Growth of Gravitational Wave Spectrum from Sound Waves in a Universe with Generic Expansion Rate

We derived here the factor $\Upsilon$, which quantifies how the gravitational wave spectrum generated by sound waves in the radiation sector grows over time, in a universe with a generic expanding rate set by another dominant energy content. When the dominant energy density satisfies $\rho \propto a^{-3(1+w)}$, we found that $\Upsilon$ has a compact analytical expression: $\Upsilon =\frac{2[1-y^{3(w-1)/2}]}{3(1-w)}$, where $y = a(t)/a(t_s)$ which is the ratio of the scale factor at a later time $t$ to that at $t_s$ when gravitational wave production from sound waves starts. This generic result reduces to that derived previously for radiation-dominated and matter-dominated cases, thus generalizing previous formulas to more general cosmological contexts and providing more accurate results. The derivation relies solely on a stationary source, implying that this generic result of $\Upsilon$ serves as an universal factor in describing the growth of the gravitational wave production and can appear beyond cosmological phase transitions.