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 in the radiation-dominated epoch. The correlator of the velocity field is evaluated in accordance with the sound shell model. In our derivation, we include the effects of the expansion of the Universe, which are relevant in particular for sourcing processes whose time duration is comparable with the Hubble time. Our results show 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 within the sound shell model. The resulting bump around the peak of the GW spectrum may represent a distinctive feature of GWs produced from acoustic motion. Nothing similar has been observed for vortical (magneto)hydrodynamic turbulence. Nevertheless, we find that the $k^9$ scaling is less extended than expected in the literature, and it does not necessarily appear. The dependence on the duration of the source, $\delta \tau_{\rm fin}$, is quadratic at small frequencies $k$, and proportional to $\ln^2 (1 + \delta \tau_{\rm fin} H_*)$ for an expanding Universe. At frequencies around the peak, the growth is suppressed by a factor $\Upsilon = 1 - 1/(\tau_{\rm fin} {H}_*)$ that becomes linear when the GW source is short. We discuss in which cases the dependence on the source duration is linear or quadratic for stationary processes. This 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, as a function of the parameters of the phase transition