= Constant-equation-of-state comoving density equation
{title2=$\Delta''+\frac{2(1-3w)}{(1+3w)\eta}\Delta'-\frac{6(1-w)}{(1+3w)\eta^2}\Delta-w\nabla^2\Delta=0$}
For a flat single-fluid background with constant <barotropic equation of state>, $w\ne-1/3$, adiabatic perturbations and zero anisotropic stress, $a\propto\eta^{2/(1+3w)}$ and $a^2\bar\rho\propto\eta^{-2}$. The scalar potential obeys $\phi''+[6(1+w)/(1+3w)]\phi'/\eta-w\nabla^2\phi=0$. Substituting the Poisson constraint $\Delta\propto\eta^2\nabla^2\phi$ gives the displayed equation. The homogeneous mode and the degenerate vacuum fluid $w=-1$ require separate interpretation.
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