Radial Acceleration Relation from Ultra-light Scalar Dark matter
aa r X i v : . [ a s t r o - ph . GA ] M a y Radial Acceleration Relation from Ultra-light Scalar Dark matter
Jae-Weon Lee, Hyeong-Chan Kim, and Jungjai Lee
3, 4 Department of electrical and electronic engineering, Jungwon university,85 Munmu-ro, Goesan-eup, Goesan-gun, Chungcheongbuk-do, 367-805, Korea School of Liberal Arts and Sciences, Korea National University of Transportation, Chungju 27469, Korea Division of Mathematics and Physics, Daejin University, Pocheon, Gyeonggi 487-711, Korea Korea Institute for Advanced Study 85 Hoegiro, Dongdaemun-Gu, Seoul 02455, Korea
We show that ultra-light scalar dark matter (fuzzy dark matter) in galaxies has a quantummechanical typical acceleration scale about 10 − ms − , which leads to the baryonic Tully-Fisherrelation. Baryonic matter at central parts of galaxies acts as a boundary condition for dark matterwave equation and influences stellar rotation velocities in halos. Without any modification of gravityor mechanics this model also explains the radial acceleration relation and MOND-like behavior ofgravitational acceleration found in galaxies having flat rotation curves. This analysis can be extendedto the Faber-Jackson relation. The baryonic Tully-Fisher relation (BTFR) [1] is a tight empirical correlation between the total baryonic mass ( M b )of a disk galaxy and its asymptotic rotation velocity v f ; M b ∼ v f . Semi-analytic models for BTFR based on baryonicprocesses in a cold dark matter (CDM) cosmology predict significant scatter from individual galaxy formation history,but observed BTFR is largely independent of baryonic processes and has small scatter [2]. There is another strongrelation called radial acceleration relation (RAR) between the radial gravitational acceleration traced by rotationcurves (RCs) of galaxies and predicted acceleration by the observed baryon distributions [3]. There are models [4]based on CDM paradigm explaining RAR, but it is unclear whether this tight relation can survive chaotic processesof galaxy formation and mergering. These relations are puzzling, because galactic halos seem to be dark matter (DM)dominated objects and RCs at outer parts of galaxies are believed to be governed mostly by DM not by baryons. Thereare other relations challenging conventional DM models such as Faber-Jackson relation or baryon-halo conspiracy [5].On the other hand BTFR and RAR are consistent with Modified Newtonian dynamics (MOND) which was proposedto explain the flat RCs without introducing dark matter [6]. According to MOND Newtonian gravitational accelerationof baryonic matter g b should be replaced by g obs = p g b g † , (1)when g b < g † ≃ . × − ms − . The value of g † can be determined from RCs of galaxies [6, 7]. However, MONDalso has its own difficulties in explaining the properties of galaxy clusters and cosmic background radiation [8].In this letter, we show that ultra-light scalar dark matter (fuzzy dark matter) has a quantum mechanical typicalacceleration scale g † , which naturally leads to dynamically established BTFR. Without any modification of gravityor mechanics this model also explains the RAR and MOND-like behavior of gravitational acceleration.Although the CDM model well explains observed large scale structures of the universe, it encounters many difficultiesin explaining galactic structures. For example, numerical studies with CDM predict cuspy DM halos and many satellitegalaxies, which are in tension with observational data [9–12]. Recently, there have been renewed interests in scalarfield dark matter [13–17] (SFDM, often also called fuzzy DM [18], ultra-light axion, BEC DM or wave DM) asa solution of these problems. In this model DM is a ultra-light scalar with mass m ≃ − e V in Bose-Einsteincondensation (BEC). Its long Compton wavelength λ c = 2 π ~ /mc ≃ . ξ of a galaxy is about the de Broglielength ξ dB rather than λ c , which helps in solving the problems of CDM.In this model, galactic halos are self-gravitating giant boson stars where gravitational force of matter balances withquantum pressure from the uncertainty principle with spatial uncertainty ξ about ξ dB . From the uncertainty principle ξmv ≃ ξm p GM c /ξ ≥ ~ one can estimate ξ ≃ ~ /GM c m , where M c is