= Solution
Here
$$
c_s^2=c^2\frac{dP}{d\rho}=\frac43\alpha c^2\bar\rho^{1/3}.
$$
Since $\bar\rho\propto t^{-2}$ and $a\propto t^{2/3}$, $c_s^2/a^2=L_0^2/t^2$ for a constant $L_0^2$. With $\dot a/a=2/(3t)$ and $4\pi G\bar\rho/c^2=2/(3t^2)$, substitution of $\delta=t^n$ gives
$$
n^2+\frac13n+L_0^2k^2-\frac23=0,
$$
so
$$
n_\pm=-\frac16\pm\sqrt{\frac{25}{36}-L_0^2k^2}.
$$
For $k\ll\widetilde k_J=5/(6L_0)$, the modes approach $t^{2/3}$ and $t^{-1}$. For $k\gg\widetilde k_J$, the exponents are complex with real part $-1/6$, giving logarithmic oscillations with decaying envelope $t^{-1/6}$. Taking $L_0\simeq50\,\mathrm{Mpc}$ would pressure-suppress structure on enormous, cosmologically important scales and is incompatible with the observed growth of smaller-scale structure.
Back to article page
