On the power spectrum of inflationary cosmologies dual to a deformed CFT

On the power spectrum of inflationary cosmologies dual to a deformed CFT

Paul McFadden
Abstract

We analyse slow-roll inflationary cosmologies that are holographically dual to a three-dimensional conformal field theory deformed by a nearly marginal scalar operator. We show the cosmological power spectrum is inversely proportional to the spectral density associated with the 2-point function of the trace of the stress tensor in the deformed CFT. Computing this quantity using second-order conformal perturbation theory, we obtain a holographic power spectrum in exact agreement with the expected inflationary power spectrum to second order in slow roll.


On the power spectrum of inflationary cosmologies dual to a deformed CFT

 

  • Perimeter Institute for Theoretical Physics,
    Waterloo, Ontario, Canada N2L 2Y5.

E-mail: pmcfadden@perimeterinstitute.ca

 

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1 Introduction

In this paper we pursue precision holographic cosmology. Starting from a three-dimensional quantum field theory – specifically, the deformation of a conformal field theory by a nearly marginal scalar operator – we compute holographically the power spectrum of the dual inflationary cosmology to second order in slow roll. Our strategy is extremely simple. First, we show that the cosmological power spectrum is inversely proportional to a certain spectral density in the dual QFT, namely that associated with the 2-point function for the trace of the stress tensor. We then compute this quantity directly in the dual QFT using conformal perturbation theory, essentially an expansion in the small parameter controlling the nearly marginal dimension of the scalar operator. From a cosmological perspective, this scalar operator is dual to the inflaton, and our procedure is analogous to an expansion in the spectral tilt. Crucially, this expansion is valid even when the dual QFT is strongly coupled, as is the case when the bulk inflationary cosmology is governed by ordinary Einstein gravity. In this sense, we are effectively able to compute on both sides of the holographic correspondence simultaneously, setting up a striking test of holographic cosmology.

Our scenario is then as follows. Driven by the nearly marginal scalar operator, the dual QFT undergoes an RG flow from the original UV CFT to a nearby IR fixed point. Since the two fixed points lie close together, the coupling of the scalar operator never becomes large along the flow, and in fact is bounded by the same small parameter controlling the dimension of the operator. Cosmologically, each fixed point of the RG flow corresponds to a de Sitter asymptotic region, since the three-dimensional conformal group (in Euclidean signature) coincides with the isometry group of four-dimensional de Sitter spacetime. We thus have an inflationary cosmology undergoing a slow-roll evolution from an early-time de Sitter phase, corresponding to the IR fixed point, to a different late-time de Sitter phase governed by the UV fixed point. The inflaton rolls from a local maximum of the potential down to a nearby local minimum, with the total distance traversed remaining small. (We will not consider how to exit inflation here, though this may be accomplished by allowing additional operators to enter the RG flow changing its fate in the UV.)

Since the slow-roll parameter vanishes at each extremum of the potential, and the separation of these extrema is moreover small, we find ourselves in a situation where the slow-roll parameters satisfy . Our holographic calculation in the dual QFT thus successfully recovers the slow-roll inflationary power spectrum up to and including the second-order and corrections, including all the correct numerical prefactors. (To recover the corrections proportional to would require working to higher order still in conformal perturbation theory.) As well as offering a fresh perspective on the inflationary power spectrum, these results provide a remarkable confirmation of the consistency of the holographic framework.

Our analysis in this paper refines and extends that of [1], which tackles both the power spectra and non-Gaussianities for the same scenario, but restricted to leading order in conformal perturbation theory. The second-order analysis we present here resolves a number of important technical issues that could be avoided at leading order, most notably the definition of the renormalised scalar operator away from criticality. Here, we define the scalar operator so that the corresponding running coupling exactly coincides with the bulk inflaton at horizon crossing. Our present approach is also significantly streamlined. By using RG-improved perturbation theory, we are able to dispense with the explicit series resummation of correlators with integrated insertions performed in [1]. We also now derive the holographic power spectrum directly in terms of the slow-roll parameters at horizon crossing, without requiring their evaluation as specific functions of momentum.

Related recent discussions of holographic cosmologies dual to a deformed CFT may be found in [2, 3, 4], while relevant earlier literature inspired by the dS/CFT correspondence [5, 6, 7] includes [8, 9, 10, 11, 12, 13, 14]. For a pure CFT calculation for the inflationary correlator of three gravitons, see [15, 16], and for other recent approaches to holographic cosmology, see [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].

