Partially massless fields during inflation

Journal of High Energy Physics, Apr 2018

Abstract The representation theory of de Sitter space allows for a category of partially massless particles which have no flat space analog, but could have existed during inflation. We study the couplings of these exotic particles to inflationary perturbations and determine the resulting signatures in cosmological correlators. When inflationary perturbations interact through the exchange of these fields, their correlation functions inherit scalings that cannot be mimicked by extra massive fields. We discuss in detail the squeezed limit of the tensor-scalar-scalar bispectrum, and show that certain partially massless fields can violate the tensor consistency relation of single-field inflation. We also consider the collapsed limit of the scalar trispectrum, and find that the exchange of partially massless fields enhances its magnitude, while giving no contribution to the scalar bispectrum. These characteristic signatures provide clean detection channels for partially massless fields during inflation.

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Partially massless fields during inflation

HJE massless elds during in ation Daniel Baumann 0 1 3 6 Garrett Goon 0 1 3 4 6 Hayden Lee 0 1 2 3 5 Guilherme L. Pimentel 0 1 3 6 Clear Water Bay 0 1 3 Kowloon 0 1 3 Hong Kong 0 1 3 and W Symmetry, Higher Spin Gravity 0 17 Oxford Street, Cambridge, MA 02138 , U.S.A 1 Princetonplein 5 , Utrecht, 3584 CC , The Netherlands 2 Institute for Advanced Study, Hong Kong University of Science and Technology 3 Science Park 904 , Amsterdam, 1098 XH , The Netherlands 4 Institute for Theoretical Physics, Utrecht University 5 Department of Physics, Harvard University 6 Institute of Theoretical Physics, University of Amsterdam The representation theory of de Sitter space allows for a category of partially massless particles which have no at space analog, but could have existed during in ation. We study the couplings of these exotic particles to in ationary perturbations and determine the resulting signatures in cosmological correlators. When in ationary perturbations interact through the exchange of these elds, their correlation functions inherit scalings that cannot be mimicked by extra massive elds. We discuss in detail the squeezed limit of the tensor-scalar-scalar bispectrum, and show that certain partially massless elds can violate the tensor consistency relation of single- eld in ation. We also consider the collapsed limit of the scalar trispectrum, and nd that the exchange of partially massless elds enhances its magnitude, while giving no contribution to the scalar bispectrum. These characteristic signatures provide clean detection channels for partially massless elds during in ation. Cosmology of Theories beyond the SM; Higher Spin Symmetry; Conformal - Partially 1 Introduction 2 Higher-spin elds during in ation De Sitter representations Free theory of partially massless elds 2.3 Couplings to in ationary uctuations 3 Imprints on cosmological correlators 2.1 2.2 3.1 3.2 3.3 Trispectrum: h Consistency relations { 1 { Introduction Observations suggest that our Universe started as de Sitter space [1] and will end as de Sitter space [2, 3]. Understanding the physics of de Sitter (dS) is therefore of particular relevance [4{6]. There exist a number of interesting features of this spacetime which do not have counterparts in at space. For example, while particles in Minkowski are either massive or massless [7, 8], the representation theory of de Sitter space allows for an extra category of partially massless (PM) particles [9{11]. At special discrete values of the mass-to-Hubble ratio, m=H, the theory gains an additional gauge symmetry and some of the lowest helicity modes of the would-be massive particles become pure gauge modes. In this paper, we revisit the theoretical and observational status of partially massless particles during in ation. Partially massless particles have a number of intriguing features that motivate us to study their e ects during in ation, despite their somewhat exotic nature and their uncertain status as interacting quantum eld theories.1 First of all, if PM particles existed during in ation, they would lead to rather distinct imprints in cosmological correlation functions. Moreover, the chances of detecting these signals may even be bigger than for massive particles, since the rate at which PM particles would be produced during in ation is larger than that for massive particles. Furthermore, while massive elds decay on superhorizon scales, the amplitude of certain PM elds can remain constant or even grow. They can therefore survive until the end of in ation and their contributions to the soft limits of in ationary correlators are unsuppressed. Another interesting feature of PM particles is that their masses are protected against radiative corrections by the gauge symmetry. Finally, since the existence of PM particles is tied to the non-zero (and nearly constant) Hubble parameter during in ation, their detection would provide further evidence for an in ationary, de Sitter-like period of expansion in the early universe. To describe the e ects of PM particles on cosmological correlators, we will construct gauge-invariant couplings between higher-spin particles and the in ationary scalar and tensor uctuations. Using these couplings, we will show that partially massless higherspin particles leave unique imprints in the soft limits of in ationary correlation functions. These soft limits are a well-known detection channel for extra elds during in ation since a symmetry xes their form in single- eld in ation [19, 20]. Most of the focus, so far, has been on the soft limit of the scalar bispectrum h i, where is the primordial curvature perturbation, which would receive characteristic non-analytic contributions from massive particles [21{27]. A strict violation of the consistency relation | that is, a modi cation of the leading term in the squeezed limit | would require extra massless scalars. An even more robust consistency relation exists for the soft limit of h i, where is a tensor uctuation. In that case, the leading term in the tensor squeezed limit cannot be altered by the addition of light scalars. Moreover, contributions from massive spin-s 1In order to write a local theory with the right number of degrees of freedom, one imposes a gauge symmetry that must be obeyed by the higher-spin Lagrangian. It turns out to be hard to nd self-consistent interacting theories of these degrees of freedom. It might be that a single light degree of freedom in the spectrum implies the existence of many other higher-spin degrees of freedom. An extreme example of this is the occurrence of in nitely many elds in the Vasiliev theory of massless higher-spin particles in (A)dS [12, 13]. Similarly, interacting theories of an in nite tower of partially massless elds were studied in [14{18]. { 2 { elds are constrained by the Higuchi-Deser-Waldron (HDW) bound in dS space, m2 1)H2 [28{30]. Achieving a strict violation of the tensor consistency relation has so far been restricted to models where a subset of the de Sitter isometries are fully broken. This includes models with large corrections to the kinetic terms of the spinning elds2 [31, 32] and models with broken spatial isometries due to position-dependent background elds [33, 34]. In this paper, we will show that the presence of PM elds during in ation can lead to a strict violation of the consistency relation for h i with a characteristic angular dependence. In addition, the exchange of PM particles creates an enhanced scalar trispectrum i, without producing a scalar bispectrum h i . These features of the cosmological correlators are rather unique and provide a clean detection channel for PM elds during in ation. h h Outline. The paper is organized as follows. In section 2, we review the higher-spin representations in de Sitter space, and derive the interaction vertices with the in ationary scalar and tensor uctuations. In section 3, we determine the imprints of partially massless elds in in ationary correlators, with particular emphasis on the tensor-scalar-scalar bispectrum i and the scalar trispectrum h i. In section 4, we present our conclusions. The appendices contain technical details of the computations presented in the main text. In appendix A, we expand on the free theory of higher-spin elds in de Sitter space. In appendix B, we derive the interaction vertices between a spin-4 eld and the in ationary uctuations, and discuss generalizations to higher spin. In appendix C, we present the calculation of h i and h i in the in-in formalism. Finally, in appendix D, we derive the scaling behavior of the collapsed trispectrum and the squeezed bispectrum using the operator product expansion and the wavefunction of the universe. Notation and conventions. Throughout the paper, we use natural units, c = ~ = 1, with reduced Planck mass Mp2l = 1=8 G. The metric signature is ( + + +), spacetime indices are denoted by Greek letters, ; ; = 0; 1; 2; 3, and spatial indices by Latin letters, i; j; = 1; 2; 3. Conformal time is , and a prime (overdot) on a eld refers to a derivative with respect to conformal (physical) time. Three-dimensional vectors are written in boldface, k, and unit vectors are hatted, k^. Expectation values with a prime, hfk1 fk2 i0, indicate that the overall momentum-conserving delta function has been dropped. The spin and depth of a eld are labelled by s and t, respectively. Partial and covariant derivatives r 1 r s . 2It is possible to evade the HDW bound by writing quadratic actions that are non-covariant, but respect the preferred slicing of the spacetime during in ation. If the non-covariant terms are comparable in magnitude to the covariant terms, then the signs of the kinetic terms for the di erent degrees of freedom of the higher-spin eld can be tuned separately. In this case, one can protect lower-helicity modes from becoming ghost-like, while allowing for mass terms that evade the unitarity bound. In other words, by assuming a large breaking of de Sitter symmetry, there is enough freedom to write ghost-free quadratic actions that violate the HDW bound. We thank Andrei Khmelnitsky for discussions on this point [31]. { 3 { Higher-spin elds during in ation In this section, we introduce the free theory of higher-spin elds during in ation and their couplings to the in ationary perturbations. We begin, in section 2.1, by reviewing the classi cation of particles in de Sitter space. We highlight the existence of a class of partially elds. The free theory of these elds will be presented in section 2.2. Finally, in section 2.3, we discuss the allowed couplings between these elds and the in ationary scalar and tensor uctuations. 