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\hfill{\sf CERN-TH/2002-209}

\vspace*{2.cm}
\begin{center}
{\bf Neutrino Spectrum Distortion  Due to Oscillations
 and its BBN Effect}   

\vspace*{0.2cm}
Daniela Kirilova$^{\dagger}$\\[0.3cm]
{\it $^\dagger$Institute of Astronomy, Sofia, Bulgaria\\
and Theory Division, CERN, Geneva, Switzerland}
\end{center}
\vspace*{0.2cm}
\begin{abstract}

We  study the  distortion of electron neutrino 
energy spectrum  due to oscillations with the sterile neutrino 
 $\nu_e\leftrightarrow \nu_s$, 
and its influence on BBN 
 for 
different initial  populations of the  sterile state  $\delta N_s$ at 
the onset of oscillations. 
 Only the case of an initially empty  sterile state was  studied in  
previous publications. 

We obtain  the primordial abundance of He-$4$ in models with 
electron--sterile neutrino oscillations for  all possible $\delta N_s$:
 $0 \le \delta N_s \le 1$ and for the model of oscillations, effective 
after electron neutrino decoupling, for which the spectrum distortion 
effects on the neutron--proton transitions are the strongest. 
 
    It is found that   the spectrum 
distortion effect may be  the dominant one  not only for small 
 $\delta N_s$,  but also for big initial population of the sterile 
state. In the resonant case it may   
play a considerable role even for very big  $\delta N_s\sim 0.8$. 

We discuss cosmological constraints on neutrino mixing for small  $\delta 
N_s$.

 


\end{abstract}
 
\vspace*{0.5cm}
\section*{Introduction}


Sterile neutrinos $\nu_s$  
 may be present at the onset of 
the nucleosynthesis epoch.   There may be different 
reasons for their  production  --- they are 
naturally produced in GUT models~\cite{brahmachari},  in models with large extra 
dimensions~\cite{extradim} and Manyfold Universe models~\cite{Manyfold}, 
in mirror matter models~\cite{mirror}. 
They may be produced also in 
 $\nu_{\mu,\tau}\leftrightarrow \nu_s$ oscillations in the 
preceding epoch according to the most popular  4-neutrino mixing neutrino 
schemes.~\footnote{In recent years there was a big development 
in 4-neutrino mixing  
schemes. The most widely discussed models of  $\nu$-mixing 
 for explanation of 
 the solar, atmospheric and
LSND neutrino anomalies are 
 2+2 and 1+3 mixing schemes~\cite{mixing}.}

Atmospheric and LSND neutrino data require oscillations with 
maximal mixings and  mass 
differences $\delta m^2_{atm}\sim 10^{-3}$ eV$^2$ and  
$\delta m^2_{LSND}\sim 10^{-1}$ eV$^2$,  which are effective  
before the  BBN epoch. And although  a pure sterile channel solution is 
excluded for any of the neutrino anomalies, 
the analysis  of neutrino oscillation experimental data allows a certain,  
small fraction of sterile neutrino to 
participate into these oscillations.
So, in  many schemes owing  to  $\nu_{\mu}\leftrightarrow \nu_s$ 
or  $\nu_{\tau}\leftrightarrow \nu_s$
oscillations,  $\nu_s$ state is partially thermalized   
before the
nucleosynthesis epoch. 
 The degree of population of $\nu_s$ may be different, depending on the 
concrete model of $\nu_s$ production. 

In recent years, strong constraints on the sterile neutrino impact in 
oscillations, explaining atmospheric and solar neutrino anomalies, were
obtained from the analysis of experimental oscillations 
data~\cite{constraints,strumia,conca}.
There also exist stringent cosmological constraints on $\nu_s$ produced in 
oscillations, 
based on BBN nucleosynthesis of  $^4\!$He. 
For the case  $\nu_{\mu,\tau}\leftrightarrow \nu_s$ see for 
example ref.~\cite{dol}  
and for the   $\nu_e\leftrightarrow \nu_s$  see ref.~\cite{NPS}.
The  constraints on electron--sterile
neutrino oscillations~\cite{res,PRD}, excluded the active--sterile LOW 
solution
to the solar neutrino puzzle, in addition to the already 
excluded LMA solution and to the pure sterile solution to the atmospheric 
neutrino anomaly from pioneer works; see for example ref.\cite{bd}.
(For more detail about constraints on neutrino oscillations from cosmology, 
see  refs.~\cite{dubnastro, dolgov}.)

