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 $I_{\chi}$ 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 $\rho_{\text{GW}}\propto a^{-4}$, and
$\Omega_{\text{GW}}\propto 1/a$ in the matter dominated era, so the oscillating
gravitational waves behave as radiation.
