= Shifted density perturbation in Newtonian gauge
{title2=$\Delta=\delta-3(1+w)\Psi$}
For one constant-$w$ perfect fluid without anisotropic stress, <Newtonian gauge> continuity gives $\Delta'=-(1+w)\theta$. Combining it with the Euler equation and the Einstein constraint yields a closed equation for the shifted density perturbation. For matter domination, the large-scale limit is $\Delta''+4\Delta'/\eta=0$, with leading modes constant and $\eta^{-3}$. This shifted variable should not be conflated with the comoving-gauge density contrast, which includes a velocity term instead.
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