Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 60 / 4 / ii / Solution

Assume matter-dominated growth over the two epochs, so $D(z)\propto (1+z{)}^{-1}$. A fixed definition of the characteristic mass selects fixed peak height ${\nu }_{\ast }$. Therefore $D(z)\sigma (M_{\ast })$ is constant. With $\sigma (M)\propto M^{-s}$, the observed ratios giveSince a spherical top-hat window function has $M\propto R^3$, the rms slope isFor a cosmological density power spectrum $P(k)\propto k^n$, the smoothed matter density variance isThe dimensionless integral converges for $-3<n<1$. Thus $m=-(n+3)/2$, andFor reference, substituting $s=1/4$ into the differential Press-Schechter halo mass function gives a low-mass prefactor $M^{-7/4}$ and an exponential argument proportional to $M^{1/2}$, so its displayed-form exponents are $\alpha =-7/4$, $\beta =1/2$. The prefactor normalization varies with epoch.

The effective galactic-scale index $-3/2$ need not equal the primordial nearly scale-invariant index one. The cosmological transfer function modifies perturbations between the primordial epoch and galaxy formation: modes entering the horizon in the radiation-dominated era grow differently from larger modes. Schematically $P_m(k,z)\propto D(z{)}^2{k}^{n_s}T(k{)}^2$, with $n_s\simeq 1$, while the cold-dark-matter transfer function decreases toward small scales. The result is a much redder late-time cosmological density power spectrum, with a scale-dependent slope. The inferred $-3/2$ is a local power-law approximation to the processed matter spectrum, not a measurement of the primordial index.