94 The hot universe
This is not surprising because we need only a small excess of electrons to compen-
sate the electric charge of the protons.
Finally, let us estimate the ratio of the number densities of neutrons and protons
when they are still in chemical equilibrium with each other and with leptons. At the
beginning of this section, we found that chemical equilibrium impliesμp−μn=
μνe−μe.Using this relation together with (3.61), one immediately obtains
nn
np
=exp(
−
mn−mp+μνe−μe
T)
exp(
−
Q
T
)
, (3.84)
whereQ≡mn−mp 1 .293 MeV and we have neglectedμνeandμein the latter
equality. The relation above will be used to set up the initial conditions for primordial
nucleosynthesis.
3.4.2 Neutrino decoupling and electron–positron annihilation
At early times, the main contribution to the energy density comes from relativistic
particles. Neglecting the chemical potentials, (3.51) and (3.53) imply that their total
energy density is
εr=κT^4 , (3.85)where
κ=π^2
30(
gb+7
8
gf)
, (3.86)
andgbandgfare the total numbers of internal degrees of freedom of all relativistic
bosons and fermions respectively. Let us calculateκ in the universe when the
only relativistic particles in equilibrium are photons, electrons, the three neutrino
species and their corresponding antiparticles.Photons have two polarizations, and
sogb= 2 .Electrons have two internal degrees of freedom, but each type of neutrino
has only one because neutrinos are left-handed. The antiparticles double the total
number of fermionic degrees of freedom and, therefore,gf = 10 .Thus, in this
case,κ 3 .537. Every extra bosonic or fermionic degree of freedom changesκby
κb 0 .329 orκf 0 .288 respectively.
Comparing (3.85) to (1.75), we find the relation between the temperature and
the cosmological time in a flat, radiation-dominated universe:
t=(
3
32 πκ