Chern-Simons Modified General Relativity
aa r X i v : . [ h e p - t h ] A ug Chern-Simons Modified General Relativity
Stephon Alexander a , c and Nicol´as Yunes b , c a Department of Physics and Astronomy, Haverford College, Haverford, PA 19041 b Department of Physics, Princeton University, Princeton, NJ 08544, USA c Institute for Gravitation and the Cosmos, Department of Physics, ThePennsylvania State University, University Park, PA 16802, USA
Abstract
Chern-Simons modified gravity is an effective extension of general relativity thatcaptures leading-order, gravitational parity violation. Such an effective theory ismotivated by anomaly cancelation in particle physics and string theory. In thisreview, we begin by providing a pedagogical derivation of the three distinct wayssuch an extension arises: (1) in particle physics, (2) from string theory and (3)geometrically. We then review many exact and approximate, vacuum solutions ofthe modified theory, and discuss possible matter couplings. Following this, we reviewthe myriad astrophysical, solar system, gravitational wave and cosmological probesthat bound Chern-Simons modified gravity, including discussions of cosmic baryonasymmetry and inflation. The review closes with a discussion of possible futuredirections in which to test and study gravitational parity violation.
Key words:
Chern, Simons, string theory, loop quantum gravity, parity violation
PACS: code, code
Email addresses: [email protected] (Stephon Alexander), [email protected] (Nicol´as Yunes).
Preprint submitted to Physics Reports 29 May 2018 ontents1 Introduction 42 The ABC of Chern-Simons and its Tools 6 Outlook 91References 92 Introduction
Over the last two decades we have experienced a wealth of observationaldata in the field of cosmology and astrophysics that has been critical in guid-ing physicists to our current fundamental theories. The combined CMBR [1],lensing [2], large scale structure [3] and supernovae data [4] all point to anearly universe scenario where at matter radiation equality, the universe wasdominated by radiation and dark matter. Soon after, the universe underwenta transition where it became dominated by a mysterious fluid, similar to acosmologcal constant (if it is not evolving), named dark energy. Both darkmatter and dark energy might be a truly quantum gravitational effect or sim-ply a modification of General Relativity (GR) at large distances.In the absence of a full quantum gravitational theory, how do we comeabout constructing a representative effective model? One of the most impor-tant unifying concepts in modern physics is the gauge principle, which in factplayed a seminal role in the unification of the strong, weak and electromag-netic interactions through the requirement that the action be invariant undera local SU (3) × SU (2) × U (1) Y gauge transformation. Given the success ofthe gauge principle, much effort has been invested in various branches of the-oretical and mathematical physics, unveiling deep connections between gaugetheories and geometry. The modern route to such gauge geometrical picture isvia the interpretation of Yang-Mills gauge fields as connections on a principalfibre bundle and the Riemann tensor as the curvature on the tangent bun-dle. Although the independent combination of the Standard Model of particlephysics and GR accounts for all four observed forces, a complete unificationis still lacking, in spite of their semblance as gauge theories.Gauge principles invariably point us in the direction of a peculiar, yetgeneric modification to GR that consists of the addition of a Pontryagin or“Chern-Simons” (CS) term to the action. Due to its gauge principle roots,such an extension connects many seemingly unrelated areas of physics, in-cluding gravitational physics, particle physics, String Theory and Loop Quan-tum Gravity. This effective theory is in contrast with other GR modificationsthat are not motivated by predictive elements of a more fundamental theo-retical framework. One of the goals of this report is to explore the emergenceof CS modified gravity from these theoretical lenses and to confront model-independent predictions with astrophysics, cosmology and particle physics ex-periments.Many have argued that studying an effective theory derived from StringTheory or particle physics is futile because, if such a correction to the Einstein-Hilbert action were truly present in Nature, it would be quantum suppressed.In fact, String Theory does suggest that the coupling constant in front of theCS correction should be suppressed at least at the electroweak scale level oreven the Planck scale level. Indeed if this were the case, the CS correctionwould be completely undetectable by any future experiments or observations.Other quantities exist, however, that String Theory predicts should also4e related to the Planck scale, yet several independent measurements and ob-servations suggest this is not the case. One example of this is the cosmologicalconstant, which according to String Theory should be induced by supersym-metry breaking. If this breakage occurs at the electroweak scale, then the valueof the cosmological constant should be approximately 10 eV , while if it oc-curs at the Planck scale it should be about 10 eV . We know today thatthe value of the cosmological constant is close to 10 − eV , no-where near theString Theory prediction.A healthy, interdisciplinary rapport has since developed between the cos-mology and particle communities. On the one hand, the astrophysicists con-tinue to make more precise and independent measurements of the cosmologicalconstant. On the other hand, the particle and String Theory communities arenow searching for new and exciting ways to explain such an observable value,thus pushing their models in interesting directions.Similarly, a healthy attitude, perhaps, is to view the CS correction as amodel-independent avenue to investigate parity violation, its signatures andpotential detectability, regardless of whether some models expect this correc-tion to be Planck suppressed. In fact, there are other models that suggest theCS correction could be enhanced, due to non-perturbative instanton correc-tions [5], interactions with fermions [6], large intrinsic curvatures [7] or smallstring couplings at late times [8,9,10,11,12,13,14,15,16,17].After mathematically defining the effective theory [Section 2], we shall dis-cuss its emergence in the Standard Model, String Theory and Loop QuantumGravity [Section 3]. We begin by reviewing how the CS gravitational termarises from the computation of the chiral anomaly in the Standard Modelcoupled to GR. While this anomaly is cancelled in the Standard Model gaugegroup, we shall see that it persists in generic Yang-Mills gravitational theo-ries. We then present the Green-Schwarz anomaly canceling mechanism andshow how CS theory arises from String Theory, leading to a Pontryagin cor-rection to the action in four-dimensional gravity coupled to Yang-Mills theory.The emergence of CS gravity in Loop Quantum Gravity is then reviewed, asa consequence of the scalarization of the Barbero-Immirzi parameter in thepresence of fermions.Once the effective model has been introduced and motivated, we shallconcentrate on exact and approximate solutions of the theory both in vac-uum and in the presence of fermions [Sections 4, 5 and 6]. We shall see thatalthough spherically symmetric spacetimes that are solutions in GR remainsolutions in CS modified gravity, axially-symmetric solutions do not have thesame fate. In the far field, we shall see that the gravitational field of a spinningsource is CS modified, leading to a correction to frame-dragging. Moreover, thepropagation of gravitational waves is also CS corrected, leading to an expo-nential enhancement/suppression of left/right-polarized waves that dependson wave-number, distance travelled and the entire integrated history of theCS coupling.With such exact and approximate solutions, we shall review the myri-5ds of astrophysical and cosmological tests of CS modified gravity [Sections 7and 8]. We shall discuss Solar-System tests, including anomalous gyroscopicprecession, which has led to the first experimental constraint of the model. Weshall then continue with a discussion of cosmological tests, including anoma-lous circular polarization of the CMBR. We conclude with a brief summary ofleptogenesis in the early universe as explained by the effective theory.This review paper consists of a summary of fascinating results producedby several different authors. Overall, we follow mostly the conventions of [18,19],which are the same as those of [20] and [21], unless otherwise specified. In par-ticular, Latin letters at the beginning of the alphabet a, b, . . . , h correspond tospacetime indices, while those at the end of the alphabet i, j, . . . , z stand forspatial indices only. Sometimes i, j, . . . , z will instead stand for indices repre-senting the angular sector in a 2 + 2 decomposition of the spacetime metric,but when such is the case the notation will be clear by context. Covariantderivatives in four (three) dimensions are denoted by ∇ a ( D i ) and partialderivatives by ∂ a ( ∂ i ). The Levi-Civita tensor is denoted by ǫ abcd , while ˜ ǫ abcd isthe Levi-Civita symbol, with convention ˜ ǫ = +1. The notation [ A ] standsfor the units of A and L stands for the unit of length, while the notation O ( A ) stands for a term of order A . Our metric signature is ( − , + , + , +) andwe shall mostly employ geometric with G = c = 1, except for a few sectionswhere natural units shall be more convenient h = c = 1. CS modified gravity is a 4-dimensional deformation of GR, postulatedby Jackiw and Pi [22] . The modified theory can be defined in terms of itsaction: S := S EH + S CS + S ϑ + S mat , (1)where the Einstein-Hilbert term is given by S EH = κ Z V d x √− gR, (2)the CS term is given by S CS = + α Z V d x √− g ϑ ∗ R R, (3)the scalar field term is given by S ϑ = − β Z V d x √− g h g ab ( ∇ a ϑ ) ( ∇ b ϑ ) + 2 V ( ϑ ) i , (4) Similar versions of this theory were previously suggested in the context of stringtheory [23,24], and three-dimensional topological massive gravity [25,26]. S mat = Z V d x √− g L mat , (5)where L mat is some matter Lagrangian density that does not depend on ϑ . Inthese equations, κ − = 16 πG , α and β are dimensional coupling constants, g is the determinant of the metric, ∇ a is the covariant derivative associatedwith g ab , R is the Ricci scalar, and the integrals are volume ones carried outeverywhere on the manifold V . The quantity ∗ R R is the Pontryagin density,defined via ∗ R R := R ˜ R = ∗ R abcd R bacd , (6)where the dual Riemann-tensor is given by ∗ R abcd := 12 ǫ cdef R abef , (7)with ǫ cdef the 4-dimensional Levi-Civita tensor. Formally, ∗ R R ∝ R ∧ R , buthere the curvature tensor is assumed to be the Riemann (torsion-free) tensor.We shall discuss in Sec. 6 the formulation of CS modified gravity in first-orderform.Unfortunately, since its inception, CS modified gravity has been studiedwith slightly different coupling constants. We have here attempted to collectall ambiguities in the couplings α and β . Depending on the dimensionalityof α and β , the scalar field will also have different dimensions. Let us forexample let [ α ] = L A , where A is any real number. If the action is to bedimensionless (usually a requirement when working in natural units), it thenfollows that [ ϑ ] = L − A , which also forces [ β ] = L A − . Different sections ofthis review paper will present results with slightly different choices of thesecouplings, but such choices will be made clear at the beginning of each section.A common choice is α = κ and β = 0, leading to [ α ] = L − and [ ϑ ] = L ,which was used in [22,27,28,29,30,31,32,7,33,34,35,36]. On the other hand,when discussing Solar System tests of CS modified gravity, another commonchoice is α = − ℓ/ β = −
1, where ℓ is some length scale associated with ϑ [37], which then implies [ α ] = L , [ ϑ ] = L − and [ β ] = 1But is there a natural choice for these coupling constants? A minimal,practical and tempting choice is α = 1, which then implies that ϑ is dimen-sionless and that [ β ] = L − , which suggests β ∝ κ . From a theoreticalstandpoint, the choice of coupling constant does matter because it specifiesthe dimensions of ϑ and could thus affect its physical interpretation. For ex-ample, a coupling of the form α ∝ κ − suggest S CS is to be thought of asa Planckian correction, since G = ℓ p , where ℓ p is the Planck length. On theother hand, if one wishes to study the CS correction on the same footing as When working in geometrized units, a dimensionless ϑ can still be achieved if β ∼ κ , but here κ is dimensionless, thus pushing all dimensions into α , which nowpossess units [ α ] = L α = κ and push allunits into ϑ . By leaving the coupling constants unspecified with α and β free,we shall be able to present generic expressions for the modified field equations,as well as particular results present in the literature by simply specifying theconstants chosen in each study.The quantity ϑ is the so-called CS coupling field , which is not a constant,but a function of spacetime, thus serving as a deformation function . Formally,if ϑ = const. CS modified gravity reduces identically to GR. This is becausethe Pontryagin term [Eq. (6)] can be expressed as the divergence ∇ a K a = 12 ∗ R R (8)of the Chern-Simons topological current K a := ǫ abcd Γ nbm (cid:18) ∂ c Γ mdn + 23 Γ mcl Γ ldn (cid:19) , (9)where here Γ is the Christoffel connection. One can now integrate S CS by partsto obtain S CS = α ( ϑ K a ) | ∂ V − α Z V d x √− g ( ∇ a ϑ ) K a , (10)where the first term is usually discarded since it is evaluated on the boundaryof the manifold . The second term clearly depends on the covariant derivativeof ϑ , which vanishes if ϑ = const. and, in that case, CS modified gravity reducesto GR.For any finite, yet arbitrarily small ∇ a ϑ , CS modified gravity becomessubstantially different from GR. The quantity ∇ a ϑ can be thought of as an embedding coordinate , because it embeds a generalization of the standard 3-dimensional CS theory into a 4-dimensional spacetime. In this sense, ∇ a ϑ and ∇ a ∇ b ϑ act as deformation parameters in the phase space of all theories. Onecan then picture GR as a stable fixed point in this phase space. Away fromthis “saddle point,” CS modified gravity induces corrections to the Einsteinequations that are proportional to the steepness of the ϑ deformation param-eter.The equations of motion of CS modified gravity can be obtained by vari-ation of the action in Eq. (1). Exploiting the well-known relations δR bacd = ∇ c δ Γ bad − ∇ d δ Γ bac (11)and δ Γ bac = 12 g bd ( ∇ a δg dc + ∇ c δg ad − ∇ d δg ac ) , (12)one finds The implications of discarding this boundary term will be discussed in Sec. 2.5 S = κ Z V d x √− g (cid:18) R ab − g ab R + ακ C ab − κ T ab (cid:19) δg ab + Z V d x √− g " α ∗ R R + β (cid:3) ϑ − β dVdϑ δϑ + Σ EH + Σ CS + Σ ϑ (13)where (cid:3) := g ab ∇ a ∇ b is the D’Alembertian operator and T ab is the total stress-energy tensor, defined via T ab = − √− g δ L mat δg ab + δ L ϑ δg ab ! , (14)where L ϑ is the Lagrangian density of the scalar field action, ie. the integrandEq. (4) divided by √− g , such that S ϑ = R V L ϑ d x . Thus, the total stress-energy tensor can be split into external matter contributions T ab mat and a scalarfield contribution, which is explicitly given by T ϑab = β (cid:20) ( ∇ a ϑ ) ( ∇ b ϑ ) − g ab ( ∇ a ϑ ) ( ∇ a ϑ ) − g ab V ( ϑ ) (cid:21) . (15)The tensor C ab that appears in Eq. (13) is a 4-dimensional generalizationof the 3-dimensional Cotton-York tensor, which in order to distinguish it fromthe latter we shall call the C-tensor . This quantity is given by C ab := v c ǫ cde ( a ∇ e R b ) d + v cd ∗ R d ( ab ) c , (16)where v a := ∇ a ϑ , v ab := ∇ a ∇ b ϑ = ∇ ( a ∇ b ) ϑ (17)are the velocity and covariant acceleration of ϑ , respectively. The last line ofEq. (13) represents surface terms that arise due to repeated integrations byparts. Such terms play an interesting role for the thermodynamics of blackhole solutions, which we shall review in Sec. 2.5.The vanishing of Eq. (13) leads to the equations of motion of CS modi-fied gravity. The equations of motion for the metric degrees of freedom (themodified field equations) are simply G ab + ακ C ab = 12 κ T ab , (18)where G ab = R ab − g ab R is the Einstein tensor. The trace-reversed form ofthe modified field equations R ab + ακ C ab = 12 κ (cid:18) T ab − g ab T (cid:19) , (19) In the original work of [22], the C-tensor was incorrectly called “Cotton tensor”,but the concept of a higher-dimensional Cotton-York tensor already exists [38] anddiffers from the definition of Eq. (16) T = g ab T ab is the trace of the total stress-energy tensor. Thus, it followsthat as in GR the modified field equations must also satisfy R = − κ T = 0 , (20)where the right-hand side holds in the absence of matter.The vanishing of the variation of the action also leads to an extra equationof motion for the CS coupling field, namely β (cid:3) ϑ = β dVdϑ − α ∗ R R, (21)which we recognize as the Klein-Gordon equation in the presence of a potentialand a sourcing term. One then sees that the evolution of the CS coupling isnot only governed by its stress-energy tensor, but also by the curvature ofspacetime. One can in fact also derive this equation from the requirement ofenergy-momentum conservation: ∇ a ( G ab + C ab ) = 12 κ ∇ a T ab , (22)where the first term on the left-hand side identically vanishes by the Bianchiidentities, while the second is proportional to the Pontryagin density via ∇ a C ab = − v b ∗ R R. (23)Equation (21) is then established from Eq. (22), provided external matter de-grees of freedom satisfy ∇ a T ab mat = 0. Alternatively, Eq. (22) also tells us that,provided the scalar field satisfies its evolution equation [Eq. (21)], then thestrong equivalence principle is satisfied since matter follows geodesics deter-mined by the conservation of its stress-energy tensor.At this junction, one might worry if this set of equations is well-posed asan initial value problem, ie. that given generic initial data, there exists a uniqueand stable solution that is continuous on the initial data (a small change inthe initial state leads to a small change in the final state). A restricted class ofmodified CS theories have already been shown to be well-posed as a Dirichletboundary value problem [39], via the construction of a Gibbons-Hawking-York boundary term (see Sec. 2.5). In principle, these results can easily begeneralized to initial or final boundaries (Cauchy hypersurfaces), by treatingthe case where the normal vector to the boundary is timelike, and to genericCS field ϑ , since the addition of kinetic or potential terms should not modifythe analysis of [40,39]. Such conclusions thus imply that given an initial state,there exists a unique final state in CS modified gravity.The issue well-posedness of the theory as an initial value problem, how-ever, remains still formally open, since the above arguments cannot necessar-ily be used to demonstrate that the theory is stable. Such instabilities are10ooted in the potential appearance of third time derivatives in the equationsof motion. As regards to these higher-order derivatives, notice first that for v a = const. such derivatives do not arise and the modified field equationsremain second-order. Moreover, notice that even for generic ϑ , third timederivatives also vanish in a linear stability analysis, as these derivatives aremultiplied by terms at least quadratic in the metric perturbation. Non-linearstabilities, however, could arise upon the full non-linear evolution of the mod-ified field equations, a topic that is currently being investigated.Formally, the CS modified equations of motion presented here [Eqs. (18)and (21)] represent a family of theories, parameterized by the couplings α and β . Of this family, two classes or formulations are particularly interesting:the non-dynamical framework ( α arbitrary, β = 0) and the dynamical frame-work ( α and β arbitrary but non-zero). These two formulations are actuallytwo distinct theories, because in the dynamical formulation the scalar fieldintroduces stress-energy into the modified field equations, which in turn forcesvacuum spacetimes to possess a certain amount of “scalar hair.” On the otherhand, such hairy spacetimes are absent from the non-dynamical formulation,but this one instead acquires an additional differential constraint that mightoverconstrain it.One of the benefits of introducing the coupling constants α and β is thatwe can easily specify if we are considering the dynamical ( α = 0 = β ) or thenon-dynamical ( α = 0, β = 0) formulation. In this review article, we shallattempt to present as many generic expressions with α and β unspecifiedas possible, but when summarizing existing results we will have to focus onone specific formulation. On average, the non-dynamical formulation has beeninvestigated much more than the fully-dynamical one, which is why the pre-sentation might seem slightly biased toward the non-dynamical theory. Oneshould remember that this bias is not because the non-dynamical theory ispreferred, but only because it is easier to study than the fully-dynamical sce-nario. It is critical then to pay close attention to the beginning of each sectionin the remainder of this review article,as we shall specify whether the resultsthat are being presented correspond to the dynamical or the non-dynamicaltheory by specifying the choice of α and β (an issue that is particularly relevantwhen discussing solutions to the modified theory in Secs. 4 and 5). The non-dynamical framework is defined by setting β = 0 at the levelof the action, such that the scalar field does not evolve dynamically, but isinstead externally prescribed. Such was the formulation introduced by Jackiwand Pi [22], with the particular choice α = κ and β = 0, which implies[ ϑ ] = L .Within this non-dynamical model, there is a particular choice of ϑ , pro-11osed by Jackiw and Pi [22], that has been used extensively: ϑ = tµ → v µ = " µ , , , . (24)where µ is some mass scale, such that [ µ ] = L − . We shall refer to Eq. (24)as the canonical CS coupling . This choice of CS scalar is popular because forcertain sufficiently symmetric line elements (eg. the Schwarzschild metric), the4-dimensional C-tensor reduces exactly to the ordinary 3-dimensional Cottontensor. Moreover, with this choice, spacetime-dependent reparameterizationof the spatial variables and time translation remain symmetries of the CSmodified action [22]. In spite of this, there is nothing truly “canonical” aboutthis choice of embedding coordinate and other interesting choices are alsopossible.Irrespective of the choice ϑ , non-dynamical CS modified gravity, is a con-strained theory, in the sense that all solutions must satisfy an additional dif-ferential condition, sometimes referred to as the Pontryagin constraint : ∗ R R = 0 . (25)This constraint arises directly from the variation of the action in Eq. (13)with β = 0. We shall see in Secs. 4 and 5 that this constraint imposes severerestrictions on the dynamics of solutions of the non-dynamical theory.What does the Pontryagin constraint really mean physically? Some in-sight can be gained by reformulating this constraint in terms of its spinorialdecomposition. Gr¨umiller and Yunes [35] have realized that the trivial relation ∗ R R = ∗ C C , (26)where C is the Weyl tensor C abcd := R abcd − δ [ a [ c R b ] d ] + 13 δ a [ c δ bd ] R (27)and ∗ C its dual ∗ C abcd := 12 ǫ cdef C abef , (28)opens the door to powerful spinorial methods that allows one to map theWeyl tensor into the Weyl spinor [41], which in turn can be characterized bythe Newman-Penrose (NP) scalars (Ψ , Ψ , Ψ , Ψ , Ψ ). Following the nota-tion of [42], the Pontryagin constraint translates into a reality condition on aquadratic invariant of the Weyl spinor, I , ℑ ( I ) = ℑ (cid:16) Ψ Ψ + 3Ψ − Ψ (cid:17) = 0 . (29)The reality condition of Eq. (29) directly implies that any spacetime ofPetrov types III , N and O automatically satisfies the Pontryagin constrain,12hile spacetimes of Petrov type D, II and I could violate it. Moreover, thisreality condition also directly implies that not only the Kerr solution but alsogravitational perturbations thereof violate the Pontryagin constraint. This isbecause, although Ψ , = 0 in this perturbed spacetime, ℜ (Ψ ) = 0 = ℑ (Ψ )generically, which violates Eq. (29) [35,43].Another reformulation of the Pontryagin constraint can be obtained fromthe electro-magnetic decomposition of the Weyl tensor (cf. e.g. [44]), given by( C abcd + i ǫ abef C ef cd ) u b u d = E ac + iB ac , (30)where u a is a normalized time-like vector, and E ac and B ac are the electric andmagnetic parts of the Weyl tensor respectively. Gr¨umiller and Yunes [35] haveshown that in this decomposition [45], the Pontryagin constraint reduces to E ab B ab = 0 . (31)Such a restriction forces certain derivatives of the Regge-Wheeler function inthe Regge-Wheeler [46] decomposition of the metric perturbation to vanish,which has drastic consequences for perturbations of the Schwarzschild space-time, as we shall discuss in Sec. 5.The electromagnetic decomposition of the Pontryagin constraint leads tothree possible scenarios: purely electric spacetimes B ab = 0; purely magneticspacetimes E ab = 0; orthogonal spacetimes, where E ab is orthogonal to B ab .In fact, Eq. (31) is a perfect analogue to the well-known electrodynamicscondition ∗ F F ∝ E · B = 0. In electromagnetism, such a condition is satisfiedin electrostatics ( B ab = 0), magnetostatics ( E ab = 0) and electromagneticwaves ( E ab B ab = 0). The Pontryagin constraint can thus be rephrased as“the gravitational instanton density must vanish,” since the quantity ∗ F F issometimes also referred to as the “instanton density.”The severe requirements imposed by the Pontryagin constraint on thespace of allowed solutions, together with the arbitrariness in the choice of CSfield, make the non-dynamical formulation rather contrived. First, differentchoices of ϑ will lead to sufficiently different solutions, each of these withdifferent observables. Without an external prescription to decide what ϑ is,one loses the predictive power of the Einstein equations and replaces it by afamily of possible solutions. Moreover, all choices of ϑ so far explored are ratherunnatural or unphysical, in that they lead to a field with infinite energy, sincethe field’s kinetic energy is constant. Such fields are completely incompatiblewith the dynamical framework, which implies that results arrived at in thenon-dynamical framework cannot be directly extended into the dynamicalscheme. Second, the Pontryagin constraint can be thought of as a selectionrule, that eliminates certain metrics from the space of allowed solutions. Sucha selection rule has been found to overconstrain the modified field equations,to the point that only the trivial zero-solution is allowed in certain cases [34].Having said this, the non-dynamical framework has been useful to qual- tatively understand the effect of the CS correction on gravitational parityviolation. Only recently has there been a serious, albeit limited effort to studythe much more difficult dynamical formulation and preliminary results seemto indicate that solutions found in this framework do share many similaritieswith solutions found in the non-dynamical scheme. The non-dynamical for-mulation should thus be viewed as a toy-model that might help us gain someinsight into the more realistic dynamical framework. The dynamical framework is defined by allowing β and α to be arbitrary,but non-zero constants. In fact, β cannot be assumed to be close to zero (ormuch smaller than α ), because then the evolution equation for ϑ becomessingular. This formulation was initially introduced by Smith, et. al. [37], withthe particular choice α = − ℓ/ β = −
1, which implies [ ϑ ] = L − . Inthis model, the CS scalar field is thus not externally prescribed, but it insteadevolves driven by the spacetime curvature. The Pontryagin constraint is thensuperseded by Eq. (21), which does not impose a direct and hard constrainton the solution space of the modified theory. Instead, it couples the evolutionof the CS field to the modified field equations.The dynamical formulation, however is not completely devoid of arbitrari-ness. Most of this is captured in the potential V ( ϑ ) that appears in Eq. (4),since this is a priori unknown. In the context of string theory, the CS scalarfield is a moduli field, which before stabilization has zero potential (ie. itrepresents a flat direction in the Calabi-Yau manifold). Stabilization of themoduli field occurs via supersymmetry breaking at some large energy scale,thus inducing an (almost incalculable) potential that is relevant only at sucha scale. Therefore, in the string theory context, it is reasonable to neglect sucha potential when considering classical and semi-classical scenarios.The arbitrariness aforementioned, however, still persists through the def-inition of the kinetic energy contained in the scalar field. For example, thereis no reason to disallow scalar field actions of the form S new ϑ = − Z V d x √− g h β ( ∂ϑ ) + β ( ∂ϑ ) i , (32)which leads to the following stress-energy tensor T new ϑab = β h β ( ∂ϑ ) i ( ∇ a ϑ ) ( ∇ b ϑ ) − g ab h β ( ∂ϑ ) + β ( ∂ϑ ) i , (33)where we have used the shorthand ( ∂ϑ ) := g ab ( ∇ a ϑ ) ( ∇ b ϑ ). The Pontryaginconstraint would then be replaced by β (cid:3) ϑ + 2 β ∇ a h ( ∇ a ϑ ) ( ∂ϑ ) i = − α κ ∗ R R. (34)Of course, the choice of Eq. (4) is natural in the sense that it corresponds14o the Klein-Gordon action, but it should actually be the more fundamentaltheory, from which CS modified gravity is derived, that prescribes the scalarfield Hamiltonian. In the string theory context, however, the moduli field pos-sess a canonical kinetic Hamiltonian, suggesting that Eq. (32) is the correctprescription [27].Another natural choice for the potential of the CS coupling is the C-tensor itself. In other words, consider the possibility of placing the C-tensoron the right-hand side of the modified field equations and treating it as simplya non-standard stress-energy contribution. Such a possibility was studied byGr¨umiller and Yunes [35] for certain background metrics, which are solutionsin GR but not in CS modified gravity. The Kerr metric is an example of sucha background, for which they found that the induced C-tensor stress-energyviolates all energy conditions. No classical matter in the observable universeis so far known to violate all energy conditions, thus rendering this possibilityrather unrealistic.
