Journal of Modern Physics, 2012, 3, 1979-1982
http://dx.doi.org/10.4236/jmp.2012.312247 Published Online December 2012 (http://www.SciRP.org/journal/jmp)
Theoretical Physics Department, University of Thessaloniki, Thessaloniki, Greece
Received July 13, 2012; revised October 25, 2012; accepted November 8, 2012
Solar, atmospheric and reactor neutrino experiments established that neutrinos are massive. It is quite natural then to
consider neutrinos as candidate particles for explaining the dark matter in halos around galaxies. We study the gravita-
tional clustering of these neutrinos within a model of a massive core and a surrounding spherical neutrino halo. The
neutrinos form a degenerate Fermi gas and a loaded polytropic equation is established. We solve the equation and we
obtain the neutrino density in a galaxy, the size of the galaxy and the galactic rotational curves. The available data favor
a neutrino with a mass around 10 eV. The consequent cosmological implications are examined.
Keywords: Neutrinos; Neutrino Mass; Gravitational Clustering; Dark Matter; Polytrope Equation
Solar, atmospheric, reactor and accelerator neutrinos have
provided compelling evidence for the existence of neu-
trino oscillations, implying non-zero neutrino masses
[1,2]. Solar neutrino oscillations indicate a mass squared
difference while atmospheric neu-
trino oscillations suggest ATM Further,
attributing the LSND anomaly to neutrino oscillations
implies the existence of a fourth neutrino, a sterile one,
with a mass above 1 eV . Cosmology offers further
insights on the neutrino mass scales [4-7]. It is inevitable
that neutrinos have participated in gravitational cluster-
ing around massive galaxies, constituting part of the dark
matter. Our purpose is to investigate the neutrino clus-
tering effect, obtaining information on the cosmological
presence of massive neutrinos.
10 - 10
The universe today exhibits structure on many scales.
Galaxies range in mass, with most bright galaxies having
. The sizes and distributions of
present-day galaxies reflect the spectrum of initial den-
sity fluctuations, seeded by random motion or cosmic
strings . We expect that each galaxy attracts the cos-
mological neutrinos in its neighbourhood, creating a neu-
trino halo of an average mass 10
Our idealized galaxy consists of a spherical massive
core (of a mass c
and a radius c of few
Kpc) surrounded by a spherical neutrino halo. Hydro-
static equilibrium prevails
where is the neutrino density, m is the neutrino
mass and is the pressure of the neutrino gas.
Considering that the neutrinos form a degenerate Fermi
gas we obtain a polytrope rquation
with the degeneracy factor (left-handed neutrinos
and right-handed antineutrinos). Equations (1) and (2)
Introducing the (dimensionless) variables p and x
Equation (3) becomes
The variable p is related to the gravitational potential
opyright © 2012 SciRes. JMP
0 is the gravitational potemtial at the outer part of
the galaxy, where the density vanishes. Indeed the nu-
merical solution of Equation (8) always provides a finite
, where 0. The radius of the galaxy is
then (Equation (6)). The total mass of the
is related to the derivative of
At the boundary condition is
Equation (8) together with the boundary conditions (10)
and (11), represents a loaded polytrope [9-13]. Similar
mathematical structures appear in the study of a galactic
nucleus, a neutron star, or in the Thomas-Fermi descrip-
tion of an atom.
We solved numerically Equation (8), using the bound-
ary condition of Equation (11) and an arbitrary positive
value for the derivative of at c
px . The nu-
merical solution provides a function which rises
up to a maximum, then decreases until it vanishes at
some point 0
0. We evaluated also the deri-
vative of in the neighbourhood of 0
sequently the left-hand side of Equation (10), thus deter-
mining the ratio c
. To obtain the desired ratio
, we repeat the numerical evaluation with a dif-
ferent value for the derivative of at c
, until we
achieve the predetermined c
ratio. The numeri-
cal study revealed the following essential features of the
1) The precise value of 0
is largely independent of
the ratio c
. More massive neutrino halos provide
larger values for the maximum of , but all neutrino
densities vanish in the vicinity of 0. Thus the
radius R of the galaxy is set up by the constant q (Equa-
tions (6) and (7)).