the halo mass scale and v is a typical rotationvelocity of a galaxy. If we identify M c ∼ M ⊙ and ξ ∼ m ≃ ~ / √ ξGM c ≃ O (10 − ) e V. Note that ξ is not a constant but almost independent of otherproperties of the galaxy except for M c . We suggest that ξ and the uncertainty principle lead to a natural accelerationscale g † = GM c /ξ ≃ ~ /m ξ ≃ O (10 − )ms − of SFDM. We will show that this acceleration scale g † from theuncertainty principle gives a hint to the aforementioned relations of galaxies.In SFDM model, DM scalar field φ is described by the action S = Z √− gd x [ − R πG − g µν φ ∗ ; µ φ ; ν − U ( φ )] , (2)where the typical potential is U ( φ ) = m | φ | + λ | φ | . For fuzzy DM λ = 0. In the Newtonian limit the Einsteinequation and the Klein-Gordon equation from the action can be reduced to the Schr¨odinger equation [27] i ~ ∂ t ψ ( r , t ) = − ~ m ∇ ψ ( r , t ) + m Φ ψ ( r , t ) (3)and the Poisson equation ∆Φ( r ) = 4 πG ( ρ d ( r ) + ρ b ( r )) (4)with a self-gravitation potential Φ and wavefunction ψ ≡ √ mφ . Here, ρ d is a DM density and ρ b is a baryonic matterdensity, both of which contribute to Φ. Since galaxies are non-relativistic, in this model a galactic DM halo is welldescribed by the macroscopic wavefunction ψ which is a solution of the Schr¨odinger equation.For simplicity we consider a spherical fuzzy DM halos. Integrating the above equation gives magnitude of totalgravitational acceleration g obs ( r ) ≡ |∇ Φ | = 4 πGr Z r ( ρ d ( r ′ ) + ρ b ( r ′ )) r ′ dr ′ ≡ g d ( r ) + g b ( r ) , (5)where g d ( r ) is the acceleration from dark matter and g b ( r ) from baryonic matter at galactocentric radius r . TheMadelung representation [20, 22] ψ ( r, t ) = p ρ d ( r, t ) e iS ( r,t ) / ~ (6)is useful to calculate g obs in a fluid approach. Substituting Eq. (6) in to the Schr¨odinger equation, one can obtain amodified Euler equation ∂ v ∂t + ( v · ∇ ) v + ∇ Φ + ∇ pρ d − ∇ Qm = 0 , (7)where v ≡ ∇ S/ m , p , and Q ≡ ~ m ∆ √ ρ d √ ρ d are a fluid velocity, the pressure from a self-interaction (if λ = 0), and aquantum potential, respectively. The quantum pressure ∇ Q/m helps fuzzy dark matter to overcome the small scaleproblems of CDM and plays an important role in this paper.By taking v = 0 and ∂ t v = 0, we find a stationary equilibrium condition g obs ( r ) = g d ( r ) + g b ( r ) = ~ m (cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) ∆ √ ρ d √ ρ d (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (8)which describes the dynamical balance between the gravitational attraction and the quantum pressure. This is thekey equation to understand the origin of RAR in our model. It is interesting that the fuzzy DM density profile ρ d andhence the wavefunction ψ traces the total gravitational acceleration not just g d . Using an approximation ∂ r ∼ /ξ in ∇ Q/m one can define the characteristic acceleration for fuzzy DM halos more precisely g † ≡ ~ m ξ = 2 . × − (cid:18) − e V m (cid:19) (cid:18) ξ (cid:19) m/s . (9)Note that this scale has a quantum mechanical origin which is a unique feature of fuzzy DM. g † defined in this way isalmost independent of ρ d . This fact might explain the universality of g † . However, in realistic situations galaxies withdifferent masses can have somewhat different ξ , and hence, g † can have some ranges in our model. Quite interestingly, m Ξ @ p c D FIG. 1. (Color online) g † as a function of ξ and m ≡ m/ − e V. The thin lines correspond to g † = (7 , , , , , , × − m / s from the left, respectively. The dashed red line represents the observed value g † = 1 . × − m/s . if we use the typical core size of the dwarf galaxies ( ∼
300 pc [28]) as ξ , one can reproduce the observed value g † = 1 . × − m/s for a favorable mass m = 1 . × − e V. Fig. 1 shows an effect of the parameter ξ on g † fora given m .Let us see how g † affects galaxies. According to precise numerical studies with fuzzy DM [29] a massive galaxy hasa soliton-like core with size ξ = O (10 ) pc surrounded by a virialized halo of granules (also with size ∼ ξ ) having aNavarro-Frenk-White (NFW) density profile. In the regions where g obs ≫ g † (as in a center of a galaxy) baryonicmatter is usually more concentrated than fuzzy DM and the gravitational acceleration mainly comes from baryon mass.On the other