Precisely what type of parity violation is induced by the CS correction?Let us first define parity violation as the purely spatial reflection of the triadthat defines the coordinate system. The operation ˆ P [ A ] = λ p A is then said tobe even, parity-preserving or symmetric when λ p = +1, while it is said to beodd, parity-violating or antisymmetric if λ p = −
1. By definition, we then havethat ˆ P h e Ii i = − e Ii , where e Ii is a spatial triad, and thus ˆ P h e aijk i = − e aijk . Notethat parity transformations are slicing-dependent, discrete operations, whereone must specify some spacelike hypersurface on which to operate. On theother hand the combined parity and time-reversal operations is a spacetimeoperation that is slicing independent.How does the CS modification transform under parity? First, applyingsuch a transformation to the action one finds that S is invariant (ie. parityeven) if and only if ϑ transforms as a pseudo-scalar ˆ P [ ϑ ] = − ϑ . Applyingsuch a transformation to the modified Einstein equations one finds that theC-tensor is invariant if and only if the covariant velocity of ϑ transforms as avector ˆ P [ v a ] = + v a , or equivalently if ϑ is as a pseudo-scalar.The transformation properties of the CS scalar are not entirely free in thedynamical formulation. Since ϑ must satisfy the evolution equation ∇ a v a ∝ ∗ R R , we see that P [ v a ] = + v a , and thus ϑ must be a pseudo-scalar. In thenon-dynamical framework, however, one is free to choose ϑ in whichever waydesired and thus the transformation properties of the action and field equationscannot be a priori determined. Of course, if one is to treat the modified theoryas descending from string theory or particle physics, then ϑ is required to bea pseudo-scalar as the dynamical theory also requires.Statements about the parity-transformation properties of a theory do notrestrict the parity-properties of the solutions of the theory. A clear exam-15le can be derived from Maxwell’s theory of electromagnetism. Propagatingmodes (electromagnetic waves) travel at the speed of light in vacuum, but inthe presence of a dielectric medium, they become birefringent, leading to Fara-day rotation. Even though the Maxwell action and field equations are clearlyparity-preserving, solutions exist where this symmetry is not respected. An-other example can be obtained from GR, where the theory is clearly paritypreserving, but solutions exist (such as the Kerr metric and certain Bianchimodels) that do violate parity.Such symmetry considerations can be used to infer some properties ofbackground solutions (ie. representations of the vacuum state) in the dynami-cal formulation. First, if one is searching for parity-symmetric solutions (as inthe case of spherically symmetric line-elements), then ∗ R R = 0, which forces θ to be constant (assuming this field has finite energy). One then sees thatparity-even line-elements will not be CS corrected. On the other hand, if oneis considering parity-odd spacetimes (such as the Kerr metric), then the Pon-tryagin density will source a non-trivial CS scalar, which will in turn modifythe Kerr metric through the field equations. Such a correction will tend to in-troduce even more parity-violation in the solution, as we shall discuss furtherin Sec. 5.Clear signals of parity violation can be obtained by studying perturba-tions about the background solutions. As in the case of Maxwell theory, CSmodified gravity has the effect of promoting the vacuum to a very specialtype of medium, in which left- and right- moving gravitational waves are en-hanced/suppressed with propagation distance. Such an effect is sometimesreferred to as “amplitude birefringence,” and it is analogous (but distinct)to electromagnetic birefringence (see Sec. 5 for a more detailed discussion ofamplitude birefringence). The modified theory then can be said to “prefer achirality,” since it will tend to annihilate a certain polarization mode. Opposite to common belief, GR does not admit a Dirichlet boundary-value problem as formulated in the previous section. This is so because thevariation of the Ricci scalar in the Einstein-Hilbert action leads to boundaryterms that depend both on the variation of the metric and its first normalderivative. In order to become a well-posed, Dirichlet boundary value prob-lem, the Einstein-Hilbert action must be supplemented by a boundary coun-terterm, so-called Gibbons-Hawking-York (GHY) term. This term cancels theaforementioned boundary terms thus yielding a well-posed boundary valueproblem.The issue of non-dynamical CS modified gravity as a well-posed boundaryvalue problem has been addressed by Grumiller, et. al. [39] with the conven-tions α = κ and β = 0. As in GR, the CS action as presented in Eq. (3) doesnot lead to a well-posed boundary value problem and counterterms must beadded. Let us then concentrate on Σ CS and, in particular, on boundary terms16nvolving normal derivatives of the variation of the metric, neglecting irrele-vant terms. In [39] and in this section, “irrelevant terms” are defined as thosethat are bulk terms but not total derivatives, or as those that are boundaryterms that vanish on the boundary.Let us then define the induced metric on the boundary as h ab := g ab − n a n b , (35)where the boundary is a hypersurface with spacelike, outward-pointing unitnormal n a . The extrinsic curvature is then K ab := h ca h db ∇ c n d , (36)which is simply the Lie derivative along n a , where ∇ a stands for the four-dimensional covariant derivative operator. Note that the variation of this quan-tity is given by δK ab = 12 h ca h db n e ∇ e δg cd , (37)up to irrelevant terms.With this machinery, one then finds that the variation of the Einstein-Hilbert and CS actions lead to the following boundary terms [39]: δS EH = − κδ Z ∂ V d x √ h K, (38) δS CS = − αδ Z ∂ V d x √ h ϑ CS ( K ) , (39)up to irrelevant terms, where one defines [39] CS ( K ) := 12 ǫ nijk K il D j K kl . (40)In Eqs. (38), (39) and (40), K := K aa is the trace of the extrinsic curvature, i, j, k and n stand for indices tangential and normal to the hypersurface respec-tively, and D i is the covariant derivative along the boundary. Equation (38) isin fact the GHY term, while Eq. (39) is analogous to this term in CS modifiedgravity. Note that Eq. (39) depends only on the trace-less part of the extrinsiccurvature, and thus, it can be thought of as complementary to the GHY term.The boundary terms introduced upon variation of the action can be can-celled by addition of the following counterterms: S bEH = 2 κ Z ∂ V d x √ hK,S bCS = 2 α Z ∂ V d x √ h ϑ CS ( K ) . (41)Again, Eq. (41) is the GHY counterterm, while Eq. (41) is a new countertermrequired in CS modified gravity in order to guarantee a well-posed boundary17alue problem. Interestingly, we could also have performed this analysis interms of the CS current, using ∗ R R ∝ v a K a . Doing so [39] α Z V d x √− g ϑ ∗ R R + 2 Z ∂ V d x √ h ϑCS ( K ) = − Z V d x √− g v a K a + Z ∂ V √ hϑCS ( γ ) , (42)up to irrelevant terms, where CS ( γ ) is given by CS (Γ) := 12 ǫ nijk Γ lim (cid:18) ∂ j Γ mkl + 23 Γ mjp Γ pkl (cid:19) (43)with Γ ijk the Christoffel connection.The CS counterterm presented above, however, only holds in an adaptivecoordinate frame, where the lapse is set to unity and the shift vanishes. Incovariant form, Grumiller, et. al. [39] have shown that the action S = κ Z V d x √− g (cid:18) R + α κ ϑ ∗ R R (cid:19) + 2 κ Z ∂ V d x √ h (cid:18) K + α κ ϑn a ǫ abcd K be ∇ c K de (cid:19) + α Z ∂ V d x √ h F ( h ab , ϑ ) , (44)has a well-posed Dirichlet boundary value problem. This action is a general-ization of the counterterms presented above, which holds in any frame. Thelast integral is an additional term that is intrinsic to the boundary and doesnot affect the well-posedness of the boundary value problem, yet it is essentialfor a well-defined variational principle when the boundary is pushed to spatialinfinity [47,48]. The first place we encounter the CS invariant is in the gravitationalanomaly of the Standard Model. In this chapter, we shall give a pedagogi-cal review and derivation of anomalies that includes the gravitational one.An anomaly describes a quantum mechanical violation of a classicallyconserved current. According to Noether’s theorem, invariance under a clas-sical continuous global symmetry group G yields the conservation of a globalcurrent j Aa , with A labelling the generators of the group G : ∂ a j aA = 0 . (45)An anomaly A A is a quantum correction to the divergence of j Aa which rendersit non-zero, ∂ a j aA = A A . 18n the one hand, gauge theories with chiral fermions usually have globalanomalies in the chiral currents, j a = ¯ ψγ a γ ψ . Such anomalies do not leadto inconsistencies in the theory, but they do possess physical consequences.Historically, precisely this type of anomaly led to the correct prediction ofthe decay rate of pions into photons, π o → γγ , by including the anomalousinteraction π ǫ abcd F ab F cd .On the other hand, gauge anomalies are also a statement that the quan-tum theory is quantum mechanically inconsistent. Gauge symmetries can beused to eliminate negative norm states in the quantum theory, but in orderto remain unitary, the path integral must also remain gauge invariant. Quan-tum effects involving gauge interactions with fermions can spoil this gauge-invariance and thus lead to a loss of unitarity and render the quantum for-mulation inconsistent. Therefore, if one is to construct a well-defined, unitaryquantum theory and if gauge currents are anomalous, then these anomaliesmust be cancelled by counterterms.A common example of a global anomaly in the Standard Model is theviolation of the U (1) axial current by a one-loop triangle diagram betweenfermion loops and the gauge field external legs. Let us then derive the anomalyusing Fujikawa’s approach in 1 + 1 dimensions [49,50], generalized to d + 1dimensions. Since amplitudes and currents can be generated from the pathintegral, Fujikawa realized that anomalies arise from the non-invariance ofthe fermionic measure in the path integral under an arbitrary fermionic fieldredefinition. For concreteness let us consider a massless fermion coupled toelectromagnetism in 3 + 1 dimensions. The action and partition function forthis theory are the following: S = Z d x ( − e F ab F ab + i ¯ ψγ a D a ψ ) (46) Z = Z DADψD ¯ ψe iS [ A a ,ψ, ¯ ψ ] , (47)where ψ is a Dirac fermion, A a is a gauge field, F ab is the electromagnetic fieldstrength tensor, e is the coupling constant or charge of the Dirac fermions and γ a are Dirac matrices, where the overhead bar stands for complex conjugation.Such a toy theory is invariant under a chiral tranformation of the form ψ → e iαγ ψ = ψ + iαγ ψ + . . . , (48)where α is a real number and γ is the chiral Dirac matrix. Such an invari-ance leads to the U (1) global Noether current j Axial a = ¯ ψγ a γ ψ . In the pathintegration approach, current conservation is exhibited by studying the Wardidentities, which can be derived by requiring that the path integral be invariantunder an arbitrary phase redefinition of the Dirac fermions.The non invariance of the fermionic measure is precisely the main ingre-dient that encodes the chiral anomaly. In order to study this effect, we mustfirst define the measure precisely. For this purpose, it is helpful to expand ψ in terms of orthonormal eigenstates of iγ a D a :19 a D a φ m = λ m φ m (49)and ψ ( x ) = X m a m φ m ( x ) , (50)where a m are Grassmann variable multiplying the c-number eigenfunctions φ m ( x ) and D a is the gauge covariant derivative. The measure is then definedas DψD ¯ ψ = Y n da n d ¯ a n . (51)Let us now demand invariance of the partition function: Z DADψD ¯ ψe iS [ a,ψ, ¯ ψ ] = Z DA ′ Dψ ′ D ¯ ψ ′ e iS ′ [ a,ψ, ¯ ψ ] , (52)where the fermions transform as follows ψ ( x ) → ψ ′ ( x ) + ǫ ( x ) , ¯ ψ ( x ) → ¯ ψ ′ ( x ) + ¯ ǫ ( x ) (53)for a chiral transformation ǫ ( x ) = iα ( x ) γ ¯ ψ ( x ). We see then that the La-grangian transforms as Z d x ( ¯ ψ ′ iγ a D a ψ ′ ) = Z d x [ ¯ ψiγ a D a ψ − ( ∂ a α ) ¯ ψiγ a γ ψ ] . (54)Assuming that the measure is invariant, integrating by parts and varying theaction with respect to α , we recover the Ward identity ∂ a < ¯ ψγ a γ ψ > = 0 , (55)which is nothing but the statement of axial current conservation.Naively, we might conclude that the classical global current carries overto the quantum one, but Fujikawa [49,50] realized that such a reasoning as-sumes that the path integral measure is invariant. A more careful analysisthen reveals that a change of variables in the measure affects the coefficientof the Dirac fermion eigenstate expansion via a ′ m = X n ( δ mn + B mn ) a n (56)where B mn = i Z d x φ † m αγ φ n . (57)Using the Grassmanian properties of a m , this transformation returns the Ja-cobian in the measure Dψ ′ D ¯ ψ ′ = [det(1 + B )] − DψD ¯ ψ, (58)where det( · ) and Tr( · ) shall stands for the determinant and trace respectively.20he key to obtaining the anomaly resides in computing det(1 + B ). Ex-panding the determinant to first order in α det(1 + B ) = e Tr[ ln (1+ B )] = e Tr( B ) , (59)and hence, [det(1 + B )] − = e − i R d xα ( x ) P n φ † γ φ n ( x ) (60)Regulating the fermion composite operator with a cut-off λ n /M X n φ + n ( x ) γ φ n ( x ) → X n φ + n ( x ) γ φ n ( x ) e λ nM (61)and using that the mode functions are eigenfunctions of iγ a D a , we can alsowrite Eq. (61) as X n φ † n γ e ( iγaDa )2 M φ n = < | tr [ γ e ( iγ a D a ) /M ] > (62)In order to simplify this expression we can use the identity ( iγ a D a ) = − D a D a +(1 / σ ab F ab where σ ab = ( i/ γ a , γ b ] which leads us to evaluate < x | Tr[ γ e ( − D +(1 / σ ab F ab ) /M ] | x > . (63)As we take the limit M → ∞ we can expand in powers of the backgroundelectromagnetic field by writing − D = − ∂ + ... . We are led to the followingexpression: < x | e − ∂ /M | x > = i Z d k E π e − k E /M = iM π . (64)The other terms that will follow arise from bringing down powers of the back-ground field.Terms with one power of the background field and with the trace of γ vanish, since Tr[ γ σ ab ] = 0. In the limit M → ∞ , terms that are second orderin the background field also vanish, leaving:Tr h γ
12 ( 12 M σ ab F ab ) i < x | e − ∂ /M | x > = − π ǫ abcd F ab F cd . (65)Putting all the non-vanishing terms above together gives us the Jacobian pref-actor: [det(1 + B )] − = e i R d xα ( x )( π ǫ abcd F ab F cd ) (66)The partition function with this change of variables becomes Z [ A ] = Z DψD ¯ ψe i R d x ( ¯ ψiγ a D a ψ + α ( x )( ∂ a j Aa +(1 / π ) ǫ acbd F ab F cd )) (67)When we vary the partion function with respect to α we get the famous ABJanomaly [51,52] ∂ a j Aa = − π ǫ abcd F ab F cd . (68)21he above derivation of the ABJ anomaly also applies for the gravita-tional anomaly. Similar to Eq. (68), if we use the Riemann curvature ten-sor instead of the field strength tensor, we will obtain the gravitational ABJanomaly: D a j Aa = − π ǫ abcd R abef R cdef . (69)Note that the right-hand side of this equation is proportional to the Pontryagindensity of Eq. (6). The gravitational ABJ anomaly can be canceled by addingthe appropriate counter term in the action, which in turn amounts to includingthe CS modification in the Einstein-Hilbert action.Recently, it has been shown that the CS action is also induced by otherstandard field theoretical means. In particular, [53] showed that the CS actionarises through Dirac fermions couplings to a gravitational field in radiativefermion loop corrections, while [54,55] showed that it also arises in Yang-Mills theories and non-linearized gravity through the proper-time method andfunctional integration. We refer the reader to [53,54,55] for more informationon these additional mechanisms that generate the CS action. In the previous section, we derived the chiral anomaly in a 3 + 1 gaugefield theory coupled to fermions. We saw that while gauge and global anoma-lies can exist, gauge anomalies need to be cancelled to have a consistent quan-tum theory. In what follows we will show how the CS modification to generalrelativity arises from the Green-Schwarz anomaly canceling mechanism in het-erotic String Theory. The key idea is that a quantum effect due to a gaugefield that couples to the string induces a CS term in the effective low energyfour dimensional general relativity.Recall that the action of a free, one-dimensional particle can be describedas the integral of the worldline swept out over a “target spacetime” X a ( τ ),parametrized by τ . The infinitesimal path length swept out is dl = ( − ds ) / = ( − dX a dX b η ab ) / (70)where η ab is a 9 + 1 D Minkowski target space-time and the action is then S = − m Z dl = − m Z dτ ( − ˙ X a ˙ X a ) / (71)We can easily extend the discussion of a point particle to a string byparametreizing the worldsheet with a target function in terms of two coordi-nates ( σ, τ ). Consider then the string, world-sheet field X a ( σ, τ ) embedded ina D -dimensional space-time, G ab that sweeps out a 1 + 1 world-sheet denotedby coordinates ( σ, τ ) and world-sheet line element ds = h AB dX A dX B . Theindices A, B here run over world-sheet coordinates. Analogous to the point22article, the free string action is described by S st = T Z ds = T Z dσdτ √− hh AB ∂ a X A ( σ, τ ) ∂ b X B ( σ, τ ) G ab , (72)where T is the string tension.This string is also charged under a U (1) symmetry and couples to theNeveu-Schwarz two-form potential, B µν , via S B = Z d σ∂ a X ( σ, τ ) ∂ b X ( σ, τ ) B ab . (73)It is this fundamental string field B ab that underlies the emergence of CSmodified gravity when String Theory is compactified to 4 D . In a seminalwork, Alvarez-Gaume and Witten [56] showed that GR in even dimensions willsuffer from a gravitational anomaly in a manner analogous to how anomaliesare realized in the last section. The low-energy limit of superstring theories are10 dimensional supergravity (SUGRA) theories. As discussed in the previoussection, a triangle loop diagram between gravitons and fermions will generatea gravitational anomaly; similarly hexagon loop diagrams generate anomaliesin 10 dimensions. Remarkably, Green and Schwarz [57,58] demonstated thatthe gravitational anomaly is cancelled from a quantum effect of the stringworldsheet B field, since the string worldsheet couples to a two form, B ab .The stringy quantum correction shifts the gradient of the B ab field by a CS3-form. This all results in modifying the three-form gauge field strength tensor H abc in 10 D supergravity. H abc = ( dB ) abc → ( dB ) abc + 14 (cid:16) Ω abc ( A ) − α ′ Ω abc ( ω ) (cid:17) (74)This naive shifting of H abc conspires to cancel the String Theory anomaly.We refer the interested reader to Vol II of Polchinski’s book [59] for a moredetailed discussion of the Green-Schwarz anomaly canceling mechanism.We begin our analysis from the compactification of the heterotic string toits 4D, N = 1 supergravity limit. For concreteness we consider the compact-ification to be on six dimensional internal space (ie. a Calabi-Yau manifold).Similar to the Kaluza-Klein idea, when we dimensionally reduce a 10 dimen-sional system to four dimensions, many fields (moduli) which characterize thegeometry emerge. These fields cause a moduli problem since their high energydensity will overclose the universe, hence, they need to be stabilized. The dis-cussion of moduli stabilization is beyond the scope of this review and we pointthe reader interested in this field of research to the work of Gukov et. al [60].In what follows, we will assume that all moduli except the axion are stabilizedand will not explicitly deal with them in our analysis.Our starting point is the 10D Heterotic string action in Einstein frame[61] and we ignore the coupling to fermionic fields since they are not relevantfor our discussion. In this theory the relevant bosonic field content is the 10D23etric, g ab , a dilaton φ and a set of two and three form field strength tensors H := H abc and F := F ab respectively. S = Z d x √ g (cid:20) R − ∂ a φ∂ a φ − e − φ H abc H abc − e − φ Tr( F ab F ab ) (cid:21) , (75)where H = dB − (cid:16) Ω ( A ) − α ′ Ω ( ω ) , (cid:17) (76) B := B ab , and where Ω ( A ) := Ω abc ( A ) and Ω ( ω ) = Ω abc ( ω ) are the gaugeand gravitational CS three-forms respectively, which in exterior calculus formare given by Ω ( A ) = T r (cid:16) dA ∧ A + 23 A ∧ A ∧ .A (cid:17) (77)We now dimensionally reduce the 10D action to 4D, N = 1 supergravity cou-pled to a gauge sector by choosing a four-dimensional Einstein frame metric, g SMN = g EMN e φ . The 10D line element splits up into a sum of four and sixdimensional spacetime line elements ds = ds + g mn dy m dy n , (78)where g mn is a fixed metric if the internal 6-dimensions are normalized to havevolume 4 α ′ . The compactified effective gravitational action becomes S D = 12 κ Z d x √− g " R − ∂ µ S ∗ ∂ µ S ( S + S ∗ ) , (79)where S = e − ψ + iθ , with ψ and θ the four dimensional dilaton and model-independent axion fields respectively. The dilaton emerges as the four dimen-sional Yang-Mills coupling constant g Y M = e ψ , which we can assume here tobe fixed, while the axion derives from the spacetime and internal componentsof B ab .Let us now focus our attention on the axionic sector of the 4D heteroticstring. The bosonic low energy effective action takes the form S d = 2 α ′ Z d x √− g (cid:16) R + A − e − φ H abc ∧ ∗ H abc − e − φ Tr( F ab F ab ) . (80)When we explicitly square the kinetic term of the three-form field strengthtensor, H abc ∧ ∗ H abc = h ( dB − (cid:16) Ω ( A ) − α ′ Ω ( ω ) (cid:17)i (81)we obtain the cross term ∗ dB ∧ Ω , where the dual to the three-form dB isequivalent to exterior derivative of the axion dθ = ∗ dB . After integrating byparts, one ends up with the sought after gravitational Pointryagin interaction[62] Z d xf ( θ ) R ∧ R (82)24here here f ( θ ) = θ V M α ′ and where V is a volume factor measuredin string units and determined by the dimensionality of the compactification.Additionally, integration by parts also unavoidingly introduces a kinetic termfor f ( θ ), which we did not write explicitly above. Alexander and Gates [27]used this construction to place a constraint on the string scale provided thatthe gravitational CS term was responsible for inflationary leptogenesis. The CS correction to the action also arises in Loop Quantum Gravity(LQG), which is an effort toward the quantization of GR through the postu-late that spacetime itself is discrete [63,64,65]. In this approach, the Einstein-Hilbert action is first expressed in terms of certain “connection variables”(essentially the connection and its conjugate momenta, the triad), such thatit resembles Yang-Mills (YM) theory [66] and can thus be quantized via stan-dard methods. Currently, there are two versions of such variables: a selfd-ual SL (2 , C ), “Ashtekar” connection, which must satisfy some reality condi-tions [67]; and a real SU (2), Barbero connection, constructed to avoid thereality conditions of the Ashtekar one [68]. Both these formalisms can be com-puted from the so-called Holst action, which consists of the Einstein-Hilbertterm plus a new piece that depends on the dual to the curvature tensor [69],but which does not affect the equations of motion in vacuum by the BianchiidentitiesAshtekar and Balachandran [70] first analyzed parity (P) and charge-parity (CP) conservation in LQG [70], which led them essentially to CS theorywith a constant CS parameter. Since LQG resembles YM theory, its canonicalvariables must satisfy a Gauss-law like GR constraint D a E aI = 0, where D a isa covariant derivative operator and E aI is the triad. This constraint generatesinternal gauge transformations in the form of triad rotations.Physical observables in any quantum theory must be invariant under bothlarge and small, local gauge transformations. As in YM theory, the latter canbe associated with unitary irreducible representations of the type e inθ , where n is the winding number and θ is an angular ambiguity parameter. Wavefunc-tions in the quantum theory must then be invariant under the action of theserepresentations, but this generically would lead to different wavefunctions ondifferent θ -sectors. Instead, one can rescale the wavefunctions to eliminate this θ dependance, at the cost of introducing a B-field dependence on the conju-gate momenta, which in turn force the Hamiltonian constraint to violate Pand CP.Asthekar and Balachandran [70] noted that this ambiguity can be related,as in YM theory, to the possibility of adding to the Einstein-Hilbert actionthe term S θ = iθ π Z d x ∗ R R, (83)which is essentially the CS correction to the action when the scalar field ϑ = θ
25s constant and pulled out of the integral. In this sense a CS-like term arisesnaturally in LQG due to the requirement that wavefunctions, and thus physicalobservables, be invariant under large gauge transformations.But the θ -anomaly is not quite the same as CS modified gravity. Afterall, the above analysis is more reminiscent to the chiral anomaly in particlephysics, discussed in Sec. 3.1. Recently, however, the connection between LQGand CS modified gravity has been completed, along a bit of an unexpectedpath. Taveras and Yunes [71] first investigated the possibility of promoting theBarbero- Immirzi (BI) parameter to a scalar field. This parameter is anotherquantization ambiguity parameter that arises in LQG and determines theminimum eigenvalue of the discrete area and discrete volume operators [72]. Ata classical level, the BI parameter is a multiplicative constant that controls thestrength of the dual curvature correction in the Holst action [69]. Taveras andYunes realized that when this parameter is promoted to a field one essentiallyrecovers GR gravity in the presence of an arbitrary scalar field at a classicallevel.Although the Holst action is attractive from a theoretical standpoint sinceit allows a construction of LQG in either Ashtekar or Barbero form, thisaction has also been shown to lead to torsion and parity violation when onecouples fermions to the theory [73,74,75]. This issue can be corrected, whilestill allowing a mapping between GR and the Barbero-Ashtekar formalism, byadding to the Holst action a torsion squared term, essentially transforming theHolst term to the Nieh-Yan invariant [76]. When one couples fermions to theNieh-Yan modified theory, then the resulting effective theory remains torsionfree and parity preserving [77].Inspired by the work of Taveras and Yunes [71], Mercuri [78,79] and Mer-curi and Taveras [80] considered the possibility of promoting the BI parameterto a scalar field in the Nieh-Yan corrected theory. As in the Holst case, theyfound that the BI scalar naturally induces torsion, but this time when thistorsion is used to construct an effective action they found that one unavoid-ingly obtains CS modified gravity. In particular, one recovers Eq. (3) with ϑ = [3 / (2 κ )] / ˜ β , with ˜ β the BI scalar and α = 3 / (32 π ) √ κ , while the scalarfield action becomes Eq. (4) with β = 1 and vanishing potential. One of the most difficult tasks in any alternative theory of gravity is thatof finding exact solutions, without the aid of any approximation scheme. Inthe context of string theory, Campbell, et. al. [23] showed that certain lineelements, such as Schwarzschild and FRW, lead to an exact CS three-form,which thus does not affect the modified field equations. In the context of CSmodified gravity, Jackiw and Pi [22] showed explicitly that the Schwarzschildmetric remains a solution of the non-dynamical modified theory for the canon-ical choice of CS scalar. Shortly after, Guarrera and Hariton [81] showed that26 ig. 1. Space of solutions of Einstein gravity E and CS modified gravity CS . In thisfigure, we have set 2 κ = 1 for simplicity of presentation. the FRW and Reissner-Nordstrom line elements also satisfy the non-dynamicalmodified field equations with the same choice of scalar, verifying the results ofCampbell, et. al. [23]. Recently, Grumiller and Yunes [35] carried out an ex-tensive study of exact solutions in the non-dynamical theory for arbitrary CSscalars, with the hope to find one that could represent a spinning black hole.All of these investigations concern vacuum solutions in the non-dynamicalframework ( β = 0), with the coupling constant choice α = κ . We shall alsochoose these conventions here. Let us begin with a broad classification of general solutions in CS mod-ified gravity. Grumiller and Yunes [35] classified the space of solutions, a 2-dimensional representation of which is shown in Fig. 1, into an
