Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 56 / 4 / i / Solution

For fixed epoch and the increasing branch $\nu >1$, increasing mass lowers the smoothed matter density variance, hence raises the halo peak height $\nu$. The rate $({\nu }^2-1)\dot{D}/D$ therefore increases and its inverse decreases. Rarer, more massive haloes have a shorter fractional abundance-growth time, even though their actual number density is much smaller. They lie farther into the exponential tail of the Press-Schechter halo mass function, so a small change in the linear growth factor causes a large fractional change. This statement compares fixed-mass bins in the same cosmology and concerns relative growth, not the time for one halo to assemble all its mass.