Advances in High Energy Physics

Volume 2017 (2017), Article ID 4025386, 14 pages

https://doi.org/10.1155/2017/4025386

## Bose-Einstein Condensate Dark Matter Halos Confronted with Galactic Rotation Curves

Institute of Physics, University of Szeged, Dóm Tér 9, Szeged 6720, Hungary

Correspondence should be addressed to L. Á. Gergely; moc.liamg@ylegreg.a.olzsal

Received 7 October 2016; Revised 15 December 2016; Accepted 25 December 2016; Published 8 February 2017

Academic Editor: Sergei D. Odintsov

Copyright © 2017 M. Dwornik et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We present a comparative confrontation of both the Bose-Einstein Condensate (BEC) and the Navarro-Frenk-White (NFW) dark halo models with galactic rotation curves. We employ 6 High Surface Brightness (HSB), 6 Low Surface Brightness (LSB), and 7 dwarf galaxies with rotation curves falling into two classes. In the first class rotational velocities increase with radius over the observed range. The BEC and NFW models give comparable fits for HSB and LSB galaxies of this type, while for dwarf galaxies the fit is significantly better with the BEC model. In the second class the rotational velocity of HSB and LSB galaxies exhibits long flat plateaus, resulting in better fit with the NFW model for HSB galaxies and comparable fits for LSB galaxies. We conclude that due to its central density cusp avoidance the BEC model fits better dwarf galaxy dark matter distribution. Nevertheless it suffers from sharp cutoff in larger galaxies, where the NFW model performs better. The investigated galaxy sample obeys the Tully-Fisher relation, including the particular characteristics exhibited by dwarf galaxies. In both models the fitting enforces a relation between dark matter parameters: the characteristic density and the corresponding characteristic distance scale with an inverse power.

#### 1. Introduction

The visible part of most galaxies is embedded in a dark matter (DM) halo of yet unknown composition, observable only through its gravitational interaction with the baryonic matter. Assuming the standard CDM cosmological model, the Planck satellite measurements of the cosmic microwave background anisotropy power spectrum support 4.9% baryonic matter, 26.8% DM, and 68.3% dark energy in the Universe [1, 2].

Investigation of mass distribution of spiral galaxies is an essential tool in the research of DM. Beside the stellar disk and central bulge, most of the galaxies harbour a spherically symmetric, massive DM halo, which dominates the dynamics in the outer regions of the stellar disk. Nevertheless there are examples of galaxies which at larger radii are better described by a flattened baryonic mass distribution (global disk model) [3].

Several DM candidates and alternatives have been proposed, the latter assuming Einstein’s theory of gravity breaking down on the galactic scale and above ([4–12]). In brane-world and -gravity models, the galactic rotation curves could be explained without DM ([13–16]).

At this moment there are strong experimental constraints for all proposed dark matter candidates. Supersymmetric dark matter has been strongly constrained by LHC [17, 18], sterile neutrinos disruled with 99% confidence level by IceCube [19], Weekly Interacting Massive Particles (WIMPs) severely bounded by the LUX [20], PandaX-II [21], and Xenon100 [22] experiments. Extra dimensional effects as dark matter substitutes have been also contained by LHC [23]. Massive Compact Halo Objects (MACHOs) with masses below 20 solar masses have been shown to give at most 10% of dark matter by microlensing experiments on the Large Magellanic Cloud [24]. There is still hope for larger mass MACHOs as dark matter candidates, revived after the spectacular first direct detection of gravitational waves [25], sourced by black holes of approximately 30 solar masses.

It is well known that hot dark matter (HDM) consisting of light ( eV) particles cannot reproduce the cosmological structure formation, as they imply that the superclusters of galaxies are the first structures to form contradicting CMB observations, according to which superclusters would form at the present epoch [26]. Warm dark matter ( keV) models seem to be compatible with the astronomical observations on galactic and also cosmological scales [27, 28]. Leading candidates for warm dark matter are the right handed neutrinos, which in contrast with their left handed counterparts do not participate in the weak interaction. The decay of these sterile neutrinos produces high amount of X-rays, which can boost the star formation rate leading to an earlier reionization [29]. The existence of sterile neutrinos was however severely constrained by recent IceCube Neutrino Observatory experiments [30]. Cold dark matter (CDM) also shows remarkably good agreement with observations over kpc scales ([31, 32]). Particular CDM candidates, like neutralinos (which are stable and can be produced thermally in the early Universe) and other WIMPs originating in supersymmetric extensions of the Standard Model were severely constrained by recent LHC results, rendering them into the range 200 GeV GeV [33]. In a Higgs-portal DM scenario the Higgs boson acts as the mediator particle between DM and Standard Model particles, and it can decay to a pair of DM particles. Very recent constraints established by the ATLAS Collaboration on DM-nucleon scattering cross-section impose upper limits of approximately 60 GeV for each of the scalar, fermion, and vector DM candidates (see Figure of [34]), within the framework of this scenario. While MACHOs of masses less than 10 solar masses (like white dwarfs, neutron stars, brown dwarfs and unassociated planets, and primordial black holes in the astrophysical mass range) were disruled either by Big Bang Nucleosynthesis constraints or microlensing experiments as dominant DM candidates, primordial black holes with intermediate mass could still be viable candidates [35, 36].

