On the gauge dependence of gravitational waves generated at second order from scalar perturbations

TL;DR

This paper examines how the calculated energy density of second-order gravitational waves from scalar perturbations depends on the choice of gauge in different cosmological models, revealing significant discrepancies.

Contribution

It provides an analytical comparison of gauge dependence in gravitational wave energy density calculations for different equations of state, clarifying previous numerical results.

Findings

Huge differences between Newtonian and comoving gauge results for $w \,\ge 0$

Newtonian and uniform curvature gauges agree for $w>0$

For $w=0$, the uniform curvature gauge differs from the comoving gauge by a factor and from the Newtonian gauge significantly.

Abstract

We revisit and clarify the gauge dependence of gravitational waves generated at second order from scalar perturbations. In a universe dominated by a perfect fluid with a constant equation-of-state parameter $w$, we compute the energy density of such induced gravitational waves in the Newtonian, comoving, and uniform curvature gauges. Huge differences are found between the Newtonian and comoving gauge results for any $w \,(\ge 0)$. This is always caused by the perturbation of the shift vector. Interestingly, the Newtonian and uniform curvature gauge calculations give the same energy density for $w>0$. In the case of $w=0$, the uniform curvature gauge result differs only by a factor from that of the comoving gauge, but deviates significantly from that of the Newtonian gauge. Our calculation is done analytically for $w=0$ and $w=1/3$, and our result is consistent with the previous numerical one.