Gauge transformation of scalar induced tensor perturbation during matter domination

Abstract

The scalar-induced secondary gravitational wave as the stochastic gravitational background is a useful tool to study the physics in the early universe. We study the scalar-induced tensor perturbations at second-order during matter domination in seven different gauges. We obtain the results in six other gauges from that in the Newtonian gauge using the gauge transformation law of the scalar-induced tensor perturbation. We find that the kernel functions ${I}_{\ensuremath{\chi}}$ in the synchronous and comoving orthogonal gauges are the same if the residual gauge modes in these two gauges are eliminated. By identifying the oscillating terms $\mathrm{sin}x$ and $\mathrm{cos}x$ in the scalar-induced tensor perturbations as the scalar-induced secondary gravitational waves, we find that its energy density is actually gauge independent. The energy density ${\ensuremath{\rho}}_{\mathrm{GW}}\ensuremath{\propto}{a}^{\ensuremath{-}4}$, or ${\mathrm{\ensuremath{\Omega}}}_{\mathrm{GW}}\ensuremath{\propto}{a}^{\ensuremath{-}1}$ in the matter-dominated era, and the scalar-induced secondary gravitational waves behave as radiation.