Constraint on dark matter central density in the Eddington inspired Born-Infeld (EiBI) gravity with input from Weyl gravity
Alexander A. Potapov, Ramil Izmailov, Olga Mikolaychuk, Nikolay Mikolaychuk, Mithun Ghosh, Kamal K. Nandi
CCONSTRAINT ON DARK MATTER CENTRAL DENSITY INTHE EDDINGTON INSPIRED BORN-INFELD (EiBI)GRAVITY WITH INPUT FROM WEYL GRAVITY
Alexander A. Potapov ,a , Ramil Izmailov ,b , Olga Mikolaychuk ,c ,Nikolay Mikolaychuk ,c , Mithun Ghosh ,d and Kamal K. Nandi , , ,e Department of Physics & Astronomy, Bashkir State University,Sterlitamak Campus, Sterlitamak 453103, RB, Russia Zel’dovich International Center for Astrophysics, M. Akmullah BashkirState Pedagogical University, Ufa 450000, RB, Russia Department of Mathematics, University of North Bengal, Siliguri 734013,WB, India a Email: [email protected] b Email: [email protected] c Email: [email protected] d Email: [email protected] e Email:[email protected] number(s): 04.50.Kd, 95.30.Sf, 04.50.1h
Abstract
Recently, Harko et al. (2014) derived an approximate metric of the galac-tic halo in the Eddington inspired Born-Infeld (EiBI) gravity. In this metric,we show that there is an upper limit ρ upper0 on the central density ρ of darkmatter such that stable circular orbits are possible only when the constraint ρ ≤ ρ upper0 is satisfied in each galactic sample. To quantify different ρ upper0 for different samples, we follow the novel approach of Edery & Paranjape(1998), where we use as input the geometric halo radius R WR from Weylgravity and equate it with the dark matter radius R DM from EiBI gravityfor the same halo boundary. This input then shows that the known fittedvalues of ρ obey the constraint ρ ≤ ρ upper0 ∝ ( R WR ) − . Using the mass-to-light ratios giving α , we shall also evaluate ρ lower0 ∝ ( α − M lum R − andthe average dark matter density (cid:104) ρ (cid:105) lower . Quantitatively, it turns out thatthe interval ρ lower0 ≤ ρ ≤ ρ upper0 verifies reasonably well against many darkmatter dominated low surface brightness (LSB) galaxies for which values of ρ are independently known. The interval holds also in the case of Milky1 a r X i v : . [ g r- q c ] A ug ay galaxy. Qualitatively, the existence of a stability induced upper limit ρ upper0 is a remarkable prediction of the EiBI theory. Key Words:
Dark matter, central density, modified gravity
I. INTRODUCTION
Early observations [1-3] on rotational data of spiral galaxies, now recon-firmed by observations extending well beyond the optical disc [4-18], indicatethat they do not conform to Newtonian gravity predictions. Hence the hy-pothesis is that there could be large amounts of non-luminous matter hiddenin the galactic haloes. The rationale is this: Doppler emissions from stablecircular orbits of neutral hydrogen clouds in the halo allow measurement oftangential velocity v tg of the clouds treated as probe particles. Contrary toNewton’s laws, where v should decay with radius r , observations indicatethat it approximately levels off with r in the galactic halo region, which inturn calls for the presence of additional non-luminous mass, the so calleddark matter. Since dark matter has not yet been directly observed, the darkmatter hypothesis is often variantly referred to as the missing mass problem.Several well known theoretical models for dark matter exist in the liter-ature but it is impossible to list all of them here (only some are mentionedin [19-34]). In its usual formulation, dark matter is a parametrization of theobserved velocity discrepancies and is not a prediction of the formulation.Some simulations require fine tuning of halo parameters to luminous pa-rameters galaxy by galaxy − a procedure that only enlarges the number ofparameters rather than reducing them (See [35], pp.32-33; see also footnote13). There exist yet another variety of halo models, which treat the miss-ing mass problem as a failure of the Newtonian theory on galactic distancescales rather than as a prediction for dark matter. Such models actuallydo not require dark matter at all for the interpretation of observed rotationdata. This class of theories include, e.g., Modified Newtonian Dynamics(MOND) developed by Milgrom [36-42], Scalar-Tensor-Vector Gravity the-ory developed by Moffat [43-45], Weyl conformal gravity implemented byMannheim and O’Brien [48]. For brevity, we call the last the MO modeland we shall use this ingredient in the sequel. A remarkable speciality of theMO model is that, using the best available galactic optical and radio data,and a standardized, non-biased, treatment for selecting appropriate galactic Weyl conformal gravity has been debated for and against in the literature. For in-stance, Flanagan [46] argues that if the source has associated with it a macroscopic longrange scalar field, breaking conformal symmetry, the theory does not reproduce attractivegravity in the solar system. However, subsequently, Mannheim [47] has counter-arguedthat Schwarzschild tests of solar gravity could still be recovered even in the presence ofsuch macroscopic fields. without the need for any dark matter whatsoever.There are various other (non-)dark matter models that are capable of ac-counting, for example, for observations of galaxy clusters and gravitationallensing or structure formation. A leading example is the cold dark matter(CDM) model, which is a part of the current standard model ΛCDM of cos-mology. These models are based on different phenomenologies such as infla-tion and nucleosynthesis [49-66]. They can successfully explain observationsof galaxy clusters [49-54], gravitational lensing [55,56] or structure forma-tion [57], to name the most important ones. These models postulate thatgalactic cores may consist of axions [58], massive gravitons [59], BEC [60] orother collisionless particles. The post-recombination fluctuation spectrumnicely explains the formation of galaxies and clusters [49-54]. The CDM isa successful paradigm accounting for the small density inhomogeneities thatseed structure formations 10 − sec after the bang and as such provides abold probe into the Early Universe [57]. Some other and recent works onCDM models are mentioned here though the list by no means is exhaustive[61-66].Recently, another alternative candidate for dark matter is also beingspeculated. This is based on the evidence of soft positron spectrum in theAMS-02 [67,68] cosmic ray data. Despite this alternative, the observed flatrotation curves are still considered as a robust proof that dark matter essen-tially is of gravitational origin described by general relativity [69-71]. Butin general relativity, matter-gravity coupling is linear, while some authorsargue (for non-minimal coupling of modified gravity with matter or otherinsights into the paradigm, see [72-77]) that there is no obvious reason asto why the coupling should be linear. Following this thought, an interestingmodification of matter-gravity coupling leading to the Eddington-inspiredBorn-Infeld (EiBI) theory has been recently developed by Ba˜nados and Fer-reira [78,79], which we shall use as another ingredient. Only in vacuum, theEiBI theory is equivalent to standard general relativity. This new, and moregeneral, theory has led to interesting observable predictions in the context ofsolar interior dynamics, big bang nucleosynthesis, neutron stars, the struc-ture of other compact stars [80-83] including the possibility of nonsingularcosmological models and alternative to inflation [84].To get a more complete view of the two ingredients of the present work,the EiBI and the Weyl gravity, it is necessary to mention that several im-portant predictions follow from the two theories. For instance, recent in-vestigation by Du et al. [85] on large scale structure formation in the EiBIgravity shows a deviation in the matter power spectrum between the EiBIgravity and the ΛCDM model, which is a testable prediction. Stability andlocalization of gravitational fluctuations in the EiBI brane system have beenstudied in Refs. [86,87]. Further, as shown by Wei et al. [88], strong gravita-tional lensing observables in EiBI are controlled by the coupling parameter3 , which is a new prediction that lends itself to future testing. Similarly,in Weyl gravity, the first order light deflection angle θ W by a