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**1. Homework Statement**

This problem concerns a simplified model of the history of a flat universe involving a period of inflation. The history is split into four periods:

(a) [itex]0<t<t_3[/itex] radiation only

(b) [itex]t_3<t<t_2[/itex] vacuum energy dominates, with an effective cosmological constant [itex]\Lambda = \frac{3}{4} t_3^2[/itex]

(c) [itex]t_2<t<t_1[/itex] a period of radiation domination

(d) [itex]t_1<t<t_0[/itex] matter domination

Give simple analytical formulae for the expansion parameter [itex]a(t)[/itex] which are approximateley true in these four phases.

**2. The attempt at a solution**

In epoch (d) I guess the ordinary solution for radiation dominated universes holds:

[tex] a(t)=a_0 \left( \frac{t}{t_0} \right) ^{2/3}[/tex]

and in epoch (c) I guess for the matter solution

[tex]a(t) = a_0 \left( \frac{t}{t_0} \right) ^{1/2}[/tex]

but here I have used the present scale factor [itex]a_0[/itex] and present time [itex]t_0[/itex] in both cases. Is that really correct?

For the epoch (b) I have used a de Sitter solution with a cosmological constant as

[tex]a(t) = A exp \left[ \left( \frac{1}{3} \Lambda \right)^{1/2} ct \right][/tex]

but I don't know if this is true.

For the first epoch I have no clue. Should I use the ordinary radiation dominated solution or should I look at more Planck scaled solutions?