Mild bounds on bigravity from primordial gravitational waves
MMild bounds on bigravity from primordial gravitational waves
Matteo Fasiello a,b and Raquel H. Ribeiro c,a,d a CERCA/Department of Physics, Case Western Reserve University,10900 Euclid Ave, Cleveland, OH 44106, U.S.A. b Stanford Institute for Theoretical Physics, Stanford University,Stanford, CA 94306, U.S.A. c School of Physics and Astronomy, Queen Mary University of London,Mile End Road, London, E1 4NS, U.K. d Perimeter Institute for Theoretical Physics,31 Caroline St N, Waterloo, Ontario, N2L 6B9, Canada
E-mails: [email protected], [email protected]
Abstract.
If the amplitude of primordial gravitational waves is measured in the near-future, whatcould it tell us about bigravity? To address this question, we study massive bigravity theories byfocusing on a region in parameter space which is safe from known instabilities. Similarly to inves-tigations on late time constraints, we implicitly assume there is a successful implementation of theVainshtein mechanism which guarantees that standard cosmological evolution is largely unaffected.We find that viable bigravity models are subject to far less stringent constraints than massive gravity,where there is only one set of (massive) tensor modes. In principle sensitive to the effective gravitonmass at the time of recombination, we find that in our setup the primordial tensor spectrum is moreresponsive to the dynamics of the massless tensor sector rather than its massive counterpart. We fur-ther show there are intriguing windows in the parameter space of the theory which could potentiallyinduce distinct signatures in the B -modes spectrum. a r X i v : . [ a s t r o - ph . C O ] M a y ontents It is well known that the pioneering work of Fierz and Pauli [1] sparked a theoretical programmeaimed at deriving a consistent fully non-linear theory of a massive spin-2 field. This search hasbeen refueled by the observation of the current accelerated expansion of the universe [2–4], anacceleration which is sometimes predicted in massive gravity models. There were, however, seriousobstructions to constructing such theories, both theoretical and observational. Of these, the presenceof the Boulware–Deser ghost [5] and the vDVZ discontinuity which prevented the massless limit fromreproducing the results of General Relativity (GR) in the regime where they have been verified, arethe most notorious.The ghost-free extension was put forward only very recently [6–8] up to fully non-linear level [9,10], and it relies on carefully chosen interactions which are ghost-free by construction and implementthe so-called Vainshtein screening [11]. Once active, this mechanism guarantees not only the recoveryof the GR limit, but also improves the stability of the theory under quantum corrections (see, forexample, Refs. [12–16]). For reviews on massive gravity see Refs. [17, 18].There has been extensive work exploring cosmological solutions in massive gravity [19–35] (andits bigravity generalization). If the graviton mass is responsible for the late time acceleration, itappears sensible to set it to order of the Hubble parameter today or even smaller. While an ob-servationally motivated choice, there are arguments [14, 15] as to why a small mass is a technicallynatural one. Most stringent constraints on the graviton’s mass arise from the physics of solar systemscales, and therefore late time observations.In the case of massive gravity, one may wonder whether there exist further bounds on thegraviton mass from early time cosmology. On the other hand, in bigravity, which one can think ofas massive gravity equipped with an additional Einstein-Hilbert piece for the reference metric, thedynamics is richer and strongly dependent on the role played by massless as well as massive tensormodes. Which observables could be more sensitive to the massive (bi)gravity dynamics? Massive– 1 –bi)gravity comes with five(seven) degrees of freedom(d.o.f.) and, in the current data-driven era, onemight want to test the presence of these additional d.o.f.s at lowest order in perturbation theory.However, the dynamics of the scalar and vector sectors can and sometimes is efficiently Vainshtein-screened, rendering the cosmological evolution almost unaltered for most part of cosmic history.This is precisely what one would want, but ought to verify, if any massive (bi)gravity theory is to beemployed to describe late time acceleration.On the other hand, measuring the amplitude of primordial Gravitational Waves (GWs) can beused to constrain various massive (bi)gravity models. This amplitude is traditionally parametrised bythe tensor-to-scalar ratio, r , which is defined as the ratio of amplitude of tensor to scalar fluctuations.Naively, a massive mode decays after crossing the horizon during inflation, lowering the tensor powercompared to the usual GR prediction. However, there are interesting subtleties related to identifyingsuch a signal in the data [36]. In bigravity in particular, the presence of the additional tensor modesis not screened in the Vainshtein sense, calling for an investigation of this additional sector.In massive gravity there are naturally two metrics (for a general approach see [37]). If onlyone of the metrics is dynamical then there is a clear cut massive eigenstate for the tensors and r is sensitive to the corresponding graviton effective mass. In bigravity, however, both metricsare dynamical [38, 39], resulting in two sets of coupled tensor modes with time-dependent masseigenstates. These will contribute to the tensor power spectrum in a non-trivial way. It followsthat the imprint of bigravity on primordial GWs is necessarily less transparent, which has motivatedfurther work in bigravity phenomenology in the early universe [40–42] and other contexts [43–53].We focus on a regime of the bigravity theory where several stability requirements are met—seeRefs. [22, 44, 52, 54], where the background cosmology was extensively studied. In agreement withthe perturbation analysis discussed in Refs. [55, 56], in that regime the Higuchi bound [49, 52, 57] issatisfied and the gradient instability [22] is pushed outside the reach of the effective theory [56]. Outline .