Abstract
We have provided a numerical study of the influence of the resonant activesterile neutrino oscillations , on the primordial production of helium4. The evolution of the neutrino ensembles was followed selfconsistently with the evolution of the nucleons, using exact kinetic equations for the neutrino density matrix and the nucleon number densities in momentum space, from the time of neutrino decoupling till the freezeout of nucleons at 0.3 MeV.
The exact kinetic approach enabled us to study precisely the neutrino depletion, spectrum distortion and neutrino mixing generated asymmetry due to oscillations at each momentum mode, and to prove that their effect on nucleosynthesis is considerable.
We have calculated the dependence of the primordially produced helium4 on the oscillation parameters for the full range of mixing parameters of the model of oscillations with small mass differences eV. We have obtained isohelium contours on the plane. Cosmological constraints on oscillation parameters, more precise than the existing ones were extracted, due to the exact kinetic approach and the proper account for the neutrino spectrum distortion and the oscillations generated asymmetry.
Cosmological nucleosynthesis and activesterile neutrino oscillations with small mass differences:
The resonant case
D. P. Kirilova^{1}^{1}1Permanent address: Institute of Astronomy,
Bulgarian Academy of Sciences,
blvd. Tsarigradsko Shosse 72, 1784 Sofia, Bulgaria
and M. V. Chizhov^{2}^{2}2Permanent address: Centre for Space Research
and Technologies,
Faculty of Physics,
University of Sofia, 1164 Sofia, Bulgaria
The Abdus Salam International Centre for Theoretical Physics,
Strada Costiera 11, 34014 Trieste, Italy
To Emmanuil, Vassillen and Rosanna
1 Overview of neutrino oscillations and primordial nucleosynthesis
Nowadays several major experiments point to the existence of neutrino oscillations. Besides, the neutrino puzzles, namely, the solar neutrino deficit and the atmospheric neutrino anomaly, which are believed to be explained in terms of neutrino oscillations, are still with us. Therefore, it seems useful to obtain more precise cosmological constraints on neutrino oscillation parameters. Moreover, the very low values of mass differences, explored in the cosmological model we will discuss, are beyond the reach of present and near future experimental constraints. In the present work we explore the effect of resonant activesterile neutrino oscillations with small mass differences eV on the primordial production of helium4 and obtain precise cosmological constraints on the neutrino oscillation parameters.
The problem of neutrino oscillations and Big Bang Nucleosynthesis (BBN) has been discussed in numerous publications [1][24]. First cosmological constraints from BBN on oscillation parameters were obtained analytically in [8]. Then numerical calculations of the oscillations effect on primordial nucleosynthesis were made in [10]. In these works an account for the depletion of the neutrino number densities due to oscillations was provided, while the neutrinoantineutrino asymmetry and the distortion of the neutrino spectrum were neglected (see also [13]). The importance of the neutrino spectrum distortion for the BBN with oscillations was first noticed for the vacuum oscillation case in ref. [6]. First precise account of both neutrino spectrum distortion and the oscillations generated neutrinoantineutrino asymmetry effects in BBN calculations and cosmological constraints on oscillation parameters were provided in ref. [19].
The problem of activesterile nonresonant neutrino oscillations and the primordial helium4 production was thoroughly investigated and completed for the model of nonequilibrium oscillations with small mass differences [6, 19] in refs. [21, 22]. There we have analyzed numerically the role of oscillations in BBN using the exact kinetic equations for the neutrino density matrix and nucleon number densities in momentum space. The exact kinetic approach enabled us to reveal the important role of the spectrum distortion and neutrino population depletion. On the other hand, for the nonresonant case it was shown that the lepton asymmetry can be neglected in case initially it was of the order of the baryon one [21], while in case it was greater than 10 it may considerably influence BBN [22]. Precise constraints on the oscillation parameters by almost an order of magnitude better than the existing ones [10, 13] concerning the neutrino squared mass differences, were obtained due to the exact kinetic approach and selfconsistent account of the evolution of the neutrinos and the nucleons. The analytical fit to the exact constraints ^{3}^{3}3 The constraints are derived for primordial helium4 . Other isohelium contours and the corresponding constraints were calculated in refs. [21, 22] is:
(1) 
for eV.
