Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 4 / 10E / Solution

With a prime denoting conformal-time differentiation, $a^{\prime }=a\dot{a}$ and $\mathcal{H}=a^{\prime }/a=\dot{a}$. Multiplying the Friedmann equation by $a^2$ gives ${\mathcal{H}}^2+k{c}^2=8\pi G\rho a^2/3$, and henceThe Friedmann acceleration equation and $P=w\rho c^2$ giveDifferentiate $\Omega -1=k{c}^2/{\mathcal{H}}^2$ and substitute this expression:For $k=0$, $\Omega =1$ identically and the same equation holds directly. The relations assume $\mathcal{H}\ne 0$, as is true on the expanding branch in the question.

For pressureless matter, $w=0$, the cosmological density parameter flow points away from $\Omega =1$: initial $\Omega >1$ increases, while $0<\Omega <1$ decreases. In the closed case the expanding branch can end at a turnaround, where $\mathcal{H}=0$ and this density parameter diverges. For vacuum domination, $w=-1$, the sign reverses: values above one decrease and positive values below one increase toward one. The plotted curves are representative conformal-time solutions with $c=1$ and separately chosen origins; inflationary future infinity occurs at a finite conformal time.

In a decelerating matter or radiation era, small deviations from flatness grow: near one, $d\log |\Omega -1|/d\log a\simeq 1+3w>0$. Observed near-flatness therefore requires extremely small earlier deviations, the flatness problem. During accelerated expansion with $w<-1/3$, this exponent is negative and curvature is driven toward negligible relative size. For $w=-1$, $|\Omega -1|$ decays approximately as $a^{-2}$. Sufficiently long cosmic inflation supplies the inflationary solution of the flatness problem before ordinary decelerating evolution resumes.