However, the cosmological constraints on  active-sterile mixing 
were obtained 
in a simple two-neutrino mixing scheme~\footnote{Except in 
ref.~\cite{bilenky}, which  discusses cosmological
constraints in a specific 
4-neutrino mixing scheme.}, 
and hence, the case when the sterile
neutrino 
state was assumed initially empty  at the epoch 
before  oscillations    
became effective  in the Universe evolution, $\delta N_s=N_{\nu}-3=0$. 
$N_{\nu}$ is the number of 
neutrino species in equilibrium. 
Since  the presence of a non-empty sterile
state before  oscillations was not considered in
previous
analyses of oscillations effects on the neutrino
spectrum  
distortion and their influence on
BBN, in this letter we will address this question.

We  omit the  assumption   $\delta N_s=0$ and explore the more 
general case  when initially $\delta N_s \ne 0$. 
\
 An initial  $\delta
N_s \ne 0$ present before $\nu_{\mu,\tau}\leftrightarrow \nu_s$   
will just lead to an earlier increase of the total energy density of the 
Universe, and it is straightforward to re-scale the existing constraints. 
In the  $\nu_e\leftrightarrow \nu_s$ oscillations case, however, the 
presence of 
$\nu_s$ at the onset of oscillations influences in addition 
  the kinetic 
effects of  $\nu_e\leftrightarrow \nu_s$ on BBN. Therefore we chose 
 $\nu_e\leftrightarrow \nu_s$  
oscillations case to explore  electron neutrino distortion due to oscillations 
and its influence on nucleons freezing and on  primordial  $^4\!$He,  
$Y_p$,  
for  $0 \le \delta N_s\le 1$.
As an application of our results on spectrum distortion effects, we 
present our preliminary results for the cases 
 $\delta N_s=0.1$ and  $\delta N_s=0.2$.


\section*{Spectrum distortion of electron neutrino \\and its kinetic 
effect}


We consider the degree of population of the sterile neutrino state  
as a free 
parameter and, varying its value in the range $[0,1]$ 
analyze the spectrum distortion of $\nu_e$. Next,  we 
calculate  the 
primordial 
abundance of  $^4\!$He and the cosmological constraints 
on the  $\nu_e\leftrightarrow \nu_s$ mixing.  
Further on we will denote by  $\delta N_s$  
the degree of population of the sterile neutrino state at  
active neutrino decoupling ($T\sim 2$ MeV).
 
The presence  of $\nu_s$ has the following two kinds of  effects on BBN:\\
(a) it leads to  {\it an increase of the energy density} of the
Universe, and \\
(b) through the depletion of electron neutrino number
density,  distortion of  the equilibrium spectrum of $\nu_e$, 
 and   production of
asymmetry
between neutrinos and anti-neutrinos, all due to oscillations,
it {\it changes the nucleons  kinetics}, which is  important for
$n/p$-freezing~\cite{dubnastro}.

(a) The first effect is usually described by  an increase
of the effective
number of
the energy
density degrees
of freedom $g=(30/\pi^2)(\rho/T^4)$. At the
BBN epoch  $g=10.75+7/4\delta N_s(T_{\nu}/T)^4$.
 Hence, this effect leads to a faster expansion rate $H\sim g^{1/2}$ and
higher freezing
temperature for nucleons $T_f\sim g^{1/6}$, when nucleons were more abundant:
$$
n/p\sim \exp\left[-\Delta m/T_f\right]
$$
 