—This note is organised as follows. In § § r would impose constraintson massive gravity and bigravity. In § B -modes imprints of massivebigravity in the CMBR signal. We summarize our work in § The ghost-free, non-linear extension of the Fierz–Pauli [1] mass term can be written L mGR = M g √− g (cid:32) R [ g ] + 2 m (cid:88) n =0 α n n ! (4 − n )! L n [ K ] (cid:33) , (2.1)where M g refers to the Planck mass associated with the metric g µν , α n are constant interactioncoefficients and the composite tensors K are defined by K µν [ g, f ] = δ µν − X µν with X µν ≡ (cid:16)(cid:112) g − f (cid:17) µν . (2.2)The scalar interaction potential is routinely and symbolically given in terms of the Levi–Civitatensors, L n [ K ] ≡ E E K n where the summation of indices is implicit— the interactions are built out– 2 –f characteristic polynomials of the eigenvalues of K , which is at the core of the ghost-free nature ofthe theory. This theory goes generally by the name of massive gravity .If the reference metric, f µν , becomes dynamical, then the theory is referred to as bigravity .In this case, the Lagrangian in Eq. (2.1) needs to be augmented by the kinetic term of the metric f µν , given by the corresponding Einstein–Hilbert term (with respective Planck mass M f ), which is,to date, the only known ghost-free, Lorentz-invariant kinetic term allowed in four dimensions [58].Therefore, in bigravity models, the Lagrangian becomes : L mBiGrav = L mGR + M f (cid:112) − f R [ f ] . (2.3)It is important to emphasise the following points about generic theories of massive bigravity.First, notice that in these theories there is only one value for the bare graviton mass, m , whichenters in the interaction potential, L . The two sets of helicity-2 modes generated by the two metrics g µν and f µν are associated with a massive and a massless spin 2-fields. The massive set of modeshas an effective mass, m eff , which is in turn related to m through the α n coefficients of the mixinginteractions L n . This m eff is the value which enters e.g. the predictions for observables such as thetensor power spectrum and r . Observational constraints are often directly sensitive only to “dressed”versions of the graviton mass, a fact we shall make use of repeatedly in this note.We also stress that bigravity theories describe two dynamical metrics while being entirely con-sistent with Weinberg’s theorem [59, 60] which argues that no two massless spin-2 fields mediatingthe long-force gravitational interaction can coexist. In bigravity there is a massive and a masslessfield, so that no violation of such theorem occurs. The task of finding viable massive (bi)gravity cosmological solutions has recently received consid-erable attention, generating a rich and interesting literature [40–42] and other contexts [43–52]. Ano-go theorem exists for FLRW solutions in massive gravity [34]: there are no FLRW solutions ina massive gravity theory with a Minkowski reference metric, f µν = η µν . Far from being a problem,this realization has lead to further work in several different directions.Within massive gravity, one may relax the exact homogeneity or isotropy assumption by intro-ducing some degree of inhomogeneities in the St¨uckelberg fields [34], consider open (closed) FLRWsolutions [21, 25, 61, 62], investigate the cosmology arising from a different choice for f [49, 63] orstudy an extended, ghost-free version of the theory preserving the same five degrees of freedom [64].Naturally, another way out of the no-go is that of considering theories with additional degrees offreedom. Massive bigravity and generic multi-metric theories, the quasi-dilaton model [65–69] allbelong to this class. Alternatively, one may choose a non-trivial coupling to matter [70].In any of the above contexts, once the background solutions have been found, a lot of caremust be exerted before declaring those solutions viable. Setting aside for a moment observationalconstraints, the study of background and perturbations alone might indeed reveal some pathologies.A healthy branch of solutions may support vanishing kinetic terms (the origin of strong-couplingissues) or it may lead to perturbations which do not satisfy the unitarity (Higuchi) constraint orpresent a gradient instability.Complementing these requirements are those stemming from observational viability: in shortone would want the “background” cosmic evolution to be essentially ascribable to GR up to the To this one generally adds minimally-coupled matter sector(s). – 3 –atest era, that of dark energy domination. One should add that, in order to pass, for example, solarsystem tests, an active and effective Vainshtein mechanism also needs to be in place.In what follows we will review the analysis and employ the notation of Refs. [55, 56]. Ourinterest lies in the flat, homogeneous and isotropic FLRW universe. We taked s g = g µν d x µ d x ν = − N ( t )d t + a ( t ) d x (2.4)d s f = f µν d x µ d x ν = −N ( t )d t + b ( t ) d x , (2.5)respectively for each metric. The background Friedmann equations read3 H g = m ˆ ρ m,g + ρ g M g , (2.6)3 H f = m κ ˆ ρ m,f + ρ f κ M g , (2.7)where H g ≡ ˙ a/aN and H f ≡ ˙ b/b N are the Hubble parameters, κ ≡ M f /M g , and ρ g and ρ f are theenergy density associated with matter coupling directly to the metric g µν and f µν , respectively. Itwill be convenient for simplicity of notation to define the following dimensionless quantitiesˆ ρ m,g ≡ U ( ξ ) − ξ U (cid:48) ( ξ ) , ˆ ρ m,f ≡ ξ U (cid:48) ( ξ ) , ˆ ρ m ≡ ˆ ρ m,g ( ξ ) − ξ κ ˆ ρ m,f ( ξ ) , (2.8)with primed variables being differentiated with respect to ξ = ξ ( t ) ≡ b ( t ) /a ( t ), and U ( ξ ) ≡ − α + 4( ξ − α − ξ − α + 4( ξ − α − ( ξ − α , (2.9)where α n ’s correspond to the interaction coefficients, cf. Eq. (2.1). To avoid a number of pathologiesdescribed in Refs. [22, 56, 71], we choose the so-called healthy branch of cosmological solutions whichcorresponds to the dynamical equation H g = ξ H f . (2.10)It will also be convenient for our subsequent analysis to define J ( ξ ) ≡ (cid:18) U ( ξ ) − ξ U (cid:48) ( ξ ) (cid:19) (cid:48) . (2.11)It follows from this definition that J ( ξ ) is an implicit function of the α n coefficients which set thestrength of the several interactions in the Lagrangian (2.1). In bigravity, the mass eigenstates corresponding to the massive modes are generically time depen-dent [52]. In order to isolate the massless and massive tensor modes at each given time one oughtto diagonalise the modes using an appropriate basis. It is a convenient feature of the region