Slowly rotating Bose Einstein condensate galactic dark matter halos, and their rotation curves
Eur. Phys. J. C
Slowly rotating Bose Einstein condensate galactic dark matter halos, and their rotation curves
Xiaoyue Zhang 1 2
Man Ho Chan 0
Tiberiu Harko 4 5 6
ShiDong Liang 3
Chun Sing Leung 7
0 Department of Science and Environmental Studies, The Education University of Hong Kong , Hong Kong , People's Republic of China
1 Department of Astronomy, Peking University , Beijing 100871 , People's Republic of China
2 School of Physics and Yat Sen School, Sun Yatsen University , Guangzhou 510275 , People's Republic of China
3 School of Physics, State Key Laboratory of Optoelectronic Material and Technology, Guangdong Province Key Laboratory of Display Material and Technology, Sun YatSen University , Guangzhou 510275 , People's Republic of China
4 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
5 School of Physics, Sun Yatsen University , Guangzhou 510275 , People's Republic of China
6 Department of Physics, BabesBolyai University , Kogalniceanu Street, 400084 ClujNapoca , Romania
7 Department of Applied Mathematics, Hong Kong Polytechnic University , Kowloon, Hong Kong , People's Republic of China
If dark matter is composed of massive bosons, a BoseEinstein condensation process must have occurred during the cosmological evolution. Therefore galactic dark matter may be in a form of a condensate, characterized by a strong selfinteraction. We consider the effects of rotation on the BoseEinstein condensate dark matter halos, and we investigate how rotation might influence their astrophysical properties. In order to describe the condensate we use the GrossPitaevskii equation, and the ThomasFermi approximation, which predicts a polytropic equation of state with polytropic index n = 1. By assuming a rigid body rotation for the halo, with the use of the hydrodynamic representation of the GrossPitaevskii equation we obtain the basic equation describing the density distribution of the rotating condensate. We obtain the general solutions for the condensed dark matter density, and we derive the general representations for the mass distribution, boundary (radius), potential energy, velocity dispersion, tangential velocity and for the logarithmic density and velocity slopes, respectively. Explicit expressions for the radius, mass, and tangential velocity are obtained in the first order of approximation, under the assumption of slow rotation. In order to compare our results with the observations we fit the theoretical expressions of the tangential velocity of massive test particles moving in rotating BoseEinstein condensate dark halos with the data of 12 dwarf galaxies and the Milky Way, respectively. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 The BoseEinstein condensate dark matter model . . . 2.1 The GrossPitaevskii equation . . . . . . . . . . 2.2 ThomasFermi approximation . . . . . . . . . . 2.3 Emergence of vortices . . . . . . . . . . . . . . . 3 Deformation of the slowly rotating BEC dark matter halos . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The energy density and the gravitational potential of the rotating BEC dark matter halos . . . . 3.2 The continuity conditions . . . . . . . . . . . . . 3.3 The firstorder corrections to density and radius . 4 Gravitational and astrophysical properties of rotating BoseEinstein condensate dark matter halos . . . . . 4.1 Mass and gravitational potential of the slowly rotating BEC halo . . . . . . . . . . . . . . . . . 4.2 Velocity dispersion of particles in slowly rotating BEC halos . . . . . . . . . . . . . . . . . . . 4.3 The logarithmic density slopes . . . . . . . . . . 4.4 Tangential velocity of test particles in slowly rotating BEC dark matter halos . . . . . . . . . . 5 Galactic rotation curves in the rotating BoseEinstein condensate dark matter model . . . . . . . . . . . . . 5.1 Fitting results . . . . . . . . . . . . . . . . . . . 6 Discussions and final remarks . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . .

Contents
Appendix A: Derivative of the density profile with
respect to kr . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
The assumption of the existence of dark matter (DM) is one of
the cornerstones of present day cosmology and astrophysics
[
1–5
]. The first evidence for its presence in the Universe
was provided by the study of the galactic rotation curves.
More exactly, the idea of DM was first suggested to explain
the rotation curves of spiral galaxies, whose rotation curves
decay far more slowly than one would expect by taking into
account the effects of baryonic matter (gas and stars) only.
This behavior is considered as evidence for the existence of a
supplementary (and exotic) missing mass component, most
likely consisting of new particle(s) lying beyond the standard
model of particle physics. The rotation curves still represent
the most powerful and convincing evidence for DM [
6–9
].
But various other astrophysical and cosmological
observations have also provided evidence for the existence of dark
matter, like, for example, the recent determination of the
cosmological parameters from the Planck satellite observations
of the cosmic background radiation [10]. These observations
have also shown that dark matter cannot be explained by
baryonic matter only, thus confirming the standard cold
dark matter ( CDM) cosmological paradigm. Other types
of observations, such as gravitational lensing also require the
existence of dark matter for a consistent interpretation of the
data [
11–13
]. A particularly strong evidence for the existence
of dark matter is provided by the observations of a galaxy
cluster called the Bullet Cluster. In this cluster the baryonic
matter and the dark matter components are separated due to a
collision of its two components that occurred in the past [
14
].
Measurements of the cosmological parameters by using the
Planck data on the cosmic microwave background radiation
indicate that the Universe is composed of 4% baryons, 22%
nonbaryonic dark matter and 74% dark energy [
10
].
Dark matter models can be divided into three types: cold,
warm and hot dark matter models, respectively, by the energy
of the particles composing them [
15
]. The main candidates
for dark matter are the WIMPs (weakly interacting massive
particles) and the axions [
15
]. WIMPs are heavy particles that
interact via the weak force [
16,17
]. Axions are bosons that
were first proposed to solve the strong CP problem [
18,19
].
If the axions form the dark matter, then at low temperature
the axion gas will form a Bose–Einstein condensate (BEC).
There are also other theories that try to explain the
observations without introducing dark matter. These theories are
based on a modification of the law of gravity at the
galactic scales. The earliest one of them is the MOND theory
(modified Newtonian dynamics) [
20
]. Modified gravity
theories have also been used extensively as an alternative to dark
matter [
21–31
]. An interesting possibility to detect the
presence of dark matter is via its possible annihilation. If such a
physical process does indeed occur, then a large number of
gammaray photons and positrons could be produced.
Observationally, some excess positron emission in our galaxy has
been detected [
32–39
]. Therefore, it may be possible that the
excess positron and gammaray emissions could be explained
by the annihilation of dark matter with mass m ∼ 10–100
GeV [
32–34
]. For a detailed discussion of this problem, as
well as of the possibility of alternative interpretations of the
observational data see [
35–39
].
Even though dark matter models can give a good
explanation of the qualitative behavior and constancy of the rotation
curves, an important contradiction did arise as a result of
the in depth comparison of the simulation results with the
observations. Data on almost all observed rotation curves
show that they rise less steeply than cosmological
simulations of structure formation in the standard CDM model
in the presence of a single pressureless dark matter
component do predict. The simulations indicate a central dark
matter density profile that behaves as ρ ∼ 1/r (a cusp) [
40
],
while the observed rotation curves indicate the presence of
constant density cores [
41,42
]. This is the socalled core–
cusp problem in dark matter physics. Another important open
question dark matter models have to face is the “too big to
fail” problem [
43,44
]. By using the Aquarius simulations it
was shown that the most massive subhalos in the dark matter
halos predicted in the CDM model are inconsistent with the
dynamics of the brightest Milky Way dwarf spheroidal
galaxies [
44
]. While the bestfitting hosts of the dwarf spheroidals
galaxies have 12 < Vmax < 25 km/s, the λCDM simulations
all predict at least ten subhalos with Vmax > 25 km/s. These
results cannot be explained in the framework of the
CDMbased models of the satellite population of the Milky Way.
The main problem emerging here is related to the densities
of the satellites, with the dwarf spheroidals required to have
dark matter halos that are a factor of ∼ 5 more massive than
is observed.