hand, a DM dominated region at a large r beyond the core usually has g obs ≤ g † , because g † representsthe typical acceleration of DM cores if they were made of only fuzzy DM. Therefore, for massive galaxies, g † acts asa parameter discriminating baryonic matter dominated regions ( r ≪ ξ ) from DM dominated regions ( r ≫ ξ ). Forbaryonic matter dominated regions such as central parts of massive galaxies g b ≫ g d , and obviously g obs ≃ g b ≫ g † ,which explains the 1:1 linear part of RAR graph in Fig. 2.On the other hand, there are three regions where g obs can be much smaller than g † ; I) Outermost edge of galaxies( r > O (10 )kpc). II) Outer parts of massive galaxies with almost flat RCs (kpc < r < O (10)kpc). III) Small dwarfgalaxies ( r < kpc).Unlike MOND, in our model if a galaxy is well isolated from others, the rotation velocity in the region I is expectedto drop off because of lack of matter. For example, the Milky way and earlier galaxies seem to have falling RCs [30, 31]in the outermost edge. However, observational data in this region is still rare and uncertain, so we ignore this regionin this letter to understand the observed RAR.Since the observational data points satisfying Eq. (1) mainly come from the region II, and BTFR also relies on theflat rotation velocity data in this region, we will first focus on the flat RCs for which g obs ∼ r − . There are many ããã ã ãã ã ã ãã ãã ã ã ãó ó ó óóó óóóóóóóó óóó ó ó óæ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - - - - - - - - - log g b l og g o b s FIG. 2. (Color online) RAR between g b and the observed acceleration g obs . The red dots represent the binned values of 2693points from SPARC database [32]. Triangles and boxes represent data for dwarf spheroidal galaxies extracted from Ref. 33.Two dashed lines are 1:1 line and p g † g b line with m = 1 . × − e V and ξ = 300 pc, respectively. The black solid curverepresents our theoretical approximation g obs ≃ g b + p g † g b with the same m and ξ . The blue horizontal line represents thetypical acceleration g for dwarf galaxies with M tot = 10 M ⊙ . attempts to obtain the flat RCs with SFDM using excited states [15, 16, 34] or specific potentials [35, 36]. To findthe RAR in the region II in fuzzy DM models we need to know ρ d . Numerical studies with only fuzzy DM indicatethat DM halos have a solitonic core with size about ξ surrounded by an NFW-like profile from virialized granules[29]. Thus, an average DM density over the granules for this quasi-stationary system can be roughly given by usinga step function Θ [37]; ρ d ( r ) ≃ Θ( r e − r ) ρ sol + Θ( r − r e ) ρ NF W . (10)Here ρ sol ∝ / (1 + ( r/r c ) ) is a soliton density, and the NFW profile is ρ NF W = ρ d r /r (1 + r /r ) with constants ρ d , r , r e and r c . Quite interestingly, however, a recent numerical work [38] found that if we include baryon (stars) inthe inner halo, the total matter density ρ tot ≡ ρ b + ρ d follows an almost isothermal profile ( ρ tot ∼ r − and g obs ∼ r − )near the half-light radius r h of the baryon matter rather than Eq. (10). The only cases exhibit this features are when ρ b is comparable to ρ d at the half mass radius, which is consistent with the arguments about g † below Eq. (9). Thatis, r h is the position where g obs ∼ g † and the DM dominance and the flat RCs start. The physical origin of thisnumerical behavior is unclear, but it seems to be a kind of averaging effect of log-slope of DM density and baryonmatter density [38]. If we accept the numerical result, in fuzzy DM model, the region where g obs ≪ g † in massivegalaxies usually corresponds to the region with almost flat RCs and r < O (10)kpc as observed.In this region, we can find a relation between g obs and g b by a simple reasoning. As r increases beyond baryondominated regions, M b ( r ) slowly approaches a total baryon mass M b = const . , and g b ( r ) decreases faster than g d ( r )does. At a point r † the acceleration g b becomes comparable to g d , and g obs approaches the typical value g † , whichmeans g b ( r † ) ≃ g † / r † is about the half-lightradius, i.e., M b ( r † ) ≃ M b ( r h ) = M b /
