) k − m a , (5.37)note that, as expected, the GR result Φ = − Ψ is recovered for m →
0. Using the aboveexpression for η in the modified Poisson equation we can define the effective gravitational coupling Y as follows: 1 + Y ≡ − k Ψ3 H a Ω M δ M = (1 − αV ) k − m k a (1 − αV ) k − m a . (5.38)Using the latter it is possible to rewrite Eq. (5.32) as:¨ δ M + 2 H ˙ δ M −
32 (1 + Y ) Ω M δ M = 0 . (5.39)Thus, within the QSA, using the value of m required for a compatible background history m ≈ H we have that the slip parameter is essentially the same as in GR, η ≈
1, while theeffective gravitational coupling is modified by the background value of the field V and gets largeras soon as nonlocality starts to drive the accelerated expansion. For example, using the referencevalue m /H ≈ . − αV ≈ .
98, and hence Y ≈ .
02, so that the effectivegravitational coupling strength is enhanced by 2% .
Lunar Laser Ranging measurements provide a constraint on the time variation of the Newtonconstant, see for example Ref. [103]. Currently, this constraint is of order:˙ GG = 7 . ± . × − , yr − = 0 . ± . × − . h H , (5.40)where h is the Hubble constant expressed in units of 100 km s − Mpc − . As argued in Ref. [82],nonlocal modifications of gravity result in field equations which can be written as: G µν + ∆ G µν = 1 M P l T µν , (5.41)where ∆ G µν accounts for deviations from standard GR. The terms which in the latter are pro-portional to G µν will generally result in a modification of the gravitational coupling, which forEq. (4.40) reduce to: G eff = G N − αV = QSA G N − αV . (5.42)where in the last equality we are considering k (cid:29) H , thus we applied the result of the previoussection whitin the QSA, which is surely a well justified assumption for scales related to theEarth-Moon distance. Taking the time derivative of the above expression we can write:˙ G eff Gef f = 2 α ˙ V − αV , (5.43)or, using the e-fold number parameter: ˙ G eff Gef f = H ˜ V (cid:48) ˜ V , (5.44) ersonal Contribution:Nonlocal gravity Section 5.2 and we conclude that Lunar Laser Ranging constraints are satisfied in VAAS only for m /H ≤ − . The latter upper bound is two order of magnitude smaller than the value required forreproducing a viable cosmological expansion, thus the LLR experiments rule out VAAS gravity.Indeed, LLR have already ruled out several nonlocal proposal, including the RR and the DWmodel, with the exception of the RT model which we mention in section 3.1.5.2. There arehowever a couple of caveats; if a static solution of the above field equations exist it would beclearly compatible with LLR test, thus one should study whether these static solutions are stableagainst time perturbations. Moreover, the calculation presented here is based on the assumptionsthat we can extrapolate the solution we found for linear cosmological perturbations all the waydown to Earth-Moon scales, which is not guaranteed a priori and very difficult to prove due tothe lack of a satisfactory geometrical description that joins cosmological and local scales. Indeed,the results of Ref. [82] rely on the use of the McVittie metric to connect the cosmological solutionwith the solar system one, but such an assumption could be too strong. As we already mention, in the VAAS model the asymptotic equation of state of the Universeapproaches the value w eff → − RR model and of the one proposed in Ref. [144]showed a similar behavior. Motivated by the latter apparent coincidence, we studied in Ref. [5]the behavior of the Late-times asymptotic effective equation of state for different Lagrangianscontaining functions of (cid:3) − R . Applying the same technique we developed in Ref. [4] for the VAASmodel we show analytically that, under a certain choice of initial conditions, all the models inwhich a term (cid:3) − R appears explicitly in the Lagrangian result in an asymptotic equation ofstate w eff → −
1. This happens because such a term will inevitably diverge in the future,reflecting the classical instability we discussed in Sec. 4.2.2. As a result, H will diverge but˙ H/H →
0. On the other hand, we found that if the function f ( (cid:3) − R ) is chosen in such a waythat | f ( ∞ ) | ≤ const , like for the standard DW model, the asymptotic equation of state will notapproach asymptotically the ΛCDM value. The general scheme presented here was developed in Ref. [4] to study the late-times behaviorof the model [3]. A sketch of the general strategy is the following: we use the fact that thesign of the first derivative of the auxiliary fields is determined by the formal solutions of the KGequations. Then, imposing initial conditions compatible with radiation and matter domination,we are able to understand qualitatively the evolution of the nonlocal fields when matter sourcesare totally diluted by imposing consistency with the first Friedmann equation. Finally, we insertthe asymptotic solution obtained for the fields and their derivatives into the acceleration equationto compute the asymptotic value of ξ . Note that the scheme presented here is only valid if wemake the crucial assumptions ξ + 2 ≥
0, which is reasonable since we fix the initial conditionsduring the radiation-dominated era, when ξ = −
2, to which follow a matter-dominated epoch ξ = − / ersonal Contribution:Nonlocal gravity Section 5.2 First, let us notice that we can extract information about the qualitative behavior of U alreadyfrom the structure of its Klein Gordon equation. Indeed, it is straightforward to realize that withthe above choice of initial conditions, see Appendix B for the explicit computation, the followinginequality for U (cid:48) holds: 0 ≤ U (cid:48) ≤ , (5.45)so that choosing non-negative initial conditions would always imply U ≥
0. Since we know that U has a definite sign, and we have constrained its first derivative, we grossly know its qualitativeevolution. Moreover, since the other auxiliary fields are generally defined in terms of U , we canconclude that it is possible to obtain a similar amount of information from their Klein Gordonequations.Another crucial information on the behavior of the system comes from the first Friedmannequation, which could be written:(1 + g ( N ) NL ) h = Ω R + Ω M + Ω NL , (5.46)where the functions g ( N ) and Ω NL are the modifications due to nonlocal terms.The initial conditions set the value g ( N i ), and since we can estimate the signs of the firstderivative of the auxiliary fields entering in g NL , using their KG equations we are able to estimatethe qualitative behavior of g ( N ) through the cosmological history, and in particular its asymptoticvalue at Late-times when matter and radiation are diluted enough. To begin with let us consider the RR field equations (4.30),(4.31) and (4.32) when matter andradiation density are negligible and define ˜ V ≡ − γV :˜ V = γU h + ˜ V (cid:48) (cid:18) U (cid:48) − (cid:19) , (5.47) ξ = 12 ˜ V (cid:20) γUh − U (cid:48) ˜ V (cid:48) + 4 ˜ V (cid:48) (cid:21) , (5.48)˜ V (cid:48)(cid:48) + ˜ V (cid:48) (3 + ξ ) = − γUh . (5.49)The formal solution of Eq. (5.49) for ˜ V (cid:48) compatible with the initial condition ˜ V (cid:48) ( N i ) = 0 is:˜ V (cid:48) = − γe − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) Uh , (5.50)where F ( N ) was defined in Eq. (5.7). From Eq. (5.50) it is straightforward to realize that ˜ V (cid:48) is always negative since U is always positive, while imposing vanishing initial conditions for thenonlocal fields at early times determines the initial value ˜ V = 1. On the other hand from the righthand side of Eq. (5.47) we see that ˜ V must be positive and so we can conclude that 0 ≤ ˜ V ≤ Note that the parameter γ can be considered as positive definite since changing its sign corresponds to switchthe sign of the nonlocal interaction term in the Lagrangian. In this case it is more convenient to change thesign of the source term in the equation of the auxiliary field U , in such a way that the product γU is positivedefinite. Here we are neglecting the radiation and matter contributions which are anyway also positive definite. ersonal Contribution:Nonlocal gravity Section 5.2 This last argument tells us then that ˜ V (cid:48) must also vanish at late-times, or it would push ˜ V tonegative values. On the basis of these considerations we can conclude that at late-times we have˜ V ∼ γU h , ˜ V (cid:48) ∼ , (5.51)and using the above results in Eq. (5.48) we get: ξ ∼ γU h h γU ∼ U → , (5.52)where the last limit holds true since U diverges. In order to study qualitatively the dynamic of the DW model at late-times we have first of allto understand qualitatively the behavior of the free function f ( U ) defined in Eq.(4.26), since itenters directly in the Friedmann equations and also rules the dynamics of the localized field V .It is straightforward to realize from Eq. (4.26) that ( − . < f < f < f → U → ∞ .The formal solution of (4.21) for V (cid:48) is: V (cid:48) = − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) (2 + ξ ) ¯ f . (5.53)Since ¯ f < ξ > − V (cid:48) >
0. Since we impose initial conditions in such a way that during the radiation-dominatedepoch V is vanishing, we also can conclude that V >
0. At late-times, when matter is completelydiluted Friedmann equations (4.18) and (4.19) become: V = − V (cid:48) (cid:18) − U (cid:48) (cid:19) + ¯ f U (cid:48) + f + 1 , (5.54)(2 ξ + 3) (1 + f − V ) = V (cid:48)(cid:48) − f (cid:48)(cid:48) + (cid:0) V (cid:48) − f (cid:48) (cid:1) (2 + ξ ) + U (cid:48) V (cid:48) . (5.55)Note that the first two terms in the right hand side of Eq. (5.54) are strictly negative since, V (cid:48) > f <
0, while f + 1 >
0. On the other hand V (cid:48) > V is a monotonic function,and we are left with two cases; either U diverges, in which case ¯ f → f → ( − . U → const , in which case U (cid:48) → f (cid:48) → f + 1 → const . In both cases consistencyrequires that V (cid:48) →
0, or V will be a decreasing function and so V (cid:48) <
0. Thus we can concludethat asymptotically: V ∼ ¯ f U (cid:48) + f + 1 . (5.56)Using the above in (5.55) we obtain finally: ξ ∼ U (cid:48) − − f (cid:48)(cid:48) ¯ f − U (cid:48) , (5.57) Note that U cannot reach a constant value since U (cid:48) = 0 is possible only for ξ = −
2, and we easily see fromEq. (5.52) that at late-times ξ > ersonal Contribution:Nonlocal gravity Section 5.2 which is in general non-vanishing. Note also that: f (cid:48)(cid:48) ¯ f = ( ¯ f U (cid:48) ) (cid:48) ¯ f = U (cid:48)(cid:48) + ¯¯ f ¯ f U (cid:48) ; (5.58)and we can conclude that if U → ∞ the term ¯¯ f / ¯ f → −∞ , while U (cid:48)(cid:48) cannot diverge since0 < U (cid:48) <
6, so in this case the asymptotic effective equation of state w eff → ∞ . On the otherhand, if U → const , we have U (cid:48) = U (cid:48)(cid:48) → ξ → −
2, in such a way that theeffective equation of state approaches one of radiation type.We have then shown that in the DW model with the distortion function given by (4.26) atlate-times w eff (cid:54) = − By studying the Late-times asymptotic equation of state for a number of nonlocal models werealized that the auxiliary localized field related to (cid:3) − R will diverge asymptotically if we imposevanishing or positive initial conditions. The divergence of the latter in turn will push H → ∞ while the ratio H (cid:48) /H → / (cid:3) R is still dynamical asymptotically, and diverges,while the auxiliary fields related to the Lagrange multipliers used to localize the theories freezeand approach a constant value. We show how the mechanism works using as an example the RR model and the DW model, since in the latter the divergence of U is hidden in the distortionfunction f ( U ), which is regular for U → ∞ . The full computation for the VAAS model and themodel proposed in Ref. [144] are reported in Appendix B.It is important to remark that our conclusions strongly depend on the choice of initial condi-tions. Indeed, our method relies on the observation that by using Friedmann equations we canconstrain the sign of the auxiliary fields, while their KG equations provide constraint on the signof the first derivatives for our choice of initial conditions. As an example, let us consider the RR model. If we choose a negative initial condition in Eq. (4.32) for the field U then V (cid:48) > w eff → /
3. However, our analysis still holds for any choiceof initial conditions with non-vanishing but positive values of U . As discussed in Ref. [141], thereare fundamental motivations that justify processes during the inflationary epoch that result ina huge non-vanishing positive values for the field U in the RD epoch. It is interesting to notethat a behavior of the type w Eff → −
Coincidence Problem [33] is less severe (if not a problem at all, depending on the personal perspective), since at somepoint of its history, independently of the initial conditions, the Universe always passes througha phase in which the matter and DE densities are of the same order and then DE starts todominate, which in the standard model in terms of cosmic time accounts for at least the last 3.5billion years. On the other hand, in the nonlocal models considered here we have to deal with adifferent sort of coincidence. Indeed, the Hubble function reaches a minimum when the nonlocalfields cosmological density starts to dominate, and the occurrence of this is roughly today. Thisoccurrence looks to us coincidental at least as the one present in ΛCDM. HAPTER 6.
