Bose-Einstein Condensate Dark Matter Halos confronted with galactic rotation curves

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.


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 ΛCDM 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].
The mass distribution of spiral galaxies is essential for investigating 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 stellar disk at the outer regions. Nevertheless there are examples of galaxies which at larger radii are better described by a flattened baryonic mass distribution (global disk model) [3].
It is well known that hot dark matter (HDM) consisting of light (m ∝ 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 [17]. Warm dark matter (m ∝ keV) models seem to be compatible with the astronomical observations on galactic and also cosmological scales [18]; [19]. 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 [20]. Cold dark matter (CDM) also shows remarkably good agreement with observations over kpc scales ( [21]; [22]). Particular CDM candidates, like neutralinos (which is stable and can be produced thermally in the early Universe) and other weakly interacting massive particles (WIMPs) originating in supersymmetric extensions of the Standard Model were severely constrained by recent LHC results, rendering them into the range 200GeV m n 500GeV [23]. 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 Fig. 4 of Ref. [24]), within the framework of this scenario.
Large N-body simulations (e.g. [25]) performed in the framework of the ΛCDMmodel (Λ being the cosmological constant) predict that CDM halos surrounding galaxies must have central density cusps [26]. The cusps appear in the Navarro-Frenk-White (NFW) DM density profile ρ N F W (r) = ρ s /(r/r s )(1 + r/r s ) 2 , where r s is a scale radius and ρ s is a characteristic density. On the observational side however, highresolution rotation curves show instead 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 [27]. On the other hand the NFW model is remarkably successful on Mpc scales. The surface number-density profiles of satellites decline with the projected distance as a power law with the slope −2 ÷ −1.5, while the line-of-sight velocity dispersions decline gradually [28]. These observations support the NFW model on scales of 50-500 kpc.
In a cosmological treatment various scalar field dark matter models are also successfully employed see Ref. [29] 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 λ T ∝ 1/ √ mT , which can be large for light bosons (m <eV) and for low temperature. Below a critical temperature (T c ), the bosons' wave packets, which are the order of λ T overlap, resulting 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 [30]. The self gravitating condensate is described by the Gross-Pitaevskii-Poisson equation system in the mean-field approximation [31], [32], [33], [10]. In the Thomas-Fermi approximation, a 2-parameter (mass m and scattering length a) density distribution of the BEC halo is obtained [see Eq. (3) below] which is less concentrated towards the centre as compared to the NFW model, relaxing the cuspy halo problem.
In Ref. [34] the authors study a model where a normal dark matter phase with an equation of state P = ρc 2 σ 2 tr condensed into a BEC. Here σ tr is the one-dimensional velocity dispersion and c is the speed of light. Requiring the continuity of the pressure at the transition point, the condensation occurs at a redshift 1 + z c ≈ 1.22 × 10 3 (m/10 −33 g) 1+σ 2 tr (σ 2 tr /3 × 10 −6 ) 1/3(1+σ 2 tr ) (a/10 −10 cm) −1/3(1+σ 2 tr ) with a critical temperature of T c ≈ 6.57 × 10 3 (m/10 −33 g) 1/3 (σ 2 tr /3 × 10 −6 ) 2/3 (a/10 −10 cm) −2/3 K. The values of the parameters m, a and σ tr are quite uncertain for dark matter particles. Adopting the numerical value σ tr = 0.0017 [34] and assuming z c > 900, then taking the scattering lengths: a = 10 3 fm, a = 10 −14 fm and a = 10 −55 fm we find the following mass ranges: m > 1 eV, m > 2 × 10 −6 eV and m > 4.57 × 10 −20 eV, respectively. 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. Galactic size stable halos can form with m > 10 −24 eV (see Fig. 3. in Ref. [35]). For the masses and scattering lengths considered above the halos are stable and they fall into the galactic size range.
