Chapter 3: THE FRIEDMANN MODELS

3.6.1 General solutions for R(τ)
Before looking at general solutions to the Friedmann equation, it is useful to look at
the asymptotic behaviour of models with general Ω. Setting Λ = 0, we may develop
the Friedmann equation, using the relations between curvature and Ω (equation 3.27)
and between R and z, as follows:
(3.42)
Thus
R&
R&
R
2
2
2
0
8π
GρR
kc
= −
3 R A
3 2
8π
Gρ0R0
= −( Ω −1)
H
3 R
2 R0
= H0
Ω0
−( Ω −1)
H
R
2
= H Ω ( 1+ z)
− Ω + 1
0
2
0 0
2
0 0
2
0 0
(3.43)
R&
R
2
2
= H ( 1 +Ω z)
2 0
0
0
Hence, if Ω 0 < 1, we can see that the Universe has had roughly constant & R (i.e. has
had undecelerated expansion) since the epoch corresponding to (1+z) ~ Ω 0
-1. At
earlier times, & R behaves as for an Ω = 1 Universe (though, note, with a different H
than the Ω 0 = 1 Universe that would have the same H 0 today).
We can see this also by looking at Ω(z), which may be calculated as follows: From
(3.43) we have
2 2 2
H = H ( 1+ z) ( 1+
Ω z)
0
0
Substituting this expression for H(z) into the definition of Ω and knowing ρ(R), we
get
Ω =
=
8πGρ
2 2
3H
( 1+ z) ( 1+
Ω z)
0
8πGρ
( 1+
z)
0
2
3H
( 1+
Ω z)
0
0
0
Thus
(3.44) Ω Ω ( 1 + z)
=
0
( 1 + Ω z)
0