Personal Contribution:Ricci Inverse Gravity
I don’t think there is a final theory ofanything. It’s theories (turtles) all theway down.
Jim Peebles
In Ref. [6] we proposed a novel class of modified gravity theories based on the inverse of theRicci tensor R µν , which we call the anticurvature tensor A µν . Taking the trace of the latter weobtain the anticurvature scalar A , which can then be used to construct a new type of Lagrangiandensities. It is interesting to note that with the anticurvature scalar is very simple to write downLagrangian densities terms with the same dimension as R , like for example A − or R A , andthus without introducing new dimensional constants. The anticurvature tensor A µν is defined as the inverse of the Ricci tensor: A µρ R ρν = δ µν , (6.1)from which, taking the trace, we obtain the anticurvature scalar A = A µν g µν . Note that it ispossible to write the inverse of any matrix in terms of its adjugate, i.e. in terms of the matrixitself and the Levi Civita symbols. In particular, for the anticurvature tensor we have: A µν = 4 R κπ R λρ R ξσ ε µκλξ ε νπρσ R αζ R βη R γθ R δι ε αβγδ ε ζηθι . (6.2)From the latter equation we can appreciate that a theory based on the anticurvature scalar isactually a strongly nonlinear, higher order theory of gravity.The field equations for a general Lagrangian f ( R, A ) are: f R R µν − f A A µν − f g µν + g ρµ ∇ α ∇ ρ f A A ασ A νσ − ∇ ( f A A µσ A νσ ) − g µν ∇ α ∇ β ( f A A ασ A βσ ) − ∇ µ ∇ ν f R + g µν ∇ f R = T µν , (6.3)where f A,R indicate derivation with respect to the Ricci or anticurvature scalars, see AppendixC for a detailed derivation of the above field equations. A code that evaluates the equations of motion for any f ( R, A ) in a given metric is made publicly available here ersonal Contribution:Ricci Inverse Gravity Section 6.2 It is well known that one can recast an f ( R ) theory in the form of a scalar-tensor theory in theEinstein frame introducing a scalar field non minimally coupled to gravity. Usually this is doneby defining a scalar field φ = df /dR and performing a Legendre transformation of the function f .Such an approach, however, fails here, because A is not a one-to-one function of R and therefore df ( A ) /dR is in general not invertible. We are interested in understand which kind of behavior could arise from Eq. (6.3) for cosmologicalimplications. Since we are mainly interested in Dark Energy phenomenology, let us begin witha de Sitter ansatz for the metric g µν . Under this assumption R µν = Rg µν /
4, and it is straight-forward to compute A µν from the inverse metric. In this case all the terms with derivatives inEq. (6.3) vanish, and taking the trace one has in vacuum: f R R − f A A − f = 0 . (6.4)Since on de Sitter background R = 12 H and A = 4 / (3 H ), the latter equation can be easilysolved for any f ( R, A ) model to check whether one gets non-trivial (i.e. H (cid:54) = 0) solutions thatcould replace a cosmological constant. For instance, if f = R − αA (where α is a constant withdimensions H ) then we see that H = ( α/ / = const .Having seen that it is in principle possible to have de Sitter solutions in this model, let usinvestigate the behavior in a flat FLRW background. In this case we have that the Ricci andanticurvature scalars can be written: R = 6 (cid:16) ˙ H + 2 H (cid:17) , (6.5) A = 2(6 + 5 ξ )3 H (1 + ξ )(3 + ξ ) , (6.6)where ξ = ˙ H/H . It is straightforward to realize from Eq. (6.6) that the anticurvature scalarbecome singular in the following cases: H → ξ → −
1, or ξ → −
3. The fact that for H → A n . On the other hand, this does notapply for Lagrangian containing negative powers A − n , which are instead singular for ξ → − / Theorem 2 (FLRW no-go)
If the cosmic evolution passes through any one of these values of ξ : ξ = − , ξ = − or ξ = − / , either A or A − , or any of their powers, develops a singularity.If during the evolution A passes through both ±∞ , then any term in the Lagrangian thatcontains A n , for n positive or negative, will blow up. This behavior will reflect into equations ofmotion that also contain a singularity at the same cosmic epochs. Observations [20, 21, 27, 19] tell us that the Universe evolved from a decelerated phase with w eff ≈ ξ ≈ − .
5) into an accelerated phase w eff ≈ − . ξ ≈ − . ξ = − ξ = − / A and A − will both be singular at some epoch between decelerationand acceleration. Consequently, any Lagrangian that contains additive terms proportional to A n (e.g. the two simplest scale-free models, f ( R, A ) = R + αA − and f ( R, A ) = R + αR A , with α a dimensionless constant) are ruled out as Dark Energy models. Notice also that R = 6 H ( ξ + 2) ersonal Contribution:Ricci Inverse Gravity Section 6.3 so no power of R can cure the singularity. In order to see some concrete realizations of the no-gotheorem in the following we will briefly illustrate the cosmological behavior for the Lagrangians f = R + αA − and f = R + αR A . R + αA − In this case equations (6.3) for a FLRW background reduce to: ρ t = 3 αH ( ξ + 3) (5 ξ + 6) − ξ (cid:48) ξ + 6) + 3 H , (6.7)which is the modified Friedmann equations, and to: w t ρ t = − αH (cid:104) (5 ξ + 6) (cid:0) ( ξ + 3) (2 ξ + 3)(5 ξ + 6) − ξ (cid:48)(cid:48) (cid:1) + 270 ( ξ (cid:48) ) − ξ + 2)(5 ξ + 6) ξ (cid:48) (cid:105) ξ + 6) − H ξ − H , (6.8)which is the ( i, i ) equation. Note that, as follows from the No-go theorem, the singularity at ξ = − / t in ρ t to indicate the totalmatter, which of course satisfy the continuity equation ρ (cid:48) t = − w t ) ρ t . From Eq. (6.7) we caneasily define the energy density associated to the anticurvature scalar:Ω A ≡ − α ( ξ + 3) (5 ξ + 6) − ξ (cid:48) ξ + 6) . (6.9)We will now consider two different cases, considering pressureless matter only and then adding acosmological constant Assuming w t = 0 we find for Eqs. (6.7) and (6.3) the following critical points:Ω m = 1 + α , ξ = − , (6.10)Ω m = 0 , ξ ± = 3( − − α ± √− α )100 + α . (6.11)The first critical point corresponds to a matter dominated Universe in which Ω A behaves asmatter. In particular, if α = −
4, it corresponds to an empty Universe wich behave as it was filledwith dust. The critical points of Eq. (6.11) instead admit solutions only for certain values of α ,see Fig. 6.1 for a graphical representation of ξ + . We see that for every α < ξ = − /
5, or equivalently w eff = − .
2. The propertiesof these solutions seem very interesting for cosmological implications. For instance, for α ≈ − w eff ≈ − .
67 and to anexpansion quite close to a matter dominated era, w eff ≈ .
06. Analogously, if α = −
4, one has w eff = 0, i.e. an exact matter era evolution without matter, in which the A energy density acts asa form of Dark Matter. The other solution, ξ + , corresponds to w eff = − .
