Primordially produced helium-4
in the presence of neutrino oscillations
D.R Kirilova
Institute of Astronomy, Bulgarian Academy of Sciences, blvd. Tsarigradsko Shosse 72, Sofia, Bulgaria
Abstract. The production of helium-4 during the cosmological nucleosynthesis in the presence of activesterile neutrino oscillations, ve «-)• V5, efficient after decoupling of electron neutrino, is analyzed. All known
oscillation effects on primordial nucleosynthesis, namely: increase of the effective degrees of freedom during
nucleosynthesis, neutrino spectrum distortion, depletion of electron neutrino number density and generation of
neutrino-antineutrino asymmetry, are precisely taken into account.
Primordially produced 4He abundance is calculated, in a self-consistent study of the kinetics of the nucleons
and the oscillating neutrinos, for the full range of parameters of the oscillation model with small mass differences:
8m2 < 10~7 eV2. A considerable relative increase of helium-4, up to 14% for non-resonant oscillations and up
to 32% for resonant ones is registered for a certain interval of oscillations parameters values. Combined isohelium contours §Yp/Yp = 3%, 5%,7% for resonant and non-resonant oscillations are presented. Cosmological
constraints on oscillation parameters and on the sterile solar neutrino solutions are discussed.
INTRODUCTION
Neutrino oscillations present an indication of physics
beyond the standard electroweak model. Hence, it is
appropriate to study precisely the influence of neutrino
oscillations on BBN and constrain neutrino oscillations
parameters.
In this work we analyze 4He primordial production
taking into account all known effects of ve +* vs oscillations on the primordial synthesis of 4He. 2
Although disfavored by the recent combined analyses of neutrino oscillations experimental data as a preferred oscillation channels for solving the solar and atmospheric neutrino problems, sterile neutrinos are worth
considering because they are inevitable for the explanation of all the oscillation experiments data including
LSND results, they are considered as a possible minor
additional channels in both the solar and the atmospheric
oscillation cases, and besides they may play essential
role in structure formation and as a dark matter candidate in case they are massive.
We discuss the case when neutrino oscillations become effective after the electron neutrino decoupling
from the plasma (i.e for 8m2 < 10~7eV2). The primor-
In this work we investigate a modification of the standard
Big Bang Nucleosynthesis with electron-sterile neutrino
oscillations v^ «-)• vs. The positive indications for oscillations, obtained at the greatest neutrino experiments (SuperKamiokande, Soudan 2, LSND, etc.) and the recent
SNO result \ turned the subject of neutrino oscillations
into one of the hottest points of astrophysics and neutrino physics. The solar neutrino problem, atmospheric
neutrino anomaly and the positive results of LSND experiments can be naturally resolved by the phenomenon
of neutrino oscillations, implying nonzero neutrino mass
and mixing. Massive neutrinos may also play the role of
the hot dark matter component, needed for a successful
large-scale structure formation in the Universe.
On the other hand, Big Bang Nucleosynthesis (BBN)
is often used to probe the physical conditions of the
early Universe. Requirements for an agreement between
theoretically predicted and inferred from observational
data primordial abundances of light elements D, 3He,
4
He, 7Li, restricts physics beyond the standard models.
1
The electron neutrino flux measured by the charged current reaction
rate by SNO is 3.3o apart from the SuperKamiokande precision value
of the flux from elastic scattering reaction rate. This is considered an
indication of non-electron flavor component in the solar neutrino flux,
and may be interpreted in terms of neutrino oscillations.
2
The effect of flavor neutrino oscillations to BBN is negligibly small
because the temperatures and hence the densities of the neutrinos with
different flavors are almost equal. On the contrary, the effects of activeto-sterile neutrinos may be considerable, because the sterile neutrinos
may differ considerably (by temperature and density) from the active
CP598, Solar and Galactic Composition, edited by R. F. Wimmer-Schweingruber
© 2001 American Institute of Physics 0-7354-0042-3/017$ 18.00
405
dial production of 4He was calculated in the non-resonant
and resonant oscillation cases. In both cases strong overproduction of 4He was found possible - up to 14% and
32%, correspondingly. Updated iso-helium contours are
presented and the cosmological constraints on the oscillation parameters are discussed.