2) The scale q, Equation (7), is very sensitive to the
mass of neutrino m. By increasing the mass of the neu-
trino by a factor 3, the size of the galaxy is reduced by a
factor 19. The overall data are well reproduced with a
neutrino mass at 10 eV.
3) Equation (7) gives then the numerical expression
For a massive core
, we obtain
, in agreement
with the data for the size of massive galaxies . It is
impressive that a scale q which is expressed in terms of
fundamental constants, such as the Planck constant, the
neutrino mass and the Newton constant, reproduces ac-
curately the galactic sizes.
4) Through Equation (5) we obtain back the neutrino
Nr . We observe that at small distances,
1at small Nr r
while the neutrino density diverges as , the total
Figure 1 shows the neutrino density as a function of
the rescaled distance x. The upper curve corresponds to
, while the lower curve cor-
responds to c11
. Near the galactic
core, the neutrino density is high as neutrinos/cm3,
and over the uniform density of the big-bang cosmology,
the gravitational clustering provides locally an increase
by a factor .
The spherical neutrino halo up to a rescaled distance x,
gives a mass
. The galactic rotational ve-
locity, due to the neutrino halo, is then
Figure 2 shows the galactic rotation curves (upper
Figure 1. The neutrino density in a spherical galaxy as a
function of the rescaled distance x. Upper curve corre-
sponds to Mv = Mc, the lower curve corresponds to Mv = Mc.
For a Mc = 1011 Mʘ, real distance is obtained by multiplying
the dimensionless x by 36 Kpc.
Copyright © 2012 SciRes. JMP
A. NICOLAIDIS 1981
Figure 2. The galactic rotational velocity, due to a spherical
neutrino halo, as a function of the rescaled distance x.
Upper curve corresponds to Mv = Mc, lower to Mv = Mc. For
a Mc = 1011 Mʘ, real distance is obtained by multiplying the
dimensionless x by 36 Kpc.
curve corresponds to 11
M, 10 c
curve corresponds to 11
there is agreement with the trends of the experimental
We may reach our findings by making appeal to an
even simpler model [15,16]. A gravitational potential
well of the size of the galactic halo is filled up with neu-
trinos, which are treated like a Fermi-Dirac gas at zero
temperature. The total number of neutrinos is related to
the Fermi momentum, itself determined by the minimum
kinetic energy required for the neutrino to escape from
the galaxy. It is found then that the radius of the galaxy is
, in full agreement with our results. In another
study, the analysis of the neutrino clustering by employ-
ing techniques of statistical mechanics favors again a
neutrino mass of few eV .
Is it possible to detect the cosmological relic neutrinos?
The gravitational clustering effect we studied here, en-
hances the local densities, however the average neutrino
energy is very small and the corresponding weak cross-
section (of the order of ) renders detection by
conventional means highly unlikely .
The few eV mass scale for the sterile neutrino, that our
study suggests, fits nicely with the findings of the reactor
and short-based neutrino oscillation experiments . On
the other hand the cosmological verdict for a few eV-
mass sterile neutrino is rather unclear. The standard
framework provides the constraint
A modified though cosmology, with additional
radiation, can accommodate a few eV neutrino . In
another direction, there is strong evidence for the exis-
tence of a 17 keV massive neutrino [20,21]. The coexis-
tence of distinct mass scales for the neutrinos, sub-eV,
eV and keV, indicates the complexity of neutrino physics.
Clearly we need further experimental information and
novel theoretical insights to decode the hidden dynamics.
In summary we presented a simplified model for the
gravitational clustering of massive neutrinos in the gal-
axies. There is no adjustable parameter in our analysis
and despite its simplicity the overall features of galactic
dynamics are reproduced.
I would like to thank Nicos Vlachos for helping me to
use the program MATHEMATICA in the numerical work.