These problems related to the physics of the dark matter
may be solvable if one goes one step beyond the standard
CDM model and assumes that the dark matter particles
may possess some forms of selfinteraction. Such a
possibility has gained some observational support after the study
of the data provided by the observations of 72 cluster
collisions, including both ‘major’ and ‘minor’ mergers, with the
observations done by using the Chandra and Hubble Space
Telescopes [
45
]. Collisions between galaxy clusters can
provide an important test of the nongravitational forces acting
on dark matter, and the analysis done in [
45
] gives an upper
limit of the ratio of the selfinteraction crosssection σDM and
of the mass m of the dark matter particle as σDM/m < 0.47
cm2/g (95% confidence level). A new upper limit on the
selfinteraction crosssection of dark matter of σDM < 1.28 cm2/g
(68% confidence level) was obtained in [
46
]. From a
theoretical point of view different selfinteracting dark matter
models were investigated in [
47–50
]. The effects of
selfinteracting dark matter on the tidal stripping and evaporation
of satellite galaxies in a Milky Waylike host were
investigated in [51]. Velocityindependent selfinteracting dark
matter models show a modest increase in the stellar
stripping effect with satellite mass, whereas velocitydependent
selfinteracting dark matter models show a large increase
in this effect towards lower masses, making observations of
ultrafaint dwarfs prime targets for distinguishing between
and constraining selfinteracting dark matter models. The
response of selfinteracting dark matter halos to the growth
of galaxy potentials using idealized simulations, each run in
tandem with standard collisionless cold dark matter (CDM)
was investigated in [
52
]. A greater diversity in the
selfinteracting dark matter halo profiles was found, as
compared to the standard CDM halo profiles. A selfinteracting
dark matter halo simulated with crosssection over mass
σDM/m = 0.1 cm2/g provides a good match to the
measured dark matter density profile of A2667, while an
adiabatically contracted CDM halo is denser and cuspier. The
cored profile of the same halo simulated with σDM/m = 0.5
cm2/g is not dense enough to match A2667. These findings
are in agreement with previous results [
45
] suggesting that
σDM/m ≥ 0.1 cm2/g is disfavored for dark matter collision
velocities in excess of about 1500 km/s. Therefore the
possibility that dark matter is a selfinteracting component of
the Universe cannot be rejected a priori, and physical
models whose component particles naturally exhibit this
property may provide valuable explanations and suggestions for
the dark matter candidates, and for their properties. From
both a fundamental theoretical point of view, as well as from
a phenomenological perspective, the physically best
motivated selfinteracting dark matter model can be obtained by
assuming that dark matter is in a Bose–Einstein condensate
phase.
The idea that at very low temperatures all integer spin
particles may occupy the lowest quantum state, at which
point macroscopic quantum phenomena become apparent,
was proposed, from a statistical physical point of view by
Bose and Einstein in the 1920s [
53–55
]. The Bose–Einstein
condensation process is determined by the quantum
mechanical correlation of the gas particles, which implies that the de
Broglie thermal wavelength is greater than the mean
interparticle distance. The transition to the condensate phase begins
when the temperature T of the boson gas is lower than the
critical one, Tcr, given by [
56–59
]
2π h¯2ρc2r/3
Tcr = ζ 2/3(3/2)m5/3kB ,
(1)
where m is the particle mass in the condensate, ρcr is the
critical transition density, kB is Boltzmann’s constant, and ζ
denotes the Riemmann zeta function.
It took around 70 years to achieve the experimental
realization of the Bose–Einstein condensation, which was first
observed in dilute alkali gases in 1995 [
60–62
]. From a
physical point of view the presence of a BEC in an experimental
framework is indicated by the appearance of sharp peaks in
both coordinate and momentum space distributions of the
particles.
Up to now, the main evidence for the existence of BECs
comes from laboratory experiments, performed on a very
small scale. However, the possibility of the presence of some
forms of condensates in the cosmic environment cannot be
rejected a priori, and the implications of the possible
existence of a condensate state of matter in a astrophysical or
cosmological background is certainly worth to investigate. It
has been hypothesized that due to their superfluid properties
in general relativistic compact objects, like neutron or quark
stars, the neutrons may form Cooper pairs, which would
condense eventually. Bose–Einstein condensate stars could have
maximum masses of the order of 2 M , maximum central
densities of the order of 0.1–0.3 ×1016 g/cm3, and minimum
radii in the range of 10–20 km, respectively. Their
interesting physical and astrophysical properties were investigated
in [
63–72
].
The idea that dark matter is in the form of a Bose–Einstein
condensate was proposed initially in [
73
], and then
rediscovered/reinvestigated, in [
74–84
]. A systematic study of the
properties of the BEC dark matter halos, based on the
nonrelativistic Gross–Pitaevskii (GP) equation in the presence
of a confining gravitational potential, was initiated in [
85
]. A
further simplification of the mathematical formalism of the
gravitationally bounded BECs can be achieved by
introducing the Madelung representation of the wave function, which
allows the representation of the GP equation in the equivalent
form of a continuity equation, and of a hydrodynamic Euler
type equation. Hence with the use of the Madelung
representation we obtain the fundamental result that dark matter can
be described as a nonrelativistic, Newtonian Bose–Einstein
condensed gas in a gravitational trapping potential, with the
pressure and density obeying a polytropic equation of state,
with polytropic index n = 1. The validity of the BEC dark
matter model was tested by fitting the Newtonian
tangential velocity equation to a sample of rotation curves of low
surface brightness and dwarf galaxies, respectively.
The thermal correction to the dark matter density profile
where obtained in [
86
]. In [
87
] it was shown that the density
profiles of the Bose–Einstein condensed dark matter
generally show the presence of an extended core, whose formation
is explained by the strong interaction between dark matter
particles. A further observational test of the model can be
obtained by computing the mean value of the logarithmic
inner slope of the mass density profile of dwarf galaxies,
and by comparing it with the observations. The study of the
properties of the Bose–Einstein condensate dark matter on
a cosmological and astrophysical scales is presently a very
active field of research [
88–125
]. The properties of the fuzzy
dark matter, assumed to be formed of a light (m ∼ 10−22
eV) boson, having a de Broglie wavelength λ ∼ 1 kpc, were
recently investigated in [126].
If the static properties of the BEC dark matter halos have
been studied extensively, their rotational properties have
attracted less attention. In [
81
] the presence of vortices in
a selfgravitating BEC dark halo, consisting of ultralow
mass scalar bosons was investigated, and it was pointed out
that rotation of the dark matter imprints a background phase
gradient on the condensate, which induces a harmonic trap
potential for vortices. A detailed study of the vortices in
rotating BEC dark matter halos was performed in [
91
], where
strong bounds for the boson mass, interaction strength, the
shape and quantity of vortices in the halo, and the critical
rotational velocity for the nucleation of vortices were found.
An exact solution for the mass density of a single,
axisymmetric vortex was also found. The effects of rotation on a
superfluid BEC dark matter halo were explored in [
92
], by
assuming that a vortex lattice forms. With finetuning of the
bosonic particle mass and the twobody repulsive interaction
strength, it was found that substructures on rotation curves
that resembles some observations in spiral galaxies could
exist. The study of the equilibrium of selfgravitating,
rotating BEC halos, which satisfy the Gross–Pitaevskii–Poisson
equations was performed in [
103
]. Vortices are expected to
form for a wide range of BEC parameters. However, vortices
cannot form for avanishing selfinteraction. The question if
and when vortices are energetically favored was also
considered, and it was found that vortices form as long as the
selfinteraction is strong enough.
It is the goal of the present paper to study the properties
of the BEC dark matter halos in the presence of rotation.