Einstein space , E , and a CS space , CS . Elements of the former satisfy the Einstein equations,while the elements of the later satisfy the CS modified field equations. Theintersection of E with CS , P := E ∩ CS , defines the Pontryagin space , whoseelements satisfy both the Einstein and the CS modified field equations in-dependently. From the above definitions we can now classify solutions in CSmodified gravity. Elements in P are GR solutions , because they satisfy theEinstein equations and possess a vanishing C-tensor and Pontryagin density.Elements in CS \ P are non-GR solutions , because they are not Ricci-flat butthey do satisfy the Pontryagin constraint and the CS modified field equations.A full analytic study of exact solutions has been possible only regardingspacetimes with sufficient symmetries that allow for the modify field equationsto simplify dramatically. For such scenarios, however, the search for CS GRsolutions have lead mostly to either Minkowski space or the Schwarzschildmetric. This can be perhaps understood by considering the vacuum sector of P , where the C-tensor becomes C ab | R ab =0 = v cd ∗ R d ( ab ) c = v cd ∗ C d ( ab ) c = 0 , (84)where C abcd and ∗ C are the Weyl tensor and its dual respectively [Eqs. (27)and (28)]. Such a condition implies the Weyl tensor must be divergenceless via27he contracted Bianchi identities, which leads to three distinct possibilities:(1) The (dual) Weyl tensor vanishes. In vacuum, elements of P are also Ricciflat, so this possibility leads uniquely to Minkowski space.(2) The covariant acceleration of ϑ vanishes. Such a restriction imposes strongconstraints on the geometry (cf. e.g. [42]), leading either to flat space orto the existence of a null Killing vector.(3) The contraction of the covariant acceleration and the dual Weyl tensorvanishes.Elements of P are very special, possessing a large number of symmetries andKilling vectors. On the other hand, elements of CS \ P cannot possess toomany symmetries, which explains why it has been so difficult to find them. Consider first the most general, spherically symmetric spacetime, whosemetric can be decomposed as the warped product of two 2-dimensional met-rics [82,83]: a Lorentzian one g αβ ( x γ ) ( α, β, . . . = t, r ) with some coordinates x γ ; and the metric on the 2-sphere Ω ij ( x i ) ( i, j, . . . = θ, φ ) with some coordi-nates x i . Such a line element can be written in the following 2 + 2 form: ds = g αβ ( x γ ) dx α dx β + Φ ( x γ ) d Ω S , (85)where d Ω S is a line element of the round 2-sphere and the warped factor isthe square of the scalar field Φ( x γ ), often called “areal” radius.A spherically symmetric line element [eg. Eq. (85)] always leads to avanishing Pontryagin density, ∗ R R = 0, and to a decoupling of the modifiedfield equations [35]: R ab = 0 , C ab = 0 . (86)For the metric in Eq. (85), the only non-vanishing components of these tensorsare R αβ , R ij and C αi , provided ϑ belongs to the generic family [35,34] ϑ = F ( x γ ) + Φ( x γ ) G ( x i ) . (87)In the non-dynamical framework, these results imply that spherically sym-metric line elements are always pushed to P . Similar conclusions also hold forspherically symmetric line-elements in non-vacuum spacetimes.In the dynamical framework, one must also solve the evolution equationfor the CS scalar, which here becomes a wave-like equation without a source.In the absence of a potential, this wave-like equation need not necessarily havewell-defined decaying solutions that will lead to finite energy contained in thescalar field. If this is the case, the scalar field is forced to be a constant, whichreduces the modified theory to GR.The study of spherically symmetric line elements naturally leads to thestudy of Birkhoff’s theorem in CS modified gravity. This theorem states thatthe most general, spherically symmetric solution to the vacuum Einstein equa-tions is the Schwarzschild line element. For spherically symmetric line elements28nd the CS scalar of Eq. (87), the non-dynamical CS modified field equationsdecouple and the C-tensor identically vanishes, which suffices to guaranteethat Birkhoff’s theorem still holds in the non-dynamical formulation [34].In spite of the clear persistence of Birkhoff’s theorem in the non-dynamicalformulation, this theorem does not in fact hold in the dynamical framework. Inthis scheme, the dynamical field equations contain a scalar-field stress-energycontribution that will unavoidingly lead to non-vacuum (ie. hairy) solutions.Due to the presence of such a dynamical scalar field, spherical symmetry neednot lead to staticity, and in fact, spherically symmetric spacetimes will in gen-eral be dynamical. Static solutions are, however, still present in dynamical CSmodified gravity provided the CS scalar is a constant [84].The study of the spherically symmetric spacetimes in CS modified gravityleaves us with two important lessons: • The existence of specific solutions depends sensitively on the choice of thescalar field. • The satisfaction of the Pontryagin constraint is a necessary but not a suffi-cient condition for the C-tensor to vanish.In fact, it is simple to construct a CS scalar, such as a trigonometric functionof spacetime, and show that for such a scalar the C-tensor does not vanish,thus rendering the Schwarzschild metric not a solution of CS modified gravity,in spite of the vanishing of the Pontryagin density.
Consider the Kerr metric in Boyer-Lindquist coordinates ( t, r, θ, φ ): ds = − ∆ − a sin ΘΣ dt − aM r sin ΘΣ dtdφ + ( r + a ) − a ∆ sin ΘΣ sin Θ dφ + Σ∆ dr + Σ d Θ (88)where Σ = r + a cos Θ and ∆ = r + a − M r . When CS modified gravitywas proposed, Jackiw and Pi [22] realized that the Kerr metric would notbe a solution of the modified theory because the Pontryagin density is notvanishing: ∗ R R = 96 aM r Σ cos Θ (cid:16) r − a cos Θ (cid:17) (cid:16) r − a cos Θ (cid:17) . (89)This statement is also true in the dynamical frameworks, because the Pon-tryagin density will induce a non-constant CS scalar that will lead to a non-vanishing C-tensor. One can show that the parity-odd quantity in Eq. (89),is also non-vanishing for the Kerr-Newman and Kerr-NUT spacetimes [81,35],but it is is satisfied in certain interesting physical limits, namely a → M → ot imply that a rotating BH solution is absent in the modified theory. Mod-ifications of the Kerr metric that do satisfy the CS modified field equationscan be obtained be either considering approximate solutions or by studyingthe dynamical framework. For example, in the far field limit, M/r ≪
1, thePontryagin constraint is satisfied to O ( M/r ) and approximate modified solu-tions can be derived. On the other hand, in the dynamical formalism Eq. (89)serves as a source term that drives the evolution of the CS scalar, which inturn sources corrections in the metric [84].An example of the latter can be found in the quantum-inspired studiesof Campbell [85,86,24], Reuter [87] and Kaloper [88]. For example, in [85,87], φ F ab ∗ F ab is added to the Einstein-Maxwell-Klein-Gordon action, with φ ascalar (axion) field and F ab the photon strength field tensor. The equations ofmotion for the scalar field acquire a source (the expectation value of a chiralcurrent), when one treats the photon quantum mechanically. Upon accountingfor the one-loop fluctuations of the four-vector potential [89,90], one essentiallyfinds Eq. (21), which on a Kerr background can be solved to find [87] φ = 58 αβ aM cos θr + O M r ! . (90)The scalar field presents a r − fall-off and a dipolar structure, identical tothe Kalb-Ramond axion [85]. This field leads to a non-trivial C-tensor thatcorrects the Kerr line element, thus disallowing this metric as a solution of themodified theory. Nonetheless, the axion in Eq. (90) could (and has) been usedto construct correction to the Kerr metric [84], as we shall discuss in Sec. 5. Let us now consider static and axisymmetric line elements in vacuum andin the non-dynamical framework. Both stationary and static, axisymmetricspacetimes possess a timelike ( ∂ t ) a and an azimuthal ( ∂ φ ) a Killing vector, butthe difference is that static metrics contain no cross-terms of type dtdφ . Themost general such metric is diffeomorphic to [20] ds = − V dt + V − ρ dφ + Ω (cid:16) dρ + Λ dz (cid:17) , (91)where V ( ρ, z ), Ω( ρ, z ) and Λ( ρ, z ) are undetermined functions of two coordi-nates, ρ and z .Consider first the canonical choice of CS scalar. The modified field equa-tions once more decouple, as in Eq. (86) and the spherically symmetric case,which implies all static and axisymmetric solutions are elements of P , iden-tically satisfying the Pontryagin constraint. Due to this, we can make chooseΛ = 1 and put the metric into Weyl class [20,42]: ds = − e U dt + e − U h e k ( dρ + dz ) + ρ dφ i , (92)30here U ( ρ, z ) and k ( ρ, z ) replace the functions V and Ω.The vanishing of the Ricci tensor reduces to∆ U = 0 , k ,ρ = ρ ( U ,ρ − U ,z ) , k ,z = 2 ρU ,ρ U ,z , (93)where ∆ = ∂ /∂ρ +1 /ρ∂/∂ ρ + ∂ /∂z is the flat space Laplacian in cylindricalcoordinates. The function k can be solved for through a line integral once U is determined [42].The vanishing of the C-tensor reduces to the vanishing of the contractionof the dual Weyl tensor and the covariant acceleration of the CS coupling field.For a canonical ϑ , one finds [35]Γ tρt ∗ C t ( ab ) ρ + Γ tzt ∗ C t ( ab ) z = 0 , (94)which leads to a set of nonlinear PDEs for U , in addition to the Laplaceequation of Eq. (93).The set of equations imposed by Eq. (86) can be solved exactly, yieldingeither flat space, the Schwarzschild solution or the following two solutions: ds = − z dt + zdz + z ( dρ + ρ dφ ) (95)and ds = − (cid:18) mz − (cid:19) dt + (cid:18) mz − (cid:19) − dz + z ( dρ + sinh ρ dφ ) (96)These solutions, however, contain undesirable, non-physical features, such asthe existence of a naked singularity at z = 0 [Eq. (95)], or the existence of aspacelike Killing vector k a = ( ∂ t ) a in the “outside” region z > m [Eq. (96)].The above results can also be obtained by noticing that the spatial sectorof C ab reduces identically to the 3-dimensional Cotton-York tensor for staticand axisymmetric line elements, which by the decoupling of the field equationsmust vanish exactly [35]. This implies the metric must be spatially conformallyflat. Luk´acs and Perj´es [91] have shown that such solutions to the Einsteinequations reduce to Minkowski, Schwarzschild or the line elements of Eqs. (95)and (96), as described above.The above results hold for any CS scalar in the family ϑ = ϑ ( t, ρ, z ) + ϑ ( ρ, z, φ ) . (97)For even more general CS scalars, as would arise for instance in the dynamicalformulation, the modified field equations do not necessarily decouple, andthus, new and interesting solutions could arise. Unfortunately, when this isthe case, the system of partial differential equations becomes too difficult tostudy analytically and has not really been analyzed in detail.31 .5 Stationary and Axisymmetric Spacetimes Consider now stationary and axisymmetric spacetimes [42]: ds = − V ( dt − wdφ ) + V − ρ dφ + Ω (cid:16) dρ + Λ dz (cid:17) , (98)where V , Ω, Λ and w depend on ρ and z , with the latter identified with angularvelocity of rotation about the Killing axis.Let us first consider the canonical choice of CS scalar, for which themodified field equations decouple once more. The Einstein equations can beused to set Λ to unity and put the line element in Lewis-Papapetrou-Weylform [42] ds = − e U ( dt − wdφ ) + e − U h e k ( dρ + dz ) + ρ dφ i , (99)where U and k replace V and Ω.The vanishing of the Ricci tensor reduces to a set of partial differentialequations similar to Eq. (93), with a non-trivial source that depends on w and an additional equation for this function. The vanishing of the C-tensor,however, does not lead to a vanishing Cotton-York tensor, and thus, it doesnot necessarily imply spatial conformal flatness. Grumiller and Yunes [35] haveargued that it seems unlikely that other non-trivial and physically interestingsolutions besides the static ones could arise, because R ab = 0 and C ab = 0is a strongly over-constrained differential system. This is an example of howthe non-dynamical theory can lead to an overconstrained system of equationswhen searching for physically relevant solutions, but a strict proof remainselusive.The argument presented above holds for any ϑ in the family ϑ = tµ + φν , (100)with constants µ and ν . When the CS coupling is not of the form of Eq. (100),then the field equations do not decouple, the arguments presented above donot hold and the system become more difficult to analyze analytically.In the non-dynamical framework, another route to exact solutions forgeneric CS scalars is through the Pontryagin constraint. For axisymmetricspacetimes, ∗ R R = 0, and this leads to a complicated set of partial differentialequations for w , U and k . Solutions of ∗ R R = 0 have been found to be ofPetrov type II [35], which correspond to the Van Stockum class [42] ds = ρ Ω dt − ρdtdφ + 1 √ ρ (cid:16) dρ + dz (cid:17) , (101)where Ω = Ω( ρ, z ) is arbitrary and there is no dφ component.The reduced Van-Stockum class of metrics leads to a complete decouplingof the modified field equations, except for the dt component. The vanishing32f the C-tensor can be achieved if ϑ = ϑ ( ρ, z ), while the dt component of thefield equations can be solved to findΩ = c, ϑ = ϑ ( ρ, z ) , (102)Ω = c + d √ ρ , ϑ = 23 √ ρ z + ˜ ϑ ( ρ ) , (103)where c and d are constants. Equation (102) is non-flat, possessing a non-vanishing Riemann tensor and a third Killing vector t∂ t − φ∂ φ + ct∂ φ , butit satisfies R ab = 0 and C ab = 0 independently (ie. it belongs to P ). Equa-tion (103) is Riemann and Ricci non-flat ( R abcd = 0, R ab = − C ab = 0),belonging to CS \ P . This last solution can be interpreted as a BH solutionin the mathematical sense only, provided Ω vanishes for some ρ and a Killinghorizon appears, because it allows for closed timelike curves outside the Killinghorizon [35].We see then that physically relevant, stationary and axisymmetry, ex-act solutions have not been found in non-dynamical CS gravity, even whenconsidering generic CS scalars fields. We say relevant solutions here, becausenon-physical ones have been found, but they either contain naked singularitiesor closed time-like curves. In fact, as an example of the latter, recently [92]has shown that the Godel line element ds = a ( dt − dx + e x dy / − dz +2 e x dtdy ), which is a subclass of the metrics considered in this section, satisfiesthe non-dynamical modified field equations with the CS scalar ϑ = F ( x, y ), forsome arbitrary function F . Such a solution is of class P because the C-tensorautomatically vanishes with such choices. All evidence currently points at themodified field equations being over-constrained by the Pontryagin conditionand the field equations. Although this evidence is strong, no proof currentlyexists to guarantee that no solution can be found. Moreover, in the dynamicalframework, the relaxation of the Pontryagin constrain suffices to allow theexistence of Kerr-like solutions [84]. Consider line elements that represent exact gravitational wave solutions(pp-waves [93]): ds = − dvdu − H ( u, x, y ) du + dx + dy , (104)where H is the only free function of u , x and y . The Aichelburg-Sexl limit [94]of various BHs is in fact an example of such a line element. This limit consistsof ultrarelativistically boosting the BH, while keeping its energy finite, bysimultaneously taking the mass to zero as the boost speed approaches that oflight [95,96].For such metrics, the Pontryagin constraint vanishes identically, which atfirst seems in contradiction with the results of Jackiw and Pi [22], who showedthat generic linear GW perturbations lead to ∗ R R = O ( h ). In fact, these two33esults are actually consistent, because the line elements considered here andin [22] are intrinsically different and cannot be related via diffeomorphism.The fact that the Pontryagin density vanishes for the metric of Eq. (104) isrelated to the fact that the CS velocity field v a = ∂ a ϑ is a null Killing vector v a v a = 0 for such spacetimes [35].The modified field equations decouple, except for the du component.Consider first solutions that live in P [Eq. (86)], which require the vanishingof the C-tensor, thus forcing ϑ = λ ( u ) v + ˜ ϑ ( u, x, y ) . (105)The vanishing of the Ricci tensor forces∆ H = 0 , (106)2 H ,yy ˜ ϑ ,xy = H ,xy ( ˜ ϑ ,yy − ˜ ϑ ,xx ) . (107)Therefore, for any H that solves the Laplace equation in Eq. (106), we canfind a ˜ ϑ such that Eq. (107) is also satisfied. Such a scheme allows one to lift any pp-wave solution of the vacuum Einstein equations to a pp-wave solutionof non-dynamical CS modified gravity of class P through a choice of ϑ thatsatisfies Eq. (105) and (107) [35].Consider now solutions that live in CS \ P . All non- uu components ofthe modified field equations decouple, and thus the vanishing of the C-tensoris satisfied by scalar field of the form of Eq. (105). Choosing λ ( u ) = 0 forsimplicity, we find that the uu component of the modified field equationsreduces to a third order PDE(1 + ˜ ϑ ,y ∂ ,x − ˜ ϑ ,x ∂ ,y )∆ H = ( ˜ ϑ ,xx − ˜ ϑ ,yy ) H ,xy − ( H ,xx − H ,yy ) ˜ ϑ ,xy . (108)Simplifying this scenario further by choosing ˜ ϑ = a ( u ) x + b ( u ) y + c ( u ), Eq. (108)reduces to the Poisson equation ∆ H = f , whose source satisfies a linear firstorder PDE bf ,x − af ,y − f = 0, with general solution [assuming b ( u ) = 0] f ( u, x, y ) = e x/b ( u ) φ [ a ( u ) x + b ( u ) y ] (109)where φ is an arbitrary function. With such a source and two supplementaryboundary conditions, we can now solve the Poisson equation and specify thefull modified pp-wave solution [35]. Thus, generic CS \ P solutions do existin non-dynamical CS modified gravity and can be found via the algorithmdescribed above. In the non-dynamical framework, axisymmetry seems to limit the exis-tence of solutions for a certain class of coupling functions. However, if eitheraxisymmetry, the non-dynamical behavior of ϑ or the vacuum content assump-tion is relaxed, it is possible that solution in fact do exist.34onsider losing axisymmetry first. The general idea here is to add newdegrees of freedom in the metric that could compensate for the overconstrain-ing of the decoupling of the modified field equations. Such idea is in factinspired from approximate far-field solutions found in non-dynamical modi-fied CS gravity, which indeed require the presence of additional, non-vanishing,gravitomagnetic metric components. Up to the writing of this review, the onlyattempts to find such solutions have failed [35], due to the incredible complex-ity of the differential system.Consider next spacetimes with matter content. The Kerr BH is a “vac-uum” solution in GR, but it does possess a distributional energy momentumtensor [97]. Moreover, in string theory and cosmology [28], CS modified grav-ity arises from matter currents, so the inclusion of such degrees of freedommight in principle be important. In the dynamical scheme, for example, onecould lift any GR solution to a solution of CS modified gravity by requiringthat R ab − g ab R = 8 πT mat ab , (110) C ab = 8 πT ϑab , (111)where T mat ab stands for the stress-energy of matter degrees of freedom (suchas the distributional one associated with the Kerr solution), while T ϑab is theenergy-momentum of the CS coupling. One would now have solve the system ofPDEs associated with Eq. (111) for the background which satisfies Eq. (110).Such a task, however, would imply also solving the equation of motion forthe CS scalar, thus reducing this analysis to the study of exact solutions indynamical CS modified gravity, which has not yet been performed. Approximate schemes have been employed to solve the CS modified fieldequations in different limits. The first attempt along this lines was that ofAlexander and Yunes [32,7], who performed a far-field, PPN analysis of non-dynamical CS modified gravity ( α = κ , β = 0) with canonical ϑ . This studywas closely followed by that of Smith, et. al. [37] who carried out a far-fieldinvestigation of non-dynamical solutions representing the gravitational fieldoutside a homogenous, rotating sphere, taking careful account of the match-ing between interior and exterior solutions. Konno, et. al. [98] investigatedthe slow-rotation limit of stationary and axisymmetric line elements in non-dynamical CS modified gravity with non-canonical CS scalar fields. Finally,gravitational wave solutions of CS modified gravity have been studied bya number of authors, both in Minkowski spacetime and in an FRW back-ground [22,28,31,29,33,36]. The last two set of studies were carried out in thenon-dynamical formalism with the choices α = κ and β = 0. Little is knownabout GW propagation or generation in dynamical CS modified gravity, al-35hough early efforts are being directed on that front [84,99]. The PN approximation has seen tremendous success to model full generalrelativity in the slow-motion, weakly gravitating regime (for a recent reviewsee [100]). This approximation is used heavily to study Solar System testsof alternative theories of gravity in the PPN framework, as well as to de-scribe gravitational waves from inspiraling compact binaries, which could beobserved in gravitational wave detectors in the near future. For these reasons,it is instructive to study the PN expansion of CS modified gravity, beforesubmerging ourselves in other approximate solutions.The PN approximation is essentially a slow-motion and weak-gravityscheme in which the field equations of some theory are expanded and solvedperturbatively and iteratively. As such, this scheme makes use of multiple-scaleperturbation theory [101,102,103,104], where the perturbation parameters arethe self-gravity of the objects (an expansion in powers of Newton’s constant G ) and their typical velocities v (an expansion in inverse powers of the speed oflight). For example, matter densities ρ are dominant over pressures p and spe-cific energy densities Π, while spatial derivatives are dominant over temporalones.The PN framework also requires the presence of external matter degreesof freedom, ie. bodies that are self-gravitating and slowly-moving. Such objectscan be described in a point-particle approximation [100], or alternatively witha perfect fluid stress-energy tensor [105]: T ab = ( ρ + ρ Π + p ) u a u b + pg ab , (112)where u a is the object’s four-velocity. In GR, the internal structure of thegravitating objects can be neglected to rather large PN order [106], and thusone can effectively take the radius of the fluid balls to zero, which reproducesthe point-particle result. This statement is that of the effacing principle [106],which is the view we shall take in the next section when we study CS modifiedgravity in the PPN framework. However, care must be taken, since the effacingprinciple need not hold in alternative theories of gravity. In fact, as we shall seelater on, the effacing principle must be corrected in CS modified gravity due tomodifications to the junction conditions [107,108,109,110,111,112,113,21,114].Perturbation theory, and thus the PN approximation, requires the use of aspecific background and coordinate system. In the traditional PN scheme, onelinearizes the field equation with the metric g ab = η ab + h ab , where η ab is theMinkowski background, since cosmological effects are usually subdominant.Moreover, a Lorentz gauge is usually chosen h ba,a = h ,b /
