88 Cosmological Models
In fact, this inequality is always true when
푘=+ 1 ,푝>−^1
3휌푐^2 ,푤>−^1
3
,휆> 0. (5.36)
Then we can neglect the curvature term and the휆term in Equation (5.17), which
simplifies to
푎̇
푎=퐻(푡)=
(
8 휋퐺
3
휌
) 1 ∕ 2
∝푎−^3 (^1 +푤)∕^2. (5.37)
Let us now find the time dependence of푎by integrating this differential equation:∫
d푎푎−^1 +^3 (^1 +푤)∕^2 ∝
∫
d푡,to obtain the solutions
푎^3 (^1 +푤)∕^2 ∝푡 for 푤≠− 1 , ln푎∝푡 for 푤=− 1.Solving for푎,푎(푡)∝푡^2 ∕^3 (^1 +푤) for 푤≠− 1 ,푎(푡)∝econst.⋅푡 for 푤=− 1. (5.38)In the two epochs of matter domination and radiation domination we know the
value of푤. Inserting this we obtain the time dependence of푎for a matter-dominated
universe,
푎(푡)∝푡^2 ∕^3 , (5.39)and for a radiation-dominated universe,
푎(푡)∝푡^1 ∕^2. (5.40)Big Bang. We find the starting value of the scale of the Universe independently of
the value of푘in the curvature term neglected above:
lim
푡→ 0푎(푡)= 0. (5.41)
Inthesamelimittherateofchange푎̇is obtained from Equation (5.37) with any푤
obeying푤>−1:
lim
푡→ 0
푎̇(푡) = lim
푡→ 0푎−^1 (푡)=∞. (5.42)
It follows from Equations (5.32) and (5.33) that an early radiation-dominated Uni-
verse was characterized by extreme density and pressure:
lim
푡→ 0
휌r(푡) = lim
푡→ 0푎−^4 (푡)=∞,
lim
푡→ 0
푝r(푡) = lim
푡→ 0푎−^4 (푡)=∞.
In fact, these limits also hold for any푤obeying푤>−1.
Actually, we do not even need an equation of state to arrive at these limits. Pro-
vided휌푐^2 + 3 푝was always positive and휆negligible, we can see from Equations (5.6)