Approximate gauge independence of the induced gravitational wave spectrum

TL;DR

This paper demonstrates that the spectrum of gravitational waves induced by scalar fluctuations is gauge-invariant under reasonable transformations, clarifying previous doubts about its gauge dependence in cosmological settings.

Contribution

The authors provide a simple formalism for tensor modes at second order and identify gauges where the induced GW spectrum is physically meaningful and gauge-invariant.

Findings

Induced GW spectrum is gauge-invariant under reasonable transformations.

The Newton gauge is suitable for calculations and interpretation.

The spectrum can be gauged away in certain gauges during active source periods.

Abstract

Gravitational waves (GWs) induced by scalar curvature fluctuations are an important source of the cosmological GW background and a crucial counterpart of the primordial black hole scenario. However, doubts have been cast on the theoretically predicted induced GW spectrum due to its seeming gauge dependence. In this paper, we shed light on the gauge dependence issue of the induced GW spectrum in general cosmological backgrounds. First, inspired by the Hamiltonian formalism we provide very simple formulas for the tensor modes at second order in cosmological perturbation theory. We also emphasize the difference between observable and gauge invariant variables. Second, we argue that the Newton (or shear-free) gauge is suitable for both the calculation of induced GWs and the physical interpretation. We then show that, most notably, the induced GW spectrum is invariant under a set of reasonable gauge transformations, i.e. physically well behaved on small scales, once the source term has become inactive. This includes the commonly used flat, constant Hubble and synchronous gauges but excludes the comoving slicing gauge. We also show that a particular solution of the GW equation in a dust dominated universe while the source term is active can be gauged away by a small change of gauge.