Superhorizon growing density mode of adiabatic matter and radiation (${\Delta }_c\propto {\tau }^2$)

An adiabatic perturbation obeys ${\Delta }_r=(4/3){\Delta }_c$. In a radiation background this makes the cold-matter density equation ${\Delta }_c^{\prime \prime }+{\Delta }_c^{\prime }/\tau =4{\Delta }_c/{\tau }^2$, whose growing solution is ${\tau }^2$. In a matter background it instead becomes ${\Delta }_c^{\prime \prime }+2{\Delta }_c^{\prime }/\tau =6{\Delta }_c/{\tau }^2$, with the same growing conformal-time exponent and a different scale-factor dependence.