galaxy, firstobtained by Edery and Paranjape [89], contains the galactic halo parame-ter γ appearing in the MO model. Bhattacharya et al. [90,91] calculatedhigher order deflection terms. Strong field lensing in the Weyl gravity hasbeen studied recently in [92], and the predictions can be verified by actualobservations in future. A remarkable feature of Weyl gravity is that itssolution, the MO model, already contains the successes of the well testedSchwarzschild gravity as a special case. All the above exemplify the currentstatus of the capabilities of the two theories in question.The possibility of perfect fluid dark matter within the framework of gen-eral relativity has already been explored in the literature [93,94]. A similarpossibility has been recently investigated within the framework of the EiBItheory by Harko et al. [95,96] and this is the model we are going to ana-lyze further in this paper. Using a tangential velocity profile [97,98] givingUniversal Rotation Curves (URC) and setting the cosmological constant tozero, they obtained, in the Newtonian approximation, a new galactic metricand theoretically explored its gravitational properties. However, the nu-merical values of the crucial parameter κ (denoted by κ = 2 R /π ) orequivalently the dark matter radius R DM , cannot be determined from thetheory alone − it has to be obtained either from the observed data or fromsome other model. It is also expected that the values of R DM would differfrom galaxy to galaxy. On the other hand, to our knowledge, apart from theobserved last scattering radii R last , the astrophysical literature still seems tolack concrete observed data on R DM for individual galaxies. Therefore, anappropriate numerical input for R DM is needed, which we take from Weylgravity, if we want to make quantitative predictions.At this point, we recall a novel idea of Edery & Paranjape [89], wherethey bridged two different metric theories by equating the same Einsteinangle θ E (caused by the luminous + dark matter) with the Weyl angle θ W (caused by the luminous matter alone), and drew useful and testable conclu-sions using the identity θ E = θ W . Motivated exactly by this idea, we equatethe same EiBI radius of dark matter R DM (caused by dark matter source)with the geometric Weyl radius of the galactic halo R WR (caused by theluminous matter alone). With the numerical input R DM = R WR , we shallquantify the relevant central densities in the EiBI theory (see footnote 8). We wish to clarify that we are not talking here of merging or mapping thetwo theories into one another per se but concentrating only on a particular We wish to clarify that the EiBI parameter κ is not a universal constant − it’s morelike a parameter of the theory that assumes different values depending on the physicalsituation. For instance, the structure of compact general relativistic star requires a value κ (cid:39) cm [82], which differs from the value κ (cid:39) cm inferred from dark matterdensity profiles. − one with dark matter source and the other without, not to mention differ-ences elsewhere. But both are metric theories capable of predicting for any given galactic sample a dark matter/halo boundary arising out of the samestability condition V (cid:48)(cid:48) < ρ consistent withthose estimated from fits to different known profiles. We shall see that itdoes.The radius R WR is to be understood as the geometric halo radius withits interior being filled with Weyl vacuum. We stress that Weyl vacuum is not a vacuum in the ordinary sense but an arena of interplay of several po-tential energies, predominantly the global quadratic potential due to cosmicinhomogeneities [48]. Thus, our input physically means that the total po-tential energy contained within the halo radius R WR of Weyl gravity equalsthe total invisible dark matter energy contained within R DM of EiBI grav-ity. The radius R DM is defined by the absence of dark matter density atthe halo boundary [95,96], while R WR is defined by the absence of stablecircular orbits at the halo boundary [100]. Since stability is an essentialphysical criterion because Doppler emissions from the halo emanate fromstable circular orbits of hydrogen gas [101], we think that R WR should beregarded as the only testable upper limit on the radius of a galactic halo.Fortunately, observed last scattering data R last so far have not surpassed thepredicted limiting value R WR for all the galaxies studied to date, therebylending excellent observational support to the MO model prediction of R WR .The purpose of the present paper is as follows: We shall concentrateon the low surface brightness (LSB) galaxies that are mostly dominated bydark matter. We show that there is an upper limit on the dark mattercentral density ρ specific to each individual galaxy, which we call here ρ upper0 , such that stable circular orbits in the EiBI are possible only whenthe constraint ρ ≤ ρ upper0 ∝ ( R WR ) − is satisfied in that galaxy. Using theWeyl gravity input R DM = R WR , we shall then quantify ρ upper0 , and showthat the central density ρ predicted from the fit to various simulationsincluding the NFW and Burkert density profiles does obey the constraint.Taking into account the range of the rather uncertain but possible mass-to-light ratios (denoted, say, by α ), we shall calculate also the lower centraldensity ρ lower0 ∝ ( α − M lum R − and find that ρ lower0 ≤ ρ ≤ ρ upper0 holdsfor some known LSB samples for which ρ is known from independent fits.Some illustrative galactic samples including the Milky Way are tabulated.The values fall within the predicted interval for each individual galaxy andwe conjecture that at least the upper limit might be generally true. Note that we are using here the terminology R WR in lieu of R maxstable of Ref.[100] onlyto bring it in line with the notation of the present analysis. ρ upper0 for stability of circular orbits in EiBI. In Sec.IV, using the input underconsideration, we shall quantify upper and lower central densities for somesamples to see if the estimated central densities ρ truly fall within theproposed interval, that is, if ρ lower0 ≤ ρ ≤ ρ upper0 . In Sec.V, we discuss thedark matter density profiles in the context of Milky Way. The results aresummarized in Sec.VI. We shall take units such that 8 πG = 1, c = 1, unlessrestored. II (a): EiBI MODEL
For easy reference, we outline only the salient features of the EiBI darkmatter model developed in [95,96]. The EiBI action is [78,79,85,102,103] S EiBI = 2 κ (cid:90) d x (cid:20)(cid:113) − | g µν + κR µν (Γ) | − λ (cid:113) − | g µν | (cid:21) + S matter , (1)where λ is a dimensionless parameter, g µν is the physical metric, R µν (Γ) isthe symmetric part of the Ricci tensor built solely from the connection Γ αβγ (cid:0) ≡ q ασ [ ∂ γ q σβ + ∂ β q σγ − ∂ σ q βγ ] (cid:1) derived from an auxiliary metric denotedby q µν . The meaning of the auxiliary metric q µν is that it partially satisfiesEddington field equations so that 2 κ (cid:112) | R | R µν = (cid:112) | q | q µν , which can berewritten as Einstein field equations if we equate q µν with g µν and κ withΛ − . For small values of κR , the action (1) reproduces the Einstein-Hilbertaction with λ = κ Λ + 1, where Λ is the cosmological constant, while for largevalues of κR , the action approximates to that of Eddington, viz., S Edd =2 κ (cid:82) d x (cid:112) | R | . For more details, see [78,79].Harko et al. [95,96] deals with dark matter modeling assuming certainrestrictive conditions such as spherical symmetry and asymptotic flaness, thelatter requiring that Λ = 0 ⇔ λ = 1. These assumptions of course limit theapplicability of EiBI theory but makes the problem at hand much simplerto handle. One spin-off is that the description of the physical behaviorof various cosmological and stellar scenarios was assumed to be controlledby the only remaining parameter κ . The galactic halo is assumed to befilled with perfect fluid dark matter with energy-momentum tensor T µν = pg µν + ( p + ρ ) U µ U ν , g µν U µ U ν = − v = v ∞ ( r/r opt ) ( r/r opt ) + r , (2)where r opt is the optical radius containing 83% of the galactic luminosity, r is the halo core radius in units of r opt , the asymptotic velocity v ∞ =6 (1 − β ∗ )(1 + r ), v opt = v tg ( r opt ), β ∗ = 0 .
