A reexamination of the effects of non-zero degeneracies on Big Bang Nucleosynthesis is made. As previously noted, non-trivial alterations of the standard model conclusions can be induced only if excess lepton numbers L sub i, comparable to photon number densities eta sub tau, are assumed (where eta sub tau is approx. 3 times 10(exp 9) eta sub b). Furthermore, the required lepton number densities (L sub i eta sub tau) must be different for upsilon sub e than for upsilon sub mu and epsilon sub tau. It is shown that this loophole in the standard model of nucleosynthesis is robust and will not vanish as abundance and reaction rate determinations improve. However, it is also argued that theoretically (L sub e) approx. (L sub mu) approx. (L sub tau) approx. eta sub b is much less than eta sub tau which would preclude this loophole in standard unified models

Olive, K. A.

Schramm, David N.

Thomas, D.

Walker, T. P.

English

NASA Technical Reports Server

Fermi National Accelerator Laboratory
FERMILAB-Pub-91/192-A
July 1991
NEUTRINO DEGENERACY AND COSMOLOGICAL -_-
NUCLEOSYNTHESIS, REVISITED *
K.A. Olive t, D. N. Schramm °, D. Thomas t _ T.P. Walker °
tSchool of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455
"The University of Chicago, Chicago, Illinois 60637 USA
and Fermi National Accelerator Laboratory, Batavia, Illinois 60510, USA
° Department of Physics, The Ohio State University, Columbus, OH 43210
/ uf
ABSTRACT
A reexamination of the effects of non-zero degeneracies on Big Bang Nucleosynthesis is
made. As previously noted, non-triviM Mterations of the standard model conclusions can
be induced only if excess lepton numbers Li, comparable to photon number densities 777-,
are assumed (where _?_. _-, 3 × 1099b). Furthermore, the required lepton number densities
(L07_-) must be different for u, than for v;, and u_-. It is shown that this loophole in the
standard model of nucleosynthesis is robust and will not vanish as abundance and reaction
rate determinations improve. However, it is a/so argued that theoretically I Le I _ i L_, I
"_1 L_- I "" rib << 9_- which would preclude this loophole in standard unified models.
* In press, Phys. £ett. B
Operated by Universities Research Association Inc. under contract with the United States Department of Energy
https://ntrs.nasa.gov/search.jsp?R=19910019796 2020-03-19T17:26:07+00:00Z
The baryon number of the Universe is a quantity of fundamental interest. Baryon
number violation is a process of considerable importance in both cosmology and parti-
cle physics. In many grand unification models, as well as in recent studies of the elec-
troweak model at high temperatures, violation of B is accompanied by violation of lepton
number L, generally with conservation of some linear combination of the two (B - L
for instance, in minimal SUs). In cosmology, the baryon to photon ratio, though small
(q -= (ns - nB)/n.y = O(10-1°)), has a major influence on the primordial abundances of
the light elements; may determine when the Universe became matter-dominated; and is
an important parameter in theories of galaxy formation and dark matter determinations.
Clearly it is a quantity whose value we would like to know. Fortunately, standard big bang
nueleosynthesis can provide us with relatively tight constraints, rh0 - q/10 -1° = 2.8-4.0
[1]. If lepton number violation is of the same order as baryon number violation then it has
a negligible effect on nucleosynthesis. However, the question remains, can nucleosynthesis
give any constraints on the violation of lepton number? To answer this requires a rela-
tively straightforward adjustment to the usual calculation of the primordial abundances of
the light elements. Although large lepton asymmetries can cause difficulties with models
for galaxy formation [2], they do not preclude the possibility of galaxy formation when
additional physics (e.g. late time phase transitions [3]) is taken into account.
Since we know from redshift-luminosity measurements that the total density of the
Universe is not much more than critical we can infer a limit on baryon number from
_Bh_O -a = 3.53 x 107r/ (1)
where the Hubble parameter H0 = h0100 km s -1 Mpc -1 and 8 is the microwave back-
ground temperature in units of 2.75 K. If we assume that 12 = 1 (_total, not 12s) then age
of the Universe arguments constrain f_h02£ 0.25 [4] and thus r/x0_ 70. More generally, we
2
can safely assume that f_Bh 2 < 1 which implies ql0_ 280. In addition, the upper limit on
the fractional excess of charge in the universe (n, - np)/np is of order 10 -18 [5] so we can
