Contents

Holography for inflation using conformal

[1ex] perturbation theory


Adam Bzowski, Paul McFadden and Kostas Skenderis


School of Mathematics, University of Southampton, UK.

Perimeter Institute for Theoretical Physics, Waterloo, Canada.

Korteweg–deVries Institute for Mathematics, Institute for Theoretical Physics, Amsterdam, Netherlands.


E-mail: a.w.bzowski@uva.nl, pmcfadden@perimeterinstitute.ca, k.skenderis@soton.ac.uk

Abstract

We provide a precise and quantitative holographic description of a class of inflationary slow-roll models. The dual QFT is a deformation of a three-dimensional CFT by a nearly marginal operator, which, in the models we consider, generates an RG flow to a nearby IR fixed point. These models describe hilltop inflation, where the inflaton rolls from a local maximum of the potential in the infinite past (corresponding to the IR fixed point of the dual QFT) to reach a nearby local minimum in the infinite future (corresponding to the UV of the dual QFT). Through purely holographic means, we compute the spectra and bispectra of scalar and tensor cosmological perturbations. The QFT correlators to which these observables map holographically may be calculated using conformal perturbation theory, even when the dual QFT is strongly coupled. Both the spectra and the bispectra may be expressed this way in terms of CFT correlators that are fixed, up to a few constants, by conformal invariance. The form of slow-roll inflationary correlators is thus determined by the perturbative breaking of the de Sitter isometries away from the fixed point. Setting the constants to their values obtained by AdS/CFT at the fixed point, we find exact agreement with known expressions for the slow-roll power spectra and non-Gaussianities.

 


 


1 Introduction

The primordial perturbations encode a great wealth of information about the early universe, and, as such, it is important to understand their structure as far as possible. In particular, it is important to understand which features are fixed by symmetries and which are a property of the specific fundamental theory that governs the universe at early times. Since the spacetime geometry during slow-roll inflation is quasi-de Sitter, one may anticipate that at least some of the properties of cosmological observables are fixed by the underlying broken de Sitter isometries, and indeed some of the recent literature is devoted to answering this question [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In this paper we use a holographic set up to address this question, and we find that essentially all spectra and bispectra, for both scalars and tensors, are fixed by the broken de Sitter isometries up to a number of constants. While our results are obtained within a class of models, we believe the answer holds more generally.

We will use the holographic framework we developed in our earlier work [13, 14, 15, 16, 17]. This framework is applicable when the spacetime is either asymptotically power-law or asymptotically de-Sitter. In previous works we presented a complete analysis of the power-law models (in particular their phenomenology [18, 19], see also [20, 21]) and here we will focus exclusively on the asymptotically de Sitter case. Earlier studies of this case include [22, 23, 24, 25, 26, 27, 28] building on the dS/CFT correspondence [29, 30, 31]111See [32, 33, 34, 35, 36, 37, 38, 39, 40, 41] for a sample of more recent works..

In our previous work, we proved that one can express all spectra and bispectra of inflationary models in terms of the correlation functions of a dual three-dimensional QFT, after appropriate analytic continuation in parameters and momenta. When gravity is weakly coupled, so that standard inflationary computations are valid, the dual QFT is at strong coupling. The proof in our earlier work assumed the validity of standard AdS/CFT duality which is used in order to compute the QFT correlators at strong coupling. Here we will relax this assumption and directly compute these correlators on the QFT side using conformal perturbation theory [42, 43]. As such our results are valid both when the QFT is at strong coupling and at weak coupling, with the latter corresponding to a universe that was non-geometric at early times.

Since the group of de Sitter isometries is the same as the (Euclidean) conformal group in one dimension less, an asymptotically de Sitter spacetime will have a QFT dual that in the UV becomes conformal. This QFT may either be a deformation of a CFT, or else a CFT in a non-trivial state that spontaneously breaks conformal invariance. It appears that the spacetimes corresponding to the second option cannot satisfy the slow-roll conditions, so in this paper we will focus on the first option. Once the CFT is deformed by a relevant operator it undergoes an RG flow, and the inverse of this evolution corresponds to cosmic evolution in the bulk. Of the various possible fates for the RG flow, we focus here on the case where the flow leads us to a new fixed point in the IR. The RG flow in the vicinity of this IR fixed point then gives rise to a red-tilted cosmological power spectrum in line with current observational preferences based on a minimal power-law CDM fit [44]. Our models will therefore have an asymptotically de Sitter epoch in the far past as well as in the far future.

