A Solution of the Constraint Equations

The Conformal Limit of Inflation

[0.5cm] in the Era of CMB Polarimetry

Enrico Pajer, Guilherme L. Pimentel, and Jaap V. S. van Wijck

Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,

Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands Institute of Physics, Universiteit van Amsterdam,

Science Park, Amsterdam, 1090 GL, The Netherlands Department of Applied Mathematics and Theoretical Physics,

Cambridge University, Cambridge, CB3 0WA, UK

Abstract
We argue that the non-detection of primordial tensor modes has taught us a great deal about the primordial universe. In single-field slow-roll inflation, the current upper bound on the tensor-to-scalar ratio, ( CL), implies that the Hubble slow-roll parameters obey , and therefore establishes the existence of a new hierarchy. We dub this regime the conformal limit of (slow-roll) inflation, and show that it includes Starobinsky-like inflation as well as all viable single-field models with a sub-Planckian field excursion. In this limit, all primordial correlators are constrained by the full conformal group to leading non-trivial order in slow-roll. This fixes the power spectrum and the full bispectrum, and leads to the “conformal” shape of non-Gaussianity. The size of non-Gaussianity is related to the running of the spectral index by a consistency condition, and therefore it is expected to be small. In passing, we clarify the role of boundary terms in the action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism.

## 1 Introduction and Summary

### 1.1 Introduction

Current cosmological observations are well-described by the CDM model. This model assumes an almost scale-invariant initial power spectrum of fluctuations in a specific scalar mode, known as the adiabatic mode. These initial conditions are elegantly derived from inflation, where the (approximate) scale invariance follows from one of the isometries of (quasi) de Sitter spacetime. One of the simplest realizations of this scenario is a single scalar field that slowly rolls towards the bottom of a potential, thus providing a physical clock for the the inflationary period. This particular model of slow-roll inflation has indeed received a great deal of attention.

An exciting signature of inflation are primordial tensor modes. However, despite much observational effort, a detection of primordial tensor perturbations has so far proven elusive, and current bounds for the tensor-to-scalar ratio give ( CL) [1]. It is important to ask what we learn from this bound and from the many future improvements thereof. Naively, besides the exclusion of a handful of models, the absence of a detection of gives us little new information about the early universe.

In this paper, we point out that, within single-field, slow-roll, canonical models, the non-detection of actually teaches us a great deal. The experimental detection of a deviation from the Harrison-Zeldovich spectrum, together with the current lower bound for the amplitude of tensor modes, implies a hierarchy between the slow-roll parameters that characterize the time dependence of the Hubble parameter during inflation. A consequence of this hierarchy is that all primordial correlators are constrained by conformal symmetry.

Using this hierarchy, which we dub the conformal limit of inflation, we can determine completely the shape of the power spectrum and bispectrum of scalar fluctuations. Our method combines conformal symmetry and a consistency condition for the squeezed limit of the bispectrum. The bispectrum contains a local shape and a “conformal” shape that has been studied in a different context in the literature [2, 3, 4, 5]. The amplitude of this bispectrum is small, as the local shape is parametrically of the size of the scalar tilt, while the conformal shape is of the size of the running of the tilt.

One nice implication of this result is that the overall size of the bispectrum is tied to observables in the scalar power spectrum, namely the tilt and its running. Hence, a new single-field consistency condition emerges in the scalar sector, which is in principle testable even for negligible tensor modes. If future experiments keep pushing the upper bounds on and , this consistency condition may turn out to be the ultimate test of the simplest single-field inflationary models.

Below we summarize our main results more quantitatively, while outlining the various sections of the paper.

### 1.2 Summary of Results

The simplest model of inflation consists of a single, canonical, minimally coupled scalar field. Assuming that gravity is described by General Relativity (GR), one is lead to consider the action

 S=∫d4x√−g[M2Pl2R−(∇ϕ)22−V(ϕ)]. (1.1)

We will focus exclusively on this action and postpone further comments on more general single-clock models to the discussion, in Sec. 6. The background solution of (1.1) is a FLRW spacetime with Hubble parameter which is approximately constant in the slow-roll regime. Small time dependence is parametrized by the two slow-roll parameters and . Within the model Eq. (1.1), in the first order slow-roll approximation, one finds the well-known expressions

 1−ns=2ε+η,r=16ε, (1.2)

where is the scalar spectral tilt and is the tensor-to-scalar ratio. Since observations tell us that [6, 1]

 1−ns=0.0355±0.005(68\% CL)andr<0.07(95\% CL), (1.3)

we conclude that

 ε<0.0044<η6≪η. (1.4)

In words, since we have observed a non-vanishing scalar spectral index but no primordial tensor modes, we have discovered a new hierarchy of the slow-roll parameters, namely .1 As CMB polarization experiments forge ahead in their quest for primordial tensor modes, in the event of no detection, the upper bound on gets stronger and this hierarchy becomes more and more pronounced. Since and have traditionally been treated on the same footing, some of the discussion of slow-roll inflation might need to be updated in light of this new hierarchy.

