Probing the inflationary particle content: extra spin2 field
Abstract
We study how inflationary observables associated with primordial tensor modes are affected by coupling the minimal field content with an extra spin2 particle during inflation. We work with a model that is ghostfree at the fully nonlinear level and show how the new degrees of freedom modify standard consistency relations for the tensor bispectrum. The extra interacting spin2 field is necessarily massive and unitarity dictates its mass be in the range. Despite the fact that this bound selects a decaying solution for the corresponding tensor mode, cosmological correlators still carry the imprints of such fossil fields. Remarkably, fossil(s) of spin generate distinctive anisotropies in observables such as the tensor power spectrum. We show how this plays out in our setup.
CERCA and Department of Physics, Case Western Reserve University, Cleveland, OH, 44106, U.S.A. \affiliation Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, N2L 2Y5, Canada. \affiliation Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK. \affiliation Department of Physics, Stanford University, Stanford, CA, 94306, U.S.A. \affiliation Department of Physics, Swansea University, Swansea, SA2 8PP, United Kingdom
1 Introduction
Cosmological inflation predicts the existence of a yetunobserved stochastic background of primordial tensor modes, whose detection represents one of the main challenges for future observational cosmology (see e.g. [2, 3, 4] for recent reviews). According to the simplest realizations of inflation, the primordial gravitational waves background is nearly scale invariant and almost Gaussian, with CMB polarization measurements being potentially one of its most sensitive probe. Nevertheless, in several scenarios, wellmotivated by particle physics, there arise the possibility that primordial gravitational waves (GW) can be detected at interferometer scales [5]. Given that primordial GW represent an additional window on the dynamics of inflation, and can offer important
clues on its ultraviolet completion, it is essential to explore all possibilities in view of a future detection [6].
Besides the GW signal due to standard vacuum fluctuations, there exist several different mechanisms that can source primordial tensor waves. One possibility stems from specific couplings (typically nonminimal) of the inflaton with additional fields (from scalars to vectors, all the way to higher spin fields, see e.g. [7]). Such models often lead to (controlled) instabilities and particle production affecting the tensor sector both at CMB and interferometer scales. Alternatively, the primordial tensor statistics can be modified by breaking spacetime symmetries during inflation, either spontaneously as in (super)solid inflation [8], or directly as in HořavaLifshitzmotivated inflationary systems (see e.g. [9]).
In addition to the power spectrum, its spectral index, and possible chiral properties, an excellent probe of extra dynamics during inflation is the squeezed (soft) configuration of the bispectrum. Indeed this observable is sensitive to the mass, the spin and the coupling of any additional particle content. For example, the effect of an extra light scalar on the predictions of minimal singlefield slowroll (SFSR) inflation is to enhance the bispectrum in the squeezed limit. From this property stems the wellknown identification of a detection as a “smoking gun” for multifield inflation [10]. Crucially, the information potentially stored in the bispectrum spectrum extends much further than that. Indeed, as one considers extra particle content of increasing spin, the corresponding bispectrum acquires a nontrivial angular dependence. At the same time, the bispectrum scaling with respect to soft (hard) momenta reveals information on the extra particle(s) mass. This latter property is rather remarkable: even if the field content beyond SFSR is massive to the level , and therefore shortlived, its imprints are fossilised [11] in the bispectrum and remain accessible today. This is especially important in view of the fact that the existence of irreducible unitary representation of the spacetime isometries group strongly constraints the parameter space of theories with extra spin2 fields.
In this work we extend the minimal inflationary scenario by coupling it with an extra spin2 field [12, 13, 14] and focus on the consequences for the tensor sector. This particle content, comprising an additional tensor with a mass range of order the Hubble parameter, is reminiscent of a wellstudied scenario known as quasisingle field inflation (QsF) [15, 16], where the additional field is a massive scalar. In this sense, our setup can be thought of as the tensor sector counterpart of QsF. We refer the reader to the literature [17, 18] (see also [19]) for related studies of (extra) higher spin fields populating inflation. In choosing the extra field to be a spin2 particle, one can rely on an extensive body of work [20] (see in particular [21, 22, 23, 24] for use in the inflationary context) spanning also models of the latetime universe. As we shall see, any (interacting) addition to the tensor sector of general relativity is necessarily massive. We plan to make use of a fullynonlinear ghostfree couplings between the tensor sectors known as dRGT interactions [13]. In doing so, we will exploit some of the model most desirable properties:

The structure of the theory is engineered to guarantee the absence of Ostrogradsky instabilities. In fact, the model of [14] is the only known example of multimetric theory consistent at the fully nonlinear level, and precisely dictates the coupling between massive and massless tensor fluctuations
^{1} . 
