**2.3. The SBBN-predicted Abundances**

The primordial abundances of D, ^{3}He, and
^{7}Li(^{7}Be) are rate limited,
depending sensitively on the competition between the nuclear reaction
rates (proportional to the nucleon density) and the universal expansion
rate. As a result, these nuclides are all potential baryometers. As
the Universe expands, the nucleon density decreases so it is useful to
compare it to that of the CBR photons:
*n*_{N} /
*n*_{}.
Since this ratio turns out to be very small, it is convenient to introduce

(6) |

where _{b}
is the ratio of the present values of the baryon and critical
densities and *h* is the present value of the Hubble parameter in
units of 100 km s^{-1} Mpc^{-1} . As the Universe
evolves (post-*e*^{±} annihilation)
this ratio is accurately preserved so that
at the time of
BBN should
be equal to its value today. Testing this relation over ten orders of
magnitude in redshift, over a timespan of some 10 billion years, can
provide a confirmation of, or pose a challenge to the standard model.

In contrast to the other light nuclides, the primordial abundance of
^{4}He (mass fraction Y) is relatively insensitive to the baryon
density, but since virtually all neutrons available at BBN are
incorporated in ^{4}He, Y does depend on the competition between
the weak interaction
rates (largely fixed by the accurately measured neutron lifetime) and
the universal expansion rate. The higher the nucleon density, the
earlier can the D bottleneck be breached. Since at early times there
are more neutrons (as a fraction of the nucleons), more ^{4}He
will be synthesized. This latter effect is responsible for a very slow
(logarithmic) increase in Y with
. Given the
standard model
relation between time and temperature and the measured nuclear and
weak cross sections and decay rates, the evolution of the light-nuclide
abundances may be calculated and the relic, primordial abundances
predicted as a function of the one free parameter, the nucleon
density or .
These predictions for SBBN are shown in Figure 1.

Not shown on Figure 1 are the relic abundances
of ^{6}Li,
^{9}Be, ^{10}B, and ^{11}B; for the same range
in , all of
them lie offscale, in the range 10^{-20} - 10^{-13}. The
results shown here are from the BBN code developed and refined over the
years by my colleagues at The Ohio State University (OSU). They are in
excellent agreement with the published results of the Chicago group
(Burles, Nollett, &
Turner 2001).
Notice that the abundances appear
in Figure 1 as bands. These reflect the theoretical
uncertainties in the predicted abundances. For the OSU code the errors
in D/H and ^{3}He/H are at the ~ 8% level, while they are much
larger, ~ 12%, for ^{7}Li.
Burles et al. (2001),
in a reanalysis of the relevant
published cross sections, have reduced the theoretical errors by roughly
a factor of 3 for D and ^{3}He and a factor of 2 for
^{7}Li. The reader may not notice the band shown for
^{4}He, since the uncertainty in Y, dominated by the very small
uncertainty in the neutron lifetime, is at only the ~ 0.2% level
(_{Y}
0.0005).

Based on the discussion above it is easy to understand the trends shown
in Figure 1. D and ^{3}He are burned to
^{4}He. The higher
the nucleon density, the faster this occurs, leaving behind fewer nuclei
of D or ^{3}He. The very slight increase of Y with
is largely due
to BBN starting earlier at higher nucleon density (more complete burning
of D, ^{3}H, and ^{3}He to ^{4}He) and higher
neutron-to-proton ratio (more neutrons, more ^{4}He). The
behavior of ^{7}Li is more interesting. At relatively low values of
_{10}
3, mass-7 is
largely synthesized as ^{7}Li [by
^{3}H(,
)
^{7}Li reactions], which is easily
destroyed in collisions with protons. So, as
increases at low
values, destruction is faster and ^{7}Li/H *decreases*. In
contrast, at relatively high values of
_{10}
3, mass-7 is
largely synthesized as ^{7}Be [via
^{3}He(,
)^{7}Be reactions], which is more
tightly bound than ^{7}Li and, therefore, harder to destroy. As
increases at high values, the abundance of ^{7}Be *increases*.
Later in the evolution of the Universe, when it is cooler and neutral
atoms begin to form, ^{7}Be will capture an electron and
-decay
to ^{7}Li.