2.1 De Sitter representations The relativistic equations of motion for particles of arbitrary spin in at space were derived by Fierz and Pauli in [35], based on the requirement of positive energy. This is equivalent to the condition that one-particle states transform under unitary irreducible representations of the Poincare group [7, 8]. These representations are characterized by the eigenvalues of the two Casimirs of the group: HJEP04(218) C1 C2 P P W W = m2 ; = s(s + 1) m2 ; where P is the four-momentum and W is the Pauli-Lubanski pseudovector. The mass m of a particle is a non-negative real number, whereas its spin s is a non-negative half integer. We distinguish between massive and massless particles. In four spacetime dimensions, massive particles carry 2s + 1 degrees of freedom (transverse and longitudinal polarizations). For the massless case, the theory gains a gauge symmetry which eliminates the longitudinal degrees of freedom, and only the two transverse polarizations remain. Similarly, particles in de Sitter space are classi ed as unitary irreducible representations of the isometry group SO( 1,4 ). The Casimirs of the de Sitter group have eigenvalues [9, 36] C1 C2 1 2 MABM AB = m2 2(s 1)(s + 1)H2 ; WAW A = s(s + 1) m2 s2 + s H2 ; 1 2 (2.1) (2.2) (2.3) (2.4) where MAB are the generators of SO( 1,4 ) with A; B 2 f0; ; 4g and WA is the vedimensional Pauli-Lubanski pseudovector, constructed out of two Lorentz generators. There is no globally timelike Killing vector in de Sitter space. This is clear when describing de Sitter space in embedding coordinates, where all isometry generators correspond to rotations or Lorentz boosts. This implies that the positivity constraints are imposed on certain combinations of mass and spin of the representations, rather than mere positivity of the mass. The (non-scalar, bosonic) representations of the de Sitter group fall into three distinct categories [10, 11]: principal series complementary series discrete series m2 H2 s correspond to masses in the discrete series. 1 is called the \depth" of the eld. Masses that do not belong in one of the above categories correspond to non-unitary representations and are not allowed in the spectrum (see gure 1). For spinning elds, the complementary series consists of a narrow range of mass values. For the representations in the discrete series, the system has gauge symmetries which remove would-be ghost degrees of freedom from the spectrum. The discrete series can be further divided into the following two subcategories: Partially massless elds. Representations of the discrete series with t 6= s 1 correspond to partially massless elds [37, 38]. These elds share some features of massive and massless elds in at space. On the one hand, they carry more than two degrees of freedom, akin to massive elds. On the other hand, their correlation functions have power-law behavior, which is similar to that of massless elds. Unlike massive elds, PM elds may survive until the end of in ation and therefore be directly observable. To derive this e ect, however, one needs to extend the de nition of PM elds beyond the de Sitter limit to general FRW cosmologies.3 This remains an open problem (but see appendix A and refs. [39{42]). In this paper, we instead consider the imprints of PM elds through their conversion to massless scalar and tensor perturbations during in ation. This doesn't require us to follow their evolution after in ation. Massless t = s elds. Representations of the discrete series with maximum depth, 1, correspond to massless elds. Interacting theories of massless higher-spin particles are notoriously di cult to construct in at space, restricted by various powerful no-go theorems [43{45] (see [46, 47] for reviews). In order for particles to remain massless, their interactions need to be protected by a gauge symmetry. Typically, this condition is su cient to uniquely determine the structure of the nonlinear theory.4 3We emphasize that the irreducible representations of the dS group carried by PM elds have no counterpart in Minkowski space. This can also be understood from the perspective of eld theory; only when m takes values that are particular multiples of H 6= 0, the action develops new gauge invariance that removes certain lower helicity components. As a result, there is a discontinuity in the (m; H)-plane at (0; 0) for s 2, since the number of degrees of freedom depends on the way we take the limit H ! 0. 4For massless spin-1 particles, the couplings must satisfy a sum rule which corresponds to charge conservation, as in electromagnetism. For a massless spin-2 particle, the couplings to other matter elds must be universal, thus implying the equivalence principle, as in general relativity. { 5 { The constraints on the couplings of massless higher-spin elds to charged matter imply that these elds cannot induce any long-range forces. The argument relies heavily on the at-space S-matrix, and one might wonder whether the same conclusion holds in (A)dS. It does not: a non-zero cosmological constant allows for the existence of interacting theories of massless higher-spin elds (for reviews, see [48, 49]). Having said that, all known examples require an in nite tower of massless higher-spin elds, and their interactions are highly constrained. It is not known if these examples exhaust the list of possibilities for theories of massless higher-spin elds in curved spacetimes. The in ationary phenomenology of massive particles in the principal and complementary series was studied in [22{24, 27, 32, 50{52], while the observational prospects were considered in [53{57]. In this paper, we will study novel e ects arising from the presence of particles in the discrete series. In particular, we will study the cubic couplings of partially massless elds to the in ationary perturbations and the resulting signatures in cosmological correlators. These give the leading contributions to the cosmological correlators that we consider in this paper, while self-interactions of PM elds do not contribute at tree level. The cubic couplings can be made gauge invariant at the desired order, and we will show an explicit construction of this coupling for PM spin-4 particles. 2.2 Free theory of partially massless elds Let us brie y review the free theory of PM elds. Further details are presented in appendix A. The on-shell equations of motion for a spin-s, depth-t eld are a generalization of the Fronsdal equations for massless particles in de Sitter space [58]: h s + 2 t(t + 1) H2i 1 s = 0 ; r 2 s = 0 ; 2 s = 0 : (2.5) These equations have a gauge symmetry that reduces the number of degrees of freedom from the naive estimate. For massless elds, this gauge invariance allows us to set all 0 2 s components to zero. In the case of PM elds, only a subset of the timelike components of 1 s can be set to zero. The remaining components are labeled by a \spatial spin" n. Moreover, each component has a helicity label, denoted by . We can therefore write the components of a PM eld of spin s and with n spatial indices as i1 in = X n;s "i1 in ; with s n j j and j j = t + 1; ; s. The conventions for the polarization tensors "i1 in , as well as explicit expressions for the mode functions n;s, can be found in appendix A. A few features of the power spectra of elds in the discrete series deserve to be highlighted. First of all, the mode function n;s takes the following schematic form: 1 1 n;s = p2k ( H )n 1 a0 + at + ( k )t e ik ; with constant coe cients an. Notice that this function is elementary rather than transcendental (as in the generic massive case), being of the form e ik multiplied by a rational { 6 { (2.6) (2.7) function in k . The fact that these mode functions are very similar to at space mode functions begs for an explanation. In appendix A, we provide a heuristic explanation for this feature of PM elds [59{61]. The argument shows, in a sense, why PM elds exist and also suggests a way to extend the de nition of PM elds away from the perfect de Sitter limit. Second, the two-point function of elds in the discrete series scales at late times as 1 s i !0 ! 2 ; (2.8) where = 1 t is the conformal dimension of a eld with spin s and depth t. We see that the two-point function freezes at late times for t = 1 and diverges for t > 1. It is unclear whether this carries any physical signi cance, since this two-point function is, of course, gauge dependent. On the one hand, the gauge-invariant curvature tensors built from these higher-spin elds have additional derivatives and their two-point functions therefore vanish at late times. On the other hand, when PM elds are minimally coupled to matter, these late-time divergences can become physical.5 We will return to this issue in section 3.3.2. 2.3 Couplings to in ationary uctuations When the in ationary uctuations are coupled to PM elds, one faces the additional challenge that the allowed interactions must respect the PM gauge symmetry responsible for protecting the masses of these elds. We nd that it is possible to write consistent couplings which generate an interaction of the form , as well as quadratic mixing terms on the in ationary background. The consistency of these couplings should be viewed as an e ective eld theory (EFT) statement. The actions written below will likely require additional degrees of freedom to remain self-consistent at higher orders in interactions, i.e. in order to maintain the gauge symmetry at each order in the uctuations. Nonetheless, if such self-consistent theories exist, and are weakly coupled, then the main contribution to in ationary correlations should come from the leading, minimal vertices presented below. We write a minimal coupling of the form6 ge Z d x 4 p g 1 s J 1 s ( ) ; (2.9) where J 1 s ( ) is a spin-s current, which depends quadratically on the in aton eld , and ge is an e ective coupling strength.7 This is analogous to the minimal coupling between the photon and charged matter in quantum electrodynamics (QED), A J . As in QED, the gauge transformation of 1 s forces the elds contained in J 1 s to be 5The late-time growth of the power spectrum of certain higher-spin elds is worrisome, as it could be the sign of an instability. In order to determine whether the backreaction on the dS geometry is large, it would be most natural to compute the contribution of the higher-spin elds to the background energy density. However, because the form of the higher-spin action on a generic curved background is unknown, the stress tensor is ambiguous. To address the issue, we will therefore consider other physical observables, namely correlation functions of and . Since the couplings to and are gauge invariant, we expect these expectation values to carry physical information about the backreaction of the higher-spin elds. 6We will only consider couplings to even-spin elds in this paper. For odd spins, the conserved currents need to involve at least two di erent elds, which can be achieved e.g. with a complex scalar. 