Linear matter perturbation growth equation ($\ddot{\delta }+2H\dot{\delta }-4\pi G{\bar{\rho }}_m\delta =0$)

For pressureless sub-Hubble perturbations, combine the linearized cosmological continuity equation, cosmological Euler equation and gravitational Poisson equation with matter as its source. The expansion contributes a $2H$ friction term and matter self-gravity contributes the negative coefficient of $\delta$. The background expansion can include radiation or dark energy; treating their perturbations as additional sources requires a more complete system.