Matter-era growing and decaying density modes ($\delta =A{\tau }^2+B{\tau }^{-3}$)

During matter domination in a flat FLRW metric, the cosmological continuity equation gives $\rho \propto a^{-3}$, and the Friedmann equation gives $a\propto {\tau }^2$ and $8\pi G\rho a^2=12/{\tau }^2$. The CDM density equation in a matter-radiation universe, with radiation neglected, is therefore ${\delta }^{\prime \prime }+2{\delta }^{\prime }/\tau -6\delta /{\tau }^2=0$. Its Euler-Cauchy equation powers satisfy $(p-2)(p+3)=0$, proving the displayed general solution. The growing density contrast is proportional to the scale factor, while the decaying mode is proportional to $a^{-3/2}$. This all-scale form uses the CDM-comoving synchronous gauge; other density slices require their corresponding gauge transformation.

In cosmic time during matter domination, $a(t)\propto t^{2/3}$ and the pressureless density contrast obeysIts growing and decaying solutions are $t^{2/3}$ and $t^{-1}$.

After neglecting the decaying mode, the linear growth between conformal times ${\tau }_1$ and ${\tau }_2$ is