Pulsar kicks and dark matter from a sterile neutrino
Alexander Kusenko\footnote{\uppercase{W}ork partially supported by the \uppercase{DOE} grant \uppercase{DE-FG03-91ER40662} and the \uppercase{NASA ATP} grant \uppercase{NAG5-13399}. }
The observed velocities of radio pulsars, which range in the hundreds kilometers per second, and many of which exceed 1000~km/s, are not explained by the standard physics of the supernova explosion. However, if a sterile neutrino with mass in the 1--20~keV range exists, it would be emitted asymmetrically from a cooling neutron star, which could give it a sufficient recoil to explain the pulsar motions. The same particle can be the cosmological dark mater. Future observations of X-ray telescopes and gravitational wave detectors can confirm or rule out this explanation.
kusenko_xxx.tex
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\begin{document}

%\preprint{UCLA/04/TEP/14} 

\title{Pulsar kicks and dark matter from a sterile neutrino }

\author{Alexander Kusenko\footnote{\uppercase{W}ork partially supported by
the \uppercase{DOE} grant \uppercase{DE-FG03-91ER40662} and the
\uppercase{NASA ATP} grant \uppercase{NAG5-13399}.  }}

\address{Department of Physics and Astronomy, UCLA, Los Angeles, CA
90095-1547\\ RIKEN BNL Research Center, Brookhaven National Laboratory,
Upton, NY 11973 }

\maketitle

\abstracts{ The observed velocities of radio pulsars, which range in the
hundreds kilometers per second, and many of which exceed 1000~km/s, are not
explained by the standard physics of the supernova explosion.  However, if
a sterile neutrino with mass in the 1--20~keV range exists, it would be
emitted asymmetrically from a cooling neutron star, which could give it a
sufficient recoil to explain the pulsar motions.  The same particle can be
the cosmological dark mater.  Future observations of X-ray telescopes and
gravitational wave detectors can confirm or rule out this explanation.  }

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\section{Introduction}



Pulsar velocities range from 100 to 1600~km/s\cite{astro,astro_1}. Their
distribution leans toward the high-velocity end, with about 15\% of all
pulsars having speeds over (1000~km/s)\cite{astro_1}. The origin of these
velocities is a long-standing puzzle. 
Pulsars are born in supernova explosions, so it would be natural to look for
an explanation in the dynamics of the supernova\cite{explosion}. However,
state-of-the-art 3-dimensional numerical calculations\cite{fryer} show that
even the most extreme asymmetric explosions do not produce pulsar
velocities greater than 200~km/s.  Earlier 2-dimensional
calculations\cite{sn_2D} claimed a maximal pulsar velocity up to 500~km/s
to be possible.  Of course, even this size of the kick was way too small to
explain the large population of pulsars with speeds above 1000~km/s.
% The only claim I heard
%that a standard supernova explosion could result in velocities above
%1000~km/s was based on incorrect physics and has apparently been abandoned
%by its author\cite{janka}.  

Given the absence of a ``standard'' explanation, one is compelled to
consider alternatives, possibly involving new physics.  One of the reasons
why the standard explanation fails is because most of the energy is carried
away by neutrinos, which escape isotropically.  The remaining energy must
be distributed with a substantial asymmetry to account for the large pulsar
velocities.  In contrast, only a few per cent anisotropy in the
distribution of neutrinos would give the pulsar a kick of required
magnitude.

Neutrinos are {\em produced } anisotropically, but they {\em escape} 
isotropically.  The asymmetry in production comes from the asymmetry in
the basic weak interactions in the presence of a strong magnetic field.
Indeed, if the electrons and other fermions are polarized by the magnetic
field, the cross section of the urca processes, such as $n+e^+
\rightleftharpoons p+ \bar \nu_e$ and $p+e^-\rightleftharpoons n+ \nu_e $,
depends on the orientation of the neutrino momentum.  Depending on the
fraction of the electrons in the lowest Landau level, this asymmetry can be
as large as 30\%, much more than one needs to explain the pulsar
kicks\cite{drt}.  However, this asymmetry is completely washed out by
scattering of neutrinos on their way out of the star\cite{eq}.

If, however, the same interactions produced a particle which had even
weaker interactions with nuclear matter than neutrinos, such a particle
could escape the star with an asymmetry equal its production asymmetry. 

It is intriguing that the same particle can the dark mater. 

