**2.4. Nonstandard BBN**

The predictions of the primordial abundance of ^{4}He depend
sensitively on the early expansion rate (the Hubble parameter *H*)
and on the amount - if any - of a
_{e} -
_{e} asymmetry
(the _{e} chemical
potential *µ*_{e} or the neutrino degeneracy parameter
_{e}). In
contrast to ^{4}He, the BBN-predicted abundances of D,
^{3}He and ^{7}Li are determined by the
competition between the various two-body production/destruction rates
and the universal expansion rate. As a result, the D, ^{3}He,
and ^{7}Li abundances are sensitive to the
post-*e*^{±} annihilation expansion rate, while
that of ^{4}He depends on *both* the pre- and
post-*e*^{±} annihilation expansion rates;
the former determines the "freeze-in" and the latter modulates the
importance of
-decay (see,
e.g.,
Kneller & Steigman
2003).
Also, the primordial abundances of D, ^{3}He, and
^{7}Li, while not entirely insensitive to neutrino
degeneracy, are much less affected by a nonzero
_{e}
(e.g.,
Kang & Steigman
1992).
Each of these nonstandard cases will be considered below.
Note that the abundances of at least two different relic nuclei are needed
to break the degeneracy between the baryon density and a possible
nonstandard expansion rate resulting from new physics or cosmology,
and/or a neutrino asymmetry.

**2.4.1. Additional Relativistic Energy
Density**

The most straightforward variation of SBBN is to consider the effect of
a nonstandard expansion rate *H'*
*H*. To quantify the
deviation from
the standard model it is convenient to introduce the "*expansion rate
factor*" (or speedup/slowdown factor) *S*, where

(7) |

Such a nonstandard expansion rate might result from the presence of
"extra" energy contributed by new, light (relativistic at BBN) particles
"*X*". These might, but need not, be additional flavors of active
or sterile neutrinos. For *X* particles that are decoupled, in the
sense that they do not share in the energy released by
*e*^{±} annihilation, it is convenient
to account for the extra contribution to the standard-model energy density
by normalizing it to that of an "equivalent" neutrino flavor
(Steigman et al. 1977),

(8) |

For SBBN,
*N*_{} = 0
(*N*_{}
3 +
*N*_{}) and for each
such additional "neutrino-like" particle (i.e. any two-component
fermion), if *T*_{X} =
*T*_{}, then
*N*_{} = 1; if
*X* should be a scalar,
*N*_{} =
4/7. However, it may well be that the *X* have decoupled even
earlier in
the evolution of the Universe and have failed to profit from the heating
when various other particle-antiparticle pairs annihilated (or unstable
particles decayed). In this case, the contribution to
*N*_{} from each
such particle will be < 1 (< 4/7). Henceforth we drop the *X*
subscript. Note that, in principle, we are considering any term in the
energy density that scales like "radiation" (i.e. decreases with the
expansion of the Universe as the fourth power of the scale factor). In
this sense, the modification to the usual Friedman equation due to higher
dimensional effects, as in the Randall-Sundrum model
(Randall & Sundrum
1999a,
b;
see also
Cline, Grojean, &
Servant 1999;
Binetruy et al. 2000;
Bratt et al. 2002),
may be included as well. The interest in this latter
case is that it permits the possibility of an apparent *negative*
contribution to the radiation density
(
*N*_{} < 0;
*S* < 1). For such a modification to the energy density, the
pre-*e*^{±} annihilation
energy density in Equation 1 is changed to

(9) |

Since any *extra* energy density
(
*N*_{} > 0)
speeds up the expansion of the Universe (*S* > 1), the
right-hand side of the time-temperature relation in Equation 3 is
smaller by the square root of the factor in parentheses in Equation 9.

