Radiation-era density modes without a pressure-gradient term
= Radiation-era density modes without a pressure-gradient term
{title2=$\Delta=C_+t+C_-t^{-1}$}
For $a\propto t^{1/2}$ and $\rho=3/(32\pi Gt^2)$, the specified equation $\ddot\Delta+2H\dot\Delta=(32\pi G\rho/3)\Delta$ reduces to $\ddot\Delta+t^{-1}\dot\Delta=t^{-2}\Delta$. Its two modes are $t$ and $t^{-1}$. This is a property of that equation; pressure-gradient and gauge choices must be specified before extending it to general radiation perturbations.