Rotation curves in Bose-Einstein Condensate Dark Matter Halos
aa r X i v : . [ g r- q c ] D ec Chapter 6 R OTAT ION CURVE S I N B OSE -E INSTEIN C ONDE NSATE D ARK M AT T E R H AL OS
M. Dwornik, Z. Keresztes, L. ´A. Gergely ∗ Department of Theoretical Physics, University of Szeged,Tisza Lajos krt 84-86, Szeged 6720, HungaryDepartment of Experimental Physics, University of Szeged,D´om T´er 9, Szeged 6720, Hungary
1. ABSTRACT
The study of the rotation curves of spiral galaxies reveals a nearly constant cored densitydistribution of Cold Dark Matter. N-body simulations however lead to a cuspy distributionon the galactic scale, with a central peak. A Bose-Einstein condensate (BEC) of light parti-cles naturally solves this problem by predicting a repulsive force, obstructing the formationof the peak. After succinctly presenting the BEC model, we test it against rotation curvedata for a set of 3 High Surface Brightness (HSB), 3 Low Surface Brightness (LSB) and 3dwarf galaxies. The BEC model gives a similar fit to the Navarro-Frenk-White (NFW) darkmatter model for all HSB and LSB galaxies in the sample. For dark matter dominated dwarfgalaxies the addition of the BEC component improved more upon the purely baryonic fitthan the NFW component. Thus despite the sharp cut-off of the halo density, the BEC darkmatter candidate is consistent with the rotation curve data of all types of galaxies.
2. INTRODUCTION
Cosmological observations provide compelling evidence that about 95% of the content ofthe Universe resides in the unknown dark matter and dark energy components [60]. Theformer resides in bound systems as non-luminous matter [57, 68], the latter in the form of azero-point energy pervading the whole Universe. Dark matter is thought to be composed ofpressureless, cold, neutral, weakly interacting massive particles, beyond those existing in ∗ E-mail address: [email protected] he Standard Model of Particle Physics, and not yet detected in accelerators or in dedicateddirect and indirect searches, excepting gravitational has been found. Therefore the possibil-ity that the Einstein (and Newtonian) theory of gravity breaks down at the scale of galaxiescannot be excluded a priori. While some studies show that the luminous matter alone canexplain the rotation in the innermost galactic regions [20,54,70], dark matter is still requiredon the larger scale. Several theoretical models, based on a modification of Newton’s lawor of general relativity, have been proposed to explain the behavior of the galactic rotationcurves [5–8, 46, 50, 51, 64, 67]. In brane world models, the galactic rotation curves can benaturally explained without introducing dark matter [24, 45, 62]. There is also a possibilitythat the rotation of galaxies in the outermost regions could be driven by magnetic fields,rather than dark matter [3].In recent cosmological models the primordial density fluctuations are generated duringan inflationary period and they are the seeds of the bottom-up structure formation model.The post-inflation regime is usually described by the Λ CDM (cosmological constant + ColdDark Matter) model which is consistent with the vast majority of the available observations,including the large scale matter distribution, the Ia type supernovae observations and thetemperature fluctuations in the cosmic microwave background radiation [55, 56, 61].However, the investigation of spiral galaxies clearly shows that the mass distribution ofgalactic-scale objects can not be explain satisfactory within the framework of the Λ CDMcosmological model. The predicted halo density profile is approximately isothermal over alarge range in radii, and it shows a well pronounced central cusp [52]. The Navarro-Frenk-White (NFW) density profile is proportional to /r close to the centre. On the observationalside however, high-resolution rotation curves show instead that the actual distribution ofdark matter is much shallower [11], presenting a constant density core. The Burkert densityprofile shows a correlation between the enclosed surface densities of luminous and darkmatter in galaxies [23]. However the astrophysical origin of this empirical density distribu-tion remains unaddressed.The knowledge of the mass distribution of spiral galaxies is a crucial step in the searchfor non-baryonic dark matter. Most of the present models assume that, beside the stellar diskand bulge, there is a spherically symmetric, massive dark matter halo, which dominatesthe total galaxy mass and also determines the dynamics of the stellar disk at the outerregions. Nevertheless Ref. [37] found that