2. Therefore, 2 g b ( r † ) ≃ GM b /r † ≃ g † . From Eq. (9) it implies r † ≃ q GM b /g † = p Gm ξ M b ~ . (11)Thereby, a bigger M b means a larger r † . Around this point g obs = | d Φ /dr | starts to be small and the rotation velocitygraph v ( r ) ≃ p | Φ( r ) | has a gentle slope, which means almost flat RCs, i.e., v ( r ) ≃ v f [38]. Using r † above one canestimate the constant rotation velocity v f ≡ p r † g † = r GM b r † ≃ ( GM b g † ) / , (12)which is just BTFR, M b = Av f with A = ( Gg † ) − = 2 m ξ G ~ = 34 . (cid:16) m − e V (cid:17) (cid:18) ξ (cid:19) M ⊙ / (km/s) . (13)Remarkably, with Eq. (9) it reproduces the observed value A = 47 ± M ⊙ km − s [39], if m = 1 . ± . × − e Vfor ξ = 300 pc. Note that r † ∼ O (kpc) is somewhat larger than ξ for a typical galaxy. One of the advantages of ourapproach is that approximate values of g † and A can be derived from the model. In our model, BTFR has a quantummechanical origin, although it is a relation among macroscopic quantities of baryonic matter. (A Tully-Fisher-likerelation between the total DM and the circular velocity was suggested for fuzzy DM in Ref. 40.) Following Ref. 41we can derive the asymptotic form of RAR from the BTFR ( M b = v f /Gg † ), g b ( r ≫ r † ) ≃ GM b r = 1 g † v f r ! = g obs g † , (14)i.e., g obs = p g b g † . This is the MOND-like behavior of g obs in the RAR graph at large radii where g b ≪ g † and v ( r ) ≃ v f . Thus, in our model MOND is just an effective phenomenon of fuzzy DM. Therefore, fuzzy DM can explainthe apparent successes of both of CDM and MOND, because it acts as CDM at super-galactic scales and as an effective MOND at galactic scales due to the finite length scale ξ . The mass discrepancy-acceleration relation (MDAR) alsoappears [42], because M tot ( r ) /M b ( r ) = g obs ( r ) /g b ( r ) ≃ p g † /g b , where M tot ( r ) is the total mass enclosed within r .We now understand how RAR behaves in our model in two extreme limits where g b ≫ g † or g b ≪ g † . An approximatefunction linking the two limits for RAR is g obs = g b + p g † g b , which is a simple sum of g b and g d ≃ p g † g b in Eq. (8)(See Fig. 2). BTFR and RAR in our model can have small scatter because these relations are from the dynamicalequilibrium condition rather than from forming history of galaxies or from baryon physics.Equation (8) seems to explain some other mysteries in massive galaxies. First, for galaxies with flat RCs we canroughly approximate the total density with a cored-isothermal one ρ obs ≃ σ / πG ( r + r † ) ≡ ρ c / (1 + ( r/r † ) ) up toa few r † as an effective core size. This leads to an universal surface density of cored galaxies [43]Σ ≃ ρ c r † ≃ σ πGr † ≃ g † πG . (15)Here σ is the stellar velocity dispersion and g † ≃ σ /r † . With Eq. (9) this reproduces the observed value [44]Σ = 141 +82 − M ⊙ pc − for m = 1 . + . − . × − e V and ξ = 300 pc. Second, for the isothermal distribution where g obs ≪ g † the wavefunction ψ in the region II should be dynamically adjusted to satisfy Eq. (8) under the smallvariation of ρ b ( r ), which explains the baryon-halo conspiracy for flat RCs [45]. Finally, we observe that Eq. (8) canbe rearranged to be an integro-differential equation for ρ d ( r ); g b ( r ) = − g d ( ρ d ( r )) + (cid:12)(cid:12)(cid:12)(cid:12) ∇ Q ( ρ d ( r )) m (cid:12)(cid:12)(cid:12)(cid:12) , (16)where g b plays a role of a source term or a boundary condition. A solution ρ d ( r ) of this wave equation at large r should be such that the right hand side approaches g b ( r ) ≃ GM b /r . For this solution details of baryon distributionat central regions except for M b are not so much relevant. This explains why g d and hence g obs are so sensitive to g b inmassive galaxies despite of variety of the galaxies and at the same time insensitive to other visible matter propertieslike luminosity.We move to the region III. In small dwarf galaxies the spatial size of baryonic matter distribution is comparableto that of DM halos, and M b can not play a role of central boundary condition as in the region II. Thereby, thearguments related to flat RCs do not hold in this region. In fuzzy DM model these galaxies are similar to the groundstate (soliton) of boson stars which has a minimum mass comparable to the quantum Jeans mass.The mass ( M tot )-radius ( R ) relation of solitonic core from the boson star theory is M tot R = β ~ /Gm , where, forexample, the constant β = 3 .