assume that the lepton number due to electrons and positrons is small. Any additional
asymmetry in the electron family must therefore be in the form of neutrino degeneracy.
The approach we take then is to allow the three known neutrinos to have chemical poten-
tials (_i =-- Izv_/T,,) up to the value 54, at which point the energy density of that species
reaches the critical value for closure, Pc = 3H02/8_rG (see equation 3 below).
The exploration of neutrino chemical potentials for big bang nucleosynthesis has been
performed on several occasions in the last few years [6-10], however a number of the
relevant reaction rates in the calculation have been updated recently as a result of improved
measurements. In particular, since the recent calculation of Terasawa and Sato [8] an
improved measurement of the neutron half-life [11] dramatically narrows the uncertainty in
the n_p rates [1,12]. In addition the rates D(d,n)3He, D(d,p)T, 4He(t,7)rLi, rLi(p,a)aHe
and rBe(n,p)rLi have changed [13]. The purpose of this paper is to investigate the effect
of these improved input parameters on the baryon and lepton asymmetries of the Universe
to accurately assess the current situation.
In addition to improvements in the numerical computation of primordial abundances,
the comparison between the predicted abundances and the observational determinations
has become more stringent. For example the older bounds on the 4He abundance Yp <
0.25 or 0.254 have now been improved to Yp = 0.23 + 0.01, and the bound on the lithium
abundance which has ranged from rLi/H < (3 - 10) x 10 -1° is now down to rLi/H_ 1.4 x
10 -1° [1]. In what follows we will examine the effect of these changes on the neutrino
degeneracy loophole. We will conclude that the situation is not qualitatively different from
the previous conclusions, despite more stringent input data. Furthermore, we will show
that loopholes in the standard big bang nucleosynthesis conclusions about baryon density
3
will always persist if one allows the introduction of different but specific combinations of
values for _, and _, and/or _,-. While the values required will seem unphysically large, as
we will discuss, this loophole will nonetheless remain.
Our calculation is based on the code of Wagoner et al [14], with the n_p rates
calculated as described in [1]. We also use the most recent reaction rates of Caughlan and
Fowler [13] and the recent data on the neutron mean-fife. The observational limits that
we use on the light elements are those found in ref. [1]:
0.22 < Yp < 0.24 (2a)
1.8 × 10 -5 < Xd
Xd + XaHe _< 10 -4
1.0 × 10 -10 _< XTLi 4- XTBe _< 1.4 x 10 -1°
(2b)
(2c)
(2d)
where Yp is the mass fraction of 4 He and X refers to number density relative to hydrogen.
The most recent calculations [1] combined with the above observational bounds indicate
that the simplest version, ie. the standard model of nucleosynthesis, is consistent with
these observations. The consistency occurs when 2.8 < rll0 < 3.3(4.0). (The higher value
up to 7710 = 4.0 is allowed when the uncertainties in key 7Li rates are taken into account,
see eg. ref. [1,15].) Through the years, there have been numerous attempts at altering the
conclusions of the standard model, by (usually) complicating the input. For example early
claims of allowing _B = 1 because of inhomogeneities or decaying particles have largely
been dispelled. The nagging alternative of including neutrino degeneracy is the one we
discuss here.
Introducing neutrino degeneracy has two effects: (1) weak reaction rates are altered by
the change in the electron neutrino distribution function, thus changing the equilibrium
4
ratio of neutron to proton densities, and (2) the energy density of neutrinos increases,
speedingup the expansionof the universe. Thus _ehas two effects,whereas_uand _ only
affect the expansionrate. Furthermore, this meansthat _, and _r are interchangeableas
far astheir effectsare concerned.
The energydensity of neutrinos (and antineutrinos) can be evaluated analytically:
where we assumethe tau neutrino is effectively masslessfor T _.. 1 MeV. For a single
neutrino species, closure density is achieved for ( = 53.8 (for a Hubble parameter of
100 km s-lMpc-1). The lepton number of the universe is then
n(l') = 12_(3--'-'_ (7r2_i + _) (4)
= n(vi)- n( i)
(where ( is the Riemann zeta function and _(3) _ 1.202). If lepton number is conserved
then _i is a constant, except during e+--e - annihilation.
Since the chemical potentials for # and r neutrinos appear only in the expression