the mass distribution for some galaxies cannotbe spherical at larger radii and instead a flattened mass distribution (global disk model)better approximates the gravitational potential.In this chapter we consider scalar field dark matter halos which have undergone a Bose-Einstein condensation (BEC) [6, 13, 14, 21, 33, 34, 63, 72]. Below a critical temperature,bosons favor joining highly populated low-energy states. At the end of this process, bosonswill occupy the same quantum ground state and form a coherent matter wave, the Bose-Einstein condensate. In this state, the bosons exhibit a repulsive interaction which preventsthe formation of central density cusps by gravitational attraction.This chapter is organized as follows. The basic properties of the Bose-Einstein con-densed dark matter are reviewed in Section 2. Then the theoretical predictions of the modelare compared with the observed rotation curve data of several types of galaxies (High Sur-face Brightness, Low Surface Brightness and Dwarf Galaxies, respectively), in Section 3.We discuss the results in Section 4. 2 . THE BOSE-EINSTEIN CONDENSATE An ideal Bose gas (a cloud of non-interacting bosons) confined in a box of volume V = L obeys the Bose-Einstein thermal distribution f = { exp [( ǫ − µ ) /k B T ] − } − , where ǫ = p / m stands for the energy of the bosons, determined by their mass m and mag-nitude p of their 3-momentum vector, µ is the chemical potential, T the temperature and k B the Boltzmann constant. The 3-momentum p = 2 π ¯ h q /L is discretized through thedimensionless vector q , with integer components. The lowest energy state has p = 0 . Thenumber of uncondensated bosons at temperature T is N T = X p =0 p / m − µ ) /k B T ] − . (1)If the energy levels follow each other densely, i.e. the the thermal energy is much largerthan the smallest energy spacing between the single-particle levels: k B T ≫ π ¯ h mV / , (2)the summation in Eq. (1) can be replaced by an integral over the momentum ( P p → V R d p / (2 π ¯ h ) ) cf. Ref. [59], thus: N T = Vλ T g / [exp ( µ/k B T )] . (3)Here λ T = ¯ h s πmk B T (4)is the thermal de Broglie wavelength and g / ( z ) = 2 √ π Z ∞ dx √ xe x /z − (5)is a special Bose function, with z = exp ( µ/k B T ) and the integral variable x = p / mk B T . At such temperatures when the ground state is vacant, the value of the chem-ical potential can be expressed from the relation N T = N , with N the total number ofbosons. If the lowest energy state ǫ = 0 is occupied, the chemical potential is given by µ = − k B T ln (cid:18) n (cid:19) ≈ − k B Tn , (6)with the ground state particle number n . In the thermodynamic limit n , N, V → ∞ , n = N/V, n /V → const, such that n /N can be expressed as n N = 1 − (cid:18) TT c (cid:19) / ( T ≤ T c ) . (7)The critical temperature T c = 2 π ¯ h mk B ng / (0) ! / , (8)3ith g / (0) = 2 . , is typically low ( T c = 3 . K for He liquid at saturated vapourpressure), and represents the temperature below which the bosons start to condensate intothe lowest energy state. By further decreasing the temperature, the relative number of par-ticles in the ground state increases. For
T > T c the ground state is vacant.The condition for the condensation T < T c can be rewritten as a relation betweenthe average distance of the bosons l = p V /N = n − / and their thermal de Brogliewavelength as l < λ T ζ / ≈ . λ T . (9)With the bosons considered a quantum-mechanical wave packet of the order of its deBroglie wavelength, the condensation occurs at the low temperatures where their wave-lengths overlap.As the thermodynamic limit is never realized exactly, corrections arising from the finitesize slightly alter the value of the critical temperature, for details see [29], [39], [40], [32].In a dense, non-ideal (self-interacting) Bose gas the particles can form molecules, andthey can reach a more stable state than a BEC. Two atoms can form a molecule if a thirdparticle takes momentum away. In a dilute gas such a scenario can be avoided. ThereforeBEC can be formed in a dilute and ultracold Bose gases. The gas is considered dilute ifthe characteristic length of the interaction l int is much smaller than the average distanceof the bosons, thus l int n ≪ . In a dilute gas the bosons are weakly interacting throughtwo-particle interactions. BEC can form in a dilute, non-ideal Bose gas, however the con-densate fraction is smaller and the critical temperature is again altered [25], [26] , [69], [15].Experimentally, BEC has been realized first by different groups in Rb ( [1], [30], [19]),and in Na ( [16], [31]), in Li ( [10]).