A recent investigation shows that a stable BEC halo can be formed as a result of gravitational collapse [36]. The model has been tested on kpc scales confronting it with galactic rotation curve observations [10]. The BEC model can explain the observed collisional behaviour of DM in the Abell 520 cluster [37] and the acoustic peaks of the cosmic microwave background (BEC could mimic the effects of the standard CDM in the CMB spectrum) [38]. It was pointed out by [39] that the effects of BEC DM should be seen in the matter power spectrum if the boson mass is in the range 15 meV < m < 35 meV and 300 meV < m < 700 meV for the scattering lengths a = 10 6 fm and a = 10 10 fm, respectively. All of the mentioned BEC particle masses are consistent with the limit m < 1.87 eV imposed from galaxy observations and N-body simulation [40].
A discrepancy was pointed out between the best fit density profile parameters derived from the strong lensing and the galactic rotational curves data [41]. However the lensing and rotational curve tests employed different galaxy samples.
In this work we propose to critically examine the BEC model against rotation curve and velocity dispersion data, investigating BEC as a possible DM candidate and 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 then employed here ( [42], [43]). The paper is organized as follows. The basic properties of the BEC DM model are reviewed in Section 2. In Section 3 the theoretical predictions of the BEC model are compared with the observed rotation curve data of three types of galaxies, the High Surface Brightness (HSB), LSB and dwarf galaxies. Measurement of the stellar velocity dispersion provides an additional method to determine the density distribution in galaxies. The Virgo Cluster, as the nearest and one of the most extensively studied galaxy clusters due to its relative proximity (16.5 M pc) offers thousands of well-resolved galaxies. In Section 4 we present the results of a combined test with velocity dispersion profiles and rotation curves of 6 Virgo cluster galaxies and derive the best fit parameters of the BEC and NFW dark matter models. The conclusions are presented in Section 5.

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 T c = 2π 2 (n/ζ) 3/2 /mk B [44]. Here n and m are the number density and the mass of the bosons, respectively, ζ = 2.612 is a constant, while and k B 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 λ T = 2π 2 / (mk B T ). The condition for the condensation T < T c can be reformulated as l < λ T /ζ −1/3 , where l is the average distance between pairs of bosons, and it occurs when the temperature, hence the momentum of the bosons decreases such that their de Broglie wavelengths overlap. The thermodynamic limit is only approximately realized, the finite size giving corrections to the critical temperature [45], [46], [47], [48]. A dilute, non-ideal Bose gas also displays BEC, however the condensate fraction is smaller than unity at zero temperature and the critical temperature is also modified [49], [50], [51], [52]. Experimentally, BEC (which could be formed by bosonic atoms, but also form fermionic Cooper pairs) has been realized first in 87 Rb [53], [54], [55], then in 23 Na [56], [57], and in 7 Li [58].
In a dilute gas, only two-particle interactions dominate. The repulsive, twobody interatomic potential is approximated as V self = λδ (r − r ′ ), with a self-coupling constant λ = 4π 2 a/m, where a is the scattering length. Then in the mean-field approximation (neglecting the contribution of the excited states) the BEC is described by the Gross-Pitaevskii equation [31], [32], [33]: where ψ(r, t) is the wave function of the condensate and ∆ is the 3-dimensional Laplacian. The probability density ρ (r, t) = |ψ(r, t)| 2 is normalized to where n 0 (t) is the number of particles and ρ (r, t) the number density of the condensate. The potential V self grav (r) /m is the Newtonian gravitational potential created by the condensate. Stationary solutions of the Gross-Pitaevskii equation can be found in a simple way by using the Madelung representation of complex wave-functions [59], [60], then deriving the Madelung hydrodynamic equations [59]. 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 [61]. In the Thomas-Fermi approximation the quantum correction potential is neglected compared to the self-interaction term. This approximation becomes increasingly accurate with an increasing number of particles [62].
The mass profile of the BEC halo is then given as The contribution of the BEC halo to the velocity profile of the particles moving on circular orbit under Newtonian gravitational force becomes [10] v 2 (r) = 4πGρ which has to be added to the respective baryonic contribution.