Large -body simulations (e.g., [39]) performed in the framework of the CDM-model ( being the cosmological constant) were compatible with CDM halos with central density cusps [40]. They are modeled by the Navarro-Frenk-White (NFW) DM density profile , where is a scale radius and is a characteristic density. Some observations support such a steep cuspy density profile [41, 42]; nevertheless certain high-resolution rotation curves instead indicate that the distribution of DM in the centres of DM dominated dwarf and Low Surface Brightness (LSB) galaxies is much shallower, exhibiting a core with nearly constant density [43]. In turn, the baryonic matter distribution may also affect the DM density profile. As shown in [44] a dark matter core within an isolated, initially cuspy dark matter halo may form due to strong supernova feedback. By contrast, adiabatic contraction of baryonic gas tends to produce even cuspier dark matter halos [45].

The surface number-density profiles of satellites decline with the projected distance as a power law with the slope , while the line-of-sight velocity dispersion declines gradually [46]. These observations support the NFW model on scales of kpc.

In a cosmological setup various scalar field DM models were also discussed ([47, 48] and references therein). A particular scalar field DM model describes light bosons in a dilute gas. The thermal de Broglie wavelength of the particles is , which can be large for light bosons () and for low temperature. Below a critical temperature (), the bosons’ wave packets, which are the order of overlap, result in correlated particles. Such bosons share the same quantum ground state, behaving as a Bose-Einstein Condensate (BEC), characterized by a single macroscopic wave function. It has been proposed that galactic DM halos could be gigantic BECs [49].

It has been shown that caustics of ring shape appear in rotating BEC models, which have an effect on rotation curves, by causing bumps [50, 51]. Such ring shaped caustics degenerate into the origin in the nonrotating BEC limit, adopted in this paper.

The self-gravitating condensate is described by the Gross-Pitaevskii-Poisson equation system in the mean-field approximation [10, 52–54]. In the Thomas-Fermi approximation, a 2-parameter (mass and scattering length ) density distribution of the BEC halo is obtained [see (3) below], which is less concentrated towards the centre as compared to the NFW model, relaxing the cuspy halo problem.

In model [55] where a normal dark matter phase with an equation of state condensed into a BEC with self-interaction ( being the one-dimensional velocity dispersion and the speed of light), the stability of the BEC halo depends on the particle mass and scattering length. For a given mass the stability occurs for larger scattering length and for given scattering length the stability appears at smaller mass. For the following scattering lengths: fm, fm, and fm, the mass of the BEC particle arises as eV, eV, and eV, respectively. Galactic size stable halos can form with eV (Figure in [56]).

A stable BEC halo can form as a result of gravitational collapse [57]. The model has been tested on kpc scales confronting it with galactic rotation curve observations [10]. It was pointed out by [58] that the effects of BEC DM should be seen in the matter power spectrum if the boson mass is in the range meV meV and meV meV for the scattering lengths fm and fm, respectively. In [59] the authors showed that the observed collisional behaviour of DM in Abell 520 cluster can also be recovered within the framework of the BEC model. All of the mentioned BEC particle masses are consistent with the limit eV imposed from galaxy observations and -body simulation [60]. A discrepancy was however pointed out between the best fit density profile parameters derived from the strong lensing and the galactic rotational curves data. In conclusion the BEC halo should be denser in lens galaxies than in dwarf spheroidals [61].

In this work we critically examine the BEC model as a possible DM candidate against rotation curve data, pointing out both advantages and disadvantages over the NFW model. Previous studies on the compatibility of the BEC model and galactic rotation curves were promising but relied on a less numerous and less diversified set of galaxies than employed here ([62, 63]). The paper has the following structure. The basic properties of the BEC DM model are reviewed in Section 2. In Section 3 a comparison is made between the theoretical predictions of the BEC model and the observed rotation curve data of three types of galaxies: the High Surface Brightness (HSB), LSB, and dwarf galaxies. The conclusions are presented in Section 4.

#### 2. The Bose-Einstein Condensate Galactic Dark Matter Halo

An ideal, dilute Bose gas at very low temperature forms a Bose-Einstein Condensate in which all particles are in the same ground state. In the thermodynamic limit, the critical temperature for the condensation is [64]. Here and are the number density and the mass of the bosons, respectively, and is a constant, while and denote the reduced Planck and Boltzmann constants, respectively. Atoms can be regarded as quantum-mechanical wave packets of the order of their thermal de Broglie wavelength . The condition for the condensation can be reformulated as , where is the average distance between pairs of bosons, and it occurs when the temperature, hence the momentum of the bosons, decreases and as a consequence their de Broglie wavelengths overlap. The thermodynamic limit is only approximately realized, the finite size giving corrections to the critical temperature [65–68]. A dilute, nonideal Bose gas also displays BEC; on the other hand, the condensate fraction is smaller than unity at zero temperature and the critical temperature is also modified [69–72]. Experimentally, BEC (which could be formed by bosonic atoms but also form fermionic Cooper pairs) has been realized first in [73–75], then in [76, 77], and in [78].