of theparameter space studied in Ref. [56] that the effective mass in this regime is actually constant, whichmakes the analytic derivation of observable quantities, such as r , much more tractable. See e.g. [21, 25, 71] for a more detailed analysis on choosing branches. – 4 –e shall start by defining the low-energy limit regime (henceforth LEL), investigated in Ref. [56],as that satisfying the following condition ρ g m M g (cid:28) ξ ρ f κ m M g (cid:28) . (2.12)Implementing both these relations as well as Eqs. (2.6) and (2.7) is of great consequence for thevalue of the bare mass m and the role of the interactions in ˆ ρ m,g . The realisation that GR providesan accurately verified description for most of cosmic history predating the dark energy era demandsthat ρ g dominates the RHS of the Friedmann equation correspondingly. This, together with theregime in Eq. (2.12) translates into the following conditions: m (cid:29) H g , ˆ ρ m,g < H g /m ⇒ ˆ ρ m,g (cid:28) α n coefficients in ˆ ρ m,g is then being put touse in taming the “large” value of the bare mass m in Eq. (2.6). We continue below navigatingthe parameter space of the LEL regime to expose its dynamical consistency and, at the same time,underline the bounds the latter demands.Combining Einstein equations with Eq. (2.10) one derives the following:ˆ ρ m ( ξ ) = − ρ g m M g + ξ ρ f κm M g ⇒ ˆ ρ m ( ξ ) (cid:12)(cid:12)(cid:12) LEL (cid:28) . (2.14)It is intuitively clear then that in the LEL a solution to Eq. (2.14), seen as a dynamical equation for ξ , is that of a constant ξ = ξ c satisfying ˆ ρ m ( ξ c ) = 0. As to the value of ξ c , stability considerationswe will touch upon later on suggest it be order unity. To fully specify the LEL regime, one needs tofurther require the following relation holds true:1 (cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) κξ c ˆ ρ m,g J ( ξ c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∼ ( κξ c ) ˆ ρ m,g ( ξ c )ˆ ρ (cid:48) m,g ( ξ c ) (2.15)This can be interpreted as follows: it corresponds to requiring | m ˆ ρ mg /m | (cid:28)
1. It holds in viewof the low-energy Higuchi bound, ( m (cid:38) H ), and Eq. (2.6) and can hold in eras preceding darkenergy domination. Condition (2.15) may be arrived at by enforcing κ (cid:28) α n coefficients within ˆ ρ m,g to enforce ˆ ρ m,g (cid:28) ˆ ρ (cid:48) m,g . Note that the latter option is notlight on the α n s because of the requirement already in place via Eq. (2.13).We note in passing that the lack of gradient instability in the scalar sector corresponds toimplementing c s ∼ c − d ln Jd ln ξ − ξ ( ρ f + P g )3 κM g m > , with m ≡ m κξ κξ Γ( ξ ) , (2.16)and where Γ( ξ ) ≡ ξ J ( ξ ) + (˜ c − ξ J (cid:48) ( ξ ) . (2.17) If the solution in the low energy regime is not such that ˆ ρ m ( ξ c ) = 0, then ˆ ρ m ( ξ c ) can always be reabsorbed into acosmological constant contribution. – 5 –he unitarity bound is automatically satisfied upon enforcing Eqs. (2.13) and (2.15). Note theappearance of the effective mass, m eff , which is the physically relevant quantity here and correspondsto the bare mass dressed by the non-linear interactions which make up the U n potentials. For lateruse we point out here that already at an intuitive level it is clear that a large effective mass wouldgive way to a dynamics dominated by the massless modes. One may borrow the familiar -integratingout d.o.f.- physical picture to guide this intuition .As we have seen, working in the LEL regime comes with a simplified dynamics. It allows forexample an expansion of all relevant quantities about ξ = ξ c , as a result of which one can write theFriedmann equation in Eq. (2.6) as3 H g (cid:39) ρ g M g (1 + κ ξ c ) + Λ eff , where Λ eff ≡ Λ (cid:18) κ ξ c ( ξ − ξ c )1 + κξ c (cid:19) , (2.18)and with Λ ≡ m ˆ ρ m,g ( ξ c ). We can thus write3 H g (cid:39) ρ g M g (1 + κ ξ c ) , (2.19)a relation we will put to use in §§ , f = ( m κ / M g ) / . Before we compute observables, we need to specify the coupling of gravity to matter. When theuniverse was 380,000 years-old, recombination occurred causing the photons to decouple from thehot plasma and free-stream up to the present time. The photons of this era are the CMBR weobserve in the microwave sky and they contain a snapshot of the physics of the early universe. Theymay then encode information about a theory of gravity which is other than GR.How does massive gravity couple to the matter sector? Classically, it was shown by Hassan &Rosen [38] that bigravity can couple covariantly to matter without reintroducing a ghostly degree offreedom. At the quantum level, one may worry that the irrelevant interactions which make up thenon-linear theory might not be under control. In bigravity or multigravity theories, one could wonder whether all the dynamical metrics coulddemocratically and covariantly couple to the same matter sector. Unfortunately this induces a ghostat an unacceptably low scale [70, 76], making the model unstable. As a result, either each dynamicalmetric couples to its own matter sector, or the coupling of gravity to matter is made through acomposite metric. In the last case, however, FLRW ansatzs for both g µν and f µν require matter towhich they couple to be identified as a dark sector, [70, 77] which can later on couple to standardmatter. This would require knowledge of this additional coupling.For simplicity, in what follows we couple matter to one of the metrics, g µν . See Refs. [49, 52, 57, 72] for more details. See also [83] on this point. In general, the quantum stability of field theories which rely on large derivative self-interactions, as is the caseof massive gravity, is not trivial. However, recent work has shed some light on the role of the quantum mechanicalrealisation of the Vainshtein mechanism in protecting the theory—see Refs. [12, 13, 16, 73–75]. – 6 –he matter Lagrangian is fully described by S matter = (cid:90) d x √− g L matter ( g, φ i ) , (2.20)where φ i labels the matter field species, including photons which encode the physics of the earlyuniverse after recombination occurs. Primordial gravitational waves are seeded by tensor modes of the primordial perturbation. In massivegravity, there is only one family of modes, which we shall denote by h µν whereas for bigravity thereare two sets of modes at play, h µν and (cid:96) µν . Therefore, the fluctuations of both metrics, g µν and f µν ,now contribute