It was noticed in ref. [19], that the resonant neutrino oscillations case is a much more complicated one, as far as rapid growth of asymmetry for certain sets of parameters is typical there [8, 16, 18, 19]. First detailed calculations of the BBN with resonant neutrino oscillations accounting for the asymmetry growth were provided in [19]. The phenomenon of the oscillationgenerated asymmetry growth was registered there and the calculations of the BBN were provided accounting both for the spectrum distortion and for the neutrino asymmetry dynamical evolution at each momentum mode. It was shown that following the behavior of the neutrinoantineutrino asymmetry at each momentum is important, particularly, when the distortion of the neutrino spectrum is considerable. As a result helium4 contour was obtained and precise constraints on the oscillation parameters were provided. They are better by almost an order of magnitude than the existing ones for the neutrino squared mass differences at large mixing angles.
In ref. [24] the effect of the neutrinomixinggenerated asymmetry was shown to be considerable – up to about relative decrease in helium4 in comparison with the case with oscillations but without the asymmetry account. Hence, a more profound study of the BBN with resonant oscillations, accounting properly for the important asymmetry effect is necessary. The purpose of the present work is to provide a more detail study of the BBN with resonant neutrino oscillations.
2 Nucleosynthesis with oscillating neutrinos
In the present work we expand and complete the original investigations [19, 24] of the asymmetry effect on primordial production of helium in the model of BBN with resonant neutrino oscillations, for the full parameter space of the nonequilibrium oscillations model [19].
We consider the case of activesterile neutrino oscillations with small mass differences, namely eV, described in detail elsewhere [19, 21], where the nonequilibrium effects are stronger and, therefore, it is less studied one till now. According to that model, oscillations proceed effectively after the active neutrino decoupling and till then the sterile neutrinos have not yet thermalized, so that their number density is negligible in comparison with the electron neutrino one. For simplicity we assume mixing just in the electron sector, ().
The set of kinetic equations describing simultaneously the evolution of the neutrino and antineutrino density matrix and and the evolution of the neutron number density in momentum space reads:
(2) 
(3) 
where , is the momentum of electron neutrino, stands for the number density of the interacting particles, is a phase space factor and is the amplitude of the corresponding process. The sign plus in front of corresponds to neutrino ensemble, while minus  to antineutrino ensemble.
The initial condition for the neutrino ensembles in the interaction basis is assumed of the form:
where .
It corresponds to the standard equilibrium distribution of active electron neutrinos, and an absence of the sterile ones. The initial values for the neutron, proton and electron number densities are their equilibrium values.
The first term in the right hand side of the equations (2) and (3) describes the effect of Universe expansion. The second term in (2) is responsible for neutrino oscillations, the third accounts for forward neutrino scattering off the medium and the last one accounts for the second order interaction effects of neutrinos with the medium. It is important for the nonequilibrium activesterile neutrino oscillations to provide simultaneous account of the different competing processes, namely: neutrino oscillations, Hubble expansion and weak interaction processes. is the free neutrino Hamiltonian. The ‘nonlocal’ term arises as an propagator effect, . is proportional to the fermion asymmetry of the plasma and is essentially expressed through the neutrino asymmetries , where and .
The neutron and proton number densities, used in the kinetic equations for neutrinos eq. (2), were substituted from the numerical calculations of eq. (3). On the other hand, and at each integration step of eq. (3) was taken from the simultaneously performed integration of the set of equations (2). I.e. we have selfconsistently followed the evolution of neutrino ensembles and the nucleons.
We account for the exact kinetics both of the neutrino and the neutronproton transition, essential for the helium4 synthesis. Besides, the equations follow neutrino evolution in momentum space, i.e. enabling to account accurately for the neutrino depletion, neutrino energy spectrum distortion and the dynamical evolution of the asymmetry.
Eq. (2) results into a set of coupled nonlinear integrodifferential equations with time dependent coefficients for the components of the density matrix of neutrinos: four equations for the components of the neutrino density matrix, and another four for the antineutrino density matrix for each momentum mode. However, due to conservation of the total neutrino number density in the discussed model, the number of the equations can be reduced to 6 equations for each momentum mode of neutrinos and antineutrinos.
The equations were integrated for the characteristic period from the electron neutrino decoupling at 2 MeV till the freezeout at 0.3 MeV. We have calculated the yields of primordially produced helium4 for the full range of the model’s parameters values, namely for ranging from to maximal mixing and eV eV. For smaller mixing parameters the effect on helium4 was shown to be negligible [19].