In the vicinity of either fixed point, the RG flow will be controlled by the most nearly marginal operator in the theory; about the UV fixed point this operator will be marginally relevant, while about the IR it will be marginally irrelevant. As we wish to construct a single-field model of inflation, we will assume this most nearly marginal operator to be a single scalar. For simplicity, we will further take this single scalar operator to control the entire RG flow from the UV to the IR fixed point. A more sophisticated holographic model incorporating the effects of reheating followed by epochs of radiation and matter domination would presumably relax this assumption, allowing other operators to enter and eventually to dominate the RG flow terminating the period of single-field inflation. Nevertheless, the simple model we study here will capture the same long-wavelength behaviour found in a more complete model.

To be able to recover standard inflationary physics from this scenario, it is necessary to be able to perform explicit calculations in the dual QFT. The strongly coupled nature of this QFT at first sight amounts to an insurmountable obstacle. Our starting point in this paper is the observation that, for an RG flow connecting two nearby fixed points, it is possible to perform perturbation theory in the small parameter controlling the separation of these fixed points.222Well-known examples of this include the and -expansions for the RG flow between the Gaussian and Wilson-Fisher fixed points in dimensions less than four. A priori, this small parameter has nothing to do with the coupling constant of the QFT, which in this case will be large. One may thus perturbatively compute correlators in terms of the CFT correlators associated with either of the fixed points. As the 2- and 3-point correlators of a CFT are universal, depending only on the dimension and OPE coefficients of the operators involved, the 2- and 3-point correlators of the QFT are also fully fixed in terms of these quantities. Ultimately, the dimension and OPE coefficient of the scalar operator dual to the inflaton will map to the slow-roll parameters of the dual cosmology.

The observed small deviation of the power spectrum from exact scale invariance plays a double role in these models: on the one hand it controls the dimension of the nearly marginal scalar operator dual to the inflaton, while on the other hand, it also controls the separation of the UV and IR fixed points in the space of couplings. More precisely, if the UV dimension of the deforming operator is , where , the IR dimension at leading order is , while the separation in coupling between the fixed points is proportional to . Cosmologically, on long wavelengths the tilt of the power spectrum is governed by the IR dimension and we obtain a red tilt , while short wavelengths probe the UV fixed point yielding a blue tilt . Since in more complete models the fate of the RG flow in the UV may be different, it is the red-tilted long-wavelength portion that is of principal interest.

Computing the 2- and 3-point correlators in the dual QFT to leading order in and inserting these in the holographic formulae derived in [13, 16, 17], we recover scalar and tensor power spectra and non-Gaussianities of exactly the form generated by a period of slow-roll inflation.333The slow-roll 3-point function for three gravitons is insensitive to the scalar deformation we study here, however, and as such may be derived from a dual QFT which is an exact CFT [3, 19]. As the treatment in these two papers is already complete we will omit this correlator from our present study. In fact, it is straightforward to systematically identify the relevant class of inflationary potentials as those deriving from a cubic polynomial superpotential. Many simple generalisations of this model are possible, both through the action of field redefinitions and through the exploration of related potentials.

We thus have a class of backgrounds for which it is possible to directly compute on both sides of the holographic correspondence. The exact agreement we find between conventional slow-roll correlators calculated through the gravity description and those calculated holographically through the dual QFT constitutes a highly non-trivial test of holographic cosmology.

The outline of this paper is as follows. Section 2 is devoted to conformal perturbation theory: we present an introduction to perturbative RG flows and explain in detail the computation of 2- and 3-point correlation functions. In Section 3, we summarise briefly the framework for holographic cosmology proposed in [13], then present the necessary holographic formulae for the cosmological power spectra and non-Gaussianities in terms of the QFT correlators computed by conformal perturbation theory. Applying these holographic formulae, we arrive at our holographic predictions for the cosmological observables. In Section 4, we identify the bulk inflationary action and solve for the background evolution and slow-roll parameters, confirming the slow-roll cosmological correlators obtained holographically. We conclude with a discussion in Section 5. Two appendices contain additional technical details concerning the Fourier transform of CFT correlators from position space to momentum space, and a list of relevant QFT Ward identities.

