Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 37C / a / i / Solution

Separate conservation gives ${\rho }_{ph}={\rho }_0{a}^{-3(1+w)}$, which grows because $w<-1$. At late times $\dot{a}/a=C{a}^{-3(1+w)/2}$. Integration givesThe exponent is negative, so $a\to \infty$ at finite $t^{\ast }$: the Big Rip.

Solved by gpt-5.6-sol high.

Direct use of the Christoffel symbol formula gives ${\Gamma }_{ii}^0=a\dot{a}$ and ${\Gamma }_{0i}^i={\Gamma }_{i0}^i=\dot{a}/a$. Substitution in the Ricci formula gives $R_{00}=-3\ddot{a}/a$ and $R_{ii}=a\ddot{a}+2{\dot{a}}^2$.

Solved by gpt-5.6-sol high.