Confronting the model with rotation curve data
In order to test the validity of our model, we confront 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 ρ s and r s are a characteristic density and distance scale, to be determined from the fit. The mass within a sphere with radius r = yr s is then given as where y is a positive dimensionless radial coordinate. Galaxy  Table 1. The distances (D) and the photometric parameters of the 9 HSB galaxy sample. Bulge parameters: the central surface brightness (I 0,b ), the shape parameter (n), the characteristic radius (r 0 ) and radius of the bulge (r b ). Disk parameters: central surface brightness (I HSB 0,d ) and length scale (h HSB ) of the disk.

HSB galaxies
In this subsection we will follow the method described in [15]. In a HSB galaxy we decompose the baryonic component into a thin stellar disk and a spherically symmetric bulge. We assume that the mass distribution of bulge component follows the deprojected luminosity distribution with a factor known as the mass-to-light ratio. We estimate the bulge parameters from a Sérsic r 1/n bulge model, fitted to the optical I-band galaxy light profiles.
The surface brightness profile of the spheroidal bulge component of each galaxy is described by a generalized Sérsic function [63] where I 0,b is the central surface brightness of the bulge, r 0 is its characteristic radius and n is the shape parameter of the magnitude-radius curve. The respective mass over luminosity is the mass-to-light ratio -for the Sun γ ⊙ = 5133 kg W −1 . The mass-to-light ratio of the bulge σ will be given in units of γ ⊙ (solar units). We will also give the mass in units of the solar mass M ⊙ = 1.98892×10 30 kg. We assume that the radial distribution of visible mass is given by the radial distribution of light obtained from the bulge-disk decomposition. Thus the mass of the bulge within the projected radius r is proportional to the surface brightness encompassed by this radius: Therefore the contribution of the bulge to the rotational velocity is where G is the gravitational constant. In a spiral galaxy, the radial surface brightness profile of the disk, decreases exponentially with the radius [64] where I HSB 0,d is the disk central surface brightness and h HSB is a characteristic disk length scale. The contribution of the disk to the circular velocity is ( where x = r/h HSB and I n and K n are the modified Bessel functions evaluated at x/2, while M HSB D is the total mass of the disk. Therefore the rotational velocity in a HSB galaxy, adds up as We confront the BEC+baryonic model with (HI and H α ) rotation curve data of 6 well-tested galaxies already employed in [15] for testing a brane-world model.  Table 2. The best fit parameters and the minimum values (χ 2 min ) of the χ 2 statistics for the HSB I and II galaxies (the first and last three galaxies, respectively). Columns 2-5 give the BEC model parameters (radius R BEC and central density ρ  (N F W )). The 1σ confidence levels are shown in the last column (these are the same for both models). For HSB I galaxies the two models give similar χ 2 min values (within 1σ confidence level), however in case of HSB II galaxies with extended flat regions, the NFW model fits better the rotation curves. The χ 2 min values in the case of BEC model are outside the 1σ confidence level for HSB II galaxies.
These were extracted from a larger sample given in [65] by requiring i) sufficient and accurate data for each galaxy and ii) spherical structure of the bulge (no rings and bars). For comparison, the NFW+baryonic model is also tested on the same sample and the respective rotation curves are plotted for both models on Figs. 1 and 2. The small humps on both figures are due to the baryonic component. For the investigated galaxies we have derived the best fitting values of the baryonic model parameters I 0,b , n, r 0 , r b , I HSB 0,d h HSB from the available photometric data. The BEC and NFW parameters, respectively were calculated (together with the corresponding baryonic parameters) by fitting these models to the rotation curve data. These are collected in Tables 1 and 2. Both the BEC and NFW DM models give comparable χ 2 min values (within 1σ confidence level) for HSB I galaxies. In case of galaxies with extended flat regions (HSB II), the NFW DM model fits better the rotation curves, nevertheless BEC model give rotational curves which fall outside the 1σ confidence level.