Rotation might be a general property of galaxies, whose
origin may be traced back to some physical processes in the
early Universe. In order to describe the Bose Einstein
condensate dark matter we adopt the Gross–Pitaevskii equation,
which gives an effective meanfield description of the
multiparticle bosonic system. The mathematical description of the
condensate is significantly simplified after introducing the
Thomas–Fermi approximation, which allows for the
description of the dark matter as a gas obeying a polytropic equation
of state, with polytropic index n = 1. By assuming a rigid
body rotation for the halo, with the use of the hydrodynamic
representation of the Gross–Pitaevskii equation, we obtain
the basic relation describing the density distribution of the
rotating condensate, which naturally generalizes the
previously obtained static profile. From the density distribution
of the rotating condensate we derive the general
representations for the mass distribution, boundary (radius), potential
energy, velocity dispersion, tangential velocity, as well as for
the logarithmic density and velocity slopes. From the general
results we obtain explicit expressions for the radius, mass,
and tangential velocity in the first order of approximation,
under the assumption of slow rotation. A comparison of our
results with the observations is performed by fitting the
theoretical expressions of the tangential velocity of massive test
particles moving in rotating Bose–Einstein condensate dark
halos with the data of 12 dwarf galaxies and of the Milky
Way galaxy, respectively.
The present paper is organized as follows. The
mathematical and physical description of the Bose–Einstein
condensate dark matter is introduced in Sect. 2, where the Gross–
Pitaevskii equation, and the Thomas–Fermi approximation
are presented. The rotating Bose–Einstein condensate dark
matter structures are investigated in Sect. 3, by using the
general approach that allows us to obtain the exact expression of
density as expressed in terms of Legendre polynomials. The
slowly rotating dark matter halo in the framework of the
firstorder approximation is considered, and the density profile
and the radius are also presented. The gravitational and
astrophysical properties of the rotating Bose–Einstein condensate
dark matter halos are studied in Sect. 4. In this section we
derive the expressions of a number of important astrophysical
quantities, like, for example, the mass distribution, potential
energy, logarithmic slopes of the density and velocity, which
could allow an in depth comparison between the theoretical
model and observations. The fitting of the theoretical model
with astronomical/astrophysical data is performed in Sect. 5,
where we compare the predicted Bose–Einstein condensate
galactic rotation curves with the observational data from 12
dwarf galaxies and from the Milky Way galaxy. We discuss
and conclude our results in Sect. 6.
2 The Bose–Einstein condensate dark matter model
In the present section we briefly introduce the fundamental
physical concepts related to the Bose–Einstein condensation,
as well as the basic equations describing the rotating
condensate. It has been shown that if the dark matter is composed
of ultralight boson particles with mass m ∼ 10−22 eV and
wavelength λ ∼ 1 kpc, then the transition temperature to
a Bose–Einstein condensate is of the order Tc ∼ 109 K
[
83
]. Hence, if dark matter is composed of Bose particles,
like, for example, the axion, it is quite natural to assume
that dark matter is in a Bose–Einstein condensate state. For
a recent discussion of the arguments from particle physics
that may motivate the existence of the ultralight dark
matter, as well as of its properties and astrophysical signatures
see [
126
].
2.1 The Gross–Pitaevskii equation
A Bose–Einstein condensate is a phase of matter in which
all the particles are localized in the ground state. The Bose–
Einstein condensation occurs for particles that have integer
spins, and thus obey Bose–Einstein statistics. We will assume
in the following that the bosons are weakly interacting and
that their interaction is described by a twobody
interparticle potential. We start our analysis by writing down first the
manybody Hamiltonian of the bosonic system,
Hˆ =
dr ˆ †(r) − 2h¯m2 ∇2 + Vrot(r) + Vext(r) ˆ (r)
1
+ 2
drdr ˆ †(r) ˆ †(r )V (r − r ) ˆ (r) ˆ (r ),
where ˆ (r) and ˆ †(r ) are the annihilation operator and the
creation operator at the position r, respectively, V denotes
the interparticle interaction potential, and m is the mass of
the particle in the condensate. In the case of a rotating dark
matter halo, the external potential Vext is the gravitational
potential, and Vrot is the effective centrifugal potential. In
the following we will adopt the comoving frame, that is, the
frame that is rotating with the same speed as the system.
To simplify the mathematical formalism, we introduce
the meanfield description, in which we decompose the field
operator in the form ˆ (r) = 0 + ˆ (r), and treat the
operator ˆ (r) as a small perturbation. Then for the meanfield
component 0 we have 0 = √N / V , where N is the total
particle number, and V is the volume. Hence 0 is equal to
the square root of the number density of the particles [
56
].
In the general timedependent case, the field operator in
the Heisenberg picture is given by
ˆ (r, t ) = ψ (r, t ) + ˆ (r, t ),
where ψ (r, t ) = ˆ (r, t ) is also called the condensate wave
function. Then for the number density of the condensate we
have ρn(r, t ) = ψ (r, t )2. The normalization condition is
N = ρn(r, t )d3 r.
In the Heisenberg representation the equation of motion
of the field operator is
∂
i h¯ ∂t ˆ (r, t ) = [ ˆ , Hˆ ] =
− 2h¯m2 ∇2 + Vrot(r) + Vext(r)
+
dr ˆ †(r , t )V (r − r ) ˆ (r , t ) ˆ (r, t ).
(2)
(3)
(4)
In the theory of the Bose–Einstein condensation one
usually assumes that the interparticle interaction is a short range
interaction, and hence we can write the interaction potential
as being proportional to a constant, which is related to the
scattering length, times a Dirac delta function [
127
]:
V (r − r) = λδ(r − r),
with
λ =
where a is the scattering length. To obtain a more
general description, in the following we introduce the function
g(ψ (r, t )2) to describe the selfinteraction term [
85
]. In
the standard approach to Bose–Einstein condensation the
selfinteraction is assumed to have a quadratic form, so that
1
g(ψ (r, t )2) = 2 λψ (r, t )4 [
56
].
With this approximation of the potential, and with the use
of the meanfield approximation, by integrating over Eq. (4)
we obtain the Gross–Pitaevskii equation, describing the main
properties of a Bose–Einstein condensate, as
∂
i h¯ ∂t ψ (r, t ) =
− 2h¯m2 ∇2 + Vrot(r) + Vext(r)
+ g (ψ (r, t )2) ψ (r, t ).
To give a more direct physical interpretation of the Gross–
Pitaevskii equation, we introduce the Madelung
representation of the wave function,
ψ (r, t ) =
ρn(r, t ) e h¯i S(r,t),
which separates the wave function into two components, its
magnitude, and a phase factor, respectively. The function
S(r, t ) has the dimensions of an action. Then in the Madelung
representation we have [
59
]
i h¯ ∂ψ ∂ S i h¯ ∂ρn 1
ψ ∂t = − ∂t + 2ρn ∂t ψ
× − 2h¯m2 ∇2 + Vrot + Vext + g (ψ 2) ψ
h¯2 ∇2√ρn
= − 2m √ρn
i h
¯
+ g (ψ 2) − 2ρn
1
+ 2m ∇ S2 + Vrot + Vext
∇ρn · ∇ S + ρn∇2 S .
(9)
Then the Gross–Pitaevskii equation is separated into two
parts. From the imaginary part we obtain
∂ρn
∂t + ∇ · (ρnv) = 0,
where v = ∇mS is the velocity of the quantum fluid. This is
the continuity equation. On the other hand, from the real part
we obtain the equation
ρn
∂v
∂t + v × (∇ × v) + (v · ∇)v
−ρn∇ Vrot − ρn∇ Vext − ∇ · σ Q ,
= −∇ P(ρn )
(5)
(6)
(7)
(8)
(10)
(11)
which is the momentum conservation, or the hydrodynamic
Euler equation [
117
]. In Eq. (11) P is the thermodynamic
pressure, which is related to the mass density ρ = ρnm of the
condensate by a barotropic type equation of state [
85,127
],
P(ρ) = g
ρ
m
ρ ρ
m − g m
.
The term σ Q is given by σ Q = − 2h¯m2 ∇√2√ρρ , and its
divergence is called the quantum stress tensor.
Therefore the two equations describing the evolution of
a Bose–Einstein condensate are the continuity and the Euler
equations of classical fluid dynamics, which describe
viscosity free flows. Also, we can see from the definition of the
velocity that the flow must be automatically irrotational. We
will discuss this issue later.
