Possible Signatures of Inflationary Particle Content: Spin-2 Fields
PPossible Signatures of InflationaryParticle Content: Spin-2 Fields
Matteo Biagetti a , Emanuela Dimastrogiovanni b , Matteo Fasiello c a Institute of Physics, Universiteit van Amsterdam, Science Park, 1098XH Amster-dam, The Netherlands b CERCA & Department of Physics, Case Western Reserve University, Cleveland,OH 44106, USA c Stanford Institute for Theoretical Physics and Department of Physics, StanfordUniversity, Stanford, CA 94306
Abstract.
We study the imprints of a massive spin-2 field on inflationary observables,and in particular on the breaking of consistency relations. In this setup, the minimalinflationary field content interacts with the massive spin-2 field through dRGT inter-actions, thus guaranteeing the absence of Boulware-Deser ghostly degrees of freedom.The unitarity requirement on spinning particles, known as Higuchi bound, plays acrucial role for the size of the observable signal. a r X i v : . [ a s t r o - ph . C O ] A ug ontents The main reason for investigating non-minimal particle content during inflation is theenergy scale at which inflationary dynamics may take place. With high-scale inflationset at about 10 GeV, there is no other comparable window on such high energyprocesses. The characterization of the signatures corresponding to different particlespecies is indispensable for a precise observation-to-theory mapping in case of detection.Cosmological probes can indeed inform our model building and even UV completionefforts. This line of reasoning is at the heart of recent literature on the subject [1–5].Additional degrees of freedom (dof) during inflation may take many shapes (mass,spin). Perhaps the most immediate indication of their impact on observables stemsfrom the value of their mass as compared to the Hubble scale H . The latter is indeeda crucial, threshold, value. The reasons are manifold. First of all, the energy scaleof inflation is itself generally of order H ; it naturally follows that too large a mass, m (cid:29) H , would justify the integrating out of any dof leaving very little in terms ofimprints (although see e.g. [6–8] for important caveats). A mass of order Hubblecan instead support distinct signatures and has been the subject of many studies,especially in the scalar case [3, 9–15]. This range of values for the mass is also naturalfrom the point of view of supersymmetric theories: if such symmetry is not broken at a– 1 –igher scale, the inflationary vacuum energy will eventually break it, thereby justifyingthe expectation that some of the fields populating the supersymmetric multiplets willhave masses of order Hubble. The behaviour of inflationary correlation functions alsodictates that particles whose Compton wavelength is much smaller than the horizon have negligible effects on late-time observables, identifying again 1 /H as a lower limit.Finally, for particles of spin s = 2 or higher, unitarity imposes lower bounds on themass in de Sitter m > s ( s − H , (1.1)which further underscores the frontier nature of the m ∼ H mass range. The unitarityor Higuchi [16–18] constraint is less stringent away from de Sitter [19, 20] but, as weshall see, it remains a powerful bound.The most efficient way to probe both the field content and inner structure of theinflationary Lagrangian is to consider N-point correlation functions, non-Gaussianities( N ≥ N + 1-point function and itsN-point counterpart.CRs stem from residual gauge diffeomorphisms of the (inflationary) theory [22–26]. One may derive non-trivial CRs when the soft mode characterizing any squeezedlimit transforms non-linearly under the residual diff. In the minimal inflationary setupthe curvature perturbation ζ and tensor mode γ transform non linearly respectivelyunder dilatation and anisotropic rescaling, thereby generating the well-known scalarand tensor CRs. In each of these two cases, the leading effect of a long scalar (tensor)mode on N short modes corresponds to the action of a residual gauge-symmetry onthe N -point function and can therefore be gauged away.In this work we focus on the case of a three-point function with a long tensor modeand two scalar short modes. In this case, the CR in single-field inflation readslim (cid:126)k → (cid:68) γ λ(cid:126)k ζ (cid:126)k ζ (cid:126)k (cid:69) (cid:48) (cid:39) (cid:68) γ λ(cid:126)k γ λ − (cid:126)k (cid:69) (cid:48) (cid:15) λij ˆ k i ˆ k j (cid:68) ζ (cid:126)k ζ (cid:126)k (cid:69) (cid:48) , (1.2)where γ and ζ are the tensor and scalar perturbations respectively, we defined thepolarization tensors (cid:15) λij as γ ij = (cid:80) λ (cid:15) λij γ λ and the prime indicates that we left implicitthe momentum-conserving delta.Whenever additional fields appear on the scene, CRs can be broken if, for ex- The equations of motion, for example in quasi-de Sitter, show that as soon as m ∼ H the massof a generic particle becomes relevant to its dynamics and leads to decaying solutions. For our purposes it suffice to have non-standard consistency relations in the sense that the varia-tion of the power spectrum is not related to the squeezed limit of one three-point function but ratherto a weighted linear combinations of at least two such observables. – 2 –mple, there exists one more independent mode that transforms non-linearly underthe diffeomorphism. It follows that the squeezed bispectrum may no longer be gaugedaway and an enhanced signal is made possible in this configuration. The power of CRsbreaking as a probe of new physics is not limited to inflationary setups and has beenwidely employed in other contexts such as large scale structure [27–36]. One shouldfurther note that the squeezed one is the best suited of configurations to reconstructthe inflationary potential as it is less sensitive to possible variations of the statistics ofthe fluctuations in sub-volumes [37].In this manuscript we will report on work where the inflationary dynamics is en-riched by a massive spin-2 field non-minimally coupled to the usual massless gravitonvia ghost-free interactions [38–40]. As a result, the standard tensor CR is modified. Wequantify the consequences of this “breaking” on the squeezed limit of the tensor-scalar-scalar bispectrum signal and show how the constraints originating from a (softened)Higuchi bound propagate all the way to observables. Despite a weakened unitarityconstraint away from dS, we find that the amplitude of the signal produced in thissetup is too weak for detection by upcoming large scale structure surveys.This work is organized as follows. In
Section 2 we introduce the model understudy comprising the inflaton field minimally coupled to a spin-2 field interacting withanother (massive) spin-2 field through dRGT-interactions. We discuss at length thefeatures of the model that will be most relevant in the inflationary context.
Section3 is devoted to outlining how consistency relations play out in our specific setup andcalculating the resulting tensor-scalar-scalar bispectrum. In
Section 4 we discuss theobservational consequences of these results. We conclude with
Section 5 , where wecomment on our findings and suggest possible directions for further work on the subject.In
Appendix A we provide a general overview of CRs and their breaking. In
AppendixB we detail on a route to modified CRs which is alternative, up to horizon scales, tothe one in
Section 3 , and include a diagrammatic proof of CRs breaking.
