Updated BBN constraints on non-equilibrium active-sterile neutrino oscillations

Big Bang Nucleosynthesis (BBN) offers impressive agreement between the predicted by theory and the observationally obtained abundances of primordially produced light elements and therefore it is used as one of the most reliable probes of the physics beyond Standard Model. The highly accurate recent determination of the primordial Deuterium and Helium-4 allows to provide updated and strengthened BBN constraints on beyond Standard Model Physics. We have provided numerical analysis of more than hundred BBN models with nonequilibrium electron-sterile neutrino oscillations with different oscillation parameters and we have constructed isohelium contours based on 1% and 3% Helium-4 uncertainty and different initial population of the sterile neutrino state. We present here updated and more strict BBN constraints on the parameters of non-equilibrium active-sterile neutrino oscillations considering several models of beyond Standard Model Physics.


Introduction
We discuss a Beyond Standard Model (BSM) of Big Bang nucleosynthesis with non-equilibrium electron-sterile neutrino oscillations.The case of late neutrino oscillations, that become effective after the decoupling of electron neutrino from the early Universe plasma is considered.The following connection between the parameters of these oscillations, namely mass difference and mixing, holds 4 2  ≤ 10 −7 (See [1] for detail description of that model of neutrino oscillations).These neutrino oscillations lead to depletion of the electron neutrino state and filling of the sterile neutrino state.Also, for many sets of oscillation parameters this type of oscillations cause considerable distortion of the electron neutrino energy distribution from its Fermi-Dirac form, relevant for the standard BBN (see refs.[2,3]).Both effects lead to reduced weak reaction rates of nucleons interactions in comparison with the standard BBN case and cause an overproduction of primordially produced He-4.Thus, using the observational data for primordial He-4 it is possible to put cosmological constraints on neutrino oscillations parameters.
The description of the evolution of the electron neutrino, participating in these non-equilibrium oscillations with depleted number density and distorted energy distribution in the expanding Universe is a complicated mathematical and numerical task.We discuss the governing equations describing the evolution of neutrino in the next section.In section 3 the contemporary model of BBN and the better precision of the determination of the primordially produced He-4 is briefly described.Then in subsection 3.1 we discuss the production of He-4 in BBN model with the non-equilibrium electron-sterile neutrino oscillations and the cosmological constraints on oscillations parameters that have been obtained as a result of our numerical analysis.Section 4 describes the shift of these constraints in case the sterile neutrino state was not initially empty.The last section presents the conclusions.

Description of  evolution in presence of late е  s
The evolution of the oscillating electron and sterile neutrino, taking into account at the same time the expansion of the Universe, neutrino oscillations and neutrino forward scattering, is described by the following equations for the ensemble of neutrino and antineutrino, correspondingly: The first term describes the expansion, the second -neutrino oscillations, the third describes forward neutrino scattering and the last term, which accounts for the weak interactions of neutrinos is negligible in this model of oscillations occurring after neutrino decoupling.We have discussed both the case of initially unpopulated sterile neutrino state, i.e. Ns=0 and partially filled sterile neutrino state 1>Ns>0.Both non-resonant m 2 <0 and resonant m 2 >0 neutrino oscillations cases were considered.We have provided numerical analysis of neutrino evolution during the BBN epoch for more than hundred sets of neutrino oscillations parameters.
The numerical analysis is hard because in case of late oscillations distortion of neutrino momentum distribution by oscillations can be considerable and should be accounted for.An accurate description of the neutrino momentum distribution is needed: 1000 bins were used to describe it in non-resonant case, and up to 10000 in the resonant case.Hence, the number of integro-differential equatins solved is multiplied by 1000 and 10000, which increases the calculation time considerably.
Moreover, in case of non-zero lepton asymmetry non-zero L term in the equations leads to coupling of the integro-differential equations and the numerical task becomes harder.L term leads to different neutrino and antineutrino evolution, as can be seen from the equations (see refs.[4,5]).
The evolution of oscillating neutrino was analyzed numerically in the temperature interval typical for the pre-BBN and BBN epoch and for all mixing angles and mass differences relevant for that model.Namely: The evolution of neutrino was studied simultaneously with the evolution of the nucleons in that pre-BBN and BBN period.Lately primordial 4 Не was obtained with more than 3% better accuracy mostly based on current observations of HeI λ10830 emission line in the brightest HII region in the extremely metal poor galaxy Leo P.An improved 4 Не abundance was derived: Yp= 0.2453±0.0034For more details see refs.[10,11,12,13].This more precisely obtained abundance of He-4 afford update and strengthening BBN constraints on beyond Standard Model physics.In particular we will discuss here the updated BBN constraints on neutrino oscillation parameters.

