Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 1 / 9D / Solution

The Hubble parameter is the fractional expansion rateHere $\rho$ is the homogeneous energy density, $k\in \{-1,0,1\}$ records the sign of the spatial curvature of an FLRW universe, and $R$ is the fixed curvature length scale used with the dimensionless scale factor $a(t)$.

Energy-momentum conservation in a homogeneous expanding universe gives the continuity equationDifferentiate the Friedmann equationUsing $\dot{a}/a=H$ and the continuity equation givesEliminating the curvature term with the Friedmann equation yieldsSince $\dot{H}+H^2=\ddot{a}/a$, this is the Raychaudhuri equation

Under the strong energy condition $\rho +3P\ge 0$, one has $\ddot{a}\le 0$. Thus $a(t)$ is concave, and for $t<t_0$ it lies below its tangent at the present time:The right side vanishes at $t=t_0-H_0^{-1}$. A positive scale factor therefore cannot extend regularly to that time; it must reach the Big Bang singularity at some $t_{\mathrm{BB}}\ge t_0-H_0^{-1}$. Equivalently,