Finite lower limit in the integral matter growth solution

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.