Past exam of the general relativity course of the University of Oxford / 2021 / 4 / d / Solution

For monotonically expanding scale factor $R$, $dt=dR/(RH)$, so the elapsed cosmic time isThe second expression uses cosmological redshift $1+z=R_0/R$; the age of the universe is the limit to its earliest specified state, if that limit is finite.

An age comparison for an initial vacuum-dominated phase needs a common normalization. Match the two histories at the end of S1, with $R=R_1$ and $H=H_1>0$, and give them identical subsequent S2 and S3 evolution, including transition radii and today's state. Below $R_1$, the history with S1 has constant $H=H_1$. The comparison history extends S2 backwards, so $\rho \propto R^{-4}$ and $H=H_1(R_1/R{)}^2$. Starting both at $R_i=ϵR_1$, where $0<ϵ<1$, their early durations areThe difference is positive becauseThe common later evolution adds the same time to both histories. Thus S1 makes this matched universe older. More strongly, as $R_i\to 0$, the exact S1 duration diverges logarithmically, whereas a radiation beginning has finite duration $1/(2H_1)$ to the match. The exact de Sitter spacetime expanding patch reaches $R=0$ only as $t\to -\infty$ for its comoving observers; a finite “age since $R=0$” does not exist in this ideal model. This does not establish past geodesic completeness of the patch.

The matching assumption is essential for a finite cutoff or realistic finite period of cosmic inflation: arbitrary changes of the present Hubble parameter or transition densities prevent a universal age comparison. The literal zero-size limit in this question is instead the infinite-duration idealization just derived.