(12)
2.2 Thomas–Fermi approximation
When the number of particles in the condensate get large
enough, the contribution to the energy of the quantum
pressure term ∇ · σ Q can be neglected except near the boundary
[
58
]. Then the equations describing the condensate become
purely classical in their mathematical form, even though their
physical interpretation must be given in the framework of
quantum statistical physics.
If we consider the Bose–Einstein condensate to be static,
or have a rigid body rotation, and we work in the corotating
frame, then v = 0. Thus from Eq. (11) we obtain
∇2 [h(ρ) + Vrot + Vext] = 0,
where ∇h(ρ) = (1/ρ) ∇ P(ρ). For the exterior potential we
assume that it is the gravitational potential, Vext = Vgrav =
V , and that it satisfies the Poisson equation,
∇2V = 4π Gρ ,
Vrot = − 21 ω2(x 2 + y2),
where G is the gravitational constant. For the rotational
potential we adopt the expression
where ω is the angular velocity of the dark matter halo. In the
case of the quadratic nonlinearity we have g (ρ) = λρ/m,
and g(ρ) = λρ2/(2m2), respectively. Thus for the equation
of state of the condensate we obtain
(13)
(14)
(15)
(16)
(17)
= 1 + 1/n = 2, it follows that the polytropic index of the
condensate is n = 1.
Hence with the use of the equation of state of the BEC
dark matter from Eq. (13) we obtain the equation describing
the variation of the density of the rotating dark matter halo as
ω2
ρ − 2π G
= 0,
.
λ a h¯2
Equation (18) is the Helmholtz equation. If the system had a
different polytropic index n = 1, instead, we will get a
general Lane–Emden equation [
85
], which will be nonlinear.
If the halo is nonrotating, ω = 0, and, under the
assumption of spherical symmetry, Eq. (18) has the solution [
85
]
sin kr
ρ(r ) = A0 kr , (20)
where A0 is an integration constant. One can obtain the radius
R of the static halo from the boundary condition ρ(R) = 0,
which gives
(18)
(19)
π
R = k = π
The central density is ρc = ρ(0) = A0. We can see from
the expression of the density that the radius only depends
on the mass and scattering length of the particles. The size
of the halo is independent of the central density. We can
determine ρc from the normalization condition, d3 rρ =
M , thus obtaining
We have already mentioned that from the definition of the
velocity v = ∇ S, the flow must automatically satisfy the
condition ∇ × v = 0, and hence the quantum BEC fluid motion
should be irrotational. However, if we expect the dark matter
halo to be rotating, we must introduce singularities of the
vorticity, and therefore the halo will contain a vortex lattice.
P(ρ) = 2 mλ2 ρ2,
giving
1 λ
ρ ∇2( P) = m2 ∇2ρ .
From the equation of state of the condensate we can see that
P ∝ ρ2, and since for a general polytropic equation of state
A2l j2l (kr ) P2l (cos θ ),
(23)
where jl (X ) are spherical Bessel functions, which are the
solutions to the radial part of the equation, while Pl (cos θ )
are the Legendre polynomials—they are the solutions to the
angular part of the Helmholtz equation. In the solution, we
have neglected all the terms of odd order, since it has been
shown that a rotating mass must be symmetric about its
equator. This result is called Lichtenstein’s theorem [
135
]. To
determine the coefficients A2l in the first order of
approximation, we will write down the solution for the gravitational
potential, and use the fact that it is continuous at the boundary
of the dark matter halo.
The gravitational potential satisfies the equations
where r0 is the radius (boundary) of the halo. We can
represent the potential in a general form as
First, let us recall the concepts of vortex and vorticity.
A vortex is a region of fluid in which the flow is rotating
around an axis line. Its vorticity is the curl of the velocity,
w = ∇ × v. If the fluid rotates like a rigid body with an
angular velocity = (0, 0, ), we have the velocity v =
× r = (− y, x , 0) and the vorticity w = ∇ × v = 2 .
A vortex can also be irrotational; if it has angular velocity
= (0, 0, αr −2), its vorticity will be 0 except at the axis line,
where the vorticity will be infinite. If a fluid is irrotational,
then although the particles have an angular velocity, they will
not rotate over themselves.
If we expect the dark matter halo to rotate like a rigid
body, it will give rise to a vortex lattice [
91,92
]. It was
already shown in laboratory experiments that such vortex
lattices arise when an asymmetry is introduced [
128,129
]. It
has been shown experimentally that vortices arise at a
critical angular velocity, at which the energy of the system is
lower if it generates vortices [
91
]. When the angular velocity
gets higher, instead of generating a bigger vortex, a lattice of
several vortices will be generated [
128
].
How a vortex will influence the properties of the dark
matter halo was studied in [
91
]. Significantly, a core appears
at the center of the vortex, and within the core the density is
zero. In [
92
], this feature was used to explain the wiggles in
the rotation curves of the galaxies.
For simplicity, we will not study the vortices in this paper.
We will assume that the halo rotates like a rigid body, we
will ignore the cores that generate inhomogeneities in the
density profile, and we will concentrate our attention on how
rotation causes the deformation of the halo, and influences
its observable physical properties.
3 Deformation of the slowly rotating BEC dark matter
halos
There have been many studies considering the problem
of rotating polytropes, using different approximations and
building different models, like, for example, in [
130,131
].
For a detailed discussion of the rotational properties of n = 1
polytropes see [132]. We will assume the halo to be
rotating slowly, and thus a firstorder approximation is
sufficient. Chandrasekhar has worked on this problem in 1933
[
133,134
], and we will mainly follow his method (for a
comparative study of the different approaches to the rotation
problem see [131]).
3.1 The energy density and the gravitational potential of
the rotating BEC dark matter halos
We have already obtained the Helmholtz equation (18)
describing the matter distribution inside a rotating BEC dark
matter halo. Its general solution is given by
A2l j2l (kr ) P2l (cos θ ) .
(27)
∞
l=1
V = V0(kr ) +
V2l (kr ) P2l (cos θ ).
Then the gravitational potential equation inside the halo
becomes
1 ∂
r 2 ∂r
r 2 ∂ V
∂r
1 ∂
+ r 2 sin θ ∂θ
sin θ
∂ V
∂θ
= 4π G
In the following we denote ξ = kr , and μ = cos θ ,
respectively. Thus Eq. (27) takes the form
ξ
2 dV0
dξ
2ω2
= k2 +
d
dμ
1 − μ
2 d Pj
dμ
A0 j0(ξ ) + constant.
+ j ( j + 1) Pj = 0, j ∈ N ,
For the higherorder terms, since the P2l satisfy the equation
(24)
(25)
(26)
(28)
(29)
(30)
(31)
A2l j2l (ξ ). (32)
Since ∇2( j2l P2l ) = − j2l P2l , and by taking into account that
the Bessel functions j2l satisfy the equation
=
=
we obtain the equation
ξ
2 dV2l
dξ
4π G
= − k2
A2l
4π G
V2l = − k2
V2l = C2l ξ 2l ,
4π G
V2l = − k2
V (ξ, μ) = V0(ξ ) +
−
2l(2l + 1)
ξ 2
V2l
A2l j2l (ξ ) + constant.
A particular solution of the above equation is
The regular solution of the equation
ξ
2 dV2l
dξ
−
where C2l are arbitrary integration constants. Hence the
general solution for the V2l is
A2l j2l (ξ ) + B2l ξ 2l
+ constant,
giving for the gravitational potential V the general solution
×
∞
l=0
= 3ωk22 ξ 2 −
+constant.