The simplest realization of this scenario is a model that adds only one
singlet fermion to the Standard Model.  The SU(2)$\times$U(1) singlet, a
sterile neutrino, mixes with the usual neutrinos, for example, with the
electron neutrino.  

For a sufficiently small mixing angle between $\nu_e$ and $\nu_s$, only one
of the two mass eigenstates, $\nu_1$, is trapped.  The orthogonal state,
%
\begin{equation}
| \nu_2 \rangle = \cos \theta_m | \nu_s \rangle + \sin \theta_m | \nu_e
\rangle , 
\end{equation}
%
escapes from the star freely.  This state is produced in the same basic
urca reactions ($\nu_e+n\rightleftharpoons p+e^-$ and
$\bar\nu_e+p\rightleftharpoons n+e^+$) with the effective Lagrangian
coupling equal the weak coupling times $\sin \theta_m$.

We will consider two ranges of parameters, for which the $\nu_e \rightarrow
\nu_s$ oscillations occur on or off resonance. First, let us suppose that a 
resonant oscillation occurs somewhere in the core of the neutron star.
Then the asymmetry in the neutrino emission comes from shift in the
resonance point depending on the magnetic field\cite{ks97}.  Second, we
will consider the off-resonance case, in which the asymmetry comes directly
from the weak processes, as described above\cite{fkmp}.

\section{Resonant, Mikheev-Smirnov-Wolfenstein oscillations}


Neutrino oscillations in a magnetized medium are described by an effective
potential\cite{magn}  
%
\begin{eqnarray}
V(\nu_{\rm s}) & = & 0  \label{Vnus} \\
V(\nu_{\rm e})& = & -V(\bar{\nu}_{\rm e}) =  V_0 \: (3 \, Y_e-1+4 \,
Y_{\nu_{\rm e}}) \label{Vnue} \\ 
V(\nu_{\mu,\tau}) & = & -V(\bar{\nu}_{\mu,\tau}) = V_0 \: ( Y_e-1+2 \, 
Y_{\nu_{\rm e}}) \ 
+\frac{e G_{_F}}{\sqrt{2}} \left ( \frac{3 N_e}{\pi^4} 
\right )^{1/3}
\frac{\vec{k} \cdot \vec{B}}{|\vec{k}|} \label{Vnumu}
\end{eqnarray}
%
where $Y_e$ ($Y_{\nu_{\rm e}}$) is the ratio of the number density of electrons
(neutrinos) to that of neutrons, $\vec{B}$ is the magnetic field, 
$\vec{k}$ is the neutrino momentum, $V_0=10 \: \rm{eV} \: (\rho/10^{14} g
\, cm^{-3} )$.  The magnetic field dependent term in equation (\ref{Vnumu})
arises from polarization of electrons and {\em not} from a neutrino
magnetic moment, which is small and which we will neglect. 


The condition for resonant MSW oscillation $\nu_i \leftrightarrow
\nu_j$ is

\begin{equation}
\frac{m_i^2}{2 k} \: \cos \, 2\theta_{ij} + V(\nu_i) = 
\frac{m_j^2}{2 k} \: \cos \, 2\theta_{ij} + V(\nu_j)  
\label{res}
\end{equation}
where $\nu_{i,j}$ can be either a neutrino or an anti-neutrino. 

In the presence of the magnetic field, the condition (\ref{res}) is
satisfied at different distances $r$ from the center, depending on the
value of the $(\vec{k} \cdot \vec{B})$ term in (\ref{res}). The average
momentum carried away by the neutrinos depends on the temperature of the
region from which they escape.  The deeper inside the star, the higher is
the temperature during the neutrino cooling phase.  Therefore, neutrinos
coming out in different directions carry momenta which depend on the
relative orientation of $\vec{k}$ and $\vec{B}$.  This causes the asymmetry
in the momentum distribution.

The surface of the resonance points is 

\begin{equation}
r(\phi) = r_0 + \delta \: cos \, \phi, 
\end{equation}
where $cos \, \phi= (\vec{k} \cdot \vec{B})/k$ and $\delta$ is determined
by the equation 
$(d N_n(r)/dr) \delta \approx 
e \left ( 3 N_e/\pi^4 \right )^{1/3} B$.
This yields\cite{ks97} 

\begin{equation}
\delta = % \left ( \frac{3 N_e}{\pi^4} \right )^{1/3} \:
%\frac{e}{2} \: B \left / \frac{dN_e(r)}{dr} \right. =
\frac{e \mu_e}{ \pi^2} \: B \left / \frac{dN_n(r)}{dr} \right. ,
\label{delta}
\end{equation}
where $\mu_e \approx (3 \pi^2 N_e)^{1/3} $ is the chemical potential of the
degenerate (relativistic)  electron gas.