LSB galaxies
LSB galaxies are characterized by a central surface brightness at least one magnitude fainter than the night sky. They form the most unevolved class of galaxies [66]. LSB galaxies were found to be metal poor, indicating their low star formation rates as compared to their HSB counterparts [67]. They exhibit a wide spread of colours ranging from red to blue [68] and are characterized by large variety of properties and morphologies. Although the most commonly observed LSB galaxies are dwarfs, a significant fraction of LSB galaxies are large spirals [69].
Our model LSB galaxy consists of a thin stellar+gas disk and a CDM component in a form of BEC. The disk component is the same as for the HSB galaxies, the surface brightness profile being [64]   where I LSB 0,d is the central surface brightness and h LSB the disk length scale. We can calculate the disk contribution to the circular velocity as where q = r/h LSB and M LSB D is the total mass of the disk while the modified Bessel functions I n and K n are evaluated at q/2. Therefore for a generic projected radius r, the rotational velocity in this combined model is written as A preliminary check confirmed that the BEC+baryonic model represents a better fit than the purely BEC model.
We confronted the BEC model with 6 LSB galaxies taken from a larger sample [70]. The data for these high quality rotation curves are based on both HI and Hα measurements. From a χ 2 -test the parameters in both the BEC+baryonic and NFW+baryonic models were identified, these are shown in Table 3 Table 3. The best fit BEC and NFW parameters of the LSB I and II type galaxies (the first and last three galaxies, respectively). For LSB I galaxies the BEC DM model gives significantly better fitting velocity curves (within 1σ confidence level) than the NFW model. However the velocity curves are outside the 1σ confidence level for LSB II galaxies.
For LSB I galaxies the BEC DM model gives significantly better fitting velocity curves (all within the 1σ confidence level) compared to the NFW model (which in two cases out of the three gives fits falling outside 1σ). For LSB II galaxies the quality of the fits is comparable, but in both models they are beyond the 1σ confidence level.

Dwarf galaxies
About 85% of the known galaxies in the Local Volume [71] are dwarf galaxies. They are defined by an absolute magnitude fainter than M B ∼ −16 mag, on the other hand they are more extended than globular clusters [72].
The formation history of dwarf galaxies is not well-understood. They formed at the centres of subhalos orbiting within the halos of giant galaxies. Five main classes are distinguished based on their optical appearance: dwarf ellipticals, dwarf irregulars, dwarf spheroidals, blue compact dwarfs, and dwarf spirals. The representants in the last type can be regarded as the very small ends of spirals [73].
All dwarf galaxies have central velocity dispersions in the range 6 ÷ 25 km/s [74]. In a typical dwarf galaxy, assuming dynamical equilibrium, the mass derived from the observed velocity dispersions is much larger than the observed total visible mass. This implies that the mass-to-light ratio is very high compared to other types of galaxies, hence they can play an important role in the study of DM distribution on small scales. Dwarf galaxies are ideal objects to prove or falsify various alternative gravity theories [75].
In order to test the BEC model, we have selected a sample of 7 dwarf galaxies for which high resolution rotation curve data is available. We fitted both the BEC+baryonic and the NFW+baryonic models, respectively, with similar baryonic components as for the LSB galaxies. As the length scales of the stellar disks are not available for this sample, they were calculated by χ 2 minimization, too.
A preliminary check showed that the addition of the BEC dark matter halo to the baryonic model improved (giving lower χ 2 min values) on the fit in all cases. By contrast, the NFW model was unable to improve on the purely baryonic fit in four out of seven cases. We note that since the data does not contain the error margins, the χ 2 min values are relatively high (beyond the 1σ confidence level in most cases). The best fit BEC and NFW parameters are shown in Table 4 and the corresponding rotation curves are represented on Fig. 5. The inclusion of the BEC DM model gives significantly (in some cases one order of magnitude in the value of χ 2 ) better fits compared to the case of NFW model. This is due to the cusp avoidance in the central density profile of the Galaxy  BEC model and the fact that dwarf galaxies do not exhibit extended flat regions in their rotation curves.