7We will sometimes set H = 1, so that ge becomes a dimensionless quantity. { 7 { charged under the PM gauge symmetry, i.e. they must transform non-trivially if the action including (2.9) is to be gauge invariant o -shell. If we were to extend the theory to higher orders in the interactions, then the combined transformations would dictate the form of additional couplings needed for consistency, as well as possible deformations of the gauge transformations. This is analogous to starting with a at space coupling between a massless spin-2 eld and a scalar, h , demanding the linear gauge symmetry for h and, due to consistency, being led directly to the fully nonlinear, di eomorphism-invariant theory of a minimally-coupled scalar in general relativity (see e.g. [62, 63]). Finding such a nonlinear completion of the PM theory is challenging, and we will content ourselves with studying the leading coupling presented above. For previous literature on the construction 1 s . For concreteness, we will study the example of a spin-4 eld, but generalizations to higher spin are, in principle, straightforward. We will consider two special cases: a massless spin-4 eld and a partially massless spin-4 eld with depth t = 1. We rst determine the conserved currents that can couple to these spin-4 elds. Under the gauge symmetry, the in aton must transform in a non-trivial fashion.8 Leaving the details to appendix B, we quote here the form of the current for a partially massless spin-4 eld J ( ) = r( r ) + ; where the ellipses represent terms that vanish on-shell when contracted with the purpose of computing quantum expectation values, we expand the in aton into the time-dependent background value and its perturbations, (t; x) = (t) + spatially at gauge, (2.10) and (2.9) imply a cubic coupling between the eld . In terms of the gauge-invariant curvature perturbation, = (H= _) , this is L / ge a8 where we have only kept the spatial components of the spin-4 eld. While this is the unique non-vanishing term for a massless eld, not all of the zero components vanish in the partially massless case. Although these non-spatial components can be included in our analysis, because they have the kinematical structure of lower-spin elds, we will opt for the full spatial components of the spinning eld to show the characteristic spin-s e ect. Similar couplings exist for general spin-s, depth-t elds, see appendix B. The mixing between the graviton and a partially massless eld can naturally be obtained by evaluating the coupling (2.9) on the in aton background and perturbing the metric. The resulting coupling is (2.10) 1 4 . For (t; x). In 1 4 and (2.11) L / ge _2 ij00 _ ij : a2 (2.12) The size of this mixing term is correlated with that of the cubic coupling. 8Since the in aton is charged under the gauge symmetry, it is possible for the in aton background (t) to a ect the quadratic structure of the higher-spin action via higher-order couplings. To analyse this requires knowing the precise form of these couplings at quadratic order in the higher-spin eld. In the present work, we assume that any such modi cations are small. We thank Paolo Creminelli and Andrei Khmelnitsky for discussions on this point. { 8 { Notice that this coupling vanishes when evaluated on the background metric. However, as shown in appendix B, it still leads to a nontrivial vertex of the form K ( ) = r + : L / he _ ij00 _ ij : a2 Due to the number of zero components, these mixing terms are only non-trivial if the higher-spin eld has depth t 1. The presence of _ in the prefactor indicates that mixing arises only when the conformal symmetry of the background is broken, consistent with the fact that h i vanishes identically when the conformal symmetry is exact. In section 3, we will study the cosmological imprints of the interactions in (2.11), (2.12) and (2.15). To estimate the allowed sizes of the couplings ge and he , we write the interaction Lagrangian in terms of canonically normalized elds Lint ge a8 | ge 1 + he {z ~ he } p r ij00 _ icj : a2 To get an enhanced coupling, it is desirable to write down an independent mixing operator. For this purpose, we consider the following alternative coupling he Z d x and he is a coupling constant. Again, we can nd the form of this coupling by demanding gauge invariance. The main di erence between (2.9) and (2.13) is that, while we expect the former to be present in the standard minimal coupling scheme, the latter is an additional allowed coupling that we can introduce in our e ective theory. For the case of spin 4, the current takes the form ge . p : (2.17) and icj where Here we are setting H = 1, so that ge and he are dimensionless and that the elds c Mpl ij have order one uctuations. We have used _ 1 and Mpl and are the scalar and tensor amplitudes, respectively, and introduced the tensor-to-scalar ratio r . This rewriting of the Lagrangian makes manifest that the strength of the cubic interaction is determined by ge and the quadratic mixing by the The regime where (2.16) remains perturbative depends on whether or not the interaccombination h~e pr. tion (2.13) is included in the action: If (2.13) is not included, then h~e = ge 1 and the quadratic and cubic couplings are both determined by ge . In order for the quadratic mixing to remain weakly coupled, we then have to impose 2, this implies ge < 1, ensuring that the cubic interaction is also perturIf (2.13) is included in the action, then the quadratic mixing depends on a combination of ge and he , which we have denoted by h~e in (2.16), while the cubic coupling is determined by ge alone. In principle, this allows a large coupling to the tensor mode without a correspondingly large coupling to the scalars.9 The requirements of weak mixing and perturbative interactions now place the following bounds on the couplings:10 ~ he . p ; 1 r ge . 1 : (2.18) 2, the couplings can only saturate (2.18) if the two contributions to h~e cancel to a high degree. The net e ect of including (2.13) in the action is to boost the possible size of both the tensor-scalar-scalar bispectrum and the scalar trispectrum; HJEP04(218) see section 3.2 and section 3.3. interactions for massive spinning elds (i.e. those belonging to the principal series) were considered in the context of the EFT of in ation [66, 67]. In that case, it was found that the quadratic mixing is always tied to the mixing , so that the weak coupling constraint of the latter induces a factor of p implying h~e . 1. In contrast, the fact that PM r on the size of the graviton coupling, elds lack a longitudinal mode means that the mixing vanishes on-shell. As a result, we expect the vertex to remain weakly coupled under a less stringent condition as in (2.18), not constrained by the size of the scalar coupling. 3 Imprints on cosmological correlators We will now study the imprints of partially massless elds in cosmological correlators. In section 3.1, we brie y review the scalar and tensor consistency relations in single- eld in ation. We then compute, in section 3.2 and section 3.3, the impact of an intermediate higher-spin eld on both the tensor-scalar-scalar three-point function and the scalar fourpoint function (see gure 2). We show that these elds can lead to a strict violation of the tensor consistency relation and a characteristic scaling and angular dependence in the four-point function, while giving no contribution to the scalar three-point function. 3.1 Consistency relations Symmetries play a crucial role in constraining in ationary correlation functions. In singleeld in ation, acts as the Goldstone boson of spontaneously broken conformal symmetries, nonlinearly realizing dilatations and special conformal transformations. The corresponding Ward identities imply consistency relations between correlation functions of di erent orders. 9It would also be interesting to consider the case of strong mixing, h~e > 1, as a way to boost the tensor power spectrum without a corresponding e ect on the scalar power spectrum. In this paper, we restrict ourselves to the more conservative case of weak mixing. 10These bounds should be viewed as order-of-magnitude estimates, since various numerical factors appear in actual computations of correlation functions. The correct perturbativity conditions are expressed as bounds on the sizes of correlation functions, see appendix C. i (left) and h For example, the Taylor expansion of the three-point function around the squeezed limit must take the form k1!0 lim h k1 k2 k3 i0 = P (k1)P (k3) X an 1 n=0 k1 k3 n ; with a0 = 1 ns, where ns = 0:968 0:006 is the scalar tilt [68]. The leading term of the squeezed limit of the bispectrum is therefore xed by the scale dependence of the power spectrum [19]. The consistency relation also constrains a1, and the model dependence in single- eld in ation only enters at quadratic order [20, 69{72]. This means that observing any non-analytic scaling behavior of the three-point function between n = 0 and n = 2 would be a clear signature of additional massive elds during in ation. Extra massless scalars would allow for a strict violation of the single- eld consistency relation, i.e. a modi cation of the coe cient a0. Similar consistency relations exist in the tensor sector. For example, the tensor-scalarscalar correlator can also be Taylor expanded around the squeezed limit kl1i!m0h k1 k2 k3 i0 = P (k1)P (k3) E2 (k^1 k^3) X bn 1 n=0 k1 k3 n ; where Es (k^1 k^3) spin-s, helicityk^i1 3 k^3is " i1 is (k^1) denotes the contraction between momenta and a polarization tensor. The two leading coe cients in this expansion are fully determined by symmetry; for example, b0 = (4 ns)=2. This assumes that the only long-lived quadrupolar anisotropies are those sourced by the graviton. Again, there can be non-analytic contributions from extra massive particles as in the scalar case, but this time only with spin greater than or equal to two [27]. However, achieving a strict violation of the consistency relation for tensors turns out to be much more di cult. In particular, it was pointed out in [73] that spin-2 particles cannot a ect the leading term in the tensor consistency relation when the de Sitter symmetries are (approximately) respected, since unitarity forbids the existence of light spin-2 particles with mass 0 < m2 < 2H2. On the other hand, the particle spectrum allowed by dS representations is much richer and exists beyond spin two, as was reviewed in section 2. It is then natural to ask whether higher-spin elds can a ect this conclusion. Another interesting kinematical limit of cosmological correlators is the collapsed limit of the four-point function. This limit probes light states that are being exchanged in the (3.1) (3.2) HJEP04(218) four-point function, and it essentially factorizes into a product of three-point functions [25, 74{77]. This is analogous to the operator product expansion (OPE) limit of conformal correlation functions. In schematic form, we can write where kI = k1 + k2 is an internal momentum. As in the OPE, the expression (3.3) involves a sum over all intermediate states, and the sum is dominated by the uctuations that decay the slowest outside of the horizon. See appendix D for more on this OPE perspective. The Suyama-Yamaguchi (SY) relation [78] bounds the size of the collapsed trispectrum in terms of the size of the squeezed bispectrum [74, 79]: (3.3) (3.4) (3.5) (3.6) ^NL lim lim 56 f^NL 2 ; f^NL ^NL k1!0 12 P (k1)P (k3) 5 h k1 k2 k3 i0 ; 1 h k1 k2 k3 k4 i0 kI !0 4 P (k1)P (k3)P (kI ) ; where we have introduced the following nonlinearity parameters and assumed that they have the same momentum scaling. The SY bound (3.4) is saturated when a single source is responsible for generating the curvature perturbations.11 As we will show below, higher-spin elds provide an interesting example which nontrivially satis es this bound. In particular, we will show that PM elds do not generate any scalar bispectrum, while sourcing a nontrivial trispectrum. 