This  
reflects  into
an overproduction of  $^4\!$He, since it strongly depends on the $n/p$-freezing 
ratio: $Y_p\sim 2 \exp(-\Delta m/T_f)/[1+ \exp(-\Delta m/T_f)]$,
where $\Delta m=m_n-m_p\sim1.3$ MeV is the neutron--proton mass 
difference.
Besides, larger $g$ leads to a shorter time for neutrons decay before
nucleosynthesis, which again leads to an increase in the produced
 $^4\!$He.
%higher \rho corresponds to smaller t - i.e. shorter decay period
This  effect on  BBN is well known~\cite{g}. The
approximate fit to the exact
calculations is:
       $\delta Y_p \sim 0.013\delta N_s$. 
 The maximum helium overproduction corresponding to $\delta N_s=1$ 
is $\sim 5\%$.


(b) The influence of the kinetic  effects of  oscillations
 on BBN is quite obvious, having in mind that
(i) oscillations take
place between
equilibrium electron neutrino and  less populated  sterile neutrino 
ensemble,
 that (ii) the oscillations probability is inversely  proportional to the
energy
of  neutrinos 
 $P \sim \delta m^2/E$, so that neutrinos with different momenta start oscillating
at different cosmic times, 
and that (iii) the proton density is bigger than the neutron one. 
 Due to
that,
the neutrino energy spectrum $n_{\nu}(E)$ may strongly deviate
from its equilibrium form~\cite{kir}. 

 In case oscillations take place after
the decoupling of active neutrinos,
a strong spectrum distortion for both  the electron neutrino and
the anti-neutrino is possible.
This spectrum distortion
  affects the kinetics of nucleons freezing - it leads to an earlier
$n/p$-freezing and an overproduction  of
 $^4\!$He  yield.

The effect  can be easily understood  having in mind that
the  distortion leads both to a {\it depletion of the active neutrino
number densities}  in favor of the sterile ones $N_{\nu}$:
$$
N_{\nu}\sim \int {\rm d}E E^2 n_{\nu}(E)
$$

\noindent and to a {\it decrease of the mean neutrino energy}.~\footnote{
The decrease of the electron neutrino energy due to
oscillations into low temperature sterile neutrinos,
has also an additional effect: Due to the threshold of
the reaction converting protons into neutrons,
when neutrinos have lower energy, protons are preferably produced,
which may lead to an under-production of  $^4\!$He~\cite{dk}.
However, this turns to be a minor effect.}
This 
reflects into  the weak rates, governing nucleons transitions during
neutrons freezing, lower than in the standard BBN model,
$\Gamma_{weak}\sim N_{\nu_e} \bar E_{\nu}^2$, and hence, to
 earlier freezing when neutrons were more abundant.
So, helium is   
over-produced.
We will denote the kinetic effects by  $\delta N_{kin}$
further on.

The kinetic effects (b) depend strongly on the initial population  of the
sterile neutrino at BBN.
Larger  $\delta N_s$  decreases the kinetic  
effects,
because the element of initial non-equilibrium between the active and the
sterile states is less expressed.
Hence for any specific value of  $\delta N_s$
it is necessary to provide a separate analysis.


In the case when $\delta N_s=1$~~~ $\nu_s$ are in
equilibrium (the sterile state is
as abundant as the electron one),   and hence 
the $n$--$p$ 
kinetics does not feel the oscillations,  $\delta N_{kin}=0$.  
 The final effect is  only due
to the energy increase, i.e.  $\delta N_{tot}=\delta N_s$.

For the case $\delta N_s=0$ the kinetic effect of oscillations  was
studied by
numerical analysis
for both  the resonant~\cite{res}  
 and non-resonant~\cite{PRD} oscillation cases.
In this case  the kinetic effects (b) for given fixed
mixing parameters
 are the highest,  $\delta N_{kin}^{max}$, as far as the non-equilibrium
element  --- the difference
between the sterile and active neutrino number densities at the beginning
of
oscillations --- is the greatest.
 The overproduction of $^4\!$He may be enormous:  up to $14\%$ in the
non-resonant oscillation case and up to $32\%$ in the
resonant one~\cite{bern}.
This corresponds effectively to more than $6$ additional neutrino states
 $\delta N_{kin}^{max}\sim 6$
 For  the case   $\delta N_s=0$ the kinetics effects are known and 
 stringent cosmological constraints on the
oscillation parameters
were obtained  on
the basis
of helium-overproduction~\cite{NPS}.


