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Gauge transformation of scalar induced tensor perturbation during matter domination

We study the scalar induced tensor perturbations at second order during matter domination in seven different gauges. Considering the obtained solution from the Newtonian gauge, we use the gauge transformation law of the scalar induced tensor perturbation to derive the solution in six other gauges. After identifying and eliminating the residual gauge modes in the synchronous and comoving orthogonal gauges, we obtain the same analytical results of the kernel function $I_χ$ for these two gauges as those obtained from the gauge transformation. For the scalar induced gravitational waves oscillating as $\sin x$ and $\cos x$, we find that $ρ_{\text{GW}}\propto a^{-4}$, and $Ω_{\text{GW}}\propto 1/a$ in the matter dominated era, so the oscillating gravitational waves behave as radiation.