3.2 We now study the e ects of higher-spin particles on the h i correlator12 and demonstrate that they violate the consistency condition (3.2). As was discussed in section 2.3, there exist two types of graviton couplings: one generated from the coupling (2.11) and another from the coupling (2.15). The former has the property that its signature in the tensor bispectrum is correlated with its e ect on the scalar sector, while the latter allows the coe cients to be independently tuned, which can lead to an enhanced signal. 11Strictly speaking, the SY bound is only saturated if the single source goes through a signi cant nonlinear classical evolution on superhorizon scales. This is because, in single- eld slow-roll in ation, the trispectrum from graviton exchange leads to a contribution ^NL = O("), which is in fact parametrically larger than f^N2L = O("2) [80]. These slow-roll-suppressed e ects are typically much smaller than the e ects that we are interested in. 12Partially massless elds do not contribute to h i at tree level, as a spinning eld without a longitudinal degree of freedom is kinematically forbidden to oscillate into a single scalar eld. Hence, h non-Gaussian correlator in which partially massless higher-spin elds can leave an imprint. i is the simplest We will use the following measure of the tensor-scalar-scalar bispectrum amplitude hNL 6 17 X and implies hNL = pr=16 for single- eld slow-roll in ation [19]. where the bispectrum is evaluated in the equilateral con guration, k1 = k2 = k3 k, with vectors maximally aligned with the polarization tensor. Our normalization agrees with [81] The size of the tensor bispectrum due to a higher-spin exchange can be estimated as hNL h i where the top and bottom cases correspond to excluding and including the term (2.13) in the action, respectively. The perturbativity requirements (2.17) and (2.18) imply, respectively, hNL . 8 The tensor bispectrum can be constrained using the hBT T i, hBT Ei, hBEEi correlators of the CMB anisotropies [81, 82]. The forecasted constraints from the CMB Stage IV 0:1 for the local-type non-Gaussianity. The coupling he is currently unconstrained, whereas the non-detection of the trispectrum puts an upper bound on ge (see section 3.3) and consequently the size of the tensor bispectrum. Shape. As we described earlier, only higher-spin elds with depths t = 0 or 1 contribute to h i at tree level. For t = 1, the tensor bispectrum induced by the exchange of a PM eld takes the following form in the soft limit k1!0 lim h k1 k2 1k3 i0 = P (k1)P (k3) Y^s ( ; ') ; (3.7) (3.8) (3.9) (3.10) where Y^s is a spherical harmonic, is an e ective coupling constant de ned below, and the angles are de ned by cos = k^1 k^3 and ei' = " k^3. The polarization tensor for the graviton "ij is built out of two polarization vectors ", " that span the plane perpendicular to k^1, which are xed up to a phase (see appendix A). We note that the bispectrum (3.10) has the same scaling as the leading term in the tensor consistency relation (3.2). Thus, elds with t = 1, s > 2 generate strict violations of (3.2) by moving the value of b0 away from its predicted form. The case t = 1, s = 2 doesn't constitute a violation, as it corresponds to the massless graviton. We dub this shape \local tensor non-Gaussianity", in analogy to the non-Gaussian shape that violates the scalar consistency relation. Due to the kinematics of the mixing with the graviton, only the helicity = 2 modes of the higher-spin eld contribute in the three-point function. Since these modes have the same amplitude, we have absorbed all of the numerical factors into than order unity in the weakly non-Gaussian regime. (The precise overall normalization as ge h~e pr, which is required to be less ˆ Y 2 +s 0 ˆ Y −200 π/2 θ π correspond to spins 4, 6, and 8, respectively. = cos 1(k^1 k^3). The solid, dashed, and dotted lines a function of the spin of the eld can be found in appendix C.) For t = 0, the bispectrum is suppressed by a factor of k1=k3 relative to the t = 1 case. The angular dependence is given by the usual spherical harmonic of degree (spin) s and order (helicity) , which can be factorized into longitudinal and transverse parts Y^s ( ; ') = E ( ; ')P^s (cos ) ; (3.11) where P^s (x) / (1 x2) =2Ps (x) is a version of the associated Legendre polynomial with a suitable normalization. On top of the usual quadrupole moment due to the external tensor mode, we see that the exchange of a higher-spin eld induces an extra longitudinal angular component. Aligning k1 with the z-axis, we can express the hard momentum in spherical polar coordinates as k^3 = (sin cos '; sin sin '; cos ). The angular dependence can then be written as a product of E only of . For example, some explicit expressions are = sin2 e i ' and the longitudinal part which is a function X = 2 Y^s ( ; ') = <> 36 (35 + 60 cos 2 + 33 cos 4 ) sin2 cos 2' (210 + 385 cos 2 + 286 cos 4 + 143 cos 6 ) sin2 cos 2' s = 8 s = 4 s = 6 : (3.12) Figure 3 shows the angular dependence as a function of the angle for ' = 0. We can read o the spin of the particle by measuring the period of the oscillations. For the purpose of data analysis, having the full bispectrum shape available would also be helpful. The expression of the tensor bispectrum for general momentum con gurations can be found in appendix C. Treating the soft tensor mode as a classical background, the squeezed tensor bispectrum would also contribute an anisotropic correction to the power spectrum in the following schematic way [83]: h h k k0 i0 q = P (k) 1 + hNL q Y^s ( ; ') i h P (k) 1 + Qi1 is ki1 ^ k^is i ; (3.13) generated by higher-spin elds.13 for a given realization of the tensor mode, with the angles and ' now de ned in terms of k and q. The external tensor mode is not directly observable, but its variance Q is, after averaging over all long momenta. It would be interesting to measure this anisotropic 2 2 hQi1 is i e ect due to PM elds in large-scale structure observations. See e.g. [73, 84, 85] for forecasts on the detection limit of Q2 from future experiments and [86] for related work on anisotropy We next consider the contribution of the interaction (2.11) to the correlator h i. We will rst focus on the special case of depth t = 1 elds which freeze on superhorizon scales, and then study the imprints of t > 1 elds which continue to grow after they exit the horizon. Fields of depth t = 0 are not addressed in detail, as they decay on superhorizon scales and are hence phenomenologically less interesting. Details of the relevant in-in calculations can be found in appendix C. 3.3.1 Depth t = 1 elds For a given spin s 2, there exist s 1 partially massless states. The depth t = 1 case is rather special, as it is the only state which freezes on superhorizon scales. In the following, we discuss the signature of such states on the h i correlator via the second diagram in We use the standard measure of the size of the trispectrum where the right-hand side is evaluated at the tetrahedral con guration with ki = k and k^i 1=3. The estimated size of non-Gaussianity induced by the exchange of a PM eld is Imposing the weak coupling constraints (2.17) and (2.18), we nd gure 2. k^j = (3.14) (3.15) (3.16) g 2 e 2 : NL . 8 <r 1 ; : 2 ; where the top and bottom cases correspond to excluding and including the term (2.13) in the action, respectively. Of course, the shape of the correlator crucially a ects the 13When higher-spin elds acquire classical background values, they can also leave statistically anisotropic imprints in higher-order correlators [87]. We thank G. Franciolini, A. Kehagias, and A. Riotto for sharing their draft with us. observability of the signal. As we will describe below, the trispectrum under consideration has a similar scaling behavior in the soft limit as the \local" trispectrum, which arises in multi- eld in ationary models (see [88] for a review). For comparison, the current observational bound on the size of the latter is NloLcal = ( 9:0 7:7) 104 [89]. This bound roughly translates into ge < 10 2. Since there is no bispectrum counterpart for this signal, our scenario is an example which satis es the SY relation in the most extreme manner. Shape. Let us describe the collapsed limit of the trispectrum, leaving the details of the full shape to appendix C. For partially massless elds with depth t = 1, we get kI !0 g 2 e lim h k1 k2 k32k4 i0 = P (k1)P (k3)P (kI ) X s j j=2 where the amplitude of each helicity mode is Es = 64 2(2s 1)!!]2s!(s + 1)!(s )!( 1 )!!(s 2)!(s + )!( + 1)! 