Inflation and deformation of conformal field theory

TL;DR

This paper develops a holographic (boundary QFT) description of inflation by identifying the bulk wave function with the boundary generating functional, $\psi_{\rm bulk} \propto Z_{\rm QFT}$, and derives explicit relations between curvature perturbation correlators and boundary operator $O$ correlators using a Weyl Ward-Takahashi identity. It analyzes a small deformation of a CFT by $O$ with dimension $\Delta$, introducing $\lambda = \Delta - d$ and a boundary beta-function, to express the primordial power, bispectrum, and trispectrum of the curvature perturbation $\zeta$ in terms of boundary correlators $\langle O\cdots O\rangle_u$, including non-local boundary vertices $W^{(n)}$. The work provides formulae linking $P(k)$, $B(k_1,k_2,k_3)$, and $T(k_1,k_2,k_3,k_4)$ to $W^{(2)}$, $W^{(3)}$, and $W^{(4)}$, respectively, and discusses the Suyama–Yamaguchi inequality within this holographic context, showing the inequality holds in generic non-singular cases and highlighting the role of boundary four-point data. The approach is manifestly independent of the bulk gravitational dynamics, enabling analysis of strongly coupled bulk phases and offering a cross-check against domain-wall/cosmology holographic methods. Potential extensions include loop corrections, tensor modes, and concrete boundary models with specified $n$-point functions of $O$.

Abstract

It has recently been suggested that a strongly coupled phase of inflation may be described holographically in terms of a weakly coupled quantum field theory (QFT). Here, we explore the possibility that the wave function of an inflationary universe may be given by the partition function of a boundary QFT. We consider the case when the field theory is a small deformation of a conformal field theory (CFT), by the addition of a relevant operator O, and calculate the primordial spectrum predicted in the corresponding holographic inflation scenario. Using the Ward-Takahashi identity associated with Weyl rescalings, we derive a simple relation between correlators of the curvature perturbation and correlators of the deformation operator O at the boundary. This is done without specifying the bulk theory of gravitation, so that the result would also apply to cases where the bulk dynamics is strongly coupled. We comment on the validity of the Suyama-Yamaguchi inequality, relating the bi-spectrum and tri-spectrum of the curvature perturbation.

Figures (2)

• Figure 1: The diagram of the bi-spectrum
• Figure 2: The diagrams of the tri-spectrum