The static configuration of N interacting scalar bosons placed in the external potential V ext in a second quantized formalism is characterized [15] by the Hamiltonian operator ˆ H = Z d r ′ ˆΨ + (cid:0) r ′ (cid:1) " − ¯ h m ∆ ′ + V ext (cid:0) r ′ (cid:1) ˆΨ (cid:0) r ′ (cid:1) + 12 Z d r ′ d r ′′ ˆΨ + (cid:0) r ′ (cid:1) ˆΨ + (cid:0) r ′′ (cid:1) V self (cid:0) r ′ − r ′′ (cid:1) ˆΨ (cid:0) r ′ (cid:1) ˆΨ (cid:0) r ′′ (cid:1) . (10)The operators (singled out by hats) are taken in the Schr¨odringer picture . The boson fieldoperators ˆΨ ( r ′ ) and ˆΨ + ( r ′ ) annihilate and create a particle at the position r ′ , while ∆ ′ isthe 3-dimensional Laplacian with respect to the coordinates r ′ . The repulsive , two-bodyinteratomic potential is V self = λδ (cid:0) r − r ′ (cid:1) (11)with λ = 4 π ¯ h am (12) The Hamiltonian operator coincides in the Schr¨odringer and Heisenberg pictures since it does not dependexplicitly on time. a .The field operator ˆΨ ( r ) is decomposed in terms of the single-particle annihilation op-erators ˆ a α as ˆΨ ( r ) = X α Ψ α ( r ) ˆ a α , (13)where Ψ α is the wave function of single-particle state | α i . The summation is taken overthe single-particle state. The functions Ψ α are orthonormal and form a complete set ofsingle-particle wave functions, i.e. X α Ψ α ( r ) Ψ ∗ α (cid:0) r ′ (cid:1) = δ (cid:0) r − r ′ (cid:1) , (14)where a star denotes the complex conjugation. Denoting the particle numbers in some state(labeled α ) by n α , the bosonic annihilation ˆ a α and creation ˆ a + α operators act on the Fockspace as ˆ a α | n , n , ..., n α , ... i = √ n α | n , n , ..., n α − , ... i , (15) ˆ a + α | n , n , ..., n α , ... i = √ n α + 1 | n , n , ..., n α + 1 , ... i , (16)and satisfy the following commutation relations: h ˆ a α , ˆ a + β i = δ αβ , [ˆ a α , ˆ a β ] = 0 , h ˆ a + α , ˆ a + β i = 0 . (17)The numbers n α are the eigenvalues of the operator n α = ˆ a + α ˆ a α . The commutation relations h ˆΨ ( r ) , ˆΨ + (cid:0) r ′ (cid:1)i = δ (cid:0) r − r ′ (cid:1) , h ˆΨ ( r ) , ˆΨ (cid:0) r ′ (cid:1)i = 0 , h ˆΨ + ( r ) , ˆΨ + (cid:0) r ′ (cid:1)i = 0 (18)for the field operators follow from (17) and (14).Since the Hamiltonian operator is time-independent, the boson field operator ˆΨ ( r , t ) inthe Heisenberg picture is ˆΨ ( r , t ) = e i ˆ Ht/ ¯ h ˆΨ ( r ) e − i ˆ Ht/ ¯ h = X α Ψ α ( r ) e i ˆ Ht/ ¯ h ˆ a α e − i ˆ Ht/ ¯ h = X α Ψ α ( r ) ˆ a α ( t ) , (19)with ˆ a α ( t ) = e i ˆ Ht/ ¯ h ˆ a α e − i ˆ Ht/ ¯ h . (20)Similar equations are valid for the respective adjoint operators ˆΨ + ( r , t ) , ˆ a + α ( t ) . The fieldoperators obeys the Heisenberg equation, i.e.: i ¯ h ∂∂t ˆΨ ( r , t ) = h ˆΨ ( r , t ) , ˆ H i = " − ¯ h m ∆ + V ext ( r ) + λ ˆΨ + ( r , t ) ˆΨ ( r , t ) ˆΨ ( r , t ) . (21)Instead Eq. (21), it is effective computationally to use a mean-field approximation . Thebasic idea [9] is to separate the BEC contribution in the field operator as ˆΨ ( r , t ) = Ψ ( r ) ˆ a ( t ) + ˆΨ ′ ( r , t ) , (22)5here the zero subscript denotes the ground-state and ˆΨ ′ ( r , t ) carries the effects of theexcited states. BEC occurs when the number of particles n ( t ) in the condensate becomesvery large, hence the states with n ( t ) and n ( t ) + 1 correspond to the same configuration.In this case ˆ a ( t ) ≈ ˆ a +0 ( t ) ≈ p n ( t ) and the expectation value of the BEC contributionis ψ ( r , t ) = q n ( t )Ψ ( r ) , (23)the wave function of the condensate. By comparison the contribution from the non-condensed part is small, therefore ˆΨ ′ ( r , t ) represents a perturbation with a negligible ex-pectation value in the leading order approximation.The probability density ρ ( r , t ) = | ψ ( r , t ) | , (24)by the choice R d r | Ψ ( r ) | = 1 is normalized to n ( t ) = Z d r ρ ( r , t ) , (25)such that ρ ( r , t ) also represents the number density of the condensate.In the leading order approximation (neglecting the contribution of the excited states)Eq. (21) becomes i ¯ h ∂∂t ψ ( r , t ) = " − ¯ h m ∆ + V ext ( r ) + λρ ( r , t ) ψ ( r , t ) , (26)known as the Gross-Pitaevskii equation [27], [28], [58] which describes the Bose-Einsteincondensate in the mean-field approximation. In order to find a solution of Eq. (26) it is worth to use the Madelung representation ofcomplex wave-functions [43], [73]: ψ ( r , t ) = q ρ ( r , t ) exp (cid:20) i ¯ h S ( r , t ) (cid:21) , (27)where the real-valued phase S ( r , t ) has the dimension of an action. The real part of Eq.(26) gives ∂S∂t + 12 m ( ∇ S ) + λρ + V ext + V Q = 0 , (28)corresponding to a generalized Hamilton-Jacobi equations with quantum correction poten-tial V Q = − ¯ h m ∆ √ ρ √ ρ . (29)The imaginary part of (26) becomes a continuity equation ∂ρ∂t + ∇ ( ρ v ) = 0 , (30)6hile the gradient of (28) gives mρ (cid:20) ∂ v ∂t + ( v ·∇ ) v (cid:21) = −∇ p − ρ ∇ V ext − ρ ∇ V Q . (31)Here we have introduced the notations v = ∇ S ( r ) m , (32)and p = λ ρ . (33)The i th component of the last term of Eq. (31) can be rewritten as [2] ρ ∇ i V Q = X j ∇ j σ Qij , (34)where σ Qij = − ¯ h m ρ ∇ i ∇ j ln ρ . (35)Eqs. (30) and (31) correspond to the usual continuity and Euler equations of fluid me-chanics, with v the classical velocity field, p the pressure and σ Qij representing a quantumcorrection to the stress tensor. Eqs. (30) and (31) are the Madelung hydrodynamic equa-tions.