∞
l=1
V2l (ξ ) P2l (μ) = 3ωk22 ξ 2 −
ω2
ρ − 2π G +
l=0
B2l ξ 2l P2l (μ)
3.2 The continuity conditions
To determine the B2l , we consider the behavior of the
pressure P, which is given by the static limit of the Euler equation
(11) as
∂ P ∂ V
∂r = −ρ ∂r + ρω2r (1 − μ2),
or, equivalently,
∂ P ∂ V 2 ρω2ξ
∂ξ = −ρ ∂ξ + 3 k2
and, for all l = 0, 1, A2l = 0. A0 is not determined by
the boundary condition of the potential at radius R, since an
arbitrary constant can always be added to the gravitational
potential. We can determine it by the boundary condition of
the matter density at the center of the dark matter halo,
dρ0(0)
dξ
ω2
= 0, ρ0(0) = 2π G + A0 = ρc.
Hence we find
ω2
A0 = ρc − 2π G .
3.3 The firstorder corrections to density and radius
With the use of the expressions for the coefficients obtained
above, after substituting ξ1 = π and ξ = kr , respectively, we
find the firstorder correction to the density ρ of the rotating
Bose–Einstein condensate dark matter halo as
After substituting the pressure with the use of the Bose–
Einstein condensate equation of state, and equating the
coefficients of the series expansions, we obtain
ω2
B2 = 12π G , B2l = 0, l = 1.
Therefore
V (ξ, μ) = 3ωk22 ξ 2 −
4π G ∞
k2
l=0
ω2
− 3k2 ξ 2 P2(μ) + constant.
To the accuracy we are working in, this potential should be
continuous with the external potential on the sphere of radius
ξ1 = k R, where R is the boundary of the nonrotating sphere,
Ve =
4π G ∞
k2
l=0
ξ C2l2+l1 P2l (cos θ ) + Constant.
To determine the coefficient A2, we will equate the potentials
and their first derivatives at ξ1,
(33)
(34)
(36)
(37)
(38)
(39)
(40)
(41)
(42)
(43)
(44)
(45)
(46)
(47)
(48)
(49)
We will describe the effect of the rotation on the
structure of the dark matter halo by using the dimensionless
parameter 2. In particular, the case of slow rotation
corresponds to values of 2 so that 2 << 1. In the limit case
2 → 0 we recover from Eq. (51) the static limit for the
halo density, ρ(r )ρc = j0(kr ). The angular momentum is
usually described by using the dimensionless spin parameter
λ = J E 1/2/G M 5/2, where J is the angular momentum,
and E is the gravitational energy of the halo [
136
]. From a
physical point of view the spin parameter λ is the ratio of the
actual angular momentum of the galaxy and of the maximum
angular momentum value needed for rotational support.
Simulations have shown that λ is in the range 0.02–0.10 [
136
].
The comparative variation of the density in the
nonrotating and rotating cases at θ = π/2 (corresponding to
the equatorial plane) is presented in Fig. 1.
To see how the halo is deformed by rotation, we need to
obtain the boundary, or the radius where ρ = 0. Since the
deformation from the spherical shape is small, in the first
approximation we write
In order to obtain the firstorder approximation we will
further expand this expression to the first order of ω2/2π G, and
we obtain
ξ =
2ω2 − 5ω2 P2(cos θ )
4Gρc
This has the same form as in Chandrasekhar’s work [
133
].
Hence the boundary of the slowly rotating Bose–Einstein
condensate dark matter halo is located at
4 Gravitational and astrophysical properties of rotating
Bose–Einstein condensate dark matter halos
In the present section we will obtain some basic gravitational
and astrophysical properties of the slowly rotating Bose–
Einstein condensate dark matter halos, which could allow
for an in depth comparison of the theoretical model with
the astronomical observations. In particular, we will consider
the mass distribution within the halo, as well as to its
gravitational potential energy. Moreover, we will concentrate on
astrophysical parameters like velocity dispersion,
logarithmic density and velocity slopes, and the tangential velocity
expression, which allows for a detailed comparison of the
model predictions with observational data.
4.1 Mass and gravitational potential of the slowly rotating
BEC halo
As we have already seen, the general solution for the matter
density distribution ρ inside the rotating halo in spherical
coordinates is given by
The boundary (radius) r0 of the halo is defined as the surface
whose points satisfy the condition ρ(r0) = 0. The equation
for the density involves infinitely many terms in general.
The mass profile m(r, θ = π ) within a radius r is given
which involves an integration of the spherical Bessel
functions, which can be done by using the relation
j2l (kr )r 2dr = √π 2−2(l+1)r03
3
l + 2
(kr0)2l
×1 F˜2 l + 2 ; l + 2 , 2l + 2 ; − 41 k2r02 ,
3 5 3
(61)
where 1 F˜2 is the is the regularized generalized
hypergeometric function p Fq (a; b; z) / (b1) · · · bq . Thus we
obtain for the total mass the expression
M (r0) =
0
×
×
3G
2 3
π ω r
0 sin θ dθ + 2π 3/2
dθ P2l (cos θ ) sin θ
0
π
√π 2−2(l+1)r 3
0
∞
l=0
A2l
3
l + 2
(kr0)2l 1
For dark matter halos located within a radius r ≤ π/ k −
3ω2/4Gρck, we can calculate the mass distribution within
radius r :
m(r ) = 2π
+ kr cos kr − sin kr = 4πk3ρc (kr )
2 sin kr
kr
− 1 −
2 cos kr +
Fig. 2 Variation of the dimensionless ratio m(ξ )/M∗, M∗ = 4πρc/k3,
as a function of ξ = kr for a rotating Bose–Einstein condensate dark
matter halo for different values of 2: 2 = 0 (solid curve), 2 = 0.01
(dotted curve), 2 = 0.0225 (short dashed curve), 2 = 0.04 (dashed
curve), 2 = 0.0625 (long dashed curve), and 2 = 0.09 (ultralong
dashed curve), respectively
As compared to the nonrotating case, a second term appears,
which is due to the presence of the rigid body type rotation.
The mass profile within radius r is bigger than in the
nonrotating case, and it depends on the central density ρc. These
results are consistent with the slower decay of the density
profile for larger values of the radial coordinate r . The variation
of the dimensionless ratio m(ξ )/M∗, where M∗ = 4πρc/ k3,
is represented in Fig. 2.
The total mass of the condensate in the first order of
ω2/2π G is
M (r0) = 2π
π
For a given galaxy, the total halo mass is fixed by the physical
processes leading to its formation, and it is unchanged due to
the presence of rotation. But, as one can see from Eq. (64), a
rotating halo is able to hold more mass than a static one, and
in this sense the rotation of the dark matter halo becomes a
stabilizing factor against gravitational collapse.
To perform the integration in the accuracy of the first order
of ω2/2π G, we first integrate over r , then expand the result
in the first order, and then we perform the integration over
μ. This procedure does not affect the final result obtained by
doing the series expansion after performing the full
integration, since ω2/2π G is independent of r and μ, and thus the
power of it is not changed after each integration.
Then we can express the central density ρc as a function
of the total mass and angular velocity as
The total volume of the halo becomes
VBEC = 2π
sin θ dμ
From the above expression of the volume it follows that due
to rotation the halo has expanded. The added volume is
proportional to ω2, and inversely proportional to ρc.
With the help of the total mass and of the volume we obtain
the mean density of the BEC halo:
Using the gravitational potential, we can calculate the
gravitational binding energy U defined as
1
U (r ) = 2
ρ(r, θ )V (r, θ )d3 r.
We can determine the constant by using the continuity of the
potential near the radius ξ1 = k R,
ω2 2
3k2 ξ1 −
2ω2
3k2 ξ1 −
4π G ω2
k2 (ρc − 2π G ) j0(ξ1) + constant =
4πk2G (ρc − 2ωπ2G ) j0(ξ1) = − 4πk2G Cξ120 .
Thus we obtain
4π G ω2
constant = − k2 ρc − k2 (π 2 − 2).
Hence the gravitational potential is given by
4π G 2ω2
V (r, θ ) = − k2 ρc [1 + j0(kr )] + k2 j0(kr )
5ω2π 2
j2(kr ) P2(cos θ )
− 3k2
ω2 ω2
+ 3k2 (kr )2 [1 − P2(cos θ )] − k2
Hence the total gravitational potential energy of the BEC
dark matter halo is given by
U (r0) = π
π
As compared to the nonrotating case, the total gravitational
energy has a second negative term, and hence for certain
values of ρc it is lower than in the nonrotating case.
