Dark Neutrinos

doi:10.4236/jmp.2012.312247

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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

Email: [email protected]

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.

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

. 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





Our idealized galaxy consists of a spherical massive

core (of a mass c



r

 and a radius c of few

Kpc) surrounded by a spherical neutrino halo. Hydro-

static equilibrium prevails

1d drP





24π

dd GmN

rmNr

r





(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

4π5







2g

(2)

with the degeneracy factor (left-handed neutrinos

and right-handed antineutrinos). Equations (1) and (2)

lead to

213

1d1 d













(3)



13 53

32 2

34πmg Gh







(4)

Introducing the (dimensionless) variables p and x

through

32 32

6π

GMp

Nr r















rqx

(5)



(6)

432 1212

3π

qgm GM













 (7)

Equation (3) becomes

232

d=.

x (8) 

The variable p is related to the gravitational potential





Vr through

A. NICOLAIDIS

1980





rV 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

, where 0. The radius of the galaxy is

then (Equation (6)). The total mass of the



0px R

Rqx





galaxy M





is related to the derivative of



px at

dxx

















rr



1.0

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



x



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

0px







and sub-

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

loaded polytrope:

1) The precise value of 0

is largely independent of

the ratio c



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

1.7 10qKpc



 (12)

For a massive core

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,







behaves as

1at small Nr r



0r

(13)

while the neutrino density diverges as , the total

neutrino mass



remains finite.

Figure 1 shows the neutrino density as a function of

the rescaled distance x. The upper curve corresponds to

c11

M10, c





, while the lower curve cor-

responds to c11

Mc,

M



. 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











, where



12 d.

My pyy











 (14)





Obviously x









. The galactic rotational ve-

locity, due to the neutrino halo, is then











. (15)

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.

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



, lower

curve corresponds to 11

M, c

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

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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