Photon and Lepton Decoupling 131γ eA
vμ vτveBFigure 6.3A system of communicating vessels illustrating particles in thermal equilibrium
(from K. Kainulainen, unpublished research). At about 3.8MeV, valve A closes so that휈휇and휈휏
decouple. At 2.3MeV, valve B closes so that휈ealso decouples, leaving only e−and훾in thermal
contact.
At decoupling, the neutrinos are still relativistic, since they are so light (Table A.3).
Thus their energy distribution is given by the Fermi distribution, Equation (6.28), and
their temperature equals that of the photons,푇휈=푇훾, decreasing with the increasing
scale of the Universe as푎−^1. But the neutrinos do not participate in the reheating
process, and they do not share the entropy of the photons, so from now on they remain
colder than the photons:
푇휈=푇훾∕ 1. 40. (6.65)The number density푁휈of neutrinos can be calculated as in Equation (6.12) using
Equation (6.10), except that the−1 term in Equation (6.10) has to be replaced by+1,
which is required for fermions [see Equation (6.28)]. In the number density distribu-
tions [Equations (6.10) and (6.28)], we have ignored possible chemical potentials for
all fermions, which one can do for a thermal radiation background; for neutrinos it
is an unproven assumption that nonetheless appears in reasonable agreement with
their oscillation parameters.
The result is that푁휈is a factor of^34 times푁훾at the same temperature. Taking
the difference between temperatures푇휈and푇훾into account and noting from Equa-
tion (6.12) that푁훾is proportional to푇^3 , one finds
푁휈=^3
44
11
푁훾. (6.66)
After decoupling, the neutrino contribution to푔∗decreases because the ratio푇푖∕푇
in Equation (6.40) is now less than one. Thus the present value is
푔∗(푇 0 )= 2 + 3
7
4
(
4
11
) 4 ∕ 3
= 3. 36. (6.67)
The entropy density also depends on푔∗, but now the temperature dependence for
the neutrino contribution in Equation (6.40) is(푇푖∕푇)^3 rather than a power of four. The
effective degrees of freedom are in that case given by Equation (6.67) if the power^43
is replaced by 1. This curve is denoted푔∗Sin Figure 6.2.