A stationary state ψ is ψ ( r , t ) = q ρ ( r ) exp (cid:18) iµ ¯ h t (cid:19) , (36)with µ = const. Then the continuity equation is automatically satisfied while Eq. (28) leadsto V ext + V Q + λρ = µ . (37)The quantum correction potential V Q has significant contribution only close to thebound [75], therefore it can be neglected as compared to the self-interaction term λρ . This Thomas-Fermi approximation becomes increasingly accurate with an increasing number ofparticles [42].If V ext ( r ) /m is the Newtonian gravitational potential created by the condensate, itsatisfies the Poisson equation: ∆ V ext m = 4 πGρ BEC , (38)where ρ BEC = mρ is the mass density of the BEC and G is the gravitational constant.The Laplacian of Eq. (37) and Eq. (38) give ∆ ρ BEC + 4 πGm λ ρ BEC = 0 . (39)7or a spherical symmetric distribution this simplifies to d ( rρ BEC ) dr + 4 πGm λ ( rρ BEC ) = 0 , (40)with the solution [75], [6]: ρ BEC ( r ) = ρ ( c ) BEC sin krkr , (41)where k = s Gm ¯ h a (42)and ρ ( c ) BEC ≡ ρ BEC (0) is a central density, determined from the normalization condition(21) as ρ ( c ) BEC = n mk π . (43)The exists of a central finite density is exactly the required feature which represents anadvantage over cuspy dark matter profiles derived from N-body simulation.At the end of this subsection we comment on the validity of the Thomas-Fermi ap-proximation. The quantum correction potential (29), rewritten in spherical coordinates andinserted into Eq. (37) generates constant and ρ − BEC terms, while the contribution of theself-interaction term is proportional to ρ BEC . Close to the boundary R BEC therefore theThomas-Fermi approximation fails. Multiplying Eq. (37) by (cid:16) ρ BEC /ρ ( c ) BEC (cid:17) , with ρ BEC given by Eq. (41), and integrating on the range r ∈ [0 , R BEC ] gives the global weight ofthese terms as ¯ h /km for the V Q term and ¯ h an for the self-interaction term, respectively.Therefore the Thomas-Fermi approximation holds valid for n ≫ /ka , a condition alsoobtained by a different method in Ref. [75]. In the following, we investigate the possibility that dark matter halos are Bose-Einsteincondensates [72].The size of the BEC galactic dark matter halo is defined by ρ ( R BEC ) = 0 , giving k = π/R BEC , i.e. R BEC = π s ¯ h aGm . (44)The mass profile of the BEC halo is then given by m BEC ( r ) = 4 π Z r ρ BEC ( r ) r dr = 4 πρ ( c ) BEC k r (cid:18) sin krkr − cos kr (cid:19) . (45)The contribution of the BEC halo to the velocity profile of the particles moving on circularorbit under Newtonian gravitational force is v ( r ) = 4 πGρ ( c ) BEC k (cid:18) sin krkr − cos kr (cid:19) . (46)This has to be added to the respective baryonic contribution.8igure 1. Best fit curves for the HSB galaxy sample. The dashed lines hold in the baryonicmatter + BEC model, while the solid lines in the baryonic matter + NFW model. In thesecond and third example the curves run very close to each other.
4. CONFRONTING THE BEC MODEL WITH ROTATIONCURVE DATA
In order to test the validity of the BEC dark matter model, we confront the rotation curvedata of a sample of 3 High Surface Brightness (HSB) galaxies, 3 Low Surface Brightness(LSB) galaxies and 3 dwarf galaxies, with both the NFW dark matter and the BEC densityprofiles.The commonly used NFW model is based on the numerical simulations of dark-matterhalos in the Λ CDM framework [52]. The mass density profile is given by ρ NF W ( r ) = ρ s ( r/r s ) (1 + r/r s ) , (47)where there are two fit parameters ρ s and r s .The mass within a sphere with radius r = yr s is then given by M ( r ) = 4 πρ s r s (cid:20) ln(1 + y ) − y y (cid:21) , (48)where y is a dimensionless radial coordinate. We follow the method described in Ref. [24]. In a High Surface Brightness galaxy wedecompose the baryonic component into a thin stellar disk and a spherically symmetricbulge. We assume that the mass distribution of the bulge component follows the deprojectedluminosity distribution, the proportionality factor being the mass-to-light ratio. We estimatethe bulge parameters from a S´ersic r /n bulge model, fitted to the optical I-band galaxy lightprofiles. 9 alaxy D I ,b n r r b I HSB ,d h HSB
Mpc mJy / arcsec kpc kpc mJy / arcsec kpcESO215G39 61.29 0.1171 0.6609 0.78 2.58 0.0339 4.11ESO322G76 64.28 0.2383 0.8344 0.91 4.50 0.0251 5.28ESO509G80 92.86 0.2090 0.7621 1.10 4.69 0.0176 11.03 Table 1. The distances ( D ) and the photometric parameters of the 3 HSB galaxy sample.Bulge parameters: the central surface brightness ( I ,b ), the shape parameter ( n ), the charac-teristic radius ( r ) and radius of the bulge ( r b ). Disk parameters: central surface brightness( I HSB ,d ) and length scale ( h HSB ) of the disk.