(10) |

In the post-*e*^{±} annihilation Universe the extra
energy density is diluted by the heating of the photons, so that

(11) |

and

(12) |

While the abundances of D, ^{3}He, and ^{7}Li are most
sensitive to the baryon density
(), the
^{4}He mass fraction (Y) provides
the best probe of the expansion rate. This is illustrated in
Figure 2 where, in the
*N*_{} -
_{10}
plane, are shown
isoabundance contours for D/H and Y_{P} (the isoabundance curves
for ^{3}He/H and for ^{7}Li/H, omitted for clarity, are
similar in behavior to that of D/H). The trends illustrated in
Figure 2 are easy to understand in the context
of the discussion above. The higher the baryon density
(_{10}),
the faster primordial D is destroyed, so the relic abundance of
D is *anticorrelated* with
_{10}.
But, the faster the Universe expands
(
*N*_{} > 0),
the less time is available for D destruction, so D/H is positively,
albeit weakly, correlated with
*N*_{}. In
contrast to D (and to ^{3}He and ^{7}Li),
since the incorporation of all available neutrons into ^{4}He is
not limited by the nuclear reaction rates, the ^{4}He mass fraction
is relatively insensitive to the baryon density, but it is very
sensitive to both the pre- and post-*e*^{±}
annihilation expansion
rates (which control the neutron-to-proton ratio). The faster
the Universe expands, the more neutrons are available for ^{4}He.
The very slow increase of Y_{P} with
_{10}
is a reflection of the fact that for a higher baryon density, BBN begins
earlier, when there are more neutrons. As a result of these complementary
correlations, the pair of primordial abundances
*y*_{D}
10^{5}(*D* / *H*)_{P} and Y_{P}, the
^{4}He mass fraction, provide
observational constraints on both the baryon density
() and
on the universal expansion rate factor *S* (or on
*N*_{})
when the Universe was some 20 minutes old. Comparing these to
similar constraints from when the Universe was some 380 Kyr old,
provided by the *WMAP* observations of the CBR polarization and
the spectrum of temperature fluctuations, provides a test of
the consistency of the standard models of cosmology and of
particle physics and further constrains the allowed range of
the present-Universe baryon density (e.g.,
Barger et al. 2003a,
b;
Crotty, Lesgourgues,
& Pastor 2003;
Hannestad 2003;
Pierpaoli 2003).

The baryon-to-photon ratio provides a dimensionless measure of the
universal baryon asymmetry, which is very small
(
10^{-9}).
By charge neutrality the asymmetry in the charged leptons must also be
of this order. However, there are no observational constraints, save
those to be discussed here (see
Kang & Steigman
1992;
Kneller et al. 2001,
and further references therein), on the magnitude of any
asymmetry among the neutral leptons (neutrinos). A relatively small
asymmetry between electron type neutrinos and antineutrinos
(_{e}
10^{-2})
can have a significant impact on the early-Universe
ratio of neutrons to protons, thereby affecting the yields of the
light nuclides formed during BBN. The strongest effect is on the
BBN ^{4}He abundance, which is neutron limited. For
_{e} > 0,
there is an excess of neutrinos
(_{e}) over antineutrinos
(_{e}), and the
two-body reactions regulating the neutron-to-proton ratio (Eq. 5) drive
down the neutron abundance; the reverse is true for
_{e} <
0. The effect of a nonzero
_{e} asymmetry on the
relic abundances of the other light nuclides is much weaker. This is
illustrated in Figure 3,
which shows the D and ^{4}He isoabundance curves in the
_{e} -
_{10}
plane. The nearly horizontal ^{4}He curves reflect
the weak dependence of Y_{P} on the baryon density, along with its
significant dependence on the neutrino asymmetry. In contrast,
the nearly vertical D curves reveal the strong dependence of
*y*_{D}
on the baryon density and its weak dependence on any neutrino
asymmetry (^{3}He/H and ^{7}Li/H behave similarly:
strongly dependent
on , weakly
dependent on
_{e}).
This complementarity between *y*_{D} and Y_{P}
permits the pair {,
_{e}}
to be determined once the primordial abundances of D and ^{4}He are
inferred from the appropriate observational data.