176 Cosmic Microwave Background
8.1 The CMB Temperature
Predictions. In 1948,Georg Gamow(1904–1968),Ralph AlpherandRobert Herman
calculated the temperature at that time of the primordial blackbody radiation which
started free streaming at the LSS. They found that the CMB should still exist today,
but that it would have cooled in the process of expansion to the very low temperature
of푇 0 ≈5K. This corresponds to a photon wavelength of
휆=
hc
푘푇 0= 2. 9 × 10 −^3 m. (8.1)This is in the microwave range of radio waves (see Table A.3). (The term ‘microwave’
is actually a misnomer, since it does not refer to micrometer wavelengths, but rather
to centimeters.)
We can now redo their calculation, using some hindsight. Let us first recall from
Equations (5.39) and (5.40) that the expansion rate changed at the moment when
radiation and matter contributed equally to the energy density. For our calculation
we need to know this equality time,푡eq, and the temperature푇eq. The radiation energy
density is given by Equation (6.41):
휀r=(푔훾+ 3 푔휈)^1
2푎푇eq^4. (8.2)The energy density of matter at time푇eqis given by Equation (6.24), except that the
electron (e−and e+) energy needs to be averaged over the spectrum [Equation (6.28)].
We could in principle solve for푇eqby equating the radiation and matter densities,
휀r(푇eq)=휌m(푇eq). (8.3)We shall defer solving this to Section 8.4.
The transition epoch happens to be close to the recombination time [푧recin redshift,
see Equation (6.70)] and the LSS [푧LSSin redshift, see Equation (6.71)]. With their
values for푡eq,푇eqand푡 0 and Equation (5.39), Gamow, Alpher and Herman obtained a
prediction for the present temperature of the CMB:
푇 0 =푇eq(푡
eq
푡 0) 2 ∕ 3
= 2 .45 K. (8.4)
This is very close to the present-day observed value, as we shall see.
Discovery. Nobody paid much attention to the prediction of Gamovet al., because
the Big Bang theory was generally considered wildly speculative, and detection of the
predicted radiation was far beyond the technical capabilities existing at that time. In
particular, their prediction was not known toArno PenziasandRobert Wilsonwho,
in 1964, were testing a sensitive antenna intended for satellite communication. They
wanted to calibrate it in an environment free of all radiation, so they chose a wave-
length of휆= 0 .0735m in the relatively quiet window between the known emission
from the Galaxy at longer wavelengths and the emission at shorter wavelengths from
the Earth’s atmosphere. They also directed the antenna high above the galactic plane,
where scattered radiation from the Galaxy would be minimal.