Gravitational Wave Backreaction in $f(R,G)$ Gravity

Abstract

We develop a complete framework for gravitational wave propagation and backreaction in $f(R,G)$ modified gravity. Using a scalar-tensor formulation with two auxiliary fields, we derive the effective energy-momentum tensor for high-frequency gravitational waves, extending the Isaacson formalism to incorporate the coupled dynamics of the two scalar degrees of freedom arising from the Ricci scalar and Gauss-Bonnet terms. Applying our formalism to the concrete model $f(R,G) = R + αR^2 + βG$ with dimensionless coupling $\tildeβ = βH_{\text{inf}}^2$, we identify three observational signatures: (i) a stochastic background $Ω_{GW}(f)$ too faint for direct detection; (ii) a frequency-dependent phase shift $Δϕ(f) \propto \tildeβ f$ detectable for $\tildeβ \gtrsim 10^{-9}$ via matched filtering of binary inspirals; and (iii) amplitude damping $δh/h \propto \tildeβ f \ln(1+z)$ reaching the percent level for $\tildeβ \sim 10^{-8}$, constrainable by multi-messenger standard sirens. These results show that $f(R,G)$ gravity makes testable predictions for next-generation observatories, improving constraints on the Gauss-Bonnet coupling by 28 orders of magnitude over current bounds. The framework developed here provides a foundation for studying modified gravity through gravitational wave observations.