One-loop gravitational wave spectrum in de Sitter spacetime

TL;DR

This paper calculates the one-loop corrected two-point function for tensor perturbations in de Sitter space, revealing invariance properties and divergences in the power spectrum at equal times.

Contribution

It provides an exact, invariant two-point function for tensor perturbations with one-loop corrections in de Sitter spacetime for arbitrary time separations.

Findings

Two-point function is invariant under de Sitter subgroup transformations.

The equal-time power spectrum diverges due to loop corrections.

The two-point function is a well-defined distribution, but the power spectrum is ill-defined at equal times.

Abstract

The two-point function for tensor metric perturbations around de Sitter spacetime including one-loop corrections from massless conformally coupled scalar fields is calculated exactly. We work in the Poincar\'e patch (with spatially flat sections) and employ dimensional regularization for the renormalization process. Unlike previous studies we obtain the result for arbitrary time separations rather than just equal times. Moreover, in contrast to existing results for tensor perturbations, ours is manifestly invariant with respect to the subgroup of de Sitter isometries corresponding to a simultaneous time translation and rescaling of the spatial coordinates. Having selected the right initial state for the interacting theory via an appropriate i\epsilon prescription is crucial for that. Finally, we show that although the two-point function is a well-defined spacetime distribution, the equal-time limit of its spatial Fourier transform is divergent. Therefore, contrary to the well-defined distribution for arbitrary time separations, the power spectrum is strictly speaking ill-defined when loop corrections are included.