The oscillation effects on BBN and the required kinetic approach for their description are discussed in the
next section. The results on 4He primordially produced
abundance in the presence of oscillations and the cosmological constraints on oscillation parameters are discussed in the last section.
COSMOLOGICAL NUCLEOSYNTHESIS
WITH NEUTRINO OSCILLATIONS
Standard Cosmological Nucleosynthesis
According to the standard BBN, during the early hot
and dense epoch of the Universe, light elements D, 3He,
4
He, 7Li were synthesized successfully. 4He was the most
abundantly produced. Only negligible amounts of D,
3
He and 7Li were formed. Because of the low density,
growing Coulomb barriers and stability gaps at A=5,
A=8, the formation of larger nuclei was postponed until
the formation of stars several billions of years later. The
most reliable and abundant data are now available for
4
He, therefore it is the traditionally used element for the
analysis of the oscillations effect on nucleosynthesis.
The contemporary values for the mass fraction of
4
He, Yp, inferred from observational data, are 0.2380.245 (the systematic errors are supposed to be around
0.007) [1].
4
He is a result of a complex network of nuclear reactions, proceeding after the neutron-to-protons freezing.
It essentially depends on the freezing ratio (n/p)/. The
reactions:
ve + n o p + e
(D
maintained the equilibrium of nucleons at high temperature (T > 1 MeV). Their freeze-out occurred when in
the process of Universe cooling the rates of these weak
processes, Fw, became comparable and less than the expansion rate H(t):
Then nucleon number densities departed from their
thermal equilibrium values, and decreased slowly due to
the weak interactions of eq.(l) and the neutrons decays
that proceeded until the effective synthesis of D began.
406
The produced 4He is a strong function of the number of the effective degrees of freedom at BBN epoch,
geff, and the neutron mean lifetime TW, parametrizing
the weak interactions strength. Besides, 4He is a logarithmic function of the baryon-to-photon ratio T|, due to
the nuclear reactions dependence on nucleon densities,
i.e. Yp(geff^n,r[). Also it depends on the electron neutrino number density and spectrum, and on the neutrinoantineutrino asymmetry, which enter through Tw. In the
standard BBN model the number of neutrino flavors
equal to three, zero lepton asymmetry and equilibrium
neutrino number densities and spectrum distribution is
assumed:
Almost all neutrons, present at the beginning of nuclear
reactions, are sucked into 4He. So, the primordially produced mass fraction of 4He can be approximated by
Yp ~ 2(n/p)f/(l+n/p)fexp(-t/'cn).
Primordially produced 4He abundance Yp, is calculated
with great precision within the standard BBN model [2],
the theoretical uncertainty is less than 0.1% (\5Yp\ <
0.0002) within a wide range of r|. The best baryometer
now is deuterium. The predicted Yp at the best fit value
of T|, obtained from deuterium measurements, r| = 5 x
10"10, is Yp = 0.2462. The recent CMB anisotropy data
is also in remarkable agreement with the baryon density
determined from deuterium measurements and BBN.
So, the predicted primordial 4He abundance Yp is in
accordance with the observational data and is consistent
with other light element abundances.
Effects of neutrino oscillations on
nucleosynthesis
Cosmological nucleosynthesis with neutrino oscillations was studied in numerous publications [3]. In case
neutrino oscillations are present in the Universe primordial plasma, they may lead to changes in the Big Bang
Nucleosynthesis, depending on the oscillation channels
and the way they proceed.
The basic idea of oscillations is that mass eigenstates
V| are distinct from the flavor eigenstates v/:
Then in the simple two-neutrino oscillation case, the
probability to find at a distance / a given neutrino type
in an initially homogeneous neutrino beam of the same
type is:
V^T
where 8m2 (the neutrino squared mass difference) and $
(the oscillations mixing angle) are the oscillation parameters. E is the neutrino energy.