It is well-known that additional degrees of freedom populating the inflationary La-grangian have the potential to modify CRs in favour of a enhanced squeezed bispectrumsignal (see
Appendix A for a general discussion). There already exist several studies onthe subject in the literature. We focus our analysis here on the case where the minimalinflationary particle content is enriched by a spin-2 field. Our discussion ought to startfrom the realization due to [54] that a theory of interacting spin-2 particles admitsat most one massless spin-2 field. As we are working from the perspective of addingparticle content to the minimal (GR + scalar field) inflationary scenario, we are thenimmediately led to introducing a new, massive, graviton.The theory of massive gravity has received enormous attention in recent years.This is due to the combination of experimental results [55–57] confirming the current Note that there is a further implicit restriction: even indirect, e.g. mediated by other fields,interactions are forbidden. – 3 –cceleration of the universe and the recent non-linear formulation [38, 39] of a ghost-free massive gravity theory known as dRGT. An infrared modification of gravity isindeed among the most studied mechanisms, along with dark energy, to explain latetime acceleration. The interest for a theory of massive gravity is of course muchwider than pertains its use for late-time cosmology. Within string theory for example,open strings have spin-2 excitations whose lowest energy state is massive at tree level.The formulation of [39] in particular, has found applications as varied as e.g. its useas a framework for translational symmetry breaking and dissipation of momentumin holography [58, 59]. Our interest is focussed on the inflationary context: here aconsistent massive spin-2 field next to GR and a scalar inflaton field takes the form ofa theory known as bigravity. This is an extension of dRGT theory that contains thesame ghost-free structure. In bigravity each of the two metrics, g and f , has its ownEinstein-Hilbert term and they interact via the dRGT potential. The action reads S = (cid:90) d x (cid:34) M P √− g R [ g ] + √− g P g ( X, ϕ ) + 2 √− g m M V + M f (cid:112) − f R [ f ] (cid:35) , (2.1)where a few details are in order: • The interaction potential V is defined as V = (cid:88) n =0 β n E n ( (cid:112) g − f ) , (2.2)where the β n are free parameters and the polynomials E n ( X ) take the form E ( X ) = 1 , E ( X ) = Tr( X ) ≡ [ X ] , E ( X ) = 12 (cid:0) [ X ] − [ X ] (cid:1) , E ( X ) = 13! (cid:0) [ X ] − X ][ X ] + 2[ X ] (cid:1) , E ( X ) = 14! (cid:0) [ X ] − X ][ X ] + 8[ X ][ X ] + 3[ X ] − X ] (cid:1) . (2.3) • Due to the properties of the E n polynomials, if it were not for the coupling tomatter (here restricted to the metric “g”) the action would be symmetric underthe exchange g ↔ f M g ≡ M P ↔ M f , β n ↔ β − n . • The mass M has already been symmetrized via M = M P M f M P + M f . (2.4)and we have introduced κ ≡ M f M P . • As a consequence of the above points, it is not strictly true that g is the masslessspin-2 field and f is the massive one. In fact, the mass eigenstate of the theoryare in general time-dependent. We can on the other hand work in a low energyconfiguration [62] where this is approximately true.– 4 – The P g ( X, ϕ ) stands for a generic matter Lagrangian and X = − g µν ∂ µ ϕ∂ ν ϕ .We will assume the potential within P g ( X, ϕ ) is driving the background dynamicsof the inflaton field ϕ , minimally coupled to g only. We refer the reader to Section2.3 for the more general case.The theory in Eq. (2.1) has 8 propagating degrees of freedom: 5 (2T+2V+1S) froma healthy spin-2, 2 more from a massless spin-2 (2T) and a scalar. In what followswe will focus our attention on the 2 × T degrees of freedom plus one scalar. Letus elaborate on the reasoning behind this choice. Non-linear massive (bi)gravity isendowed with a very efficient screening mechanism known as Vainshtein screening[60, 61]. In specific configurations the coupling to matter of the helicity-0 and helicity-1 modes is highly suppressed. This is, for example, what allows massive gravity to passa host of cosmological tests in setups (e.g. solar system scales) where observations arein exquisite agreement with general relativity: the additional degrees of freedom arescreened. We are working in a rather different, inflationary, background here; we willnevertheless assume that the non-linearities in the helicity-0 and helicity-1 part ofthe theory will generate sufficient screening so that we can focus the analysis on theremaining dofs.There exist in the literature several important studies [63–67] on the perturbationtheory of all the degrees of freedom (including matter) populating bigravity during thevarious cosmological epochs. Requiring that an efficient screening is in place in aninflationary background is tantamount to requiring that non-linearities in the theory(these studies usually stop at quadratic order in perturbation theory) will downplayany effect due to the vector and helicity-0 modes. In support of this assumption is alsothe intuition that Vainshtein screening is typically more efficient in homogeneous andisotropic backgrounds. On the other hand, we stress here that studies of perturbationslimited to quadratic order already indicate [66] that all bigravity degrees of freedomare well-behaved during inflation.The Vainshtein mechanism relies on the fact that whenever non-linearities areimportant the canonical normalization of e.g. the helicity-zero mode is redressed andso will be also the coupling with matter. The helicity-0 (usually indicated by π )coupling to matter is dictated by the structure in Eq. (2.1) and includes the term π T µµ . Whenever non-linearities in π modify the normalization of the kinetic term, thecorresponding coupling to matter will be π c T µµ = 1 Z ( π , β n ) π T µµ , (2.5)where Z is a function of background quantities and the non-linearities are weightedby the β n s. As soon as π self-interactions are important, Z (cid:29) f dynamical. Even though f does not directlycouple to matter, it does so via g and so the coupling does not undergo the same kindof suppression as the helicity-0/1 modes. On this basis we can go on and prioritize Also terms such as ∂ µ π∂ ν π T µν are allowed, see [68]. – 5 –he observational effects of the tensor degrees of freedom contained in f . What mustnot be omitted in our analysis are the consequences that purely formal consistencycriteria on the helicity-0 and vector sector have on the observables made up by theremaining dofs. The most important case in point is the helicity-0 unitarity bound,which we address next. The reader familiar with the matter and more interested inthe calculation for the bispectrum signal may want to skip the content of the rest of Section 2 and go directly to