for the energy density, introducing (_,, _r is clearly equivalent to introducing additional
neutrino flavors. We can write
30 . 2 4
N,e ff = 3 + Z _2 (_i + _i/21r2) • (5)
where N,e ff parameterizes the neutrinos in terms of massless neutrino species. Following
the Z ° results from LEP we will naturally assume that there are three neutrino species
and that the tau neutrino is light. Note that although _i "" 50 corresponds to N_eff _ 105,
one could equivalently add 105 fermion degrees of freedom which do not couple to the Z °
and hence avoid the LEP limit.
The effect of neutrino degeneracy on the helium abundance can be readily understood:
Increasing _, or _r (with either sign) raises the expansion rate before nucleosynthesis. This
leads to a higher freeze-out temperature for weak interactions and hence an increased yield
of 4He. (This of course, is the same argument as for an increased number of neutrino
flavors.) The dramatic increase in the expansion rate (due to the equivalent of ,._ 105
neutrinos) is allowable when one notes that although raising _e also affects the expansion
rate (increasing it), in addition it changes the weak reaction rates. The neutron to proton
density ratio (in equilibrium) is given by
nlp=exp(-AmlT-_,) (6)
where Am -- m,, -- mp = 1.29 MeV, so increasing _e leads to a smaller value of n/p when
the weak rates freeze out and hence a smaller yield of 4He. Hence, for fixed r/ the two
quantities _e and _,/_- can be played off against each other without grossly affecting the
4He abundance. Changing the sign of _ has the opposite effect on the weak reaction
rates, but gives the same contribution to the energy density. As r/increases so does 4He
production as the nuclear reactions producing 4He become more efficient relative to the
expansion rate. Increasing the expansion rate (by increasing _e,_,,r) and decreasing (n/p)F
(by increasing _e) can readily compensate for an increase in r/.
D+SHe and 7Li production are far less sensitive to n/p at freeze-out as their abundance
is primarily determined by competition between nuclear reaction rates and the expansion
rate. The longer the nuclear rates are in equilibrium, the more D and 3He are destroyed.
For rll0 _ 3 (£ 3) the production of mass 7 nuclei (TBe, 7Li) increases (decreases) with
increasing rI. Thus an increase in _,_,r can always compensate for an increase in 7/. At
fixed 7/ relatively large increases in _e (driving n/p ---* 0 exponentially) are necessary to
compensate increases in _,,. In the case of D+ZHe, the increase in _ shuts down D
6
and 3He production. For r/10_ 3 the increase in _ shuts off 7Be(n,p), the destruction
channel for mass 7, and results in more 7Li. For r/10_ 3 an increase in _e results in less
TLi production (via 4He(t,7)) and greater destruction (via 7Li(p,a))--the net result being
a decrease in 7Li.
In figures la-d we show the regions of the _-_, plane (taking _ = _,) allowed by
the observational limits for r/10 = 2.8, 3.3, 10,280 and for 7", = 889.6 sec. The curves in
the figures are iso-abundance curves at the observational bounds given in (2a-2d). The
allowed region is shown in bold. For r/10 < 2.7 the region in which the 4He limits are
satisfied has Xd + X3He > 10 -4.
At 7710 ---- 2.7 we are able to meet the constraints for _, = _r = 0 and _e = 0.1. As
we increase r] the yields of deuterium and 3He begin to drop and at r]10 -- 2.8 we are able
to have _ = _, = _,- = 0 (fig. la). (If we increase _, sufficiently (to ,-, 25) the increased
expansion rate reduces the time available for nucleosynthesis and brings Yp back down to
-,, 0.24 [16]. However the D and 3He abundances continue to rise, so this region is still
ruled Out. This corresponds to ,,, 18000 neutrino flavors [17].) r/10 = 2.8 corresponds to the
lower limit of ref. [1]. Here the limits on the chemical potentials are [_[ < 0.6 (if _, - _,-)
and -0.02 < _e < 0.1. The lepton numbers are then constrained by [L,,[ < 0.15 and
-0.0050 < L_ < 0.025. As 77rises further the deuterium and 3He abundances continue to
drop, while Yp begins to increase. (Raising ,7 causes nucleosynthesis to begin earlier, giving
a larger value of n/p at the onset, and hence increasing Yp.) The limits on XrLi -FXTBe also
begin to move to higher values of the chemical potentials, leaving the allowed region around
0g _e_ 0.1, 0g _,g 2 for ,710 = 3.3 (fig. lb). This is the highest value of ,7 for which the
origin falls within the allowed region. As we continue to raise ,7 the observationally allowed