The surface brightness profile of the spheroidal bulge component of each galaxy is de-scribed by a generalized S´ersic function [71] I b ( r ) = I ,b exp " − (cid:18) rr (cid:19) /n , (49)where I ,b is the central surface brightness of the bulge, r 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 being γ ⊙ =5133 kg W − . In what follows, the mass-to-light ratio of the bulge σ will be given in unitsof γ ⊙ (solar units), while the masses in units of the solar mass M ⊙ = 1 . × kg.The radial distribution of visible mass is given by the radial distribution of light obtainedfrom 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: M b ( r ) = σ N ( D ) F ⊙ π r Z I b ( r ) rdr, where F ⊙ ( D ) is the apparent flux density of the Sun at a distance D Mpc, F ⊙ ( D ) =2 . × − . f ⊙ mJy , with f ⊙ = 4 .
08 + 5 lg ( D/ , and N ( D ) = 4 . × − D − m − arcsec . (50)Therefore the contribution of the bulge to the rotational velocity is v b ( r ) = GM b ( r ) r . (51)10 .1.2. Disk contribution In a spiral galaxy, the radial surface brightness profile of the disk exponentially decreaseswith the radius [22] I d ( r ) = I HSB ,d exp (cid:18) − rh HSB (cid:19) , (52)where I HSB ,d is the disk central surface brightness and h HSB is a characteristic disk lengthscale. The contribution of the disk to the circular velocity is [22] v d ( x ) = GM HSBD h HSB x ( I K − I K ) , (53)where I n and K n are the modified Bessel functions calculated at x/ r/ h HSB and M HSBD is the total mass of the disk.
Therefore the rotational velocity in a HSB galaxy receives the following contributions v tg ( x ) = v b ( x ) + v d ( x ) + v DM ( x ) . We confront the BEC+baryonic model with ( HI and H α ) rotation curve data of 3 galax-ies already employed in Ref. [24], which were extracted from a larger sample given inRef. [54] by requiring i) sufficient and accurate data for each galaxy and ii) manifest spher-ical structure of the bulge (no visible rings and bars). For comparison, the NFW+baryonicmodel is also tested on the same sample, by plotting the respective rotation curves in bothmodels. The results are represented on Fig. 1.For the HSB galaxy sample we have derived the best fitting values of the baryonicmodel parameters I ,b , n , r , r b , I HSB ,d and h HSB from the available photometric data.The BEC and NFW parameters were calculated (together with the corresponding baryonicparameters) by fitting these models to the rotation curve data. These are collected in thetables 1 and 2.
Low Surface Brightness (LSB) galaxies are characterized by a central surface brightnessat least one magnitude fainter than the night sky. These galaxies form the most unevolvedclass of galaxies [36] and have low star formation rates as compared to their HSB counter-parts [49]. LSB galaxies show a wide spread of colours ranging from red to blue [53] andrepresent a large variety of properties and morphologies. Although the most commonly ob-served LSB galaxies are dwarfs, a significant fraction of LSB galaxies are large spirals [4].Our model LSB galaxy consists of a thin stellar+gas disk and a cold dark matter compo-nent in a form of BEC. The disk component is the same as at the HSB galaxies, the surfacebrightness profile is [22] I d ( r ) = I LSB ,d exp (cid:18) − rh LSB (cid:19) , alaxy σ (BEC) M HSBD (BEC) R BEC ρ ( c ) BEC χ (BEC) σ (NFW) M HSBD (NFW) r s ρ s χ (NFW) 1 σ ⊙ M ⊙ kpc − kg/m ⊙ M ⊙ kpc − kg/m ESO215G39 0.7 4.96 50 0.2 22.9 0.6 3.84 187 14.7 22.22 34.18ESO322G76 0.8 9.08 5 0.5 48.31 0.9 8.29 920 0.7 49.02 53.15ESO509G80 1.3 52.02 7 1.6 19.77 0.9 11 22 800 33.48 36.3
