The CMB Temperature 179one steradian of sky. In SI units this is 10−^9 Jm−^1 sr−^1 s−^1. This quantity is equivalent
to the intensity per unit frequency interval,퐼(휈). One can transform from d휆to d휈by
noting that퐼(휈)d휈=퐼(휆)d휆,fromwhich
퐼(휆)=휈^2
푐
퐼(휈). (8.9)
The relation between energy density휀rand total intensity, integrated over the spec-
trum, is
휀r=^4 휋
푐 ∫퐼(휈)d휈. (8.10)Energy and Entropy Density. Given the precise value of푇 0 in Equation (8.8), one
can determine several important quantities. From Equation (6.41) one can calculate
the present energy density of radiation
휀푟, 0 =1
2
푔∗푎S푇 04 = 2. 606 × 105 eV m−^3. (8.11)The corresponding density parameter then has the value
훺r=휀푟, 0
휌c= 2. 473 × 10 −^5 h−^2 , (8.12)using the value of휌cfrom Equation (1.31) and the compromise value퐻 0 = 0 .71. Obvi-
ously, the radiation energy is very small today and far from the value훺 0 =1 required
to close the Universe.
The present value of the entropy density is
푠=
4
3
휀푟, 0
kT=
4
3
푔∗S
2
푎S푇^4
kT= 2. 890 × 109 m−^3. (8.13)Recall [from the text immediately after Equation (6.67)] that the(푇휈∕푇)dependence
of푔∗Sis a power of three rather than a power of four, so the factor( 114 )^4 ∕^3 becomes
just 114 and푔∗Sbecomes 3.91.
The present number density of CMB photons is given directly by Equation (6.12):
푁훾=휁(^3 )
2
휋^2
(
kT
푐ℏ) 3
= 4. 11 × 108 photons m−^3. (8.14)Neutrino Number Density. Now that we know푇 0 and푁훾we can obtain the neutrino
temperature푇휈= 1 .949K from Equation (6.65) and the neutrino number density per
neutrino species from Equation (6.66),
푁휈=3
11
푁훾= 1. 12 × 108 neutrinos m−^3. (8.15)For three species of relic neutrinos with average mass⟨푚휈⟩, Equation (6.68) can be
used to cast the density parameter in the form
훺휈=3 ⟨푚휈⟩
94. 0 ℎ^2 eV