Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1

The cosmological continuity equation and $P_i=w_i{\rho }_i$ giveWith $a_0=1$, $1+z=a^{-1}$ andthe Friedmann equation becomesSpatial curvature may be included as an effective component with $w=-1/3$ and ${\Omega }_{k,0}=-k/H_0^2$.

Given the component parameters, solve the first-order equationA cosmic string network has $w=-1/3$, so $\rho \propto a^{-2}$ and $H\propto a^{-1}$. Therefore $\dot{a}$ is constant and a string-dominated expanding universe hasafter choosing the Big Bang as $t=0$.

For pressureless matter, ${\rho }_m{a}^3={\rho }_{m,0}$ when $a_0=1$. Setthen ${\dot{a}}^2=D/a-k$. Substitution of $a=A(1-\cos \theta )$ makes this identity hold whenIndeed $dt/d\theta =B(1-\cos \theta )=Ba/A$, and henceafter setting $t=0$ at $\theta =0$. This is the closed matter-dominated Friedmann solution.

The small-$\theta$ expansions areWriting $x=(6t/B{)}^{1/3}$ and reverting the second series gives $\theta =x(1+x^2/60+\cdots )$. Thereforeso $C=(A/2)(6/B{)}^{2/3}$. The leading $a\propto t^{2/3}$ is the flat matter-dominated limit.

During recollapse $H<0$, so nearby comoving galaxies are increasingly blueshifted rather than redshifted. Their physical separations and angular-diameter distances shrink, making them appear larger and generally brighter; in a closed geometry sufficiently old light can also produce repeated or strongly focused images.

The cosmic microwave background temperature obeys $T_{\gamma }\propto a^{-1}$. It therefore rises without bound as $\theta \to 2\pi$, and its photons are blueshifted. Before the formal Big Crunch, the growing temperature reionizes matter, scattering makes the universe opaque, and the idealization that old galaxies and the original last-scattering surface remain directly visible eventually fails.