Physical Foundations of Cosmology

98 The hot universe

following estimate for the luminosity-to-mass ratio:

L

Mbar



1

4

1. 1 × 10 −^5 erg

(1. 7 × 10 −^24 gm)×(3. 2 × 1017 s)

 5

erg

gm s

 2. 5

L

M

,

whereMandLare the solar mass and luminosity respectively. However, the ob-
servedL/Mbar≤ 0. 05 L/M,and therefore, if the luminosity of baryonic matter
in the past was not much larger than at present, less than 0.5% of^4 He can be fused
in stars.
The only plausible explanation of the helium abundance is that it was produced in
the very early hot universe when the fusion energy constituted only a small fraction
of the total energy. The energy released was then thermalized and redshifted long
before the universe became transparent. It is obvious that a substantial amount of
helium cannot be formed before the temperature drops below the binding energy
∼28 MeV.Indeed, primordial nucleosynthesis took place at temperature roughly
0 .1 MeV, that is, a few minutes after the big bang. The amount of helium produced
depends on the availability of neutrons at this time, which, in turn, is determined by
the weak interactions maintaining the chemical equilibrium between neutrons and
protons. These weak interactions become inefficient when the temperature drops
below a few MeV and, as a consequence, the neutron-to-proton ratio “freezes out.”
Thus, the processes responsible for the chemical abundances of primordial elements
began seconds after the big bang and continued for the next several minutes.
In this section, we use analytical methods to calculate the abundances of the light
primordial elements. Although more precise results are obtained with computer
codes, the quasi-equilibrium approximation used here reproduces the numerical
results with surprisingly good accuracy. In addition, the analytical methods allow
us to understand why and how the primordial abundances depend on cosmological
parameters.

3.5.1 Freeze-out of neutrons

We begin with the calculation of the neutron freeze-out concentration. The main
processes responsible for the chemical equilibrium between protons and neutrons
are the weak interaction reactions:

n+ν
p+e−, n+e+
p+ν. (3.96)

Hereνalways refers to the electron neutrino. To calculate the reaction rates, we can
use Fermi theory according to which the cross-sections can be expressed in terms
of the matrix element for the four-fermion interaction represented in Figure 3.5:

|M|^2 = 16

(

1 + 3 g^2 A

)

G^2 F(pn·pν)(pp·pe), (3.97)