On the gauge dependence of scalar induced secondary gravitational waves during radiation and matter domination eras

TL;DR

This paper investigates the gauge dependence of scalar-induced secondary gravitational waves during radiation and matter domination, proposing a counter-term approach to resolve divergences in energy density spectra across different gauges.

Contribution

It introduces a method to address gauge-dependent divergences in SIGWs by adding counter-terms, improving the physical consistency of the energy density spectrum calculations.

Findings

Divergences in energy density depend on gauge choice.

Counter-terms can remove fictitious divergences.

Physical origin of gauge dependence is analyzed.

Abstract

We revisit the vital issue of gauge dependence in the scalar-induced secondary gravitational waves (SIGWs), focusing on the radiation domination (RD) and matter domination (MD) eras. The energy density spectrum is the main physical observable in such induced gravitational waves. For various gauge choices, there has been a divergence in the energy density, $\Omega_{\text{GW}}$, of SIGWs. We calculate SIGWs in different gauges to quantify this divergence to address the gauge-dependent problem. In our previous studies, we had found that the energy density diverges in the polynomial power of conformal time (e.g., $\eta^6$ in uniform density gauge). We try to fix this discrepancy by adding a counter-term that removes the fictitious terms in secondary tensor perturbations. We graphically compare the calculations in various gauges and also comment on the physical origin of the observed gauge dependence.