Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 62 / 1

Let $H=\dot{a}/a$ be the Hubble parameter, and normalize the scale factor by $a(t_0)=1$. The cosmological continuity equation gives ${\rho }_M={\rho }_{M0}{a}^{-3}$ for pressureless matter, whereas the cosmological constant has constant energy density ${\rho }_{\Lambda }$. The spatially flat Friedmann equation therefore becomesThe present Hubble constant is $H_0=H(t_0)$. In terms of the present critical density ${\rho }_{\mathrm{crit},0}=3H_0^2/(8\pi G)$, the matter cosmological density parameter is ${\Omega }_M={\rho }_{M0}/{\rho }_{\mathrm{crit},0}$. Spatial flatness gives ${\rho }_{\Lambda }/{\rho }_{\mathrm{crit},0}=1-{\Omega }_M$, so

For $0<{\Omega }_M<1$, set $x=a^{3/2}$ and choose the expanding branch of the Friedmann equation. ThenSeparating variables and choosing cosmic time $t=0$ at the Big Bang givesThus the flat dust-and-vacuum expansion isSetting $a(t_0)=1$ gives the age of a flat matter-Lambda universe,The two matter-vacuum age endpoint limits follow below.

As ${\Omega }_M\downarrow 0$, use $\mathrm{arsinh}z=\log (z+\sqrt{1+z^2})\sim \log (2z)$ to obtainAt fixed Hubble constant, the age of a flat matter-Lambda universe diverges logarithmically. The limiting purely vacuum-dominated de Sitter spacetime has an exponential scale factor and no Big Bang at a finite value of cosmic time in this flat slicing.

As ${\Omega }_M\uparrow 1$, use $\mathrm{arsinh}z\sim z$ near zero. ConsequentlyThe flat dust-and-vacuum expansion tends to $a(t)=(3H_{0}t/2{)}^{2/3}$, the Einstein-de Sitter universe.

A radial light ray in the FLRW metric travels comoving radial distance $d\chi =dt/a$ in units $c=1$. The total comoving visibility radius is therefore the full available conformal time interval,Insert the Friedmann equation from part (a), then put $y=1/a$ to getFor $0<{\Omega }_M<1$ the integrand is bounded at zero and decays as $y^{-3/2}$ at infinity, so the total conformal lifetime of a flat matter-Lambda universe is finite. Equivalently, this distance is the present comoving particle horizon plus the remaining comoving distance to the cosmological event horizon. This describes causal visibility under the assumed ability to see back to the initial singularity.

The substitution $u=y[{\Omega }_M/(1-{\Omega }_M){]}^{1/3}$ also evaluates the integral using the beta function: