= Solution
An <adiabatic mode> corresponds locally to a common displacement along the background history. Its density changes obey $\delta\rho_i/\dot{\bar\rho}_i=\delta t$ for every component. The <cosmological perfect-fluid continuity equation> then implies that $\Delta_i/(1+w_i)$ is common to all fluids. Thus, in the common <comoving gauge in cosmology> relevant to the growing superhorizon mode,
$$
\boxed{\Delta_r=\frac43\Delta_c.}
$$
Equivalently, the relative entropy perturbation $\Delta_c-(3/4)\Delta_r$ vanishes. A common time-slicing shift preserves this relation because it changes both normalized density perturbations by the same amount.
The flat background <Friedmann equation> gives $4\pi Ga^2\bar\rho=(3/2)\mathcal H^2$. During <radiation domination>, $\mathcal H=1/\tau$ and the matter background is negligible. Substituting the adiabatic relation into the density equation therefore gives
$$
\Delta_c''+\frac1\tau\Delta_c'=\frac4{\tau^2}\Delta_c.
$$
With $\Delta_c\propto\tau^p$, the indicial equation is $p^2=4$, so the modes are $\tau^2$ and $\tau^{-2}$.
During <matter domination>, $\mathcal H=2/\tau$ and the radiation background is negligible. The same calculation gives
$$
\Delta_c''+\frac2\tau\Delta_c'=\frac6{\tau^2}\Delta_c,
$$
whose indicial equation is $p^2+p-6=(p-2)(p+3)=0$. Its modes are $\tau^2$ and $\tau^{-3}$. Thus the <superhorizon growing density mode of adiabatic matter and radiation> satisfies \b[$\Delta_c\propto\tau^2$ in both eras]. The common conformal-time exponent does not imply the same scale-factor dependence: it is $\Delta_c\propto a^2$ in the radiation era and $\Delta_c\propto a$ in the matter era.
Back to article page