The strong coupling scale (and therefore also the UV cutoff) of the theory is naturally much larger (see Section 2.1) than our working energy, set by the Hubble parameter . This, together with the fact that the model is automatically ghostfree at fully nonlinear level, implies that one can explore a regime in parameter space that cannot be otherwise probed using a fully general EFT approach
^{2} . 
Adding a massive spin2 to the minimal inflationary scenario comes with an extra five degrees of freedom: two tensors, two vectors and a scalar. When the nonlinearities in the dRGT interactions become important, the effective coupling to matter (including the inflaton) of the extra vectors/scalar is strongly suppressed: this is what is known as the Vainshtein screening mechanism at work [25]. On the other hand, screening does not affect the extra tensor sector, precisely the one we shall focus on in this work. The key observable will be the tensor threepoint function, the smallest point function to directly probe nonGaussianities in the theory and imprints of extra modes therein. We will study the contributions to the bispectrum due to the new nonderivative dRGT interactions and show how these modify the standard tensor consistency condition in play for SFSR inflation.
Nonderivative interactions tend to enhance the socalled local contributions to the bispectrum or lead to intermediate (i.e. interpolating between different configurations) shapes for nonGaussianities. This is wellstudied for the scalar sector (see e.g. the review in [26]), and is also expected to hold for the tensor sector. Our results provide a theoretical and phenomenological characterization of tensor nonGaussianities beyond SFSR that is essential in view of future experimental probes [27].
This paper is organized as follows: in Section 2 we introduce our nonminimal inflationary scenario and study the dynamics of tensor fluctuations around de Sitter space (a proxy for inflation) up to fourth order in perturbations. We examine the bounds on fields mass range that stem from requiring this to be a consistent setup; in Section 3 we analyse how the presence of new tensor modes affects cosmological correlators, and the consequences for tensor nonGaussianity; Section 4 is devoted to more general considerations on tensor consistency relations in light of our findings, and on observational prospects for the bispectrum resulting from modifications/violations of consistency relations; we conclude in Section 5, whilst technical details of the calculations can be found in the Appendices.
2 The model
Our setup consists of (i) the standard action
(1) 
where the inflaton (and, in general, any matter content) is minimally coupled to gravity, described by the metric . We further consider (ii) an additional spin2 field, making use of the fact that the only known system consistently coupling two spin2 tensor modes, here called and , is described by the action of [14]:
(2) 
where the combinations (with an arbitrary matrix) are defined as
(3) 
The action (2) contains the EinsteinHilbert terms for the metrics and (with
and denoting the corresponding
Planck masses), a potential term controlled by the parameter parameterizing the mass in the tensor sector, and
five dimensionless constant parameters . For
convenience – as well as to keep the identification
(4) 
As a proxy for inflation, we consider de Sitter solutions for both background metrics . The corresponding background equations of motion () are
(5) 
Note that in Eq.(5) is implicit the choice of the healthy branch [20] of solutions () and we have in particular considered the case . We write the metric fluctuations as
(6)  
(7) 
As anticipated, we will focus our analysis on the tensor sector and in particular on the traceless transverse components of the metrics . Although we shall not consider them directly, we nevertheless ought to ensure that we explore a region in parameter space for the theory where unitarity is preserved also in the helicity0 and helicity1 sector. To this aim, we will enforce on the parameters and the Higuchi bound of Section 2.1. In order to extract the observables of interest here, namely the tensor power spectrum and bispectrum up to oneloop order, it will be necessary to expand the action up to fourth order in perturbations. The new physics w.r.t. SFSR is in the nonderivative interactions making up the action that we schematically write as:
(8) 
The Lagrangian densities at each order read
(9)  
(10)  
(11)  
