Transition from radiation to matter domination
19
( T.-)
• 2
2
= 81rGN ~N T4
(1.109)
T
3 30"
where we have neglected the cosmological constant and the curvature term. as in
section 1.5. This has solution
t = ~ ( 90
2
)1/2
( 1.110)
1r 2 N.
MpT- 2
::::: 1.5IMpN;I/2 T -2.
(1.111)
If. for example. the appropriate N. for T above 100 GeV is that of the SU(3) x
SU(2) x U(1) standard model or that of the supersymmetric standard model.
then
427
or
915
N·=T
(1.112)
T
respectively.
Equations (1.54) and (1.1 06) imply the following connection between
temperature and time for the matter-dominated, universe:
T ex t- 2 / 3
for matter domination.
(1.113)
For a matter-dominated universe.
3
2 2
( T )
p(T) = 3MpHoQo To
(1.114)
where we have used (1.56). (1.1 06). (1.58) and (1.40). Using (1.108). the
Friedmann equation (1.34) may be rewritten as
( T t)2
=
(T)3
(1.115)
HJQo To
with solution
2
t = 3(HoQ~/2)-1 (~r3/2
(1.116)
1.9 Transition from radiation to matter domination
As we have seen in ( 1.49) and (1.50). the energy density of radiation decreases as
R- 4 as the universe expands whereas the energy density of matter decreases as
R-3. Thus. radiation domination gives way to matter domination at some point
in the expansion of the universe. For a matter-dominated universe. the energy
density is given by (1.114) and for a radiation-dominated universe by (1.103).
However. there is a subtlety in the interpretation of N. which must be taken into
account. We shall assume that the transition temperature is sufficiently low that