Note added: While this paper was being finalised, [45] appeared which contains some overlap with Section 3.3.1.

2 Conformal perturbation theory

2.1 Perturbative RG flows

We consider a three-dimensional Euclidean conformal field theory perturbed by the addition of a nearly marginal scalar operator of dimension , where . The action takes the form

(1)

where represents the dimensionless bare coupling at some energy scale close to the UV fixed point, and is the bare scalar operator of dimension . We have introduced a non-trivial background metric acting as a source for the stress tensor . After differentiating with respect to a sufficient number of times to bring down the appropriate insertions of we will then set . With this accomplished, correlators in the perturbed theory differ from those in the unperturbed CFT by an insertion of

(2)

where the range of integration must be cut off so that no two operator insertions are closer than . Remarkably, we will find that all terms in this sum contribute at leading order meaning the entire series must be resummed.

On a flat background, the unperturbed CFT correlators take the form

(3)

where and we use the subscript zero to distinguish correlators in the unperturbed CFT from correlators in the perturbed theory. The constant encodes the arbitrary normalisation of the 2-point function, while the constant appears in the OPE

(4)
Figure 1: When the OPE coefficient is a positive constant of order unity the -function has a UV fixed point at the origin and a nearby IR fixed point at .

The -function for the coupling may be found by requiring invariance of the partition function under changes of (see, e.g., [46], or the more recent [47]), yielding

(5)

Assuming to be a positive constant of order unity, the -function is as illustrated in Fig. 1 and we obtain an RG flow to a nearby IR fixed point at

(6)

If instead vanishes or is negative, then the nature of the IR theory will depend on the higher order coefficients in the -function; we will not consider these cases here.444 Nevertheless, we anticipate a simple generalisation to cases where , cf. footnote 13. For positive then, since is small throughout the flow, we may remove higher order terms in the -function by a suitable field redefinition . In the following, we will assume this has been accomplished and work with the purely quadratic -function. Results for the general case may then be found by undoing the field redefinition, generating corrections at subleading orders in the expansion parameter .

Expanding the quadratic -function about the IR fixed point, we find

(7)

In the IR CFT, thus has dimension while the OPE coefficient is unchanged.

The entire RG flow may be obtained by integrating the function directly, yielding

(8)

where we imposed the boundary condition for . Inverting, we find

(9)

where

(10)

Consequently, as we remove the cutoff,

(11)

allowing us to identify as the dimensionful renormalised coupling in the UV CFT.

2.2 Scalar 2-point function

Let us now compute the 2-point function of in the perturbed theory,

(12)

where is an unperturbed CFT correlator with integrated scalar insertions,

(13)

To regulate the integral the range of integration is restricted so that no two insertion points approach closer than the cutoff distance . As our intention is to work to leading order in , it is sufficient to compute only the leading singular behaviour of as . We will see by the argument to follow that in this limit; combined with the prefactor , each term in the sum (12) then makes an order one contribution.

Beginning with the integral , we may formally impose the cutoff by inserting two Heaviside step functions,

(14)

As we are not interested in contact terms in the 2-point function (12) we will assume that . If we now vary with respect to the cutoff , we pick up contributions from the two spherical shells surrounding and ,

(15)

where in the second line we used the OPE (4). As is independent of , upon integrating we obtain

(16)

where is an arbitrary constant, the dependence being fixed on dimensional grounds as no other scales are present. The ellipsis indicates omitted contributions from the remaining terms in the OPE (4). Crucially, these contributions cannot take the form of poles unless there are other terms in the OPE scaling as for some nonzero constant . As a simplifying assumption, we will therefore assume that such terms, if present at all, are of subleading order in , i.e., the associated OPE coefficient is of order or greater.555An exception to this will be the stress tensor, although as we will see in Section 2.4 its inclusion does not affect our present results. Physically, this means that at leading order is the only operator becoming marginal in the limit . The equation (16) then captures the leading behaviour in this limit.

To determine the constant of integration we require this limit to be non-singular, fixing . As we then obtain a logarithmic dependence on the cutoff signalling a Weyl anomaly,

(17)

For , on the other hand, there is no Weyl anomaly and we may safely remove the cutoff. Sending , we find

(18)

The apparently singular behaviour of this result as is simply an artefact of removing the cutoff.