2)! : We see that the overall scaling behavior for a given depth is independent of the particle's spin. As advertised before, the scaling is the same as the local trispectrum, being proportional to P (k1)P (k3)P (kI ). The amplitude of each helicity mode is uniquely determined by its spin. The angular dependence is again factorized into the transverse part E and the longitudinal part P^s . The transverse polarization tensors project the momenta k1 and k3 onto the plane perpendicular to kI . The trispectrum is therefore a function of the angles cos k1 k^I and cos 0 ^ k3 k^I between the vectors, as well as the angles ' and '0 on the ^ projected plane with respect to the polarization tensor. The values of the projection angles depend on the two chosen polarization directions on the plane, but the di erence is independent of this choice (see gure 4). We can write the full angular dependence as Y^s ( ; ')Y^s ( 0; '0) / e i Ps (cos )Ps (cos 0) ; where the factor of (1 cos2 ) =2 = sin inside the associated Legendre polynomial contributes to the transverse part. Note that, as a consequence of the addition theorem for spherical harmonics, we have [24] Esj j Y^s ( ; ')Y^s ( 0; '0) ; is the angle between k^1 and k^3. This is not quite the angular dependence that we observe, since each helicity of the higher-spin eld has a di erent amplitude, and some of the helicities are missing for the PM eld. For fs; tg = f4; 1g, we have 4 X j j=2 E4j j Y^4 ( ; ')Y^4 ( 0; '0) / 14 15 (5 + 7 cos 2 )(5 + 7 cos 2 0) sin2 sin2 0 cos 2 + 75 cos cos 0 sin3 sin3 0 cos 3 + sin4 sin4 0 cos 4 : (3.21) 105 4 the exchanged (partially) massless eld is contained in the ( ; 0; ) dependence of the trispectrum. As follows from (3.19), the helicity- component of the eld is responsible for the / cos term in (3.21). We see that each helicity mode contributes with a distinct angular dependence, with all of the amplitudes being roughly of the same size. The factorization of the angular dependence into the sum over polarizations of the intermediate particle is analogous to what happens for at-space scattering amplitudes when the intermediate particle goes on-shell, which is a consequence of unitarity. In the case of cosmological correlators, there is the precise relationship between the amplitudes of di erent helicities, which is a consequence of conformal symmetry. 3.3.2 Depth t > 1 elds We now study the impact of depth t > 1 partially massless elds on the scalar trispectrum, which includes, in particular, the massless case t = s 1. At tree level, this class of elds generates characteristic divergences in h i, when evaluated in the collapsed con guration. In section 2.2, we encountered the peculiar result that the two-point functions of depth t > 1 partially massless elds diverge at late times; cf. eq. (2.8). However, the physical content of this fact is not clear, since the two-point function is gauge dependent. The in uence of the higher-spin eld can instead be tested in a gauge-invariant manner by calculating its e ect on h i via the exchange diagram in gure 2. In the collapsed limit, this trispectrum is directly sensitive to the higher-spin two-point function14 and its scaling behavior due to the exchange of a spin-s, depth-t PM particle is given by kI !0 g 2 e lim h k1 k2 k32k4 i0 = P (k1)P (k3)P (kI ) k1k3 k2 I t 1 s X j j=2 Esj;tj Y^s ( ; ')Y^s ( 0; '0) ; (3.22) 14This fact is made manifest in the wavefunction of the universe formalism; see appendix D. with Es;t de ned in (C.29). Notice that this diverges at a rate that is faster than P (kI ) for t > 1. There are several e ects which could make the strongly divergent behavior of (3.22) less extreme. First, loop diagrams with more intermediate higher-spin elds would also lead to a singular behavior that goes as ge2n(k1k3=kI2)n(t 1), where n is the number of loops.15 Although these contributions are higher order in ge , they would be more singular than the tree-level diagram if t > 1. In that case, it does not make sense to only consider an individual diagram, but we would instead need to sum all of them. It is conceivable that the result after the resummation will behave more tamely in the collapsed limit. Second, consistency of the theory may require the introduction of many new degrees of freedom and interactions which will also contribute to the scalar trispectrum. If so, the multitude of particles may soften the collapsed limit of h i. This is the behavior claimed in [90] which calculates, via a dual description, the four-point function of a conformally-coupled scalar due to the exchange of the tower of massless higher-spin elds in the minimal Vasiliev theory. In that case, the complete correlator behaves much more softly in the collapsed limit than the individual exchange diagrams do, which have similarly divergent behavior to (3.22). 4 Conclusions In this paper, we have studied the eld theory of partially massless elds during in ation and discussed their imprints on cosmological correlators. Our main conclusions are: Partially massless elds can have a consistent linearized coupling to a scalar eld with arbitrary mass. We have constructed the corresponding conserved currents, and derived the relevant couplings between these higher-spin elds and the in ationary scalar and tensor perturbations. Partially massless elds lead to a vanishing scalar bispectrum, but a non-zero trispectrum. The trispectrum has an unsuppressed behavior in the collapsed limit and a distinct angular dependence. Partially massless elds can lead to a strict violation of the tensor consistency relation while respecting de Sitter symmetry, providing a loophole to the theorem of [73]. This local tensor non-Gaussianity is analogous to the sensitivity of the scalar bispectrum to extra light scalar species. Partially massless elds can mix quadratically with tensor modes, but not scalar modes. The reason is purely kinematical | there is no longitudinal mode that mixes with the single scalar leg. This means that we can potentially enhance the tensor power spectrum, while not altering the scalar power spectrum. To realize this intriguing possibility would require understanding the regime of strong mixing of PM elds with the in ationary tensor modes. 15This scaling behavior follows from a simple estimate of the contribution from ladder diagrams involving the cubic vertex at n loops. For the contribution due to vertices with more legs, a more detailed analysis is necessary. (top) [89, 91{93] and tensor-scalar-scalar bispectrum (bottom) [81, 82, 94, 95] for local nonGaussianity. The red and green regions correspond to the sensitivity levels of Planck [89] and forthcoming experiments, respectively. The \gravitational oor" refers to the guaranteed level of non-Gaussianity sourced by gravitational nonlinearities during in ation, while \non-perturbative" denotes the strongly non-Gaussian regime. Partially massless elds do not decay outside the horizon, making them, in principle, directly observable after in ation. Predicting the nal spectrum of PM elds is challenging, because it requires coupling these elds to the matter uctuations in the late universe, something we did not address in this paper. It also requires understanding the meaning of PM elds away from the de Sitter limit. Figure 5 is a schematic illustration of current and future constraints on the scalar trispectrum and the tensor-scalar-scalar bispectrum. We see that future surveys will probe deeper into the allowed parameter space of primordial non-Gaussianities. We also see that there are still many orders of magnitude of parameter space left to be explored before we would hit the gravitational oor. Perhaps, within this unexplored territory, there will be new surprises and a rich cosmological fossil record waiting to be discovered. Acknowledgments We thank Nima Arkani-Hamed, Matteo Biagetti, Cora Dvorkin, Kurt Hinterbichler, Austin Joyce, Andrei Khmelnitsky, Juan Maldacena, Soo-Jong Rey, David Vegh, and Matthew Walters for helpful discussions, and Dionysios Anninos, Paolo Creminelli and Antonio Riotto for comments on a draft. D.B., G.G. and G.P. acknowledge support from a Starting Grant of the European Research Council (ERC STG Grant 279617). G.G. is also supported by the Delta-ITP consortium, a program of the Netherlands organization for scienti c research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). H.L. thanks the Institute for Advanced Study for hospitality while this work was completed. G.P. acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie-Sklodowska Curie grant agreement number 751778. We acknowledge the use of the Mathematica package xAct [96]. By increasing the depth, we obtain lighter elds, whose number of degrees of freedom is smaller than those of their heavier cousins. In more detail, the depth interpolates between the number of degrees of freedom of a massive eld of spin s in at space; i.e., 2s + 1, and 2, the number of degrees of freedom of a massless eld.20 Finally, we should mention an interesting proposal to select the massless graviton action, using conformal gravity as a starting point [60] (see also [99]). By imposing Dirichlet boundary conditions on the late-time amplitudes for the uctuations,21 Maldacena showed that the wavefunction of the universe for Einstein gravity can be computed using conformal gravity. It would be interesting to study whether a similar procedure would select a partially massless eld, using (A.29) as a starting point. The procedure outlined in [60] does not work for the case of the spin-2 eld with m2 = 2H2 [100], as ghost degrees of freedom appear in the interaction vertices beyond cubic order. Nonetheless, it was shown in [100] that the spin-2 PM eld can propagate in an Einstein spacetime (see also [39]). For our purposes, having a quadratic action for PM elds in a cosmological FRW background would su ce to determine whether the power spectra of higher-spin elds leave interesting imprints in the late universe. B Spin-4 couplings In this appendix, we derive the form of the couplings of a scalar eld and the graviton to a (partially) massless spin-4 eld that were used in the main text. We also comment on the generalization of our couplings to partially massless elds of arbitrary spin. B.1 Conserved current We will construct the linearized coupling between (partially) massless spin-4 elds and a scalar eld via the standard Noether procedure.22 For concreteness, we will present expressions for dS4, but these results can easily be generalized to AdS or any number of dimensions. We consider a coupling of the form ge Z d x 4 p g 1 s J 1 s ( ) : First, we write the most general totally symmetric rank-s tensor up to s derivatives as s=2 X k=0 J 1 s = kr( 1 k r k+1 s) + ; s = 2; 4; ; Bunch-Davies vacuum. in AdS. 20In reality, the counting should begin with 2s + 2 and descend in pairs from the highest weight representation; the extra degree of freedom is ghost-like and decouples in the free theory [98]. In practice, the highest depth partially massless eld has 2s degrees of freedom, and the reduction of its number of degrees of freedom by unity is attributed to Weyl symmetry. 