to the power spectrum of primordial GWs.We start by reviewing the cosmological perturbation analysis performed by De Felice et al. [56].We write the metric in ADM variables g µν = − N d t + a ( t ) ( γ ij + h ij )d x i d x j and f ij = −N d t + b ( t ) ( γ ij + (cid:96) ij )d x i d x j (3.1)where γ ij is spatially flat, a and b are scale factors and N and N are lapse functions correspondingto each space-time metric. The quadratic action of the tensor modes becomes S (2)tensors = M g (cid:90) d x N a √ γ (cid:34) ˙ h ij ˙ h ij N + h ij a ∇ h ij + κ ˜ c ξ (cid:32) ˙ (cid:96) ij ˙ (cid:96) ij N + (cid:96) ij b ∇ (cid:96) ij (cid:33) − m Γ( ξ ) (cid:0) h ij − (cid:96) ij (cid:1) ( h ij − (cid:96) ij ) (cid:35) , (3.2)where γ can be set to unity, ∇ denotes the spatial Laplacian and˜ c ≡ N aN b . (3.3)Dotted quantities are differentiated with respect to cosmic time. The action above (3.2) featuresnon-trivial interactions between h and (cid:96) fluctuations, which makes analytical predictions far fromstraightforward. To make progress, it is be useful to find a regime in which a diagonalisation of thisaction is possible, as argued before.Considering only the transverse and traceless perturbations as the relevant degrees of freedom,it is convenient to choose a different basis for the tensor perturbations as follows [56]: H − ij ≡ h ij − (cid:96) ij and H + ij ≡ h ij + κ ξ (cid:96) ij κ ξ . (3.4)We stress that the validity of this diagonalization is limited to the LEL regime. The mass eigen-states of a bigravity theory are in general time dependent and so will be the diagonalization basis.Nevertheless, in what follows we will employ the customary inflationary normalization for the tensorwave functions whose behaviour at recombination is what determines possible CMBR imprints.Before proceeding with the analysis of tensor modes in the LEL configuration we pause here tostress that any instability in the tensor sector should not go unnoticed and it is different in nature– 7 –rom the one concerning the other d.o.f.s. Indeed, no matter how efficient the screening of scalarsand vectors, the two copies of the tensors at hand can be safely considered unscreened. The act ofnormalizing things “as usual” for inflation and considering the tensors diagonalized from the onsetof the LEL may be insensitive to part of the dynamics which goes beyond the LEL regime , as isexemplified by the work in [40, 78], which respectively focused on different branches of solutions.Working at lowest order in perturbations, the simplified action for the tensor modes becomes S (2)tensors = 18 (cid:90) d x a (cid:34) M (cid:32) ˙ H ij + ˙ H + ij + H ij + a ∇ H + ij (cid:33) + M − (cid:32) ˙ H ij − ˙ H − ij + H ij − a ∇ H − ij − m H ij − H − ij (cid:33) (cid:35) . (3.5)As we shall see shortly, it is this mass parameter m eff associated with the tensor modes whichenters observable quantities and which can be used to constrain the parameters of the theory. Thenormalisation of the kinetic terms is set by the mass scales M + and M − given by M ≡ (1 + κ ξ c ) M g and M − ≡ κ ξ c (1 + κ ξ c ) M . (3.6)In perturbation theory the matter Lagrangian (2.20) is given, to lowest order, by δS matter = (cid:90) d x h µν T µν . (3.7)The usefulness of the { + , −} basis is clear: H + ij is the massless mode whereas the massive modeis H − ij . In light of our choice of coupling to matter (3.7), photons do not couple to a massive normassless graviton. Instead, they couple to a linear combination of massive and massless modes in thediagonal basis described in Eq. (3.5). Importantly, the massive mode is associated with a redressedmass m eff , which albeit related, is not the graviton mass.In terms of the diagonalised variables of Eq. (3.4), the coupling to the matter sector is δS matter = (cid:90) d x (cid:26) H + ij + κ ξ c κ ξ c H − ij (cid:27) T ij . (3.8)This means that gravity and the history of the universe will, in principle, change. To guarantee thatthe cosmology remains invariant so as to agree with ΛCDM and to reproduce the Newtonian limit,we need to require that the Planck mass (or equivalently the Newton’s gravitational constant) isessentially the same as measured in solar system scales. Consider two mass tests described by theenergy-momentum tensor above and subject to gravity set by this matter coupling. One may easilyderive that the effective Planck mass associated with this modified gravitational force is actually thePlanck mass associated with the metric g µν : M Pl , eff ≡ M g . (3.9) This is important because the coupling among the tensor modes, although very small, in time can have significanteffect once higher order in perturbations are taken into account. This again points to the importance of the initialvalue problem as emphasized in [40]. See also [42] for a step in this direction. Note that the scales probed by [42] arefar beyond the reach of the LEL regime. – 8 –o far, all we have assumed within the LEL was that ξ c ∼ O(1), but there were no stringentconstraints on the value allowed for κ . If we demand the cosmological evolution to be the same formost of cosmic history, as well as negligible modifications to the Newton’s gravitational constant,then from the Friedmann equation (2.19), we ought to require κ (cid:28) ξ − c ∼ O(1) . (3.10)Considering the uncertainty associated with the empirical determination of the value of theNewton’s gravitational constant, we estimate κ to be 1 part in 100 , M f (cid:28) M P l . This results, in turn, in thelowering of the naive strong coupling scale ( m M f ) / ≡ Λ ,f . This worry is legitimate and we justnote here that the scale we are negotiating with is that of H r . One can easily check that even if thestrong coupling scale is as low as Λ ,f , the theory is still predictive at the time of recombination, forwhich the graviton mass can be chosen such that H r (cid:28) Λ ,f , a condition which is automatic in theLEL regime. Other d.o.f .