Past exam of the general relativity and cosmology course of the University of Oxford / 2012 / 3 / a / Solution

Normalize $a(t_0)=1$, taking $a$ dimensionless and $r$ as a length, so the source's curvature constant $k$ has units of inverse length squared. With densities in the mass-equivalent convention, the first Friedmann equation isThe sum can include dust, radiation and other material components; vacuum energy must be represented either in that sum or by $\Lambda$, without double counting. For physical energy densities $u_i$, replace ${\rho }_i$ by $u_i/c^2$. The companion acceleration equation is $\ddot{a}/a=-(4\pi G/3){\sum }_i({\rho }_i+3p_i/c^2)+\Lambda c^2/3$, with energy conservation ${\dot{\rho }}_i+3H({\rho }_i+p_i/c^2)=0$ for separately conserved components.

For an empty universe with $\Lambda =0$, $H^2=-k{c}^2/a^2$. On the expanding branch with $H_0>0$,Equivalently, in the convention where the normalized curvature label is $k=-1$ and the scale factor has length units, $a_0=c/H_0$. A rescaling of comoving coordinates changes the numerical magnitude of $k$ but not its negative sign. The degenerate empty case $H_0=0$ has $k=0$ and a constant scale factor; it does not have a finite age $1/H_0$.