The centrifugal potential of the rotating dark matter halo
is
Hence we can calculate the centrifugal potential energy as
(67)
Ucen = 2π
π
∂ < v j >
∂t
+
∂ V
= −n ∂ x j −
i
i
n < vi >
∂ nσi2j
∂ xi
∂ < v j >
∂ xi
The centrifugal potential energy is always lower than 0.
Hence the effective potential energy in the corotating
frame can be obtained:
Ueff = U + Ucen =
12π 3Gρc2
k5
,
π 2 − 2 .
where the notation <> means an average at a given point
and time (x , t ), and
2
σi j ≡ < (vi − < vi >) v j − < v j > >
= < vi v j > − < vi >< v j > .
implies ∂∂vtj
For a static spherical symmetrical system, one can further
simplify the radial Jeans equation by adopting the
assumptions: (1) steadystate hydrodynamic equilibrium, which
0, respectively; (2)
= 0, and < vr >=
< vθ >=< vφ >= 0, and σr2θ = σr2φ = σθ2φ = 0,
respectively, which follows from the spherical symmetry of the
system, and (3) a single tangential velocity dispersion for all
directions σt2t = σθ2θ = σφ2φ .
Hence when the tangential velocity dispersion tensor is
isotropic, σr2r = σt2t = σ 2, the Jeans equation reduces to
ρ1n ∂ρ∂nrσ 2 = − ∂∂Vr . (79)
By assuming that all particles have the same mass, the particle
velocity dispersion is obtained:
σ 2 = ρ1 r ∞ ρ ∂∂Vr dr. (80)
For a rotating system, we can still adopt the following set
of assumptions: (1) ∂∂vtj = 0, vr = vθ = 0; (2) vi v j =
2
0, and (3) σr2r = σφφ = σ 2 (isotropic velocity distribution),
respectively.
Thus in the presence of rotation the radial Jeans equation
becomes
∂∂r (ρnσ 2) − ρrn vr2 = −ρn ∂∂Vr . (81)
Therefore for the mean value of the square of the radial
velocity we obtain
vr2(r, θ )
1
= ρ(r, θ ) r
r0
In order to find an explicit expression for vr2(r, θ ) , we
expand the integrand to the firstorder in ω2/2π G. Hence
for r < r0 we find
vr2(r, θ )
=
2ρcGπ
k3r
ω2
sin kr + 24k5r 3 cos(kr )
×
− 12k2r 2 + 10 P2(cos θ ) 6k4r 4 + π 2(6 − 7k2r 2)
+ 12k2r 2 + 10 P2(cos θ )π 2
2 2
− 6 + k r
× cos(2kr ) + 24k3r 3 sin(kr )
+ 30k P2(cos θ )π 2r sin(2kr ) .
We define the kinetic energy K of the halo in terms of the
average velocity dispersion σv as
3
K (r ) = 2
3
K (r ) = 2 M (r )σv2,
If the velocity dispersion is a constant,
(86)
(87)
(89)
(90)
.
(91)
and at the boundary K takes the value
K (r0) =
K (r0)
U (r0) 
For the ratio of the kinetic and potential energy, after
expanding to the first order of ω2/2π G, we obtain
Comparing this expression to the nonrotating case, we can
see that rotation generates a second (negative) term in the
parentheses, and hence in the presence of rotation the ratio
of the kinetic and potential energy of particles in motion in
a rotating BEC dark matter halo is lower than in the
nonrotating case.
4.3 The logarithmic density slopes The logarithmic density slope of the rotating dark matter halo is defined by [87]
αBEC(r, θ ) = d lndρln(rr, θ ) . (88)
By taking into account the general solution of the Helmholtz
equation 18 we can calculate the logarithmic slope of the
rotating BEC dark matter halo as follows. First for the
derivative of the density with respect to the radial dimensionless
variable kr we have (for the proof see Appendix A)
dρ
d(kr ) =
∞
l=0
A2l −
,
EBC 0.20
Fig. 3 Variation of the mean value of the logarithmic density slope
αBEC for a rotating Bose–Einstein condensate dark matter halo as a
function of , for different values of the core radius Rcore: Rcore =
0.30R (solid curve), Rcore = 0.35R (dotted curve), Rcore = 0.40R
(short dashed curve), Rcore = 0.45R (dashed curve), Rcore = 0.50R
(long dashed curve), and Rcore = 0.55R (ultralong dashed curve),
respectively
By expanding Eq. (91) to the first order of ω2/2π G, we
obtain
ω2
αB EC (r, θ ) = − [1 − kr cot kr ] − 12π Gρc kr cos(kr )
×
− 6 + 6kr cot kr + 5 P2(cos θ )π 2 kr j1(kr )
− (2 + kr cot kr ) j2(kr ) .
(92)
As compared to the nonrotating case, a third term is added
to the expression of the logarithmic slope of the density. The
third term is smaller than the nonrotating value of αBEC, and
it varies with θ . This seems reasonable, since the rotation
pushes the matter outward, and the ratio of the center density
to the density at larger radii is smaller.
Since the logarithmic density slope varies with θ , it is
no longer convenient to define the core radius Rcore as
αBEC(Rcore) = 1. If we simply define it as Rcore = n R =
nπ/ k, where n is a constant which must be determined from
observations (n = 0.6 gives the value of the core radius in
the static case [
87
]), we can define the mean value of the
logarithmic density slope within the radius 0 ≤ r ≤ Rcore as
1
αBEC = Rcore 0
Rcore
αBEC(r, θ )θ=π/2 dr.
(93)
The variation of the mean value of the logarithmic slope
of the density αBEC is represented, as a function of , and
for different values of Rcore, in Fig. 3.
The core density is small as compared to the nonrotating
case at θ = 0, and larger at θ = π/2.
For a given ρc, the quantity ρc Rcore can be obtained:
π
ρc Rcore = 8R2
M −
4.4 Tangential velocity of test particles in slowly rotating
BEC dark matter halos
In the Newtonian approximation the tangential velocity of
a test particle moving in the Bose–Einstein condensed dark
matter halo is given by
Vt2g(r ) = Gmr(r ) . (96)
In the slow rotation approximation, and for r ≤ π/ k −
3ω2/4Gρck, we have
Vt2g(r ) = −
×
×
4π Gρc
k2
(kr )2
3
1 −
2
cos(kr ) −
+ cos kr −
sin kr
kr
sin(kr )
kr
sin kr
kr
=
We can see that due to rotation a second positive term is
added to the expression of the tangential velocity, so that at a
distance r , the tangential velocity of a test particle becomes
higher as compared to the nonrotating case. This is because
the mass profile within the radius is higher. The variation of
the ratio Vtg(ξ )/ V∗ as a function of ξ = kr , where V∗ =
4π Gρc/ k2, is represented in Fig. 4.
At the equatorial boundary of the halo we have
+
We can see that the second term proportional to ω2 is also
positive.
Hence in a rotating BEC dark matter halo, the tangential
velocity of a test particle is larger than in the static one. Since
the ratio of the density at larger and smaller radii is greater
(94)
(95)
(97)
(98)
Fig. 4 Variation of the dimensionless tangential velocity of test
particles Vtg(ξ )/ V∗, V∗ = 4πρc/ k2, as a function of ξ = kr , for a rotating
Bose–Einstein condensate dark matter halo for different values of 2:
2 = 0 (solid curve), 2 = 0.01 (dotted curve), 2 = 0.0225 (short
dashed curve), 2 = 0.04 (dashed curve), 2 = 0.0625 (long dashed
curve), and 2 = 0.09 (ultralong dashed curve), respectively
than in the nonrotating case, it follows that the tangential
velocity is bigger at larger radii than in the nonrotating case.