We call the model in Eq. (1.1) in the regime the conformal limit of inflation. This limit is spelled out in detail in Sec. 2.1. The conformal limit describes all viable, single-field, slow-roll canonical models. For example, small-field models with a subPlanckian inflaton displacement, are a specific case of Eq. (1.4). In fact, the Lyth bound [7] implies

 Missing or unrecognized delimiter for \right (1.5)

where is the duration of the observable part of inflation, and are evaluated at CMB scales and the last inequality follows from . Certain models with Planckian field excursion, such as Starobinsky-like inflation, are also particular cases of the conformal limit (see Sec. 2.2).

One remarkable fact about the conformal limit, which according to Eq. (1.4) might well describe our universe, is that all primordial correlators are constrained by conformal symmetry, up to small, slow-roll suppressed corrections.2 This is because, in this regime, the theory of inflaton fluctuations is approximately invariant under de Sitter isometries, which at late times, on superHubble scales, are isomorphic to the 3-dimensional Euclidean conformal group. In Sec. 3, we use these symmetries to fully fix the shape and the amplitude of the spectrum and bispectrum in the conformal limit. For instance, consider the equal-time power spectrum of inflaton perturbations around a background . On superHubble scales, the time dependence of is the same as that of the background. Dilation symmetry fixes the -dependence in terms of the time dependence, yielding

 Missing or unrecognized delimiter for \right (1.6)

where is a constant and a prime indicates that we dropped and a Dirac delta function (we present our conventions below). Notice that, for , evolves on superHubble scales, as expected. Using , we can convert to the curvature perturbations , which are conserved on superHubble scales. We choose to convert to at some late conformal time , the same for every wavenumber . We find

 ⟨ζkζ−k⟩′=~Cτ−η∗k3+η, (1.7)

where we have absorbed the -independent factor into the constant . The spectral tilt is then easily seen to be , in agreement with Eq. (1.2) for . Notice that we did not have to solve the constraint equations of GR. Also, the contribution to the observed spectral tilt , which is the largest one since , does not signal a breaking of dilation invariance during inflation. Rather, it is a precise consequence of the invariance of correlators under dilations (but not of correlators).

The bispectrum of is also constrained by de Sitter isometries; it is the sum of two different shapes with arbitrary coefficients. To fix them we calculate the squeezed limit and impose that it matches the squeezed limit of the bispectrum derived using a background-wave argument. This is analogous to Maldacena’s consistency condition in comoving gauge [14].3 This uniquely fixes the three-point function of (see Eq. (3.36)). Performing the second order gauge transformation from to curvature perturbations gives

 ⟨ζk1ζk2ζk3⟩ = (H24εM2Pl)2∗{η∗2k31+k32+k33k31k32k33+˙η∗2H∗1k31k32k33× (1.8) [(−1+γE+log(−Kτ∗))3∑i=1k3i−∑i≠jk2ikj+k1k2k3]},

where all time-dependent factors are evaluated at independently of . In Sec. 4, we show that Eq. (1.8) agrees with the direct calculation using the in-in formalism [15], and satisfies the squeezed limit consistency relation of Maldacena to next-to-leading order. An alternative derivation of the bispectrum using the wave functional formalism is presented in Sec. 5.

The non-Gaussian shape appearing in the second line of Eq. (1.8) has been previously derived for spectator fields in de Sitter in [2, 3, 4, 5]. Our new result is to point out that these shapes with specific relative coefficients follow from symmetry arguments, and describe non-Gaussian features of primordial fluctuations in single-field inflation in the conformal limit. We recognize the first term in Eq. (1.8) as the usual local shape with coefficient , which is not locally observable to leading order in derivatives [16]. The second term, proportional to , is the next-to-leading order result, first derived in [15]. We dub its -dependence the conformal shape, since it is invariant under special conformal transformations (see Sec. 3.2) and we show it in Fig. 1.

In the conformal limit of inflation, the following simple relation arises between the amplitude of the conformal shape and the running of the power spectrum ():

 fconf.NL=−2536αs. (1.9)

This is a new single-field consistency relation, which can in principle be tested within the scalar sector alone. Given the small expected size of the running, this relation will not be tested in the near future. Nevertheless, it is worth keeping in mind that the most natural, non-informative prior on is a log-flat prior extending all the way to or less. With this prior, the tensor consistency condition might well be even harder to test than this new, scalar one given by Eq. (1.9).

Finally, a few additional new results are scattered around the paper. For the convenience of the reader, we provide here an executive summary:

• We show in Sec. 4 how to derive the bispectrum in comoving gauge, without any field redefinition. This clarifies when and why the boundary terms in the action are important. In particular, they need to be included to obtain constant correlators of on superHubble scales (see also [15]).