Section 3 . The unitarity bound on the helicity-0 mode corresponds to the condition on the co-efficient of the kinetic term in the quadratic theory to be positive . This stems fromrequiring that the Hamiltonian is bounded from below so that the dynamical systemis stable. Naturally, the latter also implies that the coefficients of the mass and spatialgradient terms, if present, need also be positive. It turns out that the helicity-zeroHiguchi bound is the most restrictive on the parameter space of bigravity theory, sothat it is justified to study it more in detail. It is well-known that unitary represen-tations of the de Sitter group for spin-2 fields are massless, partially massless, andmassive, with m = 0 (GR) , m = 2 H (partially massless) , m > H (massive) . (2.6)It has been shown [20] that the existence of helicity-1 and helicity-0 interactions inthe partially massless case implies the absence of the intriguing conjectured partiallymassless symmetry for both massive gravity and bigravity, at least as long as thekinetic term is of the standard form. In the massive representation, our stepping awayfrom de Sitter and, most importantly, our dealing with (i) a fully non-linear theory of(ii) bigravity rather than massive gravity, relaxes [20] to some extent the bound on thenow dressed mass ˜ m : ˜ m (cid:32) M P H f M f H (cid:33) ≥ H (2.7)where we have defined ˜ m as˜ m ≡ m HH f M M P (cid:32) β + 2 β HH f + β H H f (cid:33) , (2.8)and the quantities H and H f are the Hubble scales associated to the FLRW solutionfor the metrics g and f respectively . These notions have immediate consequences alsofor the tensor sector and, as we shall see, for the observability of the (cid:104) γζζ (cid:105) bispectrumin the squeezed limit. In fact, we anticipate here (see Eqs.(3.5)-(3.10) in the text) See [69] for a derivation of the Higuchi bound from the perspective of the AdS/CFT correspon-dence and in particular as a consequence of reflection positivity in radially quantized CFT . Note that the absence of β , β is due to the fact that these quantities are typically fixed by thetadpole cancellation requirement. – 6 –hat the mass in the (suitably diagonalized) tensor equations of motion in the low k limit is the very same LHS of Eq. (2.7) [18]. This implies that the helicity-0 modeunitarity bound forces an effective mass on the tensor sector, and in particular one ofat least order Hubble: the corresponding tensor wavefunction will then inevitably havea component that decays outside the horizon, to the detriment of the observed signal.As for possible tachyonic and gradient instabilities, we will work in a regime where anysuch possibility is above the reach of the effective theory: the “low energy” regime of[62]. For the sake of providing a more complete picture, we find it necessary to brieflyexpound on the strong coupling scale (scs) of the bigravity theory at hand. Naturally,it is the presence of the g − f interaction term that dictates the lowest scales in thetheory. In particular, depending on the hierarchy between the two masses M g and M f ,the lowest “naive” strong coupling scale (in the helicity-0 sector) can be as low as:Λ g,f = (cid:0) m M g,f (cid:1) / . (2.9)A few comments are in order.Taking at face value the above quantities may lead to some worries about a scs per-ilously close to the energy we are working at. Indeed, although the Higuchi boundforces the dressed mass to be larger than the Hubble scale H , one typically satisfiesthis condition by requiring H f /M f (cid:29) H/M g , and allowing m (cid:46) H . This will be alsothe case for our parameter space of choice as keeping the ratio M f /M g · H /H f smallwill allow approximately constant mass eigenstates in the tensor sector [62]. However,the scs is naive in the sense that the background via the Vainshtein mechanism changesthe canonical scale and so the scale at which the fluctuations are strongly coupled. In-stead of the naive scale Λ, one should rather think of a larger quantity as the strongcoupling scale, Λ ∗ = √ Z Λ, where Z is the same quantity discussed below equation(2.5). The efficient screening we require goes then in the same direction of a largerstrong coupling scale. In writing Eq. (2.1) we have assumed the existence of a unique matter sector that isminimally coupled to the metric g only; it is in fact this coupling (together with thehierarchy between M g and M f ) that breaks the otherwise symmetric role played bythe two metrics in our setup. One might ask: is this the only possibility? Would anyother coupling lead to a ghostly (classical or quantum) degree of freedom? How aboutcoupling different matter sectors to different metrics? Naturally, coupling matter justto the f sector is just as acceptable but the g ↔ f exchange would be mirrored by aswapping of the observables we are going to calculate, and deliver no detectable advan-tage. Coupling different matter sectors to different metric fields is also not immediatelyhelpful because communication between different matter sectors will be mediated and– 7 –herefore suppressed. On the other hand, what might prove to be more intriguing forus is the coupling via the effective composite metric [70]: g eff µν = α g µν + 2 α β g µα (cid:16)(cid:112) g − f (cid:17) αν + β f µν , (2.10)where α and β are two arbitrary real dimensionless parameters. In this case the matterLagrangian in Eq. (2.1) would be replaced, in a simple example, by: L m = √− g eff (cid:104) g µν eff ∂ µ ϕ∂ ν ϕ + V ( ϕ ) (cid:105) . (2.11)This specific form for the coupling preserves the ghost-free nature of the theory bothat the classical and quantum level [70]. From the EFT perspective any ghost that isabove the cutoff scale of the theory is completely harmless and it is in this sense thatone can freely use the coupling via composite metric in Eq. (2.10). Equipped with sufficient information on our bigravity model, we now go on to tackle theobservational signal generated by the tensor-scalar-scalar three-point function (cid:104) γ g ζζ (cid:105) ,where γ g is the g -metric tensor fluctuation and ζ is a repository of scalar fluctuations.As mentioned in Section 1 and detailed in
Appendix A , non standard CRs are possiblewhenever additional field content has non-linear transformation under a gauge diff.The extra field content of choice here includes the f -metric new tensor dof. In orderto most clearly show how consistency relations are modified in our setup we providein this manuscript two different routes to non standard CRs. We present the firstone below, in Section 3.2 , and leave the second one to
Appendix B . The latter relieson an approximation that does not hold outside the horizon and cannot therefore betrusted all the way to late-time observables. Nevertheless, we find the presentation in
Appendix B very instructive and choose to include it in the manuscript. In
Section 3.2 we will solve the full coupled system of equations for the traceless transverse part of h f , h g . We will then calculate the tree level (cid:104) γ g ζζ (cid:105) diagram and show that its late timelimit is different than that of the standard single-field model as it contains a term thatdepends on the mass m . The late-time power spectra P γ g and P ζ cannot capture thismass dependence at tree level, signaling that CRs are indeed modified in our setup. Following for example [18, 20, 62], it is straightforward to derive background equationsand study the quadratic perturbation theory around FLRW solutions for the metric f and g . We denote¯ g µν d x µ d x ν = − N d t + a δ ij d x i d x j , ¯ f µν d x µ d x ν = − ˜ N d t + b δ ij d x i d x j . (3.1) This is in contradistinction with the procedure outlined in
Appendix B where an approximationof the type δ L (2) (cid:28) L (2)0 on parts of the quadratic Lagrangian is necessary. We do not include P γ f here because γ f does not couple directly with matter. However, theconclusion would not change as at tree level and in the late time limit P γ f too is unaware of the mass m . – 8 –ere N and ˜ N are background lapse functions and a and b are the scale factors for thecorresponding metrics g and f . The Friedmann equations read:3 M P H = ρ ( ϕ ) + 3 m M (cid:88) n =0 β n (3 − n )! n ! (cid:18) ba (cid:19) n , (3.2)3 M f H f = 3 m M (cid:88) n =0 β n +1 (3 − n )! n ! (cid:18) ba (cid:19) n − , (3.3)where ρ ( ϕ ) in Eq. (3.2) and in particular the potential V ( ϕ ) in it, is the leading term,driving inflation, on the RHS. We notice in passing that we will be working in theso-called healthy branch (see e.g. [62]) of bigravity solutions in order to avoid strongcoupling issues. The healthy branch is defined by the condition H/H f = b/a ≡ ξ . Upon using g µν = ¯ g µν + a h g µν , f µν = ¯ f µν + b h f µν , (3.4)one obtains from the quadratic action the equations of motion for h f , h g . Let us start with the equations of motion for γ T Tg and γ T Tf , the