Our results are based on hundreds of combinations. The spectrum distribution we have usually described by 1000 bins. Mind, however, that for some sets of parameters, where rapid growth of asymmetry occurs, even 5000 bins do not give satisfactory good description of the great spectrum distortion and the rapid sign changing behavior of the asymmetry. Fortunately, we have estimated the effect of this numerical uncertainty on the calculated production of helium4 and found that it is much less than for the full oscillation parameters range.
Therefore, we are not discussing here the asymmetry behavior, but present only the results of our study concerning nucleosynthesis which are trustable. The analysis of the precise asymmetry evolution itself deserves further investigation. We are quite convinced by our studies, that, surely, the real physical behavior of the asymmetry should not be a function of the calculational parameters, such as different error control, step size, et cetera. (See, however, the opposite point of view on that question by Shi in [18]). According to us, such kind of a dependence on the calculational parameters points only to the unsatisfactory accuracy of the numerical calculations or of the calculational methods used. The question is even more complicated, as far as we have estimated that the neutrino evolution equations at resonance have high stiffness. Hence, the usual explicit numerical approach is not applicable for the description of the asymmetry evolution, especially, if the correct account for the spectrum spread of neutrino is provided. To solve the stiff equations numerically, implicit methods should be used. For 1000 bins of the spectrum a system of 6000 equations describing the neutrino density evolution should be solved simultaneously. And this is a hopeless task with our facilities now. We will discuss this question in more detail elsewhere.
3 Results and conclusions
The major effects, of the discussed resonant oscillations with small mass differences on helium4 production, are due to the depletion of the neutrino number densities, neutrino spectrum distortion and the neutrino asymmetry growth due to oscillations.
(a) Depletion of population due to oscillations:
As far as oscillations become effective when the number densities of are much greater than those of , the oscillations tend to reestablish the statistical equilibrium between different oscillating species. As a result decreases in comparison to its standard equilibrium value due to oscillations in favor of sterile neutrinos. The depletion of the electron neutrino number densities due to oscillations into sterile ones strongly affects the reactions rates. It leads to an effective decrease in the weak processes rates, and, hence, to an increase of the freezing temperature of the ratio and the corresponding overproduction of the primordially produced .
(b) Distortion of the energy distribution of neutrinos:
Neutrinos with different momenta begin to oscillate at different temperatures and with different amplitudes. First the low energy part of the spectrum is distorted, and later on this distortion concerns neutrinos with higher and higher energies. The effect of the distortion of the energy distribution of neutrinos on helium4 production is twofold. On one hand an average decrease of the energy of active neutrinos leads to a decrease of the weak reactions rate, and hence, to an increase in the freezing temperature and the produced helium. On the other hand, there exists an energy threshold for the reaction . So, in case when the energy of the relatively greater part of neutrinos becomes smaller than that threshold the freezing ratio decreases leading to a corresponding decrease of the primordially produced helium4 [25]. The numerical analysis showed that the total effect of the distortion of the energy distribution is an increase in the produced helium.
(c) Neutrino asymmetry:
Neutrino mixing generated asymmetry effect was found to be considerable. (See also ref. [24]). It was proven numerically, that in the case of small mass differences we discussed and naturally small initial asymmetry, the growth of the asymmetry is less than 4 orders of magnitude. Hence, beginning with asymmetries of the order of the baryon one, the asymmetry does not grow enough to influence directly the kinetics of the transitions. Consequently, the apparently great asymmetry effect (as illustrated in Fig. 2) is totally due to the indirect effects of the asymmetry on BBN. The maximal asymmetry effect is around ‘underproduction’ of in comparison with the case of BBN with oscillations but without the asymmetry account.
The total effect of oscillations, with the complete account of the asymmetry effects, is still overproduction of helium4, in comparison to the standard value, although considerably smaller at small mixing angles than in the calculations neglecting asymmetry. Therefore, nucleosynthesis constraints on the mixing parameters of neutrino are alleviated considerably due to the asymmetry effect.
From the numerical integration for the full range of oscillation parameters we have obtained the primordial helium yields . Some of the iso helium contours calculated in the discussed model of cosmological nucleosynthesis with resonant neutrino oscillations are presented on the plane in Fig. 1.
At present the primordial helium values extracted from observations differ considerably [26]. Therefore, we consider it useful to provide the exact calculations for various isohelium contours up to 0.26. Knowing more precisely the primordial helium4 value from observations, it will be possible to obtain the excluded region of the mixing parameters using the results of this survey. For example, assuming the ‘low’ observational value of primordial [26], the cosmologically excluded region for the oscillation parameters is situated on the plane to the right of the curve, which gives overproduction of helium in comparison with this observational value.
