3.5 Nucleosynthesis 119deuterium reaches its final abundance, and we can substitute into (3.173) the values
ofX 3 fHeandXTfobtained previously. Forη 10 1 ,the first term on the right hand side
in (3.173) dominates and, using the estimateXTf 10 −^5 ,we obtainX 7 fLi∼ 10 −^9.
For largerη 10 ,the tritium abundanceXTf is smaller and, consequently,X 7 fLide-
creases asη 10 grows, but only until the second term on the right hand side in
(3.173) starts to dominate. The minimum final^7 Li abundance,X 7 fLi∼ 10 −^10 ,is
reached forη 10 between 2 and 3 (see Figure 3.8).Then, further increase inη 10
causes the^7 Li abundance to rise. This rise is mostly due to the temperature depen-
dence ofr; forη 10 > 3 ,the freeze-out temperature is determined by the efficiency
of the^7 Benreaction, which in turn depends on the neutron concentration. In a
universe with high baryon density, the deuterium and free neutrons burn more effi-
ciently and disappear earlier (at a higher temperature) than in a universe with low
baryon density. Therefore, the^7 Li concentration freezes out at a higher temperature
at whichris larger. Note also that, forη 10 > 5 ,the^7 Benreaction becomes ineffi-
cient before^3 He reaches its freeze-out concentration, and hence, to estimateX 7 fLi
properly we have to substitute in (3.173) the actual value ofX^3 Heat^7 Li freeze-out,
which is larger thanX 3 fHe.Numerical calculations show that after passing through a
relatively deep minimum withX 7 fLi∼ 10 −^10 ,the lithium concentration comes back
to 10−^9 atη 10 10.
In summary, the trough in theX 7 fLi−η 10 curve is due to the competition of
two reactions. In a universe withη 10 < 3 ,most of the^7 Li is produced directly in
the^4 HeT reaction. Forη 10 > 3 ,the reaction^7 Benis more important and^7 Li is
produced mainly through the intermediate^7 Be reservoir.
Beryllium-7 is not so important from the observational point of view, so, simply
to gain a feeling for its abundance, we estimate it in the range 5>η 10 > 1 ,in which
(^7) Be freeze-out occurs after that of deuterium. The quasi-equilibrium solution for
free neutrons is valid at this time and, substitutingXnX^2 D/R 1 into (3.171), we
find
X 7 fBe=
7
12
R 1 X (^4) He
(
λ (^3) He (^4) He
λ^7 Ben
)
X 3 fHe
(
XDf) 2 ∼^10
− 12 X
f(^3) He
(
XDf
) 2 , (3.174)
where the experimental values for the ratios of the relevant reactions have been used.
In this case, the product of the corresponding ratios changes by a factor of 5 over
the relevant temperature interval, so (3.174) is merely an estimate. Forη 10 = 1 ,
we haveXDf 4 × 10 −^4 ,X 3 fHe 3 × 10 −^5 and, hence,X 7 fBe∼ 2. 5 × 10 −^10.
The observed light element abundances are in very good agreement with theo-
retical predictions, thus lending strong support to the standard cosmological model.
Observations suggest that 7>η 10 >3 at 95% confidence level.