Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3 / d / Solution

For matter plus a cosmological constant,Set ${\delta }_m=Hu$. Substitution into the equation from part (c) makes the coefficient of $u$ vanish by the Friedmann relation and leavesorTherefore the two independent modes can be writtenThe first is the decaying mode. Normalizing the second to $D_{+}(a)\sim a$ at early times gives the integral linear growth factor in a matter-Lambda universe