= Nonrelativistic density equation in synchronous gauge
{title2=$\delta''+\mathcal H\delta'+(c_s^2k^2-4\pi Ga^2\bar\rho)\delta\simeq0$}
For a matter-dominated fluid with constant $c_s^2=w\ll1$, combine <stress-energy conservation> in synchronous gauge with the scalar trace Einstein equation. Discard pressure corrections to the background gravitational and velocity-damping terms while retaining the potentially large pressure-gradient term $wk^2\delta$. This produces the displayed leading nonrelativistic equation. It is not an exact equation at finite $w$. At negligible pressure the growing and decaying matter-era solutions are $\tau^2$ and $\tau^{-3}$.
// Target: cosmology.bigb
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