The outline of this paper is as follows. In Section 2, we introduce the relevant spectral representation for the stress tensor 2-point function in the dual QFT and discuss its relation to the power spectra of the corresponding cosmology. In Section 3 we turn to conformal perturbation theory. We discuss the deformation of a three-dimensional CFT by a nearly marginal scalar operator, computing the -function of the deformed theory and the RG-improved 2-point function for the trace of the stress tensor. We then extract the spectral density with the help of a dispersion relation. In Section 4 we consider the nature of the UV CFT, using simple AdS/CFT calculations on a non-dynamical AdS background to evaluate the required input data for our conformal perturbation theory. Section 5 then discusses how to choose a renormalisation scheme so that the bulk inflaton at horizon crossing maps precisely to the running coupling in the dual QFT. This choice of scheme simplifies the translation of QFT variables into cosmological parameters, with the -function for the running coupling and its derivatives mapping to the bulk slow-roll parameters at horizon crossing. Our result for the holographic power spectrum follows in Section 6, where we elaborate on the nature of the bulk cosmology. We conclude in Section 7 with a discussion of open directions. Two appendices provide additional technical information: Appendix A solves for the running coupling in the dual QFT and verifies the consistency of our results with those of [1], while Appendix B presents details of the AdS/CFT calculation of correlators summarised in Section 4.

2 Holographic power spectra from spectral functions

We begin in this first section with a general discussion of the holographic power spectrum. Our approach to holographic cosmology is based on standard AdS/CFT in combination with the domain-wall/cosmology correspondence. In [29, 30], holographic formulae were derived for the primordial scalar and tensor power spectra in terms of the 2-point function for the stress tensor in the dual QFT. (For higher-point correlation functions, see [31, 32].) This dual QFT is three-dimensional and it is convenient to take it to live in Euclidean signature.

On general grounds, in momentum space the stress tensor 2-point function takes the form

(2.1)

where the transverse traceless and transverse projectors

(2.2)

The function thus encodes the transverse traceless piece of the correlator, while encodes its trace, namely . Our double bracket notation here simply indicates the removal of the momentum-conserving delta function, i.e.,

(2.3)

Through standard AdS/CFT, the stress tensor 2-point function for a three-dimensional QFT flowing to a fixed point in the UV can be related to the asymptotic behaviour of metric perturbations in a four-dimensional asymptotically AdS domain-wall spacetime. The domain-wall/cosmology correspondence then allows these domain-wall fluctuations to be mapped to cosmological perturbations on a corresponding cosmological spacetime. In this manner the stress tensor 2-point function in the dual QFT can be related to the late-time scalar and tensor power spectra in the corresponding cosmology, and , by the following holographic formulae:

(2.4)

The imaginary parts in these expressions are taken after applying the analytic continuation , where is the momentum magnitude. (The 3-momentum in the dual QFT corresponds to the comoving 3-momentum on spatial slices in the cosmology.)

In fact, the formulae quoted in [29, 30] contain an additional minus sign associated with a continuation of the rank of the gauge group of the dual QFT. Since the only appearance of here is as an overall factor multiplying and , however, we have simply included the corresponding sign directly into the holographic formulae above.

In the following, we wish to suggest an alternative physical interpretation for these formulae in terms of the spectral representation for the stress tensor 2-point function. In three Euclidean dimensions, adopting the conventions of [33], this reads

(2.5)

where . Physically, the dimensionless spectral functions and encode the contribution to the 2-point function from spin-0 and spin-2 intermediate states of mass . Originally, interest in these spectral functions was motivated by attempts to extend Zamolodchikov’s c-theorem to higher dimensions, see, e.g., [34, 33, 35, 36].

In momentum space, the spectral representation takes the form

(2.6)

These formulae allow and to be analytically continued to the complex plane with a branch cut along the negative real axis. The spectral functions and may then be extracted by applying standard dispersion relations [33]. For example, integrating along the contour given in Figure 1, we find

(2.7)

Differentiating (2.6) directly, on the other hand, yields

(2.8)

In this fashion, we identify

(2.9)

Given the branch cut lies along the negative real axis, the imaginary parts in these formulae are precisely the same as those in the holographic formulae (2.4), and hence

(2.10)

Expressed in this form, the holographic formulae relate physical quantities in both theories – the cosmological power spectra and the spectral functions in the dual QFT – without any analytic continuations. Since we have incorporated the continuation of [29, 30] into the holographic formulae (2.4) (rather than than applying it to the dual QFT), the dual QFT here is unitary meaning the spectral functions above are positive [33].