where etc. Notice that the new interactions described by the potential term in Eq. (2) and in Eqs. (9)(11) do not contain any derivatives. Indeed, there cannot be derivative couplings among spin2 fields that are consistent at the fully nonlinear level [28]. Such couplings are of course still allowed in the purely EFT sense of e.g. [19]. Despite the compact expression for each , the basis mixes the two metrics already at second order, as in
(12) 
with indicating the standard quadratic kinetic terms for the tensors. It is convenient to use an alternative basis that decouples the fields in . In order to do so, one may first canonically normalize the spin2 kinetic terms, rescaling fields according to
(13)  
(14) 
and then considering a rotation in field space so as to diagonalize the potential:
(15)  
(16) 
with
(17) 
The quadratic Lagrangian in the new basis reads
(18) 
with given by the combination:
(19) 
Interactions between the massless field and the massive particle start now at third order, and are regulated by the following cubic Lagrangian
(20) 
The corresponding expression for the quartic nonderivative interactions is more lengthy and can be found in Eq.(66) of Appendix A. It is sufficient at this stage to point out that all its terms are proportional to a unique combination of the parameters corresponding to as per Eq. (19).
2.1 The structure of the interactions
At quadratic level we identified a basis where massless and massive tensor modes propagate independently, the mass being proportional a linear combination of the coefficients we call , see Eq. (19). At cubic and quartic order interactions among the two fields are controlled by and there are no new selfinteractions for the massless mode besides the ones already present in GR. These features can be understood quite generally within the context of the theorem in [29]: it states that, in the limit in which the massive field becomes massless (that is, ), interactions among the two massless fields must vanish. As we shall see, an implication of this result for our setup is that the new contributions to the power spectrum and bispectrum appear starting with loop diagrams, and, at leading order, depend on the dimensionless parameter .
Bounds on the parameters space
We provide here a brief discussion on the bounds that limit our analysis to a specific region of the parameter space of the theory. First, we recall [30] that the notion of particles is best defined by unitary irreducible representation of (in our case) the de Sitter isometry group. These identify, for massive
(21) 
where have used the definition in Eq. (4) for . This condition
reduces to the wellknow Higuchi [34] bound in the linear massive gravity limit ().
We note in passing that the unitarity bound on the mass of the (extra) tensor mode confines us to the same playground that defines the parameter space of the extra scalar mode in QsF inflation.
The origin of the next bound is rather different, and the corresponding inequality may in principle be relaxed depending on the setup under consideration. In obtaining the solution for the wavefunction in Eq. (31), we assume a realvalued . Were we to allow, as it is certainly possible
(22) 
(23) 
although, as mentioned above, the upper bound is not strictly necessary and may be relaxed.
Strong coupling scale
In order to expound on the strong coupling scale of the model under scrutiny, we need to make contact with the nonlinear massive gravity theory of [13]. In such context, the socalled naive strong coupling scale is . We note that it emerges most clearly in a specific scaling limit of the theory known as decoupling limit (DL). The DL analysis shows how the smallest strong coupling scale has to do with the helicity0 mode interactions. Indeed, for purely tensor modes one would expect that scale to be , just as is the case for general relativity.
On nontrivial backgrounds the nonlinearities in the massive theory actually redress the effective strong coupling scale, always in the direction of making it larger, , where measures the nonlinear terms contributions to the standard kinetic term for the helicity0 mode. In Eq. (2) both metrics are dynamical and, as a result, the naive strong coupling scale becomes with . Given the bounds on we discuss in the previous section and the fact that we assume , one may conclude we are safely away even from what is only the naive strong coupling scale in our setup. It is worth pointing out that it would be hard to put together a scale compatible with inflation, such as the one we require here, with the use of a massive graviton as a mechanism for latetime acceleration. We make no such attempt here, and view our Eqs. (1)(2) as simply describing the (healthy) field content of a minimal inflation model enriched by a spin2 particle.