The medium distinguishes between different neutrino
types due to their different interactions with fermions
present in the hot plasma of BBN epoch. This leads to
different potentials for different neutrino types. In the
adiabatic case the effect of the medium can be described
by introducing matter oscillation parameters that are expressed through the vacuum ones and through the characteristics of the medium. The matter mixing angle is then
sin2 ®m = sin2
in case they have already decoupled. This leads to faster
Universe expansion H(t) ~ geff, and to earlier n/pfreezing, Tf ~ (geff)1/6, at times when neutrons were
more abundant [4, 5]:
n/p ~ exp(-(/wn - mp)/Tf)
This effect leads to 4He overproduction. However, observational data on helium allows not more than one additional neutrino type, therefore forbids efficient production of sterile neutrinos due to oscillations.
• distorting the neutrino spectrum
[sin2 fl + (Q T L - cos2#)2
where Q = -M2r4/(8m2M2,), L - -aET3La/(5m2),
La is expressed through the fermion asymmetries of the
plasma, a and b are positive constants different for the
different neutrino types, —L corresponds to the neutrino
and +L to the antineutrino case.
Although in general the medium suppresses oscillations by decreasing their amplitude, there also exists a
possibility of enhanced oscillations transfer, in case a resonant condition between the parameters of the medium
and the oscillations parameters holds:
Then the mixing in matter becomes maximal, independently of the value of the vacuum mixing angle.
At BBN epoch with the cooling of the Universe, an interesting interplay between the two terms is observed. At
high temperatures when \Q\ > |L|, 8m2 > 0 corresponds
to a non-resonant case, while 8m2 < 0 corresponds to
a resonant case, and the resonance holds in both neutrino and antineutrino sectors. At low temperatures, when
| Q | < \L|, the resonance is possible either for neutrinos in
the case 8m2 > 0 or for antineutrinos in the case 8m2 < 0.
Oscillations are capable to shift neutrino number
densities and spectrum from their equilibrium values.
Besides, oscillations may change neutrino-antineutrino
asymmetry and excite additional neutrino types. Thus,
the presence of neutrino oscillations invalidates the main
BBN assumptions about three neutrino flavors, zero lepton asymmetry and equilibrium neutrino number densities and energy distribution.
Through these effects oscillations affect the expansion
rate and the weak interaction rates. Shifting particle densities and energy spectrum of the electron neutrinos from
their equilibrium values, oscillations directly influence
the kinetics of nucleons during the weak freeze-out.
Hence, neutrino oscillations may effect primordial nucleosynthesis by
Since oscillation rate is energy dependent F ~ 8m2 /E
the low energy neutrinos start to oscillate first, and later
the oscillations become noticeable for the more energetic
neutrinos. Due to that, the neutrino spectrum may become strongly distorted. This effect was shown to be considerable for the active- sterile oscillations in vacuum [6]
and in matter [7, 8]. The effect was proved important in
the resonant oscillations case [9] and in the non-resonant
one [10].
The neutrino spectrum distortion effect on 4He primordial abundance has two aspects:
An average decrease of the energy of electron neutrinos leads to a decrease in rw, and subsequently increases
the freezing temperature and primordially produced 4He.
On the other hand, due to the threshold for the reaction
ve + p —)• n + e+ , when due to oscillations the energy of
the greater part of the neutrinos becomes smaller than
that threshold, the (n/p)f -ratio decreases leading to a
decrease of Yp.
The total effect is an overproduction of4He primordial
abundance.
• depleting the active neutrino number densities Nv
The effect was first studied for vacuum oscillations in
ref. [6] and for matter oscillations in ref. [1 1]. It was precisely calculated with the account of spectrum spread in
ref. [8]. Electron neutrino depletion slows down the weak
rates, Tw ~ NVE2, and leads to an earlier n/p-freezing
and an overproduction of 4He yield. The evolution of
electron neutrino depletion due to ve «-»• vs oscillations
was calculated and presented for different mass differences and for different mixing angles in ref. [10].
The net effect of spectrum distortion and neutrino
depletion on the production of 4He may be considerable
(see Fig.2 from ref. [9]) and much stronger (several times
larger) than the effect due to excitation of an additional
neutrino type.