An important observational quantity is the logarithmic
slope of the tangential velocity βtg, defined by
βtg (r ) = −
d ln Vtg (r )
d ln r
.
Generally, βtg can be obtained:
d ln Vtg
βtg(r ) = − d ln r
d ln Vt2g
= − 2d ln r
r dVt2g
= − 2Vt2gdr
r
= − 2Vt2g
1
= − 2
Gm (r )
r
−
Gm(r )
r 2
m (r )
r − 1 .
m(r )
At the center, where r = 0, we have βtg(0) = −1. This gives
the same result as in the static case, since ω = 0 at the center.
By expanding the logarithmic slope of the tangential
velocity to the first order in ω2/2π G, we obtain
βtg
1 k2r 2
2 1 + kr cot(kr ) − 1
+
3 3 3kr cos(kr ) + (−3 + k2r 2) sin kr
k r
6(−kr cos kr + sin kr )2
This is smaller than the value without rotation. The variation
of the logarithmic slope of the tangential velocity is shown,
for different values of the dimensionless parameter , in
Fig. 5.
(101)
Fig. 5 Variation of the logarithmic slope of the tangential velocity βtg
as a function of ξ = kr , for a rotating Bose–Einstein condensate dark
matter halo for different values of 2: 2 = 0 (solid curve), 2 = 0.01
(dotted curve), 2 = 0.0225 (short dashed curve), 2 = 0.04 (dashed
curve), 2 = 0.0625 (long dashed curve), and 2 = 0.09 (ultralong
dashed curve), respectively
5 Galactic rotation curves in the rotating Bose–Einstein
condensate dark matter model
As a next step in our analysis we compare the predictions
of the slowly rotating Bose–Einstein condensate dark
matter model with the observational data obtained for a
sample of HSB, LSB and dwarf galaxies. From a realistic
astrophysical point of view, the matter content in a galaxy
consists of a distribution of baryonic (normal) matter, obtained
as the algebraic sum of the masses Mstar of the stars, of
the ionized gas with mass Mgas, of the neutral hydrogen
of mass MHI etc., as well as of dark matter of mass MDM.
In the following we assume that dark matter is in the form
of a slowly rotating Bose–Einstein condensate. Therefore
the total mass of the galaxy is can be obtained: Mtot =
Mstar + Mgas + MHI + MDM + · · · = MtBot + MDM, where
MtBot = Mstar + Mgas + MHI + · · · is the total baryonic mass
in the galaxy. The rotation velocity of a test particle vr ot is
given by the sum of the different matter contributions, as
2 2 2 2
vrot = vgas + vstars + · · · + vhalo,
(102)
2 2
where vgas and vstars are the contributions of the baryonic
2
gas and stars, respectively, while vhalo is the dark matter
contribution, which we assume to be given by Eq. (97). Hence
the contribution of the rotating BEC dark matter halo to the
rotational velocity can be represented as
2
vhalo = 80.563 ×
×
×
R
kpc
2
Table 1 The bestfit parameters
for the 12 considered galaxies
In Eq. (103), R is the radius of the static BEC dark matter
halo, which is fixed by the numerical values of the scattering
length a and the mass m of the dark matter particle.
5.1 Fitting results
In order to test our model, we apply it to small nearby dwarf
galaxies (size ≤ 12 kpc) and our Milky Way galaxy. We
use the data of the Spitzer Photometry and Accurate
Rotation Curves (SPARC) obtained in [
138
] for investigation. The
baryonic components (bulge, disk and gas) included in the
data are obtained by observations using the homogeneous
surface photometry at 3.6 µm [
138
]. The rotation curve
contributions of the baryonic components are only determined
by the masstoluminosity ratios ϒ ∼ 1M /L of the disk
and of the bulge. Nevertheless, the surface brightness and the
resultant rotation curves obtained may also suffer from some
systematic uncertainties, such as the irregularities in
brightness profiles, uncertainties in inclination and patchy
distribution of gas [
138
]. We choose the candidates which are Hubble
stage T = 0−6 galaxies, bulgeless galaxies, and with the
distance to the galaxy D ≤ 20 Mpc. Generally speaking, these
galaxies have less diffuse features and smaller uncertainties
in the observational data. Based on these criteria, we
consider 12 dwarf galaxies for testing the present Bose–Einstein
condensate dark matter model.
Based on Eq. (103), we have altogether four free
parameters for fitting (R, ρc, ω, ϒ ). The masstoluminosity ratio of
the disk ϒ mainly affects the central rotation curve, while the
other three control the entire shape of rotation curve. Here,
we define the reduced χ 2 as
R (kpc)
,
where Ndof is the number of degrees of freedom, vrot,i is the
calculated rotation velocity, vobs,i is the observed rotation
velocity, and σi is the observational uncertainties of the
rotation velocity. By minimizing the reduced χ 2 value, we can
obtain the bestfit values of R, ρc, ω and ϒ for each of the
considered dwarf galaxies (see Table 1).
In Fig. 6, we show the bestfit rotation curves for the 12
dwarf galaxies.
We can see that most of the bestfit values of R fall into
a small range R = 5–8 kpc. Although the bestfit R for the
UGC07151 galaxy is quite small (R = 2.9 kpc), the
accept2
able range of R is R = 2.5–6.5 kpc for χred ≤ 4. Therefore,
the results for the 12 dwarf galaxies are still consistent with
a fixed value of R ≈ 6.5.
With the help of Eq. (64), which gives the mass of the
Bose–Einstein condensate dark halo, we can predict the mass
of the considered sample of galaxies. In the limit of small
rotational values we obtain for the total mass the expression
M = 1.87043 × 107 ×
The predicted masses of the 12 dwarf galaxies are
presented in Table 2.
We also apply our model to our Milky Way galaxy. By
using the data obtained in [
139
] and the baryonic model used
in [
140
], we perform a similar fit for the Milky Way data (see
Fig. 7). Since the masstoluminosity ratios of the bulge and
disk are included in the baryonic model, we only have three
free parameters to fit. The bestfit values are R = 6.9 kpc,
150
/)
s
m
k
t(y100
i
c
o
l
e
v
n
ito 50
a
t
o
R
00
200
/)
s
km150
(
y
it
c
leo100
v
n
o
ti
ta 50
o
R
00
120
/)s100
m
k
t(y 80
i
c
leo 60
v
n
ito 40
a
t
oR20
00
5
5
5
r (kpc)
r (kpc)
r (kpc)
2
NGC0024
NGC0100
NGC2976
1
2
5
6
7
1
2
4
5
6
UGC06667
UGC07151
UGC08286
2
4
6
8
10
r (kpc)
r (kpc)
ρc = 3.0 × 10−24 g cm−3 and ω = 2.6 × 10−16 s−1 (χr2ed =
0.54). We also show the fit for R = 10 kpc (χr2ed = 1.02) for
reference in Fig. 7. Generally speaking, the bestfit values for
the Milky Way galaxy and the dwarf galaxies are consistent
with each other.
6 Discussions and final remarks
In the present paper we have investigated the possibility that
the BEC dark matter halos could be in fact rotating. Presently,
there is a common paradigm in the study of superfluidity
according to which rotational motion in a Bose–Einstein
condensate can exist only in the presence of quantized vortices.
However, in a recent numerical study [
141
] it was shown
that the merging of two twodimensional concentric Bose–
Einstein condensates with axial symmetry may lead to the
formation of a spiral dark soliton. This happens if one of
the two condensates has a nonzero initial angular
momentum. The spiral dark soliton makes possible the transfer of
angular momentum between the two condensates, and allows
the merged condensate to rotate, even in the absence of
quantized vortices. A similar physical process could have acted on
a galactic scale, since galactic collisions and merging, which
Table 2 The predicted mass of
the dark halo for the 12
considered galaxies
Fig. 7 The Milky Way rotation curve. The baryonic contributions are
represented by the red line (bulge) and green line (disk). The blue lines
are the dark matter contribution (solid: R = 6.9 kpc; dashed: R =
10 kpc). The black lines are the resultant rotation curves (solid: R =
6.9 kpc; dashed: R = 10 kpc). The gray lines and error bars are the
observational data obtained in [
139
]
occurred frequently in the early stages of the evolution of
the Universe, may have favored angular momentum transfer
between galaxies. Therefore the rotation of BEC dark
matter halos may be possible even in the absence of quantized
vortices.