— Before proceeding with the study of observables related to tensor perturbations wepause here to comment on the role played by the other degrees of freedom in our setup. Clearlymassive gravity alone already spans 5 d.o.f. whose effect on cosmological evolution up to recombina-tion must be accounted for. So far we have largely neglected the dynamics of the scalar (helicity-0)and vector (helicity-1) sector. Their dynamics has already been the subject of careful investigationsat the lowest orders in perturbation theory hinting at early(late) time instabilities. However, suchan analysis may or may not capture the full physical picture, especially if limited to low orders inperturbation theory. It has further been suggested that a way around these issues may be foundin a restricted pool of favourable initial conditions and that the so-called initial value problem forbigravity needs further study [40] (see also Ref. [81] for related work). We can only but agree onthis latter point. A step in this direction was taken in [42] where initial conditions are discussed inan inflationary context. This implicitly assumes that the effective strong coupling scale is as high as H inf . In this manuscript we take the view that, as is often the case in solar system dynamics, anefficient Vainshtein mechanism will make use of the non-linearities of the theory to milden the roleplayed by the scalar(vector) sector in the cosmological evolution, effectively screening them. Althoughsuggestive results are present in the literature, at this level this is indeed just an assumption we makehere, and not a small one at that.We now turn to the observable spectrum arising in this bigravity model. The coupling inEq. (3.8) dictates the power transmitted by the tensor modes (both from the massive and themassless spin-2 fields) to the CMBR photons. Before presenting the formula for r , we make a shortdigression into the power spectra of massive and massless tensor modes which would individually beimprinted in the CMBR. For bigravity theories, the power spectrum will be a composite measure ofboth signals. For the purposes of our estimates, it suffices to work at lowest order in the slow-rollapproximation, which is assumed to hold throughout inflation. Power spectrum of massless modes .—The analysis of the perturbation theory for massless tensormodes is well known in the literature [82]. In bigravity theories, the massless mode is represented Here we are focusing on the case when the strong coupling scale can be as low as ( m M f ) / . Being a very lowscale, this could prove difficult for the phenomenology at and above that energy scale. Incidentally, we note that an active screening in the LEL demands [55] that | dln J ( ξ ) / dln( ξ ) | (cid:29)
1, see Table 1. – 9 –y H + in Eq. (3.5) and their evolution is fixed by the time they cross the horizon, after which theiramplitude remains constant thereafter. It is therefore sufficient to determine their super-horizonevolution. On super-horizon scales, the resulting power spectrum is given by P massless ≡ (cid:104) H + ij ( k ) H + jl ( k ) (cid:105) ≡ P t , GR κξ c . (3.11)Here we adopt the usual notation that starred quantities are evaluated at the time of horizon crossingfor the mode k , and we use H ≡ H g to avoid clutter. Notice the different normalisation arising fromEq.(3.5) of the tensor spectrum from the GR one, which we have denoted by P T , GR , where the onlyscale is H . Power spectrum of massive modes .—Massive tensor modes have their evolution determined bytheir corresponding mass horizon, and they do not get frozen on super-horizon scales but ratheroscillate. These modes are denoted by H − in Eq. (3.5) and give the following power spectrum P massive ≡ (cid:104) H − ij ( k ) H − jl ( k ) (cid:105) ≡ κ ξ c κξ c P t , mGR (3.12)where P t , mGR stands for the power spectrum obtained from the standard GR action for tensorswhere there is an additional mass term (in this case m eff ). Notice again the different non-trivialnormalisation of the power spectrum (arising from Eq.(3.5)). The power transmitted by the tensor modes in this class of theories is governed by how they coupleto matter via Eq. (3.8). It follows that the primordial power spectrum of tensors is (cid:104) h ij ( k ) h jl ( k ) (cid:105) = (cid:104) H + ij ( k ) H + jl ( k ) (cid:105) + (cid:18) κ ξ c κ ξ c (cid:19) (cid:104) H − ij ( k ) H − jl ( k ) (cid:105) (3.13)with the individual power spectra given by Eqs. (3.11) and (3.12). The fact that this prediction isdifferent from that of GR is only one source of modification to the tensor-to-scalar ratio, which isalso sensitive to the scalar perturbations. Since the Friedmann equation is still slightly changed, thiswill reflect on the matter power spectrum and therefore on the primordial scalar power spectrum.In particular, from Eq. (2.19) the energy density in matter is ρ matter = ρ g κ ξ c , (3.14)which results in the scalar power spectrum changing to P s , biGrav = P s,GR κ ξ c , (3.15)where the subscript s refers to scalar. Consequently, these theories predict r BiGrav = r GR + ( κ ξ c ) r mGR , (3.16)where r GR denotes the tensor-to-scalar ratio in GR (that is, assuming a massless graviton) whereas r mGR corresponds to that in a theory of a massive graviton. Under the assumption that the cosmologyis not significantly changed and in the limit where κξ c (cid:28)
1, it follows that the prediction for r inthese theories is the same as the original for GR.– 10 – .2 Summary on bounds in parameter space To obtain the observable predictions in bigravity theories, our starting point was the region inparameter space in the theory where known instabilities are absent [56] which we have furtherrestricted due to the κ (cid:28) Stability requirement Parameters Bound
Higuchi bound m eff m > H g Absence of strongly coupled perturbations H g H g = ξ H f No gradient instabilities & active Vainshtein m, H m (cid:29) H , | dln J ( ξ ) / dln( ξ ) | (cid:29) M g κ (cid:28) ξ ∼ O(1) Table 1 . Collection of stability bounds for the phenomenology discussed in this paper. Notice that theseconditions correspond to relatively strict bounds on the α n interaction coefficients in the massive bigravityLagrangian. This can be easily seen by expressing J in terms of α n via Eqs. (2.9), (2.11). The tensor-to-scalar ratio is not the only observable which can provide constraints to this classof theories. Interesting imprints in the CMBR are also forecast which, provided they affect observablemodes, can reveal interesting physics about the early universe. We turn next to such characteristicsignatures in the B -modes signal, which were originally unveiled in Ref. [36]. We adapt their resultsto discuss their implication on the bigravity models we discussed here. The study of the CMB spectrum plays a fundamental, unparalleled role in modern