3 Observables in the tensor sector
We first quantize tensor fluctuations around de Sitter space and then investigate the properties of the power spectrum and bispectrum. Employing the ) basis the massless and massive tensor modes are decomposed in Fourier space as
(24)  
(25) 
where the polarization tensors defined in Appendix B. The operators , are written in terms of standard creation/annihilation operators, satisfying the standard commutation relations, and the mode functions as in
(26)  
(27) 
The evolution equations in de Sitter(dS) space are
(28)  
(29) 
where denotes derivatives w.r.t. conformal time, and the solutions for the mode functions read
(30)  
(31) 
with the Hankel function of the first kind, and
(32) 
In deriving solutions (30), (31) one assumes standard BunchDavies conditions for the vacuum. We now move on to observables, starting with the power spectrum.
3.1 Power spectrum
The power spectrum of each polarization of the tensor perturbations is formally given by
(33) 
Note that we are after the power spectrum of the field because it is the one metric that couples to matter. We shall, in the following calculations, express in the basis for convenience, but we will always have in mind and as the end products of our analysis on the observables. Using Eq. (30), one finds the leading power spectrum to be
(34) 
It follows that at tree level the power spectrum for has the same structure as in the minimal inflationary scenario
Let us analyze the origin of the diagrams in Fig. (1). Although the wavefunction of the field in the standard power spectrum receives contribution from both fields, only the former (massless) field survives in the latetime limit. Inspection of Eq. (20) reveals that two vertices corresponding to the same interaction contribute to diagram (a). Similarly, quartic interactions of the type and in Eq. (66) correspond to the vertex in diagram (b). As clear from the cubic and quartic Lagrangians, each interaction relevant to our setup counts one power of the ‘coupling constant’ . It follows that at one loop the power spectrum is limited to be at most proportional to and, as we shall see, the bispectrum to be at most proportional to . Since at a given perturbative order the maximum number of bispectrum vertices exceeds those in the twopoint function, it is always possible to ensure that the threepoint function is parametrically different from the power spectrum: this will play a key role in our model departure from standard consistency relations, see Section 4. We estimate below the size of the 1loop contributions to the tensor power spectrum:
(35)  
(36)  
(37) 
where , and we make use of the definition in Eq. (4).
In the previous expressions we wrote, as representative examples, the contribution corresponding to the term of the quartic Lagrangian (see Appendix A) in Eq. (36), and the contribution corresponding to terms such as and in Eq. (37). The term in has been computed in the limit
Before moving on to the bispectrum calculation, we pause here to comment on the scale dependence of the power spectrum. In General Relativity (GR) an exact scaleinvariance is a consequence of de Sitter symmetries. Since our setup has a richer dynamics than GR, and given that massive tensors are known to be
characterised by a blue spectrum, one might wonder what to expect for the index . The extra scaledependence of a massive external field occurs by means of an extra (w.r.t. to their massless counterpart) dependence on powers of at late times. However, in our case as well as e.g. quasisingle field inflation (in the scalar sector), the massive fields are not external but rather part of internal legs of Feynman diagrams, and their behaviour is integrated over with the leading contribution typically coming from the domain corresponding to the horizon region. As a result, one does not expect an additional scaling from internal massive lines. See also [35] for a discussion on this point.
3.2 Bispectrum
The main ingredients for the calculation of the 1loop threepoint function are essentially the same as for the power spectrum. The bispectrum is defined as
(38) 
Here too, in order to probe the extra dynamics and in the absence of new purelymassless interactions in the cubic (quartic) Lagrangian, the presence of massive spin2 fields forces us to consider the oneloop order. Let us begin by evaluating the amplitude corresponding to diagrams (c) and (d) in Fig. (2).
By employing the inin formalism, we schematically find (see Appendix C for details):
(39)  
(40)  
The overall amplitudes of the bispectrum contributions are parameterized as:
(41)  
(42)  
(43) 
where again, diagram arises from the contribution proportional to in the quartic Lagrangian and diagram is due to terms such as and , with .
Notice that, although we are able to extract almost all the dependence on the theory parameters from the integrals, there remains an irreducible dependence which cannot be factored out but follows from numerical integration. The value of the contribution it typically [16, 36] and, naturally, a similar functional dependence is present also in the 1loop power spectrum computation. The fact that , as well as their ratio, are functions of and not directly of will be particularly important in Section 4.