2, which allows oneto cast the field equations as a wave equation with non-trivial, non-linearsource terms. One can show that to first order in the metric perturbation, thelinearized CS modified field equations in the non-dynamical formalism, with36anonical CS scalar and in the Lorentz gauge, can be written as [32,7] (cid:3) η H ab = − π (cid:18) T ab − g ab T (cid:19) + O ( h ) , (113)where the superpotential H ab captures the CS modification to the PN expan-sion to the modified field equations and it is given by H ab := h ab + ˙ ϑ ˜ ǫ cd ( a h b ) d,c , (114)where ˜ ǫ abcd stands for the Levi-Civita tensor density. The 00 component, thesymmetric spatial part and the trace of the superpotential are equal to thatof the metric perturbation, because the Levi-Civita symbol forces the CS cor-rection to vanish. The formal solution to the modified field equations thenreduces to H ab = − π (cid:3) − η (cid:18) T ab − g ab T (cid:19) + O ( h ) , (115)where the inverse D’Alembertian operator stands for a Green function integral.This formal solution is in fact identical to that of the PN expansion of GR inthe limit ˙ ϑ → h ab = h ( GR ) ab + ˙ ϑ ζ ab + O ( h ) , (116)where h ( GR ) ab is the GR solution, which is ϑ -independent and satisfies h ( GR ) ab := − π (cid:3) − η (cid:18) T ab − g ab T (cid:19) , (117)and where ζ ab is an unknown function that is first order in ˙ ϑ . When we combineEqs. (115), (114) and (116) we find ζ ab + ˙ ϑ ˜ ǫ cd ( a ζ b ) d,c = 16 π ˜ ǫ cd ( a ∂ c (cid:3) − η (cid:18) T b ) d − g b ) d T (cid:19) , (118)which can be solved for to find ζ ab = 16 π ˜ ǫ cd ( a ∂ c (cid:3) − η (cid:18) T b ) d − g b ) d T (cid:19) , (119)where we have neglected terms second order in ϑ . The metric perturbationthen reduces to h ab = − π (cid:3) − η (cid:18) T ab − η ab T (cid:19) + 16 π ˙ ϑ ˜ ǫ kℓi (cid:3) − η (cid:18) δ i ( a T b ) ℓ,k − δ i ( a η b ) ℓ T ,k (cid:19) , (120)where i, j, k stand for spatial indices only.37 .2 Parameterized Post-Newtonian Expansion Solar system tests of alternative theories of gravity are best performedwithin the PPN framework. This framework was first proposed by Eddington,Robertson and Schiff [115,116], but it matured with the work of Nordtvedtand Will [117,118,119,120] (for a review see e. g. [115]). PPN theory proposesthe construction of a model-independent super-metric that represents the PNapproximate solution to a family of gravity theories, parameterized by PPNparameters. Solar system experiments can then measure these parameters,thus selecting a particular member of this family. Currently, many of theseparameters have been experimentally determined with tight error bars, all ofwhich are consistent with GR [121].The PPN framework allows for tests of alternative theories of gravitythrough such PPN parameters. Given an alternative theory, one must first con-struct its PN solution and then compare it to the PPN super-metric. Throughthis comparison, one can read off how the PPN parameters depend on funda-mental parameters of the alternative theory. But since PPN parameters havebeen experimental constrained, one can propagate these constraints to thefundamental parameters of the alternative theory under consideration, thusobtaining an automatic Solar System test.Solar System tests in the PPN framework require the PN solution to themodified field equations. In this framework, however, it is not sufficient toleave the solution expressed in terms of the inverse D’Alembertian operator,but instead it must be parameterized in terms of PPN potentials, which aresimply Green function integrals over the stress energy tensor. Moreover, thePN expansion must be carried out to slightly different orders in v for differentcomponents of the metric, so as to obtain a consistent Lagrangian formulationof the theory . In the PPN framework, it suffices to compute g to O ( v ), g i to O ( v ) and g ij to O ( v ). Finally, the Lorentz gauge differs slightly fromthe PPN gauge, related via an infinitesimal gauge transformation, where thelatter is perturbatively defined via [115] h jk,k − h ,j = O (4) , h k,k − h kk, = O (5) , (121)where i, j, k stand for spatial indices only in the remaining of this section, h kk is the spatial trace of the metric perturbation and the symbol O ( A ) stands forterms of O ( ǫ A ), with ǫ the standard PN expansion parameter of O (1 /c ) [32,7].With this machinery at hand, we can now perturbatively expand thetrace-reversed CS modified field equations. For the remaining of this chapter,we shall concentrate on the non-dynamical formalism of CS modified gravity,with the canonical choice of CS scalar. Expressions for the linearized Ricciand C-tensors to O (4) in an arbitrary gauge and for generic ϑ are long and For example, such order counting is necessary in order to calculate the gravita-tional deflection of light consistently to first order. ϑ , the linearized Ricci tensor becomes R = − ∇ h − h ,i h ,i + 12 h ij h ,ij + O (6) ,R i = − ∇ h i − h , i + O (5) ,R ij = − ∇ h ij + O (4) , (122)while the linearized C-tensor is given by C = O (6) ,C i = −
14 ˙ ϑ ˜ ǫ kli ∇ h l,k + O (5) ,C ij = −
12 ˙ ϑ ˜ ǫ kl ( i ∇ h j ) l,k + O (4) , (123)where ∇ = η ij ∂ i ∂ j is the flat-space Laplacian. As in Sec. 5.1, we find twodistinct corrections due to the CS modification: one to the transverse-tracelesspart of the spatial metric and the other to the vector metric perturbation.Let us now solve the linearized CS modified field equations iterativelyand perturbatively. To O (2), the 00 component of the metric is not modifiedby the C-tensor and the field equation becomes ∇ h = − πρ, (124)because T = − ρ . Equation (124) is the Poisson equation, whose solution interms of PPN potentials is h = 2 U + O (4) , (125)where U is the Newtonian potential [32,7]. We shall not present these poten-tials here, but they can be found in [115].To this same order, the 0 i component does not provide any information,while the ij sector leads to the following field equations: ∇ h ij + ˙ ϑ ˜ ǫ kl ( i ∇ h j ) l,k = − πρδ ij , (126)where δ ij is the Kronecker delta. This equation can be rewritten in terms ofthe superpotential of Sec. 5.1 as ∇ H ij = − πρδ ij , (127)whose solution is [32,7] H ij = 2 U δ ij + O (4) . (128)39ollowing the same procedure as in Sec. 5.1, we can use the decomposition ofEq. (116) to find that ζ ij + ˙ ϑ ˜ ǫ kl ( i ζ j ) l,k = 0 . (129)The second term on the left-hand side of Eq. (129) is a second order correctionin ˙ ϑ and can thus be neglected, which then renders h ij = H ij to O (2), wherethe latter is given in Eq. (128). The physical reason why the CS modificationdoes not correct the spatial sector of the metric is related to the source studiedhere, together with the PPN gauge. In fact, we shall see in Sec. 5.6 that whenone studies gravitational wave propagation in vacuum, the spatial sector of themetric is indeed modified. Moreover, if we were to study the O (4) correctionsto the spatial sector of the metric, we would probably find non-vanishing CScorrections, but such a study has not yet been carried out.To next order, O (3), the only relevant field equations are related to thegravitomagnetic sector of the metric. The field equations become [32,7] ∇ h i + 12 h , i + 12 ˙ ϑ ˜ ǫ kli ∇ h l,k = 16 πρv i , (130)which with the lower-order solution of Eq. (125) and the superpotential ofEq. (114) becomes ∇ H i + U , i = 16 πρv i , (131)whose solution is H i = − V i − W i , (132)where V i and W i are PPN vector potentials. Combining Eq. (114) with Eq. (116),we can solve Eq. (132) to find [32,7] h i = − V i − W i + 2 ˙ ϑ ( ∇ × V ) i + O (5) , (133)where ( ∇ × A ) i = ˜ ǫ ijk ∂ j A k is the standard curl operator of flat spaceFinally, to next order, O (4), we need only analyze the 00 component ofthe modified field equations. Since the 00 component of the C -tensor does notcontribute to the field equations to this order, the modified field equationsreduce exactly to those of GR, as also does their solution. We therefore findthat the PPN solution to the CS modified field equations is given by g = g ( GR )00 + O (6) ,g i = g ( GR )0 i + 2 ˙ ϑ ( ∇ × V ) i + O (5) ,g ij = g ( GR ) ij + O (4) , (134)where g ( GR ) ab is the PPN solution of GR. This solution in fact satisfies thePontryagin constraint ⋆ RR = 0 to leading order because the contraction of theLevi-Civita symbol with two partial derivatives vanishes. Thus, the equationsof motion for the fluid can be obtained directly from the covariant derivativeof the stress-energy tensor (the strong equivalence principle holds).40y comparing this PPN solution to the super-metric of the PPN frame-work, we can now read off the PPN parameters of CS gravity [32,7]. Doing so,one finds that all PPN parameters of CS gravity are identical to those of GR( γ = β = 1, ζ = 0 and α = α = α = ξ = ξ = ξ = ξ = 0), except for thenew term in g i . This CS correction cannot be mimicked by any standard PPNparameter, and thus, in order to model parity-violating theories an additionalPPN parameter χ must be introduced, namely g i = g ( P P N )0 i + χ ( M ∇ × V ) i , (135)where we have multiplied the curl operator by the total mass M , in orderto make χ a proper dimensionless parameter. In CS modified gravity, thisparameter is simply χ = 2 ˙ ϑ/M .Canonical CS modified gravity can then be tested by experimentally con-straining χ , which thus directly places a constraint on the canonical CS scalar˙ ϑ = 1 /µ . The only requirement for such a contribution to be non-vanishing isthat the PPN vector potential V i be non-vanishing. This is satisfied by non-static sources, ie. objects that are either moving or spinning relative to thePPN rest frame. Just because the correction to the gravitomagnetic sector ofthe metric is non-vanishing, however, does not imply that physical observables,such as the Lense-Thirring effect, will also be corrected. In fact, as we shallsee in Sec. 7.1, for such corrections to emerge, the curl of the gravitomagneticcorrection to the metric must be non-vanishing, which holds only for movingpoint particles.A caveat should be discussed at this point. As we already alluded to,the point-particle approximation holds in GR for many sources, such as blackholes, neutron stars or other regular stars, because the effacing principle holdsand Birkhoff’s theorem also holds. In CS gravity, however, the former does notnecessarily hold and the junction conditions are modified. One should then alsobe aware that a homogenous solution to the linearized field equations mightbe lacking from this analysis. Such a solution is most likely oscillatory innature and would not affect the average behavior of the correction, probablyacquiring importance only when the latter vanishes. In the next section weshall see how such a boundary solution arises and contributes significantly forspinning bodies, for which the homogeneous correction to the Lense-Thirringeffect vanishes. The analysis presented so far has dealt with the non-dynamical formal-ism, concentrating primarily on point particle sources. Approximate solutions,however, can be developed for extended objects as well. As already described,in GR the effacing principle guarantees that for certain sources both the ex-tended and point particle approach coincide up to rather high PN order. Thisresult somewhat relies on the junction conditions of GR, which here are mod-41fied due to the presence of the C-tensor in the field equations. Moreover, thePN scheme is not the only formalism in which far field solution can be in-vestigated. In particular, the gravitoelectromagnetic analogy [122,123] is alsouseful to obtain far field solutions, since it allows one to use machinery fromthe theory of electromagnetism.Smith, et. al. [37] performed the first detailed analysis of rotating, ex-tended objects in CS modified gravity in the gravitoelectromagnetic scheme,modeled via the stress-energy tensor T ab = 2 t ( a J b ) − ρt a t b , (136)where the current J a := ( − ρ, ~J ) and t a = [1 , , ,
0] is a time-like unit vec-tor . Such a choice of stress-energy is equivalent ot that of a pressurelessperfect fluid in the limit as the fluid balls tend to particles. Moreover, Smith, et. al. [37] focused on the dynamical formulation of CS modified gravity, withthe conventions α = − ℓ/ β = −
1. Such a study is forbidden in thenon-dynamical formalism, because the Pontryagin constraint does not allowfor rotating solutions. In the dynamical scheme, however, such solutions areallowed, provided the Pontryagin density is balanced by the dynamics of thescalar field.In spite of working in the dynamical theory, Smith, et. al. [37] chosea canonical ϑ , thus implicitly treating this field as non-dynamical. Such achoice of scalar field is formally inconsistent with dynamical CS modified grav-ity, since the non-vanishing Pontryagin constraint will force spatial variationson ϑ . Nonetheless, this choice could become a good approximation, if one isonly concerned with far-field solutions. This is because, as we have shown inEq. (89) of Sec. 4.3, the exterior Pontryagin density for a rotating source scalesas ∗ R R ≈ aM /r , which then forces β (cid:3) ϑ = − α aM r = 8 ℓ aM r , (137)with uncontrolled remainders of O ( M/r ) . Thus, spatial variations will inducemodifications to the canonical ϑ that are at most of O ( M/r ) , which is beyondthe order considered in this section. A full dynamical study of this problemhas yet to be carried out.Let us now return to the gravitoelectromagnetic formalism, in which cer-tain components of the metric perturbation are identified with a four-vectorpotential, namely A a := −
14 ¯ h ab t b , (138)where ¯ h ab is the trace-reversed metric perturbation ¯ h ab := h ab − η ab h and h = η ab h ab is the trace of the latter. In analogy with electromagnetism, one Recall that indices are raised and lowered with the Minkowski metric in the far-field approximation. E i = ∂ i A − ∂ A i , B i = ǫ ijk ∂ j A k , (139)where i, j stand for spatial indices only. The geodesic equations can then bewritten in terms of these fields as F i = − mE i − m ( v × B ) i , (140)where the cross product is that of flat Euclidean space, from which one canread off the Lorentz force acceleration a i = F i /m .We can now apply this formalism to the linearized CS modified field equa-tions. In the Lorentz gauge, A a,a = 0, the modified field equations become [37] ~ ∇ · ~B = 0 , (141) ~ ∇ × ~E = − ∂ ~B∂t , (142) ~ ∇ · ~E = 4 πG ( ρ + T ϑ ) (143) ~ ∇ × ~B − ∂ ~E∂t − m cs ✷ ~B = 4 πG ~J , (144)where m cs is a characteristic mass scale defined via m cs := 2 κα ˙ ϑ = − κℓ ˙ ϑ . (145)In the above equations, the first two arise from the definitions of the four-vector potential, while the last two (Gauss’s law in Eq. (143) and Amp`ere’slaw in Eq. (144)) are modified in CS modified gravity, the former arising fromthe time-time component of the modified field equation and the latter from thegravitomagnetic sector. Smith, et. al. [37] neglect T ϑ because they associate itwith the energy density of the scalar field, which they argue must be uniformthroughout the Solar System, not larger than the mean cosmological energydensity and negligible relative to ρ . Moreover, from Eq. (15), we see that T ϑ is at least quadratic in ˙ ϑ and can thus be neglected, provided 3 β/ (2 ρµ ) ≪ ~J = ρ [ ~ω × ~r ] Θ( R − r ) , (146)where R is the radius of the rotating body, ρ is its density, ~ω is its angu-lar velocity, r is the distance from the origin, and Θ is the Heaviside stepfunction. One can now use this mass current in Amp`ere’s law to solve thisequation for A i , imposing continuity and finiteness at the origin. The actualexpressions for the four-vector potential can be found in [37], but its associ-ated gravitomagnetic field is given by ~B = ~B GR + ~B CS , where the GR piece is43imply ~B GR = 4 πGρR (cid:16) − r R (cid:17) ~ω + 3 r R ˆ r × (ˆ r × ~ω ) , r ≤ R, R r [2 ~ω + 3ˆ r × (ˆ r × ~ω )] , r ≥ R, (147)while the CS correction is given by ~B CS = 4 πGρR { D ( r ) ~ω + D ( r ) ˆ r × ~ω + D ( r ) ˆ r × (ˆ r × ~ω ) } . (148)The function D , , ( r ) are actually functionals of spherical Bessel functions ofthe first j ℓ ( x ) and second y ℓ ( x ) kind, given explicitly in [37]. Remarkably, thegravitoelectromagnetic analogy allows for a solution to the linearized modi-fied field equations for an extended rotating source, which possesses both apoloidal and toroidal gravitomagnetic field. In other words, the metric doesnot only contain gravitomagnetic components co-aligned with the spin axisof the rotating extended body, but also along other axis, perpendicular tothe plane defined by ~ω and ~r . Such terms cannot indeed be removed via acoordinate transformation since B iφ is generically non-vanishing.The solution for the metric perturbation found in the gravitoelectro-magnetic analogy for extended sources differs slightly from that found inthe PN scheme for point particles. Some of these differences arise becausein the latter the point particles were allowed to possess non-vanishing angu-lar and linear momentum. The main difference, however, is not due to thestress-energy tensor studied, but to the fact that extended and point particletreatments are not equivalent in CS modified gravity. In the former, an ad-ditional oscillatory behavior is needed, encoded in the appearance of Besselfunctions, in order to guarantee continuity across the surface of the sphere.Such terms are necessary because the C-tensor modifies the junction condi-tions of GR [107,108,109,110,111,112,113,21,114].The gravitoelectromagnetic analogy also sheds some light on the study ofexact solutions to CS modified gravity, reviewed in Sec. 4. The CS correctionto Amp`ere’s law, Eq. (144) changes the character of the differential system,from a first-order one to a second-order one. Such a result is in fact expectedfrom the structure of the modified field equations [Eq. (18)], since the C-tensor depends on third-order derivatives of the metric. One then could alsoexpect that approximate solution to CS modified theory need not be easilyobtained as deformations of GR solutions. This is because, in principle, the CScorrection could produce dramatic changes to the dynamical behavior of thesolution (ie. the solution, for example, could be an element of CS \ P in thenomenclature of Sec. 4, which would hold little resemblance to GR solutions).CS modified theory and GR, however, have both now been sufficiently testedto allow for a perturbative treatment of the modified theory, at least in theSolar System. Whether such treatment is allowed in more non-linear, strongfield scenarios remains to be studied further.44 .4 Perturbations of the Schwarzschild Spacetime Black hole perturbation theory has been incredibly important in GR,leading for example to a better understanding of subtle issues related to BHphysics and the extraction of gravitational waves in dynamic spacetimes. More-over, through the discovery of quasinormal ringing, new tests of GR have beenproposed (see eg. [124] for a review). These accomplishments alone suggest astudy of BH perturbation theory in CS modified gravity could lead to inter-esting results. Such a study was carried out by Yunes and Sopuerta [34] in thenon-dynamical formalism with α = κ and β = 0.Let us begin by introducing the basics of BH perturbation theory, asdeveloped in GR. Consider then the following perturbed metric g ab = ¯ g ab + h ab , (149)where ¯ g ab is the background Schwarzschild metric and h ab is a generic metricperturbation. Henceforth, objects associated with the background will be de-noted with an overhead bar. The spherical symmetry of the background allowsone to expand the metric perturbation in tensor spherical harmonics, thus sep-arating the angular i = { θ, φ } and temporal-radial µ = { t, r } dependence inthe perturbed field equations and yielding a 1 + 1 PDE system. This systemcan be further simplified by distinguishing between polar and axial parity har-monics: polar (axial) modes acquire a ( − l [( − l +1 ] prefactor upon paritytransformations. The aforementioned simplification comes about because inGR the perturbative field equations decouple into two subsystem that can beclassified by their parity.With these considerations in hand, we split the metric perturbation into h ab = h a ab + h p ab , where each of these pieces is decomposed in tensor sphericalharmonics: the temporal-radial sector becomes h a µν = 0 , h p µν = X ℓ,m h p ,ℓmµν = X ℓ,m h ℓmµν Y ℓm , (150)the temporal-angular sector is given by h a µi = X ℓ,m h a ,ℓmµi = X ℓ,m h ℓmµ S ℓmi , h p µi = X ℓ,m h p ,ℓmµi = X ℓ,m p ℓmµ Y ℓmi , (151)and the angular-angular sector becomes h a ij = X ℓ,m h a ,ℓmij = X ℓ,m H ℓm S ℓmij , h p ij = X ℓ,m h p ,ℓmij = X ℓ,m r (cid:16) K ℓm Y ℓmij + G ℓm Z ℓmij (cid:17) , (152)asterisks denote components given by symmetry and where Y ℓm are standardscalar spherical harmonics [see [125] for conventions], Y ℓmi and S ℓmi are polarand axial vector spherical harmonics respectively, and Y ℓmij , Z ℓmij and S ℓmij arepolar, polar and axial, tensor spherical harmonics. Vector spherical harmonics45re defined for ℓ ≥ Y ℓmi ≡ Y ℓm : i , S ℓmj ≡ ǫ ij Y ℓmj . (153)while tensor spherical harmonics are defined for ℓ ≥ Y ℓmij := Y ℓm Ω ij , Z ℓmij := Y ℓm : ij + ℓ ( ℓ + 1)2 Y ℓm Ω ij , S ℓmij := S ℓm ( i : j ) , (154)where Ω ij and ǫ ij are the metric and Levi-Civita tensor on the 2-sphere re-spectively, while colon stands for covariant differentiation on the 2-sphere. Allmetric perturbations are functions of t and r only.The perturbative field equations can be decoupled in GR in terms ofcomplex master functions, known as the Cunningham-Price-Moncrief (CPM)master function [126] and the Zerilli-Moncrief (ZM) master function [127,128].The former is given byΨ ℓm CPM = − rλ ℓ (cid:18) h ℓmr,t − h ℓmt,r + 2 r h ℓmt (cid:19) , , (155)where λ ℓ = ( ℓ + 2)( ℓ − /
2, while the latter is given byΨ ℓm ZM = r λ ℓ ( K ℓm + (1 + λ ℓ ) G ℓm + f Λ ℓ (cid:20) f h ℓmrr − rK ℓm,r − r (1 + λ ℓ ) p ℓmr (cid:21)) , (156)where f = 1 − M/r is the Schwarzschild factor and Λ l = λ ℓ + 3 M/r .The perturbative field equations can be written in terms of these masterfunction, obtaining the so-called master equations h − ∂ t + ∂ r ⋆ − V Polar / Axial ℓ ( r ) i Ψ ℓm CPM / ZM = 0 , (157)where r ⋆ is the tortoise coordinate r ⋆ = r + 2 M ln [ r/ (2 M ) −
1] and V Polar / Axial ℓ are potentials, which depend on r , parity and harmonic number [125]. Inprinciple, the right-hand side of the master equations is not zero, but dependson some matter sources, which we neglect here since we are searching forvacuum metric perturbations. Moreover, although the master functions aregauge invariant, the analysis is simplified if one picks a gauge. Henceforth wechoose the Regge-Wheeler gauge in which H ℓm = 0 = G ℓm = p ℓmµ .With all this machinery, we can now study BH perturbation theory in CSmodified gravity. Yunes and Sopuerta [125] studied the non-dynamical for-malism, in which the Pontryagin plays a critical role. In terms of perturbationfunctions this constraint reduces to ∗ R R = 96 Mr (cid:20) h ℓmt + r (cid:16) h ℓmr,t − h ℓmt,r (cid:17)(cid:21) ℓ ( ℓ + 1) Y ℓm . (158)Although this constraint is automatically satisfied by the background, it isnot satisfied for generic metric perturbations. Remarkably, this precise com-46ination of metric perturbations can be written in terms of the CPM functionexactly: ∗ R R = − Mr ( ℓ + 2)!( ℓ − ℓm CPM Y ℓm . (159)The Pontryagin constraint then forces the CPM function to vanish for all har-monics ℓ ≥
2, which is not equivalent to requiring that all axial perturbationsalso vanish. One then arrives at the constraint h ℓmr,t = h ℓmt,r − r h ℓmt . (160)The set of allowed solutions is thus reduced by this constraint, which in facttends to lead to an overconstrained system of perturbative field equations, aswe shall see next. It is also in this sense that the non-dynamical frameworkleads to an overconstrained system of partial differential equations.Let us now concentrate on the perturbative field equations in CS modifiedgravity. In order to simplify calculations, Yunes and Sopuerta [125] chose ascalar field of the form ϑ = ¯ ϑ ( t, r ), which possesses the same symmetries as thebackground. After harmonically decomposing the perturbative field equations,one finds [125] G ℓmµν [ U ℓm Polar ] = − C ℓmµν [ U ℓm Axial ] , G ℓmµ [ U ℓm Polar ] = − C ℓmµ [ U ℓm Axial ] , (161) G ℓm [ U ℓm Polar ] = − C ℓm [ U ℓm Axial ] , H ℓm [ U ℓm Polar ] = − D ℓm [ U ℓm Axial ] , (162) H ℓmµ [ U ℓm Axial ] = − D ℓmµ [ U ℓm Polar ] , I ℓm [ U ℓm Axial ] = − E ℓm [ U ℓm Polar ] . (163)where U ℓm Polar and U ℓm Axial denote polar and axial metric perturbations respec-tively U ℓm Polar = ( h ℓmµν , p ℓmµ , K ℓm , G ℓm ) , U ℓm Axial = ( h ℓmµ , H ℓm ) , (164)and where the right hand sides of Eq. (161) and (163) also depend on deriva-tives of ϑ . The functionals in Eqs. (161) and (163) are long and unilluminating,so we shall omit them here, but they are presented in full detail in [125]. Per-haps not too surprisingly, the parity-violation induced by CS modified gravitybreaks the axial-polar decoupling expected in GR. Instead, we now find thatmodes with opposite parity are coupled and cannot in general be treated sep-arately.Due to the non-decoupling of the perturbative field equations, a genericstudy of their solution is a quixotic task. Yunes and Sopuerta [125] investigatedseveral specific cases, from which one can then extrapolate generic conclusions.Let us then first consider the canonical choice of ϑ and single-handed metricperturbations, ie. purely polar or purely axial perturbations. In either of thesecases, one can show that the Pontryagin constraint leads to an overconstrainedPDE system, and thus, if one set of perturbations vanishes, then all metricperturbations must vanish. Moreover, these conclusions do not only hold forthe canonical choice of ϑ but also for many other members of the family47 ϑ ( t, r ) [125]. However, a generic result for arbitrary ϑ or arbitrary axial andpolar metric perturbations has not yet been produced, due to the incrediblecomplexity of the perturbative equations.The overconstraints of the non-dynamical formalism are mainly due tothe Pontryagin condition, which is relaxed in the dynamical formalism. Yunesand Sopuerta [125] have studied BH perturbation theory in the dynamical for-malism, where ϑ and h ab are treated as independent perturbation parameters.Consider then the split g ab = ¯ g ab + ǫ h ab , (165) ϑ = τ ( ¯ ϑ + ǫ δϑ ) = τ ¯ ϑ + τ ǫ X ℓ ≥ ,m ˜ ϑ ℓm Y ℓm , (166)where here ǫ and τ are bookkeeping, independent perturbation parameters.The expansion for ϑ is a bivariate expansion, where ¯ ϑ is some backgroundvalue for the CS coupling and is spherically symmetric, while ˜ ϑ ℓm are harmoniccoefficients of the ϑ perturbation.In the dynamical formalism, the Pontryagin constraint is replaced byan evolution equation for the scalar field, which to leading order is simply¯ (cid:3) ¯ ϑ = 0, where we have set the potential to zero. To first order in ǫ , theevolution equations become − ǫ α δ ( ∗ R R ) = ǫτ β (cid:26) ¯ (cid:3) δϑ − (cid:20) ¯ ϑ ,ab + (cid:16) ln √− ¯ g (cid:17) ,a ¯ ϑ ,b (cid:21) h ab − ¯ ϑ ,a h ab,b + 12 h ,b ¯ g ab ¯ ϑ ,b (cid:27) . (167)where δ ( ∗ R R ) is the functional coefficient of ∗ R R to O ( ǫ ) given explicitlyin [125]. Similarly, to leading order in ǫ the perturbative field equations alsobecome modified, with corrections that arise from the expansion of the stress-energy tensor of ϑ . By relaxing the Pontryagin constraint, the PDE systemceases to be overconstrained and might allow generic metric oscillations. How-ever, since τ ≪
1, the magnitude of the CPM master function will be con-strained to be small, which could lead to interesting observational conse-quences, for example in the emission of energy by gravitational wave radi-ation. A full dynamical study of the solution to these equations has not yetbeen carried out.
