Gauge transformation of scalar induced tensor perturbation during matter domination

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 secon-dorder 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-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 sin x and 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 rho(Gw) proportional to a(-4), or Omega(Gw) proportional to a(-1) in the matter-dominated era, and the scalar-induced secondary gravitational waves behave as radiation.

Automated summary

The paragraph discusses the results of studying scalar-induced tensor perturbations during matter domination in various gauges, revealing that the kernel functions in synchronous and comoving gauges are the same and identifying oscillating terms in the perturbations as the secondary gravitational waves induced by scalars. Additionally, it is found that the energy density of these waves is gauge independent and behaves like radiation in the matter-dominated era.