Finite lower limit in the integral matter growth solution
= Finite lower limit in the integral matter growth solution
Changing the lower limit in $H(a)\int_{a_i}^{a}d\widetilde a/(\widetilde aH)^3$ 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.