Here,  accounting for all the  effects (a) and (b), we calculate 
$Y_p$,
 for different  $\delta N_s$ values and different
sets of oscillation parameters  $Y_p(\delta N_s, \delta m^2,
sin^22\vartheta)$ and present the dependence of
 $\delta N_{kin}$ on  $\delta N_s$.



We have analyzed  the self-consistent evolution of the oscillating neutrino 
and the nucleons, beginning from the neutrino decoupling at $\sim 2$ 
MeV till the 
freezing of nucleons. We have followed the line of work described in 
detail in ref.~\cite{res},    
 omitting the assumption for negligible density 
of the sterile neutrinos at the onset of   $\nu_e\leftrightarrow
\nu_s$ oscillations.   

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 and  distorting   neutrino spectrum.
Therefore, we have provided a self-consistent analysis of the evolution of 
the
nucleons number densities  
 $n_n$ and the ones of the  oscillating neutrinos  $\rho$  in the high 
-temperature 
Universe. We have solved self-consistently the set of
the following coupled integro-differential equations.
The first equation describes the kinetics of the neutrino ensembles in
terms of the
density matrix of neutrino $\rho$ and anti-neutrino $\bar{\rho}$. The
second
equation describes the kinetic evolution of the neutrons.
 

\begin{eqnarray}
&&{\partial \rho(t) \over \partial t} =
H p_\nu~ {\partial \rho(t) \over \partial p_\nu} +
\nonumber\\
&&+ i \left[ {\cal H}_o, \rho(t) \right]
+i \sqrt{2} G_F \left(\pm {\cal L} - Q/M_W^2 \right)N_\gamma
\left[ \alpha, \rho(t) \right]
+ {\rm O}\left(G_F^2 \right),
\label{kin}
\end{eqnarray}
\begin{eqnarray}
&&\left(\partial n_n / \partial t \right)
 = H p_n~ \left(\partial n_n / \partial p_n \right) +
\nonumber\\
&& + \int {\rm d}\Omega(e^-,p,\nu) |{\cal A}(e^- p\to\nu n)|^2
\left[n_{e^-} n_p (1-\rho_{LL}) - n_n \rho_{LL} (1-n_{e^-})\right]
\nonumber\\
&& - \int {\rm d}\Omega(e^+,p,\tilde{\nu}) |{\cal A}(e^+n\to
p\tilde{\nu})|^2
\left[n_{e^+} n_n (1-\bar{\rho}_{LL}) - n_p \bar{\rho}_{LL}
(1-n_{e^+})\right].
\end{eqnarray}
where $\alpha_{ij}=U^*_{ie} U_{je}$,
$p_\nu$ is the momentum of electron neutrino,
 $n$ stands for the number density of the interacting particles,
${\rm d}\Omega(i,j,k)$ is a phase-space factor, and  ${\cal A}$ is the
amplitude of the corresponding process.
The plus sign  in front of ${\cal L}$ corresponds to the neutrino
ensemble, the  minus sign - to the anti-neutrino ensemble.

Mixing just in the electron sector is assumed: 
$\nu_i=U_{il}~\nu_l$ ($l=e,s$).
The initial condition for the 
neutrino ensembles in the interaction basis
is assumed to be of the form:
$$
{\cal \rho} = n_{\nu}^{eq}
\left( \begin{array}{cc}
1 & 0 \\
0 & S
\end{array} \right),
$$
where $n_{\nu}^{eq}=\exp(-E_{\nu}/T)/(1+\exp(-E_{\nu}/T))$, while $S$ 
measures the degree of population of the sterile state. 