traceless transversecomponents of, respectively, h g and h f . Tensor perturbations of a given k-mode areseparable into two independent helicity modes with the same mode-function: γ g , γ f .Combining Eq. (2.1) and (3.4) one obtains[18]: γ (cid:48)(cid:48) g − τ γ (cid:48) g + k γ g + M H τ (cid:32) r fg r fg (cid:33) ( γ g − γ f ) = 0 , (3.5) γ (cid:48)(cid:48) f − τ γ (cid:48) f + k γ f + M H τ (cid:0) r fg (cid:1) ( γ f − γ g ) = 0 , (3.6)where M = ˜ m (cid:2) /r fg (cid:3) and we introduced the parameter r fg ≡ M f M P HH f . (3.7)In writing Eqs. (3.5)-(3.6) we have also assumed a slowly varying ξ = b/a . It turnsout to be convenient at this stage to change basis, from γ g,f to γ + , − , where To conform with standard notation we have defined the ratio of the two scale factors as ξ ≡ b/a .This is also the symbol generally used for gauge transformations. Which one is meant should be clearfrom the context. As anticipated, henceforth we shall focus on the tensor sectors perturbations. For a systematicstudy of quadratic perturbations in bigravity, we refer the reader to, for example, the work in [18]. – 9 – ± = γ g ± r fg γ f , (3.8)so that the equations of motion now read γ (cid:48)(cid:48) + − τ γ (cid:48) + + k γ + = 0 , (3.9) γ (cid:48)(cid:48) − − τ γ (cid:48) − + (cid:18) k + M H τ (cid:19) γ − − M H τ (cid:32) − r fg r fg (cid:33) γ + = 0 . (3.10)The solutions, once written back in the γ g , γ f basis are γ g = (cid:32) r fg r fg (cid:33) HM P √ i − kτ ) k / e − ikτ (3.11)+ (cid:32) − r fg r fg (cid:33) r fg HM P e iνπ (1 + i ) (cid:114) π − τ ) / H (1) ν ( − kτ ) ,γ f = (cid:32) r fg r fg (cid:33) HM P √ i − kτ ) k / e − ikτ (3.12)+ (cid:32) − r fg r fg (cid:33) r fg HM P e iνπ (1 + i ) (cid:114) π − τ ) / H (1) ν ( − kτ ) , where we have defined ν ≡ (cid:112) / − M /H .Note that the solution for the mode-functions in Eqs. (3.11)-(3.12) were found by solv-ing Eqs.(3.9)-(3.10) first and requiring that the mode functions asymptote to Bunch-Davies vacua deep inside the horizon: γ g ( k, τ ) → | kτ |(cid:29) a M P e − i k τ √ k , γ f ( k, τ ) → | kτ |(cid:29) b M f e − i k τ √ k . (3.13)It is worth mentioning the special case r fg = 1. In this particular configuration, de-spite the presence of the mass term in the equations of motion, the two solutions for γ g , γ f are independent and identical to two copies of the usual massless solution for γ g in standard single-field inflation. One concludes that in this case, at the level of thequadratic action for the traceless transverse tensor components, the theory behaves astwo copies of GR and one has to go to cubic order to start probing the massive gravityinteractions. Equipped with the solution to the quadratic theory, we can now consider the effect ofthe m driven g − f coupling on standard observables such as the tensor-scalar-scalarbispectrum. We shall be considering the contribution due to the γ ijg ∂ i ζ∂ j ζ interaction.– 10 –ote that we define the gauge-invariant comoving curvature perturbation ζ in the stan-dard way, ζ = ψ g + H δρ/ ¯ ρ (cid:48) , where ψ g is the gravitational potential identified with thediagonal part of the spatial components of the metric g (see also [63, 66]). The impor-tant question is if this quantity is conserved on scales larger than the Hubble radius.Since it is only the metric g that interacts with matter, in light of the conservationequation and of our assumption that the helicity-0 and helicity-1 modes are screened,one immediately finds that ζ (cid:48) = ψ (cid:48) g = 0 outside the horizon, and we conclude thatone can adopt for ζ the usual expression. This is consistent with e.g. the analysis in[66]. Since the γ ijg ∂ i ζ∂ j ζ interaction is already present in single-field inflation, we canalready anticipate that the imprint of the massive gravity potential is all stored in thewavefunctions and, in particular, their M -dependent parts. The formal expression forthe bispectrum in the in-in formalism is (cid:68) γ λg ( t, (cid:126)k ) ζ ( t, (cid:126)k ) ζ ( t, (cid:126)k ) (cid:69) = − i (cid:90) tt d t (cid:48) (cid:68) (cid:104) γ λg ( t, (cid:126)k ) ζ ( t, (cid:126)k ) ζ ( t, (cid:126)k ) , H int ( t (cid:48) ) (cid:105) (cid:69) (3.14)where H int ( t (cid:48) ) = −L int ( t (cid:48) ) = − M P (cid:15) (cid:90) d x a ( t (cid:48) ) γ g,ij ( (cid:126)x, t (cid:48) ) ∂ i ζ ( (cid:126)x, t (cid:48) ) ∂ j ζ ( (cid:126)x, t (cid:48) ) . (3.15)It is convenient to split the result of the bispectrum calculation in two pieces. One,which we indicate as B m , will be due to the “massless” component of γ g in theintegral of Eq. (3.15); the other one, corresponding to the remaining component of thewavefunction, will be indicated by B m . One obtains: (cid:68) γ λg ( t, (cid:126)k ) ζ ( t, (cid:126)k ) ζ ( t, (cid:126)k ) (cid:69) = (2 π ) δ (3) ( (cid:126)k + (cid:126)k + (cid:126)k ) e λij (ˆ k ) (cid:16) − ˆ k i ˆ k j − ˆ k i ˆ k j (cid:17) [ B m + B m ] . (3.16)It is straightforward (and identical to the single-field case) to derive the quantity in B m . The integral necessary to obtain an explicit expression for B m can be performedanalytically by fixing each time the value of the mass M . Let us fix it to M = √ H ,a value that saturates the Higuchi bound. The explicit expression for the bispectrumnow reads: B m ( k , k , k ) = H (cid:15) M P (cid:32) r fg r fg (cid:33) I m ( k , k , k ) , (3.17) B m ( k , k , k ) = H (cid:15) M P − r fg (cid:0) r fg (cid:1) r fg I m ( k , k , k ) , (3.18) The massless component of γ g is the first term in Eq. (3.11), which is unaware of ν = ν ( M ) andis therefore termed massless. – 11 –here I m ( k , k , k ) ≡ (cid:80) i k i + 2 (cid:80) i (cid:54) = j k i k j + 2Π i k i k k k k t , (3.19) I m ( k , k , k ) ≡ i k i (cid:18) k ( k + k ) + 3 k k + k + k k t − γ − ln [ − k t τ ] (cid:19) , (3.20)with k t ≡ k + k + k and where γ is the Euler’s constant. The logarithm in the lastexpression is handled in the standard fashion [86–88]. The presence of a log term inthe bispectrum signals a non-trivial interaction between the hard ζ modes and the softgraviton outside the Hubble radius and accounts for the fact that the massive gravitonmode, despite its decay on large scales, sources the massless graviton perturbationfrom the moment it leaves the Hubble radius till the end of inflation. Such an effect isnot found in the standard single-field scenario.We can now take the squeezed limit ( k (cid:28) k (cid:39) k ) of the expressions in (3.17)-(3.18),to get B m ( k , k , k ) k (cid:28) k (cid:39) k −−−−→ P λγ g ( k ) P ζ ( k ) , (3.21) B m ( k , k , k ) k (cid:28) k (cid:39) k −−−−→ P λγ g ( k ) P ζ ( k ) k k (cid:18) − r fg r fg (cid:19) r fg (5 − γ − − k τ ]) , (3.22)where we have defined the tensor and scalar power spectra at late times P λγ g ( k ) = lim τ → (cid:2) γ g ( k , τ ) γ ∗ g ( k , τ ) (cid:3) (cid:39) (cid:32) r fg r fg (cid:33) H M P k , (3.23) P ζ ( k ) = lim τ → [ ζ ( k , τ ) ζ ∗ ( k , τ )] (cid:39) H (cid:15) M P k (3.24)A tensor-scalar-scalar bispectrum that satisfies the consistency condition would, in thesqueezed limit, behave according to (cid:104) γ λg ( (cid:126)k ) ζ ( (cid:126)k ) ζ ( (cid:126)k ) k (cid:28) k (cid:39) k −−−−→ − (2 π ) δ (3) ( (cid:126)k + (cid:126)k + (cid:126)k ) 12 e λij (ˆ k )ˆ k i ˆ k j P λγ g ( k ) P ζ ( k ) ∂ ln P ζ ( k ) ∂ ln k . (3.25)As expected, the contribution from B m in (3.21) does indeed obey such rule. Thesqueezed limit contribution from B m in (3.22), however, does not: its expression de-pends on M whilst both the tree level scalar and tensor power spectrum do not. The M dependence in Eq. (3.22) is implicit in that the exponent of the very last term, k /k , can be thought of as proportional to 3 / − (cid:112) / − M /H . We stress that this Note that the second term in the mode-function for γ g , Eq. (3.11), generates terms scaling aspositive powers of kτ for kτ →