• In App. A, we note that the GR constraint equations are easier to solve in flat gauge. Since the conformal limit implies the decoupling limit (see Eq. (2.1)), we can use the hierarchy between the scalar and gravitational perturbations to solve the constraints in an expansion, but to all orders in the field fluctuations. The structure of the solutions to the constraint equations in comoving gauge is then a simple consequence of the change of coordinates from flat to comoving gauge.

• It is well-known that the constraint equations can be solved to first order in perturbations if one is interested in the cubic action [14]. We prove in App. A.3 that the constraint solution to order is sufficient to derive the action to order , failing for the first time only at order . This is a stronger result than the one proven in [17], where the -th order constraint solution was proved to be sufficient for the action only up to order . In particular, the known solution of the constraint equations to order 2 (see e.g. [18]) can already be used to derive the action to 5th order.

Notation and conventions We use natural units, , with reduced Planck mass . Our metric signature is . Overdots and primes will denote derivatives with respect to physical time and conformal time , respectively. The conformal time is defined by . We use the Hubble slow-roll parameters defined by

 ε≡−˙HH2,η≡˙εHε,ξn≥3≡∂lnξn−1∂N, (1.10)

where and is the number of e-foldings. We denote by the same symbol the lapse function in ADM decomposition, however the distinction between the two is always clear from the context. We use the label to indicate the potential slow-roll parameters, defined by

 εV≡M2Pl2(V′V)2,ηV≡M2PlV′′V. (1.11)

We indicate by the magnitude of the comoving wavenumber k. Our Fourier conventions are

 F(x)=∫k~F(k)eik⋅x,where we use the shorthand∫k≡∫d3k(2π)3. (1.12)

A prime on a correlator indicates that we dropped the Dirac delta function and a factor of ,

 Missing or unrecognized delimiter for \right (1.13)

## 2 The Conformal Limit of Inflation

In this section, we define the conformal limit of inflation and summarize the simplifications that it implies for the solutions of the GR constraint equations. Technical details are collected in App. A and App. B.

### 2.1 The Conformal Limit

We are interested in the simplest model of single-field inflation: a canonical, minimally coupled scalar field with an arbitrary potential , as in the action Eq. (1.1). According to observations, our universe is well described by the regime , as discussed around Eq. (1.3).

The predictions of this model can be obtained by working at zeroth order in but keeping and higher slow-roll parameters. More precisely, we start considering the de Sitter limit, namely . Since we want to keep the amplitude of primordial scalar perturbations finite, we need to demand also . Hence, we consider the limit

 ε,HMPl→0withH2εM2Pl,η,ξn,⋯\leavevmode\nobreak finite. (2.1)

One intuitive way to think about this limit is to keep constant and send to infinity. It is then clear that this is a decoupling limit, in which the metric becomes non-dynamical (i.e., a classical background). We will see that it is consistent to work in this decoupling limit in flat gauge, and convert to curvature perturbations at the end of the calculation. As we discuss shortly in Sec. 2.2, this limit includes all viable small-field models of inflation () as well as Starobinsky-like inflation.

Since gravity becomes non-dynamical in the limit Eq. (2.1), after choosing spatially flat gauge the action reduces to that of a scalar field in de Sitter. The theory for perturbations around an inflationary background is

 S=−∫d3xdte3Ht[12∂μφ∂μφ+∞∑m=2φmm!V(m)(¯ϕ)], (2.2)

where is the -th derivative of the potential with respect to . The time dependence of the background induces a time dependence of the interaction coefficients that breaks the boost and dilation isometries of de Sitter. On the other hand, this breaking of de Sitter isometries is suppressed by the slow roll parameters and can therefore be neglected to leading order. To see this, let us write in terms of Hubble slow-roll parameters, as discussed in App. B. One finds that is a polynomial in and , with potentially negative half-integer powers of . The time dependence of is then given by the time dependence of the slow-roll parameters, which is slow-roll suppressed

 ξn(N) ≈ ξn(N∗)[1+∂Nξnξn∣∣∣N∗(N−N∗)]=ξn(N∗)[1+O(ξn+1)], (2.3)

hence proving our claim.

This suppression of the breaking of de Sitter isometries is unique to a canonical scalar field.4 If the scalar field were non-canonical, e.g. some model, the time dependent background would break de Sitter isometries by an amount that is not suppressed by the slow-roll parameters. This is easy to see for example for the speed of sound . Inflaton perturbations propagate on the sound cone defined by , but this is not invariant under de Sitter boosts, which reduce to Minkowski boosts at short distances and leave only the light cone invariant.5 Therefore we need to assume that the scalar field is canonical, as in Eq. (1.1). In particular, all coefficients in the Effective Field Theory of Inflation [19] that parameterize deviations from the vanilla slow-roll, canonical inflation are assumed to be negligible. Non-canonical models are further discussed in Sec. 6.