72 + 0 . ( L/L ∗ ), L ∗ =10 . L (cid:12) . For spiral galaxies, r = 1 . L/L ∗ ) / . Under the Newtonianapproximations that the pressure p (cid:39)
0, 8 πκρ (cid:28)
1, and ( r/r opt ) (cid:29) r , theEiBI field equations yield the Lane-Emden equation with polytropic index n = 1, which has an exact nonsingular solution for dark matter densitydistribution as [95,96] ρ (0) ( r ) = ρ sin (cid:16) r (cid:113) κ (cid:17) r (cid:113) κ , (3)where ρ (0) (0) = ρ is the constant central density. Assuming that thehalo has a sharp boundary R DM , where the density vanishes such that ρ (0) ( R DM ) = 0, one has R DM = π (cid:114) κ . (4)The density profile (3) exhibits an unphysical behavior of becoming nega-tive for R DM < R < R DM , which is why one has to require a sharp haloboundary ρ (0) ( R ≥ R DM ) = 0. This quite specific behavior of the densityprofile differs from those of Navarro-Frenk-White (NFW) or Burkert densityprofiles (that decay to zero only as r → ∞ ). The mass profile of the dark matter is M ( r ) = 4 π (cid:90) r ρ (0) ( r ) r dr = 4 R π ρ [sin( r ) − r cos( r )] , (5)where the dimensionless quantity r = πr/R DM . The total mass of the darkmatter M DM and the mean density (cid:104) ρ (cid:105) in EiBI theory are given respectivelyby M DM = M ( R DM ) = 4 π ρ R , (cid:104) ρ (cid:105) = 3 ρ π . (6)The approximate physical metric has been derived in [35] as dτ = − B ( r ) dt + A ( r ) dr + r C ( r )( dθ + sin θdφ ) , (7) The solution (3) to the Lane-Emden equation and its connection to dark matter wasfirst pointed out in [96]. We thank an anonymous referee for pointing this out. The observational situation is that the mass density profiles on the logarithmic scaleclearly indicate that the density is decaying at a finite radius [104]. While NFW or Burkertdensity profiles might lead to different physical metrics of interest, the key point of thepresent investigation is that the dark matter has a finite radius following from stability considerations [100]. See also footnote 11. The role of the URC profile (2) is that it determines the metric function B ( r ) (cid:104) ≡ e ν ( r ) (cid:105) .Following Chandrasekhar [105], we know that the tangential velocity is given by v = r e ν ( e ν ) ,r . See the details in Ref.[93]. Using the URC for v , defining r/r opt = x , andintegrating, we obtain e ν ( r ) = ( x + r ) v ∞ . Restoring the original variable r/r opt , anddefining r = πr/R DM , we immediately obtain the metric function B ( r ) of Eq.(8). ( r ) = e ν (cid:34)(cid:18) R DM πr opt (cid:19) r + r (cid:35) v ∞ , (8) A ( r ) = (cid:18) R DM π (cid:19) (cid:104) − ρ r sin( r ) + ρ cos( r ) (cid:105) (cid:104) − ρ r sin( r ) (cid:105) , (9) C ( r ) = (cid:18) R DM π (cid:19) (cid:20) − ρ r sin( r ) (cid:21) , (10)where e ν is an arbitrary constant of integration (which we set to unity) andthe dimensionless quantity ρ = ρ R π .Note that the surface area of a sphere at the boundary of dark matterhalo defined by r = π , has the value S = 4 πr C ( r ) = 4 πr (cid:16) R DM π (cid:17) =4 πR , which is just the spherical surface area in ”standard coordinates”.Thus the dark matter radius R DM can be identified with standard coordinatehalo radius R HR derived below. We shall need some of the above equationsin the sequel. II (b): MANNHEIM-O’BRIEN MODEL
The unique Weyl action is S Weyl = − α g (cid:90) d x √− g (cid:104) C λµνσ C λµνσ (cid:105) = − α g (cid:90) d x √− g (cid:20) R µσ R µσ −
13 ( R αα ) (cid:21) , (11)where C λµνσ is the Weyl tensor and α g is the dimensionless gravitationalconstant. The resulting field equations are fourth order and trace free, ratherlong and complicated, so we omit them here. The exact solution of vacuumWeyl gravity for the metric ansatz dτ = − B ( r ) dt + 1 B ( r ) dr + r ( dθ + sin θdφ ) (12)was derived by Mannheim & Kazanas [43] that describes the metric outsideof a localized static, spherically symmetric source of radius r embeddedin a region with T µν ( r > r ) = 0 as follows (after suitably redefining theconstants): B ( r ) = (1 − M γ ) / − Mr + γr − kr , (13)where M, γ, k are constants of integration. Schwarzschild solution is recov-ered at γ = 0 , k = 0 as a special case of Weyl gravity.On the other hand, in Refs.[48,107], the arguments and calculations in-stead proceed from the considerations of potential . In Weyl gravity, a given8ocal gravitational source generates a gravitational potential per unit solarmass as follows ( Eq.(8) of Ref.[107]): V ∗ = − β ∗ c r + γ ∗ c r , where β ∗ and γ ∗ are constants. Then, on integrating V ∗ over the local lu-minous matter distribution, one obtains the local contribution to tangentialvelocity at r = R > R [107]: v R (cid:39) N ∗ β ∗ c R (cid:18) − R R (cid:19) + N ∗ γ ∗ c (cid:18) − R R − R R (cid:19) . (14)The meanings and values of the symbols above are as follows: R is the scalelength such that most of the surface brightness is contained in R ≤ R of theoptical disk region, N ∗ is total number of solar mass units in the luminousgalaxy obtained via the mass-to-light ratio: ( M/L ) L = M lum = N ∗ M (cid:12) .Thereafter, detailed arguments (See Refs.[48,107]) are used to introducetwo additional potentials of cosmologial origin, viz., γ c R and − kc R , thatcontribute to velocity such that v R = v R + γ c − kc R. (15)The numerical values of the constants giving best fits to rotation curves ofall the 111 galaxy samples in [107] are: β ∗ = GM (cid:12) /c = 1 . × cm, γ ∗ = 5 . × − cm − , γ = 3 . × − cm − , k = 9 . × − cm − . (16)It is evident from Eq.(15) that, in making the fits, the only parameterthat can vary from one galaxy to the other is the mass-to-light ratio ( M/L )leading to a galaxy dependent N ∗ . The mass of HI gas is known and includedin the fit. With everything else being universal, no hypothetical dark matteris needed − potential effects of cosmological origin plus the local potentialcaused by the luminous mass of a galaxy are enough to account for theobserved rotation data.For each galaxy with specific value of N ∗ and other fixed constants asin (16), v of Eq.(15) gives a finite value of R at the terminating velocity v ( R term ) = 0, where the potentials balance. To illustrate various resultsderived here, we need to consider samples and so in the following we choosethe LSB galactic sample ESO 1200211. The reason for this choice is that itis one of the samples, whose central dark matter density has been recentlyfitted to various known density profiles by Robles & Matos [108] and thusit lends itself to easy comparison with the results of the present paper. The9lot in Fig.1 shows that the rotation curve v decays to zero at a radialdistance R term = 52 .
