Primordial fluctuations in extended Liouville theory

Abstract

Liouville gravity can be used to precisely model features of 3+1 dimensional cosmology in a simplified 1+1d setting. We study primordial fluctuations in a generally covariant extension of Liouville theory, in the context of single field inflation. The scale invariant spectrum of scalar curvature perturbations is exhibited, and their three-point correlation function is computed in the slow roll approximation. We recover Maldacena's consistency relation for the three-point function, which in this context depends on a global shift symmetry of extended Liouville theory.

Inflation in Flatland

“…This metric is clearly of the form (2.8), and in fact solves the Einstein's equations everywhere [65]. 7 Furthermore, we can induce this metric via an asymptotic symmetry transformation by matching 8…”

“…6 For now the presence of the lapse and shift variables in the ADM parameterization is not important, but will play an important role in Section 3. 7 Although we are working in the context of Einstein gravity throughout most of our discussion, the kinematic considerations of this Section actually rely only upon the asymptotic structure of the metric. Even in non-Einsteinian gravity theories, so long as they are fundamentally diffeomorphism invariant and possess asymptotically (A)dS solutions, the asymptotic symmetries will map solutions to solutions and all of our arguments should go through.…”

“…2 One of our goals is to elucidate the nature of these symmetries and investigate how they act on fields during inflation. 1 Some work has been done on lower-dimensional inflation in the past: [6,7] considered the dynamics of eternal inflation and the properties of perturbations in (1 + 1)-dimensional inflation. The question of starting inflation and some aspects of perturbations in 2+1 dimensions were explored in [8,9].…”

“…This is exemplified by the effective field theory of inflation [11,12], in which inflation is seen as a spontaneous breaking of time diffeomorphism invariance. It is also useful to think of gauge-fixed inflation as a process of global symmetry breaking, where the curvature perturbation, ζ, is the corresponding Nambu-Goldstone 1 Some work has been done on lower-dimensional inflation in the past: [6,7] considered the dynamics of eternal inflation and the properties of perturbations in (1 + 1)-dimensional inflation. The question of starting inflation and some aspects of perturbations in 2 + 1 dimensions were explored in [8,9].…”