Radiation-era density modes without a pressure-gradient term ($\Delta =C_{+}t+C_{-}{t}^{-1}$)

For $a\propto t^{1/2}$ and $\rho =3/(32\pi G{t}^2)$, 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.