18
The standard model of cosmology
the expansion of the universe which is characterized by the Hubble time H -I .
As discussed in section 2.2, the pressure p, entropy density s and energy density
p due to a gas of ultrarelativistic particles (in which the temperature T is much
greater than all masses) are given by
]l'2
T4
P
(1.101)
= 90 N •
211'2
s = 45 N.T 3
(1.102)
11'2
=
4
p
30 N • T
(1.103)
where
N. = NB + iNF'
(1.104)
The numbers NB and N F ofbosonic and fermionic degrees of freedom are defined
after (2.19). The entropy S in a comoving volume R 3 (t)
S=sR 3
(1.105)
is expected to be conserved because a homogeneous universe has no temperature
differences to generate heat transfer.
(For an explicit proof of entropy
conservation, see section 3.4 of Kolb and Turner or section 15.6 of Weinberg in
the general references.) Thus, to the extent that the entropy density is dominated
by the ultra-relativistic particles
RT = constant
(1.106)
while N. is constant. Equation (1.106) is valid even for a matter-dominated
universe because it is only the particles with mass m smaller than the temperature
T that are present in thermal equilibrium with appreciable number densities and
contributing to the entropy, although all particles contribute to the energy density.
In reality, RT will show small discontinuous changes as the temperature drops
below the mass of particular particle species. Subject to this caveat, equation
(1.53) for the time dependence of the scale factor now implies the following
connection between temperature and time for a radiation-dominated universe:
T ()( ,-1/2
for radiation domination.
(1.107)
The constant of proportionality in this equation may be calculated from the
Friedmann equation. When RT is a constant,
(~Y = (tY
(1. \08)
and, using ().1 03), the Friedmann equation (1.34) may be rewritten as
The standard model of cosmology
the expansion of the universe which is characterized by the Hubble time H -I .
As discussed in section 2.2, the pressure p, entropy density s and energy density
p due to a gas of ultrarelativistic particles (in which the temperature T is much
greater than all masses) are given by
]l'2
T4
P
(1.101)
= 90 N •
211'2
s = 45 N.T 3
(1.102)
11'2
=
4
p
30 N • T
(1.103)
where
N. = NB + iNF'
(1.104)
The numbers NB and N F ofbosonic and fermionic degrees of freedom are defined
after (2.19). The entropy S in a comoving volume R 3 (t)
S=sR 3
(1.105)
is expected to be conserved because a homogeneous universe has no temperature
differences to generate heat transfer.
(For an explicit proof of entropy
conservation, see section 3.4 of Kolb and Turner or section 15.6 of Weinberg in
the general references.) Thus, to the extent that the entropy density is dominated
by the ultra-relativistic particles
RT = constant
(1.106)
while N. is constant. Equation (1.106) is valid even for a matter-dominated
universe because it is only the particles with mass m smaller than the temperature
T that are present in thermal equilibrium with appreciable number densities and
contributing to the entropy, although all particles contribute to the energy density.
In reality, RT will show small discontinuous changes as the temperature drops
below the mass of particular particle species. Subject to this caveat, equation
(1.53) for the time dependence of the scale factor now implies the following
connection between temperature and time for a radiation-dominated universe:
T ()( ,-1/2
for radiation domination.
(1.107)
The constant of proportionality in this equation may be calculated from the
Friedmann equation. When RT is a constant,
(~Y = (tY
(1. \08)
and, using ().1 03), the Friedmann equation (1.34) may be rewritten as