Assuming a black-body radiation luminosity $\propto T^4$, 
the asymmetry in momentum distribution is\cite{ks97}
%
\begin{equation}
\frac{\Delta k}{k} = \frac{4 e}{3 \pi^2} \: \left ( \frac{\mu_e}{T}
\frac{dT}{dN_n} \right) B.
\label{dk1}
\end{equation}


To calculate the derivative in (\ref{dk1}), we use the relation between the
density and the temperature of a non-relativistic Fermi gas. Finally, 
%
\begin{equation}
\frac{\Delta k}{k} = \frac{4 e\sqrt{2}}{\pi^2} \: 
\frac{\mu_e \mu_n^{1/2}}{m_n^{3/2}T^2} \ B =
0.01 \left ( \frac{B}{3\times 10^{15} {\rm G} }\right )
\label{dk2}
\end{equation}
if the neutrino oscillations take place in the core of the neutron star, at
density of order $10^{14} \, {\rm g\,cm^{-3}}$.  The neutrino oscillations
take place at such a high density if one of the neutrinos has mass in the
keV range, while the other one is much lighter.  The magnetic field of the
order of $10^{15}-10^{16}$~G is quite possible inside a neutron star, where
it is expected to be higher than on the surface.  (In fact, some neutron
stars, dubbed magnetars, appear to have surface magnetic fields of this
magnitude.)

Some comments are in order.  First, a similar kick mechanism, based
entirely on active neutrino oscillations (and no steriles) could also work
if the resonant oscillations took place between the electron and tau
neutrinospheres\cite{ks96}.  This, however, would require the mass
difference between two neutrinos to be of the order of 100~eV, which is
ruled out.  Second, the neutrino kick mechanism was criticized incorrectly
by Janka and Raffelt\cite{jr}.  It was subsequently shown by several
authors\cite{ks98,barkovich} that Janka and Raffelt made several
mistakes, which is why their estimates differ from
eq.~(\ref{dk2}).
%\cite{ks98,barkovich}.   

\begin{figure}[ht]
%\epsfxsize=10cm   %width of figure - will enlarge/reduce the figures
%\epsfbox{fig3.eps}
%\figurebox{2cm}{3cm}{} %to have a box alone 
\centerline{\epsfxsize=4.1in\epsfbox{kick_s.eps}}   
\caption{The range of parameters for the sterile neutrino mass and mixing.
  Regions 1 and 2 correspond to parameters consistent with the pulsar kicks
  for (1) resonant and (2) off-resonant transitions, respectively.  Both
  regions overlap with a band in which the sterile neutrino is 
  dark matter.}
\end{figure}



\section{Off-resonant oscillations}

For somewhat lighter masses, the resonant condition is not satisfied
anywhere inside the star.  In this case, however, the off-resonant
production of sterile neutrinos in the core can occur through ordinary urca
processes.  A weak-eigenstate neutrino has a $\sin^2\theta $ admixture
of a heavy mass eigenstate $\nu_2$.  Hence, these heavy neutrinos can be
produced in weak processes with a cross section suppressed by $\sin^2\theta
$. 

Of course, the mixing angle in matter $\theta_m$ is not the same as it is
in vacuum, and initially $\sin^2\theta_m \ll \sin^2\theta$.  However, as
Abazajian, Fuller, and Patel\cite{Fuller} have pointed out, in the presence
of sterile neutrinos the mixing angle in matter quickly evolves toward its
vacuum value.  When $\sin^2\theta_m \approx \sin^2\theta$, the production 
of sterile neutrinos is no longer suppressed, and they can take a fraction
of energy out of a neutron star.  

Sterile neutrinos escape with a sizable asymmetry due to weak interactions
of fermions polarized by the magnetic field.  (Once again, we neglect a
neutrino magnetic moment and consider only the matter fermions.)  The
resulting asymmetry can explain the pulsar kicks if the mass and mixing
angle fall inside region 2 in Fig.~1.  