5, i.e. an acceleratedsolution still marginally compatible with observations. In Fig. 6.2 we see the behavior of theHubble parameter for the particular case α −
4. A cosmic evolution that moves from one suchsolution to the other would be indeed an intriguing possibility, replacing both Dark Matter and ersonal Contribution:Ricci Inverse Gravity Section 6.4 - - - - - - - - - - α w e ff Figure 6.1.: The two real branches of the function w eff ( α ) = − ξ/ − ξ − is a stable attractor only for − ≤ α ≤
0. The critical point ξ + is a stable attractor for α ≤ ξ = − /
2. These findingsare supported by the numerical investigation shown in Fig. 6.2, so that the cosmic evolutionwill end up either at ξ + or ξ − , depending on whether the initial w eff is above or below thesingularity at w eff = − .
2. The crucial point is that no trajectory can cross the w eff = − . w eff = 0 to an accelerated one around w eff ≈ − . If a cosmological constant is present we can combine Eqs. (6.7) and (6.3) and obtain: ξ (cid:48)(cid:48) = 6(5 ξ + 6) Ω m + ξ (5 ξ + 6) (cid:0) α + 16) + ( α + 100) ξ + 6( α + 40) ξ (cid:1) + 135 α ( ξ (cid:48) ) − α (cid:0) ξ + 11 ξ + 6 (cid:1) ξ (cid:48) α (5 ξ + 6) , (6.12)which must be solved together with the continuity equation:Ω (cid:48) m = − (3 + 2 ξ )Ω m . (6.13)In this case the phase space is more complicated, ξ − is now always unstable and ξ + is a stableattractor for α < −
16. The critical point ξ = − / ξ = 0, which is a stable de Sitter attractor when − ≤ α ≤
0. However, thebottom line is the same, as can be immediately gleaned from Fig. 6.3, so the model is ruled outas a candidate for Dark Energy even when a cosmological constant is added, regardless of thevalue of α . ersonal Contribution:Ricci Inverse Gravity Section 6.4 Figure 6.2.: Numerical solutions ξ ( a ) of Eq. (6.11) in case of Ω m (cid:54) = 0, Ω Λ = 0 with w = 0 and α = −
4. The solutions ξ = − / ξ = − / ξ = − / Motivated by the interesting phenomenology which could be described within the anticurvaturescalar, we will try now to address some possible escape routes from the no-go theorem.
To begin with, we should consider possible modifications of the background geometry that allowfor a different evolution. If we relax the assumption of a spatially flat Universe the anticurvaturescalar become: A = 2(6 + 5 ξ + 3Ω k )3 H (1 + ξ )(3 + ξ + 6Ω k ) , (6.14)where we can see that the appearance of the energy density associated with the curvature shiftsthe singularity ξ = − A , while shifts the singularity ξ = − / A − .Thus, in principle the presence of curvature is able to shift the singularity of the Lagrangiansoutside the range required by the observations. However, since observations suggest that Ω k ∼ ds = − dt + a ( t ) (cid:16) e β x ( t ) dx + e β y ( t ) dy + e β z ( t ) dz (cid:17) , (6.15)where we have defined the averaged scale factor a ( t ) = (cid:113) a x ( t ) a y ( t ) a z ( t ) , (6.16)so that a i ( t ) = a ( t ) e β i , and the β i satisfies (cid:80) i β i = 0. For the sake of simplicity, let us specialize ersonal Contribution:Ricci Inverse Gravity Section 6.4 Figure 6.3.: Numerical solutions ξ ( a ) of Eq. (6.12) with matter and cosmological constant, for α = −
4. The upper curves converge toward the de Sitter attractor at ξ = 0. Thelower curves converge towards the divide line at ξ = − /
5, which is now also anattractor. The red dashed line is the ΛCDM behavior.to the case β x = − β z ≡ β and β y = 0. In this case the anticurvature scalar A reads: A = 1 H ξ + 6 + ( β (cid:48) ) ξ ) (cid:16) ξ + ( β (cid:48) ) (cid:17) + 2(3 + ξ )(3 + ξ ) − ( β (cid:48)(cid:48) + β (cid:48) (3 + ξ )) , (6.17)which for β (cid:48) = 0 reduces to the FLRW case. As we can see, the singularity ξ = − β (cid:48) ) /
6. Note that also the singularity appearing in A − , ξ = − /
5, is ingeneral shifted. For example, if β (cid:48)(cid:48) is negligible, we have that A − is singular for ξ = 24 − β (cid:48) β (cid:48) − ≈ −
65 (1 − ( β (cid:48) )
10 ) , (6.18)(the last approximate equality being valid for β (cid:48) (cid:28)
1) which recovers the FLRW case for β (cid:48) = 0,while being regular in ξ = − / β (cid:48) = 48 /
5, i.e. the two roots of Eq. (6.18) for ξ = − / β (cid:48) of order unity to move the singularity outside the observational range,which on the other hand is not likely to be compatible with experimental data.To illustrate that, let us naively estimate β from the evidence of anisotropic expansion claimedrecently in [162], emerged from X-ray observations of galaxy clusters. Here the authors findthat the highest and the lowest values observed for the universe expansion rate are H max ∼ H min ∼
66 km/s/Mpc. Assuming that the averaged Hubble factor is H ∼ β (cid:48) ∼ . , (6.19)which shows that generally β (cid:48) is constrained from the observations to be too small to shift thesingularities of A outside the observational range. ersonal Contribution:Ricci Inverse Gravity Section 6.5 Another possibility is to consider Lagrangian densities which are not singular when A or A − diverge. Still considering scale-free Lagrangians for simplicity, we can choose for example thescalar densities R + αR exp[ − β ( RA ) ] or R/ (1 + αRA ). In the former case, for example, theFriedmann equation around the critical points, i.e. assuming ξ (cid:48) = ξ (cid:48)(cid:48) = 0, becomes:3 H (cid:18) − α P ( ξ, β )( ξ + 1) ( ξ + 3) e − β ( ξ +2)2(5 ξ +6 , )2( ξ +1)2( ξ +3)2 (cid:19) = ρ m , (6.20)where P ( ξ, β ) is a polynomial of order five in ξ and linear in β . It is straightforward to realisethat the above equation is regular on the poles of the denominator due to the presence of theexponential factor.Another option is to include, as in Gauss Bonnet gravity for the Ricci tensor, scalar combina-tions of higher order in the anticurvature tensor, like A µν A µν . In FLRW background the latterlooks as follows: A µν A µν = 49 H ξ + 15 ξ + 9( ξ + 1) ( ξ + 3) . (6.21)We see that it still contains the singularities at ξ = − ξ = −
1, but remarkably it nevervanishes, and thus ( A µν A µν ) − can be used to build Lagrangians which are free of this kind ofsingularities. We have shown that it is difficult to describe Dark Energy using polynomial Lagrangians ofthe anticurvature scalar because of the no-go theorem. On the other hand, we found that aninteresting phenomenology arise already for the simplest choices of f ( R, A ), and thus Lagrangiansthat escape the no-go theorem are particularly promising for cosmological model building.It is important to realize that in this framework we are introducing higher order derivativeterms in the Lagrangian, see Eq. (6.2), and then we expect that instabilities will generally occurunless we consider degenerate Lagrangians. The above stability issues and the no-go theoremshould be taken into account when a particular form of f ( R, A ) is specified, which is the task weaddress for future works. HAPTER 7.
Personal Contribution:Strong Lensing for testing Gravity and
Cosmology
Simplicity is the touchstone in findingnew physical laws. . . If it’s elegant, thenit’s a rough rule of thumb: you’re on theright track
Kip Thorne
Lensing effects provide fertile ground for testing gravitational theories from more than a cen-tury. In particular, it is well known that weak gravitational lensing by Large Scale Structures canprovide useful insights on the nature of Dark Energy. In this chapter we will discuss instead thepotential of strong gravitational lensing in achieving a similar task. After a brief review of themain equations governing this phenomenon, we will introduce two novel drift effects proposed byus in Ref. [7], and discuss the possibility of using them for testing violations of the EquivalencePrinciple and Dark Energy models [8].
It is a well known results of geometrical optics that light rays passing through a medium willgenerally be refracted, as encoded in the Snell’s law. Considering the gravitational field in theempty space around its source as a sort of ”medium”, it is a reasonable expectation that lightrays traveling through it will be deflected. It is slightly uncomfortable to give a meaningfulexplanation of this effect within Newtonian gravity because photons have no rest mass, and thusshould be blind to the gravitational interaction. On the other hand, it is a standard approach tostudy the motion of bodies in a gravitational potential by mean of test particles, i.e. particleswhose mass is small enough to ignore their backreaction on the gravitational potential. Thus,considering photons as test particles we expect already in Newtonian gravity the deflection oflight rays close to a massive body. This result was indeed obtained by Soldner more than onecentury before Einstein’s theory of general relativity. In GR, instead, the interpretation of thegravitational potential in the empty space as a sort of medium is straightforward, and is logical toconclude that trajectories of massless particles will in general be bent because of the curvature’sgradient of the spacetime. A translation from german of the original article is available here ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.1 An effective treatment of the above phenomenon resembles the standard approach of geometri-cal optics. The source of the gravitational field that deflects light rays is then called lens , and theoverall effect gravitational lensing . We distinguish between the deflection caused by an extended,approximately continuous distribution of sources and the one caused by a single, massive object.The former is which is generally dubbed weak lensing , and causes distortions on the shape of thebackground objects, see Ref. [163] for a review of weak lensing and its cosmological applications.When the lens is instead composed by a single massive object along the line of sight between theobserver and the source we are instead in a regime of
Strong Lensing , which will be our mainsubject during the rest of this chapter.
