Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 310 / 2 / e

The Mészáros effect suppresses the growth of modes which enter the horizon during radiation domination. Modes with comoving wavenumber $k\gg k_{\mathrm{eq}}$ enter earlier, but gain only a logarithmic growth factor until matter-radiation equality; modes entering during matter domination grow as $a$ after entry. Consequently small-scale amplitudes are reduced relative to an extrapolation of the unsuppressed large-scale cosmological density power spectrum, creating a turnover near the matter-radiation equality scale $k_{\mathrm{eq}}={\mathcal{H}}_{\mathrm{eq}}$ in units $c=1$.

More quantitatively, for adiabatic primordial fluctuations with scalar spectral index $n_s$, the late-time linear spectrum can be written, up to a time-dependent normalization,The cold-dark-matter transfer function approaches a constant at $k\ll k_{\mathrm{eq}}$, whereas logarithmic small-scale matter transfer gives $T(k)\propto (k_{\mathrm{eq}}/k{)}^2\log (k/k_{\mathrm{eq}})$ at large $k$. ThusFor approximately scale-invariant initial curvature fluctuations, $n_s\simeq 1$, the large-$k$ scaling is $k^{-3}{\log }^{2}k$, rather than $k$. This discussion ignores baryonic acoustic structure and nonlinear clustering. The Mészáros equation solution basis includes both a constant and a logarithmic branch during radiation domination; the physical logarithmic growth is selected by matching the earlier radiation forcing, so not every mathematical solution grows logarithmically.