A combined test of rotational curves and velocity dispersion profiles
In this section we present the results of a complementary analysis of the projected (line of sight) velocity dispersion of 6 Virgo Cluster galaxies. The chosen sample consists of 6 early-type spiral galaxies, the data of which is taken from [76]. The rotational curve data is also available for these galaxies, therefore we are able to perform a combined χ 2 test. For the two independent data setsχ 2 denotes the sum of the two individual χ 2 values. In a spherically symmetric and isolated setup the Jeans equation for the velocity dispersion σ(r) is [77] ∂ ρσ 2 ∂r where r is the radial distance from the galaxy center, ρ is the mass density and Φ is the gravitational potential. In self-gravitating systems ρ and Φ are related by the Poisson equation. For a given density distribution Eq. (18) can be solved for σ 2 , obtaining for the two models investigated here: σ 2 N F W (r) = 2 π Grhos r − ln r rs (rs + r) 2 rs −1 + ln rs + r rs 3 ln rs + r rs × (rs + r) 2 rs −1 − 8 rs − 4 r − 2 rs 2 r + rs 3 r 2 + r 2 rs + π 2 rs + 2 π 2 r + π 2 r 2 rs − 9 rs − rs 2 r − 7 r + 6 dilog × rs + r rs (rs + r) where dilog(x) = 0 x ln(1−x) x dx denotes the dilogarithm function. The line of sight (observed) velocity dispersion σ 2 LOS is given by [78]: with R denoting the apparent distance of the star from the galactic centre in the plane perpendicular to the line of sight. For a realistic model we also include the velocity dispersion contribution of the stellar disk [79] σ Here τ is the mass to light ratio, h is the scale length and I 0d is the central surface brightness of the stellar disk. We fit the BEC and NFW models with the observed velocity dispersion and rotation curve data sets simultaneously (each data sets determined from the Doppler vcc1859 Figure 6. Best fitting line of sight velocity dispersion profiles for the investigated Virgo cluster galaxies. Solid black lines refer to the BEC model, while the dashed red curves to the NFW dark matter model. Both models fit the velocity dispersion data comparably, however the BEC model approximates the data slightly better close to the centres of the galaxies. In case of VCC1003 and VCC1253 the combined tests are completely failed because the best fit parameters are very different for the rotational and velocity dispersion curves as shown in Table 6. shift), where the subscript DM refers to either of the corresponding dark matter models (i.e. BEC or NFW). The corresponding values ofχ 2 min are given in Table 5. All combined fits are far beyond the 3σ confidence level, hence neither of the models is compatible with both datasets.
For the galaxies VCC1003 and VCC1253 the combined fits were the worst, due to the fact that the best fit parameters for the rotational curve and velocity dispersion data are very different for these galaxies. The parameters of these individual fittings are shown in Table 6. We show the best fit curves for both models resulting from the combined χ 2 -test on Figs. 6 and 7.

Discussions and final remarks
We have performed a χ 2 -test of the BEC and NFW DM models, with the rotation curves of a sample of 6 HSB, 6 LSB and 7 dwarf galaxies completed with a combined test of 6 Virgo cluster galaxy rotation curve and velocity dispersion profiles. For improved accuracy we also included realistic baryonic models in every case. For the HSB galaxy sample, both the rotation curve and the surface photometry data were available. Most of the rotation curves were smooth, symmetric and uniform in quality.
For the investigated galaxies, we decomposed in the standard way the circular velocity into its baryonic and DM contributions:  Figure 7. Best fitting rotational velocities for the investigated Virgo cluster galaxies. Solid black lines refer to the BEC model, while the dashed red curves to the NFW DM model. The BEC model gave better fits in two cases (VCC1253 and VCC1859), while for the rest of the galaxies, the predictions of the two models are comparable. In case of VCC1003 and VCC1253 the combined tests are completely failed because the best fit parameters are very different for the rotational and velocity dispersion curves as shown in Table 6.  Table 5. The best fit parameters and the minimum valuesχ 2 min of the combined χ 2 statistics for the 6 Virgo cluster galaxies. Here τ is the mass to light ratio, h is the scale length and I 0d is the central surface brightness of the stellar disk. The remaining ones are defined in Table 2. rotation curves are χ 2 best-fitted with the baryonic parameters and the parameters of the two DM halo models (BEC and NFW).