21These are the natural boundary conditions one must impose in order to compute the wavefunction of the universe. From the wavefunction of the universe one can extract in ationary correlation functions by using j j2 as a probability distribution. When we compute in ationary expectation values directly, we do not impose late time boundary conditions on the eld variables, but only that the initial state is the 22See [101{103] for similar Noether constructions of higher-spin currents for complex and conformal scalars (B.1) (B.2) where the ellipses denote terms that have contractions among the derivatives with appropriate factors of the metric. To begin with, let us construct a current for a massless spin-4 eld. This eld changes under a gauge transformation as = r( ) : r J four derivatives: Invariance of the coupling under this transformation implies that the current is conserved, = 0. We begin by writing down the most general form of a spin-4 current up to J = 1 r 4 + 2r r 3 + 3r 2 r 2 + 4gr2 r 2 + 5gr r 3 + 6H2gr r r 2 + 9H2g2r r + 10H4g2 2 sym ; where we have suppressed the indices and g stands for the metric. Under the gauge transformation (B.3), the coupling these terms, we also introduce a transformation rule for the scalar eld J will generate terms such as r r 4 . To cancel = ( 1r + 2r r + 3r r + 4r ) : We demand that the coupling is invariant under gauge transformations for both the spinning and scalar elds. We nd that a massless spin-4 eld can couple to scalar elds with arbitrary mass, at least at the linearized order.23 After many integrations by parts and dropping boundary terms, the o -shell gauge invariance xes the coe cients to be (B.3) (B.4) (B.5) 3 = 7 = 3 1 + 26 1 ; 3 where we have given the result for a massless scalar eld. These conditions leave four parameters unconstrained, while we expect them to be fully xed if the full theory were known. Nevertheless, this ambiguity does not have any observable consequences and goes away when we evaluate the coupling on-shell. In the transverse and traceless gauge, all terms in the current that are proportional to the metric drop out in the coupling. Essentially, the form of the coupling is uniquely xed on-shell (up to terms that are related by integrations by parts). The current can be put in the form J = r( r ) : (B.7) This leaves only one free parameter for the on-shell coupling, namely the coupling constant ge . Moreover, all the zero components of the spin-4 eld vanish on-shell. The covariant derivatives will lead to terms that involve ij / 0 ij , but these do not contribute since 23Further imposing the tracelessness condition of the current forces the scalar to be conformally coupled with mass m2 = 2H2. the polarization tensor of is traceless, "iijk = 0. This means that we can simply replace the covariant derivatives in the current with partial derivatives. The on-shell current thus takes a particularly simple form One can perform a similar procedure of constructing an o -shell gauge-invariant linearized coupling between massless spinning elds and a scalar eld for general spin. However, it is easy to see that the on-shell coupling will, again, take a unique form. Up to integration by parts, this is a8 J = a2s We will compute the correlation functions that arise from this coupling in appendix C. Next, let us consider the partially massless case. The on-shell gauge transformation of a spin-4, depth-1 partially massless eld is [cf. (A.10)] = r( r r ) + H2g ( r ) ; with the gauge parameter subject to the conditions (A.11).24 The coupling (B.1) is invariant under this transformation if the current satis es the condition r J + H2r J = 0 : Again, we will start with the most general ansatz for the current (B.4). Since the gauge transformation (B.10) contains more derivatives than that for massless elds, this time we require a scalar transformation involving ve derivatives. It turns out that we can recast the most general totally symmetric scalar transformation in the form = (~1H4 r + ~2H2 r + ~3r r + ~4 r r + ~5 2 r + ~6r via the use of the on-shell conditions for the gauge parameter . The coupling will be invariant under the transformations (A.10) and (B.12) if (B.8) (B.9) (B.10) (B.11) r ) ; (B.12) 3 = ~1 = 1 + 2 ~6 ; 5 = 4 1 ~ 2~4 for a massless scalar eld. Again, the form of the coupling simpli es greatly on-shell, taking the form (B.7). Evaluating the coupling in components, we can determine the coe cients of the couplings for both the spatial and non-spatial components of the PM eld. Generalizing to higher spins and focusing on spatial components, the on-shell coupling, again, uniquely takes the form (B.9) for elds of any depth. 24To ensure full o -shell gauge invariance, one needs to introduce a number of auxiliary lower-spin elds. Here we will restrict ourselves to the physical degrees of freedom of the on-shell PM eld. We are also interested in getting a interaction vertex, which would contribute to the correlator h i. This coupling can naturally be generated by evaluating the scalar elds in the coupling (B.1) on the background and perturbing the metric.25 In principle, this procedure could produce terms with and without derivatives acting on ij . Naively, the latter would violate the tensor consistency relation even when is massive, in contrast to the result of [73]. However, we will show that such terms are indeed absent. We take _ to be constant, which implies that r0 r = ijklrij rkl = ijkl 0 0 _2 ij kl = 0. Starting from (B.7), we have where the last expression follows from perturbing the metric in spatially at gauge. We see that this also produces a tadpole term iijj . Note that the spatial trace itself gives gij gkl ijkl = a 4( iijj 2 ijkk ij ) : Demanding the absence of the tadpole exactly cancels the ij term. Consequently, the vertex only involves _ ij . Using the traceless condition, we will denote this coupling for the general spin case by ge _2 ij0 0 _ ij : a2s 2 he Z d x 4 p g vertex is proportional to _2 and is correlated with the cubic vertex An alternative coupling between and is of the form where K is linearly dependent on .26 The most general form of K up to four derivatives is K = ~1r( ) + ~2g( r ) + ~3g( r ) + ~4g( g ) + ~5g( g ) The last three terms vanish identically upon using the equation of motion of . Imposing the on-shell conservation condition r K + H2r K = 0, we obtain ~1 = 16H2 ~2 : imposed, g r graviton. For example, we have While the remaining terms also vanish when the background on-shell gauge conditions are = 0, g = 0, they can still induce nontrivial couplings to the r = ijklrijkl = ijkl 0 p 0 _ ij 0k pl = a 4H2 H iijj 2H ijkk ij + 3 2 ijkk _ ij _ : 25The same procedure would also yield and interactions, which can contribute to the correlators h 26Ni oatnedthhat this coupling induces terms of order O( ; i, respectively. . The o -shell consistency of keeping both J and K types of couplings then requires introducing an extra scalar transformation rule of the form = O( ; ). This can always be done at linear order in . ) under the gauge transformations of and (B.15) (B.16) (B.17) Again, only the _ ij term remains after cancelling the tadpole. The other term gives r = g ij r ij = g gimgjn = a 4H ~ii ij 0mn _ ij ijkl ij ~ij + 1 2 ~ij _ ij _ ; (B.21) = g denotes the trace. On the other hand, the tadpole gives g gij ij = g gij ij + g gij ij + g gij ij = ~ii ij ijkk ij ~ij : (B.22) (B.23) (C.1) (C.2) (C.3) The same logic prevents couplings involving ij without a time derivative.27 We see that both terms will then lead to the generic form of the coupling _ ij0 0 _ ij ; a2 which, in contrast to (B.16), is proportional to a single factor of _ and can be independent from the cubic vertex In this appendix, we compute cosmological correlators involving an exchange of a (partially) eld. We provide details of the computation of the scalar trispectrum and the tensor bispectrum in section C.1 and section C.2, respectively. In appendix D, we will analyze the soft limits of these correlators by applying the operator product expansion to both the wavefunction of the universe and to the nal in-in correlator. Preliminaries. The expectation value of an operator Q at time 0 is computed by h jQ( 0)j i = h0j hTei R 01 d HI ( )i Q( 0) hTe i R 01 d HI ( )i j0i ; where j i (j0i) is the vacuum of the interacting (free) theory, T (T) denotes (anti-)time ordering, and HI is the interaction Hamiltonian. We use the i prescription and replace (1 + i ) in the time integrals to evaluate the expectation value in the interacting vacuum. To compute quantum expectation values, we follow the usual procedure of quantization. We promote the elds and to operators and expand in Fourier space ( ; k) = k( )ay(k) + h:c: ; i1 in (k) = i1 in (k^) s;s(k; )by(k; ) + h:c: ; where the creation and annihilation operators obey the canonical commutation relations [a(k); ay(k0)] = (2 )3 (k k0) ; [b(k; ); by(k0; 0)] = (2 ) 0 (k k0) : 3 27The tadpole cancellation is more manifest in the language of the EFT of in ation, where graviton couplings are generated through the extrinsic curvature Kij 12 _ij in comoving gauge. X j j=t+1 The mode functions of and are given by k = H _ p 2k3 (1 + ik )e ik ; k = H p Mpl 2k3 (1 + ik )e ik : (C.4) The mode functions of elds in the discrete series can be derived using the formulas given in appendix A. The tree-level diagrams that we will compute have two interaction vertices. Expanding the in-in master formula (C.1) to quadratic order gives where it is understood that each side is evaluated in the appropriate vacuum state. The result for the tree-level exchange of a spin-s eld is given by d~ hHI ( )Q( 0)HI (~)i 2Re d~ hQ( 0)HI ( )HI (~)i ; Z 1 d 1 (C.5) HJEP04(218) T1 2Re[T2] + 23 perms ; 1 2 3 4E12I E34I 1 2 3 4E12I E34I Z 1 a2s 4 3 4 I ; 1 a2s 4 3 4 I ; hQ( 0)i = 1 d 1 C.1 s X j j=t+1 (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) (C.13) (C.14) where we have suppressed the time arguments (e.g. the mode functions outside of the integrals are evaluated at = 0) and de ned Eabl s;s(ki) ; k^i1 a k^ais=2 k^is=2+1 b k^bis "i1 is (k^l) : We have also set H = 1, which can be trivially restored. These integral formulas are even applicable for massive elds in the complementary and principle series upon setting t = 1 in (C.6). Spin-4 PM particles. Before presenting a formula for arbitrary spin, let us consider the example of a fs; tg = f4; 1g eld. Using the mode functions (A.18){(A.20), the nontime-ordered integral T1 can be expressed in the following compact form T1 = I1 = ( 4 X I1 ; j j=2 E12I E34I )4Aj j k1k2k3k4kI3 Kj j(k1; k2; kI )Kj j(k3; k4; kI ) ; where we have expressed the amplitude in terms of the dimensionless power spectrum and de ned A2 = 1=32, A3 = 1=1960, A4 = 1=686000, and K2(ki; kj ; kl) = J0(ki; kj ; kl) + J1(ki; kj ; kl) K3(ki; kj ; kl) = 5J0(ki; kj ; kl) + 5J1(ki; kj ; kl) + J2(ki; kj ; kl) K4(ki; kj ; kl) = 140J0(ki; kj ; kl) + 140J1(ki; kj ; kl) + 55J2(ki; kj ; kl) + 13J3(ki; kj ; kl) ; (C.15) with ki1 is ki1 + + kis . The momentum scaling in (C.12) is correct, since Kn (or Jn) scales as 1=k. For the time-ordered integral T2, the inner layer instead consists of an inde nite integral Z kln ki2jl (1 + n; ikijl ) + kij kijl (2 + n; ikijl ) + kikj (3 + n; ikijl ) an integer where The function T2 can then be written as 1 Jn(ki; kj ; kl) d (1 + iki )(1 + iki )(ikl )ne ikijl = kln n! ki2jl + (n + 1)! kij kijl + (n + 2)! kikj ; ki3j+l n ki3j+l n (n; x) = (n n 1 1)! e x X xm : m=0 T2 = 4 X I2 ; j j=2 in terms of the function (C.16) (C.17) (C.18) (C.19) (C.20) (C.21) (C.22) (C.23) (C.24) where (n; x) is the incomplete gamma function, which takes the following form when n is I2 4 = L00 + Lf01g + L11 ; I2 3 = 25 L00 + 25 Lf01g + 5 Lf02g + +25 L11 + 5 Lf12g + L22 ; I2 2 = 19600 L00 + 19600 Lf01g + 7700 Lf02g + 1820 Lf03g + 19600 L11 + 7700 Lf12g + 1820 Lf13g + 3025 L22 + 715 Lf23g + 169 L33 ; with Lfmng Lmn = ( Lmn + Lmn and 4 E12I E34I k1k2k3k4kI3 1 not very illuminating. Although it is possible to obtain a closed-form expression for Lmn, the result is lengthy and In the collapsed limit, kI k1 k2, the expressions simplify dramatically. This is because the spin-4 mode function in the long-wavelength limit simpli es to d (1 + ik1 )(1 + ik2 )(ikI )m Ln(k3; k4; kI ) ei(kI k1 k2) : 4;4(k ! 