Paul McFadden and Kostas Skenderis
Institute for Theoretical Physics, Gravitation and Astro-Particle Physics Amsterdam,
Korteweg-deVries Institute for Mathematics,
Science Park 904, 1090 GL Amsterdam, the Netherlands.
E-mail: P.L.McFadden@uva.nl, K.Skenderis@uva.nl
We investigate the non-Gaussianity of primordial cosmological perturbations within our recently proposed holographic description of inflationary universes. We derive a holographic formula that determines the bispectrum of cosmological curvature perturbations in terms of correlation functions of a holographically dual three-dimensional non-gravitational quantum field theory (QFT). This allows us to compute the primordial bispectrum for a universe which started in a non-geometric holographic phase, using perturbative QFT calculations. Strikingly, for a class of models specified by a three-dimensional super-renormalisable QFT, the primordial bispectrum is of exactly the factorisable equilateral form with , irrespective of the details of the dual QFT.
A by-product of this investigation is a holographic formula for the three-point function of the trace of the stress-energy tensor along general holographic RG flows, which should have applications outside the remit of this work.
- 1 Introduction
- 2 Perturbed domain-walls and cosmologies
- 3 The cosmological bispectrum
- 4 Holographic 3-point functions
- 5 Holographic formulae for cosmology
- 6 Holographic non-Gaussianity
- 7 Discussion
- A Gauge-invariant variables at second order
- B Constraint equations
Primordial cosmological perturbations and their properties provide some of the best observational clues to the physics of the very early universe. Acting as the seed for structure formation, these initial inhomogeneities left behind an imprint in the cosmic microwave background (CMB), and in the distribution of large-scale structure, through which their properties may be directly inferred. To date, there is no compelling evidence for any departure from Gaussianity: the different Fourier modes of the perturbations appear to be uncorrelated, with random phases. This implies that all higher correlation functions of the primordial perturbations may be expressed in terms of the 2-point function, or equivalently its Fourier transform, the power spectrum . In particular, the 3-point function and all odd higher-order correlators should vanish.
Nevertheless, the significant improvement in observational data expected in the very near future may change this situation. Quantitatively, the Fourier transform of the 3-point function of curvature perturbations, the scalar bispectrum, may be parameterised by an overall amplitude, , along with a momentum dependence or ‘shape’ function (see [1, 2, 3, 4] for reviews). From the WMAP -year data , the observational constraints on , for two specific choices of shape function, are
where the first value corresponds to the ‘local’ shape [6, 7, 8] and the second corresponds to the ‘equilateral’ shape . In just a few years’ time, the results from the Planck satellite are expected to further reduce the uncertainty on to approximately . Constraints deriving from observations of large-scale structure may also in future be competitive with those from the CMB .
In view of its power to elucidate features of the mechanism through which the primordial perturbations were generated, any future detection of primordial non-Gaussianity will be of paramount importance for cosmology. In the context of inflation, the leading candidate for such a mechanism, primordial non-Gaussianity reveals details of the dynamical interactions present during the inflationary epoch.
In the simplest models of inflation, based on a single scalar field slowly rolling down a potential, these interactions are suppressed by powers of the slow-roll parameters giving rise to an of first order in slow-roll (i.e., ) [6, 11, 12, 13, 14]. More elaborate inflationary models including, e.g., multiple scalar fields [15, 16, 17, 18, 19], non-canonical kinetic terms [20, 21, 22], inhomogeneous reheating , features in the potential [24, 25, 26], or initial state modifications [27, 28, 29] all give rise to a considerable range of values, as well as different predictions for the shape function. Since there is often little to distinguish models at the level of the power spectrum, non-Gaussianity provides a powerful means of observationally discriminating between the various inflationary candidates, as well as serving to constrain other non-inflationary scenarios . Note for these purposes it is necessary to distinguish the primordial non-Gaussianity generated during the inflationary epoch from that generated in later stages of cosmological evolution (e.g., by the nonlinear evolution of perturbations after they re-enter the horizon in the matter and radiation eras, and by nonlinearities in the relation between metric fluctuations and temperature fluctuations in the CMB). With a sufficiently accurate understanding of their physics [31, 32], these latter sources of non-Gaussianity may be subtracted off to leave the primordial component of principal interest for constraining cosmological models.
