Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 48 / 3 / c

Let $u={\delta }_m/H$. The supplied equation has integrating factor $(aH{)}^3$:Consequently the integral linear growth factor in a matter-Lambda universe gives the two independent solutionsThe first is the conventional matter-era decaying mode. A finite lower limit in the second fixes a decaying admixture; adding a multiple of $H$ can choose a pure growing normalization.

During matter domination, write $H=h{a}^{-3/2}$. The integral solution isRemoving the second term by the independent $H$ solution gives the matter-era linear growth factorFor positive cosmological constant at sufficiently late times, $H\to H_{\Lambda }>0$. The integrand approaches $H_{\Lambda }^{-3}{a}^{-3}$, whose tail is integrable. ThereforeIts remaining approach has leading $a^{-2}$ behavior for the usual matter-plus-Lambda background. Growth of structure freezes rather than continuing as $a$. The basis function $H$ also approaches a constant; subtracting a suitable multiple of it isolates a genuinely decaying late-time solution $\propto a^{-2}$. The name “decaying mode” for $H$ refers to its matter-era behavior.