Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 9B (Cosmology)

Differentiate the Friedmann equation $H^2=8\pi G\rho /3$ and use the continuity equation $\dot{\rho }=-3H(\rho +P)$:so $\dot{H}=-4\pi G(\rho +P)$. Since $H=\dot{a}/a$,Hence the Raychaudhuri equation is

For the equation of state $P=w\rho$ with positive energy density,The expansion is accelerated for $w<-1/3$, coasting for $w=-1/3$, and decelerated for $w>-1/3$.

At equality, let ${\rho }_{\mathrm{cc}}={\rho }_{\mathrm{dust}}={\rho }_{\ast }$. The cosmological constant has $P_{\mathrm{cc}}=-{\rho }_{\ast }$, while nonrelativistic matter has $P_{\mathrm{dust}}=0$. ThereforeThe scale factor consequently has $\ddot{a}>0$: the expansion is accelerated at $a=a_{\mathrm{eq}}$.

The cosmological constant energy is constant, whereas the matter density scales as $a^{-3}$. HenceThe transition occurs when $\rho +3P={\rho }_{\mathrm{dust}}-2{\rho }_{\mathrm{cc}}=0$, so