${\cal H}_o$ is the free neutrino Hamiltonian.
The `non-local' term $Q$ arises as a $W/Z$ propagator effect,
$Q \sim E_\nu~T$.
${\cal L}$ is proportional to the fermion asymmetry of the plasma
and is essentially expressed through the neutrino asymmetries
${\cal L} \sim 2L_{\nu_e}+L_{\nu_\mu}+L_{\nu_\tau}$,
where
$L_{\mu,\tau} \sim (N_{\mu,\tau}-N_{\bar{\mu},\bar{\tau}})/ N_\gamma$
and $L_{\nu_e} \sim \int {\rm d}^3p (\rho_{LL}-\bar{\rho}_{LL})/N_\gamma$.

The equations are for the neutrino and neutron number densities in
momentum space. This allows us to account precisely for the spectrum
distortion effects due to oscillations.
They  provide a simultaneous account of the different
competing processes,
namely: neutrino oscillations  (entering through  $\rho_{LL}$ and
$\bar{\rho}_{LL}$), Universe expansion, nucleons transformations. 

The analysis was performed  for the full
set of oscillations parameters of the model: for all mixing angles
$\vartheta$ and
 mass differences
$\delta m^2 \le 10^{-7}$ eV$^2$.
The analyzed temperature interval was $[2.0, 0.3]$ MeV, because 
at  temperatures higher than $2$ MeV the
deviations from the standard BBN model without oscillations are
negligible in the discussed model of oscillations. We calculate precisely 
the $n/p$-freezing, which is the essential for the
production of helium, to temperature $0.3$ MeV, and account adiabatically
for the following decays of neutrons till the start of nuclear reactions, 
at about $0.1$ MeV.

In Figs. 1a-c the dependence of the energy spectrum distortion of the 
electron neutrino on the initial population of the sterile state is shown. 

%\begin{figure}[thb]
\hbox{\vspace{-0.2cm}}
\mbox{\hspace{2cm}}\epsfig{figure=danisp10.eps,width=10cm}\\

\hbox{\vspace{-1.5cm}}
{\bf Figure 1a:} {\small The figure illustrates the degree of 
distortion
of the electron neutrino energy spectrum $x^2\rho_{LL}(x)$, where
$x=E/T$  at a characteristic temperature  $1$ MeV,
caused by resonant oscillations with mass difference
$\delta m^2=10^{-7}$ eV$^2$ and $\sin^22\vartheta=0.1$
for different  initial sterile neutrino populations, correspondingly
 $\delta N_s=0$ (the lower curve),  $\delta N_s=0.5$
 and  $\delta N_s=0.8$ (the upper curve). The
dashed curve gives the
equilibrium neutrino spectrum for comparison.}
\ \\
%\end{figure}


%\begin{figure}[thb]
\hbox{\vspace{-1cm}}
\mbox{\hspace{2cm}}\epsfig{figure=danisp07.eps,width=10cm}\\
%\ \\
%\end{figure}

%\begin{figure}[thb]
\hbox{\vspace{-1.2cm}}
\mbox{\hspace{2cm}}\epsfig{figure=danisp05.eps,width=10cm}\\
\hbox{\vspace{-1.5cm}}
{\bf Figures 1b,c:} {\small Distortion
of the electron neutrino energy spectrum at a temperature $0.7$ MeV
(Fig.1b) and  $0.5$ MeV (Fig.1c) for the same
parameters as for Fig.1a}.
%\ \\
%\end{figure}





The oscillations parameters are
$|\delta m^2|=10^{-7}$ eV$^2$ and $\sin^22\vartheta=0.1$.
For illustration of the
evolution of the spectrum distortion we have presented it at  
characteristic
temperatures  $1$ (Fig.1a),  $0.7$ (Fig.1b) and  $0.5$ MeV (Fig.1c).
The characteristic behavior of the spectrum distortion due to oscillations is 
observed. Namely, since
oscillation rate is energy dependent $\Gamma \sim \delta m^2/E$ the low energy
part of the spectrum is distorted first (as far as low energy neutrinos 
start to oscillate first) 
and later the
distortion  becomes noticeable for  the more energetic part of the spectrum. 
The neutrino energy spectrum $n_{\nu}(E)$ may strongly deviate
from its equilibrium form during all the period of interest 
($2$ MeV -- $0.3$ MeV), 
and hence constitute the dominant effect on the overproduction of  $^4\!$He. 

