Physical Foundations of Cosmology

26 Kinematics and dynamics of an expanding universe

with the Newtonian estimate obtained by ignoring gravity. (HintUse (1.70) to fix
one of the constants of integration.)

The range of conformal timeηin flat and open universes is semi-infinite,+∞>
η>0, regardless of whether the universe is dominated by radiation or matter. For
a closed universe,ηis bounded:π>η>0 and 2π>η>0 in the radiation- and
matter-dominated universes respectively.
Finally, we consider the important case of a flat universe with a mixture of matter
(dust) and radiation. The energy density of matter decreases as 1/a^3 while that of
radiation decays as 1/a^4. Therefore, we have

ε=εm+εr=

εeq

2

((

aeq

a

) 3

+

(aeq

a

) 4 )

, (1.78)

whereaeqis the value of the scale factor at matter–radiation equality, whenεm=εr.
Equation (1.71) now becomes

a′′=

2 πG

3

εeqaeq^3 (1.79)

and has a simple solution:

a(η)=

πG

3

εeqaeq^3 η^2 +Cη. (1.80)

Again, we have fixed one of the two constants of integration by imposing the
conditiona(η=0)= 0 .Substituting (1.78) and (1.80 ) into (1.70), we find the
other constant of integration:

C=

(

4 πGεeqa^4 eq/ 3

) 1 / 2

.

Solution (1.80) is then

a(η)=aeq

((

η

η

) 2

+ 2

(

η

η

))

, (1.81)

where

η=

(

πGεeqaeq^2 / 3

)− 1 / 2

=ηeq/(

√

2 −1) (1.82)

has been introduced to simplify the expression. (The relation betweenηandηeq
immediately follows froma(ηeq)=aeq.) Forηηeq, radiation dominates and
a∝η. As the universe expands, the energy density of radiation decreases faster
than that of dust. Hence, forηηeq, dust takes over and we havea∝η^2.