96 The hot universe
we have
σeνO( 1 )α^2 w
MW^4 ,Z(p 1 +p 2 )^2 , (3.90)whereαw 1 /29 is the weak fine structure constant andp 1 , 2 are the 4-momenta of
the colliding particles. The neutrinos decouple from the electrons when the collision
time,
tν(σeνne)−^1 O( 1 )αw−^2 MW^4 T−^5 , (3.91)becomes of order the cosmological timet, which, in turn, is related to the tem-
perature via (3.87).When deriving (3.91) we have assumed that the electrons are
relativistic and hence (p 1 +p 2 )^2 ∼T^2 andne∼T^3 .Comparing (3.91) to (3.87),
one finds that the electron neutrinosνedecouple at temperature
TνeO( 1 )αw−^2 /^3 MW^4 /^3. (3.92)The exact calculation shows that the numerical coefficient in this formula is not
much different from unity and henceTνe 1 .5 MeV.
At temperatures of order MeV, the number densities ofμ- andτ-leptons are
negligibly small and the only reactions enforcing thermal contact betweenμ- and
τ-neutrinos and the rest of matter are the elastic scatterings ofνμ,τon electrons
(eνμ,τ→eνμ,τ); these are entirely due toZ-boson exchange. As a consequence,
the cross-sections for these reactions are smaller than the total cross-section of
theeνeinteractions and theμ- andτ-neutrinos decouple earlier than the electron
neutrinos.
The most important conclusion from the above consideration is that all three
neutrino species thermally decouple before the electron–positron pairs begin to an-
nihilate atT∼me 0 .5 MeV. After decoupling, the neutrinos propagate without
further scatterings, preserving the Planckian spectrum. Their temperature decreases
in inverse proportion to the scale factor and is not influenced by the subsequente±
annihilation. The energy released in the electron–positron annihilation is thermal-
ized and as a result the radiation is “heated.” Therefore, the temperature of radiation
must be larger than the neutrino temperature. Let us calculate the radiation-to-
neutrino temperature ratio. After decoupling the neutrino entropy is conserved sep-
arately. The total entropy of the other components, which is dominated by radiation
and the electron–positron plasma, is also conserved. Hence the ratio
sγ+se±
sν