Table 2. The best fit parameters and the minimum values ( χ ) of the χ statistics for the3 HSB galaxies. Columns 2-5 give the BEC model parameters (radius R BEC and centraldensity ρ ( c ) BEC of the BEC halo) and the corresponding baryonic parameters (mass-to-lightratio σ ( BEC ) of the bulge and total mass of the disk M HSBD ( BEC ) ). Columns 7-10 givethe NFW model parameters (scale radius r s and characteristic density ρ s of the halo) andthe corresponding baryonic parameters (mass-to-light ratio σ ( N F W ) of the bulge and totalmass of the disk M HSBD ( N F W ) ). The 1 σ confidence levels are shown in the last column(these are the same for both models). For all galaxies χ are within the 1 σ confidencelevel. Both model fittings give similar χ values.Figure 2. Best fit curves for the 3 LSB galaxies (the dashed lines show the combinedBEC+baryonic model, the solid lines show the combined NFW+baryonic profiles).where I LSB ,d is the central surface brightness and h LSB is the disk length scale. We cancalculate the disk contribution to the circular velocity as v d ( r ) = GM LSBD h LSB x ( I K − I K ) , (54)similarly to the case of HSB galaxies.Therefore for a generic projected radius r , the rotational velocity in this combinedmodel can be written as v tg ( r ) = v d ( r ) + v DM . We follow the analysis of the BEC model with the rotation curves of 3 LSB galaxiestaken from a larger sample [17]. These high quality rotation curve data are based on both HI and Hα measurements. From a χ -test we have determined the model parameters inboth the BEC+baryonic and NFW+baryonic models, these are shown in Table 3. The fittedcurves are represented on Fig. 2. Dwarf galaxies are the most common galaxies in the observable Universe. About 85% ofthe known galaxies in the Local Volume are dwarfs [38]. Dwarf galaxies are defined as12 alaxy
D h
LSB M LSBD ( BEC ) ρ ( c ) BEC R BEC χ ( BEC ) M LSBD ( NFW ) ρ s r s χ ( NFW ) σMpc kpc M ⊙ − kg/m kpc M ⊙ − kg/m kpc NGC 4455 6.8 0.7 0.236 1.4 5.6 9.4 0 62 39 30.37 18.11UGC 1230 51 4.5 27 0.1 27.7 19.86 26.2 2 291 19.84 8.17UGC 10310 15.6 1.9 0.412 1 7.7 2.68 0 3 660 28.78 13.74
Table 3. The best fit BEC and NFW parameters of the 3 LSB galaxies.Figure 3. The best fit curves for the dwarf galaxy sample. The BEC+baryonic model(dashed curves) gives a better fit in all cases then the NFW+baryonic model (solid lines).In both cases the fit was performed with the same baryonic model.galaxies having an absolute magnitude fainter than M B ∼ − mag, and more extendedthan globular clusters [74].The formation history of dwarf galaxies is not well-understood. According to [76],dwarf galaxies formed at the centers of subhalos orbiting within the halos of giant galaxies.Five main classes of dwarf galaxies are distinguished based on their optical appearance:dwarf ellipticals, dwarf irregulars, dwarf spheroidals, blue compact dwarfs, and dwarf spi-rals. The last one type can be regarded as the very small end of spirals [48].All dwarf galaxies have central velocity dispersions ∼ ÷ km/s [47]. If the systemsare in dynamic equilibrium, the mass derived from these velocity dispersions is much largerthan the derived stellar mass. Therefore the dwarf galaxies are among the darkest objectsever observed in the Universe, hence they play an important role in the study of dark matterdistribution on small scales. As dwarf galaxies are supposed to be dark matter dominated atall radii, they are ideal objects to prove or falsify various alternative gravity theories [12].To test the BEC model, we have selected a sample of 3 dwarf galaxies for which highresolution rotation curves are available. We performed the rotation curve fitting with theBEC+baryonic and the NFW+baryonic models, respectively. The baryonic componentswere the same as in the case of LSB galaxies. However, the length scales of the stellar disksare not available for this sample, therefore they are calculated from χ minimization, too.This comparison allows us to test the viability of our model.For the investigated dwarf galaxies the best fit BEC and NFW parameters are shown inTable 4 and the fitted rotation curves are represented in Fig. 3.13 alaxy h dwarf ( BEC ) M dwarfD ( BEC ) ρ ( c ) BEC R BEC χ ( BEC ) h dwarf ( NFW ) M dwarfD ( NFW ) ρ s r s χ ( NFW ) σkpc M ⊙ − kg/m kpc kpc M ⊙ − kg/m kpc IC 2574 1.2 0.1122 0.4 13 68.47 7.9 28.44 0 0 714.73 44.74HoI 0.2 0.0107 3.6 1.9 95.26 0.9 0.533 0 0 241.30 20.27M81dwB nor 0.9 1.023 3.7 0.7 6.19 0.7 0.705 0 0 8.4 10.42
Table 4. The best fit BEC+baryonic and NFW+baryonic parameters for the dwarf galaxysample.