Figure 1: To derive the dispersion relation (2.7) we integrate over the contour in the complex plane shown above. The contribution from the circular arc at infinity vanishes, since in the case of interest, as with . The discontinuity across the branch cut is then related to the residue at .

3 Conformal perturbation theory

Our goal in this section is now to compute the spectral function for a CFT perturbed by a nearly marginal scalar operator, using RG-improved conformal perturbation theory. Later we will substitute our result into the holographic formula (2.10) to recover the slow-roll inflationary power spectrum for scalar perturbations. In this paper we will focus solely on the scalar power spectrum, since the tensor power spectrum is simply a constant at the order to which we will work [1].

We have split this section into three parts. First, in Section 3.1, we discuss the calculation of correlators in the deformed CFT, our choice of renormalisation scheme, and the computation of the -function. Then, in Section 3.2, we use the renormalisation group to improve our perturbative calculation of 2-point functions in the deformed CFT, to recover the correct scaling behaviour in the IR. Finally, in Section 3.3, we extract the spectral function using the dispersion relation (2.9). Our treatment of conformal perturbation theory is modelled on that in [37], see also [38, 39].

3.1 Deforming a CFT

Consider a three-dimensional Euclidean CFT deformed by a slightly relevant scalar operator,

(3.1)

where has dimension with . Correlators in the perturbed theory are given by the perturbative expansion

(3.2)

where the zero subscript indicates the correlator evaluated in the original undeformed UV CFT. Transforming to momentum space, the integrated insertions become insertions at zero momentum, and we obtain

(3.3)

where from the dilatation Ward identity111In general this identity may be anomalous, but this is not the case here; see Section 4 and Appendix B. we have

(3.4)
(3.5)
(3.6)

Our double bracket notation once again indicates the removal of the momentum-conserving delta function, e.g.,

(3.7)

The are numerical coefficients characterising the UV CFT: in particular, encodes the normalisation of , while encodes the OPE coefficient of with itself. We will return to evaluate these coefficients in Section 4, however we leave them unspecified for now.

In the limit where becomes marginal, we typically encounter singularities. For example, if we convert the standard position-space expression for a CFT 3-point function to momentum space, then send keeping the OPE coefficient fixed, we find . To cure these divergences we introduce the dimensionless renormalised coupling and the renormalised operator

(3.8)

defined so that correlators of are finite in the limit with fixed. Note here that the renormalisation factor introduces a dependence into .

To define our renormalisation scheme, we set

(3.9)

at some momentum scale , where the power of on the right-hand side is fixed by the engineering dimension of the correlator. (Since is dimensionless, has engineering dimension three.) As usual, there is a degree of arbitrariness in any given choice of scheme: we will return to exploit this freedom later in Section 5. Solving for the renormalisation factor, we find

(3.10)

The -function for the renormalised coupling, , is defined by

(3.11)

where is the trace of the stress tensor. In the UV CFT, on the other hand, the trace of the stress tensor is

(3.12)

Being a conserved current, the stress tensor does not renormalise, so and hence

(3.13)

Integrating the -function, requiring that as , then fixes

(3.14)

Inverting and substituting back into (3.13), we obtain

(3.15)

where

(3.16)

We will be interested in the case where and are of order unity, which requires and with a cancellation of the leading term in , as we will find in Section 4. If is positive, moreover, we obtain an RG flow as illustrated in Figure 2 to a new IR fixed point located at

(3.17)

With and of order unity, and the IR fixed point lies close to the UV fixed point at the origin. The scaling dimension of at the IR fixed point is

(3.18)
Figure 2: For is positive and of order unity, the -function has a perturbative IR fixed point located at .

3.2 RG-improved correlators

Re-expressing (3.3) in terms of the renormalised coupling, we obtain

(3.19)

This expression, however, manifestly fails to capture the expected scaling behaviour about the IR fixed point. To rectify this defect we turn to RG-improved perturbation theory, implemented by solving the Callan-Symanzik equation. In effect, this serves to automatically resum an infinite number of terms in the perturbative expansion.222By contrast, in Section 2.2 of [1], this resummation was performed manually after recursively constructing the coefficient of each term in the series in powers of . The differential equation that was used to construct these coefficients is equivalent to that obtained by inserting a series solution in powers of into the Callan-Symanzik equation here.