In the present paper, we initiate the investigation of primordial non-Gaussianity for our recently proposed holographic description of inflationary cosmology [33, 34, 35]. The key to this description is the holographic framework depicted in Figure 1, which connects four-dimensional inflationary universes with three-dimensional non-gravitational QFTs. The basic ingredients are ordinary gauge/gravity duality (corresponding to the uppermost arrow in the figure), combined with the domain/wall cosmology correspondence [36, 37, 38] (lefthand vertical arrow), a simple analytic continuation relating cosmologies to domain-wall spacetimes describing holographic RG flows. This bulk analytic continuation may be re-expressed in the language of the dual QFT (righthand vertical arrow), whereupon it takes correlators of the dual QFT to correlators of the so-called ‘pseudo’-QFT, which we propose is dual to the original cosmology (lower dashed arrow).
On the basis of this framework, cosmological observables may be re-expressed in terms of correlators of the dual QFT. At the level of linear perturbation theory, exact formulae have been derived relating the cosmological scalar and tensor power spectra with the 2-point function for the stress-energy tensor of the dual QFT [33, 34]. The first major goal of the present work will be to extend this correspondence to quadratic order in perturbation theory. Focusing on the scalar bispectrum, the cosmological observable most relevant to the present and forthcoming observational data, we will show how this quantity may be naturally re-expressed in terms of the 3-point function of the dual stress-energy tensor111For earlier work in a similar spirit, see .. Analogous formulae for other non-Gaussian cosmological observables may be derived by similar methods and will be reported elsewhere . The Hamiltonian holographic renormalisation method we develop for computing 3-point functions is based on [41, 42], and may well be of utility in a wider holographic context222Earlier work on the computation of 3-point functions for holographic RG flows may be found in [43, 44]..
One striking feature of holographic dualities is that they are strong/weak coupling dualities, meaning that when one description is weakly coupled, the other is strongly coupled, and vice versa. In the regime where the dual QFT is strongly coupled then, the gravitational description is weakly coupled and our holographic formulae should (and indeed they do) reproduce the results of standard single-field inflation. In this situation, the application of our holographic framework offers a fresh perspective, and may lead to new insights, but offers no new predictions.
In the regime in which the dual QFT is weakly coupled, however, the corresponding gravitational description is instead strongly coupled at very early times. We emphasize that by ‘strongly coupled’ gravity we do not mean that the perturbative fluctuations around the background FRW spacetime are strongly coupled, but rather, that the description in terms of metric fluctuations is itself not valid. This is a non-geometric ‘stringy’ phase. A geometric description emerges only asymptotically, and at late times one recovers a specific accelerating FRW spacetime (to be matched to conventional hot big bang cosmology), along with a specific set of inhomogeneities. Crucially, these inhomogeneities are not linked with a perturbative quantisation around the FRW spacetime as in conventional inflation, but rather, they originate from the dynamics of the dual weakly coupled QFT. Holography thus suggests a natural generalisation of the inflationary mechanism to strongly coupled gravity, in which the properties of cosmological perturbations may be determined through three-dimensional perturbative QFT calculations.
In order to perform such calculations, it is necessary to specify more precisely the nature of the dual QFT. Ideally, one would be able to deduce this from first principles via some string/M-theoretic construction. In the absence of such a construction, we will instead pursue a (holographic) phenomenological approach. As with other known holographic dualities, the dual QFT will in general involve scalars, fermions and gauge fields, and it should admit a large limit. The question is then whether one can find a theory which is compatible with current observations.
An additional guiding principle is to consider QFTs of the type that feature in the description of holographic RG flows. This holographic description is well understood for two classes of domain-wall spacetimes, namely, those that are asymptotically anti-de Sitter, and those with asymptotically power-law scaling. Under the domain-wall/cosmology correspondence, these correspond respectively to asymptotically de Sitter, and to asymptotically power-law cosmologies. The first class of domain-wall solutions describe QFTs that are either deformations of CFTs, or else CFTs in a nontrivial vacuum state, while the second class describes QFTs with a single dimensionful parameter in the regime in which the dimensionality of the coupling constant drives the dynamics . Examples of such dualities are provided by considering the near-horizon limit of the non-conformal branes [46, 47]. The detailed holographic dictionary for these theories has been worked out only relatively recently [48, 45, 49]. These theories are characterised by the fact that they have a ‘generalised conformal structure’ [50, 51, 52, 45]. In particular, all terms in the Lagrangian have the same scaling dimension, which is however different from the spacetime dimension.