04 kpc, but this is still not the halo radius! The actualhalo radius R WR , defined by the stability inequality (21) below, will alwaysbe less than the value of R term for reasons of stability, as will soon be workedout. The MO prescription for v in Eq.(15) leads to the Mannheim-Kazanasmetric (13) of Weyl gravity in the limit of large distances away from thegalactic core in which we are interested. It can be seen as follows. Thegeodesic for a single test particle yields the tangential velocity of materialcircular orbits at the arbitrary radius r = R as [107] v = (cid:0) Rc / (cid:1) B (cid:48) , where prime denotes derivative with respect to R . Integrating, we obtain B ( R ) = 1 − N ∗ β ∗ R + ( N ∗ γ ∗ + γ ) R − kR +3 R N ∗ γ ∗ R + 15 R N ∗ γ ∗ − R N ∗ β ∗ R . (17) Note that there are two specific profiles used in the paper: One is the URC velocityprofile in Eq.(2) that is never zero at any radius, v | URC (cid:54) = 0, while the other is theMO velocity profile, Eq.(15), that can become zero at a finite radius r = R term such that v | MO ( R term ) = 0 . Such different behavior might call into question the validity of theidentification R DM = R WR used as an input in this paper.The physical meaning of this input is explained on p.4. Here we point out some addi-tional grounds justifying the input. Note that it is quite logical that two different theoriescan predict the same measurable quantity, say, light deflection. Similarly, despite dif-ferences in the behavior, the profiles still provide fitted values of the same quantity ρ .Ideally, the values should exactly be the same, which is not the case, but they are com-parable at least by order of magnitudes. In the present case, it is the assumed equality ofthe radii R DM and R WR provided by the two profiles that is in question. We have equatedthem on the ground that, observationally, there has to be only one dark matter radiusfor each sample, no matter how it is defined. So the equality in a way suggests itself.The other ground is that we have been motivated by the approach originally proposed byEdery & Paranjape [89], where they equated the Einstein and Weyl angles. Actually, suchidentifications are justified only a posteriori , when they are found to yield results that areindependently known to be true. For instance, using the input θ E = θ W for the deflectionof light by galaxies, Edery & Paranjape [89] obtained the value of γ in the metric (18)that is reasonably close to that obtained independently in [48,107] by rotational data fitsto samples. On the other hand, we also know that Weyl and Einstein theories, givingrespectively θ W and θ E , are very different from each other − the former involves fourthorder equations and is conformally invariant, while the latter shares none of these. For thisreason, at the very outset in our paper, we clarified that we were not talking of mergingor mapping the two theories entirely into one another (p.4) but concentrating only on aparticular common prediction.Similarly, despite differences in the EiBI theory and the MO model, our identification R DM = R WR leads to constraints on ρ that are found to be remarkably compatible withthe independent NFW or Burkert data fits − that is, the fitted central density values dofall within the stability induced limits. This is the a posteriori justification for using theidentity R DM = R WR . v of Eq.(15) with the observed rotational data [107] reveals that theconstant γ ( ≡ N ∗ γ ∗ + γ ) is of the order of 10 − cm − since N ∗ ≈ (roughly, the number of stellar units in a galactic luminous mass). Also,the estimates covering all samples in [44] reveal that the luminous masseslie approximately in the range M = N ∗ β ∗ ≈ − cm and from thefitted value of γ ∗ = 5 . × − cm − , it follows that N ∗ γ ∗ ≈ − cm − . Thus at halo distances (
R > R ), we can ignore the last two terms in (18)by order of magnitudes. In the same manner, we see that M γ << − M γ ) / (cid:39)
1. Thus the theoreticalmetric (13) and the fitted metric (18) coincide at the form B ( r ) (cid:39) − Mr + γr − kr (18)at halo distances R > R . This is also the form of the Weyl solution usedin Refs. [48,89]. Rephrasing, we can say that v can be arrived at by differ-entiating the metric (17), which in turn approximates to the Weyl solution(18). That’s the relevance of the Weyl solution in the fitting program.In the asymptotic limit, Eq.(15) gives v R → N ∗ β ∗ c R + N ∗ γ ∗ c γ c − kc R, (19)in which one recognizes the Schwarzschild-like potential V β ∗ = N ∗ β ∗ c /R ,two linear potential terms, viz., a local V γ ∗ = N ∗ γ ∗ c R/ V γ = γ c R/ V k = − kc R is induced by inhomogeneities in thecosmic background. Note that the last three potentials are new inputs intothe MO model [48] designed to interpret the rotation curve data.The radial geodesic in the metric (12) is given by (cid:18) drdt (cid:19) = B ( r ) − a B ( r ) r − bB ( r ) , (20)where a and b are constants of motion. The right hand side of the aboveequation is the ”effective potential” V , and V (cid:48)(cid:48) MO ≡ f ( R ) involves the deriva-tives of B ( R ) that, in turn, contains the so called quadratic potential V k (= − kc R ) introduced in Ref.[48]. [The subscript ”MO” is used here todistinguish it from the potential V EiBI to be defined in Eq.(22)]. This po-tential V k is responsible for providing a finite radius R WR .The main reason is the negative sign in V k needed for the data fit by MO.Because of this, it is quite evident from the plot of V (cid:48)(cid:48) MO [Fig.(2)] that thesample ESO 1200211 has a maximally allowed finite halo radius ∼ . V k has apositive sign (in which case no fit to data) or is altogether removed from11 ( R ), hence from V (cid:48)(cid:48) MO , it can be graphically verified that there will be no finite stable radius R WR for the halo. This shows the crucial role of V k .Thus, it is the requirement of fitting to data that indirectly limits the halosize to R WR . As to the physical reason, we see that the repulsive potential V k balances the remaining attractive potentials at r = R term [see Eq.(15)]but stability further demands that R WR < R term , as is evident from Figs.1& 2. Therefore, we can say that, at r = R WR , attractive potentials prevailover the repulsive potential V k constraining the gas on the circular orbit, asit should.The right hand side of Eq.(20) and its first derivative with respect to r ,both must vanish at the circular radius r = R giving a = R B (cid:48) ( R )2 B ( R ) , b = 2 B ( R ) − RB (cid:48) ( R )2 B ( R ) . The condition for stability is that the second derivative of the “effectivepotential” V with respect to r must be negative at the circular radius r = R . The resultant expression with values of a, b plugged in leads to thegeneric stability criterion for the MO model: V (cid:48)(cid:48) MO = 2 B (cid:48) ( R ) − B ( R ) B (cid:48)(cid:48) ( R ) − B ( R ) B (cid:48) ( R ) /R < . (21)This inequality graphically predicts a finite, stable, maximum halo radiusthat we call R WR caused solely by the quadratic potential V k ( R ) = − kc R .Interestingly, the predicted R WR lends itself to observational testing in thenear future as its value does not often much exceed the R last for manysamples.We shall apply the above algorithm to many samples but for illustration,we display V (cid:48)(cid:48) MO vs R for the same sample ESO 1200211 in Fig.2. The valueof N ∗ can be found from N ∗ = M lum /β ∗ , where M lum = [( M/L ) stars × L B + M HI ] × M (cid:12) . All necessary components can be read off from the entriesin the Table IV in [107]. The value of N ∗ (= 5 . × ) together withother constants in (15), when plugged into the inequality (21), immediatelygraphically yields R = R WR = 39 .
