Matter growth equation as a function of scale factor
= Matter growth equation as a function of scale factor
In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives
$$
\frac{d^2\delta_m}{da^2}
+\left(\frac{d\log H}{da}+\frac3a\right)\frac{d\delta_m}{da}
-\frac{3\Omega_{m,0}H_0^2}{2a^5H^2}\delta_m=0.
$$