The assumptions of a canonical scalar field plus the limit Eq. (2.1) define the conformal limit of inflation. In this limit, de Sitter isometries are unbroken (acting naturally in flat gauge) and all correlators of the inflaton perturbations (and of the graviton) must be de Sitter invariant.

For later reference, let us write the background equations of motion for the attractor FLRW solution, which will be quasi-de Sitter space in our case. They are

 ˙¯ϕ2=−2M2Pl˙H, V(¯ϕ)=M2Pl(3H2+˙H)≈3M2PlH2\leavevmode\nobreak \leavevmode\nobreak and (2.4) ¨¯ϕ+3H˙¯ϕ+V′(¯ϕ)=0.

We see that in the limit of Eq. (2.1) the potential overwhelms the kinetic energy of the background field. Nonetheless, the kinetic energy is still finite, which is why the scalar fluctuations transform into curvature perturbations under the appropriate gauge transformation.

### 2.2 Starobinsky Inflation as an Explicit Example

The skeptical reader might wonder whether it is consistent to send to zero but keep fixed, as we advocate in the conformal limit Eq. (2.1). After all and one might worry that soon becomes sizable. There are various ways to convince oneself that this is not an issue. First, consider the following explicit class of examples

 ε=ε0(−N)β⇒η=β(−N),ξ=ξn≥3=1(−N), (2.5)

where , denotes the number of e-folds () going from around at CMB scales to at the end of inflation and is a small parameter. For one indeed finds at CMB scales . It is therefore perfectly natural for a general model of inflation to have a hierarchy between the first slow-roll parameter and all the others. In fact, if we furthermore impose that the spectral tilt takes the measured value, we select (with small corrections due to the uncertainty in the duration of inflation). The examples in Eq. (2.5) actually describe two very popular classes of models: Starobinsky-like inflation [20] (and its modifications, e.g. attractors [21]) and small-field models. Consider, for example, the Starobinsky-like potential

 V=V0(1−e−ϕ/M)2, (2.6)

where is some mass scale. The slow-roll parameters in this model are precisely as in Eq. (2.5) with and . Therefore, for , one finds , parametrically. For small-field models, with subPlanckian field displacement, we cannot provide a general formula, but the estimate using the Lyth bound in Eq. (1.5) also leads to the same scaling as in Eq. (2.5) with . Notice that, in the cases discussed above, one finds the stronger condition . This implies that can be neglected even when including the first subleading order correction in , namely the order . Moreover, one can always choose parameters (in a natural way) such that for .

Alternatively, let us just Taylor expand around some arbitrary

 ε(N)−ε(N∗) = ∂ε∂N∣∣∣N∗(N−N∗)+∂2ε∂N2∣∣∣N∗(N−N∗)22+O(∂3Nε) (2.7) = ε[η(N−N∗)+ηξ3(N−N∗)22+O(η3,η2ξ3,ηξ3ξ4,ε)],

where all the slow-roll parameters are evaluated at . In the second line we see that the evolution of is itself suppressed by . So, as long as , it is consistent to neglect it.

Summarizing, one should keep in mind that Starobinsky inflation and its modifications, as well as all viable small-field models are specific cases of the conformal limit we study in this paper, satisfying the stronger condition . This ensures that the subleading correction to the bispectrum (1.8) is still large compared to deviations from conformal symmetry, expected at order .

### 2.3 Constraint Equations in the Conformal Limit

In this subsection, we discuss some interesting results about the constraint equations in the conformal limit. The derivations and a detailed discussion are collected in App. A.

In the conformal limit, the constraint equations simplify considerably. Written in flat gauge, they read

 M2Pl(R(3)−6H2−N−2hijhkl(EikEjl−EijEkl))− −M0Pl(N−2(˙¯ϕ+˙φ−Ni∂iφ)2+(−˙¯ϕ2+2V′(¯ϕ)φ+⋯)+hij∂iφ∂jφ)=0, (2.8) M2Pl(∇a(N−1(habEbi−δaihbcEbc)))+M0Pl(N−1∂iφ(Nj∂jφ−˙¯ϕ−˙φ))=0. (2.9)

An obvious observation is that, in the limit, the constraint equations admit the trivial solutions

 N=1+O(M−2Pl),\leavevmode\nobreak \leavevmode\nobreak Ni=O(M−2Pl). (2.10)

In fact, we can use as a small expansion parameter, rather than the field perturbations , as is usually the case. This allows us to solve the constraint equations to all orders in to subleading order in . We present the solution in App. A.

If we write the constraint equations in comoving gauge, cancels out, so there is no perturbation theory in the inverse Planck mass. The decoupling limit corresponds to the de Sitter limit, so if we take , one would be tempted to guess that the constraint equations have solutions as simple as Eq. (2.10). This is not the case. Even in the limit, the constraint equations have rather non-trivial solutions in comoving gauge. Nonetheless, we can obtain them by changing coordinates from flat gauge (with the background FLRW metric, due to Eq. (2.10)) to comoving gauge. Finding the coordinate transformation from to gauge is much easier than tackling the constraint equations, so this is a more economical route to solving the constraint equations. We checked explicitly that, in the conformal limit, to second order in , the solutions to the constraint equations are given by a change of coordinates from the flat FLRW metric to the comoving coordinates.