Instead of studying arbitrary perturbations, let us now concentrate onperturbations that represent slow-rotation deformations of the Schwarzschildspacetime. By searching for such solutions, we might gain some insight on howto extend the Kerr metric in CS modified gravity. In the previous section, wesaw that a CS scalar of the form ϑ = ϑ ( t, r ) will lead to an overconstrainedsystem of equations, so in this section we shall explore more general choices.The slow-rotation limit of stationary and axisymmetric line elements innon-dynamical CS modified gravity ( α = κ and β = 0) was first studied48y Konno, et. al. [98]. Consider then the following general, stationary andaxisymmetric line element ds = − f [1 + h ( r, θ )] + f − [1 + m ( r, θ )]+ r [1 + k ( r, θ )] h dθ + sin θ ( dφ − ω ( r, θ ) dt ) i (168)where f = 1 − M/r is the Schwarzschild factor, h, m, k, ω are unknownfunctions of r and θ . The metric perturbations are assumed to be linear inthe perturbation parameter: J/M , the ratio of the angular momentum of therotating compact object to the squared of its mass. With this line element,the Pontryagin constraint leads to a condition on the function ω , namely ω ,rθ + 2 cot θω ,r = 0, which leads to the solution ω = ¯ ω ( r ) / sin θ . Such a resultis so far independent of the choice of CS scalar.The modified field equations can be now linearized in the unknown func-tions and solved, given some choice for ϑ . When a canonical scalar field ischosen, Konno, et. al. [98] showed that the linearized equations force thesefunctions to vanish and thus a rotating solution cannot be found to first or-der. Note that this result is not in disagreement with the discussion in Sec. 5.2and 5.3, since there the far-field solutions found cannot be put in the form ofEq. (168).A solution to the linearized CS modified field equation can in fact beobtained in the slow-rotation limit, provided we explore other choices for ϑ .Konno, et. al. [98] showed that with the scalar ϑ = r cos θ/λ , (169)a solution can indeed be found and it is given by h ( r, θ ) = m ( r, θ ) = k ( r, θ ) = 0 , ¯ ω ( r ) = D r f + D r h r − M r − M + 4 M rf ln( r − M ) i (170)which leads to the Schwarzschild metric, plus a new term in the tφ sector ofthe metric, namely, g tφ = D f + D r h r − M r − M + 4 M rf ln( r − M ) i , (171)where D i are constants of integration that are assumed linear in J . Note, how-ever, that if D = 0, the gravitomagnetic sector of the metric naively looks as ifit could diverge in the limit as r → ∞ increases. On closer inspection, however,one finds that invariants and physically relevant observables do not diverge.For example, the scalar invariant R abccd R abcd ∝ M /r − D / ( r sin θ ),which indeed vanishes at spatial infinity, where the divergence at θ = 0 or π presumably arises due to the first-order linear perturbation scheme [129]. Thequantity λ in ϑ is a constant with units of inverse length, which curiously49oes not appear in the solution for ¯ ω . This is because the embedding coordi-nate can be factored out and does not enter into the linearized modified fieldequations to leading order in the angular momentum.Interestingly, the above solution cannot be interpreted as a small defor-mation of the Kerr line element. That is, there is no choice of D i for which g tφ can be considered a small deformable correction to Kerr. Such an observationimplies that the frame-dragging induced by such a metric will be drasticallydifferent from that predicted by the Kerr line element, in fact sufficientlyso to allow for an explanation of the anomalous velocity rotation curves ofgalaxies which we shall discuss further in Sec. 7. At the same time, however,Solar System experiments have already measured certain precessional effectsin agreement with the GR prediction [130,131,132], and thus a drastically dif-ferent frame-dragging prediction might be in contradiction with these SolarSystem tests.Recently, Yunes and Pretorius [84] have extended and generalized thisresult. They showed that in fact the solution in Eq. (170) is preserved for any ϑ in the family ϑ gen = A + A x r cos φ sin θ + A y r sin φ sin θ + A z r cos θ, (172)where A i are constants. In fact, we can rewrite this CS coupling field as ϑ = δ ab A a x b , where x a = [1 , x, y, z ] and δ ab is the Euclidean metric. Note,however, that the stress-energy tensor associated with any member of thisfamily [including Eq. (169)] is constant, and thus the energy associated withsuch a field is infinite. Because of this, the solution found here cannot beextended to the dynamical framework.Moreover, Yunes and Pretorius [84] also found another solution to theslow-rotation limit of the modified field equations, if one considers a genericCS scalar field in the non-dynamical framework: ϑ = ¯ f ( r, φ ) + r ¯ g ( φ ) + r ¯ h ( C φ − t ) + r ¯ k ( θ, φ )+ r Z drr (cid:20) − ∂ r ¯ f ( r, φ ) + 1 r ¯ f ( r, φ ) + 1 r ¯ j ( r ) (cid:21) , ¯ ω = − C r f, (173)where ¯ f , ¯ g , ¯ h , ¯ j and ¯ k are arbitrary functions and C is another integrationconstant. This arbitrary function can be chosen such that the new CS scalarpossesses a sufficiently fast decaying stress-energy tensor with non-infiniteenergy. For example, if ¯ f = ¯ g = ¯ h = ¯ k = 0 and ¯ j = − j /r , then ϑ = j /r , for constant j , and thus Eq. (173) is compatible with the dynamicalframework.The existence of two independent solutions to the modified field equationsin the non-dynamical framework and in the slow-rotation limit suggests thatthere is a certain non-uniqueness in the framework encoded in the arbitrari-ness of the choice of ϑ . This scenario is to be contrasted with the dynamical50ramework, where the CS scalar is uniquely determined by its evolution equa-tion and there is no additional freedom (except for that encoded in initialconditions).In view of this problems with the non-dynamical framework, Yunes andPretorius [84] studied the same scenario but in the full dynamical framework.A new approximation scheme is employed on top of the slow-rotation require-ment, which essentially demands that the CS correction be a small deformationof the Kerr line element, ie. the CS coupling is assumed small relative to theGR one. In this way, one finds the solution [84]: ds = ds + 54 α βκ ar Mr + 2710 M r ! sin θ dφdt,ϑ = 58 αβ aM cos( θ ) r Mr + 18 M r ! , (174)where ds is the slow-rotation limit of the Kerr metric, M is the BH massand J = a · M is the BH angular momentum to leading order. This solutionis valid to second order in the slowly-rotation expansion parameter a/M , aswell as in the strength of the coupling α/ ( βκM ). Notice that this solution,derived under the dynamical formulation, is perfectly well-behaved at spatialinfinity, remaining asymptotically flat.Equation (174) is the first rotating BH solution in dynamical CS modi-fied gravity, and can be thought of as a small deformation of a Kerr black holewith additional CS scalar “hair” of finite energy. Although this is a “hairy”solution, Sopuerta and Yunes [99] have shown that the solution is still entirelydescribed by the mass and the angular momentum of the source. The no-hairtheorem is, however, violated in that the relation between higher-multipolesand the mass quadrupole and current dipole is CS modified at ℓ = 4 multipoledue to the CS correction in Eq. (174) of the gravitomagnetic sector.The solution found in the dynamical theory presents interesting parityproperties. Since the dynamics of the CS scalar field are determined by thePontryagin density, this field is parity-violating (ie. it is a pseudo-scalar). Boththe Kerr metric and the CS correction to it are also parity-violating, but thelater is induced by a curvature-scalar field interaction, instead of due to theKerr distributional stress-energy. In fact, the parity violation introduced bythe CS correction becomes dominant in regions of high curvature.Moreover, the dynamical solution presented above shows remarkable sim-ilarities with some of the far-field results found in the non-dynamical frame- Notice that the solution for ϑ is identical to that found by Campbell [85] andReuter [87] and discussed in Eq. (90), except that there the backreaction of thisfield on the metric was ignored. Shortly after publication of this result, Konno, et. al. employed slightly differentmethods to verify that the solution found by Yunes and Pretorius indeed satisfiesthe modified field equations [133].
GW solutions have been studied by a large number of authors [22,28,29,33,36,81,99],but mostly in the non-dynamical theory, which we shall concentrate on here.The first GW investigation in non-dynamical CS modified gravity was car-ried out by Lue, Wang and Kamionkowski [134], who studied the effect ofGWs in the cosmic microwave background (see Sec. 8). Jackiw and Pi [22]also studied GWs in CS modified gravity, concentrating on the generation ofsuch waves and the power carried by them in the modified theory. Shortlyafter, such waves were used to explain baryogenesis during inflation [28] andto calculate the super-Hubble power spectrum [29]. The generation of GWswas also studied in the dynamical formalism [81] through the constructionof an effective stress-energy tensor and the Isaacson scheme [99]. Recently,GW tests have been proposed to constrain CS gravity with space-borne [33]gravitational wave interferometers.We shall here discuss GW solutions in non-dynamical CS modified gravityand postpone any discussion of GW generation to the next section. Moreover,we shall not discuss in this section cosmological power spectra, since thesewill be summarized in Sec. 8. GW propagation in CS gravity has only beenstudied in the non-dynamical formalism with β = 0 and α = κ . Let us beginwith a discussion of GW propagation in a FRW background. Consider thenthe background ds = a ( η ) h − dη + ( δ ij + h ij ) dχ i dχ j i , (175)where a ( η ) is the conformal factor, η is conformal time and χ i are comovingcoordinates. The quantity h ij stands for the gravitational wave perturbation,which we take to be transverse and traceless (TT), h := h ii = δ ij h ij = 0 and ∂ i h ij = 0. One can show that a coordinate system exists, such that the gravi-tational wave perturbation can be put in such a TT form. For the remainderof this section, i, j, k stand for spatial indices only.With such a metric decomposition, one can linearize the action to findthe perturbed field equations. In doing so, one must choose a functional formfor the CS scalar and we shall here follow Alexander and Martin [29], whochose ϑ = ϑ ( η ). One can show that the linearized action (the Einstein-Hilbertpiece plus the CS piece) to second order in the metric perturbation yields52 EH + S CS = κ Z V d x h a ( η ) (cid:16) h ij,η h j i,η − h ij,k h j i,k (cid:17) − ϑ ,η ˜ ǫ ijk ( h qi,η h kq,jη − h qi,r h kq,rj ) i + O ( h ) (176)where α = κ , commas in index lists stand for partial differentiation, η isconformal time and ˜ ǫ ijk = ˜ ǫ ηijk . Variation of the linearized action with respectto the metric perturbation yields the linearized field equations, namely [29]¯ (cid:3) h j i := 1 a ˜ ǫ pk ( j h ( ϑ ,ηη − H ϑ ,η ) h i ) k,pη + ϑ ,η ¯ (cid:3) h i ) k,p i , (177)where ¯ (cid:3) is the D’Alembertian operator associated with the background, namely¯ (cid:3) f = f ,ηη + 2 H f ,η − δ ij f ,ij , (178)with f some function of all coordinates and the conformal Hubble parameter H := a ,η /a . One could have, of course, obtained the same linearized fieldequations by perturbatively expanding the C-tensor.One can see from Eq. (177) that the evolution of GW perturbations isgoverned by second and third derivatives of the GW tensor. Jackiw and Pi [22]were the first to point out that for the canonical choice of ϑ the GW evolution isgoverned by the D’Alembertian of flat space only, if we neglect corrections dueto the expansion history of the Universe (ie. if this vanishes, then the linearizedmodified field equations for the GW perturbation are satisfied to linear order).Such a result implies there are two linearly independent polarizations thatpropagate at the speed of light.Let us now concentrate on gravitational wave perturbations, for whichone can make the ansatz h ij = A ij a ( η ) exp h − i (cid:16) φ ( η ) − κn k χ k (cid:17)i , (179)where the amplitude A ij , the unit vector in the direction of wave propagation n k and the conformal wavenumber κ > A ij = A R e Rij + A L e Lij (180)where the circular polarization tensors e R,Lij are given in terms of the linearones e + , × ij by [21] e Rkl = 1 √ (cid:16) e + kl + ie × kl (cid:17) (181) e Lkl = 1 √ (cid:16) e + kl − ie × kl (cid:17) . (182)53hese polarization tensors satisfy the condition n i ǫ ijk e R,L kl = iλ R,L (cid:16) e j l (cid:17) R,L , (183)where λ R = +1 and λ L = − (cid:20) iφ R,L ,ηη + (cid:16) φ R,L ,η (cid:17) + H ,η + H − κ (cid:21) − λ R,L κϑ ,η a ! = (184) iλ R,L κa ( ϑ ,ηη − H ϑ ,η ) (cid:16) φ R,L ,η − i H (cid:17) Before attempting to solve this equation for arbitrary ϑ ( η ) and a ( η ), it isinstructive to take the flat-space limit, that is a →
1, and thus, ˙ a → H ,one finds that the above equation reduces to [29,33] (cid:16) i ¨ φ R,L + ˙ φ − k (cid:17) (cid:16) − λ R,L k ˙ ϑ (cid:17) = iλ R,L k ¨ ϑ ˙ φ R,L , (185)where k is the physical wavenumber 3-vector, t stands for cosmic time andoverhead dots stand for partial differentiation with respect to time. Let usfurther assume that the GW phase satisfies φ ,tt /φ ,t ≪
1, which is the standardshort-wavelength approximation, as well as ¨ ϑ = ϑ = const. Then the aboveequation can be solved to first order in ϑ to find φ ( t ) = φ + kt + iλ R,L ϑ kt + O ( ϑ ) , (186)where φ is a constant phase offset and the uncontrolled remainder O ( ϑ ) stands for terms of the form k ˙ ϑ or k ˙ ϑ ¨ ϑ . The imaginary correction to thephase then implies an exponential enhancement/suppression effect of the GWamplitude, as this propagates in CS modified gravity. Recall that here we areinterested in the propagation of GWs, which is why the right-hand side ofEq. (185) implicitly omits the stress-energy tensor. We shall see in the nextsection that if there is a stress-energy tensor, then the CS correction dependsboth on the first and second derivatives of the CS scalar. Lastly, if we had notassumed that ¨ ϑ = const. then the solution would have become φ ( t ) = φ + kt + iλ R,L k ˙ ϑ ( t ) + O ( ϑ ) , (187)which still exhibits the exponential suppression/enhancement effect.The exponentially growing modes could be associated with instabilities inthe solution to gravitational wave propagation. One must be careful, however,to realize that the results above have been obtained within the approximation k ˙ ϑ ≪ ≫ k ˙ ϑ ¨ ϑ . Thus, provided ˙ ϑ is smaller than the age of the universe,54hen the instability time scale will not have enough time to set in. For largervalues of ˙ ϑ , the approximate solutions we presented above break down andone must account for higher order corrections. A final caveat to keep in mindis that these results are derived within the non-dynamical formulation of thetheory; GW solutions in the fully dynamical theory are only now being activelyinvestigated.Let us now return to the field equation for the phase in an FRW back-ground [Eq. (184)]. The solution to this equation is now complicated bythe fact that the scale factor also depends on conformal time, and thus,one cannot find a closed form solution prior to specifying the evolution of a ( η ). Let us then choose a matter-dominated cosmological model, in which a ( η ) = a η = a / (1 + z ), where a is the value of the scale factor today and z is the redshift. It then follows that the conformal Hubble parameter is simplygiven by H = 2 /η = 2(1 + z ) / . With this choice, one can now compute theCS correction to the accumulated phase as the plane-wave propagates fromsome initial conformal time to η , namely [33]∆ φ (R,L) = iλ R,L kH Z η " ϑ ,ηη ( η ) − η ϑ ,η ( η ) dηη + O ( ϑ ) , (188)which one can check reduces to the flat space result of above in the right limit.Note that the exponential enhancement/suppression effect now depends on anintegrated measure of the evolution of the CS scalar and the scale factor.The CS correction to the GW amplitude derives from a modification tothe evolution equations of the gravitational perturbation, but it also leadsto important observational consequences. One of these can be understood byconsidering a GW generated by a binary black hole system in the early inspiralphase. The GW produced by such a system can be described as follows h R,L = √ M d L M k ( t )2 ! / (1 + λ R,L cos ι ) exp [ − i (Ψ( t ) + ∆ φ R,L )] , (189)where d L = a η (1 + z ) is the luminosity distance to the binary’s center ofmass, Ψ( t ) is the GW phase described by GR, k ( t ) is the instantaneous wavenumber of the gravitational wavefront passing the detector and M is the co-moving chirp mass, which is a certain combination of the binary mass com-ponents. The inclination angle ι , the angle subtended by the orbital angularmomentum and the observer’s line of sight, can be isolated as h R h L = 1 + cos ι − cos ι exp " k ( t ) H ζ = 1 + cos ¯ ι − cos ¯ ι , (190)where we have defined ζ := H Z η " ϑ ,ηη ( η ) − η ϑ ,η ( η ) dηη . (191)55e see then that from an GW observational standpoint, the CS correctionleads to an apparent inclination angle ¯ ι , which effectively modifies the actualinclination angle by a factor that depends on the integrated history of the CScorrection:cos ¯ ι = sinh (cid:16) k ( t ) ξ ( z ) H (cid:17) + cosh (cid:16) k ( t ) ξ ( z ) H (cid:17) cos ι cosh (cid:16) k ( t ) ξ ( z ) H (cid:17) + sinh (cid:16) k ( t ) ξ ( z ) H (cid:17) cos ι ∼ cos ι + k ( t ) ξ ( z ) H sin ι + O (cid:16) ξ (cid:17) . (192)We see then that the CS correction effectively introduces an apparent evolutionof the inclination angle, which tracks the gravitational wave frequency.The interpretation of the CS correction as inducing an effective inclina-tion angle should be interpreted with care. In GR, if a gravitational wavepropagates along the line of sight, such that the actual inclination angle is( eg. π ), then the amplitude is a maximum. In CS gravity, however, theamplitude can be either enhanced or suppressed, depending on whether thewave is right- or left-circularly polarized. When the CS effect suppresses theGW amplitude, one can think of this as an effective modification of the incli-nation angle away from the maximum. However, when the CS effect enhancesthe amplitude, there is no real inclination angle that can mimic this effect( ie. the effective angle would have to be imaginary).The evolution equation for the gravitational wave perturbation dependssensitively on the scale factor evolution [see, eg. Eq. (177)]. Alexander andMartin [29] have investigated gravitational wave solutions when the scale fac-tor presents an inflationary behavior. Suffice it to say in this section thatEq. (177) can be recast in the form of a parametric oscillator equation, witha non-trivial effective potential. In certain limits appropriate to inflation, onecan solve this differential equation in terms of Whittaker functions, which canbe decomposed into products of trigonometric functions and exponentials. Inessence, the solutions present the same structure as that of a matter-dominatedcosmology. Once the gravitational wave modes have been computed, one canproceed to calculate the power spectrum, but these results will be discussedfurther in Sec. 8.1. The issue of GW generation by dynamical matter sources in CS modifiedgravity was first studied by Jackiw and Pi [22]. Once more, this problem hasbeen treated only in the non-dynamical formalism ( β = 0 and α = κ ) andwith the canonical choice of ϑ , although very recently the much more difficultproblem of GW generation in dynamical CS modified gravity has begun to beinvestigated [99].In the presence of a stress-energy tensor, the modified field equations56inearized about a Minkowski background ( g ab = η ab + h ab ) become (cid:3) η h ji + ˙ ϑ ˜ ǫ pk ( j (cid:3) η h i ) k,p − ¨ ϑ ˜ ǫ pk ( j ˙ h i ) k,p = − κ T ij , (193)where t is the standard time coordinate of Minkowski spacetime, ˜ ǫ pjk = ˜ ǫ pjk ,the D’Alembertian operator is (cid:3) := − ∂ t + δ ij ∂ i ∂ j and overhead dots standfor partial differentiation with respect to time. One can derive this equationsimply from Eq. (177) by taking the limit a → H → T ij , which must now be TT, since so is h ij implicitly.Although the explicit solution to the above equation has not yet beencomputed for a general CS scalar field, this problem has been studied for thecanonical choice of ϑ . With the assumption that ¨ ϑ = 0, we then obtain, tofirst order, the solution found in Eq. (120), which for a TT stress energy canbe recast as (cid:3) η h ij = − κ ¯¯ T ij , (194)where the effective stress-energy has been defined as¯¯ T ij := T ij − ˙ ϑ ˜ ǫ kl ( i T j ) l,k . (195)We see then that GWs generation in non-dynamical CS gravity with a canoni-cal ϑ is nothing but GR GWs in the presence of such an effective stress-energytensor [22].For concreteness, let us assume the stress-energy tensor represents amonochromatic source, with definite frequency ω , which is radiating GWsin the ˆ z -axis with wave-vector k . The only non-vanishing components of thestress-energy tensor are then T xx = − T yy and T xy = T yx and it’s Fourier trans-form shall be denoted T + and T × respectively. We then find that the effectivestress-energy becomes˜ T ij = T + − ik ˙ ϑT × T × + ik ˙ ϑT + T × + ik ˙ ϑT + − T + + ik ˙ ϑT × , (196)which exhibits the natural mixture of polarizations of CS GWs.A natural next step is to compute the power carried by such CS modifiedGWs per unit angle d Ω. Since CS GW theory for a canonical ϑ is identicallyequivalent to GR with an effective stress-energy tensor, it follows that the GWpower is given by dP R,L d Ω = 16 κG ω ˜ T ∗ ij ˜ T ij . (197)One is tempted to insert the solution for the effective stress-energy found inEq. (196) into Eq. (197). This effective stress-energy, however, is insufficient,since it is only a solution to first-order in ˙ ϑ and the expression for the poweremitted requires a second-order solution. Jackiw and Pi [22] have found this57olution, which reduces to Eq. (196) multiplied by (1 − k ˙ ϑ ) − . One then findsthat the power emitted is given by [22] dP R,L d Ω = 32 κG ω (cid:16) − k ˙ ϑ (cid:17) − h(cid:16) k ˙ ϑ (cid:17) (cid:16) T + T × (cid:17) + 2 ik ˙ ϑ (cid:16) T + T ∗× − T × T ∗ + (cid:17)i , (198)which when linearized to second order in ˙ ϑ becomes dP R,L d Ω ∼ κG ω h(cid:16) − k ˙ ϑ (cid:17) (cid:16) T + T × (cid:17) + 2 ik ˙ ϑ (cid:16) T + T ∗× − T × T ∗ + (cid:17)i . (199)The power carried by circularly polarized GWs ( T + = iT × ) is corrected by CSgravity only to second order in ˙ ϑ [22].The results presented above hold only for the canonical choice of ϑ . Hadwe allowed the second derivative of the CS scalar to be non-vanishing, wewould have found a linear correction to the power carried by CS GWs. Inthe dynamical formalism, this effect is more clear as the CS scalar also carriesenergy-momentun away from the system [99]. Such modifications to radiation-reaction would affect the inspiral and merger of binaries, as suggested in [99],which leads to a powerful test of the dynamical formulation that we shalldiscuss in Sec. 7. The first-order formulation of CS modified gravity was first studied byCantcheff [135], who realized that the modified theory would lead to non-vanishing torsion for a canonical CS scalar. Such an idea was later generalizedto arbitrary θ in the non-dynamical formalism and the torsion tensor wasspecialized to Earth’s gravitational field [6]. Such a formulation naturally al-lows for the coupling of fermions to the modified theory, thus permitting thestudy of non-vacuum spacetimes [6]. We shall summarize these results here,beginning with a description of the first-order formalism and Einstein-Cartantheory, following mainly [19,136,137]. We then continue with a discussion ofthe first-order formulation of CS modified gravity. Consider a 4-dimensional manifold M with an associated 4-dimensionalmetric g ab . At each point on this manifold, let there be a tetrad e Ia , so thatthe metric can be recast as g ab = e Ia e Jb η IJ , where η IJ is the Minkowski metric.Internal indices range I, J = (0 , , , e I . Wecan raise or lower spacetime and internal indices with the metrics g ab and η IJ respectively. 58he introduction and differentiation of internal versus spacetime indicesis crucial to the first order formalism. Riemannian fields, like the metric tensor,exist on the base manifold M and have a finite dimension, but gauge fieldscan be infinite dimensional and so they must exist on a different vector space.For example, the tetrad e I and the spin connection ω KL are 1-forms on thebase manifold, while the curvature tensor associated with it, F KL , is a 2-formon the base manifold. On the other hand, the fermion field ψ is a 0-form onthe internal vector space. A fiber bundle is defined as the union of the basemanifold and the internal vector space, with each fiber a different copy of theinternal vector space. One can think of the Lie group associated with the fiberbundle as glueing all fibers and the base manifold together [19].The recovery of spacetime indices is sometimes achieved via the wedgeproduct operator, defined via( A ∧ B ) ab := ( p + q )! p ! q ! A [ a ...a p B b ...b q ] (200)with A and B p - and q -forms respectively. Note that we shall here followthe convention that spacetime indices always appear after internal ones. Thewedge product operator satisfies the following chain rule D ( ω ) ( A ∧ B ) = (cid:16) D ( ω ) A (cid:17) ∧ B + ( − p A ∧ (cid:16) D ( ω ) B (cid:17) , (201)and the following commutativity relation A ∧ B = ( − pq B ∧ A. (202)Also note that since the wedge product acts on spacetime indices only, it existson the base manifold and not on the internal space.Now that the metric and tetrad have been defined, let us introduce thegeneralized covariant derivative operator D and the spin connection ω IJ . Givena tensor A KLa we can define the covariant derivative as D ( ω ) A KL := dA KL + ω KM ∧ A M L + ω LM ∧ A K M , (203) D ( ω ) A KL := dA KL − ω K M ∧ A ML − ω LM ∧ A KM , (204)where the exterior derivative operator d acts on spacetime indices only: dA KL := 2 ∂ [ a A KLb ] . (205)The commutator of covariant derivatives allow us to define the curvature ten-sor associate with ω IJ , namely F KL = dω KL + ω K M ∧ ω ML , (206)59hich reduces to the Riemann tensor if the spin connection is metric compat-ible and torsion-free, ie. if the connection is the Christoffel one. One can thenshow after some calculation that δ ω F IJ = D ( ω ) δω IJ . (207)The spin connection has a certain degree of freedom that can be fixed bydemanding that it be internally metric compatible D ( ω ) η IJ = 0. This conditionforces the connection to be completely antisymmetric on its internal indices ω ( IJ ) = 0. We can then define the torsion tensor as T I := D ( ω ) e I = de I + ω I M ∧ e M , (208)which is equivalent to T I ab = 2 D [ a e Ib ] . If we reinstate spacetime indices, wefind T abc = 2 C a [ bc ] , (209)where C abc is the antisymmetric part of the connection, or contorsion ten-sor. Note that internal metric compatibility is not equivalent to a torsion-freecondition.The contorsion tensor can be defined purely in terms of wedge products.Let us then decompose the spin connection into a symmetric and tetrad com-patible piece Γ I J and an antisymmetric piece C I J , the contorsion tensor. Thetorsion tensor is then given in terms of the contorsion via T I = C I J ∧ e J , (210)which can be inverted to find C IJK = −