03 kpc, which is less than R term calculatedabove. III. UPPER LIMIT DENSITY FOR STABILITY
To analyze the stability of circular orbits, one needs to analyze the secondorder derivative of the concerned potential, which we wish to do here. Tofind the potential V , note that the four velocity U α = dx σ dτ of a test particleof rest mass m moving in the halo (restricting ourselves to θ = π/
2) followsthe equation g νσ U ν U σ = − m that can be cast into a Newtonian form inthe dimensionless radial variable r (= πr/R DM ) as (cid:18) drdτ (cid:19) = E + V EiBI ( r ) (22)12hich gives, for the metric Eqs.(7)-(10) of Sec.II(a), the EiBI potential V EiBI ( r ) = (cid:20) E (cid:26) AB − (cid:27) − L ACr − A (cid:21) (23) E = U m , L = U m , (24)where the constants E and L , respectively, are the conserved relativisticenergy and angular momentum per unit mass of the test particle. Circularorbits at any arbitrary radius are defined by r = R = constant, so that drdτ | r = R = 0 and, additionally, dVdr | r = R = 0. From these two conditions followthe conserved quantities as under: L = XZ (25)and using it in V EiBI ( R ) = − E , we get E = YZ , (26)where X ≡ − κ R v ∞ ( R − ρ sin R ) (27) Y ≡ (cid:16) κR + 2 r r (cid:17) (cid:32) r + κR r (cid:33) v ∞ (cid:0) ρ R cos R + ρ sin R − R (cid:1) (28) Z ≡ (cid:110) κR (cid:0) − v ∞ (cid:1) + 2 r r (cid:111) ρ sin R + (cid:16) κR + 2 r r (cid:17) ρ R cos R − Rr r − κR (cid:0) − v ∞ (cid:1) . (29)Putting the expressions for L and E in Eq.(23), we find the completeexpression for V EiBI . The orbits will be stable if V (cid:48)(cid:48) EiBI ≡ d Vdr | r = R < V (cid:48)(cid:48) EiBI >
0. The expression for V (cid:48)(cid:48) EiBI is13 (cid:48)(cid:48)
EiBI (cid:0) R ; κ, ρ , r , r opt , v ∞ (cid:1) = v ∞ (cid:0) R + ρ R cos R − ρ sin R (cid:1) R (cid:16) κR + 2 r r (cid:17) Z × (cid:104) R r r + 8 κR (cid:0) − v ∞ (cid:1) + 6 ρ r r (cid:16) R (cid:17) + κρ R (cid:16) − v ∞ + 3 R (cid:17) − R (cid:110) r r + κR (cid:0) − v ∞ (cid:1)(cid:111) ρ cos R + (cid:110) (cid:16) R − (cid:17) r r − κR (cid:16) − v ∞ − R (cid:17)(cid:111) ρ cos (cid:0) R (cid:1) − (cid:110) r r + 4 R r r + 6 κR + 2 κR − κR v ∞ (cid:111) ρ R sin R + (cid:110) r r + κR − κR v ∞ (cid:111) ρ R sin (cid:0) R (cid:1)(cid:105) . (30)From the above expression, it is absurd to straightforwardly draw anyconclusion about stability or otherwise of the circular orbits. Clearly, muchwill depend on the parameter ranges chosen on the basis of physical con-siderations. While other parameters can be reasonably assigned, the as yetunknown parameters are the dark matter radius κ (= 2 R /π ) and thedimensionless central density ρ (= 8 ρ R /π ), again depending only on κ .In the first order approximation, the density distribution in the dark matterhas been assumed in [95,96] to be low such that 8 πGκρ (0) /c <<
1, but thecentral density ρ could still be large since | sin( x ) /x | ≤ ρ imposed by the stability criterion?The answer is in the affirmative and can be found graphically. We findthat V (cid:48)(cid:48) is indeed very sensitive to changes in ρ leading to different upperlimits ρ upper0 for different galactic samples such that stable circular orbits arepossible only when ρ ≤ ρ upper0 . The reason is that R DM changes from sam-ple to sample, as it should, and thereby leads to different (though not toodifferent) values for κ and ρ upper0 . Let us again consider the previous sampleESO 1200211, a low surface brightness galaxy with a halo/dark matter ra-dius R WR ≡ R DM = 39 .
03 kpc that corresponds to κ = 308 .
74 kpc . With κ thus fixed, we fix other parameters respecting the Newtonian approxima-tion , e.g., r = 0 . v ∞ = 0 . r opt = 8 kpc, with the dimensionless The range of r and r opt is chosen so as to ensure the Newtonian approximation r/r opt (cid:29) r , while the approximate value of v ∞ is an observed fact. The formula for r for spiral galaxies is r = 1 . × ( L B / . L (cid:12) ) / , which evaluates to r = 0 .
61 for thesample ESO 1200211, where r opt (cid:39) R , L B = 0 . × L (cid:12) , R = 2 kpc. Data takenfrom [107]. R (= πR/R DM ) chosen in the range R ∈ [0 . π, π ] corresponding tocoordinate radii R ∈ [19 .
52 kpc, 39 .
