 Solution

During matter domination, the radiation source can be neglected irrespective of whether the mode is inside or outside the horizon in the supplied linear cosmological density perturbation equation. The Friedmann equation gives $\mathcal{H}=2/\tau$ and $4\pi G{a}^2{\rho }_C=6/{\tau }^2$, just as in part (a). ThereforeSubstituting ${\tau }^q$ gives $(q-2)(q+3)=0$, so for every Fourier modeThus the matter-era growing and decaying density modes have growing behaviour ${\delta }_{C,\mathbf{k}}\propto {\tau }^2\propto a$ for all $k$. There is no pressure-gradient term introducing a $k$-dependent growth rate for cold dark matter. The coefficients retain their different earlier histories; common growth does not erase the spectral shape. A specially chosen pure decaying solution is of course possible, so the growing-mode statement does not assert that every initial condition grows.