Infrared consistency of scalar-tensor induced gravitational waves

Abstract

Abstract The interpretation of gravitational-wave spectra generated beyond linear order depends on how infrared metric perturbations are separated from genuine radiative production. We study this question in the scalar--tensor channel of cosmological induced gravitational waves, where a long scalar perturbation with momentum $q$ couples to a much shorter tensor mode with momentum $k$. In the standard radiation-era second-order kernel, the helicity-preserving soft sector contains an apparent enhancement as $q/k$ tends to zero. We show that the leading part of this enhancement is not a local source of gravitational-wave energy. It is the fixed-order expansion of the eikonal, or Shapiro, propagation phase accumulated by the short tensor wave in the long scalar metric. We extract the leading eikonal sector from the complete third-order scalar--scalar--tensor source decomposition and identify the nested $h^{(2)}\Phi$ propagation ladder as the unique term with two uncompensated long-time integrals. Its coefficient supplies the $P_{13}$ companion that cancels the phase-square contribution contained in $P_{22}$. We then formulate the residual response in a relational light-time observable, where constant and gradient long-mode effects cancel among the metric, endpoints, tetrads and null path, leaving the first observable scalar response at tidal order. The result gives an infrared-safe interpretation of the scalar--tensor induced spectrum and clarifies which part of the long-mode response can be used as a physical gravitational-wave observable.