Updated BBN constraints on е  s oscillations parameters
In BBN with electron-sterile neutrino oscillations the distortion of neutrino spectrum and the electron neutrino number density may differ considerably from the values in standard BBN and this may lead to different nucleon kinetics and modified He-4 production.We have numerically studied the evolution of nucleons during pre-BBN epoch with the presence of electron-sterile neutrino oscillations.Further the primordially produced He-4 was calculated for different sets of oscillation parameters.
We have constructed combined iso-helium contours with 1 and 3% overproduction of 4 Не for initially empty sterile state population (Ns=0) for non-resonant m 2 <0 and resonant m 2 >0 cases.All oscillations effects on BBN have been accounted for.In our calculations we have used the recent values on baryon density and the neutron life time.We have derived cosmological constraints on oscillations parameters m 2 and sin 2 2.
Figure 1 illustrates our results for iso-helium contours for 1-3% 4 Не overproduction.It clearly shows that BBN constraints become more stringent for 1% helium overproduction and they are about a half of order of magnitude strengthened than constraints for 3% helium overproduction at maximum mixing.
These BBN constraints are more stringent than experimental ones and update and strengthen the previously existing cosmological constraints (see refs.[14,15]).They excluded totally the large mixing angle and low mixing angle electron-sterile solutions to the solar neutrino problem.

The change of BBN constraints on е  s oscillations in case of non-zero initial population of the sterile neutrino
Initially present sterile neutrino has two effects on neutrino oscillations, a dynamical effect -it leads to the increase of the expansion rate of the Universe, and a kinetic effect -it suppresses the kinetic effects caused by the non-equilibrium neutrino oscillations.Hence, due to the interplay between these dynamical and kinetic effects additional inert population may increase or decrease overproduction of He-4, depending on which effect dominates.Therefore, BBN bounds have non-trivial behaviour regarding Ns: In case the dynamical effect dominates, the overproduction of 4 Не enhances and BBN constraints strengthen.When the kinetic effect dominates 4 Не overproduction decreases with Ns increase and constraints from BBN get relaxed.For more details see refs.[16,17,15].
We have implemented numerical analysis of BBN with late е  s oscillations in case of non-zero initial population of the sterile neutrino Ns=0.2,corresponding to the contemporary BBN and Cosmic Microwave Background (CMB) cosmological constraints on Ns.Namely, BBN does not allow a fully thermalized light inert state, Neff < 3.2.We have obtained BBN constraints on electron-sterile neutrino oscillations parameters corresponding to 1% He-4 overproduction and Ns=0.2.Preliminary results concerning that model were presented in ref. [18].
As illustrated in figure 2 the cosmological constraints are strengthened in comparison with the ones corresponding to initially empty sterile neutrino state, as it was expected on the basis of our previous analysis [16,15].
In figure 3 we present a comparison between isohelium contours for 1% 4 He overproduction with Ns=0.2 and 3% 4 He overproduction with Ns=0.5 respectively.As could be seen the initially partially populated sterile neutrino state bounds on е  s oscillations parameters become stronger with the improvement of primordial 4 Не determination accuracy.
These results are interesting also in connection with the suggested by neutrino oscillations experiments existence of eV sterile neutrino, which can be thermalized by oscillations with the electron neutrino [19,20,21,22].Figure 3.More stringent BBN constraints on electron-sterile neutrino oscillations parameters corresponding to Ns≠0 and contemporary precision of He-4 determination.The upper dash dotted curve corresponds to 3% He-4 overproduction and Ns=0, the dashed curve corresponds to 3% He-4 overproduction and Ns=0.5, the dotted curve corresponds to 1% He-4 overproduction and Ns=0 and the solid curve corresponds to 1% He-4 overproduction and Ns=0.2.

Conclusions
The mathematical and numerical description of late non-equilibrium electron-sterile neutrino oscillations and their cosmological influence is a difficult task mainly due to the distortion of the energy distribution of the electron neutrino caused by these oscillations.We have provided precise numerical analysis of these oscillations and their effect on BBN for hundreds sets of oscillation parameters.
The highly accurate recent determination of the primordial He-4 allows to provide updated and strengthened BBN constraints on beyond Standard Model Physics, and in particular on neutrino characteristics.We analyzed the model of BBN with non-equilibrium e  s neutrino oscillations and derived stringent constraints from Big Bang Nucleosynthesis on parameters of the neutrino oscillations corresponding to 1% accuracy of produced primordially He-4.
BBN constraints on the parameters of neutrino oscillations depend in complicated way on the sterile neutrino population in the Universe.We have analyzed BBN with non equilibrium e  s neutrino oscillations and initially not empty sterile neutrino Ns=0.2.This initial population of the sterile state is chosen because it corresponds to contemporary limits from BBN and CMB.It leads to an increase of the overproduction of He-4 and, hence, it strengthens the BBN constraints.We have calculated the shift of the BBN constraints on neutrino oscillation parameters in that case.
[6,7,8,9,1,3]ameters of BBN, namely baryon-to-photon ratio, the number of effective degrees of freedom during BBN and the neutron half life time, are now determined with good accuracy.Besides precise observational data on light elements D, He, Li produced primordially exist.There is an outstanding agreement between theoretically predicted and observationally derived abundances of primordially produced light elements.Therefore, contemporary BBN is among the most reliable precision probes for new physics during BBN epoch.In particular He-4, which is the most sensitive element to the expansion rate and to the nucleons kinetics, and the neutrino properties is traditionally used for obtaining constraints on BSM physics in the neutrino sector.For pioneer papers about BBN constraints see for example refs.[6,7,8,9,1,3].
He-4 production in presence of late е  s Contemporary Big Bang Nucleosynthesis is theoretically, experimentally and observationally well established model.Precise data on nuclear processes rates from laboratory experiments, at the energies relevant for BBN, are available.