0) / H3k3=2 4 : e ik In this case, the kI dependence can be pulled out of the integrals, after which it is easy to see that the trispectrum scales as kI 3 in the limit kI ! 0. In this limit, we nd I1 + I2 = 25( )4Cj j E1(1kI1Ek333kIIjs)=34 ; with coe cients C2 = 1=32, C3 = 5=392 and C4 = 1=35. Di erent helicity modes contribute with di erent but roughly similar amplitudes. The angular dependence now becomes E11I = Es (k^1 k^I ) = E (k^1 k^I )Ps (k^1 k^I ) ; factorizing into the transverse and longitudinal parts. The nal expression for the trispec P (k1)P (k3)P (kI ) X Cj j Y^s ( ; ')Y^s ( 0; '0) ; (C.27) Arbitrary spin. The general formula for the exchange of a spin-s, depth-t eld in the lim h k1 k2 k32k4i0 = P (k1)P (k3)P (kI ) X Ft Esj;tj Y^s ( ; ')Y^s ( 0; '0) ; k1k3 k2 I trum in the collapsed limit is then with angles de ned in section 3.2. collapsed limit is where g 2 e ;s( ! 0) = Zs 21=2 t(2t)! e ik i p t! ( k ) +t 1 n+1;s( ! 0) = 1 + t + n k n;s( ! 0) : times as relation 4 j j=2 t 1 s j j=2 (2 Es;t s!(s + t)!(s t 1)! 2t+2 t! 1)!!(2t)! 2 : denotes the amplitude for each helicity and Ft(k1; k3; kI ; 0) is a function that can depend logarithmically on some of its arguments, with an IR cuto 0. The cases without any log dependences are F0 = O(kI3), F1 = 25 and F3 = 36. Since this function is just constant for t = 1; 3, the scaling in kI for these cases is given by (C.28). On the other hand, we see that there is an extra suppression for t = 0 compared to the naive scaling that is suggested by the late-time behavior of the intermediate eld. For general depths, the functional form of Ft becomes more complicated as t increases. However, the overall scaling behavior in kI is always xed by t as in (C.28) (with a few exceptions including t = 0). The trispectrum therefore becomes singular in the collapsed limit for t > 1, in the sense that it diverges at a rate faster than P (kI ) does. The derivation of (C.28) can be found in the insert below. Derivation of (C.28). By taking the collapsed limit, we probe the late-time behavior of the intermediate eld. The spin-s, helicitymode with spatial spin n = behaves at late The late-time behavior of the n = + 1 mode can then be obtained using the recursion (C.25) (C.26) (C.28) (C.29) (C.30) (C.31) where the numerical prefactor given by (C.29) xes the amplitude of the helicitymode. In the small kI limit, the kI dependence drops out of the integrals, giving the overall scaling behavior as in (C.28). The form of Ft can be determined by computing the integral ik1 0)2(1 + ik3 0)2e2i(k1 k3) 0 Z 0 d 1 1 t (1 + ik1 )2e 2ik1 Z 0 d~ 1 ~1 t (1 ik3 ~)2e2ik3 ~ ; (C.34) This means that the n = s mode will behave as It can then be shown that the two-point function in the late-time limit becomes e ik p t! ( + t)! ( k )s+t 1 : s;s s;s( and then taking 0 ! 0 limit and multiplying by an appropriate symmetry factor. C.2 The tensor-scalar-scalar correlator with a general spin-s eld exchange is given by X 0= 2 h k1 k2 k3 i0 2ge he Re[B1 B2 B3] + 5 perms ; B1 = _3(k1k2)s=2 E1203["ij (k^3)"ij0 (k^3)] 1 2 3 E1203["ij (k^3)"ij0 (k^3)] 1 2 3 E1203["ij (k^3)"ij0 (k^3)] 1 2 3 Z 0 Z 0 Z 0 1 1 d d d a2s 4 1 2 3 a2s 4 1 2 3 0Z 0 0Z Z 1 1 as 3 ( 3 )0 ~3 0 ; (C.36) as 3 ( 3 )0 ~3 0 ; (C.37) 1 as 3 ( 3 )0 ~3 0 1 a2s 4 1 2 3 0 ; (C.38) 2;s(ki) denotes the mode function with n = 2 spatial components. These integral formulas are valid for elds with t = 1; 0; 1.28 When summing over the PM eld helicities, note that some of the integrals that involve taking the contraction of the transverse polarization tensors of the same helicity will vanish. This is because transverse polarization tensors are built out of two polarization vectors "i , which are null, "i "i = 0, and satisfy ("i ) = "i . This means that the only combination = 0 to B3. Our normalization for polarization tensors "is1 is "i1 is = 2s s gives a factor of 4 from the contraction. Spin-4 PM particles. function for n = = 2 is given by Again, let us rst specialize to the case fs; tg = f4; 1g. The mode (C.32) (C.33) (C.35) (C.39) ~k=2 = i p e ik k2(1 + ik ) 2k3 10p70 2H3 28Recall that t > 1 lacks a helicity-2 degree of freedom and that t = 1 indicates that the eld belongs to either the complementary or principal series. B1 = B2 = B3 = 3 2 3 2 Z 0 0 0 3 3 2 E123 where we have suppressed the arguments (k1; k2; k3) of the functions Kn, Jn, and Pn, the latter of which is de ned by Pn(ki; kj; kl) i (1 + ikl )Ln(ki; kj; kl) = (n + 1)! n(n5ki + kjl)(2n3kj2 + n4kl2 + n42kjkl) + ki2 2n3ki + 2(n42 1)kj + 3(n52 Not surprisingly, this has exactly the same structure as the graviton mode function. The only di erence is the spin-dependent normalization constant and extra powers of the scale factor due to the (conformal) time components of the PM With the mode functions being simple algebraic functions, it is lengthy but straightforward to compute the integrals (C.36){(C.38) as in the previous section. Skipping the details of the computation, we nd that the integrals can be expressed in the following compact form (C.40) (C.41) (C.42) (C.43) 3)kl o ; (C.44) (C.45) When the external tensor mode becomes soft, k1 k2 k3, a bunch of terms becomes unimportant and the bispectrum takes a considerably simpler form. First, we note that Pn scales as k1=k3 relative to Kn and Jn in the squeezed limit, which implies that we can neglect B3. Moreover, only J0 survives in the squeezed limit and Jn>0 is subleading in k1. Taking the permutations for which the PM eld carries the soft momentum, the nal result is given by k1!0 ~ lim h k1 k2 k3i0t=1 = P (k1)P (k3) Y^4 ( ; ') ; 1 denotes an e ective coupling strength, with angles de ned in section 3.2. Imposing the perturbativity bounds of section 2.3, this parameter naively needs to be much smaller than unity. However, the smallness of the overall numerical factor is due to the normalization of the 00ij mode function in the vertex, which should be taken into account when setting a bound on h~e . The correct perturbativity condition of the above correlator is then ~ . 1. Arbitrary spin. The tensor squeezed-limit bispectrum for an arbitrary spin-s eld with depth t 2 f0; 1g is k1!0 1 lim h k1 k2 k3i0 = P (k1)P (k3) k1 k3 1 t Y^s ( ; ') ; where we de ned Ns;t ge h~e pr, with 405 s!(s + t)! Ns;t = 4t+2 (2s 1)!!(s + 1)!(s + 2)!(t + 2)! t! (2t)! 2 : (C.46) Unlike in the case of the trispectrum, only the helicities = 2 contribute. As a result, only t 1 elds can contribute, and the t = 0 eld leads to an extra suppression in the squeezed limit relative to the t = 1 case. The derivation of the spin-dependent amplitude can be found in the insert below. This amplitude would be di erent for di erent types of vertices. Typically, the higher the spatial spin of , the larger the amplitude. In other words, Ns;t would be larger for interactions with more number of spatial derivatives, e.g. ijk @k _ ij . Nonetheless, the weak coupling constraints imply . 1 for all types of interaction vertices. Derivation of (C.45). Taking = 2 in (C.32), we get HJEP04(218) ! 0) = Z =2 21=2 t(2t)! (s + t)! s i p t! (2 + t)! ( k )s+t 1 e ik : 2;s = Z =2 21=2 t(2t)! e ik 2 s i p t! ( k )t+1 s;s 22;s( 2 4 N~s;t k2t s+3H2(s 1) s+2t ; while (C.30) gives In the squeezed limit, the product of these mode functions becomes relevant (C.47) (C.48) (C.49) (C.50) where the N~s;t is related to (C.46) by some numerical factors. In the squeezed limit, B1 B2 becomes proportional to K4(k3; k3; k1 ! 0) + 140 J0(k3; k3; k1 ! 0) = 280 J0(k3; k3; k1 ! 0) = 350 k1k3 : Also, the integration involving the vertex gives a factor of 34 . Combining with other numerical and momentum-dependent factors in (C.36) and (C.37), we arrive at (C.45). D Operator product expansions In this appendix, we use the operator product expansion (OPE) to argue for the form of cosmological correlators in various limits. In particular, we will show how these arguments x the form of the scalar trispectrum in the collapsed limit and the tensor-scalar-scalar bispectrum in the squeezed limit. The OPE is used in two di erent ways in the derivation of the collapsed trispectrum. First, it is applied directly to the collapsed limit of h i 4 . Second, we assume that the coe cients in the wavefunction of the universe have a good OPE and construct the collapsed limit of h 4i given the restricted forms of these coe cients. Both methods yield the same result, but only when su cient care is taken in deriving the OPE in momentum space. We start, in section D.1, with a review of the wavefunction of the universe, and its connection to correlation functions in conformal eld theory (CFT). The behaviour of cosmological correlators in the soft limits can be recast in terms of the OPE in momentum space, which we consider in section D.2. We will address certain subtleties in performing the Fourier transform of more standard OPE expressions in position space. In section D.3, we calculate the collapsed limit of h 4i using the two methods mentioned above. Finally, the soft limit of h i is analyzed in section D.4, using similar techniques. D.1 Wavefunction of the universe Consider a theory of a scalar and a spin-s eld29 on dS4. The wavefunction for this system is determined semiclassically as exp(iScl), where Scl is the action evaluated on the classical solutions which interpolate from the Bunch-Davies vacuum at early times to the indicated value at late times: k( ) k and k( ) classical action is then expanded as a function of the late-time eld values !0 [ k; k] exp d3k1d3k2 hOk1 Ok2 i k1 k2 + h k1 k2 i k1 k2 d3k1d3k2d3k3 hOk1 Ok2 k3 i k1 k2 k3 d3k1d3k2d3k3d3k4 hOk1 Ok2 Ok3 Ok4 i k1 k2 k3 k4 + ; (D.1) and kij ki + kj . The wavefunction of the universe calculation is similar to AdS/CFT, a fact which we will return to below. As in standard holographic computations, two conformal weights are associated to each bulk eld: = 3 2 r 9 4 m2 H2 s 3 2 s s 1 2 2 m2 H2 ; 29Indices on , and related quantities will often be suppressed in this and following sections. where the quantities in angled brackets are simply functions of the indicated momenta. Expectation values are then calculated by integrating the desired elds against j [ k; k]j2, as in quantum mechanics. For instance, this yields the late-time two-point functions h k ki0 = 1 2 RehOkO ki0 h k ki0 = 1 2 Reh k ki0 ; and the four-point function with h k1 k2 k3 k4 i0 = hO4i0A + hO4i0B Qj4=1 2 RehOkj O kj i0 ; 4 0 hO iA 4 0 hO iB Reh k12 k12 i0 k12 i0 Reh k12 Ok3 Ok4 i0 + 2 perms ; 2 RehOk1 Ok2 Ok3 Ok4 i0 ; 2 (D.2) (D.3) (D.4) (D.5) (D.6) for the scalar and spin-s eld, respectively. In the limiting case where massless, spin-s, depth-t eld, the weights s reduce to + t : The weights + and the bulk elds themselves are assigned the remaining weights, and s for and , respectively. More explicitly, the quantities in (D.7) have the following scalings: Reh k + + RehOkO ki0 / k 2 + 3 ki0 / k2 s+ 3 h k h k 2 ki0 / k ki0 / k2 s 3 ; 3 ; where the relation = 3 was used, justifying the weight assignments. D.2 OPE in momentum space In order to analyze the in ationary correlators in the kinematical regimes of interest, we must review some features of the OPE in momentum space. We begin with a discussion of the momentum space OPE between two scalar operators. This is a subtle object,30 as the process of Fourier transforming, in general, does not commute with taking the OPE limit [104, 105]. As an illustrative example, consider the O for three scalar operators with respective weights (a) (b) O ! O a, b and 1 (c) position space OPE channel c. It is of the form where xab = xa xb. Fourier transforming both sides of (D.10) gives lim xa!xb O (a)(xa)O(b)(xb) / xaba+ b c O (c)(xb) ; (a) (b) qli!m0 Ok q=2O k q=2 / k a+ b c d Oq(c) ; in d-dimensions. The result (D.11) is suspicious as it involves integrating an expression which only holds at close (but separated) points (D.10) over all possible separations, and indeed (D.11) is not in general correct. The proper momentum-space OPE can instead be derived by directly Fourier transforming the explicit position space expression for the correlator hO The Fourier transform which isolates the desired momentum con guration is (a)(xa)O(b)(xb)O(c)(xc)i. Z (a) (b) hOk q=2O k q=2Oq(c)i0 = ddx ddy eik x+iq y hO (a)(x=2)O(b)( x=2)O(c)(y)i ; and careful treatment [104, 105] reveals distinct results depending on the weight correct form of this OPE channel is (a) (b) qli!m0 Ok q=2O k q=2 / Oq(c) k a+ b c d 8 >1 > < ln q >>:(q=k)d 2 c c < d=2 c = d=2 : c > d=2 30We would like to thank Matthew Walters for discussions on this point. (D.7) (D.8) (D.9) (D.10) (D.11) (D.12) c. The (D.13) Hence, the naive result (D.11) is only correct in the regime of small weights, c We will require the same computation for the OPE channel from two scalars to a traceless, symmetric, spin-s operator of weight s: O O ! Oi1 is . The end result is analogous to (D.13). The calculation requires the knowledge of the corresponding position space correlator, which is uniquely xed by conformal invariance [106]: hO (a)(x1)O(b)(x2)Os(x3)i / x(12 a+ b z x13 x223 s+s)=2x(13 a+ s z x23 x213 s b+s)=2x(23 b+ s a+s)=2 : (D.14) Above, we have contracted all loose indices of the spin-s operator with an auxiliary null vector zi to turn it into the index-free operator Os. In order remove the auxiliary vectors, one repeatedly acts on the above with a particular derivative operator whose detailed form we will not need. We then Fourier transform (D.14) in the con guration (D.12) following [104]. The process is straightforward and results in31 >1 > < ln q >>:(q=k)d 2 s (a) (b) qli!m0 Ok q=2O k q=2 / Oi1 is (q) k^i1 k^is k a+ b s d In deriving (D.15), we have made use of the momentum space two-point function for spinning operators: hOi1 is (q)Oj1 js ( q)i0 / q2 s d i1 isj1 js (q^) ; where i1 isj1 js (q^) is a symmetric, traceless tensor structure.32 The result for is the result which one would obtain from naively Fourier transforming the position space OPE of two scalar operators,33 while the cases of larger dimension are di erent, completely analogously to the scalar result (D.13). The di erent behavior for large and small operator weights will be crucial for nding agreement between results derived via wavefunction of s < d=2 s = d=2 : s > d=2 (D.15) (D.16) s < d=2 the universe and the in-in computations. D.3 Collapsed trispectrum We now use the OPE to derive the form of the collapsed limit of h discussed above. We specialize to the case of massless and take 4i using the two methods to be a PM eld of spin s and depth t. First, we use the OPE to calculate the collapsed limit of h 4i directly. The calculation is relatively straightforward. The and elds have weight assignments = 0 and 31Where the weights are such that the Fourier transform does not converge, these results are de ned by 33See [75], for instance, for the form of this OPE. analytic continuation. 32The form of P jZs j2" i1 is "j1 js . (q^) can be found using the arguments in appendix A of [24]. It is also related to the polarization vectors and normalization factors in appendix A via a completeness relation i1:::isj1:::js / t, respectively, and hence the s < d=2 branch of the OPE applies, as d = 3 here operator, we nd qli!m0 k+q=2 k+q=2 = X Cs;t i1 is (q)k^i1 k^is kt 4 Performing the contractions and only keeping the contribution of a single spin-s, depth-t qli!m0h k1 q=2 k1 q=2 k2+q=2 k2+q=2i0 / 1 kt 4kt 4 2 q2t+1 ^s (q^) k^s2 ; where (q^) is the tensor structure de ned in (D.16). This reproduces the scaling behavior HJEP04(218) in (3.17). The angular dependence can also be matched when the tensor structure is expanded into the helicity basis. Next, we turn to the wavefunction of the universe. Wavefunction coe cients arise as analytic continuations of AdS/CFT calculations, simply because in both cases one is computing the on-shell action as a function of boundary data [19, 73, 107]. As the AdS quantities are CFT correlators which have a good OPE, it is feasible that the OPE can also be applied to wavefunction coe cients. We now apply this logic to derive OPE limits of coe cients which in turn determine the collapsed limit of h 4i. The following arguments are intended to be more heuristic than rigorous and indeed we will nd some technical disagreements between parts of the OPE prediction and concrete wavefunction calculations, though the discrepancies don't a ect the predicted overall scaling of h i The four-point function is determined by hO4i0A and hO4i0B, de ned in (D.4). In the collapsed con guration (D.18), the hO4i0A term is dominated by just one of its three 4 . 4 0 hO iA lim qOk2+q=2Ok2+q=2i0 ; Reh q qi0 due to the strong scaling of the denominator with q, via the Reh q simply the PM power spectrum h q qi0, which is proportional to q 1 2t. The terms in the numerator are evaluated using the OPE (D.15) with s > d=2, as opposed to (D.17): qi0 1 factor. This is qli!m0 Ok+q=2O k+q=2 = X Cs;t Oi1 is (q)k^i1 s;t k^is kt+2q 2t 1 (D.18) (D.19) (D.20) (D.21) (D.22) yielding Hence, in the collapsed limit, we nd q!0 lim RehOk q=2O k q=2 qi0 / kt+2 : hO4i0A / 1 kt+2kt+2 2 q2t+1 ^s 1 (q^) k^s2 ; where we restored the tensor structure. The case of graviton exchange is calculated in [108] and the result is consistent with the scaling in (D.22). We have also calculated the result for massless spin-4 exchange and we again nd agreement with (D.22). The OPE (D.20) can also be used to analyze the collapsed limit of hO4i0B, and here we nd tension with concrete calculations. Performing the contractions and keeping only the contribution from spin-s, depth-t operators, one nds the same scaling as in (D.22), hO4i0B / 1 kt+2kt+2 2 q2t+1 ^s (q^) k^s2 : (D.23) However, unlike (D.22), the result (D.23) is not in agreement with the explicit calculations for massless spin-234 or spin-4 particles, where this contribution is instead found to be subleading as q is taken soft. While this nding doesn't a ect the it does imply that the OPE cannot be naively applied35 to wavefunction coe cients and nal scaling for h i 4 , that this second OPE application is only a rough argument. In the end, after using (D.3), wavefunction heuristics give the same scaling result for the collapsed trispectrum as directly applying the OPE to (D.18). D.4 Soft tensor bispectrum We can apply similar methods to analyze the contribute of to the following correlator: qli!m0h q k q=2 k q=2i0 : However, in order for the answer to be non-trivial, we need to assume that the conformal symmetry is broken. Applying the OPE (D.17) directly and keeping only the channel contribution, one nds qli!m0h q k q=2 k q=2i0 / kt 4Y^s ( ; ') h q qi0 ; where we have expanded i1 is into the helicity basis and contracted the momenta, with the angles de ned by cos = k^ q^ and cos ' = " k^. The mixed two-point function is only non-trivial if the eld has a spin-2 component, corresponding to the restriction t the conformal symmetry is preserved, then h q qi0 = 0. However, if we assume that scale symmetry holds, but special conformal symmetries are softly broken, then it is possible to have a non-zero two-point function: h q qi0 = qt+2 ; where is a small parameter characterizing the breaking of special conformal symmetry. Inserting (D.26) into (D.25) and assuming that the OPE (D.17) holds up to O( ) corrections, we nd the leading order scaling: qli!m0h q k q=2 k q=2i0 / q3k3 q 1 t Y^s ( ; ') ; t 1 ; consistent with (C.44). Similar results follow from a wavefunction analysis. 34See [108] for more on the subdominance of hO4i0B to hO4i0A in the spin-2 case. 35We can make an interesting speculation for the origin of the mismatch: in the putative dual theory, there may be more than one operator associated to . In particular, if there is not only the operator of weight that the OO ! for example. s+, but also the associated shadow operator ~ of weight s in the spectrum, then it's possible of the shadow operator ~ leaves hO4i0A una ected. The cancellation can only happen for ~ exchange, as follows from the (D.15). 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Daniel Baumann, Garrett Goon, Hayden Lee, Guilherme L. Pimentel. Partially massless fields during inflation, Journal of High Energy Physics, 2018, 140, DOI: 10.1007/JHEP04(2018)140