In Fig. 2 a comparison between the curves, corresponding to helium abundance , obtained in the present work and in previous works [10, 13], analyzing the resonant activesterile neutrino oscillations, is presented. In [10] the excluded regions for the neutrino mixing parameters were obtained from the requirement that the neutrino types should be less than 3.4: . The depletion effect was considered, while the neutrinoantineutrino asymmetry was neglected, and the distortion of the neutrino spectrum was not studied, instead the kinetic equations for neutrino mean number densities were used.
The dashed curve, presenting our results, in case the asymmetry effect was neglected, is in a good accordance with the results of Enqvist et al. [10], where asymmetry was neglected. The difference between the two curves shows explicitly the effect of the proper account of the neutrino spectrum spread and spectrum distortion, which was provided in our work. On the other hand, the difference between our curves, the solid and the dashed one, presents the net asymmetry effect.
The results of [13] differ both from the ones of ref. [10] and from our results. We consider them not correct. Our conclusion is not only based on the discrepancy between these results and those of other studies, but on the very fact that they are not consistent even between themselves concerning resonant and nonresonant case. As is well known from the analytical formulae the results for the resonant case coincide with those for the nonresonant one at maximal mixing. This fact is illustrated in Fig. 3 of resonant and nonresonant oscillations for all studies, except ref. [13].
As is seen from the isohelium contours for , for large mixing angles we exclude mass differences eV, which is an order of magnitude stronger constraint than the previously existing. This more stringent constraint for mass differences, obtained in our work for the region of large mixing angles is due to the more accurate kinetic approach we have used and to the precise account of neutrino depletion and energy distortion. On the other hand, at small mixing angles the account of the oscillations generated asymmetry leads to an alleviation of the BBN constraints in comparison with the previous works [10, 13]. It is easy to understand, as far as the asymmetry growth results in suppression of oscillations and, hence, less strongly pronounced overproduction of helium4 due to oscillations than in the case without the asymmetry account.
In conclusion, we have shown that both the spectrum distortion and
neutrino mixing generated asymmetry
should be accounted for properly in models of BBN with oscillations,
as far as their effect is considerable. We have calculated different
isohelium
contours for the resonant case of neutrino oscillations with small mass
differences. The cosmological constraints obtained are better by an order of
magnitude than the existing ones due to the exact kinetic approach both to the
neutrino evolution and to the nucleons freezeout.
4 Acknowledgements
We highly acknowledge the hospitality and the support of ICTP, Trieste, during the preparation of this work. We are grateful to S. RandjbarDaemi for the opportunity to work at ICTP. D.K. thanks D. Sciama for the possibility to participate into the astrophysics program this summer, which was essential for the successful completing of this survey.
D.K is glad to thank V. Semikoz for stimulating discussions and encouragement at the beginning of her research on matter oscillations and BBN in 1994. We would like also to express our gratitude to A. Dolgov for fruitful discussions and P. Christensen for the overall help. Part of the numerical calculations were provided using computational powers of the Theoretical Astrophysics Center, Copenhagen. This work was supported in part by the Danish National Research Foundation through its establishment of the Theoretical Astrophysics Center.
We are obliged also to Emmanuil, Vassillen and Rosanna for their great patience during our work on this theme.
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Figure captions
Figure 1. On the plane isohelium4
contours , 0.245, 0.25, 0.255 and 0.26,
calculated in the discussed model of BBN with activesterile
resonant neutrino oscillations are shown.
For fixed primordial helium4 value,
the area to the left of the corresponding curve gives
the allowed region of the oscillation parameters.
Figure 2. In the figure a comparison between the results concerning
primordial helium4 production, obtained in the present work and
previous works [10, 13], is presented.
The dashed curve shows our results in case without
asymmetry effect account. It is in a good accordance with the
results of Enqvist et al. [10],
where asymmetry was neglected.
The difference between the two curves shows explicitly the effect of
the proper account of the spectrum spread of neutrino,
which was provided in our work.
On the other hand, the difference between our curves, the solid and
the dashed one presents the net asymmetry effect.
The artistic curve of Shi et al. [13] is obviously inconsistent
with the results of other works and we will leave it without
a comment.
Figure 3. Combined isohelium contours , for the resonant oscillations, , and the nonresonant ones, , calculated in previous studies [10, 13, 21] and in this work, are presented. The discontinuity of the curve of Shi et al. [13] reveals the discrepancy between their own results for the resonant and nonresonant case.