Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 67 / 3

Normalize the scale factor to $a(t_0)=1$, use $c=1$, and define the instantaneous critical density by ${\rho }_{\mathrm{cr}}(t)=3H(t{)}^2/(8\pi G)$. Separately conserved pressureless matter and radiation in cosmology obey the cosmological continuity equation,The present cosmological density parameters areThe cosmological constant has equivalent energy density ${\rho }_{\Lambda }=\Lambda /(8\pi G)$. Substituting into the Friedmann equation yieldsAt the present epoch this gives $k=H_0^2({\Omega }_0-1)$, or curvature contribution ${\Omega }_{K0}=-k/H_0^2=1-{\Omega }_0$. Dividing each component by $H_0^2$ provesSpatial curvature is not included in the total physical-density parameter ${\Omega }_0$. In particular, positive vacuum density does not have the radiation or matter scaling.

For $\Lambda =0$, use the instantaneous cosmological density parameterInsert the Friedmann equation from the preceding part and multiply numerator and denominator by $a^4$:This holds on an expanding or contracting branch with $H\ne 0$; in a recollapsing model the density parameter diverges at turnaround.

During radiation domination, while curvature is small, $|\Omega -1|\simeq |{\Omega }_0-1|a^2/{\Omega }_{R0}$. During matter domination the corresponding behavior is $|\Omega -1|\simeq |{\Omega }_0-1|a/{\Omega }_{M0}$. Thus the exactly flat solution $\Omega =1$ is unstable against relative curvature during decelerating expansion: a small departure grows rather than being dynamically driven to zero. The flatness problem is that a universe with modest present curvature must have had an extremely tiny initial curvature-to-density ratio at an early radiation epoch. The fact that every such solution tends to $\Omega =1$ when extrapolated to $a\to 0$ does not supply a mechanism selecting that small departure at a specified finite initial epoch. An early period of cosmic inflation can instead shrink the comoving Hubble radius and suppress $|\Omega -1|=|k|/(aH{)}^2$.

Write $H_{\Lambda }=\sqrt{\Lambda /3}$ and $\kappa =\sqrt{-k}>0$. With matter and radiation absent, the Friedmann equation isChoose the expanding branch and put its coordinate origin at $a(0)=0$. Separating variables,so the open slicing of de Sitter spacetime isFor $a(t_0)=1$ the equivalent normalized form isIt has $H(t)=H_{\Lambda }\coth (H_{\Lambda }t)$ and $H_0^2=H_{\Lambda }^2-k$, as required. In terms of a prescribed present Hubble parameter,At early times $a\simeq \sqrt{-k}t$ resembles the Milne universe, while at late times it grows exponentially. The empty model still has vacuum energy density; the vanishing of $a$ at the open-patch origin is not a curvature singularity of de Sitter spacetime.