= Gravitational potential evolution of a barotropic fluid
{title2=$\Phi_k''+3(1+w)\mathcal H\Phi_k'+wk^2\Phi_k=0$}
For a flat background dominated by one constant-$w$ <barotropic equation of state>, assume $\delta P=w\delta\rho$ and no <scalar anisotropic stress>. The <Einstein field equations> then reduce the common Newtonian-gauge potential to the stated <Fourier transform> equation, with $\mathcal H=2/[(1+3w)\tau]$. For radiation the regular solution is $3\Phi_0(\sin x-x\cos x)/x^3$, $x=k\tau/\sqrt3$, and oscillates with a decaying envelope after sound-horizon entry. For pressureless matter the two solutions are a constant and $\tau^{-5}$. A non-adiabatic pressure perturbation would supply an additional source, so constant background $w$ alone does not justify the homogeneous equation.
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