Finally, we point out (see App. A.3) that solving the constraint equations up to a certain order in perturbation theory goes a long way in finding the perturbative action. Namely, if we solve the constraint equation to order in perturbations, that is enough to determine the perturbative action to order . This result is stronger than the one derived in [17], where it was proven that the -th order solution of the constraints is sufficient for the action at order .

## 3 The Spectra from Symmetries

As discussed in Sec. 2, in the conformal limit Eq. (2.1), inflaton correlators are invariant under de Sitter isometries at leading order in slow-roll. Six of them are manifest. Invariance under spatial translations and spatial rotations tells us that the three-point function is proportional to a momentum conserving Dirac delta function and that it only depends on scalar products of the momenta. The consequences of the additional four isometries are less obvious, so we review them here. If we write the de Sitter line element in flat slicing,

 ds2=−dτ2+dx2(Hτ)2, (3.1)

it is easy to verify that these isometries are given by (in infinitesimal form)

 Missing dimension or its units for \hskip (3.2) Missing dimension or its units for \hskip (3.3)

where is a real infinitesimal parameter and an infinitesimal, 3-dimensional vector. The consequences of these symmetries for scalar and tensor correlators have already been studied in the literature [22, 8, 10, 9, 4, 11, 23, 5, 24, 25, 12], and we will use some of these results.

In practice we are always interested in the late-time correlators of inflaton perturbations , when all the modes are outside the horizon. In the limit we can neglect the term in Eq. (3.3), and transformations of are isomorphic to the infinitesimal generators of dilations and special conformal transformations of the conformal group acting on the spatial slice. For massive fields the transformation of can be taken into account by assigning a scaling dimension to the operator ; it is fixed by looking at the late time dependence . Therefore, in the late time limit, dS invariant correlators must have the same form as correlators of some CFT where fields have conformal weight fixed by the inflaton mass. This imposes strong constraints on the power spectrum and the bispectrum. In this section we show how their shape is entirely fixed by this symmetry.

### 3.1 Power Spectrum

Let us first focus on the power spectrum and re-derive the spectral tilt by converting inflaton perturbations into curvature perturbations . We can treat the inflaton as a free scalar field in de Sitter with mass . At late times de Sitter isometries impose that the power spectrum has the following form [12]

 Missing or unrecognized delimiter for \right = H2k31π[Γ(32−Δ−)2(−kτ)2Δ−+(Δ−→Δ+)], (3.4)

where . We are interested in the limit where the inflaton mass is small and we want to keep only the growing mode. At leading order in slow-roll is given by

 Δ−=V′′3H2+O((V′′)2H4)≃ηV≃−η2. (3.5)

Using , the power spectrum simplifies to

 Missing or unrecognized delimiter for \right = H24k3(−kτ)−η. (3.6)

We now need to make a gauge transformation from to , which is conserved on superHubble scales. This is achieved by where the slow-roll parameter is evaluated at a chosen time . We make two different choices and show that they give the same result. First, we can do the conversion at the Hubble crossing of each mode, namely at . The result is the usual expression

 ⟨ζkζ−k⟩′=H24εM2Pl∣∣∣H.c.1k3, (3.7)

where “H.c.” indicates that the time-dependent quantities and should be evaluated at Hubble crossing. This procedure has the advantage that one does not need to know the slow-roll suppressed time dependence of perturbations on superHubble scales, since the term (see Eq. (3.4)) becomes unity. On the other hand, is now -dependent and therefore the spectral tilt of the power spectrum hides inside the time dependence of and . This is a disadvantage since the spectral tilt remains implicit.

In our discussion, de Sitter isometries have already fixed the time dependence to be . Therefore, we might as well proceed in a different but equivalent way. We evolve each mode until some late fixed time , the same for every mode and such that for every of interest. Using we find

 ⟨ζkζ−k⟩′=H24εM2Pl∣∣∣∗1k3+ητ−η∗, (3.8)

where, as in Eq. (3.7), this is the asymptotic, time-independent value of the power spectrum. This formulation has the advantage of making the tilt explicit, since now, the factor is a constant that does not depend on . The tilt in Eq. (3.8) agrees with the standard result in the conformal limit, for .

### 3.2 Conformally Covariant Shapes of the Bispectrum

Let us turn to the discussion of how de Sitter isometries, to which we also refer as conformal symmetry, constrain the bispectrum of . We find that the bispectrum is fixed up to two multiplicative constants. As we then show in Sec. 3.4, these constants can be fixed using the consistency condition for the squeezed limit of the three-point function which we derive in the following section.

