Growth of the gravitational wave spectrum from sound waves in a universe with a generic expansion rate

We derive a compact analytical expression for the growth factor, 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 a^-3 (1+w), we obtain =21-{y^3 (w-1) /2}3 (1-w), where y=a (t) /a (tₒ) is the ratio of the scale factor at a later time t to that at tₒ 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 a universal factor that captures gravitational wave growth in various scenarios---not limited to phase transitions.