Shifted density perturbation in Newtonian gauge ($\Delta =\delta -3(1+w)\Psi$)

For one constant-$w$ perfect fluid without anisotropic stress, Newtonian gauge continuity gives ${\Delta }^{\prime }=-(1+w)\theta$. 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 ${\Delta }^{\prime \prime }+4{\Delta }^{\prime }/\eta =0$, 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.