12 ( T IJK + T JKI + T KJI ) . (211)The contorsion tensor is fully antisymmetric on its first two indices, while thetorsion tensor is fully antisymmetric on its last two indices. Equation (209)can also be obtained from Eq. (208) in spacetime indices, if we use the trans-formation law from spin to spacetime connection: ω abe = e eI ω K I a e Kb − e eI ∂ a e Ib , (212)which can be established from D (Γ) e I = 0. Eq. (212) is sometimes referred toas “the tetrad postulate”.The curvature tensor can be expressed purely in terms of the Riemanntensor (that depends only on Γ I J ) and terms that depend on the contorsiontensor: F IJ = R IJ + D (Γ) C IJ + C I M ∧ C MJ , (213)where D (Γ) is the connection compatible with the symmetric connection. TheBianchi identities in first-order form become D ( ω ) T I = R I K ∧ e K , D ( ω ) R IJ = 0 . (214)60 .2 First-Order Formulation of CS Modified Gravity Let us now apply the formalism of the previous section to GR and to themodified theory. The Einstein-Hilbert action can be recast in terms of formsas S EH = κ Z V ǫ IJKL e I ∧ e J ∧ F KL . (215)One can convert this into Eq. (2) by rewriting the curvature tensor as F IJ =(1 / F IJ KL e K ∧ e L and using the identity e I ∧ e J ∧ e K ∧ e L = − ˜ σ ǫ IJKL , (216)where ˜ σ = √− g d x , as well as the Kronecker-Delta relations ǫ abcd ǫ abef = − δ [ ce δ d ] f , (217)˜ η abcd ǫ abef = +4 √− g δ [ ce δ d ] f , (218)˜ η abcd ˜ η abef = +4 δ [ ce δ d ] f . (219)Note that full internal index contractions are identically equal to spacetimeindex contractions, if the quantities contracted are tensors.Similarly, we can attempt to recast the CS action in terms of forms via S CS = α Z V ϑ F ∧ F, (220)where the integrand reduces to R ∧ R = R IJ ∧ R IJ for a symmetric connection.Again, Eq. (220) can be converted into Eq. (3) in the same way as above, whichthen establishes that ˜ σ ∗ R R = 2 R ∧ R for a symmetric connection. We can nowintegrate by parts to obtain the CS action in terms of the Pontryagin current,but to do so we first must realize that F ∧ F = d Ω , where the Chern-Simons3-form is defined as [28,27,31]Ω = ω IJ ∧ dω IJ + 23 ω I J ∧ ω J K ∧ ω K I . (221)We then find that Eq. (220) can be rewritten as S CS = − α Z V dϑ ∧ Ω , (222)where we have neglected the boundary contribution. Converting this expres-sion to spacetime indices, one can check that one recovers Eq. (10) if theconnection is symmetric. In fact, the dual to this 3-form is actually the Pon-tryagin current in Eq. (9) in disguise, while ∗ ( d Ω ) = ∗ F F/ F ab − g ab F = 2 κ T ab , (223)61ince the CS current does not depend on the tetrad, but only the spin con-nection. Equation (223) resembles the Einstein equations, except that thequantity F ab = F cacb and F = F aa are not the Ricci tensor and scalar, butcontractions of the curvature tensor.The full curvature tensor can be reconstructed once one solves for thecontorsion tensor via the torsion condition of the modified theory. This con-dition is arrived at by requiring that the variation of the action with respectto the spin connection vanishes. Let us then rewrite the CS 3-form viaΩ = ω I J ∧ F J I − ω I J ∧ ω J K ∧ ω KI . (224)such that its variation with respect to ω KL reduces to F KL . The variation ofthe action with respect to ω KL reduces to δS = − κ Z V D (cid:16) ǫ IJKL e I ∧ e J (cid:17) δω KL − α Z V dϑ ∧ F KL δω KL , (225)where we have set β = 0 for simplicity and we have integrated the first termby parts. Cantcheff [135] has shown that δS == 0 is equivalent to the torsionconstraint D (cid:16) ǫ IJKL e I ∧ e J (cid:17) = − ακ dϑ ∧ F KL , (226)which is nothing but the CS modified second Einstein-Cartan structure equa-tion. This equation can also be rewritten in terms of the torsion tensor as ǫ IJKL T I ∧ e J = − α κ dϑ ∧ F KL . (227)The torsion condition of Eq. (227) shows that in CS modified gravity theconnection must be torsionfull. Solving for the torsion in Eq. (227) is non-trivial, due to the implicit appearance of torsion itself through the curvaturetensor on the right-hand side of this equation. However, if we replace thistensor by the Riemann ( torsionless ) tensor, then we can solve for the torsiontensor exactly to find . T I = − α κ ǫ IJKL v J R KL , (228)where here we have used the first Bianchi identity R [ IJK ] L = 0. Equation (228)should be interpreted as an approximate solution to the torsion constraint,where we have neglected higher than second powers of θ [135,6].One can now relate this torsion tensor to the gravitational field of acompact binary in the PPN formalism [6]. One can show that this tensor isproportional to the product of v a with ∇ × V and ~ ∇ U , where V i and U are This solution is the same as that found by [135] with α = 2 and shortly afterby [6] with α = κ . We have further checked that Eq. (228) is indeed a solution tothe torsion constraint of Eq. (227) θ forwhich all components of the torsion tensor vanish. Since this tensor affectsthe motion of point particles (in particular the frame-dragging effect [6]),one concludes that non-dynamical CS modified gravity generically leads tomodified precession, irrespective of θ . Care must be taken, however, since inspite of generically being non-zero, the torsion tensor is after all proportionalto the dual to the Riemann tensor. Thus, Earth-based experiments that searchfor non-vanishing torsion [138,139,140] cannot measure this effect, since theRiemann tensor on Earth is prohibitively small.Are the first and second-order formulations of CS modified gravity equiv-alent? The modified field equations of Eq. (223) resemble the Einstein equa-tions, except that the curvature tensor contains torsional pieces. When Eq. (228)is used as the torsion, one can show that all these torsional pieces conspireto produce the C-tensor of Eq. (18) [135]. One then finds that the first andsecond-order formulations of CS gravity are equivalent if and only if the CSaction in first-order form is defined in terms of the torsion-free curvature ten-sor (ie, in terms of the symmetric connection), such that Eq. (228) is the exactsolution to the torsional condition.The CS 3-form, however, has been here introduced in terms of the gener-alized spin connection, for which the torsion condition is non-linear, dependingexplicitly on the curvature tensor. For such a torsion condition, the torsiontensor in Eq (228) is still formally valid but only to linear order in θ , and thus,the first-order formalism is still equivalent to the second-order one but onlyto O ( ϑ ). The O ( ϑ ) corrections to Eq. (228) will modify the field equations,and thus, force them to not be equivalent to those of the second-order formal-ism. Cantcheff [135] has further shown that line elements that are solutionsin the second-order formulation of CS gravity ( eg. the Schwarzschild metric,which leads to a vanishing C-tensor) are not necessarily solutions to the fieldequations of the first-order formulation if higher-order in θ terms are included.The inequivalence between the first- and second-order formalism thus de-pends on quadratic or higher powers of the CS scalar. The CS modified actionconsidered in Sec. 2.1, however, only considers linear terms in θ . In principle,there will be θ and higher-order terms in the action that one could have toinclude, since these will also generically break parity [eg. ϑ ( ∗ R R ) ]. A con-sistent comparison between first- and second-order formalisms thus requiresthat such terms be taken into account, if one wishes to define the CS actionin first order form in terms of the torsionfull curvature tensor, instead of theRiemann tensor. One of the advantages of the first-order formalism is that it allows for theinclusion of fermions and bosons in the action. Let us then considering the63nclusion of the following piece to the full action of Sec. 2.1 S D = ǫ Z d x √− g (cid:16) i ¯ ψγ I e aI D a ψ + c.c. (cid:17) , (229)where c.c. stands for complex conjugation, ψ is a Dirac spinor, γ I are gammamatrices and ǫ is a coupling constant. Fermions are here represented by Diracspinors, which are gauge field that live naturally in SU (2). The tetrad andthe spin connection of GR, on the other hand, are fields that live in SO (1 , ψ , one can deduce how the generalized SU (2)covariant derivative acts on Dirac spinors: D a ψ := ∂ a ψ − (1 / ω IJ a γ I γ J ψ ,where we shall here follow the sign conventions of [73]. Variation of the fullaction with respect to Dirac fermions then leads to the massless Dirac equa-tion [6]: γ a D (Γ) a ψ = 14 e aM C aKL γ M γ K γ L ψ, (230)where notice that we have not included a mass term for the fermions forsimplicity. We see then that the Dirac equation is modified in CS modifiedgravity by a source term that depends on the contorsion tensor. Equation (230)implies that the CS effects might be enhanced in spacetime regions where themomentum of Dirac fermions is large.Before we can vary the full action with respect to the connection it isconvenient to recast it in first-order form. Doing so, Eq. (229) becomes S D = ǫ Z ǫ IJKL e I ∧ e J ∧ e K ∧ (cid:16) i ¯ ψγ L D ψ + c.c. (cid:17) . (231)Upon variation of the action with respect to the connection, Eq. (227) becomes ǫ IJKL T I ∧ e J = − α κ dϑ ∧ F KL + ǫ κ e K ∧ e L ∧ J (5) , (232)where J L := ¯ ψγ γ L ψ is the fermion axial current, e is the determinant of thetetrad field and we have used the identity γ I γ [ J γ K ] = − iǫ IJKL γ γ L + 2 η I [ J γ K ] . (233)This equation can be solved for the torsion tensor if we once again replace thecurvature tensor by the Riemann tensor on the right-hand side. Doing so, wefind [6] T I = − α κ ǫ IJKL v J R KL − ǫ κ ǫ I JKL J L e J ∧ e K , (234)which essentially is an approximate solution that neglects quadratic and higherpowers of θ . From this, the contorsion tensor can be calculated using Eq. (211)to find C IJ = 3 α κ v N ǫ [ I NML R JK ] ML + ǫ κ ǫ IJKL J L e K . (235)64ince the first- and second-order formalism are equivalent to leading orderin θ , one can compute the interacting action be reinserting the torsion solutionin the full action. Doing so, one finds [6] S = 3 ǫ κ Z V ˜ σJ a J a + ǫα κ Z V ˜ σ (cid:16) J a v b R ab − J a v a R (cid:17) (236)where we have neglected other terms that are either higher-order in θ or inthe gravitational coupling constant. The first term is the standard 4-fermioninteraction that arises in Einstein-Cartan theory coupled with fermions. Thesecond term is a new CS contribution that depends on the embedding co-ordinate, as well as the Ricci tensor and scalar. This new term represents a2-fermion interaction and it is not suppressed by the gravitational couplingconstant. We can then conclude that, at least to linear order, fermion currenttend to enhance the CS correction, some of the implications of which shall bediscussed in Sec. 7. All tests of CS modified gravity to date have been performed with as-trophysical observations and concern the non-dynamical framework. AfterAlexander and Yunes [7,32] realized that the modified theory predicts ananomalous precession effect, Smith, et. al. [37] tested the non-dynamical modelwith canonical CS scalar with LAGEOS [130,131,132] and Gravity ProbeB [141,142] observations, placing the first, albeit weak, bound on the CSscalar. In view of these results, Konno, et. al. [129] proposed that the CScorrection could be used to explain the flat, rotation curves of galaxies , whichin turn could yield yet another constraint on the non-dynamical theory fornon-canonical ϑ . Recently, Yunes and Spergel [143] used double binary pulsardata to place a bound on the non-dynamical model with canonical CS scalarthat is eleven orders of magnitude stronger than the Solar System one.The dynamical model remains untested today, mainly due to the difficultyin calculating observable quantities in a consistent way. The only possibleavenue to perform such a test seems currently to be through gravitational waveobservations [33]. Cosmological tests of the modified theory will be discussedin the next section. The non-dynamical modified theory has been so far only tested throughSolar System, frame-dragging experiments. Smith, et. al. [37] studied theanomalous precession inherent to the non-dynamical model with the conven-tions α = − ℓ/ β = 1 and we shall summarize these results here. Withthese conventions, ϑ has units of inverse length or mass and m CS is a charac-teristic mass scale defined in Eq. (145). When testing CS modified gravity, we65hall in fact place bounds on this parameter, which can be trivially related to ϑ by the inversion of Eq. (145).Only Solar System tests that sample the gravitomagnetic sector of thegravitational field can be used to test CS modified gravity in the non-dynamicalmodel. As shown in Sec. 5, the non-dynamical modified theory possesses thesame PPN parameters as GR, except for the gravitomagnetic potential. Thisimplies, in particular, that the perihelion shift of Mercury or light deflectionby the Sun cannot be used to constrain the modified theory. The only physi-cal effect of the non-dynamical CS modification is the induction of anomalousprecession effects.Precession is a generic term used to address the change in the rotation3-vector of some spinning object, ie. a non-vanishing 4-gradient of the spinangular momentum. Two types of precession can be distinguished: torque-free and torque-induced . The former corresponds to situations in which the spinangular momentum is not coaligned with the axial Killing vector. The latter,also known as gyroscopic precession , occurs in situations where there is anadditional torque (such as that of a gyroscope) that pushes on the spin an-gular momentum vector, forcing it to wobble. Gyroscopic precession can bestudied in a Newtonian framework, but relativity adds three additional correc-tions: Thomas precession , an additional special relativistic correction due tothe observer’s non-inertial rotating frame; de Sitter or geodetic precession , aGR effect that accounts for Schwarzschild-like deviations from flat spacetime;
Lense-Thirring precession , a GR correction due to the gravitomagnetic sectorof the Kerr metric.The CS modification can correct several different types of precession, de-pending on the physical scenario under consideration. Torque-free precessioncan occur if one considers the far-field expansion of a non-dynamical CS spin-ning black hole metric, where the axis of rotation seems not to be co-alignedwith the axial axis of symmetry. In this one-body scenario, in the absenceof external torques, the spin angular momentum of the black hole precessesaround the symmetry axis in a wobbling fashion. The evolution of the wob-ble angle requires the determination of the Killing axis, from which one canobtain the frequency of precession, via ω f ≈ J/I to Newtonian order, where I is the moment of inertia about the symmetry axis [144]. Since such an ar-rangement is asymmetric, one expects GW emission leading to spin-down andalignment. In the non-dynamical formalism, then, spinning black holes wouldtend to “unspin” themselves via interactions with the CS scalar, thus relax-ing to the Schwarzschild solution. If so, observations of spinning black holescould be used to constrain the magnitude of the canonical scalar [145]. Sucha possibility has not yet been studied in detail.Torque-induced precession is also modified by the CS correction throughthe correction to the gravitomagnetic sector of the metric. Consider first themotion of a test body in the external field of a CS spinning source. The far-fieldsolution for such a source was summarized in Sec. 5 for extended bodies [37]. Insuch a field, the orbital elements of a test body will experience Lense-Thirring66recession [146], which will be different in CS gravity relative to GR. Smith, et. al. [37] studied the secular time variation of the longitude of the ascendingnode ˙Ω orb [147] in the non-dynamical modified theory and found it to be givenby ˙Ω orb = Ω GR + Ω CS , (237)where the GR Lense-Thirring drag is given by˙Ω GR = 2 GJa (1 − e ) , (238)with eccentricity e , the magnitude of the spin angular momentum of the centralbody J and the CS correction ˙Ω CS given by˙Ω CS = 15 a R j ( m CS R ) y ( m CS a ) , (239)with the semi-major axis a , Earth’s radius R , and the spherical Bessel func-tions of the first and second kind j ℓ ( · ) and y ℓ ( · ).The LAGEOS satellites have measured ˙Ω and found it in agreement withGR up to experimental error, which thus allows for a test of non-dynamicalCS gravity [37]. Figure 2 shows the ratio of the GR and CS predictions asa function of the characteristic CS mass [37], where the shaded region cor-responds to a 1 σ deviation from the experimentally measured value. The re-gion where the CS prediction is in agreement with experiment is then all m CS & . × − km − . A 2 σ or 3 σ constraint would increase the shadedregion by roughly a factor of two or three, thus forcing m CS & × − km − or m CS & . × − km − . Taking the most conservative estimate translatesinto a bound for the CS scalar of | ˙ ϑ | ≤ κ/α ) km with more than 99%confidence.Another type of torque-induced precession is also affected by CS modifiedgravity, namely that experienced by a gyroscope. Consider then a gyroscopewith spin angular momentum S in circular orbit around the Earth. Neglectinggeodetic precession, which is unaffected in the non-dynamical modified theory,the rate of change S is given by˙ S i = 2 ǫ ijk B j S k , (240)where B i is the gravitomagnetic field. The precessional angular velocity is then The relation in Eq. (239) is non-monotonic, and thus, many isolated islands existfor which the observed precession is consistent with the CS prediction. Nonetheless,it is always true that for the range of values quoted here the observed precession isalways consistent with the CS prediction. The bound quoted by Smith, et. al. aresomewhat more stringent, thus including more island but also including regions ofthe parameter space that are inconsistent with observations [148] ig. 2. This figure shows the ratio ˙Ω CS / ˙Ω GR as a function of CS characteristicmass for the LAGEOS satellites, with semi-major axis a ≈ ,
000 km. The shadedregion corresponds to a 10% experimental error bar with a 1 σ detection confi-dence [130,131,132]. given by ˙Φ := | ˙ S i | / | S i | , which is corrected in CS by˙Φ CS ˙Φ GR = 15 a R j ( m CS R ) [ y ( m CS a ) + m CS ay ( m CS a )] , (241)where ˙Φ GR is the GR prediction [37]. In Eq. (241), R is the distance from thecenter of the Earth to the gyroscope (roughly 7000km for Gravity Probe B)Given an experimental verification of the Lense-Thirring gyromagnetic ef-fect, one could then place a constraint on the CS scalar, as explained previouslywith the LAGEOS satellites. Gravity Probe B [141,142] was designed to mea-sured precisely this effect to percent accuracy, but since its launch it has facedcertain difficulties that might degrade its accuracy [149]. Smith, et. al. studiedthe possibility that Gravity Probe B could place a stronger constraint thanthe LAGEOS satellites, but this was found to not be the case [37].The estimates of Smith, et. al. [37] presented above assume a sphericalEarth, where its internal structure is neglected. The Earth, however is anoblate spheroid with layers of different density. These non-spherical correctionsseem not to matter since the CS correction is affected by them in the same wayas GR. Thus, the relative difference between the GR and CS effects remainsroughly the same. Lastly, the estimates described above assume the Earth isnot moving in the barycenter frame. In the point particle case, we saw thatif their velocity is non-vanishing then the CS modification leads to a non-boundary correction to the gravitomagnetic field. These corrections remain tobe studied further. 68 .2 Binary Pulsar Test Non-dynamical (and possible dynamical) CS modified gravity has beenshown to modify only the gravitomagnetic sector of the metric, which doesnot influence most astrophysical processes. This is particularly true outside theSolar system, where stars inside some galaxy will possess randomly orientedvelocities that will lead to a vanishing averaged CS correction. On cosmologicalscales, the CS effect is also mostly irrelevant since, for example, the equationsof structure formation are not corrected, except for observations of the cosmicmicrowave background, which we shall discuss in the next chapter.Some astrophysical process, however, are CS modified. One example ofthis is the formation of accretion discs around protoplanetary systems, butalthough non-zero the CS correction would be difficult to measure becauseit would be greatly suppressed by the almost negligible compactness of suchsystems. Another, perhaps more interesting astrophysical scenario are doublebinary pulsars, such as PSR J0737 − . M ⊙ , while its radius is on the orderof 10 km, which implies their compactness is roughly 1 /
5. Such a large com-pactness leads to strong gravitational fields that can be used to test GR to anunprecedented level [151].Consider then double binary pulsars and let us model their orbital evolu-tion via a geodesic study of a compact object in the background of a rotating,homogeneous sphere. Under these assumptions, we can employ the gravito-magnetic field found by Smith, et. al. [37], where the motion is determined bythe four-acceleration ~a = − ~v × ~B , v is the velocity of one of the binary com-ponents and ~B is the gravitomagnetic field of the other [Eqs. (147) and (148)].To leading order in ˙ θ , the CS correction to the gravitomagnetic field is ~B CS = c r cos[ ξ ( r )] h ~ J − tan ξ (cid:16) ~ J × ˆ r (cid:17) − (cid:16) ~ J · ˆ r (cid:17) ˆ r i , (242)with ξ ( r ) = 2 rκ/ ( ˙ θα ), c = 15 α ˙ θ/ (4 κR ) sin[ ξ ( R )], ˆ r = ~r/r , ~ J = ~J /R and( · , × ) the Euclidean dot and cross products.Let us now parameterize the trajectory with equatorial coordinates, where [152]ˆ r = [cos u, cos ι sin u, sin ι sin u ] , ˆ t = [ − sin u, cos ι cos u, sin ι cos u ] , ˆ n = [0 , − sin ι, cos i ] , (243)are the radial, transverse and normal unit vectors relative to the comovingframe in the orbital plane. In Eq. (243), ι is the inclination angle, w is the Employing ˙ θ as an expansion parameter is not formally valid, since this quantitycan be dimensional. Corrections are actually proportional to O ( ˙ θ/R ) or O ( ˙ θ/a ),both of which can be shown to be much less than unity here. u = f + w with f the true anomaly and Ω = 0 is theright ascension of the ascending node, where here this is co-aligned with theˆ x vector [152].The variation of the Keplerian orbital elements is determined by the per-turbation equations, which can be obtained by projection of the geodesic accel-eration onto the above triad. To leading order, the radial projection a r := ~a · ˆ r is the only one CS modified, with [143] a CS r = − c ˙ u J { cos ι cos [ ξ ( r )] + sin ι cos u sin [ ξ ( r )] } . (244)The precession of the perigee is given by the perturbation equation ˙ w = − a r / ( n a e ) cos f , where the mean motion n = q M/a and e is the eccentric-ity. We shall consider next only the averaged rate of change of orbital ele-ments, and in particular < ˙ w > . Averaging over one orbital period [143] < ˙ ω > := Z T ˙ ωP dt = Z π ˙ ω (1 − e ) / π (1 + e cos f ) df, (245)during which we assume the pericenter is constant and the motion is Keplerian,such that˙ u ∼ ˙ f = n (1 + e cos f ) (cid:16) − e (cid:17) − / ,r = a (cid:16) − e (cid:17) (1 + e cos f ) − . (246)This last assumption is justified since in the weak field the CS-corrected motionof test particles about any background remains CS unmodified. [99].The CS correction to the averaged rate of change of the perigee is then,to linear order in the eccentricity [143], h ˙ w i CS = 152 a e JR ˙ θR X sin (cid:18) κRα ˙ θ (cid:19) sin (cid:18) aκα ˙ θ (cid:19) , (247)where X := a sin ι is the projected semi-major axis. Note that the limit e → a ˙ θ/R , since < ˙ ω > GR ∼ J/a . We see then that a Solar System test willnaturally be weak since a/R ∼
1, while for binaries a/R ∼ O (10 ).Formally speaking, the sole calculation of perigee precession is not suffi-cient to constrain an alternative theory with binary pulsar observations [124].In principle, at least two other quantities must be computed and measuredin order to break the degeneracy in the determination of the individual com-ponent masses. In CS gravity, however, other binary pulsar-relevant quanti-ties will not be corrected, because C identically vanishes for the canonicalchoice of CS scalar, or the correction becomes subleading, as in the case of70he quadrupole gravitational wave emission formula. The only post-Keplerianparameters that is CS corrected to leading order is ˙ ω in the non-dynamicaltheory with canonical CS scalar and all parameters can be obtained from [153].Yunes and Spergel [143] used this calculation to place a strong bound onthe non-dynamical framework. Using the relevant system parameters of PSRJ0737 − θ . × − km , → m CS &
100 meV (248)which is 10 times stronger than current Solar System constraints. A similaranalysis in the full, strong-field dynamical formalism is still lacking. In partic-ular, the coupling of fermions to the CS term might be relevant for calculationsinvolving neutron stars [6]. The only galactic study that has been currently carried out is relatedto the flat-rotation curves of galaxies [129]. Consider a collection of stars in aspiral galaxy and measurements of their orbital velocity v and their distance tothe galaxy center r , through the shift of spectral lines. The plot of this velocityas a function of distance is the so-called galaxy rotation curve , which accordingto Newtonian mechanics should obey a square-root fall-off v ∝ r − / . For alarge number of such galaxies, with different luminosities, Rubin and FordJr. [154,155] have found that the galaxy rotation curve flattens with distance,which implies the existence of additional non-visible, or dark, matter.Konno, et. al. [129] have attempted to explain the flatness of rotationcurves through non-dynamical CS modified gravity ( α = − lκ and β = 0, where l is a coupling constant) with a non-canonical CS scalar. We have alreadydiscussed the form of the metric obtained in the slow-rotation limit, whenassuming ϑ ∝ r cos θ [see eg. Eq. (169) and Eq. (171)]. With this solution,Konno, et. al. [129] studied circular, equatorial geodesic motion, and foundthat the orbital angular velocity v := rdφ/dt = − Lf / ( rE ), where E and L and the conserved energy and angular momentum, which for large radiusbecomes v = ± s Mr + C O ( J ) , (249)and we recall that C is a constant that depends on the spin angular mo-mentum. This result is to be contrasted with the Kerr metric, for which v = ( M/r ) / − J/r .The solution found by Konno, et. al. is interesting in that it sheds lighton some of the effects of non-dynamical CS modified gravity on certain ob-servables with a non-conventional CS scalar, but before victory can be claimedover the rotation curves one must consider the solution more carefully. In do-ing so, one discovers that this solution possesses a few drawbacks that renderit rather unphysical as a true spinning, BH solution. One of main problem is71ooted in that the solution was found in the non-dynamical formulation, whichas we have argued is quite contrived, arbitrary and probably not well-posed.One then hopes that an embedding of this solution in the dynamicalframework can be found, but as we explain below this is impossible. Thedynamical formulation requires that a well-defined scalar field couple to thefield equations via a stress-energy tensor. The scalar fields studied in the non-dynamical formulation, however, all possess the feature of leading to infinitetotal energy in the scalar field. This issue cannot be bypassed by simply statingthat the stress-energy contribution is second-order in the slow-rotation param-eter, since the energy contribution is not large, but infinite. Thus, the solutiondiscussed above is not a self-consistent one in the dynamical framework.Moreover, on closer inspection, Yunes and Pretorius [84] have found ad-ditional solutions in the non-dynamical formulation for other choices of CSscalar [see eg. Eq. (173)]. This family of solutions is better-behaved in thatthey do not contain any logarithmic divergences in the metric, but they leadto an orbital angular velocity v φ ∼ s Mr − Mr , (250)in the far-field limit M/r ≪
1, which cannot explain the flat rotation curves.At this juncture, one could make the argument that the above analy-sis is nothing but evidence that Nature somehow selects the solution foundby Konno, et. al. , but we shall here argue precisely the opposite. The non-dynamical framework does not suggest that either of the two solutions dis-cussed here is more valid or less valid than the other. In fact, it is the im-mense freedom in the choice of CS scalar the leads to two different observablesand points at an incompleteness of the framework. This observation, coupledwith the overconstraining feature of the non-dynamical framework, casts somedoubt as to validity of suggesting the CS correction as an explanation of theflatness in galactic rotation curves.