To begin with, let us consider gravitational lensing by point masses. A fairly standard configu-ration is given in Fig. 7.1, where θ is the apparent angular position of the source as seen by theobserver, and θ S indicate the actual position of the source that would be observed in case of nolensing. The above quantities are related by the lens equation in the thin lens approximation [164]: θ E ≡ θ ( θ − θ S ) = 4 G N M D L D LS D S = 4 G N M (1 + z L ) (cid:18) χ L − χ S (cid:19) , (7.1)where the D ’s are the angular diameter distances: D i ≡ a i χ i = 11 + z i (cid:90) z i dz (cid:48) H ( z (cid:48) ) , (7.2)in which the subscript i = L, S refers to the lens or to the source, while D LS is given by: D LS ≡ a S ( χ S − χ L ) = 11 + z S (cid:90) z S z L dz (cid:48) H ( z (cid:48) ) . (7.3)In the above equations z L,S indicates the redshifts and χ L,S the comoving distances of the lensand the source respectively. Using the above definitions it is possible to rewrite the lensingequation as: θE = 4 G N M (1 + z L ) (cid:18) χ L − χ S (cid:19) . (7.4)Being quadratic in θ , the lens equation (7.1) has the following two roots: θ = θ S ± (cid:114) θ S θ E , (7.5)which implies that, because of the lens, the original image of the source is split into two. Noticethat θ S is time-independent because of the cosmological principle. In other words, observer, lensand source form a triangle whose sides increase due to the Hubble flow, but whose angles remainunchanged, and therefore dθ S /dt = 0. In the thin lens approximation the lens equation (7.1) for a general mass distribution is :( β − α ) = ∇ θ ψ ( β ) , (7.6)where β = ( β , β ) and α = ( α , α ) are the position in the sky of the image and the sourcerespectively, and ∇ θ is the two-dimensional angular gradient. The quantity ψ ( β ) appearing in From now on we will restrict the use of the θ i notation for point mass lenses, while the α and β notation for ageneral lens profile ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.2 dzdt = 1 a e da dt a e da e dt e = (1 + z ) H H ( z ) , (6)where the Hubble constant H and the Hubble parameter H ( z ) have appeared. Measuring dz/dt would allow todetermine H ( z ), thereby providing precious informationson the energy content of the universe.In the next sections we shall obtain formulae similarto the one in Eq. (6), but in the context of strong gravi-tational lensing. III. ANGULAR DRIFT
Let’s consider the lens equation in the thin-lens ap-proximation [31]: ✓ ( ✓ ✓ S ) = 4 GMd L d LS d S , (7)where ✓ is the angular position of the image, ✓ S is theangular position of the source, M is the lens mass andwith d we denote angular-diameter distances. In Fig. 1we display the lensing scheme. O L S ✓ S ✓ b FIG. 1. Scheme for gravitational lensing. The deflection angleis = 4 GM/b , where b is the impact parameter. In particular, the angular-diameter distance to thesource is written as: d S ⌘ a S S = 11 + z S Z z S dz H ( z ) , (8)with S the comoving distance to the source, a S is thescale factor at the emission time and z S = 1 /a S d LS ⌘ a S ( S L ) = 11 + z S Z z S z L dz H ( z ) , (9)and the angular-diameter distance to the lens is the fol-lowing: d L ⌘ a L L = 11 + z L Z z L dz H ( z ) , (10)with L being the comoving distance to the lens and z L the redshift of the lens. With these definitions, let’s rewrite the lens equation as follows: ✓ ( ✓ ✓ S ) ⌘ ✓ E = 4 GM (1 + z L ) ✓ L S ◆ , (11)where ✓ E is the Einstein ring radius, if ✓ S = 0. From thefirst part of the the above equation we have the followingtwo solutions for ✓ : ✓ ± = ✓ S ± r ✓ S ✓ E , (12)i.e. two solutions ✓ ± for the images, given the high sym-metry of a point-like lens. Combining the two solutions,we get: ✓ S = ✓ + + ✓ , ✓ E = ✓ + ✓ , (13)where notice that ✓ is negative. Notice also that ✓ S is time-independent because the Hubble flow moves lensand source radially with respect to the observer (us),thereby leaving ✓ S unchanged, i.e. d✓ S /dt = 0 and thus d✓ + /dt = d✓ /dt . The time derivatives of the ob-served angles are thus directly related to the time deriva-tive of ✓ E as follows, using Eq. (13):2 d✓ E dt = ✓ d✓ + dt ✓ + d✓ dt = ( ✓ + ✓ ) d✓ + dt . (14)We can eliminate ✓ or ✓ + from the above equation, byusing again Eq. (13), and obtain:1 ✓ d✓ dt = 2 ✓ E ✓ + ✓ E ✓ E d✓ E dt . (15)Having established this correspondence between the twoderivatives, we now proceed with the calculations of thetime drift of ✓ E . Since the comoving distance is time-independent, the derivative with respect to the observertime of the Einstein radius is the following, from Eq. (11):2 ✓ E d✓ E dt = 4 GM dz L dt ✓ L S ◆ . (16)Using Eq. (6), one can write the relative variation of theangle ✓ E in the following form:2 ✓ E d✓ E dt = H H L z L . (17)Not unexpectedly, this formula is very similar to the onein Eq. (6). If H L > H (1 + z L ), we expect the Einsteinradius to shrink. This is the case for a matter-dominateduniverse, for example. IV. TIME DELAY DRIFT
The time-delay is usually divided into a geometric con-tribution and a potential one [31]. The former is due tothe bending of the trajectory of the photon, whereas the
Figure 7.1.: Scheme for strong gravitational lensing induced by a point mass lens L . θ is theapparent angle at which the source is located, θ S the true one. The deflection angleis δ = 4 GM/b , where b is the impact parameter, i.e. the distance on the lens planebetween the incoming light ray and the lens itself. Image taken from Ref. [7]Eq. (7.6) is the lensing potential and is defined as: ψ ( β ) ≡ c D LS D L D S (cid:90) β dλ Φ , (7.7)where Φ is the standard Newtonian gravitational potential and the integral is taken along thepath of the light ray, which depends on β and is parametrized by λ . Taking the divergence ofEq. (7.6), as long as the extent of the lens is small compared to cosmological distances, we canuse the Poisson equation to relate the Laplacian of the lensing potential to the mass distributionof the lens: ∇ θ ψ ( β ) = 8 πG N c D L D LS D S Σ( β ) , (7.8)where we have defined the surface mass density:Σ( β ) ≡ (cid:90) β dλ ρ , (7.9)in which appears the mass distribution of the lens ρ . For a detailed derivation and an explanationon the assumptions behind Eqs. (7.6), (7.7), (7.8) see for example Ref. [163]. As we saw in the previous section, in a strongly lensed system are present several images of thesame source. Thus, of course, a first important observable is the angular separation betweenthe various images. The entity of this variation is typically of the order of few arcseconds. Forexample, using data from Ref. [165], the quasar QSO0957 + 561 at redshift z S = 1 .
41 lensed bya cluster at z L = 0 .
31 displays two images separated by 6.1”. ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.2 On the other hand, even if coming from the same source, these several images at a givenmoment of time are not necessarily identical. Indeed, the optical path of the photons of eachimage is in general different, and thus some of the photons will require more time to reach theobserver. This effect is called
Time Delay , which we indicate with ∆, and could be divided intotwo different contributions [164]: ∆ = ∆ geo + ∆ pot , (7.10)where ∆ geo is the geometrical Time Delay and ∆ pot is the potential Time Delay. The former isdue to the bending of the trajectory of the photon, whereas the latter is due to the motion intothe lens gravitational field. For a single image the geometric ∆ t geo induced by a point mass lensis given by: ∆ t geo = (1 + z L ) (4 GM ) θ (cid:18) χ L − χ S (cid:19) , (7.11)from which we obtain ∆ geo = ∆ t geo ( θ + ) − ∆ t geo ( θ − ), where θ ± are the two roots of Eq. (7.1). Thepotential Time Delay between the two images due to a point mass in the thin lens approximationis given by: ∆ pot = ∆ t pot ( θ + ) − ∆ t pot ( θ − ) = 2 GM (1 + z L ) ln θ − θ + . (7.12)For an extended lens profile the Time Delay between two images ∆ ij is given by, see Ref. [166]: ∆ ij = D ∆ t c (cid:32) ( β i − α ) − ( β j − α ) ψ ( β j ) − ψ ( β i ) (cid:33) , (7.13)where it was defined the Time Delay distance: D ∆ t ≡ (1 + z L ) D L D S D LS . (7.14)It is straighforward to separate in the right hand side of Eq. (7.13) the contributions from thegeometrical and the potential Time Delay. Indeed, the first two terms inside the brackets aregiven by the differences between the apparent and the true position of the source, and are thusof geometrical nature. The potential Time Delay is instead given by the difference between thelensing potential at the two apparent positions.From Eq. (7.13) we can already understand the importance of precise Time Delay measure-ments for cosmological implications. Indeed, if one is able to know the position of the source α ,measures with enough precision the position of the images β i , and is able to infer the positionof the lens and its lensing potential, all the cosmological information is contained in the TimeDelay distance (7.14) through the angular diameter distances (7.2). Thus, from Time Delaymeasurements (if a reliable description of the lensing profile is given), one is able to reconstructthe Hubble factor without assuming any particular cosmological model. This is precisely the goalof the H0LiCOW collaboration , which within the COSMOGRAIL program employed TimeDelays measurements collected over the last decade to constrain the value of the cosmological pa-rameter H to a few percents level [167, 168, 169], with competitive precision with respect to othercosmological probes. Moreover, Time Delay measurements can also be used to put constraintson the Post-Newtonian parameter γ P P N , as discussed in Refs. [170, 171, 172, 173]. Furthermore,with optimistic assumptions on the surveys, in the next years the precision of observations willbe enough to provide a smoking gun for Dark Energy [174]. https://shsuyu.github.io/H0LiCOW/site/index.html ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.4 In the next sections we will discuss two new observables proposed by us in Ref. [7]: the
