Characterization of the gravitational wave spectrum from sound waves within the sound shell model

We compute the gravitational wave (GW) spectrum sourced by the sound waves produced during a first-order phase transition during radiation domination. The correlator of the velocity field is evaluated according to the sound shell model. In our derivation, we include the effects of the expansion of the Universe, showing their importance, in particular for sourcing processes with time duration comparable to the Hubble time. From the exact solution of the GW sourcing integral, we find a causal growth at small frequencies, $\Omega_{\rm GW} \sim k^3$, possibly followed by a linear regime $\Omega_{\rm GW} \sim k$ at intermediate $k$, depending on the phase transition parameters. Around the peak, we find a steep growth that approaches the $k^9$ scaling found in the sound shell model. This growth causes a bump around the GW spectrum peak, which may represent a distinctive feature of GWs produced from acoustic motion, since nothing similar has been observed for vortical turbulence. Nevertheless, we find that the $k^9$ scaling is much less extended than expected in the literature, and it does not necessarily appear. The dependence on the duration of the source, $\tau_{\rm fin} - \tau_*$, is quadratic at small frequencies $k$, and proportional to $\ln^2 (\tau_{\rm fin} {\cal H}_*)$ for an expanding Universe. At frequencies around the peak, the growth is suppressed by a factor $\Upsilon = 1 - 1/(\tau_{\rm fin} {\cal H}_*)$, which becomes linear for short duration. We discuss the linear or quadratic dependence on the source duration for stationary processes, which affects the amplitude of the GW spectrum, both in the causality tail and at the peak, showing that the assumption of stationarity is a very relevant one, as far as the GW spectral shape is concerned. Finally, we present a general semi-analytical template of the resulting GW spectrum, which depends on the parameters of the phase transition.