0, therefore they do not contribute to the power spectrum at latetimes. – 12 –s specific to the case M = √ H and in contradistinction to the case where a similar(except for an angular dependence) suppression arises from a different diffeomorphismaltogether in standard single-field inflation where CRs are conserved [89]. Note alsothat we have not included the contribution from γ f to the late time tensor powerspectrum. This is because this contribution is suppressed in light of γ f not couplingdirectly to matter. Nevertheless, our conclusions would not change as the late-timelimit of γ f is unaware of M and therefore cannot cancel off the B m bispectrum con-tribution in the squeezed limit. As we shall see more at length in Section 4 , the smallratio k /k in Eq. (3.22) and the proportionality to r fg play a key role in determiningthe detectability of the observational signal. As we have remarked above, a smaller M would lead to a smaller exponent for k /k and thus a larger signal. We shall nowelaborate on different strategies, within the same model, to enhance the signal. The Higuchi bound we inherit from the helicity-0 mode is strictly the one in Eq. (2.7)only in the low k limit, at or outside the horizon, in inflationary terms. From Ref. [18]we know that the scalar sector of the perturbations develops an instability (see Fig. 2)unless4 / k + 6 k H a H M + k − H − k H (9 a H M + k − H ) + a M − k − H ≥ , (3.26)where H = aH . The above implies that for, k > a H , there is effectively no boundon M . In other words, up to what corresponds approximately to the horizon for thelong mode, M can be much smaller than H so that the corresponding exponent ofthe ratio k /k would be much smaller and the corresponding signal in Eq. (3.22)enhanced. For this to be true we would need a time-dependent M , something notallowed in de Sitter but possible in quasi-dS. The crucial point however is that, uponstudying the k dependence of Eq. (3.26), one can see immediately that the inequalityabruptly requires M > H before we cross the k = a H threshold, with k being thelong mode. This implies that a small and time-dependent M would be in place onlywhen short modes are still well inside their horizon and the long mode is about to exit.It is well-known that in this configuration the contribution from the integral in Eq.(3.15) is in fact suppressed. We conclude then that a suitably adjusted time-dependent M can do little to improve the signal. As stated, the k /k suppression in Eq. (3.22) is due to the non-zero M -dependent termin the tensor wavefunction. Of course, simply considering the wavefunction masslesscomponent leads to the standard CR. The next logical step is to consider the masslesscomponent of the wavefunction and switch on new cubic interactions originating fromthe massive gravity potential E ( (cid:112) g − f ). In light of the fact that these interactionsdiagram(s) will be inserted within the (cid:104) γζζ (cid:105) three-point function, it is clear that wewill be dealing with one-loop diagrams. Loops would typically lead to a ( H/M P ) – 13 – igure 1 . Left: the LHS of Eq. (3.26) as a function of the physical k for M = 0. One cannotice that a k ph ≥ √ H is enough for a positive B H , regardless of the value of M . Right:the LHS of Eq. (3.26) as a function of the physical k for M ∼ √
2; for small k the value of M needs to surpass this threshold in order for the Higuchi bound to be satisfied. suppression but in bigravity we have a little more freedom in the form of the M P /M f hierarchy.To ensure non-standard CRs, we also need to verify that the resulting squeezedbispectrum cannot be expressed as in Eq. (3.25) by considering the correspondingpossible one-loop contributions in P γ or P ζ . The diagrams below should be consideredas indicative of the typical behaviour in our setup. In view of the general result, it is notnecessary to calculate the exact numerical value of the interactions and it shall sufficeto give an estimate. We will also temporarily relax the assumption that matter couplesonly to the metric g , we do so relying on couplings such as those in Eq. (2.10). Weresort here to direct coupling of f to matter because it allows us to construct diagramsinvolving the new spin-2 degrees of freedom, have γ g and two ζ s as external fields, andstay at one loop order. Two-loops would be necessary without direct coupling between f and matter. One possible diagram is shown in Fig. 2. Figure 2 . Diagrammatic representation of a typical 1-loop interaction: – the left vertexshould be thought of as originating from a “new” m M -type interaction; – the right vertexstems from standard-type direct minimal coupling of f with matter. The black (blue) wigglyline represents γ g ( γ f ), the solid ones stand for ζ . The first vertex from the left should be though of as originating from the new m M -proportional interactions whilst the second vertex is “standard” whenever matter is– 14 –oupled directly to f as well. A quick estimate of the amplitude of this contributionreturns B m ∼ H (cid:15) f M P · H M P (3.27)where we have imposed a H ∼ H f , M f (cid:28) M P hierarchy. This result is to becompared with Eq. (3.22). As expected, employing the massless components of thewavefunctions guarantees the absence of the k /k factor. However, the price to payis a Planck suppression squared. One can check that, regardless of the hierarchy, asuppression at least as strong as the typical loop scaling remains. Notice also that theoutcome does not qualitatively change for the special configuration r fg = 1 discussedin Section 3.2 . If a coupling exists between a long-wavelength tensor and two short-wavelength scalarfluctuations, different scalar Fourier modes will appear to be correlated with one an-other. These off-diagonal correlations can be employed to estimate the amplitude ofthe tensor fluctuation. Note that such an observable is only sensitive to sub-horizontensor modes, as it is not possible to resolve scalar modes that differ by an amountsmaller than H − . An optimal unbiased estimator for the tensor modes amplitudebuilt from off-diagonal correlations was proposed in [72, 73], it reads:ˆ A γ = σ γ (cid:88) (cid:126)K,λ (cid:0) P λf ( K ) (cid:1) P λn ( K )) (cid:32) | ˆ γ λ ( (cid:126)K ) | V − P λn ( K ) (cid:33) , (4.1)where V ≡ (2 π/k min ) is the survey volume, P λf ≡ P λγ ( K ) /A γ is a fiducial powerspectrum for the tensor modes, ˆ γ λ ( (cid:126)K ) is the optimal estimator for a Fourier modeamplitude defined asˆ γ λ ( (cid:126)K ) ≡ P λn ( K ) (cid:88) (cid:126)k B λ ( (cid:126)K, (cid:126)k, (cid:126)K − (cid:126)k ) /P γ ( K )2 V P tot ( k ) P tot ( | (cid:126)K − (cid:126)k | ) δ ( (cid:126)k ) δ ( (cid:126)K − (cid:126)k ) . (4.2)In Eq. (4.2) P λn stands for the variance P λn ( K ) ≡ (cid:88) (cid:126)k | B λ ( (cid:126)K, (cid:126)k, (cid:126)K − (cid:126)k ) /P γ ( K ) | V P tot ( k ) P tot ( | (cid:126)K − (cid:126)k | ) − . (4.3)The quantity P tot is the total scalar power spectrum (signal+noise) and B λ is theeffective (i.e. after subtraction of the consistency-conditions-preserving part) tensor-scalar-scalar bispectrum. The variance associated with (4.1) is given by σ − γ = 12 (cid:88) (cid:126)K,λ (cid:2) K P λn ( K ) (cid:3) − , (4.4)– 15 –nd can be used to quantify, for a given survey size and for given model parameters,the smallest detectable tensor-mode amplitude.Using the expression we derived in Eq. (3.22) for the non-trivial contribution to thebispectrum in the squeezed limit, one finds (cid:0) P λn ( K ) (cid:1) − ≈ β π K k max , (4.5)where k max is the maximum wave-number accessed with a given survey and the bis-pectrum amplitude β in the squeezed limit has been defined, from Eq. (3.22), via (cid:107)B m (cid:107) ≡ β P λγ ( K ) P ζ ( k ) Kk , hence β = O (1) × [ r fg | (1 − r fg ) | ] / (1 + r fg ). The variance forthe tensor power spectrum amplitude is then found to be σ − γ = 2 β π √ π (cid:18) k max k min (cid:19) / . (4.6)From Eq. (4.6) it is clear that, for an amplitude A γ close to the current observationalbounds (i.e. 10 − ) to be within reach of upcoming 21-cm surveys, one would need β (cid:38) O (10 ), assuming k max /k min ∼ r fg (cid:28)