The implications of the covariance under de Sitter isometries in momentum space can be derived from transformations in Eqs. (3.2)–(3.3). For the three-point correlation function of they simplify to [10, 24]

 [−3(Δ−2)+3∑a=1ka∂∂ka]⟨φ(k1)φ(k2)φ(k3)⟩′!= ‘‘local", (3.9) 3∑a=1[b⋅ka(∂2∂k2a−2(Δ−2)ka∂∂ka)]⟨φ(k1)φ(k2)φ(k3)⟩′!= 0. (3.10)

We keep the dependence in explicit for now, but for inflation we will eventually be interested in . Eq. (3.9) constraints the overall momentum scaling of the three-point function to

 ⟨φ(k1)φ(k2)φ(k3)⟩′!=k3(Δ−2)1F(k2k1,k3k1,log(k1+k2+k3)), (3.11)

with an arbitrary function with no overall momentum scaling. The term “local” on the right hand side of the dilation equation requires some explanation. Introducing a logarithm in implies introducing an arbitrary scale in the correlator. This follows [10, 26] from the assumption that we are always allowed to integrate by parts in performing Fourier transforms, which is necessary to substitute . If the Fourier transform is marginally convergent, we might miss a boundary term, which produces an anomalous (local) violation of dilation invariance. A useful example to keep in mind is that of the two point function of the stress tensor in a CFT in even dimensions. The dilation constraint would imply that , which is a contact term. The correct two point function is given by . Now, the action of the dilation operator will produce a , which is a contact term. In summary, dilation should fix the overall scaling of the correlator up to logarithms. The action of the dilation operator on the three-point function produces a term that corresponds to a contact term in the putative dual CFT. In the case at hand, this is the local non-Gaussian shape Eq. (3.19).

To constrain the three-point function using the special conformal isometry, let us use our freedom in picking to write Eq. (3.10) in a more convenient form. If we choose such that , then . This simplifies Eq. (3.10) considerably, and we obtain

 [Ka−Kb]⟨φ(k1)φ(k2)φ(k3)⟩′!=0. (3.12)

with and

 Ka≡[∂2∂k2a−2(Δ−2)ka∂∂ka]. (3.13)

The solutions to both Eq. (3.9) and Eq. (3.12), up to a multiplicative constant, are given by [24]

 ⟨φ(k1)φ(k2)φ(k3)⟩′ = (k1k2k3)Δ−3/2 (3.14) ×∫∞0dx\leavevmode\nobreak x3/2−1%KΔ−32(k1x)KΔ−32(k2x)KΔ−32(k3x).

This solution is known as the triple K integral. Only modified Bessel functions with half-integer give us expressions for in terms of elementary functions. This limits the number of exact calculations we can perform.

The integral in Eq. (3.14) might not converge and some regularization scheme is necessary. When all variables are real, Eq. (3.14) converges for

 32>3∣∣∣Δ−32∣∣∣+2. (3.15)

To construct the three-point correlation function from symmetries for , we employ the following regularisation scheme [24]:

 3→3+2δ,\leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak \leavevmode\nobreak Δ→Δ+δ. (3.16)

Note that is just an expansion parameter, as is done in dimensional regularization6. For the massless case in 3 (spatial) dimensions, we have . The modified Bessel functions then simplify to

 K3/2(x)=K−3/2(x)=√π2(1+x)e−xx32. (3.17)

Using the regularisation scheme in Eq. (3.16), we obtain

 limδ→0∫∞0dx\leavevmode\nobreak x32−1+δK−3/2(k1x)K−3/2(k2x)K−3/2(k3x)= (3.18) =limδ→0⎡⎢⎣(π2)32(k1+k2+k3)−δ(k1k2k3)32(−2+δ)⎛⎝3∑i=1k3i+δ⎛⎝3∑i≠jk2ikj−k1k2k3⎞⎠+δ2k1k2k3)⎞⎠Γ(−3+δ)⎤⎥⎦.

At leading order in , we obtain

 ⟨φ(k1)φ(k2)φ(k3)⟩′=Clock31+k32+k33k31k32k33, (3.19)

where, for future convenience, we introduced an arbitrary constant which is not fixed by conformal symmetry. Eq. (3.19) is the well-known local shape of the bispectrum. At the next to leading order in , we obtain

 ⟨φ(k1)φ(k2)φ(k3)⟩′=Cconlog(K/k∗)∑3i=1k3i−∑i≠jk2ikj+k1k2k3k31k32k33, (3.20)

where . We dub Eq. (3.20) the conformal shape of the bispectrum. To keep the argument of the log dimensionless, we subtracted a purely local term with coefficient , which we can always do by redefining the coefficient of the local term in Eq. (3.19). In the next section, we will find that can be interpreted as the Hubble scale at some late time during inflation, in Eq. (4.7). It is straightforward to use Eq. (3.12) to check that both the local and the conformal shapes are separately invariant under special conformal transformations. In particular, note that neither times the local shape nor the other two terms in the conformal shape Eq. (3.20) are separately invariant. It is only the sum of the three terms that is invariant under special conformal transformations.

