Paper 3, Section I, E
Consider a finite sphere of zero-pressure material of uniform density which expands with radius , where is an arbitary constant, due to the evolution of the expansion scale factor . The sphere has constant total mass and its radius satisfies
where
with constant. Show that the scale factor obeys the equation
where is a constant. Explain why the sign, but not the magnitude, of is important. Find exact solutions of this equation for when
(i) and ,
(ii) and ,
(iii) and .
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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