1, therefore given the above definition for β we conclude that the amplitude of thetensor-scalar-scalar correlation predicted by this model is too weak for detection byupcoming large scale structure surveys. Coupling f directly to matter as well, viathe composite metric, does allow for r fg (cid:29)
1. However the consistency/stabilityrequirements comprising background equations, the Higuchi bound, and the fact thatit is the matter sector that ought to drive inflation, will also change the definition of β so that a high r fg will not enhance the signal in the way the above expression of β suggests.A non-trivial squeezed-limit tensor-scalar-scalar bispectrum introduces also a localquadrupolar anisotropy in the scalar power spectrum which, unlike the off-diagonalcorrelation, is sensitive to super-horizon tensor modes [74] P ζ ( (cid:126)k ) | γ λ ( (cid:126)K ) = P ζ ( k ) (cid:104) Q λij ( (cid:126)K )ˆ k i ˆ k j (cid:105) , (4.7)where Q λij ( (cid:126)K ) ≡ B ( K, k, | (cid:126)K − (cid:126)k | ) P γ ( K ) P ζ ( k ) γ λij ( (cid:126)K ) . (4.8)The variance of the quadrupole reads Q ≡ π (cid:68) Q λij Q λ,ij (cid:69) = 1615 π (cid:90) k min k min K dK (cid:34) B ( K, k, | (cid:126)K − (cid:126)k | ) P γ ( K ) P ζ ( k ) (cid:35) P γ ( K ) . (4.9)where k min is the longest-wavelength tensor mode produced during inflation and k min isthe smallest observable wavenumber. Replacing the results for our model, Eqs. (3.22),(3.23) and (3.24), one arrives at Q ∼ π (cid:18) HM P (cid:19) r fg (1 − r fg ) (cid:0) r fg (cid:1) (cid:18) k min k (cid:19) (cid:46) (cid:18) HM P (cid:19) r fg (1 − r fg ) (cid:0) r fg (cid:1) . (4.10)– 16 –iven the current limits on the quadrupolar anisotropy, Q (cid:46) − [75–82], Eq. (4.10)shows that the model allows for ample room for those to be satisfied. Surveying the possible signatures of the inflationary particle content is crucial to makethe best possible use of this cosmological window on high energy physics. The squeezedlimit (and generalizations thereof) of N ≥ N + 1-point correlatorsand their N -point counterpart may be modified as a result of the additional dynamicsand reveal information about the mass and spin [1, 2, 72] of the extra degrees of free-dom. In this work we have studied the case of the minimal inflationary scenario, GR +scalar, coupled through so-called dRGT-interactions to a (massive) spin-2 field. This isa non-trivial yet economic choice going up the particle spin ladder as we capitalize onthe vast existing literature on massive spin-2 fields in the context of late-time cosmicacceleration. We employ a model which is ghost-free at the fully non-linear level andshow how consistency relations are modified in this setup.The property that is most consequential for the purposes of detection turns outto be the fact that massive unitary representations of the dS group come with a lowerbound on the effective mass of the graviton m ≥ H . Particles whose Comptonwavelength is smaller than 1 /H will typically decay outside the horizon. This reflectson the impact that modified consistency relations have on observables in our setup:the contribution of each mode comes only from an integrated-over-time effect. Theresulting signal for the tensor-scalar-scalar correlation is too weak for detection byupcoming large scale structure surveys. One can exploit the fact that the unitaritycondition is softened in FLRW backgrounds and matter may be coupled to both spin-2fields. However, demanding consistency of the overall setup leads to similar conclusions.In order to rescue the massive tensor from decay, the most natural mechanism isthat of a non-minimal coupling to the matter sector [84], where special care needs tobe exerted so as not to excite ghostly degrees of freedom both at the linear and fullynon-linear level. Many of these features characterize also the intriguing case of higherspin fields which will be the subject of future work. Acknowledgments
We are delighted to thank Alex Kehagias and Toni Riotto for collaboration at theearly stages of this work. We thank especially T. Riotto without whose input thispaper would not have been possible. M.B. thanks Daniel Baumann, Garrett Goon,Hayden Lee and Guilherme Pimentel for helpful discussions and comments and LuigiPilo for useful correspondence. E.D. and M.F. are grateful to Kurt Hinterbichler forfruitful conversation and especially to Claudia de Rham and Andrew J. Tolley forilluminating discussions and comments. M.B. acknowledges support from Delta ITPconsortium, a program of the Netherlands Organisation for Scientific Research (NWO)– 17 –hat is funded by the Dutch Ministry of Education, Culture and Science (OCW). E.D.acknowledges support by DOE grant DE-SC0009946. M.F. is supported in part byNSF PHY-1068380.
A Consistency relations: general considerations
In Appendix A and B we will adopt the notation of [24, 25]. The key requirement forconsistency relations to be in place between an N + 1-function with at least one longmode and its N -point counterpart is the existence of gauge diffeormophisms (diff) inthe action (or the equations of motion) of the physical system at hand. In cosmologicalsetups (e.g. spatially-flat FLRW background) it is often said that certain gauges, forexample unitary gauge, completely fix gauge invariance. This is indeed true for dif-feomorphisms that vanish at spatial infinity. Crucially, the residual gauge symmetriesthat CRs rely upon do not vanish at infinity [41]. It can be shown [25] that thesetransformations can be smoothly extended to configurations that do fall off at infinityand therefore corresponds to the action of adiabatic modes. The key identification isbetween these gauge transformations and the soft limit of transformations that falloff at infinity and are therefore physical. In the inflationary context, depending onthe specific gauge diff in question, the transformation of a given observable O can beidentified with the action of a long scalar or tensor mode on the same quantity O .For the CR to be non-trivial it is essential that the long mode transforms nonlinearly under the symmetry. All CRs originate from the invariance of the action underspecific residual space diffs: x i → x i + ¯ ξ i ( x ) (A.1)where the ¯ on top of ξ indicates that the gauge mode ξ must satisfy equation ofmotions similar to that of physical modes. Let us consider the example of dilations,which are defined as x (cid:55)→ (1 + λ ) x , (A.2) ζ (cid:55)→ ζ + λ (1 + x · ∂ x ζ ) . (A.3)This symmetry of the action is associated with a conserved current J µ d and a corre-sponding charge Q d = (cid:82) d xJ , such that locally we can write the transformation ofthe curvature fluctuation under dilations as δ d ζ = i [ Q d , ζ ] = − − x · ∂ x ζ. (A.4)A similar relation holds for the tensor fluctuation γ under anisotropic spatial rescaling,¯ ξ i ar = S il x l with S il = S li ; S ll = 0 = ˆ q i S il ( (cid:126)q ) . (A.5)Let us very schematically track the action of a gauge diff s on a generic observable (cid:104)O(cid:105) = (cid:104) ζ ..ζ l , γ l +1 ..γ m , ..