Focusing on this second class, we will consider here super-renormalisable theories that contain one dimensionful coupling constant. A prototype example is three-dimensional Yang-Mills theory coupled to a number of scalars and fermions, all transforming in the adjoint of . Theories of this type are typical in AdS/CFT where they appear as the worldvolume theories of D-branes. A general such model that admits a large limit is
where we consider gauge fields (, minimal scalars (, conformal scalars ( and fermions (. Note that has dimension one in three dimensions. In general, the Lagrangian (2) will also contain dimension-four interaction terms (see ). We will leave these interaction unspecified, however, as they do not contribute to the leading order calculations we will perform here.
As shown in [33, 34, 35], it is straightforward to find holographic models of this form that yield cosmological predictions compatible with current observations. Generically, we obtain a nearly scale-invariant spectrum of small amplitude perturbations, where the overall amplitudes of the power spectra scale as , and their deviation from scale invariance is of order the dimensionless effective coupling, , where is a typical momentum. In particular, the small observed amplitude of the scalar power spectrum implies consistent with the large limit, while the smallness of the observed deviation from scale invariance is consistent with the assumed weak coupling limit where . Through appropriate choice of the field content of the dual QFT, it is further possible to satisfy the current observational upper bounds on the ratio of tensors to scalars.
A distinctive prediction of these holographic models is a well-defined running of the spectral indices: in the scalar case, for example, the running is given by minus the deviation from scale invariance. This prediction is markedly different from conventional slow-roll inflation, where the running is heavily suppressed, and may potentially be excluded by the forthcoming Planck data .
Having established a precise holographic formula linking the cosmological bispectrum to the 3-point function for the dual stress-energy tensor, our second major goal will be to use this formula to predict the cosmological non-Gaussianity arising from a weakly coupled dual QFT of the form (2). These predictions will complement those for the power spectra discussed above, and may potentially reveal further distinctive observational signatures of holographic models.
The remainder of this paper is organised as follows. In Section 2, we discuss perturbation theory for domain-walls and cosmologies at quadratic order: after defining the metric fluctuations and the gauge-invariant curvature perturbation , we evaluate the cubic interaction Hamiltonian and set up the domain-wall/cosmology correspondence. We also introduce response functions relating the curvature perturbation to its corresponding canonical momentum; these response functions will play a central role in our subsequent holographic analysis. In Section 3, we summarise the calculation of the cosmological bispectrum and show how to re-write it in terms of response functions. We then proceed, in Section 4, with the holographic calculation of the 3-point function for the trace of the dual stress-energy tensor. After a brief introduction to the radial Hamiltonian holographic renormalisation methods we use, we compute the holographic 3-point function for both asymptotically AdS and asymptotically power-law domain-walls. Finally, in Section 5, we combine these results to show how the cosmological observables may be expressed in terms of correlation functions of the dual QFT, and in Section 6, after performing the relevant QFT calculations, we arrive at a holographic prediction for primordial non-Gaussianity.
2 Perturbed domain-walls and cosmologies
2.1 Defining the perturbations
Domain-walls and cosmologies may be described in a unified fashion via the ADM metric
where the perturbed lapse and shift functions may be written to second order as
with for a Euclidean domain-wall333A Lorentzian domain-wall can be obtained by continuing one of the coordinates to become time . The continuation to a Euclidean domain-wall is convenient, however, since the QFT vacuum implicit in the Euclidean formulation maps to the Bunch-Davies vacuum on the cosmology side. Other choices of vacua may be accommodated using the real-time formalism of . This is an interesting extension that we leave for future work. (whereupon becomes the transverse radial coordinate) and for a cosmology (whereupon becomes the cosmological proper time). The spatial indices run from to , and we have assumed (for simplicity) the background geometry to be spatially flat.