Let us note that for the invariance under dilation, Eq. (3.9), of the local shape is manifest, while that of the conformal shape is spoiled by the presence of the logarithmic term. This looks like a paradox and requires some explanation. Under the action of the dilation operator, the conformal shape produces precisely the local shape. Like in the discussion after Eq. (3.9), we have an anomaly due to neglecting the boundary contribution to the Fourier transform of the conformal shape. This is particularly clear if we interpret these three-point functions as coming from a putative CFT. To obtain the CFT three-point function from our shape functions, we strip the factors in the denominators of both of them. The conformal shape will give a CFT three-point function, and the action of dilations on this three-point function will give a sum of terms that only has support when two points coincide; in other words, dilations are not violated when all three points are separated. The local shape gives a CFT three-point function that has zero support when all three points are separated. In position space, it corresponds to terms of the form , etc.

The other way to see that there is no paradox is to restore the dependence in the logarithm and note that, for , the de Sitter isometries corresponding to dilations act on the equal-time correlation functions as

 δλ⟨φ(k1)…φ(kn)⟩′=λ[τ∂τ−3(n−1)−n∑a=1kai∂kai]⟨φ(k1)…φ(kn)⟩′=0. (3.21)

This operator follows from Eq. (3.2) and it takes into account the fact that the rescaling of coordinates must be accompanied by a shift in the conformal time. As the dependence of the bispectrum is only appearing explicitly in the term, we see that the dilation anomaly precisely cancels with the action of the term. This is exactly what we expect, because the three point-function is de Sitter invariant.

Before proceeding, we mention that a similar calculation can be done for the case , corresponding to . Using again the regularization scheme in Eq. (3.16), we obtain

 limδ→0∫∞0dx\leavevmode\nobreak x−1+δK−1/2(k1x)K−1/2(k2x)K−1/2(k3x)= (3.22) =limδ→0⎡⎣(π2)32(k1+k2+k3)−δ√k1k2k3Γ(δ)⎤⎦.

At leading order in , we find

 ⟨φ(k1)φ(k2)φ(k3)⟩′=C1k1k2k3. (3.23)

Similar to the case, we can also expand Eq. (3.2) to next to leading order, where one obtains

 ⟨φ(k1)φ(k2)φ(k3)⟩′=C2log(K/k∗)k1k2k3. (3.24)

It is interesting to note that while both of these shapes are allowed by symmetry, in the explicit in-in calculations for massive fields with it turns out that .

### 3.3 A Soft Inflaton Consistency Relation

So far we were able to show how symmetries fix the bispectrum to be a sum of two different shapes with amplitudes and . The next step is to fix these constants. This is possible due to a consistency condition that relates the squeezed limit bispectrum and the power spectrum. This condition is well known for correlators. Here we re-derive it for the bispectrum following the standard approach [27].

When taking the squeezed limit of a correlation function, i.e. taking one of the external momenta to be very small, one of the modes has a much smaller wavenumber than the others. This mode leaves the Hubble radius much earlier. To leading order in derivatives, the short modes perceive it as a change in the background7. In this way, we can rewrite the squeezed limit of a correlation function as a field shift

 ⟨φl(x,τ)φs(x1,τ)...φs(xn,τ)⟩=⟨φl(x)⟨~φs(x1,τ)...~φs(xn,τ)⟩φl⟩, (3.27)

where the subscript and refer to small and long wavelength modes, respectively. Note that on the LHS of Eq. (3.27) we have a ()-point function and on the RHS we have a -point function. The last term in Eq. (3.27) is evaluated in the background of a mode. We can Taylor expand the right hand side of Eq. (3.27)

 ⟨~φ(x1,τ)...~φ(xn,τ)⟩∣∣φl=⟨φs(x1,τ)...φs(xn,τ)⟩+φl(x)[δδφl(x)⟨φs(x1,τ)...φs(xn,τ)⟩]φl=0+O(φ2l). (3.28)

The second term gives the leading non-vanishing contribution to Eq. (3.27), which becomes

 ⟨φl(x1,τ)φs(x2,τ)φs(x3,τ)⟩=⟨φl(x1,τ)φl(z,τ)⟩[δδφl(z)⟨φs(x% 2,τ)φs(x3,τ)⟩φl]φl=0, (3.29)

where . In Fourier space this becomes

 ⟨φ(k1,τ)φ(k2,τ)φ(%k3,τ)⟩′k1≪k2,k3≈⟨φl(k1,τ)φl(−k1,τ)⟩′δδφl⟨φs(k2,τ)φs(k3,τ)⟩′∣∣φl=0. (3.30)

