100 The hot universe

the cross-section is reduced by the factor

1 −ne=

[

1 +exp(−e/T)

]− 1

to account for the Pauli exclusion principle. Given this factor, the substitution of
(3.99) and (3.97) into (3.98) gives

σnν

1 + 3 g^2 A

π

G^2 Fe^2 ve

[

1 +exp(−e/T)

]− 1

. (3.100)

Because the number density of the nucleons is negligible compared to the number
density of the light particles, the spectra of neutrinos and electrons are not signif-
icantly influenced by the above reactions and always remain thermal. Hence, the
nνinteractions occurring within a time intervaltin a given comoving volume
containingNnneutrons reduce the total number of neutrons by

Nn=−

(

∑

ν

σnνnνvνgν

)

Nnt, (3.101)

where

nν=

[

1 +exp(ν/Tν)

]− 1

is the neutrino occupation number andgνis the phase volume element (see
(3.26), whereV=g=1). The velocity of neutrinosvνis equal to the speed of
light:vν= 1.
It is useful to introduce the relative concentration of neutrons

Xn=

Nn

Nn+Np

=

nn

nn+np

. (3.102)

Taking into account that the total number of baryons,Nn+Np,is conserved, and
substituting (3.100) into (3.101), we find that the rate of change ofXndue to the
nνreaction is
(
dXn
dt

)

nν

=−λnνXn=−

1 + 3 g^2 A

2 π^3

G^2 FQ^5 J(1;∞)Xn, (3.103)

where

J(a;b)≡

∫b

a

√

1 −

(me/Q)^2

q^2

q^2 (q− 1 )^2 dq

(

1 +e

TQ

ν(q−^1 )

)(

1 +e−

QTq), (3.104)

and the integration variable is

q≡(ν/Q)+ 1 =e/Q.