5.1. Degenerate BBN
There is another alternative to SBBN which, although currently
less favored, does have a venerable history: BBN in the presence
of a background of degenerate neutrinos. First, a brief
diversion to provide some perspective. In the very early
universe there were a large number of particle-antiparticle
pairs of all kinds. As the baryon-antibaryon pairs or,
their quark-antiquark precursors, annihilated, only the
baryon excess survived. This baryon number excess,
proportional to ,
is very small (
10-9).
It is reasonable, but by no means compulsory, to assume
that the lepton number asymmetry (between leptons and
antileptons) is also very small. Charge neutrality of the
universe ensures that the electron asymmetry is of the same
order as the baryon asymmetry. But, what of the asymmetry
among the several neutrino flavors?
Since the relic neutrino background has never been observed
directly, not much can be said about its asymmetry. However,
if there is an excess in the number of neutrinos compared to
antineutrinos (or, vice-versa), "neutrino degeneracy", the
total energy density in neutrinos (plus antineutrinos) is
increased. As a result, during the early, radiation-dominated
evolution of the universe,
' >
,
and the universal expansion rate increases (S > 1).
Constraints on how large S can be do lead to some weak
bounds on neutrino degeneracy (see, e.g.
Kang & Steigman 1992
and references therein). This effect occurs for degeneracy in
all neutrino flavors
(
e,
µ, and
).
For fixed baryon density, S > 1 leads to an increase in
D/H (less time to destroy D), more 4He (less time to transform
neutrons into protons), and a decrease in lithium (at high
there is less time to produce 7Be). Recall that for S = 1
(SBBN), an increase in
results in
less D (more rapid
destruction), which can compensate for S > 1. Similarly,
an increase in baryon density will increase the lithium yield
(more rapid production of 7Be), also tending to compensate
for S > 1. But, at higher
, more
4He is produced,
further exacerbating the effect of a more rapidly expanding universe.
However, electron-type neutrinos play a unique role in BBN,
mediating the neutron-proton transformations via the weak
interactions (see eq. 2.24). Suppose, for example, there
are more e than
e. If
µe is
the
e chemical
potential, then
e
µe / kT is the
"neutrino degeneracy parameter"; in this case,
e >
0. The excess of
e will
drive down the neutron-to-proton ratio, leading to a reduction in
the primordial 4He mass fraction. Thus, a
combination of three adjustable parameters,
,
N
, and
e may
be varied to "tune" the primordial abundances of D, 4He, and
7Li. In Kneller et al.
(2001; KSSW),
we chose a range of primordial abundances similar to those adopted here
(2
105(D/H)P
5; 0.23
YP
0.25;
1
1010(Li/H)P
4) and explored the
consistent ranges
of
,
e
0, and
N
0.
Our results are shown in Figure 16.
It is clear from Figure 16 that for a large
range in ,
a combination of
N
and
e
can be found so
that the BBN-predicted abundances will lie within our adopted
primordial abundance ranges. However, there are constraints
on
and
N
from the
CMB temperature fluctuation spectrum (see KSSW for details and further
references). Although the CMB temperature fluctuation spectrum is
insensitive to
e, it
will be modified by any changes in the universal expansion rate. While SBBN
(
N
= 0) is
consistent with the combined constraints from BBN and the
CMB (see Section 4.5) for
10
5.8
(
B
h2
0.021), values of
N
as large
as
N
6 are also
allowed (KSSW).