Gauge transformation of scalar induced gravitational waves

TL;DR

This paper derives a gauge-invariant formula for scalar induced gravitational waves (SIGWs), compares results across six gauges, and clarifies gauge effects on SIGW energy density evolution.

Contribution

It provides a general gauge transformation formula for SIGWs and analyzes their gauge dependence in various cosmological gauges.

Findings

Newtonian, uniform curvature, synchronous, and uniform expansion gauges give consistent SIGW energy densities.

Pure gauge modes are identified and removed in the synchronous gauge.

SIGW energy density evolves differently in various gauges, e.g., increasing as η^2 or η^6.

Abstract

The gauge dependence of the scalar induced gravitational waves (SIGWs) generated at the second order imposes a challenge to the discussion of the secondary gravitational waves generated by scalar perturbations. We provide a general formula that is valid in any gauge for the calculation of SIGWs and the relationship for SIGWs calculated in various gauges under the coordinate transformation. The formula relating SIGWs in the Newtonian gauge to other gauges is used to calculate SIGWs in six different gauges. We find that the Newtonian gauge, the uniform curvature gauge, the synchronous gauge and the uniform expansion gauge yield the same result for the energy density of SIGWs. We also identify and eliminate the pure gauge modes that exist in the synchronous gauge. In the total matter gauge and the comoving orthogonal gauge, the energy density of SIGWs increases as $\eta^2$. While in the uniform density gauge, the energy density of SIGWs increases as $\eta^6$.