925 for the half mass radius of DM [21]. Therefore, using the mass-radius relation thecore of DM dominated dwarf galaxies has a typical acceleration g = GM tot R ≃ G m M tot β ~ ≥ Gm γ M J β ~ , (17)which gives 4 . × − ms − for m = 1 . × − e V and M tot = 10 M ⊙ . Here we identify γM J to be the minimumgalaxy mass from the quantum Jeans mass M J ( z ) = π (cid:18) ~ G m (cid:19) ¯ ρ ( z ) , (18)where γ ≃ . ρ ( z ) is the background matter density at redshift z . Since relevant mass here is the total mass M tot = M b + M d , g obs is insensitive to the fraction of baryonic matter aslong as M b ≪ M tot . This explains the flattening and large scatter of the RAR curve for small dwarf galaxies where g b < − m/s (See Fig. 2). Note that g has a minimum value from the quantum Jeans mass M J .Regarding galaxy formation, fuzzy DM has only two free parameters, the particle mass m and the backgroundmatter density ¯ ρ ( z ). If we represent ξ with these parameters, we can fully determine g † and A from the model. Fromthe boson star mass-radius relation [15, 46], a natural candidate for ξ is suggested [47, 48] to be ξ = β ~ GM tot m = 3 β ~ / π / γ ( Gm ¯ ρ ( z )) / , (19)which is about 2 kpc for M tot = 10 M ⊙ and m = 1 . × − e V. This size is somewhat larger than the ob-served core size r c ∼ O (10 ) pc for a massive galaxy, although the profile of the core is quite similar to theground state of boson stars. According to numerical studies with fuzzy DM, the smallness of r c is attributed tothe nonlocal uncertainty principle applied to r c and velocity dispersion σ , i.e., r c σ ∼ ~ /m [49]. More precisely, r c = 1 . a / (10 − e V/m)(10 M ⊙ /M h ) / kpc, where M h is a halo mass [49] and a is the scale factor of the universe.It gives ξ ≃ r c = 300 pc for typical halos with M h = 10 M ⊙ and m = 1 . × − eV at present ( a = 1). Fromthe r c formula we expect g † ∝ a − / ∝ (1 + z ) / . Since r c is a slow function of M h , ξ is almost independent ofproperties of massive galaxies such as luminosity. However, in this case, g † ∼ ξ − ∼ M h depends on the halo mass.Another possibility is that the self-interaction with λ can give a fixed length scale ξ ∼ √ λm p /m with the Planckmass m p [16].Our analysis can be easily extended to the Faber-Jackson relation [50], which is an empirical relation L ∝ σ between the luminosity L and the central stellar velocity dispersion σ of elliptical galaxies. If we assume baryon massto light ratio Υ b ≡ M b /L ≃ M ⊙ /L ⊙ is almost constant for elliptical galaxies [51] and σ ∼ v f , BTFR in Eq. (13)implies L = M b Υ b ≃ . σ Υ b (cid:16) m − e V (cid:17) (cid:18) ξ
300 pc (cid:19) M ⊙ / (km/s) , (20)which is comparable to the observed value L ≃ L ⊙ σ / (km/s) [50]. Due to differences in Υ b for individual galaxies,we expect larger scatter in the Faber-Jackson relation than in BTFR as observed.In our simple model with fuzzy DM g † are not so universal. Interestingly, a recent observation implies dwarf discspirals and Low Surface Brightness galaxies have different RAR curves and g † [52]. There are many studies on thecharacteristic mass and length scale in SFDM models, however little attention has been given to the characteristicacceleration so far [53]. The acceleration scale of fuzzy DM related to the scaling laws such as BTFR and Faber-Jackson relations can play an important role in evolution of galaxies and deserves further studies. These relationsand observed MOND-like phenomenon in galaxies seem to add another support for fuzzy DM. In theoretical pointof view, the value of ξ for galaxy mass scale M c is almost the same as the crossover distance due to dark matter inquantum theory of gravity [54]. This work will provide an avenue in understanding the nature of quantum gravitybecause the properties of characteristic length scale is related to those in emergent quantum gravity.Authors are thankful to Scott Tremaine for helpful comments. [1] S. S. McGaugh, J. M. Schombert, G. D. Bothun, and W. J. G. de Blok, Astrophys. J. , L99 (2000).[2] F. Lelli, S. S. McGaugh, and J. M. Schombert, The Astrophysical Journal Letters , L14 (2016).[3] S. S. McGaugh, F. Lelli, and J. M. Schombert, Phys. Rev. Lett. , 201101 (2016).[4] A. D. Ludlow et al. , Phys. Rev. Lett. , 161103 (2017).[5] S. Trippe, Z. Naturforsch. A69 , 173 (2014).[6] M. Milgrom,
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