Isaacson energy--momentum tensor of high-frequency gravitational waves in general $f(R,G)$ gravity

Abstract

We construct the averaged energy--momentum tensor of high-frequency gravitational waves, in the sense of Isaacson, for a general function of the Ricci scalar and the Gauss--Bonnet invariant on de~Sitter, extending a recent $f(R)$ result. At linear order, every such theory behaves like $f(R)$ with a renormalised second derivative. At second order, tensor waves carry the Isaacson stress times the effective Planck mass, while scalar waves pick up an unsuppressed, traceless and non-null Gauss--Bonnet term. In the degenerate classes $F(R+\lambda G)$ and $R-2\Lambda +F(G)$, its size does not depend on $F$. For an isotropic ensemble, it lowers the energy that scalar waves contribute to the background by a factor between $1/3$ and $1$; for a single beam, the averaged energy violates the dominant energy condition. In wave packets, conservation fixes an accompanying gradient term. Both terms are gauge invariant at leading order, and the main result is confirmed in four ways: two independent implementations of the field equations, an exact special case and a direct variation of the action. Off de~Sitter, theories with a non-degenerate Hessian have no high-frequency regime. The degenerate class keeps the de~Sitter structure with ordinary slow-roll corrections. Still, stability of its scalar mode in the radiation and matter eras makes the Gauss--Bonnet correction negligible at late times.

arXiv commentv6: new title and a new analysis; supersedes v1-v5. 14 pages. Code: doi:10.5281/zenodo.23811207