Galaxy
The analysis of the HSB I galaxies showed a remarkably good agreement for both DM models with observations. The quality of the fits of the BEC and NFW models with the rotation curve data was comparable. However the rotation curves of the HSB II type galaxies are significantly better described by the NFW model.
It was previously known that for LSB galaxies and without including the baryonic sector, the BEC model gave a better fit than the NFW model [42]. We additionally found that including the baryonic component improves on the fit of [42]. Our detailed  Table 6. The best fit parameters and the minimum valuesχ 2 min of the χ 2 statistics for the VCC1003 and VCC1253 galaxies. The first two lines show the best fit parameters based on the rotation curves data, while the last two lines related to the pure velocity dispersion data fitting. The parameters are the same as in Table 5 analysis showed a significantly better performance of the BEC model for LSB type I galaxies, while comparable fits for LSB type II galaxies. These latter fits were however outside the 2σ confidence level.
The unsatisfactory large distance behaviour of the BEC model for both the HSB and LSB galaxies of type II originates in the sharp cutoff of the BEC DM distribution and clearly indicates that a modification of the BEC model on large distances would be desirable, also to comply with the very distant behaviour of the universal rotation curves (URCs) [80].
A possible alternative is including vortex lattices into the halo ( [81]; [44]), as when a BEC is rotated at a rate exceeding certain critical frequency, quantized vortices can be formed. This vortex lattice in principal can influence the galactic rotation curve and provide a flat velocity profile with oscillatory structure ( [82]; [42]). One such suggestion takes into account the effects of the finite DM temperature on the properties of the DM halos. An enhanced BEC model which takes into account excited states as suggested in Refs. ( [83]; [84]) could significantly modify the DM halo density profiles, such investigations fall however outside the scope of the present paper.
From the above analysis of HSB and LSB galaxies it is also obvious that (while on large distances the BEC model suffers from problems due to the sharp cutoff) close to the it works overall better than the NFW model. This is also supported by our fit of both the BEC+baryonic and NFW+baryonic DM models with rotation curve data of a sample of 7 dwarf galaxies. Since dwarf galaxies are DM dominated, they provide the strongest test of the compared models. The results are shown in Fig. 5. We also note that the NFW DM improved over the pure baryonic fit in four cases out of seven, while including the BEC component improved over the fit with the baryonic component in all cases.
For all cases we have determined the BEC parameters ρ (c) BEC , R BEC , given in Tables 2, 3, 4 and 5. The averages of the radii R DM of the BEC halos for the HSB, LSB and dwarf galaxies are R HSB BEC ≈ 4.06kpc, R LSB DM ≈ 6.48kpc and R dwarf DM ≈ 5.94kpc, respectively. The values of R DM are consistent within the order of magnitude with the halo radii of 59 other galaxies determined from weak lensing ( [85]).
The relation among the mass m of the BEC particle, its coherent scattering length a and the radius of the DM halo R DM can be written as ( In order to constrain the values of scattering lengths, we chose the lower bound of the mass range for the axion, 10 −6 eV as the particle mass. This gives the following scattering lengths for the three types of galaxies; a HSB ≈ 5.4 × 10 −14 fm, a LSB ≈ 1.37 × 10 −13 fm and a dwarf ≈ 1.15 × 10 −13 fm. These values are consistent with the results of [85], which are based on a statistical analysis of 61 DM dominated galaxies. The total energy of the BEC halo is negative with these scattering lengths and particle mass, meaning the halo is stable (see Fig. 3. of [35]). We intended to shed more light on the applicability of these models for describing DM halos through a fit with the velocity dispersion profiles of 6 Virgo cluster galaxies for which the rotation curve data was also available. A combined χ 2 minimization test, fitting the corresponding rotation curve and velocity dispersion data simultaneously showed however quite deceivingly that the combined χ 2 values fall outside the 3σ confidence level for both models, therefore neither of them seems versatile enough to comply with the combination of both type of measurements.