Proceeding now to the general integral , we first introduce step functions to regulate the separation between all possible pairs of insertion points enforcing , and for all such that . Differentiating with respect to the cutoff, in place of (2.2) we now obtain

(19)

where the combinatorial factor counts the number of step functions we had initially, i.e., the number of pairs we can form by bringing together the insertion points , either amongst themselves or with either or , namely

(20)

Note that in writing (19) we have effected a dilute gas approximation in which contributions to the integral from configurations in which more than two insertion points coincide are neglected. This approximation is justified since the phase space associated with these configurations is comparatively small while the value of the integrand is comparable.

Prior to integrating (19), it is useful to trade for

(21)

so that

(22)

To fix the arbitrary constant of integration, we then require that as , i.e., the constant is chosen so as to cancel the leading pole as . Divergences due to subleading poles will be cancelled by the subleading terms we omitted in (19). Given that as , we find that at each order the constants arising from integration with respect to vanish, yielding

(23)

with as given in (6).

The 2-point function in the perturbed theory may now be evaluated at leading order in courtesy of (12),

(24)

Sending (taking note of the -dependence of in (9)), we may re-express this result as a sum of exact CFT 2-point functions with shifted dimensions:

(25)

where is the renormalised coupling defined in (10) and

(26)

Summing up the binomial series, we find

(27)

Finally, to transform to momentum space, starting from (25) we use the result

(28)

and then resum to find

(29)

Here, and throughout the paper, our notation indicates momentum space correlators in which the overall delta function from momentum conservation has been removed. The result (29) is correct to leading order in after expansion in the renormalised coupling .

2.3 Scalar 3-point function

In the case where all the are of comparable magnitude, i.e., when there is effectively a single scale , our arguments above may be straightforwardly generalised to yield the leading order 3-point function for separated insertions,

(30)

The power of minus six appearing in this result arises because here we are summing a binomial series derived from the combinatorial factor in place of (20), encoding the presence of an additional fixed scalar insertion. Note also that, since the dimension of differs from the spatial dimension only by , the general CFT correlator with an arbitrary number of integrated scalar insertions is invariant under special conformal transformations at leading order in , constraining the arbitrary functions arising from integrating with respect to the cutoff to be of the form .

If instead the are no longer all of comparable magnitude, we find ourselves in a limit where the OPE is applicable, and utilising our previous result (27) we find, e.g., when ,

(31)

We may then combine (30) and the three limiting cases of the form (31) into a single result applicable at leading order for all configurations,

(32)

Expanding out the binomial series, the leading order 3-point function in the perturbed theory may alternatively be expressed as a sum of exact CFT 3-point functions with shifted dimensions,

(33)

where

(34)

This expanded form of the 3-point function is useful for performing the Fourier transform to momentum space, as we must do in order to ultimately connect with standard inflationary results. We discuss the details of this Fourier transform in Appendix A.1. At leading order in , the result is

(35)

where the reference momentum is chosen as the largest of the three momenta and hence is implicitly nonzero (see Appendix A.1 for details).

As a check on our calculations, note that in the squeezed limit where we take one of the remaining momenta to zero, we obtain

(36)

This result is of precisely the expected form, since

(37)

and hence the zero-momentum limit

(38)

Inserting our earlier result (29) for the 2-point function in momentum space, we recover precisely (36).

In fact, on dimensional grounds for some function , since when vanishes the 2-point correlator in the perturbed theory must reduce to that of the exact CFT, for which . This yields the Callan-Symanzik equation

(39)

which, when combined with (38), gives

(40)

This result will be useful in Section 3.3.1 when we discuss the inflationary consistency relation for the scalar bispectrum [31].

2.4 Introducing the stress tensor

In this subsection, we now generalise our discussion of conformal perturbation theory to include the stress tensor. After first establishing the definition of this operator in the perturbed theory, we return to the unperturbed CFT to consider the form of 3-point correlators with mixed scalar and stress tensor insertions. From these correlators we may read off the corresponding OPEs, and hence understand the behaviour of stress tensor insertions in correlators of the perturbed theory.