σ N (cid:105) , which we take to be the N -point function made up by– 18 –ny combinations of fields populating the Lagrangian and consider σ as a placeholderfor any type of particle content. The transformation s acts according to δ s (cid:104)O(cid:105) (cid:12)(cid:12)(cid:12) connected ∝ D · (cid:104)O(cid:105) , (A.6)where we have been deliberately agnostic about the rules of the s transformationexcept for indicating by D· a generic differential operator. Crucially, only the linearcomponent of δ s acting on the fields is relevant for connected diagrams . The verysame action of the s diff can be expressed in an alternative but equivalent fashion andit is the equivalence of the two that sits at the heart of consistency relations. Theeffect of the δ s transformation is associated to the conserved current, and consequentlyto its corresponding charge so that δ s (cid:104)O(cid:105) ∼ (cid:104) Q s |O(cid:105) . At this stage we are certainly atliberty to introduce the identity operator in between (cid:104) Q s |O(cid:105) = (cid:88) n (cid:104) Q s | n (cid:105)(cid:104) n |O(cid:105) , (A.7)where the | n (cid:105) represent mutually orthogonal independent states. Setting up this basisrequires no work in the minimal inflationary scenario as naturally (cid:104) ζγ (cid:105) = 0. In thecase of anisotropic rescaling, the tensor field transformation has both a linear and anon-linear component while the scalar mode only the former. It follows that Q ar actsnon-trivially only on γ : (cid:104) Q ar | γ (cid:105) ∼ (cid:104) γ (cid:105) + c number , (A.8)the first term on the RHS being zero . With no additional field content present, itfollows that: D · (cid:104)O(cid:105) ∝ (cid:104) Q ar | γ (cid:105)(cid:104) γ |O(cid:105) (A.9)or, rather less schematically, the CR reads: − (cid:15) λij (ˆ k ) (cid:88) a =2 (cid:18) k ia ∂∂k ja (cid:104) ζ k ζ k (cid:105) (cid:48) c (cid:19) = lim k → (cid:104) γ λ k ζ k ζ k (cid:105) (cid:48) c P γ ( k ) (A.10)where “ c ” stands for the connected part of the diagrams and the prime symbol is areminder that the momentum-conserving delta has been removed. Albeit admittedlyvery schematic, the above account is sufficient to see that one obvious route to non-standard CRs is the presence of an additional field whose component orthogonal to the Note here that the action of the linear component of the δ s transformation cannot always, strictlyspeaking, be put in the clean form of the RHS of Eq. (A.6), see e.g. [25]. For example, the lineartransformation of the curvature fluctuation ζ under a generic diff comprises a term proportional tothe tensor mode γ ij . However, this technical consideration is in no way relevant to the discussionin this section and the simplified form is indeed in place for the specific case of anisotropic rescalingacting on two hard scalar modes. The ket | γ (cid:105) describes the action of the operator ˆ γ ij on the Bunch-Davies vacuum of the theory.For a more general and detailed treatment see, for example, the work in [25]. – 19 – | γ (cid:105) , | ζ (cid:105)} basis transforms non-linearly under anisotropic rescaling and whose N + 1-point function with O is non-trivial : D · (cid:104)O(cid:105) ∝ (cid:104) Q ar | γ (cid:105)(cid:104) γ |O(cid:105) + (cid:104) Q ar | σ ⊥ (cid:105)(cid:104) σ ⊥ |O(cid:105) , (A.11)where σ stands here for a generic new field. This will be precisely the case of theadditional (traceless transverse component of) tensor field that will populate our non-miminal inflationary setup. Of course, there are other ways to non-standard or evenbroken consistency relations: one option is to have a different symmetry breakingpattern or no residual diffs altogether [42–50]; another possibility is to have excitedinitial states [51]. The reader might wonder how would our brief discussion change inthe latter case. In the above we have been rather casual about defining the free fieldseigenstates | ζ (cid:105) , | γ (cid:105) and the implicit vacuum wavefunction; the definition is modified inthe case of non Bunch-Davies initial conditions (see [25] for a more thorough discussionon this and e.g. [52, 53] for examples of excited initial states wavefunctionals). B Alternative route to CRs breaking
In this section we shall work under the approximation that the quadratic Lagrangian(Hamiltonian) of the model, Eq. (2.1), can be written down as L (2)full = L (2)0 + δ L (2) f − g where the last piece, even though quadratic, is handled just like an interaction in thein-in formalism, with its own interaction vertex. This treatment is allowed so long as δ L (2) f − g (cid:28) L (2)0 and we refer the reader to [9] for a well-studied example within the quasi-single-field inflation paradigm. This approximation is the basis for an “alternativeroute” to CRs in our setup which, we stress it here, will be mostly qualitative.We anticipate that treating the f − g coupling in the quadratic Lagrangian as aninteraction is an approximation that is no longer valid when the long mode crossesthe horizon. In the approach outlined in Section 3 , one solves for the fully coupledequations in the quadratic Lagrangian so that all of the information on the (quadratic) f − g coupling is stored in the mode-functions.The quadratic action for the traceless transverse part of the tensors h f , h g is schemat-ically as follows: L (2) ∼ M P ( ∂γ g ) + M f ( ∂γ f ) + m M Γ( H/H f )( γ g + γ g γ f + γ f ) (B.1)where Γ is a specific function of the β n s (one can think of them as order one) and the H/H f ratio. The latter is not too relevant here as the M f /M P ratio will play the mainrole. We want to show that the γ g − γ f mixing can be treated as a perturbation on topof the remaining free Lagrangian. To this end, we recall that the usual normalizationfactor for the wavefunctions is γ g ∝ HM P and γ f ∝ H f M f , so that the Einstein-Hilbertcontributions to the quadratic Lagrangian go rispectively like H , H f and we takethe former as being of the same order or smaller than the latter. It follows that the What we mean by non-trivial here is rather more than requiring a non-zero N + 1-point function,as will be clarified below. – 20 –ondition for each term proportional to M in Eq. (B.1) to be smaller than the E-Hterms is, in order: m H M M P Γ (cid:28) , m H H f H M M f M P Γ (cid:28) , m H H f H M M f Γ (cid:28) , (B.2)the most relevant being of course the second inequality because it stems from requiringthat the γ g − γ f mixing term can be treated as an interaction. It is important to checkat this stage the compatibility of the second inequality with the Higuchi bound aswell as the Friedman equations. For convenience, we report the Higuchi bound as inEq. (2.7) and assume the term regulated by β to be the leading one. The bound tobe compared with the conditions in Eq. (B.2) reads:14 m H H f H M M f ≥ . (B.3)Upon inspection, it is immediate that the first two requirements in Eq. (B.2) aresatisfied if the hierarchy M f (cid:28) M P is in place ; we shall assume it for the remainingparts of the manuscript . This in particular implies that the γ g − γ f mixing canbe treated as an interaction (see Fig. 3) without conflicting with the all-importantunitarity bound. For consistency, we shall also require that the leading part in theRHS of Eq. (3.2) is also due to the inflationary potential and not the m -proportionalpart. To that end, and again focussing the attention on the ∝ β contribution, weought to require that: m H HH f M M P (cid:28) , (B.4)which is compatible with all of the above. Finally, Eq. (3.3) requires m H H f H M M f ∼ , (B.5)also consistent. Reassured by these checks, let us turn to consistency relations in thisapproximation scheme.We want to provide evidence that tensor CRs in our setup take a non standard form.We have reviewed in Section A the origin of the CRs relevant for observables such as thetensor-scalar-scalar three-point function. We are interested in the action of anisotropic As we shall see, the first one is a byproduct of the second one in our parameter space and we willnot need the third inequality to be satisfied at all. We must stress here that, at this stage, this is not a necessary assumption. However, the com-bination of Higuchi bound, the requirement on the Friedman equation to be dominated by matterdriving inflation, and having a slowly varying ξ , leads to M f (cid:28) M P . See [62, 71] for further details. The third requirement in