Paper 3, Section I, E

Cosmology | Part II, 2014

Consider a finite sphere of zero-pressure material of uniform density ρ(t)\rho(t) which expands with radius r(t)=a(t)r0r(t)=a(t) r_{0}, where r0r_{0} is an arbitary constant, due to the evolution of the expansion scale factor a(t)a(t). The sphere has constant total mass MM and its radius satisfies

r¨=dΦdr\ddot{r}=-\frac{d \Phi}{d r}

where

Φ(r)=GMr16Λr2c2,\Phi(r)=-\frac{G M}{r}-\frac{1}{6} \Lambda r^{2} c^{2},

with Λ\Lambda constant. Show that the scale factor obeys the equation

a˙2a2=8πGρ3Kc2a2+13Λc2,\frac{\dot{a}^{2}}{a^{2}}=\frac{8 \pi G \rho}{3}-\frac{K c^{2}}{a^{2}}+\frac{1}{3} \Lambda c^{2},

where KK is a constant. Explain why the sign, but not the magnitude, of KK is important. Find exact solutions of this equation for a(t)a(t) when

(i) K=Λ=0K=\Lambda=0 and ρ(t)0\rho(t) \neq 0,

(ii) ρ=K=0\rho=K=0 and Λ>0\Lambda>0,

(iii) ρ=Λ=0\rho=\Lambda=0 and K0K \neq 0.

Which two of the solutions (i)-(iii) are relevant for describing the evolution of the universe after the radiation-dominated era?

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