 Solution

During matter domination, the spatially flat background has $a\propto {\tau }^2$ and $\mathcal{H}=2/\tau$. Let ${\theta }_m=\nabla \cdot {\mathbf{v}}_m$ denote the scalar velocity divergence. Linear stress-energy conservation in synchronous gauge givesThe first equation follows from the perturbed volume element $a^3(1+h/2)$; the second states that peculiar dust velocity redshifts as $a^{-1}$. Differentiating the first and substituting the trace of the Einstein field equations, with $w_m=0$ and ${\Omega }_m=1$, yieldsThereforeTrying ${\delta }_m\propto {\tau }^p$ gives $p(p-1)+2p-6=(p-2)(p+3)=0$. The complete solution of this density equation isThese are the matter-era growing and decaying density modes, proportional to $a$ and $a^{-3/2}$. In a dust-comoving synchronous gauge, ${\theta }_m=0$ and $h^{\prime }=-2{\delta }_m^{\prime }$. That rest-frame choice distinguishes physical mode coefficients from changes of density induced by residual synchronous-gauge freedom.