Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 3

For each Fourier mode, the cosmological Poisson equation givesBecause nonrelativistic matter has ${\bar{\rho }}_m\propto a^{-3}$, this implies $\Phi \propto {\Delta }_m/a$. Write $\Phi =C{\Delta }_m/a$ and use $\mathcal{H}=a^{\prime }/a$. Direct differentiation in the stated potential equation givesThe supplied Friedmann equation identity therefore yieldsSince $d/d\tau =ad/dt$, this becomes the linear growth equation

During matter domination, $a\propto t^{2/3}$, $H=2/(3t)$, and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. Substituting ${\Delta }_m\propto t^s$ into the linear growth equation giveswhose roots are $s=2/3$ and $s=-1$. ThereforeThe growing one of the matter-dominated density-perturbation modes is

Increasing the dark energy density makes dark-energy domination begin earlier. The larger Hubble parameter strengthens the $2H{\dot{\Delta }}_m$ friction term while the clustering source $4\pi G{\bar{\rho }}_m{\Delta }_m$ continues to dilute, so less large-scale structure of the universe can grow from a fixed primordial amplitude.

Deep in cosmological-constant domination, $H\simeq H_{\Lambda }$ is constant and ${\bar{\rho }}_m$ is negligible. The growth equation reduces toand henceThe surviving mode is constant: matter growth freezes.

The background remains matter dominated, so $H=2/(3t)$ and $4\pi G{\bar{\rho }}_m=2/(3t^2)$. For the reduced gravitational clustering strength, substitute ${\Delta }_m\propto t^s$ into the modified equation:The growing root is the matter-era growth with reduced gravitational clustering strengthIt satisfies $0<s(f)<2/3$ for $0<f<1$. Between $t_f$ and the onset $t_{\Lambda }$ of dark-energy domination, standard growth supplies $(t_{\Lambda }/t_f{)}^{2/3}$ whereas the modified model supplies $(t_{\Lambda }/t_f{)}^{s(f)}$. Subsequent growth freezes in both, so

A larger cosmological constant changes both the expansion history and the growth history: it alters cosmological distances, the baryon acoustic oscillation scale as observed in angle and redshift, and the time at which growth begins to freeze. The new interaction in part (d) leaves the background Friedmann equation unchanged but changes clustering abruptly from $t_f$ onward.

Redshift-resolved measurements of the linear growth factor, weak gravitational lensing, galaxy clustering, and cosmological distance measures can therefore compare geometry with growth. Equal present amplitudes do not imply equal $D(z)$, and only the higher-dark-energy model changes the distance-redshift relation. This breaks the degeneracy.