Superhorizon enhancement of the hydrodynamic GW tail in Standard Model embedded warm inflation

Hydrodynamic fluctuations in the primordial plasma generate a gravitational-wave (GW) background that grows like $f_0^3$ in terms of the current frequency. However, within the ($10_{}^{-4}\mathrm{.}\mathrm{.}.10_{}^4)$Hz band of upcoming interferometers, the modes likely re-entered the Hubble horizon when the hydrodynamic noise was no longer at its maximal value. To address this situation, we define a model-independent transfer function for GWs generated by hydrodynamic (or other white-noise) fluctuations at any time. Employing appropriate variables, we show how this transfer function, together with the simpler vacuum one, can be determined numerically for a general expansion history. The results are illustrated for Standard Model embedded warm inflation scenarios, including both the vacuum and hydrodynamic parts. Following previous estimates, we find that a pre-exit-generated hydrodynamic tail becomes almost flat below $(10_{}^5\mathrm{.}\mathrm{.}.10_{}^7)$Hz. Its small-$f_0^{}$ intercept remains finite, and can then moderately influence observational constraints through the tensor-to-scalar ratio ($r$).