03 kpc], the dimensionless density pa-rameter chosen in the range ρ ∈ [0 . π, . π ] and plot V (cid:48)(cid:48) EiBI vs R using theexpression (30).Graphical analysis shows that, while V (cid:48)(cid:48) EiBI is not much sensitive to thevariation of the other parameters within the Newtonian approximation, it is greatly sensitive to the variation of the remaining parameter ρ . Figs. 3 and4 respectively show that, for values of ρ > .
94, there is instability in theentire or partial range of the halo radii R , while Fig. 5 tells us that there isan upper limit occurring at ρ upper0 = 0 .
94 = λ upper π , where λ upper = 0 . ρ ≤ ρ upper0 , all circular orbits in the entire chosen radial rangefor R are stable. It can be verified that this value of λ upper surprisinglyremains the same for values for κ across the entire range of 111 samples(some tabulated here), so ρ upper0 is quite a reliable limit.Rewriting in terms of ρ , we have ρ upper0 = ρ upper0 π R = λ upper κ . (31)This by itself is an interesting prediction of EiBI theory. However, if wewant to quantify ρ upper0 for a given galaxy, we need to use the value of R DM but since concrete observed values are yet unavailable, we choose to use theinput R DM = R WR . It turns out that this choice, though not mandatory,works well giving definitive values for ρ upper0 for all samples. Plugging in thevalues of λ upper and κ , we find that the constraint ρ ≤ ρ upper0 immediatelytranslates into a generic constraint such that for ρ ≤ ρ upper0 , (32)all circular orbits in the chosen range for R are stable. Thus, using thevalue of κ as above in Eq.(31), the sample ESO 1200211 quantitativelyyields ρ upper0 = 5 . × M (cid:12) kpc − . In general, as long as ρ of any galaxyobeys the stability induced constraint (32), the circular material orbits inthe halo region will be stable up to a maximum radius R = R WR . IV. CENTRAL AND MEAN DARK MATTER DENSITY
So far, the algorithm has been as follows: Take any galactic sample, find R WR for that sample using the method of Sec.II(b). Then, from the identity R DM = R WR , find R DM (hence κ ) and using Eq.(31), find ρ upper0 . However,we still do not know the values of ρ for all the samples observed to date andcannot ascertain whether or not they satisfy the stability induced constraint ρ ≤ ρ upper0 . On the other hand, some notable dark matter simulations andprofiles for several samples show values for ρ that do respect this constraint(Table II). This success then prompts us to ask if there is any lower limiton ρ such that ρ lower0 ≤ ρ ≤ ρ upper0 holds.15ortunately, there is a way to find the values of ρ lower0 , once we are ableto estimate the total mass of dark matter M DM using the observed mass-to-light ratios. Fitted data are available for the luminous mass-to-light ratios( M lum /L B ) in solar units Υ (cid:12) ( ≡ M (cid:12) L (cid:12) ): M lum L B = γ Υ (cid:12) . The luminous mass ( M lum ) of a galaxy is contributed mostly by stars andgases excluding dark matter. The stellar mass-to-light ratios γ for 111 sam-ples in [107] are seen to lie between 0 . (cid:12) suggested by thepopulation synthesis models [109]. On the other hand, there is no detectabledark matter associated with the galactic disk, most of the dark matter isdistributed in the halo [110-114]. So we can write M tot = M lum + M DM , (33)then M tot L B = β Υ (cid:12) where β must be larger than γ , if there is dark matter (observed mass-to-light ratios are still uncertain). We can thus write, following Edery &Paranjape [89]: M lum M tot = 1 α , (34)which gives β = αγ and α should be so chosen as to make β > γ . Ingeneral, one takes α > M tot > M lum thereby accommodatingthe presence of dark matter.Assuming that the halos must be substantially larger than the last mea-sured point R last , the dark to luminous mass within R last then gives anupper limit through f b < M lum M tot and therefore f b < (cid:16) M DM M lum (cid:17) − . For somegalaxies, f b < .
08, as reported in de Blok and McGaugh [115]. Thus, using(34), we have f b < α and f b < .
08. Certainly, these inequalities do notconstrain α in any way. One of the infinity of options to ensure that bothhold simultaneously is to assume that α = 0 . ⇒ α = 12 .
5. While α canbe varied at will unless it is definitively fixed by independent concrete ob-served data, we shall for the moment choose the value α = 12 . α later. The current choice would imply β ∈ [2 . , γ ∈ [0 . , β ∼
100 is enough to account for the largedark matter content of LSB galaxies (i.e., large mass-to-light ratios M tot L B )such as DDO154. Currently favored Burkert density profile can provide anexcellent mass model for the dark halos around disk systems up to 100 times16ore massive than small dwarf galaxies for which the profile was initiallyintended [116,117].The ratio M lum /M tot then gives the connection between M DM of EiBItheory and the luminous mass M lum of galaxies via Eq.(33): M DM = ( α − M lum . (35)Using R WR ≡ R DM , Eq.(6) can be rewritten as ρ lower0 = π ( α − M lum R , (cid:104) ρ (cid:105) lower = 3 ρ lower0 π . (36)The superscript ”lower” indicates that it is the lower limit of the dark mattercentral density ρ because R WR is the maximum allowed halo radius (seeFig.2), where V (cid:48)(cid:48) < R ≤ R WR . Evidently, ρ lower0 isproportional to the as yet unknown parameter α . We are free to raise thevalue of α arbitrarily, but then the consequent larger values of β wouldlead to too large an amount of dark matter comparable to that existing ingalactic clusters. To illustrate the order of magnitudes involved for ρ lower0 , hence for (cid:104) ρ (cid:105) lower , we again consider the low mass LSB sample ESO 1200211 forwhich M lum = 5 . × M (cid:12) , γ = 0 . R WR = 39 .
03 kpc, κ = 308 .
74 kpc , R term = 52 .
04 kpc. Using Eqs.(36),and allowing for a fairly large amount of dark matter over luminous mat-ter corresponding to α = 12 .
5, we find ρ lower0 = 8 . × M (cid:12) kpc − and (cid:104) ρ (cid:105) lower = 2 . × M (cid:12) kpc − . To compare these values of density, we con-sider several dark matter density profiles: (i) the Bose-Einstein [108] con-densate ( ρ BEC ), (ii) Pseudo-Isothermal [120] profile ( ρ PI ), (iii) the NFW[121,122] profile ( ρ NFW i ), (iv) Burkert [116] profile ( ρ BP ), (see Sec.V belowfor the exact forms). They yield values as follows: (i) ρ BEC = 1 . × M (cid:12) kpc − , (ii) ρ PI = 4 . × M (cid:12) kpc − and (iii) ρ NFW = 2 . × M (cid:12) kpc − for ESO 1200211 (see the entries in Table I of [108]). In Sec.III,we already found for the present sample the value ρ upper0 = 5 . × M (cid:12) kpc − . We hence see that the central densities from different profiles fallwithin the stability induced limits, that is, ρ lower0 < ρ BEC, PI or NFW < ρ upper0 holds. If we take α < .
