Logarithmic soft theorems and soft spectra

Using universal predictions provided by classical soft theorems, we revisit the energy emission spectrum for gravitational scatterings of compact objects in the low-frequency expansion. We calculate this observable beyond the zero-frequency limit, retaining an exact dependence on the kinematics of the massive objects. This allows us to study independently the ultrarelativistic or massless limit, where we find agreement with the literature, and the small-deflection or post-Minkowskian (PM) limit, where we provide explicit results up to \( \mathcal{O} \)(G⁵). These confirm that the high-velocity limit of a given PM order is smoothly connected to the corresponding massless result whenever the latter is analytic in the Newton constant G. We also provide explicit expressions for the waveforms to order ω⁻¹, log ω, ω(log ω)² in the soft limit, ω → 0, expanded up to sub-subleading PM order, as well as a conjecture for the logarithmic soft terms of the type ωⁿ⁻¹(log ω)ⁿ with n ≥ 3.