Eq. (B.2) is not necessarily satisfied: the ratio can be order one andtranslates into a massive term for γ f that must be contained within L (2)0 and is not a part of δ L (2) .The same ratio appears e.g. in Eq. (B.5). – 21 – igure 3 . Diagrammatic representation of the quadratic γ g - γ f interaction. rescaling on long tensor modes defined in Eq. (A.5). Recall now from Eq. (A.11) thata non-standard CR demands at least two terms in the RHS of (cid:104) Q ar | ζζ (cid:105) = (cid:88) n (cid:104) Q ar | n (cid:105)(cid:104) n | ζζ (cid:105) . (B.6)If at least two fields (or rather two states) {| (cid:105) ≡ | γ g (cid:105) , | (cid:105) , . . . } , once orthonormalized,transform non-linearly under Q ar then we will have a non-standard CR: (cid:104) Q ar | γ g (cid:105)(cid:104) γ g | ζζ (cid:105) + (cid:104) Q ar | (cid:105)(cid:104) | ζζ (cid:105) + ... ∼ (cid:104) [ Q ar , ζζ ] (cid:105) ∼ D(cid:104) ζζ (cid:105) (B.7)In standard single-field inflation the only field non-linearly transforming under ananisotropic rescaling is γ g , hence the standard CR. In the setup of Eq. (2.1), can γ f , the traceless and transverse part of the fluctuation of the f metric, suitably or-thogonalized, serve as a non zero contribution from the | (cid:105) “state”? To answer thisquestion in the affirmative we ought to show that0 (cid:54) = (cid:104) Q ar | γ ⊥ f (cid:105)(cid:104) γ ⊥ f | ζζ (cid:105) (cid:54) = (cid:104) Q ar | γ g (cid:105)(cid:104) γ g | ζζ (cid:105) , (B.8)the last condition guaranteeing that the action of the two long modes γ f and γ g is notidentical. Note also that we have orthogonalized γ f w.r.t. γ g in the standard way: | γ ⊥ f (cid:105) = N (cid:104) | γ f (cid:105) − | γ g (cid:105)(cid:104) γ g | γ f (cid:105) (cid:105) , (B.9)with N a normalization factor. The RHS of Eq. (B.7) can now be written moreexplicitly as (cid:88) n (cid:104) Q ar | n (cid:105)(cid:104) n | ζζ (cid:105) = (cid:104) Q ar | γ g (cid:105)(cid:104) γ g | ζζ (cid:105) + (cid:104) Q ar | γ ⊥ f (cid:105)(cid:104) γ ⊥ f | ζζ (cid:105) = c g (cid:104) γ g | ζζ (cid:105) + (cid:104) Q ar | N (cid:104) | γ f (cid:105) − | γ g (cid:105)(cid:104) γ g | γ f (cid:105) (cid:105) · N ∗ (cid:104) (cid:104) γ f | − (cid:104) γ f | γ g (cid:105)(cid:104) γ g | (cid:105) | ζζ (cid:105) = c g (cid:104) γ g | ζζ (cid:105) + | N | (cid:104) ( c f − c g I g f ) · (cid:0) (cid:104) γ f | ζζ (cid:105) − I ∗ g f (cid:104) γ g | ζζ (cid:105) (cid:1) (cid:105) , (B.10)where we have assumed for simplicity that | γ (cid:105) has norm one. We have also defined thequantities c g ≡ (cid:104) Q ar | γ g (cid:105) , c f ≡ (cid:104) Q ar | γ f (cid:105) , and I g f ≡ (cid:104) γ g | γ f (cid:105) . Note that the latter stemsprecisely from the interaction in Fig. 3. Non-standard CRs now correspond to bothfactors between square brackets in the last line of Eq. (B.10) being non-zero. In orderto show that is indeed the case, let us write down the gauge transformation properties– 22 –f γ f , γ g . These, in our language, are stored into c f , c g . For the gauge parameter, ξ µ ,the transformation law reads: δγ g µν = a − ( ξ α ∂ α ¯ g µν + ¯ g αν ∂ µ ξ α + ¯ g µα ∂ ν ξ α ) ,δγ f µν = b − (cid:0) ξ α ∂ α ¯ f µν + ¯ f αν ∂ µ ξ α + ¯ f µα ∂ ν ξ α (cid:1) . (B.11)The reasoning proceeds as follows. The gauge transformation of γ f , γ g are the sameexcept for a factor of b/a , the ratio of the two metrics scale factors, a quantity alsoequal to H/H f in the branch of the theory we are considering. From here, one should inprinciple solve the constraint equations and derive the transformation law specificallyfor the symmetric traceless transverse part of γ f , γ g . On the other hand, what mattersfor us is that the following term from Eq. (B.10) is non zero: c f − c g I g f (cid:54) = 0. It isimmediate to see, for example from the function Γ in Eq. (B.1), that the quadraticinteraction term I g f contains different powers of b/a = H/H f and is also a function of β , β , β so that c f − c g I g f does not in general vanish.What remains to be proven now is that (cid:104) γ f | ζζ (cid:105) − I ∗ g f (cid:104) γ g | ζζ (cid:105) (cid:54) = 0 which is, writtendifferently, (cid:104) γ f | ζζ (cid:105) (cid:54) = (cid:104) γ f | γ g (cid:105)(cid:104) γ g | ζζ (cid:105) . Perhaps the best way to prove this last part isdiagrammatically, as shown in Fig. 4, where the grey colored circles stand for genericinteractions. It will be enough to find an internal diagram corresponding to (cid:104) γ f | ζζ (cid:105) that cannot be expressed as (cid:104) γ f | γ g (cid:105)(cid:104) γ g | ζζ (cid:105) . Let us give a few examples of interactions Figure 4 . Diagrammatic representation non standard CRs condition. that do and do not lead to a modified CR. On the left side of Fig. 5 one can see adiagram that can be drawn as in the RHS of Fig. 4 with dashed vertical lines just tothe left of where the gray vertices would go.On the other hand, the diagram on the right side of Fig. 5 cannot be put in that formand leads therefore to a non-standard consistency relation.However, it must be noted that this requires a direct coupling between γ f and ζ . Thediagram in Fig. 6 is instead a loop effect and delivers a non-standard CR regardlessof the coupling of γ f . We consider this last one as the clearest example of a modifiedconsistency relation among those we present in this Section B . As anticipated, wewill not discuss in detail the value of the bispectrum generated by these diagrams,we postpone a quantitative analysis of modified CRs to
Section 3.2 . One reason, aswe shall see shortly, is that the approximation scheme justifying the use of δ L (2) , orin other words the diagram in Fig. 3, breaks down at the horizon. Nevertheless, wefind it worthwhile to present a diagrammatic proof of CRs “breaking” in this setup.The existence of a quadratic interaction lends itself nicely to such proof, it shows very In this approximation γ f is massive and will therefore decay at late times. – 23 – igure 5 . Left: contribution to (cid:104) γ g ζζ (cid:105) consisting of two “quadratic” γ g − γ f vertices and theusual tree-level tensor-scalar-scalar interaction. Right: A γ g − γ f vertex and a three-vertexwhich is there only if γ f couples directly with ζ . clearly why CRs are indeed broken, and is very reminiscent of the one that applies to,among others, quasi-single-field inflation models. Figure 6 . A loop-diagram contribution to (cid:104) γ g ζζ (cid:105) . This diagram will be “Planck” suppressedby the H/M p and H f /M f in the wavefunctions normalizations. Let us briefly show how the approximation breaks down at the horizon. The reason isquite simple, the quadratic interaction term is proportional to m a ( τ ) h . In a quasi-de Sitter phase it is true that a (cid:39) − H τ . At late times (starting at horizon exit) thisterm becomes of the same order and eventually much larger than the Einstein-Hilbertterms we have placed in L (2)0 , thereby invalidating the approximation. Considering thatmost of the contribution to cosmological correlation functions is known to originate atand after horizon exit it is advisable to seek another scheme whose validity extendsfurther than this “alternative route”. We provided such a treatment in Section 3.2 . References [1] A. Kehagias and A. Riotto, Fortsch. Phys. , 531 (2015) [arXiv:1501.03515].[2] N. Arkani-Hamed and J. Maldacena, [arXiv:1503.08043].[3] E. Dimastrogiovanni, M. Fasiello and M. Kamionkowski, JCAP , 017 (2016)[arXiv:1504.05993].[4] H. Lee, D. Baumann and G. L. Pimentel, JHEP , 040 (2016) [arXiv:1607.03735].[5] D. Baumann and L. McAllister, [arXiv:1404.2601].[6] A. Achucarro, J. O. Gong, S. Hardeman, G. A. Palma and S. P. Patil, JHEP (2012) 066 [arXiv:1201.6342].[7] C. P. Burgess, M. W. Horbatsch and S. P. Patil, JHEP (2013) 133[arXiv:1209.5701]. – 24 –
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