5. CONCLUSION
We presented here a comprehensive introduction into the theory of Bose-Einstein conden-sates (BEC), with emphasis on a spherically symmetric self-gravitating BEC in the Thomas-Fermi approximation.In the BEC model, large-scale structures like clusters or superclusters of galaxies formsimilarly as in the Cold Dark Matter model with cosmological constant, thus all predictionsof the standard model at large scales are well reproduced [44]. Moreover the BEC modelcan explain the collisions of galaxy clusters [41] and the acoustic peaks of the cosmicmicrowave background [66].Such a BEC has been proposed recently as a galactic dark matter model and has beenpartially tested by confronting with rotation curve data in Refs. [18, 33, 65].Here we have performed a much thorough test of this BEC dark matter model by con-fronting with a sample of 3 HSB, 3 LSB and 3 dwarf galaxies and also comparing the modelpredictions with those of the widely accepted NFW dark matter model. We incorporated inall cases realistic baryonic models, taking into account the particularities of the respectivegalaxy types. Beside the rotation curve data for the HSB galaxies, the surface photometrydata was also available. The galaxies have different luminosities, disk length-scales andsurface brightnesses. Most of the rotation curve data were densely distributed and uniformin quality.We have also fitted the rotation curves of a sample of 3 dwarf galaxies with both theBEC+baryonic and NFW+baryonic dark matter models. Since dwarf galaxies are supposedto be dark matter dominated, they provide the strongest test of the compared models. Theresults are shown in Fig. 3. In all cases, the BEC dark matter model gave better results thanthe NFW dark matter model.For all galaxy types we have determined the BEC parameters ρ BEC , R
BEC , given inTables 2, 3, and 4.For the investigated galaxies, we decomposed the circular velocity into its baryonic anddark matter contributions: v model ( r ) = v baryonic + v DM . The dark matter contribution to thevelocity is given by Eq. (46). Then the rotation curves are χ best-fitted with the baryonicparameters and the parameters of the two dark matter halo models (BEC and NFW).For the sample of HSB galaxies we found a remarkably good agreement of both darkmatter models with the observations. The quality of the fits of the BEC and NFW modelswith the rotation curve data was comparable.For the sample of
LSB galaxies , the BEC model gave a slightly better fit than the NFWmodel. We additionally found that adding the baryonic component results in a better fit thanthe one presented in [65] for the pure BEC model.14n: Book TitleEditor: Editor Name, pp. 15-18 ISBN 0000000000c (cid:13)
PACS
Keywords:
Dark matter halos.
References [1] Anderson, M.H., Ensher, J.R., Matthews, M.R., Wieman, C.E., Cornell, E.A., 1995,Science , 198[2] Barcelo, C., Liberati, S., Visser, M., 2011, Class. Quant. Grav. , 1137[3] Battaner, E., Garrido, J.L., Membrado, M., Florido, E., 1992 Nature, , 6405, p.652[4] Beijersbergen, M., de Blok, W.J.G., van der Hulst, J.M., 1999, A&A , 903[5] Bertolami, O., Boehmer C.G., Harko, T., Lobo, F.S.N., 2007, Phys. Rev. D , 104016[6] Boehmer, C.G., Harko, T., 2007b, JCAP , 025[7] Boehmer, C.G., Harko, T., Lobo, F.S.N., 2008, Astropart. Phys., , 386[8] Boehmer, C.G., Harko, T., 2007a, MNRAS , 2007[9] Bogoliubov, N., 1947, J. Phys. USSR , 23[10] Bradley, C.C., Sackett, C.A., Tollett, J.J., Hulet, R.G., 1995, Phys. Rev. Lett. , 168711] Burkert, A., 1997, ”Aspects of Dark Matter in Astro-and Particle Physics”[12] Capozziello, S., Cardone, V.F., Troisi, A., 2007, MNRAS , 1423[13] Chavanis, P.-H., 2011, Phys. Rev. D , 043531[14] Chavanis, P.-H., Delphini, L., 2011, Phys. Rev. D , 043532[15] Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S., 1999, Rev. Mod. Phys. , 463[16] Davis, K.B., Mewes, M.