This research was supported in part by the Templeton
Foundation and the EU program “Human Capital and
 A. Romanino, “Neutrino Physics,” CERN Yellow Report
CERN-2012-001, pp. 153-182.
 A. Strumia and F. Vissani, “Neutrino Masses and Mix-
ings and…,” arXiv:hep-ph/0606054.
 C. Giunti, “Phenomenology of Sterile Neutrinos,” arXiv:
 N. Mavromatos, “Neutrinos and the Universe,” Invited
Paper to NUFACT11, arXiv:1110.3729 [hep-ph].
 C. Tao, “Astrophysical Constraints on Dark Matter,” arXiv:
 H. de Vega and N. Sanchez, “Highlights and Conclusions
of the Chalonge Workshop 2011,” arXiv:1109.3187 [as-
 A. Dolgov, “Cosmological Implications of Neutrinos,”
Surveys High Energy Physics, Vol. 17, No. 1-4, 2002, pp.
 M. Zombeck, “Handbook of Space, Astronomy and As-
trophysics,” Cambridge University Press, Cambridge, 1990.
 S. Chandrasekhar, “Stellar Structure,” University of Chi-
cago Press, Chicago, 1939.
 L. Landau and E. Lifshitz, “Statistical Physics,” Addison-
Wesley Publishing Co., Boston, 1969.
 J. Huntley and W. Saslaw, “The Distribution of Stars in
Galactic Nuclei: Loaded Polytropes,” Astrophysical Jour-
nal, Vol. 199, 1975, pp. 328-335. doi:10.1086/153695
 R. Viollier, F. Leimgruber and D. Trautmann, “Halos of
Heavy Neutrinos around Baryonic Stars,” Physics Letters
B, Vol. 297, No. 1-2, 1992, pp. 132-137.
 R. Viollier, D. Trautmann and G. Tupper, “Supermassive
Neutrino Stars and Galactic Nuclei,” Physics Letters, Vol.
306, No. 1-2, 1993, pp. 79-85.
CDM V. Rubin, W. Kent Ford Jr. and N. Thonnard, “Extended
Rotation Curves of High-Luminosity Spiral Galaxies. IV-
Systematic Dynamical Properties, SA through SC,” As-
Copyright © 2012 SciRes. JMP
Copyright © 2012 SciRes. JMP
trophysical Journal, Vol. 225, No. 1, 1978, pp. L107-
 R. Cowsik and J. McClelland, “Gravity of Neutrinos of
Nonzero Mass in Astrophysics,” Astrophysical Journal,
Vol. 180, 1973, pp. 7-10. doi:10.1086/151937
 R. Cowsik, “The Neutrino Hypothesis for Dark Matter in
Dwarf Spheroidals,” Journal of Astrophysics and Astron-
omy, Vol. 7, No. 1, 1986, pp. 1-6.
 T. Nieuwenhuizen, “Do Non-Relativistic Neutrinos Con-
stitute the Dark Matter?” Europhysics Letters, Vol. 86,
No. 3, 2009, Article ID: 59001.
 P. Langacker, J. Leveille and J. Sheiman, “On the Detec-
tion of Cosmological Neutrinos by Coherent Scattering,”
Physical Review D, Vol. 27, No. 6, 1983, pp. 1228-1242.
 J. Hamann, S. Hannestad, G. Raffelt and Y. Wong, “Ster-
ile Neutrinos with eV Masses in Cosmology—How Dis-
favoured Exactly?” Journal of Cosmology and Astropar-
ticle Physics, No. 9, 2011, arXiv:1108.4136.
 H. de Vega and N. Sanchez, “Model-Independent Analy-
sis of Dark Matter Points to a Particle Mass at the keV
Scale,” Monthly Notices of the Royal Astronomical Soci-
ety, Vol. 404, No. 2, 2010, pp. 885-894.
 P. Biermann and A. Kusenko, “Relic keV Sterile Neutri-
nos and Reionization,” Physical Review Letters, Vol. 96,
No. 9, 2006, Article ID: 091301.