Physical Foundations of Cosmology

114 The hot universe

It is convenient to rewrite these equations using a temperature variable instead of a
time variable (see (3.88)). Substituting the explicit value forλDDfrom (3.127) then
gives

dXn

dTMeV

=αη 10

(

R 1 Xn−X^2 D

)

, (3.151)

dXD

dTMeV

= 4 αη 10

(

X^2 D+R 2 XD−^12 R 1 Xn

)

, (3.152)

where

α≡α(T)= 0. 86 × 105 K(T)

and the coefficientK(T)describes the temperature dependence of〈σv〉DD.Its value
changes from 1 to 0.5 when the temperature drops from 0.09 MeV to 0.04 MeV.
Over the same temperature interval the coefficientsR 1 andR 2 are

R 1 ≡ 4 Xp

λpn

λDD

(3–8)× 10 −^3 , R 2 ≡ 2 Xp

λpD

λDD

( 2 .5–2. 3 )× 10 −^5 , (3.153)

where the experimental value for the ratio of the corresponding reaction rates has
been used. The system of equations (3.151) and (3.152) has attractor solutions.
First we consider the initial stage of nucleosynthesis whenXDXn.It turns
out that in this case the deuterium concentration satisfies the quasi-equilibrium
condition and we can setdXD/dT≈0 in (3.152). SinceR 2 R 1 ,the termR 2 XD
is small compared toR 1 Xn,and it follows from (3.152) and (3.151) that

XD=

√

R 1 Xn

2

[

1 +O

(

XD

Xn

)]

. (3.154)

This solution is valid after the deuterium concentration reaches its maximal value
of order 10−^2 and begins to decrease (Figure 3.7). It fails as soon asXndrops to
XDand, at this time,Xn∼XD∼R 1. Note that, according to (3.154), the maximal
concentration of deuterium is equal toXD 10 −^2 forXn 0. 12 .This is in agree-
ment with the naive estimate derived earlier by comparing the rates ofpnand DD
reactions. Substituting (3.154) into (3.151), we obtain

dXn

dTMeV

^12 αη 10 R 1 Xn. (3.155)

In this regime the neutrons determine their own fate and also dictate the quasi-
equilibrium concentrations to the other elements, including deuterium. In other
words, they regulate the shut-off valves between the reservoirs in Figure 3.6. At
the beginning of nucleosynthesis, atT=TMeV(N),most of the neutrons are still free,