Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 53 / 4 / i / 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,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 $4\pi G{a}^2\bar{\rho }=(3/2){\mathcal{H}}^2$. During radiation domination, $\mathcal{H}=1/\tau$ and the matter background is negligible. Substituting the adiabatic relation into the density equation therefore givesWith ${\Delta }_c\propto {\tau }^p$, 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 giveswhose 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 ${\Delta }_c\propto {\tau }^2$ 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.