As expected, the spectrum distortion is less expressed when increasing 
the degree of population of the sterile neutrino state $\delta N_s$. 
Correspondingly, the effect 
on primordial nucleosynthesis decreases, as illustrated on Fig.2.   

In Fig.2 we present  the frozen  neutron number density relative to 
nucleons $X_n^f=N_n^f/N_{nuc}$ 
 for a mass difference $\delta m=\pm 10^{-7}$ eV$^2$  
and mixing angle $\sin^2 2\theta=10^{-1}$ as a function of  
the sterile neutrino content  at neutrino decoupling. 
As far as $\delta Y_p/Y_p=\delta X_n^f/X_n^f$,  it is representative of  
the overproduction of 
 primordially produced helium. 

%\begin{figure}[thb]
%\hbox{\vspace{-1cm}}
\mbox{\hspace{1cm}}\epsfig{figure=dani2.eps,width=10cm}\\
{\bf Figure 2: }{\small The  solid curves present frozen  
neutron number density relative to
nucleons $X_n^f=N_n^f/N_{nuc}$ as a function  of
the sterile neutrino initial population. The dashed curves present only the 
kinetic effect, while the dotted curve presents the effect due to the 
energy 
density increase.}
\ \\
%\end{figure} 

The dotted line presents only the  effect (a), due to the energy 
density
increase $X_n^f=f(\delta N_s)$, the dashed curves present the pure 
kinetic
effects (b)  $X_n^f=f(\delta N_{kin})$,
 while  the solid lines give the total effect.
The upper curves correspond to the
resonant case, the  lower ones to the non-resonant one.



The analysis for these concrete oscillation parameters, shows that the 
overproduction
of helium is strongly 
suppressed with the increase of 
$\delta N_s$ for the resonant case, while in the non-resonant case it 
increases with 
$\delta N_s$. This is a result of the fact that,  
in the resonant 
case, the kinetic effects (b) due to the spectrum distortion are the 
dominant contribution to the overproduction of helium,   
 even for very large degree of population of the sterile state, while in 
the non-resonant case the main contribution  
comes from the increase of degrees of freedom already at very small 
$\delta N_s$. 
An  empirical approximation formula for the concrete mixing parameters 
is: 
\begin{center}
$\delta Y_p = 0.013[\delta N_{kin}^{max}(1-\delta N_s)+\delta N_s]$,\\
\end{center}
\noindent  where $\delta N_{kin}^{max}$ is the value calculated for an 
initially  empty sterile state,  
i.e. $\delta N_{tot}=\delta N_{kin}^{max}(1-\delta N_s)+\delta N_s$. 
It is a good approximation for the non-resonant case and a  rather rough 
one for the resonant case: the deviation from  the exactly 
calculated helium may be up to $\delta Y_p/Y_p\sim 0.8\%$. 
Still, it can give some idea of the trend of $\delta Y_p/Y_p$ dependence 
on $\delta N_s$.

However, for large mixing angles, the oscillation effects in the non-resonant 
case can be also considerable, as shown in ref.\cite{bern}, the kinetic 
effect can be as high as $\delta N_{kin} \sim 3$ for initially empty sterile 
state. Hence, in the non-resonant case the spectrum distortion effects may be 
the dominant one even for much larger  $\delta N_s$ than in the case illustrated 
on Fig.2. 