TimeDelay drift and the angular drift , which could be used to reconstruct H ( z ) and also to constrainviolations of the Equivalence Principle [8]. The redshift drift is the time variation of the redshift of a source due to the Hubble flow, seeRef. [164]. In an expanding universe described by the FLRW metric, one straightforwardly obtainsthe result that a photon is redshifted, and the redshift is given in terms of the scale factor asfollows: 1 + z = a a e , (7.15)where a is the scale factor evaluated at present time (which is the time of observation) and a e is the scale factor evaluated at the emission time.The derivative of the redshift with respect to the observation time t is the following: dzdt = 1 a e da dt − a a e da e dt = 1 a e da dt − a a e da e dt e dt e dt . (7.16)It is not difficult to show that: dt e dt = a e a , (7.17)and therefore, the redshift drift formula (7.16) becomes: dzdt = 1 a e da dt − a e da e dt e = (1 + z ) H − H ( z ) , (7.18)where the Hubble constant H and the Hubble parameter H ( z ) have appeared, so that measuring dz/dt would allow to study the evolution of the Hubble factor.Since the lens equation contains the angular diameter distances, we realize in Ref. [7] that theapparent positions of the source will acquire a time dependence due to the Hubble flow. Thus,applying Eq. (7.18) to the lens equation we predict the following angular drift and the TimeDelay drift for a point mass:2 ˙ θ E θ = H − H ( z L )1 + z L , ˙∆∆ = K (cid:18) H − H ( z L )1 + z L (cid:19) , (7.19)where a dot indicates derivation with respect to the observer time ˙ = d/dt , and K is a factor oforder unity given by: K = ln − θ − θ + + θ − + θ + θ − − θ + ln − θ − θ + + θ − θ − θ + θ − . (7.20)The above equations tell us that the entity of these drifts is of order H . Since H ∼ − s − , we conclude that the angular drift is of order of 10 − arc seconds per year, which with thecurrent precision of observation, ∼ . ∼ yr to be detected. TheTime Delay drift is instad of order ∼ − arc seconds per year, which would require 10 yearsto accumulate a drift detectable by current experiments, with sensitivity of order of days. ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.4 As we saw in the previous section, drift effects from Strong Lensing in GR are too small tobe detected with current experiments, being of order ∼ H . On the other hand, the situationcould change if we move to different theories of gravity, where the presence of a modified Poissonequation will change the time dependence of the lensing potential. From the phenomenologicalpoint of view, many alternative theories of gravity modify the strength of the gravitationalinteraction by inducing an effective gravitational coupling G eff , replacing the Newton constant.For example, we already saw that in VAAS gravity the time dependence of G eff is given byEq. (5.43). In f ( R ) theories of gravity instead we have G eff ∼ G N / f R , so that G eff acquires atime dependence from the term ( df /dt ) ˙ R − . However, in the expression for the deflection angle,we have a degeneracy between the mass of the lens and the gravitational coupling G . Thus, itwould not be possible in principle to distinguish between a stronger coupling or a heavier mass.On the other hand, the situation is different if we consider drift effects. In Ref. [8] we studiedthe changes induced on the Time Delay drift by a time dependent G , and analyzed for illustrativepurposes the entity of the constraints on ˙ G/G which can be extrapolated with the precision ofcurrent experiments.For the case of a point mass, Eqs. (7.19) become:2 ˙ θ E θ E = H − H ( z L )1 + z L + ˙ GG , (7.21)˙∆∆ = K (cid:32) H − H ( z L )1 + z L + ˙ GG (cid:33) , (7.22)where it is understood that G is the gravitational coupling at the redshift of the lens. Theabove equations show explicitly that drift effects in the context of Strong Lensing, contrary towhat happens with spectroscopic measurements, are sensitive to variations of the gravitationalcoupling. Since a signal of order H − H L / (1 + z L ), assuming a realistic cosmological evolution, isbeyond the sensitivity of current observations, we can convert the bounds on the drifts in upperbounds on the variation of G .Since Time Delay measurements are generally more precise than angular ones we need to obtainan expression for the time derivative of Eq. (7.13) for an extended lens profile. Time Delay drift for extended lenses
To begin with let us compute the time derivative of the lensing potential:˙ ψ ( β ) = 2 c D LS D L D S (cid:18) z L ) dz L dt (cid:90) β dλ Φ + ddt (cid:18)(cid:90) β dλ Φ (cid:19)(cid:19) . (7.23)The time dependence of the second term in the right hand side of the latter equation comesfrom the change in time of the light ray path due to a variation of β and from the Newtonianpotential. The former is difficult to compute exactly because it is in general difficult to describeprecisely the curve identified by the light ray path. On the other hand, it is reasonable to assumethat the induced variation of the curve is small and does not contribute to the support of theintegral in Eq. (7.7). In particular, this is true if we evaluate the above integral within the From now on we will drop the subscript on G eff to unburden the notation. ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.4 Born approximation, i.e. along the unperturbed light path, as it is customary in Strong Lensingapplications where Φ /c (cid:28) ψ ( β ) = ψ ( β ) (cid:18) H − H ( z L )1 + z L (cid:19) + ψ ( β ) (cid:82) β dλ ˙Φ (cid:82) β dλ Φ . (7.24)The above equation can be further simplified if we consider a static distribution of matter.Indeed, in this case the only time dependence of the Newtonian potential is through the effectivegravitational coupling G , so that we have:˙ ψ ( β ) = ψ ( β ) (cid:32) ˙ GG + H − H ( z L )1 + z L (cid:33) . (7.25)In concrete Time Delay measurements we have to take into account corrections due to thepresence of mass distributed along the line of sight. This is done by introducing a parametercalled external convergence , κ ext , and defining the real Time Delay distance as: D real ∆ t ≡ D ∆ t − κ ext , (7.26)see for example Ref. [166]. If we assume that the external convergence has a time dependencethis would be inherited by the Time Delay distance (7.14).Combining the latter and Eq. (7.25) we can finally write down the Time Delay drift for anextended lens with a static distribution of mass:˙∆ ij ∆ ij = (cid:32) ˙ GG + H − H ( z L )1 + z L (cid:33) (cid:34) D ∆ t ( β i − α ) c ∆ ij − D ∆ t ( β j − α ) c ∆ ij (cid:35) − ˙ κ ext − κ ext . (7.27) Estimating the variation of the Time Delay from current data
We now illustrate how the data can put constraints on the variation of the Time Delay. To thisend, we use the package PyCS3 from the COSMOGRAIL program , see Refs. [175, 176]. Thedata used are the simulated light curves used in [176], produced in the context of the blind TimeDelay measurement competition named Time Delay Challenge 1 (TDC1) [177], and from thequasar DES J0408-5354 [178]. We split the total time of observations in two equal time periods.Each period consists of 658 days for the trial curves, and of 93 days for DES J0408-5354. TheTime Delay between each image is then calculated for each period. The Time Delay estimatesare shown in App. D and summarized in Table 7.1. From it we can readily estimate the relativevariation (7.27) as: ˙∆ ij ∆ ij = ∆ ij ( t + δt ) − ∆ ij ( t ) δt ∆ ij ( t ) = ∆ I + IIij − ∆ Iij ∆ Iij δt , (7.28)We display the results in Table 7.2. Available here. ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.5 ∆ AB ∆ AC ∆ AD ∆ BC ∆ BD ∆ CD Trial I − . +4 . − . − . +6 . − . − . +9 . − . − . +7 . − . +4 . +9 . − . +17 . +10 . − . Trial I+II − . +2 . − . − . +1 . − . − . +9 . − . − . +1 . − . − . +7 . − . − . +8 . − . DES 0408 WFI I − . +15 . − . − . +50 . − . − . +34 . − . − . +38 . − . − . +25 . − . − . +19 . − . DES 0408 WFI I+II − . +6 . − . − . +5 . − . − . +11 . − . − . +8 . − . − . +11 . − . − . +14 . − . Table 7.1.: Time Delay ∆ between four images from a simulated quasar and the DES J0408-5354quasar [178]. Each image is labeled from A to D . The numbers I and I+II indicatethat ∆ was measured over the first half of the period of observations, or over thewhole period, respectively. All values are given in days. | ˙∆ / ∆ | AB | ˙∆ / ∆ | AC | ˙∆ / ∆ | AD | ˙∆ / ∆ | BC | ˙∆ / ∆ | BD | ˙∆ / ∆ | CD Trial ( × − ) X 27 . ± . . ± . × − ) 61 . ± . . ± . . ± . . ± . day − . The X’s represent values with uncertainty biggerthan the central value, and are thus omitted. Estimated constraint on ˙ G/G
Through Eq. (7.27) it is possible to relate constraints on the relative time variation of ∆ ij toupper bounds on the variation of ˙ G/G . The external convergence time dependence is difficult toevaluate. We expect it to depend explicitly on ˙
G/G , similarly to the potential generated by thelens, with the two contributions having the same sign, but we will assume that it is negligible withrespect to the precision of current observations. We want to stress however, as we discussed in theprevious sections, that it is in principle possible to disentangle such contribution by consideringdifferences of Time Delays drifts of multiple images. As we already saw the drift due to theHubble flow is of order of H ∼ − s − , so we will neglect it as well. Another effect thatcould be of relevance is the time variation of the Time Delay due to the peculiar velocity of thelens galaxy and its transverse motion. On the other hand, according to Refs. [179, 180], thiscontribution is estimated to be of the order of a few seconds per year, so that the effect, eventhough it is bigger than the cosmological one due to the Hubble flow, is still not appreciable withcurrent precision. Under these assumptions, and roughly estimating the term inside the squarebracket of Eq. (7.27) to be of order 1, the values of Table 7.2 can be directly converted into upperbounds on the time variation of the effective gravitational coupling. The results are reported inTable 7.3. | ˙ G/G | AB | ˙ G/G | AC | ˙ G/G | AD | ˙ G/G | BC | ˙ G/G | BD | ˙ G/G | CD Trial ( × − ) X 1 . ± . . ± . × − ) 2 . ± . . ± . ± . . ± G/G in yr − from the simulated quasar andDES J0408-5354. The X’s represent values with uncertainty bigger than the centralvalue, and are thus omitted. ersonal Contribution:Strong Lensing for testing Gravity and Cosmology Section 7.5 We have introduced two novel observables in the contest of Strong Lensing, i.e. the Time Delaydrift and the angular drift. We have shown that measuring these effects one is able to estimate H ( z ), and thus reconstruct the cosmological evolution and the effective equation of state parame-ter of the Universe w eff . We have shown that in a modified gravity framework where the effectivegravitational coupling become time dependent, those drifts earn a contribution ˙ G/G , and thustheir measurements could be used to constraint both violations the Equivalence Principle andDE models. Unfortunately, the current precision of the observation is not enough to detect thesedrifts so that the constraints we obtain are currently not very competitive. On the other hand,they could improve already in the near future with upcoming data, see for example Ref. [181],or simply increasing the observational time. Moreover, as discussed in Refs. [179, 180, 182], if inthe future strongly lensed repeating FRB will be detected, they will provide Time Delay mea-surements of such extremely high precision, nominally of the order of seconds, that even redshiftdrift effects due to the Hubble expansion will be appreciable. In this scenario, the impact forcosmological implications of the drifts we proposed in this chapter is very promising, and we keephigh expectations for the future. HAPTER 8.