cosmology. Itessentially represents an open window over the past of the Universe, taking one all the way backto the recombination epoch. It is then clear that, in the study of how a massive(massless) gravitonmay affect the CMB radiation, the crucial scale one will have to negotiate with is the size of theHubble radius at recombination, H − r . As we shall see below, a simple description emerges forthe primordial tensor sector contributions to B-modes: one can think of two main regimes for theeffective tensor mass m eff , with the transition value for m eff being H r itself.In deriving the results for this section we will heavily rely on the work in [36]. In there, theauthors present a phenomenological approach aimed at placing bounds on the graviton mass which This condition specifies the so-called healthy branch. This last assumption is to be understood within the LEL [56]. The contribution from reionisation is relevant for small (cid:96) . In the massless case (cid:96) >
20 is sufficient to ignore thiscontribution. The numerical analysis of [36] supports the approximation scheme neglecting the reionization contributionalso for a large range of effective mass values. – 11 –an be adapted to our case. Their results are quite general and hold true for the tensor sector aslong as the tensor wavefunction satisfies the usual tensor e.o.m., just as the one that follows fromEq. (3.5). As per
Section
2, the model under scrutiny here is a bigravity theory, it counts sevendegrees of freedom which include massive and massless tensor modes.As documented in Eq.(3.4), both the massless and the massive tensor modes, decoupled fromone another in the + / − basis, directly couple with matter. The crucial realization in what followsis that there exist important regions in the parameters space where, to a good approximation, theresult of the two contributions is additive not just at the level of the action , but it remains soup the source term | ψ | and therefore eventually propagates all the way to the expression for thecoefficients C TBB,l . We will argue in particular that this is the case in the mass regime which wouldin principle generate the most prominent effect in the CMB.As we have seen, it turns out that in the LEL, a regime we chose in order to avoid knowninstabilities, the relative coefficient regulating the contribution of the massive tensor modes (asopposed to the massless ones) is very small because it depends linearly on κ (cid:28)
1. This will resultin a very hard-to-detect massive tensor sector. We will see below that in a specific mass range amassive graviton actually enhances the gravitational signal by two orders of magnitude as comparedto the massless case. However, in our case this enhancement is no match for the ∝ κ suppression. Wenevertheless provide a detailed discussion also in the hope that it will be later applied to a scenariowhere the relative coefficient is order one so that “massive” imprints would be more conspicuous. A clear qualitative understanding of the dynamics of tensor modes may be arrived at by classifyingthem (see Fig.1 and [36]) according to whether or not and to where these modes are relativistic (inthe sense of satisfying q /a ( τ ) (cid:29) m ): • Modes relativistic at recombination belong to
Region I • Modes which are non-relativistic already early on, before entering the horizon, reside in
RegionII • Momenta entering the horizon as relativistic but turning non-relativistic by the recombinationepoch populate
Region III
This pictorial view refers of course to the massive modes but in the additive regime one may reason-ably expect that the exactly massless modes will generate the usual imprints to be superimposed,weighted by a relative coefficient, to the massive modes signatures.Note that the presence of the fractional relative coefficient in front of the massive contribution to h µν , e.g. in Eq. (3.8), cannot alter the description in Fig.1 . Indeed, the latter is obtained byjudiciously comparing among each other the value over time of q/a ( t ) , H ( t ) , m eff . These quantitiessquared all appear in the massive tensor modes equation of motion and their relative strength,unaffected by the fractional coefficient, signals the regime one is working at (i.e. (non)relativistic, m > ( < ) , inside/outside the horizon). This is trivially true as we linearly couple gravity to matter in Eq. (3.7). What it does alter is the relative weight of the massive tensor sector in determining the gw signal. – 12 – igure 1 . Above is the effective mass m eff in units of the Hubble rate today, H , versus multipole momenta (cid:96) . The upper limit of Region
I is obtained by requiring that the multipole (cid:96) , for which (cid:96) ∼ q ( τ − τ r ),coincides with (cid:96) , that is the multipole corresponding to a physical momentum the size of the mass m eff atrecombination. The border between Region
II and
Region
III is obtained by identifying the multipole (cid:96) m corresponding to the momentum that becomes non-relativistic at the same time it enters the horizon [36].Two horizontal lines show qualitatively different regimes for the effective mass: m > stands for mass largerthan H r and, complementarily, m < . Let us now discuss how one may generate Fig.1 and then the consequences it can possibly entailfor bigravity CMB signatures. Region I denotes modes relativistic at recombination. For a genericmultipole moment one has, in conformal time, that (cid:96) ∼ q ( τ − τ r ). The role of the ( τ − τ r ) factor isclearly that of accounting for the time evolution to the present day. In our setup this evolution canin principle depart from that of ΛCDM.Working in the LEL regime and implementing Eq.(2.19) (with κ (cid:28) eff ∼ Λ = m ˆ ρ m,g [ ξ c ]so that the form of the Friedmann equation reduces, formally, to the standard one) then, in multipolelanguage the non-relativistic threshold is reached [36] at recombination whenever: (cid:96) ≤ m eff a ( τ r )( τ − τ r ) ≡ (cid:96) ∼ m eff H (1 + z r ) (cid:90) z r ) − da (cid:112) Ω Λ eff a + Ω m a + Ω r ∼ . m eff H (1 + z r ) , (4.1)In the numerical calculation we have set Ω Λ eff ∼ Ω Λ ∼ .
73 because, as we have seen, thebackground evolution of the LEL regime mimics that of ΛCDM in our setup.It is also worth pointing out that the result of a (4.1) without a c.c. term would amount to changingthe numerical factor in front from 3 . .
