The Weyl tensor correlator in cosmological spacetimes

TL;DR

This paper derives a general expression for the Weyl tensor two-point function in cosmological spacetimes, demonstrating its divergence-free nature and connection to the tensor power spectrum, with applications to inflation.

Contribution

It provides a gauge-invariant, divergence-free formula for the Weyl tensor correlator in FLRW spacetimes, including slow-roll inflation, and links it to the tensor power spectrum.

Findings

Weyl tensor correlator is free of infrared divergences.

The formula reduces to known results in de Sitter space.

Connection established between Weyl correlator and tensor power spectrum.

Abstract

We give a general expression for the Weyl tensor two-point function in a general Friedmann-Lema\^itre-Robertson-Walker spacetime. We work in reduced phase space for the perturbations, i.e., quantize only the dynamical degrees of freedom without adding any gauge-fixing term. The general formula is illustrated by a calculation in slow-roll single-field inflation to first order in the slow-roll parameters $\epsilon$ and $\delta$, and the result is shown to have the correct de Sitter limit as $\epsilon, \delta \to 0$. Furthermore, it is seen that the Weyl tensor correlation function does not suffer from infrared divergences, unlike the two-point functions of the metric and scalar field perturbations. Lastly, we show how to recover the usual tensor power spectrum from the Weyl tensor correlation function.