Infrared consistency of scalar–tensor induced gravitational waves

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)Φ propagation ladder as the unique term with two uncompensated long-time integrals. Its coefficient supplies the P13 companion that cancels the phase-square contribution contained in P22. 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.