By using the Thomas–Fermi approximation, the basic
evolution equations of a rotating BEC halo reduce to the
case of the rotating polytropic spheres with index n =
1. Many mathematical methods have been developed for
the investigation of such systems. For example, in [
131
]
the first exact analytic solution for an arbitrarily
rotating gaseous polytrope under the assumption of the oblate
spheroidal shape was derived. To obtain the solution the
authors have adopted oblate spheroidal coordinates (ξ , η, φ),
which are related to the Cartesian coordinates by means
of the transformation x f
1 − η2 cos φ,
1 + ξ 2
=
x = f
1 + ξ 2
1 − η2 sin φ, z = f ξ η, where f =
R2
e − Rp2/ Re, where Re and Rp are the equatorial and polar
radii, respectively. In [
130
] index n = 1 polytropes were
studied under the assumption that the shape of such a
rotating fluid sphere is spheroidal. By introducing the spheroidal
coordinates (x , η), defined as κ 2r 2 = x 2 + c2 1 − η2 ,
and κ z = x η, where κ and c are constants, one can express
the gravitational potential by using spheroidal wave
functions.
The effect of the rotation of the dark matter halo in the
framework of the BEC dark matter model has also been
previously investigated. Two classes of models for rotating
halos were investigated in [
103
], in order to analyze their
stability with respect to vortex formation. In the first model
halos were modeled as homogeneous Maclaurin spheroids,
while in the second one an n = 1 polytropic
RiemannS ellipsoid was considered. Generally, it was shown that
BEC halos in the polytropic Thomas–Fermi regime typically
form vortices. The dynamics of the rotating Bose condensate
galactic dark matter halos, made of an ultralight spinless
boson gas was investigated in [
104
]. The basic approach in
this study was to obtain the numerical solution of the
system of the coupled Gross–Pitaevskii and Poisson equations,
i h¯∂ /∂ t = − h¯2/2m + V + 2π h¯2a/m2  2
and V = 4π Gm 2, respectively, in Cartesian
coordinates. It was found that ultralight spinless boson dark matter
candidates can describe well the galactic astrophysical
properties at local scales with the addition of angular momentum
to halos.
In our study we have considered the case in which the
halo has an overall rigid body rotation, and we have studied
the astrophysically relevant properties that such a rotating
halo may have. As a first (and basic) result we have obtained
the rotational corrections to the halo density profile, due to
rotation. In our investigations we have followed the approach
initiated in [
133
], and we have considered the rotation
problem in spherical coordinates. While [
133
] considers
polytropic systems with arbitrary n, in the present paper we have
systematically investigated the n = 1 configurations. In this
case the general expression of the density involves an infinite
summation over a set of radial Bessel functions, and angular
Legendre polynomials. We have restricted our investigations
of the astrophysical properties of the BEC halos to the slow
rotation case, when the deformation of the halo is small, and
the rotation parameter 2 satisfies the condition 2 << 1. In
this case explicit and simple expressions of the density of the
rotating halo can be obtained. The knowledge of the density
distribution is the first step in the investigation of the physical
properties of BEC dark matter halos. The mass distribution
inside the halo can then be obtained, and the knowledge of the
mass profile leads to the expression of the tangential velocity
of test particles, following circular orbits around the
galactic center. In our approach we have obtained the rotational
velocity in the first approximation of 2, an approximation
that may allow for the investigation of the effects of slow
rotation on the halo.
The tangential velocity and the density profile essentially
depend on three physical parameters: the central density of
the dark matter, the radius of the static density profile, and the
angular velocity of the galaxy, respectively. In order to obtain
these parameter, we have compared our theoretical results
with a (small) set of observational data, and we have fitted
our BEC rotating model with 13 observed rotation curves.
The sample consisted of 12 dwarf galaxies and the Milky
Way galaxy. The fittings of the 12 dwarf galaxies provided
a range of central densities of ρc = (0.7–6) × 10−24 g/cm3,
indicating a relatively inhomogeneous distribution of the
central densities. The angular velocities also present a relatively
large spread, ranging from 0.1 × 10−16 s−1 to 5 × 10−16 s−1,
presenting an order of magnitude variation. The predicted
masses of the halos also did present a large variation, from
0.17 × 1010 M to 4.12 × 1010 M , implying a difference
by a factor of about 24 between the smallest and the highest
galactic mass.
One of the important parameters in the theoretical models
of BEC dark matter is the radius R of the static (nonrotating)
halo. The value of R is determined by the scattering length
a and the mass m of the dark matter particle. As such, R
must be a universal constant, and its constancy, as proven
by the observations, could represent a strong argument in
favor of the condensate dark matter models. In the considered
sample of 12 dwarf galaxies R did vary between 2.9 and 8.2
kpc, respectively, by a factor of around 3. However, for this
sample of considered galaxies, a fixed value of R ≈ 6.5
kpc can give a good fit to all considered observational data.
Moreover, such a value of R can give a good fit even to the
Milky Way rotation curve, which extends up to 30 kpc, a
result which, taking into account the numerous uncertainties
in the data and in the (baryonic) fitting model, suggests that
the assumption of an universal value of R cannot be ruled
out by present day observations.
Our theoretical results have also shown that introducing
an angular velocity gives a smaller central density ρc, and a
larger radius at the boundary of the halo. Also, slow rotation
slightly increases the total mass of the angular halo. On the
other hand, a good knowledge of the baryonic mass
distribution in the galaxies and of the total central mass density
can give, via the fitting of the galactic rotation curves, a good
indication on whether the halo is rotating, or not, and the
value of its angular velocity.
In the present analysis we did not include the contribution
to the gravitational potential of the baryonic galactic matter
when we derived the mass distribution of the dark matter
halo. This requires one to modify the Poisson equation to
V = 4π G (ρ + ρb), where ρb is the density of the
baryonic matter, and to systematically include the effect of the
baryonic matter in the model. In particular, the Helmholtz
type equation describing the density distribution of the dark
matter would also depend on the density of the baryonic
matter. For more luminous galaxies, it may be necessary to take
this effect into account.
The analysis of the galactic rotation curves alone cannot
determine the basic physical properties of the condensate
dark matter. Alternative physical effects must be taken into
account to fully determine the properties of the dark matter
particle. One of such physical effects would concern
gravitational lensing by BEC halos. It was already shown in [
85
]
that the BEC dark matter gives a very different prediction for
gravitational lensing from other models of dark matter. One
might also include the effects of the vortices in the theoretical
analysis, and such an inclusion might explain the wiggles in
the rotation curves [
92
].
The observation of the possible rotation of the galactic
BEC dark matter halos would lead to a deeper understanding
of the physics of these complex systems, as well as to some
constraints on the nature and physical characteristics of the
dark matter particles. In the present investigations we have
developed some basic theoretical tools that could help in
discriminating between the standard dark matter and condensate
dark matter models.
Acknowledgements ShiDong Liang acknowledges the support
of the Natural Science Foundation of Guangdong Province (no.
2016A030313313) and the Project (XJZDL12) of Foreign Expert in
Sun Yatsen University.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Derivative of the density profile with respect
to kr
The derivative with respect to the radial variable kr of the
density profile of the Bose–Einstein condensate dark matter
halo, given by the general solution of the Helmholtz equation,
can be obtained as follows:
dρ d ∞
d(kr ) = d(kr ) l=0
A2l (−kr )2l
1 d
kr d(kr )
2l
=
∞
l=0
A2l
π
2kr J2l+ 21 (kr )P2l (cos θ )
1
A2l − 2
π
2(kr )3 J2l+ 21 (kr )
π
2kr
A2l
A2l
J2l− 21 (kr ) −
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