Physical Foundations of Cosmology

3.5 Nucleosynthesis 101

Before electron–positron annihilation the temperatures of the electrons and neu-
trinos are equal, that is,T=Tν.To estimate the integral in (3.104) we note that
(me/Q)^2  0 .15 and expand the square root in the integrand, keeping only the first
two terms. Furthermore, ignoring the Pauli exclusion principle for the electrons or,
equivalently, neglecting the second term in the denominator, we can calculate the
resulting integrals and obtain

J(1;∞)

45 ζ( 5 )

2

(

Tν

Q

) 5

+

7 π^4

60

(

Tν

Q

) 4

+

3 ζ( 3 )

2

(

1 −

1

2

m^2 e

Q^2

)(

Tν

Q

) 3

. (3.105)

It is quite remarkable that this approximate expression reproduces the exact result
with very good accuracy at all relevant temperatures. For example, forTν/Q> 1 ,
the error is about 2%,improving to 1% or better forTν/Q< 1 .Substituting (3.105)
together with the values ofGFandQinto (3.103), and converting from Planckian
to physical units, we find

λnν 1. 63

(

Tν

Q

) 3 (

Tν

Q

+ 0. 25

) 2

s−^1. (3.106)

Further simplifications made to obtain this last expression do not spoil the accuracy;
at the temperatures relevant for freeze-out ,Tν≥ 0 .5 MeV, the error remains less
than 2%.

Problem 3.15Verify that the reaction rate forne+→pν ̄is equal to

λne=

1 + 3 g^2 A

2 π^3

G^2 FQ^5 J

(

−∞;−

me

Q

)

, (3.107)

whereJis the integral defined in (3.104). Check that ifTν=TandT>me,then
λneλnν.Consider the inverse reactionspe−→nνandpν ̄→ne+.Show that
forTν=Ttheir rates can be expressed through the rates of the direct reactions:

λpe=exp(−Q/T)λnν,λpν ̄=exp(−Q/T)λne. (3.108)

Freeze-outThe inverse reactions increase the neutron concentration at a rate
λp→nXp.The balance equation forXnis therefore

dXn

dt

=−λn→pXn+λp→nXp=−λn→p

(

1 +e−

QT)(

Xn−Xeqn

)

, (3.109)

whereλn→p≡λne+λnνandλp→n≡λpe+λpν ̄are the total rates of the direct
and inverse reactions respectively, and

Xeqn =

1

1 +exp(Q/T)

(3.110)