We should also address the momentum dependence of the bispectrum. Standard tree level contributions of massless scalar (tensor) fields can interpolate between the local configuration () and the equilateral profile (with a maximum for ). One typically moves from the local to the equilateral shape as the interactions considered contain an increasing number of derivatives acting on the fields [37, 38, 39, 26, 40, 41]. For massless fields, nonderivative interactions such as the ones we consider here, peak in the local configuration. In our setup we are instead dealing with a contribution from massive fields running in the loop. As mentioned above, the effect of massive particles at tree level is a factor in front of what would otherwise be a purely local shape [16, 19]. We therefore expect a similar mildening of the local shape in our case.
Tensor nonGaussianities
Let us turn to providing an estimate of the various contributions to the simplest tensor nonGaussianities, the bispectrum. We will, as ever, focus on the nonderivative contributions as we are after the effect of an extra massive spin2 field during inflation. We estimate the size of the tensor bispectrum by means of the dimensionless parameter defined as the ratio between the threepoint function and the tensor twopoint correlator squared
(44)  
(45) 
where we have used Eq. (4). Now, for the purposes of providing an overall consistent estimate of the bispectrum amplitude, we remind the reader of Eq (23): such inequalities limit the bispectrum contributions to being respectively of order
(46) 
where in providing both estimates we have taken and used as a place order for the “Planck” masses and combinations thereof. The significance of Eq. (46) is intuitively clear: a loopsuppression is slightly lessened by the possibility
We have specifically chosen in this work to focus on the simplest, nonlinearly consistent theory of a (necessarily massive) extra spin2 field, sure in the knowledge that ours is a ghostfree model. We leave the exploration of a related model with nonminimal coupling to future work [43].
4 Tensor Consistency Relation
4.1 The standard case
Before specializing to our case, let us report here the standard tensor consistency relation (CR) [44, 45]:
(47) 
As we have done throughout the text, we restrict our considerations to the nonderivative contributions to the 1loop bispectrum and 1loop power spectrum. Consistency of perturbation theory dictates that we consider the diagrams in Fig. (3), that is we compare the 1loop bispectrum with a product of the tree level and 1loop twopoint function.
For convenience, we reproduce here the results in Eq (42) of the previous section, the amplitude of the oneloop contributions to the bispectrum:
(48) 
where one need not specify the form of the functions, but we do underscore their dependence. In order to probe the consistency relation, the squeezed limit of the bispectrum ought be compared with the power spectrum. We stress here that it is not the purelytreelevel bispectrum nor, correspondingly, the purelytreelevel power spectrum we are after: both observables are insensitive to the presence of new interactions mediated by massive modes and will therefore be connected by the standard CR. In order to capture departures from standard tensor CRs, one ought to compare the (squeezed) oneloop bispectrum to one loop contributions to the squared power spectrum:
(49) 
Comparing the results of Eqs. (48)(49), one realizes that the consistency relation in Eq. (47) does not hold in general, since the overall dependence on the parameter is different in the two cases: up to for the bispectrum, but only up to for the power
spectrum. It is important to note that the functions cannot make up for the “missing” power: we know by inspection that is not a linear (or polynomial) function of simply the variable and therefore a ratio of different functions cannot restore the consistency relation.
In our setup then, the standard consistency relation does not hold. This should not come as a surprise in a multifield model; after all, it is a very wellknown result that a large would be, if detected, a smoking gun for multi(scalar)field inflation. One might well argue that in multifield models the extra scalar is light, , whilst in our case the extra field is massive. There is however a scalar correspondent for precisely our case, namely quasisinglefield inflation, where for the extra field one has . Consistency relations are indeed modified in QsF as well. We find it worthwhile to provide, alongside the “brute force” proof of modified consistency relation just above, a general presentation on the conditions that grant CR modification/violation and place our result in a wider context. We do so in Section 4.2, 4.3 and point out similarities and subtle differences with respect to the QsF case along the way.