Since and is independent of the RG scale , we have the Callan-Symanzik equation

(3.20)

where the anomalous dimension

(3.21)

(Another way to arrive at this relation is to note that the stress tensor 2-point function has zero anomalous dimension.)

On dimensional grounds, we then have

(3.22)

The solution takes the form

(3.23)

where the running coupling is defined by

(3.24)

The function then follows from our renormalisation scheme (3.9),

(3.25)

hence

(3.26)

We can check that the UV behaviour of this result is consistent with (3.2) by solving for the running coupling about the UV fixed point as a series in . Likewise, solving for the running coupling about the IR fixed point as a series in , we now also obtain the correct scaling behaviour in the IR.

3.3 Spectral function

We turn now to the extraction of the spectral function . Multiplying (3.26) by , the RG-improved result for the trace of the stress tensor 2-point function is

(3.27)

The spectral function then follows from the dispersion relation (2.9),

(3.28)

If an explicit solution of (3.24) for the running coupling is available (see Appendix A) then this formula can be evaluated directly. For our present purposes, however, it is sufficient to write and Taylor expand:

(3.29)

Note that since the running coupling is at most of order , we have , while and . It then follows that

(3.30)

from which we obtain the spectral function

(3.31)

This result can be checked against that obtained using the explicit solutions for the running coupling discussed in Appendix A.

In both (3.30) and (3.31), the error term is a shorthand notation collecting together terms of the form . Expanded out in full, we have

(3.32)

4 Characterising the UV CFT

At the order to which we work, the properties of the dual QFT computed above depend only on the three parameters , and characterising the correlators (3.4)-(3.6) in the UV CFT. Our task in this section is now to evaluate these parameters for the CFT we are interested in, namely the CFT dual to 4d Einstein gravity with a minimally coupled scalar field and a potential, as per the simplest inflationary models.

Since these parameters describe correlators in the UV CFT itself, rather than in the deformed theory, they can be evaluated simply by perturbatively solving the scalar field equation on a fixed four-dimensional AdS geometry. An overview of these elementary AdS/CFT calculations is supplied in Appendix B; here we simply summarise the results as we need them.

In general, the parameters , and are related to the coefficients appearing in the Taylor expansion of the potential to quartic order,

(4.1)

where we work in units in which the AdS radius is set to one and . The linear term in this expansion has been removed by shifting so that the AdS critical point where occurs at the origin, while for the UV dimension of to be we require . The information in this Taylor expansion can equivalently be repackaged in terms of a superpotential , obeying

(4.2)

where333We restrict our interest here to solutions for which a critical point of the potential where corresponds to a critical point of the superpotential, . Holographic RG flows of this type possess ‘fake’ supersymmetry and are known to be stable both perturbatively and non-perturbatively, see [40].

(4.3)

In terms of these latter coefficients, the parameters characterising the UV CFT are found in Appendix B to be

(4.4)
(4.5)
(4.6)

where

(4.7)

with the Euler-Mascheroni constant. Restored to the right-hand sides of these equations, the factor of is proportional to , where is the rank of the gauge group of the dual QFT.

Using (3.16), we can then immediately read off the coefficients of :

(4.8)
(4.9)

With and of order unity, and are also of order unity so the IR fixed point indeed occurs at .

5 Calibrating to the cosmology

In this section, as a final preparation before computing the holographic power spectrum, we refine our choice of renormalisation scheme so as to set the running coupling in the dual QFT equal to the value of the inflaton at horizon crossing in the corresponding cosmology.

To begin, let us specify more precisely the nature of this inflationary cosmology. As directed by the domain-wall/cosmology correspondence [29, 30, 41, 42], we consider the inflationary cosmology associated with the inverted domain-wall potential

(5.1)

The Taylor expansion of in (4.3) now describes an expansion about a de Sitter fixed point. Since by assumption our dual QFT lives on a flat metric, we consider spatially flat four-dimensional FRW backgrounds of the form

(5.2)

We will further assume the inflaton profile is monotonic (or at least piecewise so), as befits an RG flow. The inflationary action

(5.3)

then yields the first-order equations of motion [43]

(5.4)

Introducing the slow-roll parameter

(5.5)

the log derivative of , the value of the inflaton at horizon crossing (defined as the time at which ), is given by

(5.6)

where is again our reference momentum scale. The QFT running coupling, on the other hand, satisfies

(5.7)

with and as identified above in (4.8)-(4.9).