Working henceforth in renormalised perturbation theory, the action takes the form

(41)

and we will assume the two sources and are functionally independent of one another. Now, in a completely general theory, the renormalised scalar operator may depend on either of the sources and . Such a dependence, however, generally introduces additional nearly marginal scalar operators into the spectrum,666Unless it is possible to re-express these operators in terms of and at higher order in . e.g., with dimension or with dimension . From a bulk perspective, this would then correspond to the introduction of additional light scalar fields besides the inflaton resulting in a multi-scalar model. Since our present aim is to concentrate on the single-field case, we will assume that is independent of the sources and , at least to the leading order in at which we work.

In this case, the stress tensor in the perturbed theory is related to the stress tensor in the unperturbed CFT according to

(42)

It follows that the transverse traceless piece of these stress tensors is then identical. Defining the transverse traceless projector

(43)

where is the transverse projector on a flat background, we have where and is defined similarly. In momentum space, where , we may equivalently write where the helicity projection

(44)

and similarly for . Here, the factor of one half arises because our helicity tensors satisfy the identities

(45)

where the helicities take values and repeated indices are to be summed.

In an exact CFT, the 2-point stress tensor correlator takes the form [48]

(46)

where

(47)

while the 3-point correlators we will need are

(48)
(49)

In these formulae,777The specific coefficients appearing here derive from solving (3.6) and (6.20) in [48] at leading order in , where in (3.4) of [48] equals here.

(50)
(51)

and we note in particular that the overall normalisation of the 3-point correlator with a single stress tensor insertion is fixed by the trace Ward identity [49].

Expanding out the 3-point correlators in the limit when two insertion points coincide, we obtain the following OPE contributions with scaling dimensions close to three,

(52)
(53)

where888To rewrite as a well-defined distribution over one may use differential regularisation as discussed in [48]. The remaining OPE coefficients and already have well-defined Fourier transforms.

(54)
(55)
(56)

Our first task is now to verify that the presence of the stress tensor in the OPE does not modify our earlier computations of the scalar 2- and 3-point functions. Fortunately, this is indeed the case. To illustrate this, let us consider the regulated integral given in (14). Varying with respect to the cutoff and using the OPE (53), the r.h.s. of our earlier result (2.2) acquires a new contribution

(57)

Now, in this particular example, the correlators out the front happen to vanish, but in general this will not be the case when we start from the integral containing integrated scalar insertions. Rather, the point is that the residual correlators factor out leaving the integral over a spherical shell of the OPE coefficient . From (56), this OPE coefficient is isotropic and traceless, and so its integral over a spherical shell simply vanishes. We thus obtain no new corrections to our earlier results for the scalar 2- and 3-point functions.999In fact, the isotropy of alone would be sufficient to establish this, since we would obtain residual correlators involving the trace of the stress tensor which vanish by the trace Ward identities, noting that in the regulated correlators none of the insertion points are coincident.

Let us next consider how to evaluate correlators in the perturbed theory involving one or more fixed insertions of the stress tensor. Repeating our above argument, we cannot contract an integrated scalar insertion with a fixed stress tensor to generate a fixed scalar insertion, since the OPE coefficient in (54) is likewise isotropic and traceless. Even if were not isotropic and traceless, its scaling as means that any resulting correlator would be suppressed by a factor of . Specifically, recall that each integrated scalar insertion obtained by expanding carries a factor of which contributes one power of . In the case we considered in Section 2.2 (namely, contracting a fixed scalar insertion with an integrated scalar insertion to generate a fixed scalar insertion), this factor of was offset by a factor of arising from integrating the OPE coefficient over a spherical shell, as we saw in (2.2) and (16). In the present case, however, the scaling of as means that we do not acquire this compensating factor, hence any fixed scalar insertion obtained from the contraction of an integrated scalar insertion with a fixed stress tensor will be suppressed by a factor of relative to leading order. For this same reason we may also ignore operators in the OPE with scaling dimensions not close to three.

