# Inflationatory universe

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) $0<t<t_3$ radiation only
(b) $t_3<t<t_2$ vacuum energy dominates, with an effective cosmological constant $\Lambda = \frac{3}{4} t_3^2$
(c) $t_2<t<t_1$ a period of radiation domination
(d) $t_1<t<t_0$ matter domination

Give simple analytical formulae for the expansion parameter $a(t)$ 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:

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

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

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

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

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

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

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?