Final Considerations
I managed to get a quick PhD — thoughwhen I got it I knew almost nothing aboutphysics. But I did learn one big thing:that no one knows everything, and youdon’t have to.
Steven Weinberg
Summary
Research on DE is a tricky business. In principle, one should just try to answer to the questionof what is causing the accelerated expansion of the Universe, but in practice this is just the firstdomino tile of a long chain. The Cosmological Constant Λ seems to be the most reasonable andsimplest candidate of Dark Energy, in particular because together with the CDM paradigm isable to safely match a number of different observations. However, with the outstanding precisionreached in the last years in observational cosmology, the concordance model seems to be notso concordant, with the most uncomfortable question being what is the value of the Hubbleparameter now. Curiously, this seems to be an evergreen question in cosmology since from itsbirth, and it is inspiring the tenacity shown by the scientific community through the last hundredyears in looking for a satisfactory answer. We tried to give an overview of the ΛCDM model inchapter 2 of this work; our main goal was to highlight the pillars on which it is built on and thekind of questions that it is able to answer. Of course our presentation is by no means exhaustiveor complete by itself, but we hope it is able at least to address a satisfactory number of referencesfor the curious reader.DE is so interesting because it could be a window towards new physics or, depending on one’spersonal perspective, it is already a manifestation of it. The community working on this topichas undoubtedly shown a fair amount of creativity in the last two decades, both in boosting ortrying to contain the proliferation of possible candidates of DE. In chapter 3 we tried to give areasonable classification of the kind of models that were proposed in the last years based on theLovelock theorem. Keeping into account that a proper treatment of the topic would have requiredmore than a textbook, for each class of models we presented some examples in an attempt oftransmitting the flavor and the potential of these proposals.The ultimate goal of this thesis is to present the results of our research from the last fouryears. Most of it was oriented on models of DE which does not introduce in the playground newdegrees of freedom, but instead attempt to ascribe the accelerated expansion of the Universe to inal Considerations Section 8.0 geometrical modifications of the gravitational interaction.An interesting phenomenology arises if in such geometrical description we relax the assumptionof locality for the field equations. Ultimately, the main motivation for considering these kindsof models is that quantum mechanics seems to be nonlocal at fundamental level, and such couldalso be a quantum description of gravity. In chapter 4 we review the theoretical grounds and themathematical formalism of nonlocal models of gravity based on the inverse d’Alembertian of theRicci scalar (cid:3) − R .In chapter 5 we present the results of our research on nonlocal gravity models, in particular [4,5]: • We found that the Late-times asymptotic equation of state has a common behavior inthose models which contain explicitly a term (cid:3) − R in the Lagrangian. In particular, evenif the Hubble factor does not approach a constant value, we found that ˙ H/H → H → ∞ due to the divergence of (cid:3) − R . We also show that this does not happen for theDW model if the distortion function f ( (cid:3) − ) R is chosen in such a way that f ( ∞ ) (cid:54) = ∞ . • We studied VAAS gravity within a dynamical system approach and found no stable criticalpoints. On the other hand, a qualitative study of its field equations show that it is indeedpossible to produce at background level an evolution history compatible with ΛCDM fromthe radiation dominated epoch until today. • Still in VAAS gravity, we have shown that on local scales the effects of the nonlocal modi-fications is encoded in a slip parameter and in a modification of the effective gravitationalcoupling η, Y (cid:54) = 1. • We checked consistency of the model with LLR constraints, which are not passed. How-ever, we found that the field equations admit spherically symmetric static solutions which,if realized and stable, would satisfy trivially LLR constraints and for which one of the grav-itational potential closely resemble the solution one would obtain in standard GR for theSchwarzschild-deSitter solution.In chapter 6 we discuss a geometrical theory of gravity containing higher order derivative termswhich we proposed in Ref. [6]. The theory is based on the introduction of the anticurvature scalar A , which is the trace of the inverse of the Ricci tensor A µν = R − µν . We derive the general fieldequations for an f ( R, A ) theory and assess their potential for cosmological implications. Wefound that an interesting phenomenology arise already for the simplest choices of f , but alsoa no-go theorem which claims that polynomial Lagrangians are not able to reproduce a viablecosmological history. Finally, we present some choices of f that evades the no-go theorem andthus are worth further investigation.Finally, in chapter 7 we present the results of our works [7, 8] about the potential of StrongLensing observables to study cosmology and DE. In particular, we introduce two new quantities,the angular drift and the Time Delay drift , which are a manifestation of the Hubble flow ina lensed system. These drifts, if measured, would allow for a reconstruction of H ( z ) at theredshift of the lens, and thus potentially very helpful for estimating the effective equation ofstate parameter of the Universe w eff . We also show that in a modified theory of gravity witha time dependent gravitational coupling, a similar effect arise, and we can use Strong LensingTime Delay observations to constrain violations of the Equivalence Principle. The entity of thesedrifts is unfortunately beyond the sensitivity of current experiment, and results in very weakconstraints on ˙ G/G . However, we also discuss some possible future perspectives that would makethem appealing. inal Considerations Section 8.0 Outlooks
Most of the nonlocal models of gravity presented in the literature seems to be ruled out by LLRconstraints. On the other hand they are motivated by fundamental physics, and as we saw inRef.[5] in most cases they provide a useful mechanism to trigger an accelerated expansion of theUniverse only at Late-times. For these reasons we are currently trying to understand if they canbe used, together with a Cosmological Constant Λ, to alleviate or solve the H tension by changingthe rate of expansion of the universe today without affecting CMB measurements or violatingLLR constraints. Regarding VAAS, a perturbative analysis beyond the small scales regime isstill lacking, and is crucial for properly assess the appealing of the model with respect to ΛCDM.We address this analysis for future works. Inverse Ricci gravity offers a completely new class ofmodified gravity theories, and as we saw it can introduce a very interesting phenomenology. Onthe other hand, containing higher order derivatives, one expect that it is in general unstable unlessthe Lagrangian is degenerate. We address to future works a research of which kind of f ( A, R ) willresult in a degenerate Lagrangian capable of escaping the no-go theorem. Finally, the increasingprecision of Strong Lensing measurements motivate a deeper investigation of the cosmologicalinformation we can extract from them. Moreover, being equivalent to an interferometer with armsof astrophysical size, a strongly lensed system could be sensitive to the passage of a gravitationalwave. This could be very interesting for multi-messenger astronomy, and we address to futureworks a quantitative estimation of the GW impact on Time Delay measurements to assess thepotential of this idea. PPENDIX A.
Critical points in VAAS gravity
We want to study the critical points of the system (5.1) satisfying the constraint (5.2).