-O., Andrews, M.R., van Druten, N.J., Durfee, D.S., Kurn,D.M., Ketterle, W., 1995, Phys. Rev. Lett. , 3969[17] de Blok W.J.G., Bosma A., 2002, A&A , 816[18] Dwornik, M., Keresztes, Z., Gergely, L. ´A., 2013, [arXiv:1301.6614][19] Ernst, U., Marte, A., Schreck, F., Schuster, J., Rempe, G., 1998, Europhys. Lett. , 1[20] Evans, N.W., 2001, Proc. 3rd InternationalWorkshop on the Identification of Dark-Matter.World Sci., Singapore, p. 85[21] Fukuyama, T., Morikawa, M., 2006, Progress of Theoretical Physics , 1047[22] Freeman, K.C., 1970, ApJ , 811[23] Gentile, G., Famaey, B., Zhao, H., Salucci, P., 2009, Nature , 627[24] Gergely, L. ´A., Harko, T., Dwornik, M., Kupi, G., Keresztes, Z., 2011, MNRAS ,3275[25] Giorgini, S., Pitaevskii, L., Stringari, S., 1996, Phys. Rev. A , R4633[26] Glaum, K., Pelster, A., Kleinert, H., Pfau, T., 2007, Phys. Rev. Lett. , 080407[27] Gross, E.P., 1961, Nuovo Cimento , 454[28] Gross, E.P., 1963, J. Math. Phys. , 195[29] Grossmann, S., Holthaus, M., 1995, Zeit. f. Naturforsch. , 921; Phys. Lett. A ,188[30] Han, D.-J., Wynar, R.H., Courteille Ph., Heinzen D.J., 1998, Phys. Rev. A , R4114[31] Hau, L.V., Busch, B.D., Liu, C. Z. Dutton, Burns, M.M., Golovchenko, J.A., 1998,Phys. Rev. A , R54[32] Haugerud, H., Haugset, T., Ravndal, F., 1997, Phys. Lett. A , 18[33] Harko, T., 2011a, JCAP , 022[34] Harko, T., 2011b, MNRAS , 3095[35] Harko, T., Madarassy, E. J. M., 2011 [arXiv:1110.2829v1]1636] Impey, C., Bothun, G., 1997, ARA&A. , 267[37] Jalocha, J., Bratek, L., Kutschera, M., Skindzier, P., 2010, MNRAS , 2805-2816[38] Karachentsev, I. D., Karachentseva, V. E., Huchtmeier, W. K., Makarov, D. I., 2004,AJ , 2031[39] Ketterle, W., van Druten N.J., 1996, Phys. Rev. A , 656[40] Kristen, K., Toms, D.J., 1996, Phys. Rev A , 4188[41] Lee, J.W., Lim, S., Choi, D., 2008, [arXiv:0805.3827v1][42] Lieb, E.H., Seiringer, R., Yngvason, Y., 2000, Phys. Rev. A , 043602[43] Madelung, E., 1926, Zeitschrift f¨ur Physik , 322[44] Magana, J., Matos, T., Robles, V. H., Suarez, A., 2012, Proceedings of the XIII Mex-ican Workshop on Particles and Fields [arXiv:1201.6107v1][45] Mak, M. K., Harko, T., 2004, Phys. Rev. D , 024010[46] Mannheim P. D., 1997, AJ , 659[47] Mateo, M., 1998, ARA&A , 435[48] Matthews, L. D. and Gallagher, J. S., 1997, AJ , 5[49] McGaugh, S.S., 1994, ApJ , 135[50] Milgrom, M., 1983, APJ , 365[51] Moffat, J.W., Sokolov, I.Y., 1996, Phys. Lett. B , 59[52] Navarro, J.F., Frenk, C.S., White, S.D.M., 1996, ApJ , 563[53] O’Neil, K., Bothun, G.D., Schombert, J., Cornell, M.E., Impey, C.D., 1997, AJ ,244[54] Palunas, P., Williams, T.B., 2000, AJ , 2884[55] Padmanabhan, T., 2003, Phys. Repts. , 235[56] Peebles, P. J. E., Ratra, B., 2003, Rev. Mod. Phys. , 559[57] Persic, M., Salucci, P., Stel, F., 1996, MNRAS , 27[58] Pitaevskii, L.P., 1961, Zh. Eksp. Teor. Fiz. , 646 [Sov. Phys. JETP , 451 (1961)][59] Pitaevskii, L. P., Stringari S., 2003, Bose-Einstein Condensation, Oxford UniversityPress Inc., New York.[60] Planck collaboration, 2013, [arXiv:1303.5076]178 Gergely et al.[61] Planck collaboration, 2013, [arXiv:1303.5075][62] Rahaman, F., Kalam, M., DeBenedictis, A., Usmani, A.A., Saibal, R., 2008, MNRAS , 27[63] Rindler-Daller, T., Shapiro, P., 2011, [arXiv:1106.1256][64] Roberts, M.D., 2004, Gen. Rel. Grav. , 2423[65] Robles, V. H., Matos T., 2012, MNRAS , 282-289[66] Rodriguez-Montoya, I., Magana, J., Matos, T., Perez-Lorenzana, A., 2010, ApJ ,1509[67] Sanders, R.H., 1984, A&A , L21[68] Salucci, P., Persic, M., 1999, MNRAS , 923[69] Sch¨utte, M., Pelster, A., 2008, Critical Temperature of a Bose-Einstein Condensatewith 1/r Interactions , Proceedings of the 9th International Conference, 23-28, Septem-ber, 2007, Dresden, Germany, Eds. Janke W. and Pelster A., World Scientific Publish-ing Co. Pte. Ltd., 2008. ISBN , 3650[73] Sonego, S., 1991, Found. Phys. , 1135[74] Tammann, G. A., 1994, Dwarf Galaxies in the Past, in Dwarf Galaxies, ESO Confer-ence and Workshop Proc No. 49, Eds. G. Meylan and P. Prugniel, Paris, p. 3[75] Wang, X.Z., 2001, Phys. Rev D , 124009[76] Wu, X., 2007, submitted to ApJ[77] Yu, R. P., Morgan M. J., 2002, Class. Quantum Grav.19