Past exam of the astrophysics course of the University of Oxford / 2015 / 3 / d / Solution

A power spectrum describes the variance of Fourier amplitudes as a function of wavenumber. In a statistically homogeneous continuum convention,The paper's $⟨|{\Delta }_M(k){|}^2⟩$ is the corresponding finite-volume or normalization-suppressed statement. For statistical isotropy only $k=|\mathbf{k}|$ remains. The dimensionless variance per logarithmic interval is $k^3{P}_M(k)/(2{\pi }^2)$, a different quantity.

The response changes around $2k^2\sim 9{\mathcal{H}}^2$, or $k\sim 2.1\mathcal{H}$: the velocity contribution to the relativistic constraint and the effective growth equation change relative importance across the horizon-scale transition. Large-scale ${\Delta }_M$ is approximately constant for its regular mode, while short-scale density perturbations grow gravitationally. A smooth initial potential spectrum can therefore acquire a different $k$ dependence in the density spectrum. For illustration, a constant matter-era potential gives ${\Delta }_M=-(5+k^2{\eta }^2/6)\Psi$, so sufficiently small-scale $P_M$ has four additional powers of $k$ relative to $P_{\Psi }$, while the very large-scale limits have the same power dependence.

This identifies a dynamical transition, not a universal prediction of the primordial spectrum or an exact spectral break at $2\mathcal{H}$ for arbitrary initial data. The familiar late-time matter-spectrum turnover also retains the history of horizon entry around radiation–matter equality; it is not generically tied to the current Hubble scale. Initial conditions and the intervening expansion history are needed for an actual spectrum.