In principle, the contraction of an integrated scalar insertion with a fixed stress tensor insertion may generate a fixed stress tensor insertion, since the OPE coefficient in (55) scales as and the integral over the spherical shell will not in general vanish. In fact, however, (and hence ) vanishes for the CFT dual to Einstein gravity. To see this, note first that is a property of the UV CFT alone (as opposed to the full perturbed theory), and so may be extracted from an AdS/CFT calculation on an exact AdS background. For the CFT correlator to be nonzero we would require a nonvanishing graviton-graviton-scalar coupling in the expansion of the bulk action about this background. Given a bulk action of the form

(58)

perturbing about an AdS background involves setting , where is a constant. A graviton-graviton-scalar vertex may then only come from the expansion of , yet since the background is a solution of the scalar field equation of motion this term is a tadpole and vanishes.101010This may also be seen from the explicit calculation of the cubic interaction terms in [31]. When the time derivative of the background scalar field vanishes we must use the gauge (3.2); the graviton-graviton-scalar vertex is then given by the first line of (3.17), which vanishes after taking into account (3.11). For the CFT dual to Einstein gravity then, the correlator and hence must vanish. In consequence one cannot generate a fixed stress tensor insertion from contracting an integrated scalar insertion with a fixed stress tensor insertion, at least at leading order in . (At higher order, however, this should still be possible in order to generate a nontrivial momentum dependence in the tensor 2-point function, and hence a nonvanishing tilt in the inflationary tensor power spectrum through the holographic formula (82).) We stress also that this conclusion is specific to Einstein gravity; for more general bulk actions it may be possible to obtain a nonvanishing which would allow integrated scalars to contract with fixed stress tensors. It would be interesting to explore this further in specific models.

In summary then, for Einstein gravity at leading order in we find that insertions of the stress tensor are essentially inert. Contractions of integrated scalars with fixed stress tensors produce no contribution, and contractions of integrated scalars cannot generate stress tensor insertions. Thus, only scalar insertions participate in the leading order resummation process.

2.5 Stress tensor correlators

Having ascertained the rules of the resummation process when stress tensor insertions are included, it remains to evaluate the specific 2- and 3-point correlators that will appear in our holographic formulae. No new methods are required for this analysis, only a straightforward application of those developed above for correlators with scalar insertions.

Beginning with the 2-point functions, at leading order in we find that

(59)
(60)

The first of these results is a straightforward reflection of the fact that integrated scalar insertions yield no contribution when contracted with fixed stress tensor insertions. As for the second result, although we may contract each of the integrated scalar insertions against the single fixed scalar insertion, each contraction simply returns a fixed scalar insertion and so, after resumming the binomial series to obtain the middle expression, we arrive at a pure CFT 2-point function of two operators with mismatched dimensions which vanishes.

Let us now consider the conversion of these results to momentum space. The trace and diffeomorphism Ward identities imply that the stress tensor 2-point function of a CFT is both transverse and traceless, and hence must take the form

(61)

where the transverse traceless projector is defined in (43). Projected into a helicity basis, this expression is equivalent to

(62)

To identify the coefficient explicitly, we Fourier transform the position space expression (46) with contracted indices to give

(63)

Thus, in momentum space the results (59) read

(64)
(65)

Focusing next on the 3-point functions, in the case where all the are comparable, at leading order in we find

(66)

The power featuring in the prefactor is the same as that for the scalar 2-point function since in both cases we are resumming the binomial series resulting from contracting against two fixed scalar insertions. Through consideration of the various limiting cases using the OPEs (52) and (53) as well as the 2-point results (27), (59) and (60), we may further refine this to

(67)

To convert this result to momentum space, it is useful to first re-express it as a sum of exact CFT 3-point functions with shifted dimensions. From the explicit form of the CFT correlator in (48), it follows that to leading order

(68)

The result (67) may then be cast in the desired form,

(69)

where . After evaluating the Fourier transform of the exact CFT correlator (48), the perturbed correlator (67) in momentum space may be obtained by summing this series.

The general result, derived in Appendix A.2, may take one of two forms according to our choice of nonzero reference momentum (namely, whether we select the momentum associated with , or else one of the momenta associated with , say ). Each form covers one (but not both) of the two distinct squeezed limits and , as well as the common case where all the are comparable, with both forms agreeing in this latter case. For comparison with the standard inflationary results, however, it is sufficient to examine each of these various cases separately.

Firstly, for the quasi-equilateral case in which all three momenta are comparable, namely

(70)

we find that

(71)

where

(72)

with the elementary symmetric polynomials

(73)

In a helicity basis this may be re-expressed as111111See Appendix A of [19] for explicit formulae converting between tensor and helicity bases.

(74)

where