7, a mere 10%, reflecting the fact that we owe most of theevolution since recombination to the content in Ω m , Ω r .A recombination redshift of z r (cid:39) m eff (cid:46) H , the value of (cid:96) will be so small that the B-mode spectrum willbe unaffected by non-relativistic modes . Incidentally, for m eff = H r one finds that (cid:96) ∼
64, whichis the only point shared by all three regions in Fig.1. This is true exactly only if the contribution at reionization is neglected. The fact that non-relativistic modes playno active role is intuitively clear upon noticing that the mass m eff ∼ H corresponds to the size of the visibleuniverse at recombination. – 13 –he first, most straightforward, realization is that modes which are relativistic at recombinationare entirely insensitive to the presence of the effective mass m eff . On the other hand, one can alwaysprobe the non-relativistic regime through modes satisfying q/a ( t ≤ t r ) < m eff .In the m < range there are only two possible configurations: modes relativistic at recombination(Region I), or non-relativistic momenta which only re-enter the horizon after recombination (RegionII). In this mass range modes are indeed forbidden from stepping in the horizon as relativistic andslowing down to non relativistic before recombination. Indeed, consider a q/a ( t < t r ) > m eff ; thismode can easily transition into non-relativistic after some time. On the other hand, because H ( t )has a steeper time dependence than a − ( t ) and becomes H = H r > m eff at recombination, it mustbe that if the mode q in question is outside the horizon it will stay out past the time it becomes non-relativistic. This dynamics then fits the description of modes populating Region II. The signatures ofmassive tensor modes in the m < range become distinct from those of the massless case only for long,outside the horizon, wavelengths. The source term | ψ | which feeds the expression for the multipolecoefficients depends on the time derivative of the primordial tensor perturbations ˙ h ij , which is in turnrelated to the term q + m eff 2 a ( t r ) . Clearly, at very large wavelength (from the onset of what wecall the non-relativistic regime) it is the mass contribution to dominate and provide an enhancementwith respect to the massless case. This is precisely the low- (cid:96) plateau found in [36].For completeness we report that, as one raises the value of the effective mass, a new qualitativelydifferent possibility emerges: relativistic modes might enter the horizon and slow down to becomenon-relativistic before recombination. This is the dynamics which characterizes Region III. We referthe reader to Ref. [36] for details on how to derive the border between Regions II and III. Our interestis focused on the modes of Region II which, as we have briefly reviewed, contribute to a plateau inthe CMB tensor spectrum.Having seen how Fig.1 provides an understanding on the dynamics of massive tensor modes, wepause here to note that this understanding is necessarily qualitative in nature and serves its purposequantitatively only in the asymptotics. We have shown in detail below Eq.(4.1) that a very smalleffective mass, below 10 H , will witness most modes being relativistic and, as for the non-relativisticones, those will not correspond to a high enough (cid:96) so as to leave any marks on the B-modes CMB.It is safe to say that such a small effective mass would not be detected .Navigating Region II for larger m eff will eventually lead to the “large wavelength outside-the-horizon” enhancement mentioned above, an effect propagating all the way to the B-modes spectrumand shaping up as a the low- (cid:96) plateau in the C TBB,(cid:96) multipole coefficients (for a fixed m eff of thissize and higher (cid:96) s one would step into Region I). Determining exactly the onset of this enhancementis a task best performed through the use of software such as CAMB [ ? ], see [36]. The result isthat a plateau starts emerging at about m eff ∼ . × H , its effect being most striking (twoorders of magnitude larger than the standard massless tensor signal of GR) at m eff ∼ . × H only to weaken and eventually become suppressed with respect to the massless signal as soon as m eff ∼ × H .We stress here that the plateau is a clear-cut effect that, if detected, would represent the mostprominent CMBR signature of a massive theory of gravity. Most importantly for our analysis, thefact that this effect generically takes place in the low- (cid:96) regime will, as we shall see, guarantee that abigravity model may also lead to such an imprint in the same effective mass range. Another thing tokeep in mind is that in our case, as opposed to the analysis in [36], the unitarity bound sets a strong This statement is all the more appropriate in a bigravity setup where the massive tensor modes do not have thefull weight they enjoy in a purely massive gravity theory . – 14 –pper bound on m eff , of the order H r . This requirement stems from the helicity-0 mode analysis andis therefore not necessarily present in e.g. Lorentz breaking theories of massive gravity that inspiredthe work in [36].Proceeding with the analysis at larger effective mass values one will see a suppression of thesignal. The reason for the asymptotic suppression at m eff (cid:29) H r is that non-relativistic modes willstart oscillating sooner and sooner outside the horizon with increased frequency m eff leading to anaveraging-out which amounts to a strong suppression of the signal. Determining exactly by whatvalue of m eff this effect will take over the enhancement is a task beyond the scope of the presentwork. The additive regime and the bigravity signal .— The reasons Region II is of particular interestfor us are manifold: besides being responsible for an intriguing low- (cid:96) plateau in massive gravity, thisis an area in ( q, m eff )-space whose contribution to multipole coefficients can be treated as “additive”to a good approximation in the case of bigravity.As mentioned above, where we part ways with the work in [36] is in considering two sets H + / − of tensor modes. As a consequence, our source term | ψ | will consist also of cross terms. Crucially, ina low- q and outside-the-horizon range such as Region II, the contribution of the massive tensor modesfar surpasses (assuming the contributions are equally weighted) that of its massless counterpart. Asa matter of principle then, not just massive gravity, but also bigravity can lead to distinct imprintsin the B-modes spectrum for the appropriate m eff range.In bigravity though, the nature of the signal will also depend on the value of the relativecoefficient between the massive and massless tensor modes coupling to matter via e.g. Eq.(3.4).Schematically: Signal ∼ (cid:104) ( H +source ) | m =0 + C rel ( H − source ) | m eff (cid:105) . (4.2)The first thing to keep in mind is that the unitarity bound immediately sets the dynamics inthe m eff (cid:38) H r region. A C rel (cid:29) (cid:96) enhancement at the specified m eff range and a suppression inthe asymptotics of a very large m eff . For C rel ∼ C rel (cid:38) /
10 onewould still be able to see the low multiple enhancement for the appropriate m eff ; this is because themaximum signal enhancement due to a mass (when m eff ∼ × H ) is two orders of magnitudelarger than the would-be massless signal. As for the m eff > H r range in this configuration somesuppression is to be expected but, again, this regime is best understood by running the software .Finally, the C rel (cid:28) /