{\it So, the neutrino spectrum distortion effect  is 
very strong and 
should be accounted 
for accurately, even when there is a  considerable 
population of the sterile neutrino state  before the 
beginning of the electron--sterile oscillations.}
It always gives positive $\delta N_{kin}$, which for a large range of 
initial sterile population values, is bigger than $1$. Hence, owing to these 
effects, the cosmological constraints are considerably strengthened, 
in comparison with the ones considering only effect  (a). The kinetic 
effects are 
the  strongest for $\delta N_s=0$: 
 $Y_p^{max}(\delta N_s,\delta m^2,\sin^22\vartheta)$ $=$ 
$Y_p(0,\delta m^2,\sin^22\vartheta)$.   
They disappear for $\delta N_s=1$, when 
$\nu_e$ and  $\nu_s$
states are in equilibrium, and the total effect reduces to the SBBN with 
an additional neutrino. 

%\begin{figure}[thb]
%\hbox{\vspace{-1cm}}
\mbox{\hspace{1cm}}\epsfig{figure=dani1.eps,width=10cm}\\
{\bf Figure 3: }{\small BBN constraints on oscillation parameters 
for the resonant
(r.h.s.) and the non-resonant neutrino oscillations and for initial
degree of population of the sterile neutrino state $\delta N_s=0.1$.
The dashed contours present the constraints for  $\delta N_s=0$.}
\ \\
%\end{figure}

We have also calculated  the isohelium contours
corresponding to  $3\%$ overproduction of  $^4\!$He~\footnote{Hopefully future
understanding of the big statistical error
of the
observed  $^4\!$He and its decrease may help us to have even  $1\%$
constraints.
However, at present we think that  $3\%$ overproduction is a reliable
one from the observational viewpoint.}
 for  $\delta N_s=0.1$ and $\delta N_s=0.2$. In Fig.~3 we present the
cosmological constraints on oscillation parameters  for $\delta N_s=0.1$,
for the
resonant (to the right) and the non-resonant
(to the left) oscillation cases. The region upwards of the curve is
excluded.
$\delta N_s=0$ constraints are  given for comparison by the dashed contours.




In both the  $\delta N_s=0.1$ and the  $\delta N_s=0.2$ cases  
 the cosmological constraints are slightly
 changed. In the resonant 
case they are slightly weakened, while in the non-resonant case the 
change is negligible. As a whole they remain 
stringent, as before.  
On the basis of the provided analysis of the dependence of the kinetic effects
  on the 
initial population of steriles, we expect  considerable change of the 
constraints for higher population of the sterile state. 
The exact form of the constraining contours depends on the two 
competing effects (a) and (b), and hence a detailed numerical analysis 
is necessary for it defining for each $\delta N_s$ value. 

It is straightforward to obtain a rough indicative cosmological 
constraint for the sterile neutrino, participating eventually into 
solar neutrino oscillations:  
%the mixing angle constraint from experimental data is now  
%$sin^2(2\theta)<0.16$,  
 for $\delta N_s\le 0.2$  and  
for $\delta m=10^{-7}$ eV$^2$, i.e. LOW mass range,  
$\sin^2(2\theta)<0.03$; 
for $\delta m=10^{-5.5}$   eV$^2$, i.e. LMA,  we expect 
 $\sin^2(2\theta)<0.006$. 

\section*{Conclusion}


The presence of a {\it  non-empty} sterile  
state before   $\nu_e\leftrightarrow \nu_s$  oscillations was not considered in
previous
analyses of   $\nu_e\leftrightarrow \nu_s$ oscillation effects on the neutrino
spectrum
distortion and their influence on
BBN. In this letter we have addressed this question.

The effect of the neutrino spectrum distortion  may be 
very strong,  even for a  considerable initial 
population of the sterile neutrino state  before the
beginning of the electron--sterile oscillations.


We have provided a numerical analysis,  investigating
how the presence of the sterile neutrino state, partially populated before  
oscillations, will influence  the production of  $^4\!$He in the model
of BBN with electron--sterile oscillations effective after neutrino 
decoupling. For the cases $\delta N_s=0.1$ and $\delta N_s=0.2$ the 
corresponding cosmological constraints on 
oscillation parameters have been obtained.

