= Jeans modes of a four-thirds polytropic cosmological fluid
{c}
{title2=$\delta\propto t^{(-1\pm\sqrt{25-24q})/6},\quad q=(k/k_J)^2$}
For a nonrelativistic <polytropic equation of state> $P=K\rho^{4/3}$ during <matter domination>, $c_s^2\propto a^{-1}$ and the comoving <Jeans wavenumber> is constant. The leading density equation becomes $\ddot\delta+4\dot\delta/(3t)+[2(q-1)/(3t^2)]\delta=0$, giving the displayed powers. A growing mode exists for $q<1$; at $q=1$ one mode is constant. At $q=25/24$ the repeated root has the second solution $t^{-1/6}\ln t$. For $q>25/24$ both real solutions oscillate in $\ln t$ with envelope $t^{-1/6}$. Thus the instability threshold and the onset of oscillatory solutions are distinct in an expanding background.
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