100 configuration is bound to generate a signal with almost complete overlapwith the GR profile.In general then, the difference between a bigravity theory and its massive gravity limit wouldmost clearly manifest itself as a lesser enhancement in the low- (cid:96) plateau for the former and cor-respondingly a less dramatic suppression in the large m eff region of the bigravity parameter space.From the perspective of CMBR signatures one may summarize these findings as evidence that bigrav-ity theories generate imprints which are overall less sharp than those originating from pure massive We anticipate however that in the m eff (cid:29) H r regime the source function | ψ | might well receive an importantcontribution from what we call the -cross terms- of the two tensor sectors and therefore one may not rely on themassless/massive modes additivity any longer. On the other hand, it is important to stress that a more detailedanalysis of the dynamics for which cross terms play a leading role (a purely bigravity effect this one) is bound to leadto an interesting characterization of further signatures of bigravity theories. We leave this to future work. – 15 –ravity. This is justified already at an intuitive level because the bigravity theories space spans cor-ners of pure massive gravity (e.g. the M f → ∞ limit) but is also endowed with two additional degreesof freedom which account for the dynamics of the additional massless tensor modes. In principle,depending on the relative coefficient C rel of the H + / − modes contribution one might make bigravityimprints as sharp as those for massive gravity; on the other hand, those signatures can be “watereddown” towards the purely massless spectrum by a different judicious use of the same coefficient.Zooming in the LEL regime, one can see that C rel ∼ O (1) · κ (cid:46) − thus concluding that theimprints of bigravity in the LEL are very hard to probe and the CMBR signal is expected to mimicGR. What led to our restricted parameter space were, in addition to our working in a low-energyregime, the requirement of an almost ΛCDM background evolution combined with the constraintson the effective Planck mass. In turn, this resulted in a very small relative weight for the massivetensor modes at recombination.Our setup should by no means be thought of as the only cosmologically viable option. Muchmore work is needed in this direction. It is indeed quite possible that another stable region maybe found in the future whose domain includes a massive tensor sector generating more prominentsignature in the CMB. The observed accelerated expansion of the universe has reignited the research aimed at findingcompelling theories modifying GR in the IR. Theories of massive gravity are a sure candidate for thepart. Their phenomenology is currently under intense scrutiny. Most studies so far have focused onthe late-time cosmological dynamics in these theories. In this paper we took a different approach andasked how early-time cosmology dynamics and constraints would reflect on the bigravity parameterspace.Our starting point has been a theory of bigravity, where both metrics are taken to be FLRWand matter only couples to one of them. We studied the predictions for the tensor-to-scalar ratio, r , in bigravity in a specific low-energy limit. In this regime, the two copies of tensor modes, whichare generally coupled, can be diagonalized in a time-independent fashion and solved for. It is thenpossible to write r as a linear combination of contributions from the massless and massive modes. Wefind that in this regime, once additional constraints are imposed, the contribution from the massivetensor modes is suppressed.We further showed that, although the massive sector of the theory can in general leave quite adistinctive imprint on the B-modes profile, the region in the parameter space we have been probingsupports a suppression of the massive tensor modes contribution to the overall signal in favour of aGR-like profile.We are led to conclude that a very efficient Vainshtein screening under specific conditionsleads to a hard-to-detect, as far as CMB data is concerned, bigravity imprint. One could say thatthe massive tensor sector would bypass, rather than pass, the cosmic background radiation test bymeans of a small κ . A posteriori this is not surprising: the assumed strong Vainshtein screens thenon-tensorial d.o.f.’s and κ (cid:28) H + ij , the massless modes, to h ij and suppresses H − ij . This is morally very close to GR. On the other hand, the κ (cid:28) Although, as pointed out in [40], an in depth study of the initial data problem is needed in order to decide whatreally is tuned and what is not. – 16 –tself more at solar systems scales where an analysis of the screening mechanism is more accessiblevia the decoupling limit.Further investigation is required to ascertain the degree to which an active Vainshtein canscreen at the decoupling limit and, possibly, away from it. The latter task is especially interestingand important for the analysis presented here, but also especially complicated. Indeed, in our setupa small M f is expected and it corresponds to a small naive strong coupling scale Λ ,f (cid:28) Λ . Weleave this to future work. Relation to recent works .—The phenomenology of bigravity theories has seen increased recentinterest. Some works have some overlap with our own. Comelli et al. [71], Koennig et al. [80],Lagos et al [40], Cusin et al. [78] and Amendola et al [41] employed linear perturbation theory tounderstand what pathologies may arise in each sector of bigravity. These works vary depending onthe branch and α n region they focus on but overall they span several branches of solutions includingthe one employed here. A common conclusion is that further investigation is needed on the initialvalue problem, as already stressed in [40]. A step in this direction is represented by the work in [42]by Johnson et al. In particular, [42] identified a background which generates GR-like results undera large pool of initial conditions during inflation.The low-energy regime (LEL), employed throughout this manuscript was introduced in de Feliceet al. [56]. To the parameter space therein we superimposed additional constraints (e.g. κ (cid:46) − )that lead to a reduced space where the massive tensor sector is suppressed. In this sense then, wenaturally make contact with the work by Akrami et al. in [83].Our working in a restricted LEL regime means the scale at which one imagines initial conditionsare set is far beyond the reach of our approximation and above our setup strong coupling scale sothat our contact with the results in [42] is an indirect one.While this work was nearing completion, a preprint by Sakakihara et al [84] appeared in whichthe tensor power spectrum was derived in a specific bigravity model, corresponding, in the notationof Eq. (2.1), to α n (cid:54) =2 = 0. The Authors considered the so-called healthy branch of solutions and, inas much as there is overlap with our work, they reached conclusions not unlike ours.A number of earlier works also focused on massive gravity signatures in the CMBR. Ref. [36], aswe have seen, pointed out the existence of a B-modes enhancement in the low multipoles for a specificmass range. The study in Ref. [28] concerned a general analysis of the profile and detectability ofthe gravitational wave signal arising from a time-dependent mass term in massive gravity. Acknowledgements
It is a pleasure to thank Claudia de Rham and Andrew J. Tolley for collaboration at early stages ofthis work and for illuminating conversations. We are grateful to Matthew Johnson, Adam R. Solomonand Alexandra Terrana for fruitful discussions. We also thank C. de Rham, M. Johnson andA R. Solomon for very useful comments on a draft version of this paper. The work of MF wassupported in part by grants DE-SC0010600 and NSF PHY-1068380. RHR acknowledges the hospi-tality of DAMTP at the University of Cambridge and the Perimeter Institute of Theoretical Physics.RHR’s research was supported by a Department of Energy grant de-sc0009946, the Science andTechnology Facilities Council grant ST/J001546/1 and in part by Perimeter Institute for Theoret-ical Physics. Research at Perimeter Institute is supported by the Government of Canada throughIndustry Canada and by the Province of Ontario through the Ministry of Economic Development &Innovation. – 17 – eferences [1] M. Fierz and W. Pauli,
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