region moves to higher values of _e and _l,. When we reach ,710 = 280 (where fib = 1
for H0 = 100 km s-lMpc -1) we find we require _e ,_ 1.6,_, _ 40 in order to satisfy the
observational constraints.
In fig. 2 we show the same contours as a function of qa0 and _e for _, = _ = 0. Here
it is clear that for _e = 0 the limits on the baryon to photon ratio are 2.8< r/10 < 3.3 in
agreement with ref. [1]. However a value of _e between -0.01 and +0.09 is still permissible.
We have seen therefore that the standard nucleosynthesis bound r/10 _< 3.3(4.0) can be
bypassed by introducing two new parameters and in fact allow for f_B = 1. It is amusing
to note that in addition to the limit on r/a0, the parameters _e and _,/_r are sufficient
for obliterating the cosmological bound on the number of neutrino flavors. Indeed by
choosing _, = _r = 0 and _e at its limiting value of -_ 1.6, the cosmological bound becomes
Nv_ 1.8 x 105. A further twist arises if for some reason one were compelled to take
_,,r "_ O(_e). In that event, an extra source of energy density comparable to that given
by the bound on Nv, Nv "-_O(105), is needed to account for the necessary expansion.
Despite the baroque nature of the preceeding discussion, the nucleosynthesis loophole
based on non-vanishing neutrino chemical potentials does not lead to any direct inconsis-
tencies, per se. Furthermore it is one that we do not forsee as disappearing, either because
of improved cross-section measurements or abundance determinations. This can be seen by
an examination of figs. la-d. The closest observational bound that could close the loophole
is the upper bound on (D+3He)/H. For (D+3He)/H£ 8 the window in the 7710 = 280 plot
closes, but then the standard model is also ruled out for any value of r/a0.
We do not however wish to leave the reader with the impression that we are advocat-
ing this route for achieving a large value for f/B, or that one should ignore the cosmological
bound on N_,. In principle it may be possible to have Li (_i) be non-zero, but any realistic
theory to date would only predict Li "_ O(r/). It would also be peculiar to have L_, or L_
much different from Le as would be required to have nucleosynthesis go through the loop-
hole [18]. Furthermore, unless the baryon asymmetry is generated very late, electroweak
baryon number violation which conserves B - L would require that the total lepton asym-
metry L = Le + L _ + Lr = r/ ( L = n L / n._ = n n / nT ) [19]. Hence, degenerate nucleosynthesis
which requires L_, L_,,_- >> _? and lL_,,rl > Le would necessitate a very special cancellation
to achieve L = r1. Thus while we cannot categorically eliminate large neutrino degeneracies
as a loophole for cosmological nucleosynthesis it is clear that they are exceedingly unlikely
on general theoretical grounds.
Acknowledgements
We would like to thank H.S. Kang, E.W. Kolb, G. Steigman and M.S. Turner for useful
discussions. This work was supported in part by the NSF (AST88-22595) amd NASA
(NAGW-1321) at Chicago, by the DOE and NASA (NAGW 1340) at Fermilab, and by
the DOE (DE-AC02-83ER40105) at Minnesota. The work of KAO is also supported in
part by an NSF PYI Award.
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I0
FIGURE CAPTIONS
la. Limiting contours in the _,-_e plane for 710 = 2.8, _r = _, and v, = 889.6 sec. The
limits shown are those of equations 2. The region of the plane allowed by all limits is
shown in bold.
lb. As fig. la but with 710 - 3.3.
lc. As fig. la but with 710 = 10.
ld. As fig. la but with 710 = 280 and on a larger scale.
2. The same limits in the _e-?10 plane with _. = _, = 0.
11
_e 1.o
o
I
D+3He =10 -4
H
I
7Li
-- =(1.0-1.4)x10
H
I I 1 f 1 1 I
-0.4
-0.6
-0.8
-1.0
0
111 0=2.8
I I I 1 I I I I I
1 2 3 4 5 6 7 8 9 10
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
0
I I 1
D+3He
=1 0 -4
H
I I I 1
1 2 3 4
l Ij
7Li =(1.0-1.4)x10 10
H
I 1 I I I I I
TI10=33
1 I I I I
5 6 7 8 9 10
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
D =1.8x10 5
H
0 2 3 4
1
5
D+3He =10 4
H
q10=10
I
7
I
9
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
I I t I
D =1.8x10 5
H
I I l
5 10 15 20
;7L--_t=(1.0-1.4)xl 0"
H
D+3He =10 -4
H
1] 10=280
1
25 30 35 40
1
45 50
_e o.1° _ _ ; t _ _ _ _ _ i
\ / _" D+3He -' "-4
7Li 10 _ _ ! _ _ -] U
_ t_ '_J yH _
0.08 _ =1.0x10
0.06
0.04 - 1 !
O.O2
o.oo ,I I I-
-0.02
-0.04 _ _'- "_ =1"8x105
-0.06
-0.08 _p,,,_=0 "'-
-0.10 I_ I I I I I I
I 3 I[''10 10


Neutrino degeneracy and cosmological nucleosynthesis, revisited

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