In a dilute gas, only two-particle interactions dominate. The repulsive, two-body interparticle potential is approximated as , with a self-coupling constant , where is the scattering length. Then in the mean-field approximation (in case when we neglect the contribution of the excited states) the BEC is described by the Gross-Pitaevskii equation [52–54]:where is the wave function of the condensate and is the 3-dimensional Laplacian. The probability density is normalized towhere is the number of particles in the ground state and is the number density of the condensate. The potential (**r**)/*m* is the Newtonian gravitational potential produced by the Bose-Einstein Condensate.

Stationary solutions of the Gross-Pitaevskii equation can be found in a simple way by using the Madelung representation of complex wave functions [79, 80], then deriving the Madelung hydrodynamic equations [79]. Madelung’s equations can be interpreted as the continuity and Euler equations of fluid mechanics, with quantum corrections included. However, the quantum correction potential in the generalized Euler equation contributes significantly only close to the boundary of the system [81]. In the Thomas-Fermi approximation the quantum correction potential is neglected compared to the self-interaction term. This approximation becomes more accurate as the particle number increases [82].

Assuming a spherically symmetric distribution of the condensate the following solution was found [10, 81]:where andThe central density is determined from normalization condition (2) as The Thomas-Fermi approximation remains valid for [81].

The BEC galactic DM halo’s size is defined by , giving , that is, The mass profile of the BEC halo is then given as The BEC halo contributes to the velocity profile of the particles which are moving on circular orbit as dictated by the Newtonian gravitational force [10]. This can be taken into account by the following equation:which needs to be added to the baryonic contribution, respectively.

#### 3. Confronting the Model with Rotation Curve Data

The validity of our model was tested by confronting the rotation curve data of a sample of 6 HSB, 6 LSB, and 7 dwarf galaxies, with both the NFW DM and the BEC density profiles. For reasons to become obvious during our analysis, we split both the HSB and LSB data sets into two groups (type I. and II.), based on the shapes of the curves. In the first group the rotational velocities increase over the whole observed range, while in the second set the rotation curves exhibit long flat regions.

The commonly used NFW model has the mass density profile where and are a characteristic density and distance scale, to be determined from the fit.

The mass within a sphere with radius is then given as where is a positive dimensionless radial coordinate.

##### 3.1. HSB Galaxies

In this subsection we will follow the method described in [15]. In HSB galaxy the baryonic component was decomposed into a thin stellar disk and a spherically symmetric bulge. It was assumed that the mass distribution of bulge component follows the deprojected luminosity distribution with a factor known as the mass-to-light ratio. The bulge parameters were estimated from a Sérsic bulge model, which was obtained by the fitting of the optical -band galaxy light profiles.

Each galaxy’s spheroidal bulge component has a surface brightness profile which is described by a generalized Sérsic function [83];wherein is the central surface brightness of the bulge, is the characteristic radius of the bulge, and the magnitude-radius curve’s shape parameter is denoted by .

The mass-to-light ratio for the Sun is kg . The mass-to-light ratio of the bulge will be given in units of (solar units). We will also give the mass in units of the solar mass kg. We assume that the radial distribution of visible mass follows the radial distribution of light derived from the bulge-disk decomposition. Accordingly the mass of the bulge inside the projected radius can be derived from the surface brightness observed within this radius: where is the apparent flux density of the Sun at a distance Mpc, mJy, with 4.08 + + 25 mag, and The rotational velocity related to the bulge is where is the gravitational constant.

In case of a spiral galaxy, the radial surface brightness profile of the disk decreases exponentially as a function of the radius [84];where is the central surface brightness of the disk and is a characteristic disk length scale. The disk contributes to the circular velocity as follows ([84]): where and and are the modified Bessel functions evaluated at , while is the total mass of the disk.

Accordingly in HSB galaxy the rotational velocity adds up as

In order to validate the BEC+baryonic model, we confront it with rotation curve data of 6 well-tested galaxies (which were already employed in [15] for testing a brane-world model). The data was obtained from a sample given in [37] and meets the following criteria: (i) it has to be among the best accuracies obtained from the sample and (ii) the bulge has to be spherically symmetric. As a check we also fitted the NFW + baryonic model with the same data set. The respective rotation curves are plotted for both models on Figures 1 and 2. The small humps on both figures are due to the baryonic component. From the available photometric data the best fitting values were derived for the baryonic model parameters , , , , and . By fitting BEC and NFW models to the investigated rotation curve data, the parameters for these models (as well as the corresponding baryonic parameters) were calculated. The parameter values are indicated in Tables 1 and 2.