Physical Foundations of Cosmology

3.5 Nucleosynthesis 105

is the binding energy of the deuterium. We have parameterized the baryon-to-photon
ratio by

η 10 ≡ 1010 ×

nN

nγ

. (3.120)

This parameter is related to (^) b,the baryon contribution to the current critical
density, via
(^) bh^275  6. 53 × 10 −^3 η 10. (3.121)
At temperatures of orderBD,the abundanceXDis still extremely small and
even atT∼ 0 .5 MeV,for example, it is only about 2× 10 −^13 .One of the reasons
for this is the large number of energetic photons with>BD,which destroy the
deuterium. The number of such photons per deuterium nucleus is
nγ(>BD)
nD

∼

BD^2 Te−BD/T

nNXD

∼ 1010

1

η 10 XD

(

BD

T

) 2

e−BD/T, (3.122)

which becomes less than unity only atT< 0 .06 MeV. Therefore, we expect that
deuterium can constitute a significant fraction of baryonic matter only if the tem-
perature is about 0.06 MeV. In fact, according to (3.119), forη 10 ∼O( 1 )the equi-
librium deuterium abundance changes abruptly from 10−^5 to of order unity as the
temperature drops from 0.09 MeV to 0.06 MeV.
The rates of reactions converting deuterium into heavier elements are propor-
tional to the deuterium concentration and these reactions are strongly suppressed
untilXDhas grown to a substantial value. This delays the formation of the other
light elements, including^4 He.In fact, because of the large binding energy of^4 He
(28.3 MeV), theequilibriumhelium abundance would already be of order unity at
temperature 0.3 MeV. However, this does not happen and the helium abundance
is still negligible atT 0 .3 MeV because the rate of the deuterium reactions,
responsible for maintaining helium in chemical equilibrium with the nucleons, is
much smaller than the expansion rate at this time. As a result, the heavier elements
are chemically decoupled and present in completely negligible amounts despite
their large binding energies. Only protons, neutrons and deuterium are in chemical
equilibrium with each other. This situation is usually referred to as the “deuterium
bottleneck.”

Problem 3.17Derive the formula for the equilibrium concentration of^4 He and
verify that it is of order unity atT∼ 0 .3 MeV.

Let us determine when the deuterium bottleneck opens up. This occurs when the
main reactions converting deuterium into heavier elements,

(1) D+D→^3 He+n, (2) D+D→T+p, (3.123)