Bessel density modes during exponential expansion ($\nu =2/3$)

For constant positive Hubble parameter $\mu$ and a four-thirds polytropic equation of state, a linear cosmological density perturbation obeys $\ddot{\delta }+2\mu \dot{\delta }+{\eta }^2{e}^{-3\mu t}\delta =0$. Put $\delta =e^{-\mu t}y$ and $x=2|\eta |e^{-3\mu t/2}/(3\mu )$. The resulting Bessel differential equation has order $2/3$; use ordinary Bessel functions for ${\eta }^2>0$ and modified Bessel functions for ${\eta }^2<0$. Their small-$x$ expansions give one constant limiting mode and one $e^{-2\mu t}$ mode. The constant limit need not exceed the initial amplitude without an initial-velocity condition.

If ${\eta }^2=-s^2<0$, $\delta (0)>0$ and $\dot{\delta }(0)\ge 0$, the integrating factor identity $(e^{2\mu t}\dot{\delta }{)}^{\prime }=s^2{e}^{-\mu t}\delta$ proves that $\delta$ stays positive and increases. Its finite limit satisfies ${\delta }_{\infty }=\delta (0)+\dot{\delta }(0)/(2\mu )+s^2{\int }_0^{\infty }{e}^{-3\mu t}\delta (t)dt/(2\mu )>\delta (0)$. The initial sign condition is essential: the positive decaying modified Bessel function mode has limit zero.