264
17 Earlier Times and Radiation
But we also know from Sect. 14.4 that the radiation energy density is proportional
to the inverse fourth power of the scale factor, as in (14.16). This relation and (17.1)
tell us that the temperature of the radiation must be proportional to the inverse of the
scale factor, or
T
T 0
=
a 0
a
.
(17.2)
As usual the a 0 refers to the scale factor at the present time, which we often take to be
equal to 1. This relation for the behavior of the temperature as the universe expands
is remarkably simple. It has important consequences for the CMB spectrum and also
for the behavior of the constituents of the universe at early times, long before the
present LCDM universe. It clearly shows that the big bang was a hot big bang.
It should be emphasized that the temperature of the matter in the present universe
is clearly not the same as that of the CMB since the two are not in thermal equilibrium.
Clearly there is a great diversity of temperatures in the present universe, for example
in the hot interior of stars and the cold of space. Equilibrium or lack of it is an
interesting question for ealier times.
Having worked out the dependence of the temperature of the CMB on the scale
factor let us consider how the spectrum of the CMB behaves during the expansion
of the universe. Recall the Planck distribution law for black body radiation; it tells
us that in a thermalized system at temperature T the number of photons in volume
V , with frequency between ν and ν + dν, is given by
dN =
8πν
2 dν
c 3 (e hν/kT − 1)
V.
(17.3)
Here h is Planck’s constant and k is Boltzmann’s constant. This famous distribution
is sketched in Fig. 17.1.
The CMB spectrum was accurately measured by the cosmic background explorer
satellite (COBE) and is in extremely good agreement with the Planck distribution, to
about a part in 10
5 . This leaves little doubt that the CMB is indeed thermal radiation
at 2.725 K (Fixsen 1993).
Fig. 17.1 Qualitative sketch of the Planck distribution, the spectrum of black body radiation
17 Earlier Times and Radiation
But we also know from Sect. 14.4 that the radiation energy density is proportional
to the inverse fourth power of the scale factor, as in (14.16). This relation and (17.1)
tell us that the temperature of the radiation must be proportional to the inverse of the
scale factor, or
T
T 0
=
a 0
a
.
(17.2)
As usual the a 0 refers to the scale factor at the present time, which we often take to be
equal to 1. This relation for the behavior of the temperature as the universe expands
is remarkably simple. It has important consequences for the CMB spectrum and also
for the behavior of the constituents of the universe at early times, long before the
present LCDM universe. It clearly shows that the big bang was a hot big bang.
It should be emphasized that the temperature of the matter in the present universe
is clearly not the same as that of the CMB since the two are not in thermal equilibrium.
Clearly there is a great diversity of temperatures in the present universe, for example
in the hot interior of stars and the cold of space. Equilibrium or lack of it is an
interesting question for ealier times.
Having worked out the dependence of the temperature of the CMB on the scale
factor let us consider how the spectrum of the CMB behaves during the expansion
of the universe. Recall the Planck distribution law for black body radiation; it tells
us that in a thermalized system at temperature T the number of photons in volume
V , with frequency between ν and ν + dν, is given by
dN =
8πν
2 dν
c 3 (e hν/kT − 1)
V.
(17.3)
Here h is Planck’s constant and k is Boltzmann’s constant. This famous distribution
is sketched in Fig. 17.1.
The CMB spectrum was accurately measured by the cosmic background explorer
satellite (COBE) and is in extremely good agreement with the Planck distribution, to
about a part in 10
5 . This leaves little doubt that the CMB is indeed thermal radiation
at 2.725 K (Fixsen 1993).
Fig. 17.1 Qualitative sketch of the Planck distribution, the spectrum of black body radiation