To explore this relation in the conformal limit, we need to compute how a shift in the background by affects the power spectrum. The leading interaction in the slow-roll expansion is given by the cubic term in the potential, which couples long and short modes. The linear order effect of a long mode is to generate an effective mass term for

 m2eff≡V′′′(¯ϕ)φl(τ), (3.31)

as can be seen from

 −V′′′6φ3→−V′′′6(φ3s+3φlφ2s+3φ2lφs+φ3l). (3.32)

Since evolves on superHubble scales, the effective mass of is time dependent. The general power spectrum for a field with a time dependent mass is hard to calculate, since we cannot solve the equation of motion for . On the other hand, we are interested only in the linear effect of , so we can treat perturbatively. Since we know the solution of the equations of motion for a scalar field in de Sitter, we know the Green’s function of the equation of motion for when . The linear effect of is then obtained from an integral of the Green’s function, which can be done analytically. We collect all the details of the calculation in App. C and only quote the final result here. The squeezed bispectrum one obtains by substituting the zeroth and first order mode functions Eq. (C.11) and Eq. (C.25) into Eq. (3.30) is

 Missing or unrecognized delimiter for \left (3.33)

As mentioned before, the consistency relation Eq. (3.29) is valid for any value of the mass , but for a generic we are not able to compute the derivative analytically, since we cannot perform the integral that gives the linear order correction to the power spectrum of . Besides the case , which we just discussed, there are two more cases we are able to treat analytically: the case of a small mass, , which is relevant for inflation, and the case of , which is not. For the latter, the squeezed bispectrum one obtains by substituting the zeroth and first order mode functions Eq. (C.11) and Eq. (C.26) into Eq. (3.30) is

 ⟨φ(ks,τ)φ(ks,τ)φ(%kl,τ)⟩′=πH2V′′′(¯ϕ)8τ3k2skl+O(k2lk2s). (3.34)

As we will see, both Eq. (3.33) and Eq. (3.34) correctly reproduce the squeezed limit of the bispectrum computed with the in-in formalism [15], Eq. (4.7) and Eq. (4.8).

In the case of a small mass, , discussed in App. C.3, we find Eq. (C.28) for the squeezed limit bispectrum. This expression is rather involved and describes a subleading slow-roll correction, suppressed by , to what is already a small non-Gaussianity, so we do not discuss it further here.

### 3.4 The Primordial Bispectrum

We have now all the ingredients to fix the full shape of the bispectrum. By taking the squeezed limit of Eq. (3.19) and Eq. (3.20) and comparing it with Eq. (3.33) we find that

 Cloc = H2V′′′12(−1+γE),Ccon=H2V′′′12. (3.35)

Note that the presence of the logarithmic term in the conformal shape Eq. (3.20) is what allows us to compute two constants from a single squeezed limit. Had the logarithmic term not been there, only the sum of and would have been fixed, but not their difference. We therefore find the full, equal-time bispectrum of from symmetries to be

 ⟨φ(k1,τ)φ(k2,τ)φ(k3,τ)⟩′ = H2k31k32k33V′′′(¯ϕ)12 (3.36) ×⎡⎣(−1+γE+log(−Kτ))3∑i=1k3i−∑i≠jk2ikj+k1k2k3⎤⎦,

where we made the time dependence explicit to emphasize that this bispectrum is time-dependent. This is expected since, even for , the cubic interaction makes the inflaton perturbations evolve on superHubble scales.

We can also fix the normalization of the massive case. Comparing Eq. (3.23) to Eq. (3.34) determines and leads to the bispectrum

 ⟨φ(k1,τ)φ(k2,τ)φ(k3,τ)⟩′=πH2V′′′(¯ϕ)8τ3k1k2k3. (3.37)

In the next section, we will confirm both Eq. (3.36) and Eq. (3.37), obtained from symmetries, with an explicit calculation using the in-in formalism.

Eventually, we are interested in the correlators of curvature perturbations , which, unlike those of , are conserved on superHubble scales in single-field inflation. As discussed in Sec. 3.1, we have at least two distinct options to convert to fluctuations. We can convert at some -dependent time so that the tilt hides in the time-dependent factors in Eq. (3.36). It is evident that the expression simplifies considerably if we convert at “perimeter crossing” time (p.c.), defined by the solution of

 −1+γE+log(−Kτp.c.)=0. (3.38)

Using the triangle inequality, one finds for every . In particular for the largest of the three momenta. Using this we can check that

 −kmaxτp.c.≥e1−γE2≃0.8, (3.39)

which ensures that the conversion is performed after all the three modes have crossed the Hubble radius. For the bispectrum, we need the gauge transformation between and at second order. This can be computed directly from the gauge transformation, which we review in App. A.2.2, or using the formalism, which we review in App. D. Neglecting all spatial derivatives, which are subleading on superHubble scales, the second order gauge transformation is

 ζ=−1√2εMPlφ