Gauge dependence of gravitational waves generated at second order from scalar perturbations

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 (≥ 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. *

Gauge independence of induced gravitational waves

“…Here, the "late time" limit means the subhorizon limit: time much after the horizon entry of the relevant scalar source modes as well as the tensor mode. In addition, the calculation in the uniform curvature gauge (flat gauge) coincides with the Newtonian-gauge result too [40]. The main reason for the agreement in these gauges is because the tensor perturbations are finally dominated by the freely propagating tensor perturbations (GWs) in all of the three gauges.…”

“…We also mention the interpretation of the gauge dependence reported in Ref. [40] and an important role played by the diffusion damping effect at the end of this paper.…”

“…(65) in Ref. [40]. This indicates that the gauge dependence appears only in the tensor perturbations coupling with scalar perturbation, not in the gravitational wave.…”

“…The gauge independence of the induced GWs can also be expected from the coincidence of the GWs calculated in the Newtonian gauge and the uniform curvature gauge (also called the flat gauge), shown in Ref. [40] (see the case of w > 0 in the reference). In the present paper, we revisit the calculation in Ref.…”

“…On the other hand, the tensor perturbations could be different in the comoving gauge even in the subhorizon limit during a RD era, according to Ref. [40]. We can find that the difference between the Newtonian gauge and the comoving gauge comes from the terms whose oscillations are of the form cos (u ± v)kη/ √ 3 or sin (u ± v)kη/ √ 3 , which are not the oscillations of gravitational waves, cos(kη) or sin(kη); see Eq.…”