Physical Foundations of Cosmology

106 The hot universe

become efficient. Within the relevant temperature interval, 0.06 MeV to 0.09 MeV,
the experimentally measured rates of these reactions are

〈σv〉DD1=( 1 .3–2. 2 )× 10 −^17 cm^3 s−^1 ,

〈σv〉DD2=( 1 .2–2)× 10 −^17 cm^3 s−^1 , (3.124)

respectively. Due to reactions (3.123), the number of deuterium nuclei in a comoving
volume containingNDnuclei decreases during a time intervaltby

ND=−〈σv〉DDnDNDt. (3.125)

Rewriting this equation in terms of the concentration by weight,XD= 2 ND/NN,
we obtain

XD=−^12 λDDX^2 Dt, (3.126)

where

λDD=(〈σv〉DD1+〈σv〉DD2)nN 1. 3 × 105 K(T)TMeV^3 η 10 s−^1. (3.127)

The functionK(T)characterizes the temperature dependence of the reaction rate
and it changes from 1 to 0.6 as the temperature drops from 0.09 MeV to 0.06 MeV.
A substantial amount of the available deuterium is converted into helium-3 and
tritium within a cosmological timetonly if

|XD|

( 1

2

)

λDDXD^2 tXD. (3.128)

It follows that the deuterium bottleneck opens up when

X(bn)D 

2

λDDt



1. 2 × 10 −^5

η 10 TMeV

(

XD(bn)

), (3.129)

where we have used the time–temperature relation (3.88) withκ 1. 11 .From
(3.119), one can express the temperature as a function ofXD:

TMeV(XD)

0. 061

(

1 + 2. 7 × 10 −^2 ln(XD/η 10 )

). (3.130)

Substituting this expression into (3.129) and solving the resulting equation forX(bn)D
by the method of iteration for 10>η 10 > 10 −^1 ,we find

X(bn)D  1. 5 × 10 −^4 η− 101

(

1 − 7 × 10 −^2 lnη 10

)

. (3.131)