• neutrino-antineutrino asymmetry growth
• exciting additional degrees of freedom
Active-sterile oscillations may keep sterile neutrinos in
thermal equilibrium [4] or bring them into equilibrium
407
The idea of neutrino-antineutrino asymmetry generation
during the resonant transfer of neutrinos was first pro-
posed in ref.[12]. Dynamically produced asymmetry exerts back effect to oscillating neutrino and may change its
oscillation pattern [7, 13]. Thus it may effect indirectly
BBN, even when its value is not sufficiently high to have
a direct kinetic effect on the synthesis of light elements.
For the case of small mass differences it was proven that
even very small asymmetries L « 0.01 considerably influence nucleosynthesis through oscillations, and therefore asymmetry effect on nucleosynthesis should be accounted for during asymmetry's full evolution.
Dynamically produced asymmetry suppresses oscillations at small mixing angles, leading to less overproduction of 4He. The effect of the oscillations generated asymmetry on 4He was analyzed for hundreds of
8m2 — $ combinations in ref. [9]. In the resonant case the
asymmetry effect on BBN was numerically analyzed and
shown to be considerable - up to about a 10% relative decrease in 4He compared with the case without asymmetry
account.
The required kinetic approach
It is impossible to describe analytically, without
some radical approximations, the non-equilibrium picture of active-sterile neutrino oscillations, producing
non-equilibrium neutrino number densities, distorting
neutrino spectrum and generating neutrino-antineutrino
asymmetry.
We have provided self-consistent analysis of the evolution of the nucleons and the oscillating neutrinos in the
high temperature Universe. Exact kinetic equations for
the nucleons and for the neutrino density matrix in momentum space [8] were used. This allowed to describe
precisely the spectrum distortion, neutrino depletion and
neutrino asymmetry and its back effect at each neutrino
momentum.
The equation for the neutron number densities in momentum space nn reads:
tions (entering through p/z and PLL), Hubble expansion
(first term) and weak interaction processes (next terms).
The numerical analysis was performed for the temperature interval [2 MeV,0.3 MeV].
HELIUM-4 OVERPRODUCTION DUE
TO VE ++ vs NEUTRINO OSCILLATIONS
Main results
We have calculated precisely the w/p-freezing, which
is essential for the production of 4He, till temperature
0.3 MeV. The analysis was provided for the non-resonant
case in ref. [10] and for the resonant case in ref. [9]. Hundreds of 8m2 — $ combinations were explored. The neutron decay was accounted adiabatically till the beginning
of nuclear reactions at about 0.09 MeV.
The overproduction of the primordial 4He, §Yp =
yosc — Y in the presence of v ++ v oscillations was
I
p
e
s
P
calculated for the full set of oscillations parameters of
the model: all mixing angles $ and mass differences
Sw2 < 10~7 eV2.
The neutron-to-nucleons freezing ratio Xj[ = (n/(n +
p))f as a function of neutrino mass differences for different mixings is shown in Fig.l for 8m2 > 0. In the
(dnn/dt) = Hpn (dnn/dpn) +
FIGURE 1. Neutron number density relative to nucleons as
a function of the mass differences at different mixing angles.
-nnpLL(l - ne-)}
-ne+)].
(3)
where dQ(/, j, k) is a phase space factor and & is the amplitude of the corresponding process, p/z and p/z at each
integration step of eq. (3) are taken from the simultaneously performed integration of the set of equations for
neutrino density matrix.
The equation provides a simultaneous account of the
different competing processes, namely: neutrino oscilla-
408
non-resonance case the oscillations effects become very
small (less than 1%) for small mixings: as small as
sin22^ = 0.1 for 8m2 = 10~7 eV2, and for small mass
differences: 8m2 < 10~10 eV2 at maximal mixing. For
very small mass differences 8m2 < 10~n eV2, or at very
small mixing angles sin2 2$ < 10~3, the effect on nucleosynthesis is negligible.
The effect of oscillations in the non-resonant case is
maximal at maximal mixing and greatest mass differences. In Fig. 2 (the lower curve) the maximal relative increase in the primordial 4He as a function of neu-
40
35-
log((5m2[eV2])
FIGURE 2. Maximum primordial helium-4 abundance for
the resonant (upper curve) and the non-resonant oscillation case
(lower curve), as a function of the neutrino mass differences.
