The Weyl tensor correlator in cosmological spacetimes

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.