= Linear matter density equation in Newtonian gauge
{title2=$\delta_m''+\mathcal H\delta_m'$}
For <pressureless matter> in <Newtonian gauge in cosmology> with a common metric potential $\Phi$, let $\theta=\nabla\cdot\mathbf v$. The linear equations $\delta_m'=-\theta+3\Phi'$ and $\theta'=-\mathcal H\theta-\nabla^2\Phi$, where primes denote <conformal time>, yield
$$
\delta_m''+\mathcal H\delta_m'=\nabla^2\Phi+3(\Phi''+\mathcal H\Phi').
$$
This follows by differentiating the first equation and eliminating $\theta$. On deeply subhorizon scales with a slowly varying matter potential, the metric time derivatives are small compared with its spatial <Laplacian>.
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