Several tests have recently been proposed of the modified theory withGWs. All such tests have so far concentrated on waves generated by binarysystems in the early inspiral phase, where CS correction arises due to thepropagation of the wave (instead of the mechanism). As we reviewed in Sec. 5,the main effect of the CS correction on the propagation of GWs is an amplitudebirefringence, characterized by the parameter ζ defined in Eq. (191). Thus, ifa GW detection can constrain the magnitude of ζ , one can derive a bound onthe CS scalar.Alexander, et. al. [33] have proposed a GW test of non-dynamical CSmodified with a generic CS scalar through the space-borne GW detectorLISA [156,157,158,159,160]. The sources in mind are supermassive black hole72inaries located at redshifts z <
30. In principle, in order to determine howgood of a constraint LISA could place on CS modified gravity, one wouldhave to carry out a full covariance (Fisher) matrix analysis, including all har-monics in the signal amplitude, since the CS correction affects precisely thisamplitude.One can obtain an order-of-magnitude estimate, however, by making thefollowing assumptions. First, let there be a GW detection in two GW detectors,such that one can reconstruct both the right- and left-polarized amplitudes.Second, let us model the noise as white, with one-sided spectral noise density S . Third, focus attention on the Fisher matrix Γ ij related to the parametersthat affect the amplitude of the GW signal, neglecting the phase parameters,namely [161,162] Γ ij = X k =R,L S Z t f t i ℜ ∂m k ∂λ i ! ℜ ∂m k ∂λ j ! dt, (251)where the observation period is ( t i , t f ), λ i = ( D , χ, ζ ) is a vector of parametersthat affect the amplitude (the natural logarithm of the luminosity distance tothe source D = ln d L and the cosine inclination angle χ = cos ι ) and m R,L isthe scalar detector response. The ensemble average co-variance is then simply ν ij = (Γ − ) ij , where the diagonal components are a measure of the accuracyin the determination of the ii parameter.For a GW signal detected on plane ( χ = 0), the accuracy in the determi-nation of ζ is given by [33] ν ζζ = ( M H ) ρ I IK − J (252)where M = m m / ( m + m ) is the chirp mass, with binary mass components m , , and ρ is the squared signal-to-noise ratio ρ = 1 S Z t f t i dt (cid:16) A + A (cid:17) , (253)with A R,L the GW amplitudes. In Eq. (252), the noise moments are definedvia I := Z t f t i k ( t ) M ! / dtS = − / M S
564 ( k M ) − / (cid:12)(cid:12)(cid:12) k max k min (254) J := Z t f t i k ( t ) M ! / dtS = − / M S
532 ( k M ) − / (cid:12)(cid:12)(cid:12) k max k min (255) K := Z t f t i k ( t ) M ! / dtS = 2 / M S k M ) / (cid:12)(cid:12)(cid:12) k max k min . (256)For a binary black hole system with total mass M = m + m = 10 M ⊙ (1 + z ) −
73t redshift z , a GW one year before coalescence has a wavelength in the range λ = (10 , ) c Hz − . For such a system [33] ρ = 10 h z − √ z − Hz − S ! / (257) ν ξξ = 3 . × − (cid:18) S − Hz − (cid:19) (cid:16) z − √ z (cid:17) , (258)where h = H /
100 s Mpc / km. Such a result implies a 1 σ upper bound on ξ of order 10 − for a LISA GW detection at z = 15. With the canonical choiceof ϑ , this translates to a constraint on the CS scalar of approximately α ˙ ϑ/κ < − km. Not only is the possible constraint on ˙ ϑ five orders of magnitudebetter than with Solar System tests, a GW detection also constrains a differentsector of the modified theory, since it samples the temporal evolution of theCS scalar, instead of its local value. This is because a GW detection reallyconstrains the evolution of the CS scalar from the time of emission of the GWto its detection on Earth.Perhaps a more interesting test of CS modified gravity can be performedusing GWs emitted by extreme-mass ratio inspirals or binary black hole merg-ers [99]. These systems sample the strong-gravitational regime of spacetime, inwhich the CS correction is enhanced, as shown by the CS modified Kerr solu-tion [84]. The generation of GWs would then also be CS modified, not only dueto modifications in the trajectories due to corrections to the background met-ric, but also due to the fact that the CS scalar must carry energy-momentumaway from the system. Even ignoring the latter, Sopuerta and Yunes [99] haveshown that the background modifications lead to extreme- and intermediate-mass ratio inspirals, whose waveforms are sufficiently distinct from their GRcounterpart to allow for a test of the radiative sector of the dynamical theoryover a few-month integration period. A more detailed data analysis study iscurrently underway to determine the accuracy to which the theory can betested. After the quintessence model was proposed to account for the accelerationseen from Type Ia supernovae [4], Carroll proposed that the quintessencefield can generically couple to parity violating terms in the electromagneticsector [163,164]. This was used as a constraint on the quintessence dark energymodels since such parity violating, “birefringent” couplings are generic. Soonafter, the cosmological study of CS modified gravity was first proposed byLue, Wang and Kamionkowski [134] as a way to search for parity-violatingeffects from the GW sector of the CMB polarization spectrum. This was sobecause, in CS modified gravity, an asymmetry of left and right-handed GWsleads to an anomalous cross correlation in the GW power spectrum. Such a74ioneering study was then followed up by Saito, et. al. [165], who improvedon their calculation.The possibility of parity violation in the gravitational sector led Alexan-der, Peskin and Sheikh-Jabbari (APS) [28,31] to propose an explanation forthe observed matter-antimatter asymmetry in the universe. The lepton asym-metry is generated from one simple and generic ingredient: the CS correctionto the action, which leads to an asymmetric left/right production of GWs,and due to the gravitational ABJ anomaly [51,52], chiral leptons are pro-duced. APS were able to account for a correct amount of lepton asymmetryusing the bounds placed on the GW power spectrum amplitude and the scaleof inflation. In what follows, we shall review the inflationary production ofGWs with the CS term present. We show how the GW power spectrum ismodified and apply the GW solutions to the inflationary leptogenesis mecha-nism of Alexander, Peskin and Sheikh-Jabbari. Following this, we will reviewthe LWK analysis of constraining parity violation in the CMB.
In this chapter we shall consider CS modified gravity in the context ofinflation. The idea of inflation is quite simple: a period ( t i , t f ) in which thescale factor a ( t ) ∼ e H ( t f − t i ) grows exponentially. During such an epoch, theFRW equations govern the dynamics of the scale factor a ( t ) via (cid:18) ˙ aa (cid:19) = 8 π Gρ + ka and 2¨ aa + ˙ a a + ka = − πGρ, (259)where H := ˙ aa is the Hubble parameter. We see that inflation, a ( t ) ∼ e tH ,occurs when there is a fluid of negative pressure p = − ρ .In scalar field theories, this requirement can be obtained if the scalar fieldis slowly rolling down its false vacuum potential, often referred to as slow-roll inflation. In what follows, we shall assume slow-roll inflation (see eg. [166] forfurther details) and we shall study the power spectrum of CS birefringent GWsduring the inflationary epoch. We will then rederive the consistency relationbetween the scalar to tensor ratio and the slow-roll index of inflation followingRef. [28]. Consider inflation driven by a pseudoscalar field φ with a standard kineticterm and a potential V ( φ, λ ). Consider also the Einstein-Hilbert action coupledto the gravitational CS term, as given by Eq. (3) with the choices α = 2 κ , β = 0 and f ( φ ) a functional of the inflation, which is identified with the CSscalar.Let us now linearize the action in an FRW background, as explained inSec. 5. The field equations for the GW become essentially Eq. (177) and,concentrating on plane-wave solutions, we find the evolution equation for the75hase in a left/right-basis [Eq. (184)]. Let us now introduce the quantity z s via z s ( η, k ) ≡ a ( η ) s − λ s k f ′ a (260)and the new amplitude µ s k ( η ) defined by µ s k ≡ z s h s k , where s = { R, L } , λ R,L = ± and the subscript k reminds us of the wavenumber dependance.The evolution equation for the phase then becomes a parametric oscillatorequation for µ s k : ( µ s k ) ′′ + k − z ′′ s z s ! µ s k = 0 . (261)where primes stand for derivatives with respect to conformal time η . The effec-tive potential z ′′ s /z s depends on time, on polarization but also on wavenumber,which distinguishes this CS effect from the standard GR case, where the ef-fective potential depends on conformal time only.Let us study the effective potential in more detail. Using Eqs. (260)and (261), this potential is given exactly by z ′′ s z s = a ′′ a − H λ s k ( f ′ /a ) ′ − λ s k ( f ′ /a ) − λ s k f ′ /a ) ′′ − λ s k ( f ′ /a ) − ( λ s k ) h ( f ′ /a ) ′ i [1 − λ s k ( f ′ /a )] . (262)For convenience, we choose specify the CS coupling functional f [ φ ] via [28,31] f = N π M Pl φM Pl . (263)where M Pl ≡ m Pl / √ π is the reduced Planck mass and N a number that canbe related to the string scale. In terms of the slow-roll parameters ǫ ≡ − ˙ H/H , δ ≡ − ¨ φ/ ( H ˙ φ ) and ξ ≡ ( ˙ ǫ − ˙ δ ) /H (an overhead dot means a derivative withrespect to cosmic time), we have at leading order in these parameters (see alsoRef. [167]) a ( η ) ∼ ( − η ) − − ǫ , φ ′ ≃ − M Pl H √ ǫ . (264)From these expressions, one deduces that to leading order f ′ a = − N π M Pl H a √ ǫ ≃ N π H inf M Pl ! √ ǫη (265)because H ≃ − (1 + ǫ ) /η .The equation of motion is still in a rather involved form, but it can besimplified further by introducing the characteristic scale k C and Θ parame-ter [28,31] k C ≡ k N π H inf M Pl ! √ ǫ = k Θ16 , Θ ≡ N π HM Pl ! √ ǫ . (266)76et us further introduce the variable x ≡ Θ kη/ <
0, such that Eq. (261)takes the form d µ d x + (cid:20) − f s ( x ) (cid:21) µ = 0 , (267)with f s ( x ) = 2 + 3 ǫx + λ s x (1 − λ s x ) −
14 1(1 − λ s x ) , (268)Using slow-roll perturbation theory and the functional relations for f ( φ ), theequation of motion can be maximally simplified intod µ L k d τ + (cid:20) −
14 + i Θ16 τ + 14 τ (cid:21) µ L k = 0 . (269)where we have defined τ ≡ i (1 + x ) / Θ. We recognize Eq. (269) as theWhittaker equation [see eg. Eq. (9.220.1) of Ref. [168]], whose corresponding,normalized solution is µ L k = − √ πℓ Pl √ k e ikη i e − π Θ / W i Θ / , " i (1 + x )Θ , (270)where W κ,µ ( z ) is the Whittaker function. Let us now concentrate on super-horizon scales, ie. x ∼
0, such that wecan make contact with CMB observables. The effective potentials can then beapproximated via f s ( x ) ≃ ǫx + λ s x − , (271)where the first term gives the standard slow-roll term behavior, while thesecond term represents the birefringent CS correction. With this potential,the equation of motion becomesd µ s k d x + " + 14 − λ s x − ǫx µ s k = 0 . (272)Using results from Ref. [169], we define y ≡ i s x , κ ≡ iλ s q / Θ , ξ ≡
32 + ǫ . (273)and simplify the equation of motion in the limit 256 / Θ ≫
1, which corre-sponds to large scale behavior, such that Eq. (272) becomesd µ s k d y + " −
14 + κy + (cid:18) − ξ (cid:19) y µ s k = 0 . (274)77he exact general solution to this equation is given in terms of Whittakerfunctions µ s k ( η ) = C s ( k ) W κ,ξ ( y ) + C s ( k ) W − κ,ξ ( − y ) , (275)where C s ( k ) and C s ( k ) are two constants of integration.Initial conditions are determined in the divergence-free region − < x ≪
0, where we assume µ s k has a plane wave behavior [169] C s ( k ) = − √ πℓ Pl √ k e iqη i exp − λ s π Θ32 ! , C s ( k ) = 0 , (276)where ℓ Pl is the Planck length and we have used that W κ,ξ ( y ) ∼ e − y/ y κ as y → + ∞ . This assumption is equivalent to requiring that non-linear phenom-ena occurring near the divergence of the effective potential does not affectthe initial conditions in the region x > −
1. Indeed, such an assumption isalso made in inflationary cosmology, where the vacuum is assumed to be thecorrect initial state, in spite of the fact that modes of astrophysical interesttoday originate from the trans-Planckian region [170]. A possible weaknessof the above comparison is that, in the case of the trans-Planckian problemof inflation [170,171], one can show that (under certain conditions) the finalresult can be robust to changes in the short distance physics [170,171]. In thepresent context, however, it is more difficult to imagine that the non-linearitieswill not affect the initial conditions. On the other hand, in the absence of asecond-order calculations and as a first approach to the problem, this seemsto be quite reasonable.
The power spectrum can be calculated either as the two-point correlationfunction in the vacuum state or as a classical spatial average. We take herethe latter view, since a fully consistent quantum formulation of the presenttheory is not yet available. The power spectrum is then defined as h h ij ( η, x ) h ij ( η, x ) i = 1 V Z d x h ij ( η, x ) h ij ( η, x ) , (277)with V = R d x is the total volume. Using the properties of the polarizationtensor, straightforward calculations show that h h ij ( η, x ) h ij ( η, x ) i = 1 π X s =L , R Z + ∞ d kk k | h s k | , (278)from which we deduce the power spectrum k P sh ( k ) = k π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ s k a ( η ) q − λ s kf ′ /a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (279)78he power spectrum is usually proportional to 2 k /π , where here the factorof 2 is missing because we have not summed over polarizations.The CS corrected power spectrum could be calculated exactly in termsof the Whittaker function, but only the large scale behavior is needed so, inthis regime, one has k P sh = 16 π ℓ Pl a k − ǫ ξ Γ (2 ξ ) | Γ (1 / ξ − iλ s Θ / | e − λ s π Θ / . (280)where Γ[ · ] is the Gamma function and α is the value of the scale factor today.Eq. (280) can be expanded to first order in the slow-roll parameter to obtain k P sh ( k ) = 16 H inf πm Pl A s (Θ) " − C + 1) ǫ − ǫ ln kk ∗ − ǫ B (Θ) , (281)with, A s ≡ − λ s π
16 Θ + π − ! Θ + O (cid:16) Θ (cid:17) , B ≡ − , (282)where Ψ is related to the derivative of the Gamma function via Ψ = Γ ′ / Γ.The amplitude of the CS corrected right-polarization state is reduced whilethe one of the left-polarization state is enhanced. Moreover, at leading orderin the slow-roll parameter, the spectral index remains unmodified, since n s T =d ln ( k P sh ) / d ln k = − ǫ for each polarization state.Let us now compute how the tensor to scalar ratio T /S in the modifiedtheory. In CS theory, the scalar power spectrum is not modified (see alsoRef. [172]) and reads [167] k P ζ = H inf πm Pl ǫ " − ǫ − C (2 ǫ − δ ) − ǫ − δ ) ln kk ∗ , (283)while the tensor power spectrum is given by Eq. (281). The T/S ratio is thengiven by TS ≡ k P ζ ) X s =L , R k P sh (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = k ∗ = 16 ǫ × h A L (Θ) + A R (Θ) i (284) ≃ ǫ × " π − ! Θ , (285)where we see that the linear corrections in Θ has canceled out, and one is leftwith a second-order correction only. Alternatively, we can express the aboveresult as the fraction CS correction( T /S ) Θ =0 ( T /S ) Θ=0 ≃ . × Θ , (286)79rom which it is clear that the CS correction is not observable, since one hasassumed here that Θ . − (ie. for the divergence of the effective potentialto be in the trans-Planckian region).The super-Hubble power spectrum exhibits two interesting regimes: alinear and non-linear one. The non-linear regime occurs when kη ∼ Θ − ,because the effective potential controlling the evolution of the linear pertur-bations diverges and linear cosmological perturbation theory becomes invalid.This divergence occurs for all modes (ie. for all comoving wavenumber k ) butat different times. The full non-linear regime has not yet been investigated.The linear regime is compatible with the stringy embedding of inflationarybaryogenesis [27], part of which we discussed in Sec. 3.2. In this context, Θis enhanced, possibly leading to resonant frequencies that could be associatedwith the observed baryon asymmetry. Since Θ is completely determined by thestring scale and string coupling in a model-independent fashion, one obtainsa direct link between stringy quantities and CMB anisotropies:( T /S ) Θ =0 ( T /S ) Θ=0 ≃ . (cid:18) H inf M (cid:19) g s ǫ , (287)where we have used that N = π r g s M Pl M ! , (288) M is the ten-dimensional fundamental scale and g s is the string coupling.For reasonable values of string coupling (ie. weak) and the string scale set to10 GeV, Θ ∼ − , but the stringy embedding admits much larger values ofΘ, forcing the analysis into the non-linear regime.Large values of Θ (eg. Θ & − ) require a non-linear calculation, throughwhich one could hope to obtain a significant modification to T /S that mightlead to an observable CMB signature. Unfortunately, it is precisely in thisobservable regime where technical difficulties have prevented a full analysis ofthe cosmological perturbations.