Critical points at finite distance
Consider first the case where 2 αV = 1, which is special because it eliminates ˙ H from the fieldequations. From the Klein Gordon equations we see that V = 1 / (2 α ) = constant only if m = 0.The latter is a parameter of the theory so it is not necessarily vanishing and thus, unless m = 0, V cannot be constant. With V = 1 / (2 α ) = constant, then ˙ V = Y = 0 and thus from Eq. (5.2)we get ρ = 0. We are just left with the Klein Gordon equation for U while H is completelyarbitrary, since we have lost the equations ruling its dynamics. Since providing a suitable H isone of the objectives of the model, we do not consider this possibility anymore.Now we consider 2 αV (cid:54) = 1. The right hand sides of the last three equations of the dynamicalsystem (5.1) vanish when X = Y = 0 and m = 0. Again, the latter is a parameter of the theoryso it is not necessarily vanishing. Thus if m (cid:54) = 0 we can already conclude that there are nocritical points at finite distance.Let us consider now the subclass of theories for which m = 0, i.e. with R f = 0. Demandingthe vanishing of the right hand sides of the first three equations of system (5.1) and from Eq. (5.2)we have, taking into account X = Y = m = 0:3 H (1 − αV ) = ρ − P M P l , (A.1) − H ( ρ + P ) = 0 , (A.2) H (1 − αV ) = P − ρ M P l , (A.3)3 H (1 − αV ) = ρM P l . (A.4)Now, from the second equation above we either have that H = 0 or P = − ρ , i.e. a vacuumenergy equation of state is required. If our fluid model has not such equation of state, then theonly possibility is H = 0 and thus ρ = P = 0. This critical point represents Minkowski space.Note that U and V may assume whatever constant value, except V = 1 / (2 α ).On the other hand, let us assume that indeed the fluid content satisfies a vacuum energyequation of state, i.e. P = − ρ . In this case, the above system becomes:3 H (1 − αV ) = ρM P l , (A.5) H (1 − αV ) = − ρM P l . (A.6) ritical points in VAAS gravity Section A.0 Summing the two equations we arrive at:4 H (1 − αV ) = 0 . (A.7)Since 2 αV (cid:54) = 1, we have again H = 0 and thus the same Minkowski critical point as before.There is a caveat here. When m = 0 the evolution of U is disentangled from the one of H .Therefore, we can reduce our dynamical system (5.1) to:˙ H = 11 − αV (cid:20) ρ − P M P l + 2 αHY (cid:21) − H , (A.8a)˙ ρ = − H ( ρ + P ) , (A.8b)˙ Y = − HY , (A.8c)˙ V = Y . (A.8d)From this system is not difficult to see that there is a critical point P = − ρ = − M P l (1 − αV ) 3 H = constant , (A.9)which represents a de Sitter phase. With H > U becomes:¨ U + 3 H ˙ U = − H , (A.10)which has the special, non-constant solution U = − Ht . Critical points at infinite distance
In order to investigate the critical points at infinity, we build the Poincar´e hyperspherein thevariable space plus one dimension. The equation of the sphere is the following: h + r + p + x + y + u + v + z = 1 , (A.11)where: H ≡ hz , ρ ≡ rz , P ≡ pz , X ≡ xz , Y ≡ yz , U ≡ uz and V ≡ vz . (A.12)System (5.1) thus becomes: z ˙ h = A (1 − h ) − h ( rB + xC + yD + uE + vF ) , (A.13a) z ˙ r = B (1 − r ) − r ( hA + xC + yD + uE + vF ) , (A.13b) z ˙ x = C (1 − x ) − x ( hA + rB + yD + uE + vF ) , (A.13c) z ˙ y = D (1 − y ) − y ( hA + rB + xC + uE + vF ) , (A.13d) z ˙ u = E (1 − u ) − u ( hA + rB + xC + yD + vF ) , (A.13e) z ˙ v = F (1 − v ) − v ( hA + rB + xC + yD + uE ) , (A.13f) z ˙ z = − ( hA + rB + xC + yD + uE + vF ) , (A.13g) ritical points in VAAS gravity Section A.0 where the equation for ˙ z is obtained from Eq. (A.11) and the terms A , B , C , D , E and F aredefined as follows: A ≡ zz − αv (cid:20) z ( r − p )2 M P l + m z z − u ) + 2 αhy (cid:21) − h , (A.14a) B ≡ − h ( r + p ) , (A.14b) C ≡ − hx − h − zz − αv (cid:20) z ( r − p )2 M P l + m z z − u ) + 2 αhy (cid:21) , (A.14c) D ≡ m z α − hy , (A.14d) E ≡ zx , (A.14e) F ≡ zy . (A.14f)The critical points of the above system corresponding to z = 0 are critical points at infinity. Inorder to find them, let us first define a new time parameter such that z ddt ≡ ddτ ≡ (cid:48) and thefunctions: G ≡ h (cid:0) h + γr + x + y + 2 hx − (cid:1) , (A.15)˜ G ≡ h (cid:0) h + γr + x + y + 2 hx − γ (cid:1) , (A.16)where we have assumed a barotropic equation of state linking pressure to density: P = ( γ − ρ . (A.17)The dynamical system (A.13) for z = 0 can thus be written as: h (cid:48) = hG , (A.18a) r (cid:48) = r ˜ G , (A.18b) x (cid:48) = xG − h , (A.18c) y (cid:48) = yG , (A.18d) u (cid:48) = u ( G + h ) , (A.18e) v (cid:48) = v ( G + h ) . (A.18f)It is important to emphasise that the solutions of the above system must be compatible with theFriedmann equation, which in terms of the variables on the Poincar´e sphere provides: (cid:16) − α vz (cid:17) h z + m u z − αy z (6 h + x ) − r zM P l = 0 . (A.19)Multiplying the above equation for z and then considering z = 0, we obtain: αvh = 0 . (A.20)So that at infinity Friedmann equation imposes that at least one among α , v , h is vanishing. It iseasy to see that when h = 0 the function G vanishes identically, so we have a critical hyperplanein the variables space corresponding to Minkowski spacetime.We will discuss later the stability of this critical hypersurface, and focus now on the only otherinteresting case, v = 0. In this case, Friedmann equation becomes: h − αy (cid:16) h + x (cid:17) = 0 . (A.21) We do not consider α = 0 since it simply turns off the nonlocal interacting term. ritical points in VAAS gravity Section A.0 Since h (cid:54) = 0, then from Eqs. (A.18a) and (A.18d) we have that: y = h + K , (A.22)where K is an integration constant. From Eq. (A.21) we then have: x = 3 h αy − h = 3 h α ( h + K ) − h . (A.23)This result allows us to rewrite Eq. (A.18c) as follows: x (cid:48) = G (cid:20) h α ( h + K ) − h (cid:21) − h , (A.24)and we see that it is impossible to have both (A.18a) and (A.24) vanishing without G = h = 0,and so there are no critical points, but h = 0, at infinite distance. Linearisation and stability of the critical point
In order to calculate the stability of the critical points at h = 0, let us linearise system (A.18)around the critical point h = 0, i.e, h = h + (cid:15) , r = r + η , x = x + χ , y = y + ϕ , u = u + λ and v = v + σ . Consequently, the linearised dynamical system is given by: (cid:15) (cid:48) = 0 , (A.25a) η (cid:48) = r (cid:0) γr + x + y − γ (cid:1) (cid:15) , (A.25b) χ (cid:48) = x (cid:0) γr + x + y − (cid:1) (cid:15) , (A.25c) ϕ (cid:48) = y (cid:0) γr + x + y − (cid:1) (cid:15) , (A.25d) σ (cid:48) = u (cid:0) γr + x + y (cid:1) (cid:15) , (A.25e) λ (cid:48) = v (cid:0) γr + x + y (cid:1) (cid:15) . (A.25f)Usually the stability of the critical point is studied by means of the Jacobian matrix, but un-fortunately it is degenerate in our case. However, we easily recognise that in the above systemof equations the perturbation (cid:15) is constrained to be a constant by (A.25a). Note also that thecombinations (cid:0) γr + x + y − (cid:1) and (cid:0) γr + x + y − γ (cid:1) are constant as long as we assume atime-independent equation of state. Thus all the perturbations with the exception of (cid:15) , which isconstant, go linearly with time showing an unstable behavior. PPENDIX B.
Qualitative dynamic of nonlocal models
Qualitative behavior of U The Klein Gordon equation for U in FLRW backround, defining X ≡ U (cid:48) and using N = log a astime coordinate is: X (cid:48) + (3 + ξ ) X + 6 (2 + ξ ) = 0 . (B.1)The formal solution for X (cid:48) is: X ( N ) = C e − F ( N ) − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) [2 + ξ ( ¯ N )] , (B.2)which choosing vanishing initial conditions give: X ( N ) = − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) [2 + ξ ( ¯ N )] . (B.3)It is straightforward to show that 0 ≤ X ≤
6, indeed rewrite the solution for X as follows: X ( N ) = − e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) [3 + ξ ( ¯ N )] + 6 e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) . (B.4)The first integral can be cast as: X ( N ) = − e − F ( N ) (cid:90) NN i d ¯ N d ( e F ( ¯ N ) ) d ¯ N + 6 e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) , (B.5)and thus: X ( N ) = − e − F ( N ) + 6 e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) . (B.6)Being the second and third terms on the right hand side strictly positive, we have then that X ( N ) > − Qualitative behavior of VAAS
During the radiation-dominated epoch one has ξ = − X = 0 , (B.7)This implies that U is a constant, and this constant must be zero, because of our initial condition.On the other hand, Y ( N ) = − e − ( N − N i ) (cid:90) NN i d ¯ N e ( ¯ N − N i ) m H Ω r e − N = − m H Ω r e − N (cid:0) e N − e N i (cid:1) , (B.8) ualitative dynamic of nonlocal models Section B.0 from which: ˜ V = − m H Ω r e N − m H Ω r e N i − N + C . (B.9)Since ˜ V ( N i ) = 1, we then have:˜ V = − m H Ω r e N − m H Ω r e N i − N + m H Ω r e N i + 1 . (B.10)This is very small for N large and negative, so one can basically take ˜ V = 1.During the matter-dominated epoch we have ξ = − / X = − e − N/ (cid:90) N ˜ N i d ¯ N e N/ = − e N i / , (B.11)where ˜ N i is some new initial value, chosen in the matter-dominated epoch. Being this large andnegative, we neglect the exponential with respect to − U follows: U = C − N , (B.12)with C integration constant. For Y we have a solution similar to the one we found for theradiation-dominated case. We simplify it a little bit, writing:˜ V = 1 − m H Ω m e N . (B.13)We have neglected all the exponentials contributions containing the initial e-folds number because,even in the matter-dominated epoch, it is very small. It is only at late times that ˜ V starts togrow different from one.In vacuum, i.e. when matter and radiation dilute, the first Friedmann equation is:3 ˜ V = − m U H − Y X ) + ρM P l H . (B.14)Since U < Y < X > − ρ >