\section{Sterile neutrinos as dark matter; observational consequences}

The parameter space allowed for the pulsar kicks\cite{fkmp} overlaps nicely
with that of dark-matter sterile neutrinos\cite{Fuller,dw}.  Sterile
neutrinos in this range may soon be discovered\cite{aft}.  Relic sterile
neutrinos with mass in the 1-20~keV range can decay into a lighter neutrino
and a photon.  The X-ray photons should be detectable by the X-ray
telescopes.  Chandra and XMM-Newton can exclude part of the parameter
space\cite{aft}.  The future Constellation-X can probably explore the
entire allowed range of parameters.

In the event of a nearby supernova, the neutrino kick can produce gravity
waves that could be detected by LIGO and LISA\cite{loveridge,cuesta}. 

Active-to-sterile neutrino oscillations can give a neutron star a kick.
However, if a black hole is born in a supernova, it would not receive a
kick, unless it starts out as a neutron star and becomes a black hole
later, because of accretion. (The latter may be what happened in SN1987A,
which produced a burst of neutrinos, but no radio pulsar.) If the central
engines of the gamma-ray bursts are compact stars, the kick mechanism
acting selectively on neutron stars and not black holes could probably
explain the short bursts as interrupted long bursts\cite{k_semi}.

Since one does not expect a significant correlation between the 
magnetic field inside a hot neutron star (while this field is, presumably,
growing via the dynamo effect) and the eventual exterior field of a radio
pulsar, the neutrino kick mechanism does not predict any $B-v$ correlation. 

To summarize, the nature of cosmological dark matter is still unknown.  We
know that at least one particle beyond the Standard Model must exist to
account for dark matter.  This particle may come as part of a ``package'' if
supersymmetry is right.  However, it may be that the dark matter particle
is simply an SU(2)$\times$U(1) singlet fermion, which has a small mixing
with neutrinos.  In the latter case, the same dark-matter particle would be
emitted anisotropically from a supernova with an asymmetry sufficient to
explain the pulsar kick velocities.



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Z.~Arzoumanian, D.~F.~Chernoff and J.~M.~Cordes,
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\bibitem{explosion} I.~S.~Shklovskii, Sov. Astr., {\bf 13}, 562 (1970)
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%  Tech. Univ. M\"unchen, Garching, 1998 (atro-ph/9801320), pp.~60-65.  

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%%CITATION = HEP-PH 9701311;%%

\bibitem{fkmp}
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J.~C.~D'Olivo, J.~F.~Nieves and P.~B.~Pal, {\it ibid.}, 3679 (1989);
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K.~Abazajian, G.~M.~Fuller and M.~Patel,
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%%CITATION = ASTRO-PH 0101524;%%

\bibitem{dw} S.~Dodelson and L.~M.~Widrow, Phys. Rev. Lett. {\bf 72}, 17
(1994); 
%
%\bibitem{fs} 
X.~d.~Shi and G.~M.~Fuller,
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Phys.\ Rev.\ Lett.\  {\bf 82}, 2832 (1999)
%[arXiv:astro-ph/9810076].
%%CITATION = ASTRO-PH 9810076;%%
A.~D.~Dolgov and S.~H.~Hansen,
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%%CITATION = HEP-PH 9606428;%%

\bibitem{jr}
H.~T.~Janka and G.~G.~Raffelt,
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%%CITATION = ASTRO-PH 9811144;%%

\bibitem{barkovich} 
M.~Barkovich, J.~C.~D'Olivo, R.~Montemayor and J.~F.~Zanella,
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%[arXiv:astro-ph/0206471].
%%CITATION = ASTRO-PH 0206471;%%

\bibitem{aft}
K.~Abazajian, G.~M.~Fuller and W.~H.~Tucker,
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%\cite{Loveridge:2003fy}
\bibitem{loveridge}
L.~C.~Loveridge,
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%%CITATION = ASTRO-PH 0309362;%%


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H.~J.~Mosquera Cuesta,
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%%CITATION = ASJOA,544,L61;%%
%
%``Back-reaction of Einstein's gravitational-waves as the origin of  natal
%pulsar kicks,'' 
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%%CITATION = PHRVA,D65,061503;%%
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%%CITATION = ASTRO-PH 0312399;%%


\end{thebibliography}

\end{document}

astro-ph0404483.extracted_bib
\begin{thebibliography}{0}

\bibitem{astro} See, {\it e.\,g.}, A.~G.~Lyne and D.~R.~Lorimer, Nature
  369 (1994) 127; J.~M.~Cordes and D.~F.~Chernoff,
  Astrophys. J. {\bf 505}, 315 (1998); B.~M.~S.~Hansen and E.~S.~Phinney,  
  Mon. Not. R. Astron. Soc. {\bf 291}, 569 (1997);
  C.~Fryer, A.~Burrows, and W.~Benz,  Astrophys. J. {\bf 496}, 333 (1998).

