Matter growth equation as a function of scale factor

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to $a$ gives

The growing solution of the matter perturbation equation in a spatially flat matter-plus-cosmological-constant universe isnormalized so that $D_{+}(a)\sim a$ during early matter domination.

When $H\to H_{\Lambda }>0$, the growth integral has an integrable $a^{-3}$ tail. Its product with $H$ tends to a constant, with leading correction proportional to $a^{-2}$ in a matter-plus-Lambda background. Both displayed basis functions can tend to constants; a linear combination cancels that constant and isolates the truly decaying late-time $a^{-2}$ behavior.

Changing the lower limit in $H(a){\int }_{a_i}^{a}d\tilde{a}/(\tilde{a}H{)}^3$ adds a constant multiple of $H(a)$, the independent matter-era decaying solution. Thus a finite lower limit defines a useful independent solution but can include a decaying admixture. A pure growing normalization is selected by an appropriate linear combination.

If a perturbation has amplitude $A\ll 1$ at decoupling, it needs a growth factor of at least $A^{-1}$ before cosmological-constant domination freezes its growth. The order-of-magnitude requirement isup to the collapse threshold and the gradual transition encoded by the exact growth factor.