Cosmological matching conditions for gravitational waves at second order

TL;DR

This paper derives the conditions for matching second-order gravitational wave perturbations across cosmological transitions, showing that inflation-generated gravitational waves are suppressed in the radiation era.

Contribution

It provides a detailed analysis of second-order tensor perturbation matching conditions in different gauges and applies these to inflation-to-radiation transition scenarios.

Findings

Tensor perturbations are continuous in the Poisson gauge across transitions.

Second-order gravitational waves from slow-roll inflation are suppressed in the radiation era.

Matching conditions depend on the gauge and nature of perturbations.

Abstract

We compute the second-order matching conditions for tensor metric perturbations at an abrupt change in the equation of state. For adiabatic perturbations on large scales the matching hypersurface coincides with a uniform-density hypersurface. We show that in the uniform-density gauge both the tensor perturbation and its time-derivative are continuous in this case. For non-adiabatic perturbations, the matching hypersurface need not coincide with a uniform-density hypersurface and the tensor perturbation in the uniform-density gauge may be discontinuous. However, we show that in the Poisson gauge both the tensor perturbation and its time-derivative are continuous for adiabatic or non-adiabatic perturbations. As an application we solve the evolution equation for second-order tensor perturbations on large scales for a constant equation of state and we use the matching conditions to evolve the solutions through the transition from an inflationary era to a radiation era. We show that in the radiation era the resulting free part of the large-scale tensor perturbation (constant mode) is slow-roll suppressed in both the uniform-density and Poisson gauges. Thus, we conclude that second-order gravitational waves from slow-roll inflation are suppressed.