= Bessel density modes during exponential expansion
{c}
{title2=$\nu=2/3$}
For constant positive <Hubble parameter> $\mu$ and a four-thirds <polytropic equation of state>, a <linear cosmological density perturbation> obeys $\ddot\delta+2\mu\dot\delta+\eta^2e^{-3\mu t}\delta=0$. 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.
Back to article page