The metric perturbation is then
where here, and in the remainder of the paper, we adopt the convention that repeated covariant indices are summed using the Kronecker delta (in contrast, an index is raised or lowered by the full metric). The remaining perturbations may be decomposed into scalar, vector and tensor pieces according to
where the vector perturbations and are transverse, and the tensor perturbation is transverse traceless. We similarly decompose the inflaton into a background piece and a perturbation ,
These formulae are understood to hold to second order in perturbation theory.
We define , the curvature perturbation on uniform energy density slices, so that in comoving gauge where vanishes, the spatial part of the perturbed metric reads
where is transverse traceless444We have chosen this definition so as to coincide with most of the recent literature on non-Gaussian perturbations, in particular . Note that in our previous articles [33, 34] we defined at linear order to be instead the comoving curvature perturbation, which differs by a sign.. This definition may then be straight-forwardly recast into the general gauge-invariant form (see Appendix A for details)
where . Here, and throughout, we use dots to denote differentiation with respect to and we set . The transverse projection operator is defined as
The physical significance of is that it is conserved on super-horizon scales, in the absence of entropy perturbations. This holds to all orders in perturbation theory , and serves to connect the behaviour of modes as they exit the horizon during the inflationary epoch to their initial conditions at horizon re-entry in the subsequent radiation- and matter-dominated eras.
2.2 Equations of motion
In the ADM formalism, the combined domain-wall/cosmology action for a single minimally coupled scalar field takes the form
where , is the extrinsic curvature of constant- slices, and we have taken the scalar field to be dimensionless. In this expression, the spatial gradient and potential terms appear with positive sign for Euclidean domain-walls and with negative sign for Lorentzian cosmologies, as indeed they should.
We will restrict our consideration to background solutions in which the evolution of the scalar field is (piece-wise) monotonic in . For such solutions, can in principle be inverted to , allowing the Hubble rate to be re-expressed as a function of , i.e., . The complete equations of motion for the background then take the simple form
In cosmology, this first-order formalism dates back to the work of , where it was obtained by application of the Hamilton-Jacobi method. For domain-walls, this formalism has been discussed from variety of standpoints (gravitational stability, Hamilton-Jacobi method, fake supersymmetry) in [37, 55, 56, 57, 58]. In this context, the function is the ‘fake superpotential’ (i.e., when the domain-wall solution is a supersymmetric solution of a supergravity theory, is the true superpotential).
Turning now to the perturbations, following Maldacena , the cubic action for may be derived by solving the Hamiltonian and momentum constraints and backsubstituting into the Lagrangian. Keeping careful track of the sign , we find
where (note we do not use the slow roll approximation) and .
In the Hamiltonian formalism, one then has the quadratic free Hamiltonian
is ( times) the canonical momentum conjugate to , and the cubic interaction Hamiltonian
where in this expression .
Passing to momentum space, one finds
where here, and in the remainder of the paper, we will use the shorthand notations
The coefficients , , and may be written as
where the magnitudes and the symmetric polynomials .
Writing , etc., Hamilton’s equations then read
Note that , as defined in (2.2), implicitly depends on through the overall delta function expressing momentum conservation.
2.3 Response functions
Given a perturbative solution of the classical equations of motion, we may formally expand in terms of to any given order in perturbation theory. At quadratic order, we may thus write
where we will refer to the functions and defined by this equation as response functions. (Note we have made use here of the translation invariance of the background 3-geometry). In momentum space, we then have
Rotational invariance and momentum conservation (which in particular implies ) imply that and are scalar functions of the magnitudes such that and . Thus, in the following, we will simply write
it follows that
is a solution of (27), and that
The solution for is then
where we will leave the lower limit in the integral unspecified for the time being.
2.4 The domain-wall/cosmology correspondence
Examining (11), (2.2) and (2.2) closely, we see that a perturbed cosmological solution expressed in terms of and analytically continues to a perturbed domain-wall solution expressed in terms of and , where
The first continuation serves to reverse the sign of the potential in (11) (taking, for example, dS to AdS), while the second ensures that , accounting for the necessary sign changes in the equations of motion (2.2) and (2.2) (specifically, in the coefficients , , and , as well as in the first term on the r.h.s. of (2.2)). The choice of branch cut we made in this latter continuation (i.e., rather than ) is determined by the necessity of mapping the cosmological Bunch-Davies vacuum behaviour, as (where ), to the domain-wall solution that decays smoothly in the interior, as , as required for the computation of holographic correlation functions.
