We derive a compact analytical expression for the growth factor $\mathrm{\Upsilon }$, which characterizes how the gravitational wave spectrum sourced by sound waves evolves in a universe with a generic expansion history. Assuming the dominant energy density scales as $\rho \propto a^{-3(1+w)}$, we obtain $\mathrm{\Upsilon }=\frac{2[1-y^{3(w-1)/2}]}{3(1-w)}$, where $y=a(t)/a(t_s)$ 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 general result reduces to known forms in radiation- and matter-dominated eras, thereby extending previous formulas to a broader class of cosmological backgrounds. The derivation assumes only that the source is stationary, making $\mathrm{\Upsilon }$ a universal factor that captures gravitational wave growth in various scenarios—not limited to phase transitions.

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• Gravitational waves
• Phase transitions