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)
Dark Neutrinos
A. Nicolaidis
Theoretical Physics Department, University of Thessaloniki, Thessaloniki, Greece
Received July 13, 2012; revised October 25, 2012; accepted November 8, 2012
ABSTRACT
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 [3]. 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.
25
610 eV

2
232
210 eV.m 
10 12
10 - 10
SOL
m
The universe today exhibits structure on many scales.
Galaxies range in mass, with most bright galaxies having
masses of
M
. The sizes and distributions of
present-day galaxies reflect the spectrum of initial den-
sity fluctuations, seeded by random motion or cosmic
strings [8]. We expect that each galaxy attracts the cos-
mological neutrinos in its neighbourhood, creating a neu-
trino halo of an average mass 10
310
M
M
 
Our idealized galaxy consists of a spherical massive
core (of a mass c
.
M
M

r
 and a radius c of few
Kpc) surrounded by a spherical neutrino halo. Hydro-
static equilibrium prevails
2
1d drP


24π
dd GmN
rmNr
r


Nr

Pr
(1)
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
23 2
53
3
4π5
h
PN
gm



2g
(2)
with the degeneracy factor (left-handed neutrinos
and right-handed antineutrinos). Equations (1) and (2)
lead to
2
213
1d1 d
dd
N
rN
rr rN






(3)

13 53
32 2
34πmg Gh  

(4)
Introducing the (dimensionless) variables p and x
through
32 32
2
22
2
6π
GMp
gm
Nr r









rqx
(5)

(6)
23
3
432 1212
3π
42
qgm GM






 (7)
Equation (3) becomes
232
2
d=.
d
pp
x (8) 
x
The variable p is related to the gravitational potential


Vr through
C
opyright © 2012 SciRes. JMP

A. NICOLAIDIS 1980


0
=
c
rV V
GM

V
p (9)
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
0
x
, where 0. The radius of the galaxy is
then (Equation (6)). The total mass of the

0px R
0
Rqx

c
MM

galaxy M


0
is related to the derivative of

px at
x
by
0
c
c
d
dxx
M
M
xM
p
x








c
rr

1.0
c
px x

(10)
At the boundary condition is
(11)
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
x
x

px . The nu- merical solution provides a function which rises
up to a maximum, then decreases until it vanishes at
some point 0
x



0. We evaluated also the deri-
vative of in the neighbourhood of 0
0px



px
x
and sub-
sequently the left-hand side of Equation (10), thus deter-
mining the ratio c
M
M

. To obtain the desired ratio
c
M
M

, we repeat the numerical evaluation with a dif-
ferent value for the derivative of at c

px
x
, until we
achieve the predetermined c
M
M

ratio. The numeri-
cal study revealed the following essential features of the
loaded polytrope:
1) The precise value of 0
x
is largely independent of
the ratio c
M
M

px
2.0x

. 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
5
13
1.7 10qKpc

11
10
c
 (12)
For a massive core
M
M
36qKpc70RKpc
, we obtain
and therefore

, in agreement
with the data for the size of massive galaxies [14]. 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
density


Nr . We observe that at small distances,


Nr

behaves as
32
1at small Nr r
r

0r
(13)
while the neutrino density diverges as , the total
neutrino mass
M

remains finite.
Figure 1 shows the neutrino density as a function of
the rescaled distance x. The upper curve corresponds to
c11
10
M
M10, c
M
M


, while the lower curve cor-
responds to c11
10
M
Mc,
M
M
7
10
5
10

. 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


x



, where

32
12 d.
x
My pyy

 


 (14)


0
M
Obviously x
 


. The galactic rotational ve-
locity, due to the neutrino halo, is then
12
Gr
ur

 


. (15)
Figure 2 shows the galactic rotation curves (upper
x
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.

A. NICOLAIDIS 1981
x
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
10
c
M
M, 10 c
M
M

, lower
curve corresponds to 11
10
c
M
M, c
M
M

) and
there is agreement with the trends of the experimental
data.
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
23
3Rq
53 2
10 cm

CDM0.5 eVm
, 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 [17].
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 [18].
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 [3]. 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 [19]. 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.
Acknowledgements
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
Mobility”.
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