Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 15E (Cosmology)

In a flat matter-dominated universe, $a\propto t^{2/3}$, $H=2/(3t)$ and $4\pi G{\bar{\rho }}_c=2/(3t^2)$. Substitution of ${\delta }_k\propto t^{\beta }$ gives $\beta (\beta -1)+(4/3)\beta -2/3=0$, whose roots are $2/3$ and $-1$. HenceDuring radiation domination, $a\propto t^{1/2}$ while ${\bar{\rho }}_c/{\bar{\rho }}_{\mathrm{total}}\propto a\ll 1$. The matter self-gravity term is suppressed relative to $H^2$, leaving approximately ${\ddot{\delta }}_k+t^{-1}{\dot{\delta }}_k=0$, with solutions constant and $\log t$. Thus there is no growing power-law mode comparable to the later $t^{2/3}$ growth. The elementary estimate in part (ii) neglects this slow logarithmic growth of matter perturbations during radiation domination.

Normalize $a(t_0)=1$. In the matter era, horizon crossing $c{t}_H=2\pi a(t_H)/k$ with $a=(t/t_0{)}^{2/3}$ gives $t_H/t_0=(k_0/k{)}^3$. In the radiation era, match $a=(t_{\mathrm{eq}}/t_0{)}^{2/3}(t/t_{\mathrm{eq}}{)}^{1/2}$ to obtainThe boundary mode therefore has $k_{\mathrm{eq}}=k_0(1+z_{\mathrm{eq}}{)}^{1/2}$. For $k<k_{\mathrm{eq}}$, the growing mode multiplies the primordial amplitude by $(t_0/t_H{)}^{2/3}=(k/k_0{)}^2$. For $k>k_{\mathrm{eq}}$, growth is approximately frozen until equality, then multiplied by $(t_0/t_{\mathrm{eq}}{)}^{2/3}=1+z_{\mathrm{eq}}=(k_{\mathrm{eq}}/k_0{)}^2$. Squaring these factors gives the matter power spectrumThe large-$k$ estimate omits the logarithmic radiation-era transfer correction, consistently with the stipulated approximation.