5, which should also be quite acceptable for many At much larger scales of galactic clusters, the value of β could be ≥
120 [118,119] sothat M tot L B ≥
120 in solar units. We are not contemplating galactic clusters here. Once again, a question of compatibility might be phrased as follows: The EiBI densityprofile has ρ ( R DM ) = 0 and remains zero beyond r > R DM , while, in contrast, the NFWand Burkert profiles have ρ ( r → ∞ ) = 0. Since the asymptotic behavior of latter densitydistributions are different from that of EiBI, determining whether the data obtained byfitting to NFW or Burkert profiles fall within the stability induced limits from EiBI theorycalls into question the issue of compatibility of the EiBI with those profiles.We wish to clarify that it is the central density ρ , a parameter distinctly appearing in ρ lower0 will only be further lowered and of course theinterval will be well supported.If we increase α to an (unlikely) mammoth value, say α = 300 so that M tot = 300 M lum implying β = 60 so that M tot L B = 60Υ (cid:12) in the consideredsample, then ρ lower0 ∼ . × M (cid:12) kpc − , and we notice that the proposedlimits are still not violated! If we exclude NFW profile, then the values α can be increased even further. This testifies to the validity of Eq.(36) aswell as the limits. V. Milky Way
As for our Milky Way galaxy, the latest reported estimates are the follow-ing: Using the gas terminal velocity curve, Sgr A ∗ proper motion, an oblatebulge + Miyamoto-Nagai disc and NFW halo, Kafle et al. [123] estimatedthe luminous (disc + bulge) mass to be M lum ∼ . × M (cid:12) , so that N ∗ = 1 . × and the virial mass inclusive of dark matter M vir = M tot ∼ . × M (cid:12) so that α = M tot /M lum = 7 .
68. (We have not consideredthe data/fit uncertainties). For alternative but not too different values of M tot , see [124-131]. Using the scale length R = 4 . N ∗ in the inequality (21), we find that R WR = R DM = 111 .
90 kpc.With this value of R WR , Eq. (31) then yields a value ρ upper0, MW = 6 . × all density distributions, that is under present investigation. No matter what the profileis, the target is always the same: to find information about ρ for any given galacticsample. We know that NFW profile is cuspy ( ρ NFW ∝ r − ), while others such as that ofBurkert are cored ( ρ BP ∝ r ). Despite this radical difference in behavior at the origin ,both profiles are quite well accepted though per se they are different. One could rephrasethis difference as incompatibility. The main thing however is that the fitted values ofcentral density from the two profiles should approximately be the same, at least by orderof magnitude, which actually is the case [108,132]. (A brief account of comparisons as towhich profile fits the data better is given at the very last paragraph of our paper). Inthe same vein, despite differences in the asymptotic behavior of EiBI profile and otherdensity profiles, the information about the common parameter ρ should approximatelybe the same. It would be fair to say that the derived bound for the central density ρ is indeed in agreement with the astrophysical data but that it is yet to be determinedif the EiBI density profile is compatible with future N-body simulation featuring higheraccuracy allowing for testing the outskirts of dark matter halos.Also, it is very unlikely that the attractive dark matter is spread all the way to infinity,where repulsive dark energy takes over. The finite extent of dark matter is supportedby the observed rapid decline of velocity dispersion after a certain radius [143]. It couldindeed be interesting to take this fact as an empirical input for R DM only if we knewthe exact radius at which the decline ended. Pending this knowledge, we used theoreticalinputs R WR for R DM , which yielded the interval for ρ confirmed by data fits so far. In [124], it is reported that M tot = (1 . ± × M (cid:12) from tidal effects on globularclusters and M tot = (1 . ± × M (cid:12) from globular cluster radial velocities. Theremarkable similarity between two completely independent determinations of mass maybe taken as a strong empirical signature for the existence of dark matter around theMilky Way. In [125], the reported estimate is M tot = (1 . − . × M (cid:12) , again not toodifferent. (cid:12) kpc − characteristic of the Milky way, which does not exceed the maxi-mum density ( ∼ M (cid:12) kpc − ) proposed in this paper.The density profiles considered here are of the forms: ρ BP ( r ) = ρ BP r ( r + r )( r + r ) = ρ BP (cid:18) − rr + r r + ... (cid:19) [116](37) ρ NFW ( r ) = ρ NFW ( r/r S )[1 + ( r/r S ) ] = ρ NFW (cid:18) r S r − rr S + r r S + ... (cid:19) [121,122] (38) ρ PI ( r ) = ρ PI r/r c ) = ρ PI (cid:18) − r r c + r r c + ... (cid:19) [120] (39) ρ BEC ( r ) = ρ BEC (cid:20) sin ( ξr ) ξr (cid:21) = ρ BEC (cid:18) − ξ r ξ r
120 + ... (cid:19) [108] (40)where ρ is the galactocentric density, ρ NFW is related to the density of theuniverse at the moment the halo collapsed and r , r S , r c are core, scale,characteristic radii respectively, while ξ = (cid:112) Gm / (cid:125) a in which m is themass of the dark matter particle and a is the scattering length [108]. Thefitted latest data on Milky Way dark matter central density are ρ BP = 4 . × M (cid:12) kpc − (Burkert profile) and ρ NFW = 1 . × M (cid:12) kpc − (NFWprofile) [132]. In both cases, we see that these fitted values are four ordersof magnitude less than ρ upper0,MW , as desired. From Eq.(36), for α ∼ .
68, weget ρ lower0,MW ∼ . × M (cid:12) kpc − and (cid:104) ρ (cid:105) lower0 , MW ∼ . × M (cid:12) kpc − .Once again, we see that the densities satisfy ρ lower0 < ρ BP, NFW < ρ upper0 . Alarger value of α does not disallow the inequalities at either end.Apart from the galactocentric density ρ , the local (at R (cid:12) = 8 . ρ (cid:12) provides a strong basis for the experimental endeav-ors for indirect detection of the dark matter. Though there is broad consen-sus, different groups have come up with somewhat different conclusions re-garding the local density of dark matter. For example, Kuijken and Gilmore[111-114] find a volume density near Earth ρ ⊕ (cid:39) .
01 GeV/cm = 2 . × M (cid:12) kpc − . Other reported values are the following. Bahcall et al. [133] finda best-fit value of ρ (cid:12) = 0 .
34 GeV/cm = 8 . × M (cid:12) kpc − , Caldwell andOstriker [134] find ρ (cid:12) = 0 .
23 GeV/cm = 6 . × M (cid:12) kpc − , while Turner[135] calculates ρ (cid:12) = 0 . − . = 7 . × − . × M (cid:12) kpc − .For a more comprehensive discussion on the distribution of dark matter, see[136]. The local dark matter energy density, consistent with standard esti-mates, is ρ (cid:12) = (0 . ± .
