Balance flux laws beyond general relativity

Balance flux laws of asymptotic symmetries in general relativity provide fully nonperturbative constraint equations on gravitational strain. They have proven useful for constructing numerical gravitational waveforms and for characterizing gravitational memory. As the precision of current and future detectors continues to improve, such constraints become increasingly important for high-precision tests of gravity, including searches for deviations from general relativity. This motivates a systematic understanding of analogous balance laws in theories beyond general relativity. In this work, we investigate the existence and structure of flux laws at null infinity in diffeomorphism-invariant extensions of general relativity. Our analysis is based on the covariant phase space formalism and the definition of conserved quantities, as presented by Wald and Zoupas [Phys. Rev. D 61, 084027 (2000)]. For a particularly relevant class of Horndeski theories, we derive a general expression for the flux and formulate the corresponding balance equation via the associated nonconserved charges. We cross-check our general results by comparing with previous studies of Brans-Dicke gravity. Furthermore, we demonstrate that the employed methods extend straightforwardly to scalar-Gauss-Bonnet gravity and provide a conjecture of the flux balance laws for full massless Horndeski theory. The null part of the resulting flux laws associated with null memory is compared with and validated against the alternative derivation based on the Isaacson approach to gravitational radiation. Beyond the specific results obtained, this work is intended to serve as a practical guide for computing balance laws in generic diffeomorphism-invariant theories of gravity and paves the way for an in-depth comparison between the Isaacson approach and the covariant phase space formalism.