\bibitem{astro_1} 
Z.~Arzoumanian, D.~F.~Chernoff and J.~M.~Cordes,
Astrophys. J. {\bf 568}, 289 (2002).
%``The Velocity Distribution of Isolated Radio Pulsars s,''
%arXiv:astro-ph/0106159.
%%CITATION = ASTRO-PH 0106159;%%

\bibitem{explosion} I.~S.~Shklovskii, Sov. Astr., {\bf 13}, 562 (1970)
[Astr.  Zh. {\bf 46}, 715 (1970)]. 


\bibitem{fryer}
C.~L.~Fryer,
%``Neutron star kicks from asymmetric collapse,''
Astrophys.\ J.\  {\bf 601}, L175 (2004)
%[arXiv:astro-ph/0312265].
%%CITATION = ASTRO-PH 0312265;%%

\bibitem{sn_2D} 
A.~Burrows and J.~Hayes,
Phys. Rev. Lett. {\bf 76}, 352 (1996); L.~Scheck {\em et al.},
arXiv:astro-ph/0307352.
%%CITATION = ASTRO-PH 0307352;%%

%\bibitem{janka}
% H.-Th.~Janka, in {\it
%  Proceedings of  ``Neutrino Astrophysics''}, ed. M. Altmann {\it et al.},
%  Tech. Univ. M\"unchen, Garching, 1998 (atro-ph/9801320), pp.~60-65.  

\bibitem{drt}
O.~F.~Dorofeev, V.~N.~Rodionov and I.~M.~Ternov, Sov. Astron. Lett. {\bf
11}, 123 (1985).

\bibitem{ks97}
  A.~Kusenko and G.~Segr\`e, 
%``Neutral current induced neutrino oscillations in a supernova,''
Phys.\ Lett.\ B {\bf 396}, 197 (1997)
%[arXiv:hep-ph/9701311].
%%CITATION = HEP-PH 9701311;%%

\bibitem{fkmp}
G.~M.~Fuller, A.~Kusenko, I.~Mocioiu, and S.~Pascoli,
%``Pulsar kicks from a dark-matter sterile neutrino,''
Phys.\ Rev.\ D {\bf 68}, 103002 (2003). 
%%CITATION = ASTRO-PH 0307267;%%


\bibitem{magn} 
J.~F.~Nieves and P.~B.~Pal, Phys. Rev. {\bf D40} 1693 (1989);
J.~C.~D'Olivo, J.~F.~Nieves and P.~B.~Pal, {\it ibid.}, 3679 (1989);
S.~Esposito and G.~Capone, Z.~Phys. {\bf C70} (1996) 55;
J.~C.~D'Olivo, J.~F.~Nieves and P.~B.~Pal, Phys. Rev. Lett., {\bf 64}, 1088
(1990); 
H.~Nunokawa, V.~B.~Semikoz, A.~Yu.~Smirnov, and
J.~W.~F.~Valle, Nucl. Phys. {\bf B 501}, 17 (1997). 



\bibitem{Fuller}
K.~Abazajian, G.~M.~Fuller and M.~Patel,
%``Sterile neutrino hot, warm, and cold dark matter,''
Phys.\ Rev.\ D {\bf 64}, 023501 (2001)
%[arXiv:astro-ph/0101524].
%%CITATION = ASTRO-PH 0101524;%%

\bibitem{dw} S.~Dodelson and L.~M.~Widrow, Phys. Rev. Lett. {\bf 72}, 17
(1994); 
%
%\bibitem{fs} 
X.~d.~Shi and G.~M.~Fuller,
%``A new dark matter candidate: Non-thermal sterile neutrinos,''
Phys.\ Rev.\ Lett.\  {\bf 82}, 2832 (1999)
%[arXiv:astro-ph/9810076].
%%CITATION = ASTRO-PH 9810076;%%
A.~D.~Dolgov and S.~H.~Hansen,
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%\cite{Loveridge:2003fy}
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%%CITATION = ASJOA,544,L61;%%
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\bibitem{k_semi}
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\end{thebibliography}