The transverse-traceless gauge and the gauge problem of second order gravitational waves

TL;DR

The paper tackles the gauge problem for second-order gravitational waves in cosmology by extending the transverse-traceless (TT) gauge to cosmological backgrounds and introducing a vacuum condition $\bar{\rho}\Delta=\Pi=\delta q_i=0$ that guarantees the existence of the TT gauge and makes the first-order TT tensor $h_{ij}^{(1)\mathrm{TT}}$ gauge invariant at second order. It shows that under this vacuum condition, several common gauges (Poisson, uniform curvature, synchronous, total matter) become equivalent to the TT gauge, and that in the sub-horizon limit $k\gg\mathcal{H}$ these gauges approximate TT—providing a unified explanation for why different gauges yield consistent results for scalar-induced GWs in radiation era and clarifying the role of the vacuum condition in higher-order gauge choices. The analysis yields explicit rates at which each gauge approaches TT in the asymptotic vacuum regime and explains observed equivalences in the literature, while indicating a natural path to extend the TT gauge to arbitrary higher orders. Overall, the work offers a coherent framework to address second-order GW gauge ambiguities and to interpret gauge-dependent results in cosmology with a clear connection to physically propagating TT modes.

Abstract

The gauge problem arises in the second order gravitational waves due to the mode mixing. Here, we introduce the transverse-traceless (TT) gauge to cosmological backgrounds, and find that if we choose the TT gauge at first order, the second order tensor mode would be gauge invariant. Analogous to the Ricci flat spacetime, the vacuum condition is the key to guarantee the existence of the TT gauge on cosmological backgrounds. When we have the vacuum condition, the Poisson gauge, the uniform curvature gauge, the synchronous gauge and the total matter gauge are all equivalent to the TT gauge. Once the vacuum condition is approximately satisfied, the Poisson gauge would reduce to the TT gauge at the same order of approximation. With the sub-horizon limit, the vacuum condition could be obtained approximately, and the Poisson gauge, the uniform curvature gauge and the synchronous gauge are all approximated TT gauge. Our findings explain several existing results in the literature and indicate that the proposed TT gauge is useful to discuss higher order gravitational waves.