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Holography for inflation using conformal perturbation theory

Adam Bzowski, Paul McFadden, Kostas Skenderis

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.

Holography for inflation using conformal perturbation theory

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.

Paper Structure

This paper contains 27 sections, 184 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformal perturbation theory
  3. Perturbative RG flows
  4. Scalar 2-point function
  5. Scalar 3-point function
  6. Introducing the stress tensor
  7. Stress tensor correlators
  8. Holographic calculation of inflationary correlators
  9. Holographic formulae
  10. Power spectra
  11. Non-Gaussianities
  12. Three scalars
  13. Two scalars and a graviton
  14. Two gravitons and one scalar
  15. Identifying the dual cosmology
  16. ...and 12 more sections

Figures (2)

  • Figure 1: When the OPE coefficient $C$ is a positive constant of order unity the $\beta$-function has a UV fixed point at the origin and a nearby IR fixed point at $\varphi=\varphi_1\ll 1$.
  • Figure 2: The relevant portion of the inflationary potential (left-hand plot) is from the first hilltop at $\phi=\phi_1$ to the origin. The corresponding domain-wall potential (right-hand plot) is simply minus that of the cosmology. The factor $\ell_{IR}^{-2}$ is given in \ref{['IRquantities1']}.