For the response functions, we see likewise that if we define and to be the cosmological response functions with , then the domain-wall response functions and are given by the simple analytic continuation
(Note that the response functions, as defined here, are independent of ).
In the remainder of this paper, we will use the unbarred variables , and the response functions and when performing cosmological calculations, and the barred variables , and response functions and for domain-wall calculations. To analytically continue the results from domain-walls to cosmologies, and vice versa, we use (34) and (35).
Finally, let us note the analytic continuations (34) may equivalently be expressed in terms of QFT variables as
where is the rank of the gauge group of the QFT dual to the domain-wall spacetime, and is the rank of the gauge group of the pseudo-QFT dual to the corresponding cosmology. These relations follow from (34), noting that in the standard holographic dictionary , working in units where the AdS radius has been set to unity555In fact, in our later results we will see explicitly that holographic correlation functions calculated from the gravity side of the correspondence appear with an overall prefactor of . On the QFT side of the correspondence, this prefactor corresponds to the overall prefactor of in correlators arising from the trace over gauge indices.. Our choice of branch cut in the continuation of ensures that the dimensionless effective QFT coupling, , does not change when we analytically continue from QFT to pseudo-QFT. This is important because the QFT correlators may in general be non-analytic functions of at large [59, 60].
3 The cosmological bispectrum
3.1 Computation using response functions
In this section we compute the 3-point function of cosmological curvature perturbations in terms of the second-order response function .
We begin by quantising the interaction picture field such that
(recalling that plays the role of proper time here), or equivalently, in momentum space,
The creation and annihilation operators obey the usual commutation relations
In these expressions, the mode function is a solution of the linearised equation of motion (30), with initial conditions specified by the Bunch-Davies vacuum condition.
At tree level, the 3-point function in the in-in formalism may then be evaluated according to the standard formula 
where, to ensure convergence, a suitable infinitesimal rotation of the contour of integration is understood. The lower limit represents some very early time (corresponding to very large and negative) at which the interactions are assumed to be switched on. Note that both the operators appearing in the commutator in this formula are taken to be normal ordered as indicated.
Inserting the operator equivalent of (16) for in the above formula, we may now proceed to evaluate the commutator explicitly, noting that for the cubic terms in we may replace
In this manner, we find the full 3-point function
where in the last line we have used (33). The lower limit of integration in (33) should thus be identified with the lower limit in (40). With this choice, we find as , consistent with the expected behaviour and prior to the switching on of the interactions.
Introducing the notation
the bispectrum of curvature perturbations then satisfies
where, as usual, the 2-point function is
Noting that the linearised mode functions obey the Wronskian normalisation condition
we may express the 2-point function in terms of the response function ,
Equations (47) and (44) are the main result of this section: they express the power spectrum and the bispectrum in terms of response functions. As we will see in the next section, these response functions (after analytic continuation) are directly related with 2- and 3-point functions of strongly coupled QFT via standard gauge/gravity duality.
3.2 An example: slow-roll inflation
As an illustration, let us use our results above to calculate the bispectrum to leading order in the slow-roll approximation. As noted by Maldacena , this calculation is most easily performed using the field redefinition
where the dots indicate terms that vanish outside the horizon or are of higher order in slow roll. The cubic action (2.2) may then be rewritten to leading order in slow roll as
Comparing with (2.2), we see that the field redefinition makes manifest the fact that the interaction is really of second order in the slow-roll parameter . The interaction Hamiltonian for then has only the single term
To evaluate the late-time value of the response function , we use the formula (33), substituting in the de Sitter solution
for the linearised mode functions. Here, the asterisk indicates taking the value at the time of horizon crossing (where ), while the conformal time . We also use the fact that, to leading order in slow roll, the linear response function
since and time derivatives of and are of higher order in slow roll.
We thus find
where the integration contour has been suitably rotated so as to ensure convergence of the lower limit and the subscript zero, both here and below, denotes the value in the late-time limit . Then,