 Solution

With negligible cold-dark-matter velocity, the synchronous perfect-fluid density equation gives ${\delta }_C^{\prime }=-h^{\prime }/2$. Differentiate and use the trace Einstein equation, taking the cold matter sound speed to vanish and its density to dominate:In a spatially flat matter-dominated universe, $a\propto {\tau }^2$, $\mathcal{H}=2/\tau$, and the conformal Friedmann equation gives $4\pi G{a}^2{\rho }_C=3{\mathcal{H}}^2/2=6/{\tau }^2$. HencePut ${\delta }_C\propto {\tau }^s$. The indicial equation is $s(s-1)+2s-6=(s-2)(s+3)=0$. These two independent modes span the solution space, soEquivalently the growing mode is proportional to $a$ and the decaying mode to $a^{-3/2}$. The coefficients are set by initial data. Since synchronous gauge has residual coordinate modes, interpreting the density contrast physically requires fixing its residual freedom consistently, for example by the cold-matter rest frame and metric tensor integration convention; the mode equation alone is not a gauge-invariant observable.