One of the major issues in the standard model of particle physics is theorigin of parity violation in the weak interactions. While we know that theother gauge interactions respect parity, it may be the case that the thereis a definite handedness on cosmological scales. The polarization pattern inthe CMB fluctuations can leave an imprint of parity violation in the earlyuniverse through a positive measurement of cross correlation functions thatare not parity invariant.The measurement of parity violation from CMB polarization was first dis-cussed by Lue, Wang and Kamionkowski [134]. They realized that the presenceof the CS term naturally leads to a rotation of the plane of polarization as a80MB photon travels to the observer. It was later realized by Alexander [30]that gravitational backreaction of parity violating modes can lead to loss ofpower for parity-odd spherical harmonics, which lacking a systematic expla-nation, is observed in the CMB for low multipole moments.Such considerations can be understood by studying the polarization stateof light as described through the Stokes parameters. Let us consider a classicalelectromagnetic plane-wave with electric field given by the following compo-nents: E ( t ) = a sin( ωt − ǫ ) and E ( t ) = a sin( ωt − ǫ ) (289)where we assume, for simplicity, that the wave is nearly monochromatic withfrequency ω , such that a , a , ǫ , and ǫ only vary on time scales long comparedto ω − . The Stokes parameters in the linear polarization basis are then definedas I ≡ D ( a ) + ( a ) E , Q ≡ D ( a ) − ( a ) E , (290) U ≡ h a a cos δ i , V ≡ h a a sin δ i , (291)where δ ≡ ǫ − ǫ and the brackets signify a time average over a time long com-pared to ω − . The I parameter measures the intensity of the radiation, whilethe parameters Q , U , and V each carry information about the polarizationof the radiation. Unpolarized radiation (so-called natural light ) is describedby Q = U = V = 0. The linear polarization of the radiation is encoded in Q and U , while the parameter V is a measure of elliptical polarization withthe special case of circular polarization ocurring when a = a and δ = ± π/ V as the measure of circular polarization,which is technically correct if Q = 0.While I and V are coordinate independent, Q and U depend on theorientation of the coordinate system used on the plane orthogonal to the light’sdirection of propagation. Under a rotation of the coordinate system by an angle φ , the parameters Q and U transform according to Q ′ = Q cos(2 φ ) + U sin(2 φ ) , and U ′ = − Q sin(2 φ ) + U cos(2 φ ) , while the angle defined by Φ = 12 arctan UQ ! , goes to Φ − φ following a rotation by the angle φ . Therefore, Q and U onlydefine an orientation of the coordinate system and not a particular directionin the plane: after a rotation by π they are left unchanged.Physically, such transformations are simply a manifestation of the oscil-latory behavior of the electric field, which indicate that Q and U are part ofa second-rank symmetric trace-free tensor P ij , i.e. a spin-2 field in the plane81rthogonal to the direction of propagation. Such a tensor can be representedas P ij = P − P , (292)in an orthonormal eigenbasis, where P = ( Q + U ) / is ususally called themagnitude of linear polarization. For example, in two-dimensional, sphericalpolar coordinates ( θ, φ ), the metric is Ω ij = diag(1 , sin θ ) and the polarizationtensor is P ij (ˆ n ) = Q (ˆ n ) − U (ˆ n ) sin θ − U (ˆ n ) sin θ − Q (ˆ n ) sin θ , (293)where we recall that ( i, j ) run over the angular sector only.The temperature pattern on the CMB can be expanded in a completeorthornmal set of spherical harmonics: T ( ˆn ) T = 1 + ∞ X l =1 l X m = − l a T( lm ) Y ( lm ) ( ˆn ) (294)where a T( lm ) = 1 T Z d ˆn T ( ˆn ) Y ∗ ( lm ) ( ˆn ) (295)are the coefficients of the spherical harmonic decomposition of the tempera-ture/polarization map and T is the mean CMB temperature. Likewise, we canalso expand the polarization tensor in terms of a complete set of orthonormalbasis functions for symmetric trace-free 2 × P ij ( ˆn ) T = ∞ X l =2 l X m = − l h a G( lm ) Y G( lm ) ij ( ˆn ) + a C( lm ) Y C( lm ) ij ( ˆn ) i , (296)where the expansion coefficients are given by a G( lm ) = 1 T Z d ˆn P ij ( ˆn ) Y G ij ∗ ( lm ) ( ˆn ) , a C( lm ) = 1 T Z d ˆn P ij ( ˆn ) Y C ij ∗ ( lm ) ( ˆn ) , (297)The basis functions Y G( lm ) ij ( ˆn ) and Y C( lm ) ij ( ˆn ) are given in terms of covariantderivatives of spherical harmonics by Y G( lm ) ij = N l (cid:18) Y ( lm ): ij −
12 Ω ij Y ( lm ): kk (cid:19) , (298)and Y C( lm ) ij = N l (cid:18) Y ( lm ): ik ǫ kj + Y ( lm ): jk ǫ ki (cid:19) , (299)where ǫ ij is the completely antisymmetric tensor on the 2-sphere, a colon inan index list stands for covariant differentiation on the 2-sphere, and N l ≡ l − / ( l + 2)! is a normalization factor. Since the Y ( lm ) ’s provide a complete82asis for scalar functions on the sphere, the Y G( lm ) ij and Y C( lm ) ij tensors providea complete basis for gradient-type (G) and curl-type (C) STF tensors, re-spectively. This G/C decomposition is also known as the scalar/pseudo-scalardecomposition, which is similar to the tensor spherical harmonic decomposi-tion of Sec. 5.Integration by parts transforms Eqs. (297) into integrals over scalar spher-ical harmonics and derivatives of the polarization tensor: a G( lm ) = N l T Z d ˆn Y ∗ ( lm ) ( ˆn ) P ij : ij ( ˆn ) , (300) a C( lm ) = N l T Z d ˆn Y ∗ ( lm ) ( ˆn ) P ij : ik ( ˆn ) ǫ kj , (301)where the second equation uses the fact that ǫ ij : k = 0. Given that T and P ij arereal, all of the multipoles must obey the reality condition a X ∗ ( lm ) = ( − m a X( l, − m ) ,where X = { T , G , C } . The spherical harmonics Y ( lm ) and Y G( lm ) ij have parity( − l , but the tensor harmonics Y C( lm ) ij have parity ( − l +1 .The two-point statistics of the temperature/polarization map is thengiven via C XX ′ l ≡ h a X ( lm ) ( a X ′ ( lm ) ) ∗ i , (302)where the averaging is over all 2 l + 1 values of m and over many realizationsof the sky. This two-point statistic is thus completely specified by the six( T T , GG , CC , T G , T C , and GC ) sets of multipole moments. If the temper-ature/polarization distribution is parity invariant, then C T Cl and C GCl mustvanish due to the symmetry properties of the G/C tensor spherical harmonicsunder parity transformations.Parity conservation, however, is a theoretical bias . Lue, Wang and Kamionkowski [134]provided the first time physical scenario where C TC l = C GC l = 0 due to par-ity violation in the GW power spectrum of the CMB . This physical sce-nario consisted of GWs sourced by the CS interaction term in Eq. (3) with( α, β ) = (1 ,
0) and ϑ = f ( φ ) some polynomial function of the inflaton field φ .As we have discussed, CS modified gravity leads to amplitude birefringence inGW propagation, which in turn leads to an excess of left- over right-cicularlypolarized GWs that lead to a non-vanishing C T Cl [134].In order to understand this, consider GWs during the inflationary epoch.These waves stretch and become classical at wavelengths on the order of λ ∼ /µ , where µ ∼ /f ′ is some CS energy scale, until they eventually freezeas they become comparable to the Hubble radius. When the waves exit thisradius, the fraction of the accumulated discrepancy between left- and right-polarized GWs can be estimated through the index ǫ ∼ ( M p /µ )( H/M P ) ( ˙ φ/H ) , (303) Lue et. al noticed that the CS term violates both parity and time reflections.Thus, since gravity is insensitive to charge, CPT is conserved. H is the Hubble scale and f ′′ ∼ /µ . The factor H / ˙ φ ∼ − isassociated with the amplitude of scalar density perturbations, while H/M P < − is related to the amplitude of tensor perturbations [134].Since long wavelength GWs produce temperature anisotropies of curltype, an excess of left- over right-polarized GWs produces a nonzero C T Cl .This is because the multipole coeficients a T,C ( lm ) will be non-vanishing (right) forcircularly polarized GW a T ( lm ) = ( δ m, + δ m, − ) A Tl ( k ) even l (+) , − i ( δ m, − δ m, − ) A Tl ( k ) odd l ( × ) , (304) a C ( lm ) = ( δ m, + δ m, − ) A Cl ( k ) even l ( × ) , − i ( δ m, − δ m, − ) A Cl ( k ) odd l (+) , (305)where A T,Cl are temperature brightness functions and (+ , × ) stand for plus orcross, linear GW polarizations (see eg. [173]). Likewise the multipole coeffi-cients for the gradient component of the CMB polarization are similar, withthe replacement of A T,Cl for polarization brightness functions. For a left, cir-cularly polarized GW, the sign of the even- l moments is reversed. The aboveequations allow one to understand why parity is not violated with linearlypolarized GWs. For example, let us assume that only + modes are present,then C T Cl by construction. However, a right or left, circularly polarized GWpossesses both + and x modes, and thus the cross-correction is non-vanishing: C T Cl = 2(2 l + 1) − A Tl ( k ) A Cl ( k ) (306)Lue, Wang and Kamionkowski [134] conclude that the parameter ǫ inEq. (303) could in principle be measured by a post-Planck experiment witha sensitivity of 35 µK , a result that was later confirmed by the more detailedstudy of Saito, et. al. [165]. Such results have aroused interest in the polar-ization detection community, pushing them to improve their detection sen-sitivities to measure CS-inspired, CMB parity violation. For example, Keat-ing et al [174] have considered non-vanishing parity violating correlations in-duced by a class of telescope-beam systematics, which can mimic the birefrin-gence effect. Furthermore, other authors have generalized the parity violationcross-correlations to CPT violating cross correlations in the photon sector[175,176,177,178,1]. Collider experiments have established a symmetry between matter andanti-matter, confirming the prediction of baryon number conservation in theStandard Model (SM) of elementary particle physics. In the visible Universe,however, there is an excess of matter over antimatter supported by the re-cent CMB determinations of the cosmological parameters, in particular by the84MAP experiment [1]. In view of this, one of the major puzzles in cosmologyand particle physics is to understand why and how the matter asymmetry wasgenerated during the course of the evolution of the Universe starting from asymmetric “soup” of matter and antimatter soon after the Big Bang. In thissection we will show that if CS gravity is active during the inflationary epoch,a novel mechanism of leptogenesis is generic.Quantitatively, the baryon asymmetry can be expressed in terms of theratio of baryon density excess to photon density excess [179] n B n γ = (6 . ± . × − , (307)where n B = n b − n ¯ b is the difference in number density of baryons and an-tibaryons and n γ is the number density of photons. This ratio is time indepen-dent, as the evolution of the n b and n γ with the cosmic Hubble expansion areidentical. Such a large baryon excess cannot be explained within the SM [180],because baryon number violating interactions are here loop suppressed. Theonly SM source of CP violation in the hadronic sector is in the Dirac phaseof the CKM mixing matrix, which is not enough to explain the asymmetry ofEq. (307).One can map the problem of baryon asymmetry to that of lepton asym-metry. This is because the SM weak interactions contain processes, mediatedby sphalerons ( SU (2) instantons), which interconvert baryons and leptons andare thermally activated at temperatures greater than 1TeV. Thus, a baryonasymmetry can be generated through the generation of net lepton number athigh temperature through out-of-equilibrium and CP-asymmetric processes[181,182]. Such scenarios are commonly referred to as leptogenesis .Leptogenesis is a valid route to explain the baryon asymmetry, providedone can construct a model that fulfills the so-called Sakharov conditions forleptons [183]:(1) Baryon number violating interactions should be present.(2) Charge and parity (CP) should be violated.(3) CP and baryon number violating interactions should be active when theUniverse is out of thermal equilibrium.These three requirements constitute model-independent, necessary conditionsto generate a baryon asymmetry dynamically from symmetric initial condi-tions. Within the SM, however, there are no such leptogenesis models, andhence one is forced to associate the observed baryon asymmetry to physicsbeyond the SM. A mechanism for the creation of baryon asymmetry, associated with grav-itational inflationary fluctuations, was presented by Alexander, Peskin andSheikh-Jabbari (APS) [28]. The key to this mechanism is CS modified gravity,where we shall here follow APS and set ( α, β ) = (8 κ, θ = f ( φ ) and associate85 with the inflaton field. Referring back to Sec. 2.4, the function f ( φ ) mustbe odd in φ , thus implying that f ( φ ) is odd under P and CP transformations,and allowing the parameterization of Eq. (263).Let us first spell out how the three Sakharov conditions are realized inthe APS model of baryon asymmetry, so-called gravi-leptogenesis :(1) Lepton number violation is generated here via the triangle anomaly,discussed in Sec. 3.1. In the SM, the lepton number current (and hencethe total fermion number density), has a gravitational anomaly [56]: ∂ a J aℓ = N π ∗ R R (308)where the lepton number current is given by J aℓ = P i = L,R ¯ ℓ i γ a ℓ i + ¯ ν i γ a ν i , N = N L − N R equals three in the SM, γ a are Dirac gamma matrices,and ℓ , ν denote lepton and neutrino species respectively. In the SM, theanomaly is thus a consequence of an imbalance between left- and right-handed leptons.(2) CP violation manifests itself naturally in CS modified gravity. Here,lepton number is generated due to a non-vanishing vacuum expectationvalue of the Pontryagin density in the evolution equation of the CS scalar,which is associated with the inflaton. In turn, this density is genericallynon-vanishing during inflation due to GW perturbations.(3)
Out-of-equilibrium conditions arise due to the (exponential) growth ofthe background spacetime. Such a growth leads to lepton number pro-duction that is naturally out of equilibrium.The APS leptogenesis model can be naturally realized if the inflaton fieldis associated with a complex modulus field. In such a case, one must guaranteethat the inflaton potential is sufficiently flat, such that the slow-roll conditionsare satisfied. The simplest model of this kind is that of single-field inflation anda pseudo-scalar φ as the inflaton, known as natural inflation. Such a modelcan be expanded to include multiple axions, as in N-inflation models [184].Such models fit into extensions of the SM and in string-inspired inflationarymodels .In the remainder of this chapter, we shall use particle physics notation andconventions, where h = c = 1 and M Pl = (8 πG ) − / ∼ . × GeV is thereduced Planck mass. In particular, we shall follow [186] and choose α = 8 κ , β = 0, and θ = f ( φ ), although we shall often work with the dimensionlessfunctional F ( φ ) := f ( φ ) /M . Although GW solutions have already been discussed in Sec. 5, it is in-structive to present the expansion of the Lagrangian to second order in themetric perturbation. In the TT gauge, assuming a GW perturbed FRW metric For a concise review of string-inspired inflation see [185] L = − ( h L ✷ h R + h R ✷ h L )+ 16 iF ( φ ) " ∂ ∂z h R ∂ ∂t∂z h L − ∂ ∂z h L ∂ ∂t∂z h R ! + a ∂ ∂t h R ∂ ∂t∂z h L − ∂ ∂t h L ∂ ∂t∂z h R ! + Ha ∂∂t h R ∂ ∂t∂z h L − ∂∂t h L ∂ ∂t∂z h R ! + O ( h ) (309)where t stands for cosmic time dt = a ( η ) dη and ✷ = ∂ t + 3 H∂ t − ∂ z /a . Asis clear from Eq. (309), if h L and h R have the same dispersion relation , ∗ R R vanishes, while otherwise “cosmological birefringence” is induced.From this Lagrangian, the equations of motion for h L and h R become h L = − i Θ a ˙ h ′ L , h R = +2 i Θ a ˙ h ′ R , (310)where dots denote time derivatives, and primes denote differentiation of F with respect to φ . The quantity Θ ∼ F ′ H ˙ φ/M P l , which with the functionalform of F ( φ ) becomes Θ = √ ǫ/ (2 π ) H /M P l N , where ǫ = ˙ φ / HM Pl ) − isthe slow-roll parameter of inflation [186]. We have here used the fact that theinflaton is purely time-dependent and we have neglected terms proportionalto ¨ φ by the slow-roll conditions.Gravitational birefringence is present in the solution to the equations ofmotion. Let us focus on the positive frequency component of the evolution of h L and adopt a basis in which h L depends on ( t, z ) only. In terms of conformaltime, then, the evolution equation for h L becomes d dη h L − η ddη h L − d dz h L = − i Θ d dηdz h L , (311)which is a special case of Eq. (177). If we ignore Θ for the moment and let h L ∼ e ikz , this becomes the equation of a spherical Bessel function, for whichthe positive frequency solution is h + L ( k, η ) = e + ik ( η + z ) (1 − ikη ) . (312)Let us now peel-off the asymptotic behavior of the solution with non-zeroΘ. Let then h L = e ikz · ( − ikη ) e k Θ η g ( η ) (313)where g ( η ) is a Coulomb wave function that satisfies d dη g + " k (1 − Θ ) − η − k Θ η g = 0 , (314)87hich in turn is the equation of a Schr¨odinger particle with ℓ = 1 in a weakCoulomb potential. For h L , the Coulomb term is repulsive, while for h R thispotential is attractive. Such a fact leads to attenuation of h L and amplificationof h R in the early universe, which is equivalent to the exponential enhance-ment/suppression effect discussed in Sec. 5.As we shall see, generation of baryon asymmetry is dominated by modesat short distances (sub-horizon modes) and at early times. This corresponds tothe limit kη ≫
1. In this region, we can ignore the potential terms in Eq. (314)and take the solution to be approximately a plane wave. More explicitly, g ( η ) = exp[ ik (1 − Θ ) / η (1 + α ( η ))] , (315)where α ( η ) ∼ log η/η . Let us now compute use Eq. (313) to compute the expectation value of ∗ R R in an inflationary spacetime. This expectation value is dominated by thesub-horizon, quantum part of the GW evolution. Hence, to compute h ∗ R R i we only need the two point (Green’s) function h h L h R i : G ( x, t ; x ′ , t ′ ) = h h L ( x, t ) h R ( x ′ , t ′ ) i = Z d k (2 π ) e ik · ( x − x ′ ) G k ( η, η ′ ) . (316)For k parallel to z , the Fourier component G k satisfies Eq. (311) with a delta-function source " d dη −
2( 1 η + k Θ) ddη + k G k ( η, η ′ ) = i ( Hη ) M δ ( η − η ′ ) . (317)Let us first consider the case where Θ = 0. The solution to Eq. (317) is G k ( η, η ′ ) = ℵ h + L ( k, η ) h − R ( − k, η ′ ) η < η ′ ℵ h − L ( k, η ) h + R ( − k, η ′ ) η ′ < η , (318)where ℵ ≡ ( H /M P l ) k − / h − L is the complex conjugate of Eq. (312), and h + R , h − R are the corresponding solutions of the h R equation. Since here Θ = 0, h + , − R = h + , − L . The leading-order effect of Θ = 0 is to introduce an exponentialdependence from Eq. (313). The generic solution to Eq. (317) is then G k = e − k Θ η e + k Θ η ′ G k (319)for both η > η ′ and η < η ′ .The Green’s function in Eq. (319) can now be used to contract h L and88 R and evaluate the quantum expectation value of ∗ R R . The result is h ∗ R R i = 16 a Z d k (2 π ) H k M ( kη ) · k Θ (320)where we have neglected subleading corrections in kη ≫
1. This expectationvalue is nonzero because inflation is producing a CP asymmetry out of equi-librium. The above result and computations seem to depend on the choice ofvacuum state and the form of the Green’s function, but in fact this is not thecase as one can verify by recalculating Eq. (320) using a different method,such as fermion level-crossing [187].
The leptogenesis model can be completed by computing the net lepton/anti-lepton asymmetry generated during inflation. Inserting Eq. (320) into Eq. (308)and integrating over the time period of inflation, we obtain n L = Z H − dη Z d k (2 π ) π H k η Θ M , (321)where n L is the lepton number density. The k -integral runs over all of mo-mentum space, up to the scale µ = 1 /F ′ at which the effective Lagrangiandescription breaks down, but it is dominated by modes at very short dis-tances compared to the horizon scale. The η -integral is dominated by modesat large η and it represents a compromise between two competing inflationaryprocesses: exponential expansion to large k and exponential dilution of thegenerated lepton number through expansion. The dominant contribution tolepton generation then arises from modes in the region kη ≫ n L = 172 π (cid:18) HM Pl (cid:19) Θ H (cid:18) µH (cid:19) . (322)The factor ( H/M Pl ) is the magnitude of the GW power spectrum. We shouldstress that the usual GW power spectrum comes from the super-horizonmodes, while the main contribution to n here comes from sub-horizon modes.The factor of Θ is a measure of the effective CP violation caused by birefringentGWs, while H is the inverse horizon size at inflation, giving n appropriateunits. The factor of ( µ/H ) is an enhancement over one’s first guess due to theuse of strongly quantum, short distance fluctuations to generate ∗ R R , ratherthan the super-horizon modes which effectively behave classically.The significance of the lepton number density can be understood by com-paring it to the entropy density of the Universe just after reheating, or equiv-alently to the photon number density. Recall that almost all of the entropy ofthe Universe is generated during the reheating time and it is carried by themassless degrees of freedom, ie. photons. Let us then employ one of the sim-plest (and at the same time most naive) reheating model: instant reheating.89e shall assume that all of the energy of the inflationary phase ρ = 3 H M P l is converted into the heat of a gas of massless particles ρ = π g ∗ T /
30 in-stantaneously. The entropy of this gas is s = 2 π g ∗ T /
45, where T is thereheating temperature, g ∗ is the effective number of massless degrees of free-dom, s = 1 . g ∗ n γ [186], and we have assumed an adiabatic post-reheatingevolution. With these assumptions we obtain the photon number density n γ = 1 . g − / ∗ ( HM P l ) / . Recalling that the ratio of the present baryon number to the lepton num-ber originally generated in leptogenesis is approximately n B /n L = 4 /
11 [181],in this model we obtain n B n γ = 4 . × − g / ∗ (cid:18) HM P l (cid:19) / Θ (cid:18) µH (cid:19) , (323)A less naive approach could be to follow the dilution of n L and ρ with theexpansion of the universe to the end of reheating. The final result is the same(see, however, [188] for a comment on this point). With the adiabatic expansionassumption, Eq. (323) can be compared directly to the present value of n B /n γ given in Eq. (307). In order to answer this question, one must numerically estimate n B /n γ .Substituting for Θ, Eq. (323) becomes n B n γ ≃ . × − g / ∗ √ ǫ N (cid:18) HM P l (cid:19) − / (cid:18) µM P l (cid:19) . (324)Clearly, this ratio depends on five dimensionless parameters: g ∗ , H/M P l , µ/M
P l and the slow-roll parameter ǫ and N .Some of these parameters are already constrained or given by theoreticalconsiderations. For example, within the usual supersymmetric particle physicsmodels, g ∗ ∼
100 is a reasonable choice. Moreover, WMAP data, throughthe density perturbation ratio δρ/ρ (for a single field inflation), leads to anupper bound on
H/M
P l ratio [1] of
H/M
P l . − and ǫ . .
01, or H . GeV . Finally, the factor N ≃ − is inferred from the string theorycompactification and is proportional to the square of the four dimensional M P l to the ten dimensional (fundamental) Planck mass [27] (see also the Appendixin [27]). If the gravi-leptogenesis model is to be viable, we then have thatΘ ≃ − − , (325)assuming that H saturates its current bound.The physically viable range for the parameter µ depends on the details ofthe underlying particle physics model on which our gravi-leptogenesis is based.90or example, within the SM plus three heavy right-handed neutrinos (and theseesaw mechanism), µ could be of the order of the right-handed neutrino mass,in which case µ . GeV. If we do not restrict the analysis to the seesawmechanism, µ could be larger, up to perhaps approximately the energy scaleat which the effective field theory analysis breaks down. Therefore, dependingon the details of the model10 − . µM P l . − − − . (326)Combining all these considerations, one finds that the gravi-leptogenesismodel predicts n B n γ ≃ +5 Θ (cid:18) µM P l (cid:19) . (327)Gravi-leptogenesis thus possesses a large parameter space to explain the ob-served baryon asymmetry of the Universe, but its final prediction stronglydepends on the ultraviolet cutoff of the effective theory [189,190]. In turn, thiscutoff dependence arises because lepton number is generated by graviton fluc-tuations, which we know are non-renormalizable in GR. This issue and thatof initial conditions are currently being investigated. We have provided a comprehensive review of CS modified gravity from aparticle physics, astrophysical, and cosmological perspective. From the particlephysics point of view, the presence of the CS term leads to a gravitationalanomaly-cancellation mechanism in the standard model. In cosmology, thechiral anomaly works together with inflation to amplify the production ofleptons, leading to a viable model of leptogenesis. The parity violation presentin the Pontryagin density leads to direct modifications to GW generation andpropagation.While all of these avenues are promising directions of research, there isstill much to understand. One important issue that needs to be addressed inmore detail is that of the coupling strength and the potential of the CS field.In string theory, the perturbative evaluation of this coupling suggests a Plancksuppression, while in LQG it is related to a quantum (Immirizi) ambiguity.The potential of the CS field, on the other hand, cannot be evaluated withcurrent mathematical methods. Thus, new non-perturbative techniques needto be developed to fully evaluate these terms. In lack of a concrete, accurate es-timate for these terms, we have taken an agnostic view in this review: to studythe observational consequences of the CS term and its possible constrains as afunction of the coupling strength. Such a view allows the possibility to searchfor CS effect, a detection of which would provide theorists with a strong mo-tivation to develop the necessary tools to evaluate the CS coupling strengthand the exact potential of the CS dynamical field.91nother issue that remains open is that of backreation and propagationeffects of fermions in the modified theory. Massive sterile neutrinos may be aviable candidate for dark matter. In spiral galaxies, dark matter is expectedto be at the center of the galaxy where a supermassive black hole mightreside. In the context of CS gravity, the trajectory of sterile neutrinos wouldbe controlled by the modified Dirac equation [6], which could now be studied inthe new CS-modified, slow-rotating background of Yunes and Pretorius [84]. Amodification in the behavior of fermions in highly-dense, strongly-gravitatingsystems could change the rate of black hole formation, such as in the thesymbiotic supermassive formation mechanism [191].A further avenue of future research that is worth exploring is that ofan exact solution that represents the exterior gravitational field of rapidly-rotating black holes and neutron stars in Dynamical CS modified gravity.The slow-rotating solution found by Yunes and Pretorius [84] has providedsome hints as to what type of modification is introduced to the metric by thedynamical theory, but an exact rapidly rotating solution is still missing. Sucha solution would allow for the study of extreme-mass ratio inspirals and theirassociated GWs in the modified theory, which could lead to interesting GWtests via observations with Advanced LIGO or LISA. Moreover, such a solutioncould also be used to study modifications to the location of the innermoststable circular orbit and the event horizon, which could have implications inX-ray astrophysics.A final avenue worth exploring is that of the connection of the dynamicalformulation and GW theory. The former could serve as a platform upon whichto test well-defined and well-motivated alternative theories of gravity withGW observations. An excellent source for such waves are binary systems, thedetection of which requires accurate templates. GW tests of dynamical CSmodified gravity would thus require the construction of such templates duringboth the inspiral and merger. Inspiral templates could be built through thepost-Newtonian approximation, while the merger phase will probably have tobe modeled numerically. Such studies would then allow GW observatories toexplore, for the first time, the non-linear quantum nature of spacetime.
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