4.2 General Perspective on Consistency Relations
In what follows we will cast the modifications/violations of consistency relations found in our setup in a wider context and then briefly draw a parallel with the case of CRs in QsF inflation. Consistency relations stem from a residual gauge symmetry in a physical system description
(50) 
where, for the sake of generality, we do not yet specify the form of the transformation. It suffices here to stress that, upon acting on the field , the transformation may generate a linear and also a nonlinear component in the field and it is the latter that will be crucial for CRs. The action of the transformation “a” on the power spectrum built out of gamma fields will be, schematically,
(51) 
where the first term on the RHS vanishes
(52) 
where we have inserted the identity operator . Note that we have explicitly singled out the state to underline that the right hand side of Eq. (52) has at least one nontrivial contribution. Putting together Eqs.(51, 52), one finds that
(53) 
Switching to Fourier space, and using the invariance of the vacuum under the linear component of the transformation “a”, one finds that the gauge transformation obeys , and using the overall momentum conservation:
(54) 
If the field is the only one transforming nonlinearly under , each term of the sum over on the right hand side vanishes and the reader will recognize in Eq.(54) the standard form of the consistency relation between the squeezed (soft) limit of the threepoint function (RHS) and the variation of the twopoint function (LHS) of the corresponding two hard modes. Crucially, whenever (i) there exist at least one additional field
(55) 
something that would amount to a trivial redefinition of the standard consistency relation. The immediate, observational, consequence of a modified CR is that the squeezed limit of a specific threepoint function can no longer be absorbed by a gauge transformation: it is only a specific linear combination
The power of the perspective we have just outlined on CR is best illustrated in the setup of quasisinglefield inflation, see [46]. We shall refer the reader to existing literature on QsF, with the exception of providing here the specific example of the gauge transformation acting on the scalar field in the case of QsF:
(56) 
The transformation “d” at the heart of the scalar CRs, is the dilatation symmetry (see the work in [46]), and the associated charge is . It is straightforward to prove that the orthogonalized component of the extra massive scalar field in QsF, , has both (i) a nonlinear transformation under dilatation and (ii) nontrivial (self)interactions guaranteeing the condition in Eq.(55) is satisfied. Does the same apply to our setup? We address this next.
4.3 Extra massive spin2 field during inflation
The first condition we have outlined for a CR modification is the existence of two fields nonlinearly transforming under the symmetry generating the consistency relation. In the case of tensors, it is a wellknown result that the gauge transformation in question is anisotropic rescaling [50]. The fluctuations of tensor fields in the basis do transform also nonlinearly,
(57) 
and so do their traceless transverse components.
Note that is the parameter of the most general gauge transformation and the scale factors for the metrics respectively. Since we are after the CR involving the soft limit of , the next step is to require that there exist mediated interaction(s) that contribute to the bispectrum through a diagram that cannot be written as the RHS of Eq.(55) upon identifying and . The existence of the cubic and quartic selfinteractions detailed in Section 2 suggests that this is indeed the case.
All ingredients seem to be in place for us to be able to declare a modification of tensor consistency relations in our setup. There is, however, one more piece to the puzzle with respect to the QsF case: there, a functionally independent cubic selfinteraction guarantees that the contribution on the LHS of Eq.(55), when mediated by , cannot be obtained by means of  interactions and selfinteractions. In our case things are not as straightforward given that all tensor (mixed and self)interactions share a similar dependence on the theory parameters , s. It is then necessary to resort to the more detailed calculation presented in Section 4, where we have provided a proof that the CR is indeed modified parametrically.
Another line of reasoning that arrives at the same conclusion goes as follows. Consistency relations during inflation stems from the theory being invariant under space diffs [51]. It is wellknown that a massive graviton necessarily breaks such diffs and this is a sufficient condition for CRs breaking. One may always restore diff invariance by introducing extrafields
4.4 Quadrupolar tensor anisotropy
An inflationary correlation between a longwavelength () and two shortwavelength () modes induces a modulation of the power spectrum by the longwavelength fluctuation. If the longwavelength mode is a tensor perturbation, the induced modulation is of a quadrupolar type, as it has been shown for density fluctuations [54, 53, 55] (see also [56]):