0, we can conclude that ˜
V >
0. On the otherhand,
Y < V always decreases. So, in order for ˜ V to decrease from one to zero,without becoming negative, we need that m /H ≥ H cannot tend to zero in the far future, for large N , but it must increasein order to guarantee that m /H (cid:28)
1. On the basis of this argument, we can conclude that atlate-times, when the matter is completely diluted:3 ˜ V ∼ − m U H , (B.15)this however provided that X does not diverge, otherwise we cannot neglect the product XY ingeneral.Combining the two Friedmann equations, we have that: ξ = − V (cid:20) − Y + m (1 − U )2 H + ρ − P M P l H (cid:21) . (B.16) ualitative dynamic of nonlocal models Section B.0 For non-exotic fluids one has ρ − P > ξ > −
3. At latetimes, according to our previous discussion, we have that: ξ ∼ − − V m U H ∼ . (B.17)Hence, the effective equation of state always tends to −
1. We can check now directly that X does not diverge from its solution, computed with ξ = 0. One obtains: X = − , (B.18)i.e. a U growing indefinitely negative, say: U = C − N . (B.19)With this solution, we are left with two equations: Y + 3 ˜ V = − m H U , (B.20) Y (cid:48) + 3 Y = − m H . (B.21)The derivative of the left hand side of the first equation is equal to the left hand side of thesecond equation. Hence, one finds that: ξ = − C − N = − U , (B.22)and we have the solution for H : H = C | C − N | / = 3 | U | / . (B.23)The solution for H (cid:48) is instead: H (cid:48) = − C C − N | C − N | / . (B.24) (cid:3) − R model This model was proposed in Ref. [144] and is motivated from studies of nonperturbative latticequantum gravity. The Lagrangian is: L = L EH − M (cid:3) R , (B.25)and is localized introducing the auxiliary fields: (cid:3) U = − R , (B.26) (cid:3) S = − U , (B.27) (cid:3) Q = − , (B.28) (cid:3) L = − Q . (B.29) ualitative dynamic of nonlocal models Section B.0 The field equations are: h = γ (cid:2) V + W U + h (cid:0) Z + 6 Z (cid:48) − U (cid:48) Z (cid:48) − V (cid:48) W (cid:48) (cid:1)(cid:3) + Ω R e − N + Ω M e − N , (B.30) ξ = 12 (cid:0) − γZ (cid:1) (cid:20) − R e − N − M e − N h + 32 γ (cid:18) Wh − Z (cid:48) + U (cid:48) Z (cid:48) + V (cid:48) W (cid:48) (cid:19)(cid:21) , (B.31) U (cid:48)(cid:48) + (3 + ξ ) U = 6 (2 + ξ ) , (B.32) V (cid:48)(cid:48) + (3 + ξ ) V (cid:48) = Uh , (B.33) W (cid:48)(cid:48) + (3 + ξ ) W (cid:48) = 1 h , (B.34) Z (cid:48)(cid:48) + (3 + ξ ) Z (cid:48) = Wh . (B.35)defining ˜ Z = 1 − γZ we can rewrite the late-times Friedmann equations (B.30) (B.31), whenmatter is completely diluted, as:˜ Z = γ h ( U W + V ) − ˜ Z (cid:48) (cid:18) − U (cid:48) (cid:19) − γW (cid:48) V (cid:48) , (B.36) ξ = 12 ˜ Z (cid:20) γW h + 4 ˜ Z (cid:48) − ˜ Z (cid:48) U (cid:48) + 3 γV (cid:48) W (cid:48) (cid:21) , (B.37)while the KG equation (B.35) for ˜ Z is:˜ Z (cid:48)(cid:48) + (3 + ξ ) ˜ Z (cid:48) = − γW h , (B.38)whose formal solution for ˜ Z (cid:48) is given by:˜ Z (cid:48) = − γe − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) W h . (B.39)We write for convenience also the formal solutions for V (cid:48) and W (cid:48) : V (cid:48) = e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) Uh , (B.40) W (cid:48) = e − F ( N ) (cid:90) NN i d ¯ N e F ( ¯ N ) h . (B.41)Since we chose initial conditions W i = 0 and since W (cid:48) > W >
0. Thisimplies that ˜ Z (cid:48) <
0; on the other hand Eqs. (5.5) imply V (cid:48) >
0, 0 < U (cid:48) < U >
0. In orderto understand the behavior of ˜ Z let us define the function T : T ≡ U W + Vh − W (cid:48) V (cid:48) , (B.42)taking its time derivative and using Eqs. (B.34) and (B.33) we are able to set up a differentialequation for T : T (cid:48) + 2 ξX = U (cid:48) Wh + 6 V (cid:48) W (cid:48) . (B.43) ualitative dynamic of nonlocal models Section B.0 The formal solution of Eq. (B.43) is given by: T ( N ) = 1 h ( N ) (cid:90) NN i d ¯ N (cid:0) U (cid:48) W + 6 h V (cid:48) W (cid:48) (cid:1) − C X h ( N ) , (B.44)where C X is an integration constant. Since X ( N i ) = 0 we can conclude from Eq. (B.44) that T ( N ) >
0. This in turns implies that the right hand side of Eq. (B.36) it’s always positive, andwe can conclude that, asymptotically, ˜
Z > Z is positive definite, we must have asymptotically ˜ Z (cid:48) →
0. It is straightforward to realizefrom Eq. (B.39) that this is possible only if h is a monotonic growing function that grows fasterthan W/
2. On the other hand, W is also a monotonic growing function since W (cid:48) >
0. Inparticular, since h grows faster than W , it also grows faster then a constant, and so we concludefrom Eq. (B.41) that W (cid:48) →
0. Using the latter in Eq. (B.36) we are left with:˜ Z ∼ γ h ( V + W U ) . (B.45)Using the above result in Eq. (B.37) we finally obtain: ξ ∼ VW + U ∼ , (B.46)then once again we have w eff → − PPENDIX C.
Field equations in Inverse Ricci gravity
To begin with let us consider the following action:Consider first the basic Action S = (cid:90) √− gd x ( R + αA ) , (C.1)where the anticurvature scalar A is the trace of A µν A µν = R − µν . (C.2)By differentiating Eq. (6.1), we see that δA µτ = − A µν ( δR νσ ) A στ . (C.3)We have then δS = (cid:90) d x ( Aδ √− g + √− gA µν δg µν + √− gg µν δA µν )= (cid:90) d x √− g ( 12 Ag µν δg µν + A µν δg µν + g µν δA µν ) , (C.4)and since δR αβ = ∇ ρ δ Γ ρβα − ∇ β δ Γ ρρα , (C.5)we obtain δA µν = − A µα ( ∇ ρ δ Γ ρβα − ∇ β δ Γ ρρα ) A βν = − A µα ( g ρλ ∇ ρ ( ∇ α δg βλ + ∇ β δg λα − ∇ λ δg αβ ) − g ρλ ∇ β ( ∇ α δg ρλ + ∇ ρ δg λα − ∇ λ δg αρ )) A βν = − A µα g ρλ ( ∇ ρ ∇ α δg βλ − ∇ ρ ∇ λ δg αβ − ∇ β ∇ α δg ρλ + ∇ β ∇ λ δg αρ + [ ∇ β , ∇ ρ ] δg λα ) A βν . (C.6)Using integration by parts, this becomes g µν δA µν = − g µν g ρλ ( δg βλ ∇ α ∇ ρ ( A µα A βν ) − δg αβ ∇ λ ∇ ρ ( A µα A βν ) − δg ρλ ∇ α ∇ β ( A µα A βν )++ δg αρ ∇ λ ∇ β ( A µα A βν )) + δg αρ ∇ λ ∇ β ( A µα A βν ))= 12 δg ικ ( − g ρι ∇ α ∇ ρ A µα A κµ + ∇ ( A µι A κµ ) + g ικ ∇ α ∇ β ( A µα A βµ )) . (C.7) ield equations in Inverse Ricci gravity Section C.0 So finally the variation is δg µν ( 12 Ag µν + A µν + 12 ( − g ρµ ∇ α ∇ ρ A σα A νσ + ∇ ( A σµ A νσ ) + g µν ∇ α ∇ β ( A σα A βσ ))) . (C.8)Together with the variation of the standard Hilbert-Einstein Lagrangian δg µν ( − Rg µν + R µν ) = − δg µν ( − Rg µν + R µν ) , (C.9)we obtain finally the equations for the Action (C.1) R µν − Rg µν − αA µν − αAg µν + α (cid:0) g ρµ ∇ α ∇ ρ A ασ A νσ − ∇ A µσ A νσ − g µν ∇ α ∇ ρ A ασ A ρσ (cid:1) = T µν , (C.10)where we used the fact that A ασ A νσ = A ατ g τσ A σν = A ατ A ντ = A ασ A νσ = A νσ A ασ and we employedunits in which 8 πG = 1. It can be show that the left-hand side of Eq. (C.10) is divergenceless,as it should be in order to satisfy the Bianchi identities.The extension to any Lagrangian f ( R, A ) is quite straightforward: δS = (cid:90) d x √− g ( − f ( R, A ) g µν δg µν + f A A µν δg µν + f A g µν δA µν + f R R µν δg µν + f R g µν δR µν ) , (C.11)where f R = ∂f /∂R and f A = ∂f /∂A . Then we have f R R µν − f A A µν − f g µν + g ρµ ∇ α ∇ ρ f A A ασ A νσ − ∇ ( f A A µσ A νσ ) − g µν ∇ α ∇ β ( f A A ασ A βσ ) − ∇ µ ∇ ν f R + g µν ∇ f R = T µν . (C.12) PPENDIX D.
Estimating Time delay Uncertainties withPyCS3
In this appendix, we display the time delays and their uncertainties obtained with PyCS3.Uncertainties are obtained by simulating light curves close to the data, and randomizing thetrue time delay applied to each curve, see Sec. (3.2) of Ref. [181] for more details. The finalmarginalization is done performing a hybrid approach between the “free-knot spline” and the“regression difference” estimators as explained in Sec. (3.3) of Ref. [181]. The parameter τ thresh =0 indicates the marginalization is done over the two estimators. In each figure, the top panelsshow the final time delay estimates marginalizing over the two estimators. The middle figureshows the residuals for the spline fit to the data. The top row of the bottom panels show thedistribution of data residuals for mock curves (in gray) and data (in colors), whereas the bottompanels show their normalization over the number of runs z r . The time delay estimates for thesimulated curves over the whole period of observations ( ≈ .
316 days) is shown in Fig. D.1 whileFig. D.2 is over half the total period. Similarly, Fig. D.3 ( ≈
189 days) and Fig. D.4 show timedelays for the object DES J0408-5354. stimating Time delay Uncertainties with PyCS3 Section D.0 Figure D.1.: Time delay estimates for the simulated quasar over the full observation period. stimating Time delay Uncertainties with PyCS3 Section D.0 Figure D.2.: Time delay estimates for the simulated quasar over half of the observation period. stimating Time delay Uncertainties with PyCS3 Section D.0 Figure D.3.: Time delay estimates for the quasar DES J04078-5354 over the full observationperiod. stimating Time delay Uncertainties with PyCS3 Section D.0 Figure D.4.: Time delay estimates for the quasar DES J04078-5354 over half of the observationperiod. ibliography [1] A. G. Riess et al. Observational evidence from supernovae for an accelerating universe anda cosmological constant . Astron. J. arXiv:astro-ph/9805201 . Cited onpages 1, 3, and 10.[2] S. Perlmutter et al.
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