This work is a  step toward  
generalizing and precising the  cosmological constraints 
on oscillation parameters of neutrino.
The results of this analysis may be of interest for 
mixing  schemes in which a 
portion of $\nu_s$ have been brought into equilibrium before 
neutrino decoupling, due to  $\nu_{\mu}\leftrightarrow \nu_s$ or
$\nu_{\tau}\leftrightarrow \nu_s$ oscillations.
In case the $\nu_s$ presence is
due to the much earlier (at atmospheric mass difference scale, or LSND)
oscillations of $\nu_{\mu,\tau}\leftrightarrow\nu_s$,
 $\delta N_s$  may be  directly connected with the available constraints
on the sterile
neutrino fraction, deduced
from the neutrino oscillations experimental data analysis.
So, we hope that the results   may help to choose among 
the different  possibilities for the explanation of neutrino anomalies.

Cosmological constraints now receive back the reliability they deserve,
after  it was recently realized that:\\
% in the neutrino oscillation community.
 (1) for  oscillation parameters values 
fixed from the solar and atmospheric neutrino data, no great
neutrino--antineutrino  asymmetries
are allowed,  for any neutrino flavor from BBN
considerations~\cite{pastor}.
 Besides, \\
 (2) that given the data from neutrino oscillations experiments, 
in any of the 4-neutrino mixing neutrino schemes, no large electron asymmetry
generation is possible~\cite{DiBari}, and consequently,  
 cosmological
constraints on   $\nu_e\leftrightarrow \nu_s$ oscillations
cannot be weakened by
the presence of a big neutrino
asymmetry,  produced in an 
alternative channel, which  reduces the   $\nu_e\leftrightarrow \nu_s$ oscillations
effect on
nucleosynthesis and, hence, weakens  the cosmological constraints.

Therefore, a 
more detailed study of  BBN constraints in  4-neutrino mixing neutrino 
schemes seems appropriate, although much more complicated. 
Meanwhile,  encouraged 
 by the stimulating 
discussions with teams providing the global analysis of the neutrino 
oscillation data,  
we decided to present the results of this analysis, which seems  
 indicative and helpful.


\ \\

I thank A. Dolgov for useful comments on the draft version, 
A. Strumia for stimulating discussions during
the preparation of this work and M. Chizhov for his  continuous scepticism 
and help. I am 
grateful to my parents for their help with the kids, while I was working  on
this  paper.

This work has been done while I held Corresponding Associate position at CERN TH. 
I appreciate also the  Regular Associateship  at the Abdus Salam ICTP, Trieste.

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The analytical fit to  constraints, for the non-resonant and the
resonant
oscillation case, correspondingly  is:
$\delta m^2 (\sin^2 2\vartheta)^4\le 1.5\times 10^{-9}$ eV$^2$.
 $\delta m^2 \sim 10^{-8}$ eV$^2$, at large mixings.



{\it So, the neutrino spectrum distortion effect in the resonant case is
very strong and
should be accounted
for accurately even when there is a  considerable
population of the sterile neutrino state  before the 
beginning of the electron--sterile oscillations.}
It always gives positive $\delta N_{kin}$ which for large range of
initial sterile population value is bigger than $1$. Hence, due to these
effects, the cosmological constraints are considerably strengthened, when
a proper account of the kinetic effect is provided.


 In Fig. 2 the evolution of the neutron number density is presented as
a function of  $\delta N_s$ and for different sets of oscillation
parameters, namely   
$\delta m^2=10^{-7}$ eV$^2$, $\sin^22\vartheta=10^{-1}$ and
$\delta m^2=10^{-7}$ eV$^2$, $\sin^22\vartheta=10^{-0.65}$ (which is the
resonant angle for this mass difference).
The kinetic effects depend on the concrete oscillation parameters.


\begin{figure}[thb]
\caption{Neutron number density at freeze-out  as
a function of  $\delta N_s$ and different oscillation parameters.}
\end{figure}   



For $\vartheta$ far from the resonant value, the dependance $Y_p(\delta N_s)$ is
almost linear for a concrete mixing parameters set. It is possible to
account for the non-zero population of the sterile neutrino, by  empirical
formulae  as the one given above.