The non-resonant case is calculated at maximum mixing, while
in the resonant case the helium abundance is calculated at the
resonant mixing angle for the corresponding mass difference.
trino mass differences at maximal mixing: SY^/Yp =
(Y0T - Yp)/Yp = /(Sm2)|e^/4 is presented.
In the resonant case for a given 8m2 there exists some
resonant mixing angle, at which the oscillations are enhanced by the medium, and hence, the overproduction
of 4He is greater than that corresponding to the vacuum
maximal mixing angle. This behavior of the helium production on the mixing angle is illustrated on the r.h.s. of
Fig.3. The figure presents a combined plot (for the res35
onant and the non-resonant oscillation case) of 57n dependence on the mixing angle for 8m2 = 10~7 eV and
8m2 - 1(T8 eV2.
The upper curve in Fig.2 shows the maximal relative
increase SY^/Yp in the resonant oscillations case as
a function of mass differences, i.e. each maximum 4He
value corresponds to the resonant mixing angle for the
concrete mass difference: 7^*(Sw2,$g^). As can be
seen from Figs.2 and 3, a considerable overproduction
can be achieved: in the resonant case up to 32% and in
the non-resonant one - up to 14%.
The kinetic effect of v^ «-)• vs neutrino oscillations
comprises a major portion of the total effect - i.e. Yp overproduction is mainly a result of electron neutrino depletion and spectrum distortion due to neutrino oscillations,
which effect can be larger than the one corresponding to
an additional degree of freedom.
Cosmological constraints on oscillation
parameters
Observational data on primordial 4He abundance limit
the allowed oscillation parameters. Having in mind the
large systematic uncertainty in the observational values for Yp, we have calculated several iso-helium contours. The combined iso-helium contours for the nonresonant and the resonant case of electron-sterile oscillation parameters are shown in Fig. 4 corresponding to
different values of relative increase of helium-4, namely
§Yp = (Yosc-Yp)/Yp = 3%,5%,7%.
Assuming the conventional observational bound on
$Yp/Yp = 3%, the cosmologically excluded region is
situated above the 3% contour. The analytical fits to the
-7.0
30-
-7.5 -
2520-
-8.0 -
15-
-8.5 -
105-
-9.0 -3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Iog(tan2i?)
-9.5
FIGURE 3. The dependence of the relative increase of primordial helium on the mixing angle for the resonant (r.h.s.)
and non-resonant (l.h.s.) oscillation case. The upper curve corresponds to 5m2 = 10~7 eV2, the lower one to 5m2 = 10~8
eV2.
409
-1
log(sin
FIGURE 4. The iso-helium contours corresponding to 3%,
5% and 7% overproduction of primordial helium abundance.
LOW sterile solar solution is given by the closed dashed curves.
exact constraints on oscillation parameters are:
I am obliged to the Organizing Committee of the
SOHO-ACE Workshop, hosted by the Physikalisches Institut of the Universitat Bern, Bern, Switzerland, for the
financial support of my participation in the Workshop.
This work was finalized at the Abdus Salam International
Centre for Theoretical Physics, Trieste.
5m2 (sin2 2#)4 < 1.5 x 10~9eV2
5m2 > 0,
2
10
2
2
|5m < 8.2 x l(T eV
5m < 0, large fl.
LOW electron-sterile solar solution, obtained from the
analysis of the 1258 days SuperKamiokande experimental data [14], is shown in the figure by the closed dashed
curves.
These constraints on ve «-)• vs neutrino oscillations exclude the active-sterile LOW solution to the solar neutrino puzzle [15] in addition to the LMA solution, excluded in the pioneer works. Even in the case of very
high primordial helium-4 §Yp/Yp = 1%, sterile LOW solution is still partially excluded.
This result is in agreement with the recent global
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7.
CONCLUSIONS
We have analyzed the primordial production of 4He in
the presence of ve «->• vs oscillations with small mass differences. Enormous overproduction of 4He (up to 32%)
8.
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