= Bessel density modes of a polytropic expanding universe
{c}
{title2=$\nu=\gamma-4/3,\quad\lambda=5/(6\nu)$}
For a flat Newtonian expanding background $R\propto t^{2/3}$ with <polytropic equation of state> $p=K\rho^\gamma$, a comoving <density contrast> satisfies $\ddot\delta+4\dot\delta/(3t)+[\Lambda^2t^{-2\nu-2}-2/(3t^2)]\delta=0$. Here $\Lambda=kc(t_0)t_0^{\gamma-1/3}$ and $\nu=\gamma-4/3$. For $\nu>0$, substitution $z=\Lambda/(\nu t^\nu)$ and $\delta=t^{-1/6}y(z)$ gives the <Bessel equation> of order $\lambda$. A universally valid basis uses the <Bessel function of the first kind> and <Bessel function of the second kind>, $t^{-1/6}J_\lambda(z)$ and $t^{-1/6}Y_\lambda(z)$. When $\lambda$ is not an integer, $J_{-\lambda}$ can replace $Y_\lambda$. For $4/3<\gamma<5/3$, the early oscillatory envelope is $t^{(3\gamma-5)/6}$; late modes have leading powers $t^{2/3}$ and $t^{-1}$. Late growth requires a nonzero coefficient of the growing branch.
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