Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 62 / 4 / e

First consider $k{\tau }_{\mathrm{eq}}<1$. Such a mode remains outside the horizon through matter-radiation equality, grows as ${\tau }^2$ on both sides, and keeps the same matter-era growth after entry. Squaring this growth preserves the shape of the HPZ spectrum:up to a scale-independent matching coefficient.

For $k{\tau }_{\mathrm{eq}}\gg 1$, entry occurs in radiation domination at ${\tau }_H\sim k^{-1}$, with $P_H\sim \mathcal{C}{k}^{-3}$. The density amplitude then grows only logarithmically until equality. ThusDuring matter domination, the matter-era linear growth factor of the density is $(\tau /{\tau }_{\mathrm{eq}}{)}^2$, so its square multiplies the matter power spectrum:The turnover at $k\sim {\tau }_{\mathrm{eq}}^{-1}$ records whether a mode entered during radiation domination. Earlier-entering small-scale modes miss the rapid ${\tau }^2$ growth and receive only the logarithmic Mészáros effect. This is the origin of the logarithmic small-scale matter transfer. The formula is a large-$k$ asymptote: the additive constant in the logarithmic solution and smooth equality matching must be retained near $k{\tau }_{\mathrm{eq}}=1$, where the spectrum does not vanish.

Multiplying by $k^3$ gives the small-scale mass-fluctuation varianceThis gives the requested estimate for modes entering before equality, with the logarithm replaced by a smooth order-one factor in the transition region. On comoving scale $L\sim k^{-1}$, unit variance today requiresFor a numerical illustration faithful to this question's flat, matter-plus-radiation background, take $h=0.7$ and ${\Omega }_r=4.2\times 10^{-5}{h}^{-2}$, hence ${\Omega }_C=1-{\Omega }_r\simeq 1$ rather than the $0.3$ appropriate to a vacuum-dominated model. Part 1(b), now with ${\Omega }_m={\Omega }_C$, gives ${\tau }_{\mathrm{eq}}\simeq 33\mathrm{Mpc}$. In this two-component model the same integral is valid all the way to today, givingWith $L=3\mathrm{Mpc}$, ${\tau }_0/{\tau }_{\mathrm{eq}}\simeq 258$ and $\ln ({\tau }_{\mathrm{eq}}/L)\simeq 2.39$. Therefore $\mathcal{C}\sim [258^4(2.39{)}^2{]}^{-1}\simeq 4\times 10^{-11}$, corresponding to a horizon-entry root-mean-square fluctuation of roughly $6\times 10^{-6}$. A very small primordial amplitude, of order $10^{-10}$ in this schematic normalization, can reach unit small-scale variance by today. Factors from the Fourier convention, smoothing window and equality matching preclude a precise normalization here. The calculation marks the onset of nonlinear structure formation, where extrapolating the linear solution ceases to be valid. For a universe with late vacuum domination, one must replace the matter-only growth law by its actual growth factor before making a numerical inference.