Comparing these equations, we see that at leading order in the running coupling coincides with the value of the inflaton at horizon crossing, if we choose . (Note in our conventions the inflaton and the running coupling are both dimensionless.) To extend this convenient identification to higher order, we modify our renormalisation scheme by re-defining the renormalised coupling:

(5.8)

Under this change of scheme, the -function becomes

(5.9)

where

(5.10)

For and of order unity, this produces the subleading corrections to the -function that we require in order to match with (5.6). Specifically, introducing the running coupling defined by

(5.11)

in order to identify

(5.12)

we must set

(5.13)
(5.14)

Setting in addition , the running coupling is now equal to the value of the inflaton at horizon crossing , working444In fact, this identification can be extended to all orders in by using the exact expressions for , and (see Appendix B) to obtain and in (4.8)-(4.9) exactly. One then solves for the and in (5.10) required to identify (5.9) with (5.6) to all orders in . All we require here however are the leading terms as given in (5.13)-(5.14). to order .

Under the change of renormalisation scheme (5.8), the stress tensor 2-point function, and hence the spectral densities, are invariant. It will be useful, nevertheless, to re-express the spectral function in terms of and its derivatives. To do so, starting from (3.3) we use (5.8) and (5.9) in the form

(5.15)
(5.16)

along with (5.10), (5.13) and (5.14), plus the value of from (4.4). The spectral function then takes the form

(5.17)

where and primes denote derivatives with respect to the running coupling .

6 The holographic power spectrum

Finally we are ready to determine the holographic power spectrum. Plugging the spectral function (5.17) into the holographic formula (2.10), we obtain

(6.1)

where again . To put this result in recognisable form, we introduce the slow-roll parameters

(6.2)

in addition to as defined earlier in (5.5). Since we have identified with , it follows that

(6.3)

The holographic power spectrum (6.1) is thus

(6.4)

This is precisely the standard second-order slow-roll power spectrum, as we see by comparing with the conventional inflationary calculation in [44], noting that here

(6.5)

since , and . In writing (6.4) we have also used , as follows from (5.4) and (4.3) noting that is at most of order . Our choice of units in which the de Sitter radius is one originated in (4.3).

The entire holographic cosmology is thus controlled by the -function of the dual QFT. We start by solving (5.11) for the running coupling , which in our renormalisation scheme is equal to the value of the inflaton at horizon crossing . The RG scale plays the role of the cosmological pivot scale, with being the value of the inflaton when this pivot scale crosses the horizon, . The slow-roll parameters at horizon crossing , and follow from (6.3), and the power spectrum is given by (6.4). The inflaton potential can also be reconstructed from the -function: using (5.1) and , we find

(6.6)

where

(6.7)

setting the error terms to those in (5.12). If desired, the background solution for and can similarly be reconstructed from the superpotential,

(6.8)

using the equations of motion (5.4). (See [1] for the case where .)

We emphasise that our choice of renormalisation scheme makes no impact on the final result for the cosmological power spectrum, since the spectral function is scheme-independent. Rather, the choice of scheme is simply a matter of convenience, determining the relationship between the running coupling and its -function in the deformed CFT and their counterparts in the cosmology, the inflaton and the slow-roll parameters at horizon crossing. Here we chose to identify the running coupling with the inflaton at horizon crossing, yielding the relation (6.3) between the slow-roll parameters at horizon crossing and the -function of the running coupling. An alternative choice, discussed in [9, 11, 13], would be to identify , where is the renormalised coupling. In this case, however, the relation between the running coupling and the inflaton at horizon crossing would be nontrivial, with . This relation would then have to be taken into account in going from the spectral function to the slow-roll power spectrum. While these two schemes are effectively equivalent at the order to which we currently work, they will be distinct at higher order.

Figure 3: The inflaton potential comprises a hilltop (corresponding to the IR fixed point at ) with a nearby local minimum at the origin (corresponding to the UV fixed point). The decrease in height of the potential is of order .

The form of the inflaton potential is sketched in Figure 3. The UV fixed point of the dual QFT corresponds to the local minimum at the origin, while the IR fixed point corresponds to the local maximum at . Each fixed point is associated with a CFT, corresponding to a de Sitter asymptotic region in the cosmology. Cosmic time evolution corresponds to inverse RG flow, since the IR CFT controls the early-time behaviour and the UV CFT the late-time behaviour of the cosmology.

The spectral index is given by555We assume here that allowing us to drop a term proportional to . When comparing with [44], this corresponds to the case where