1) GeV/cm = 7 . × M (cid:12) kpc − [137]. Bergstrom,Ullio and Buckley [69] find local dark matter densities acceptable in a some-what broad range 0 . − . . The fitting with Burkert profile yields ρ BP (cid:12) = 0 .
487 GeV/cm = 1 . × M (cid:12) kpc − and fitting with NFW profileyields ρ NFW (cid:12) = 0 .
471 GeV/cm = 1 . × M (cid:12) kpc − [132]. About sys-19ematic uncertainties in the determination of local density of dark matter,see [139]. Overall, one could fairly say that ρ (cid:12) ∝ − M (cid:12) kpc − .We use the above local values as a constraint to estimate the centraldensity ρ BEC given in Eq.(40) that approaches a constant value ρ BEC as r → ρ (0) ( r ) in Eq.(3) as the two are essentially ofthe same form). This behavior is consistent with the currently favored corebehavior at the galactic center, as opposed to the NFW cusp. To estimatethe values of ρ BEC for the Milky Way, we constrain the BEC profile such thatit coincides with the local value ρ BP (cid:12) = 1 . × M (cid:12) kpc − at R (cid:12) = 8 . ρ BEC = 1 . × M (cid:12) kpc − . This is quite an acceptable central density value for the MilkyWay. (We could as well use the same boundary condition using ρ NFW (cid:12) , butit does not lead to a much different value for ρ BEC ). Having determinedthe value of ρ BEC , we compare it with the corresponding values from NFWand Burkert profiles using data from [132] and observe the following: (i) ρ BEC is quite comparable with ρ BP and ρ NFW , (ii) the NFW cusp and thePI, Burkert core behavior are evident from Fig. 6. (iii) Identifying theBEC constant ξ ≡ πR DM , we see that the ρ BEC profile shows a much slower monotonic decline from its central value ρ BEC , coasting along almost flatall the way up to a finite R DM (= 111 .
90 kpc, in the present case), whereit vanishes, (iv) adopted values from the Burkert profile has allowed us topredict ρ BEC , which is seen also to be included in the limiting interval, ρ lower0 < ρ BEC,BP,NFW < ρ upper0 for the Milky Way.Returning to the profiles (39) and (40), it is remarkable that the PI andBEC profiles have the same behavior up to second order in r provided weidentify r c = √ ξ but they begin to differ in the higher order coefficientsthereafter. Also, it is known that the large majority of the high-resolutionrotation curves prefer the PI core-dominated halo model, which provide abetter description of the data than the cuspy ( ρ NFW ∝ r − ) NFW profile[140]. In this sense, the EiBI model could be a competing candidate to PImodel. It would be our future task to investigate where these two modelsagree and where they disagree. VI. CONCLUSIONS
The present paper, based on a pivotal input from Weyl gravity, viz., R DM = R WR (motivated by [89]), offers a new alternative analytical win-dow, different from the standard data-fit approaches, to look at the physicalgalactic parameters. While the latter approaches are technically more elab-orate, and the current EiBI analytic approach is not, the value of the paperlies in the fact that it can still make quantitative predictions about the limitson central dark matter density ρ . Many samples for which the values of ρ are available are shown to satisfy the inequality ρ ≤ ρ upper0 ∝ R − ∼ M (cid:12) kpc − . Only some samples are tabulated here. Table I shows the20alo/dark matter radius, the velocity terminating radius R term and the cor-responding coupling parameter κ .Going a step further, we also calculated ρ lower0 ∝ ( α − M lum R − thatdepends on a certain parameter α equal to the the ratio of luminous tototal (dark matter included) mass of a galaxy. Definitive estimates of suchratios are yet unavailable. Nevertheless, it is shown that (Table II), forreasonably wider values of α ( ≥ .
5) accounting for huge quantities of darkmatter in individual samples, profile dependent values of ρ still fall insidethe EiBI predicted interval ρ lower0 ≤ ρ ≤ ρ upper0 . These limits cover a largeclass of galaxies and indicate an interesting facet of the EiBI theory. Weespecially point out that the maximal value ρ upper0 ∝ R − ∼ M (cid:12) kpc − is purely a stability induced constraint on all galaxies with dark matter,while ( α − M lum R − ∝ ρ lower0 ≤ ρ is not, due to uncertainties in α . Thus,we would particularly advocate a practical verification of ρ upper0 ( ∝ /κ )rather than ρ lower0 . If verified, it would also mean that we have a clearcuttheoretical algorithm, applicable to all galactic samples, that provides adefinitive, falsifiable information on the radius R DM of dark matter/halo − something that seems rather scarce in the astrophysical literature.A special merit of the foregoing analyses is that the only informationneeded to calculate the above limits are those of the fitted luminous M lum values and the measured total mass M tot . Note that a small change in M lum would lead to a large change in R WR . For instance, there is an argument[117,141] for an upper mass limit indicated by the sudden decline of thevisible baryonic mass function of disk galaxies at M maxdisc = 2 × M (cid:12) .Tentatively assuming that the luminous part of the Milky Way mass M lum is 2 × M (cid:12) instead of 1 . × M (cid:12) , then the resultant R WR would jumpto 177 .
94 kpc from 111 .
90 kpc. Similarly, R term would jump to 238 kpc from150 .
17 kpc. Thus, for reliable values of R WR , the luminous mass data M lum should be as accurate as possible.We have verified that quantitative upper limit ρ upper0 ∼ M (cid:12) kpc − is respected by all the samples collected in [107], some of which are givenin Table II. The reason for such consistency is not accidental − it stemmedfrom the fact that the Weyl radius R WR has a solid foundation: The rotationcurve is a prediction of Weyl gravity MO solution containing constants ( γ ,γ ∗ , k ) that are universally applicable to all the galaxies , LSB or HSB,and that the R WR is a straightforward result from V (cid:48)(cid:48) <
0. In fact, thereported data on R last for individual samples have so far been found to obey The claim is grounded to the fact that a single set of universal constants ( γ , γ ∗ , k ) andthe ( M/L ) ratio of individual samples, all a priori known, are enough to nicely predictall the rotation data − no adjustable free parameters are needed. In contradistinction,NFW, Burkert, PI or other profiles only give the generic shapes of halos, and leave thevalues for ρ and r as free parameters to be finely tuned to data galaxy by galaxy. Thisprocedure then quickly generates large numbers of such values as more and more galacticrotation curves are considered (For details of such ”fine tuning”, see [35], pp. 32,33). last < R WR . So we conjectured that this radius R WR just might be the dark matter radius R DM specific to individual galaxies.As we saw, the constraint ρ ≤ ρ upper0 ∼ M (cid:12) kpc − is a necessarycondition for stability of circular orbits. Whether it is also a sufficient condi-tion, that is, whether there are no stable orbits in the halo if this constraintis violated, is a matter of independent practical verification. If sufficiencyturns out to be true, then we might expect to observe galaxies with no in-formation on dark matter due to lack of stable circular orbits. It may benoted that our ρ upper0 ∼ . × M (cid:12) kpc − for Milky Way is remarkablyconsistent with the local upper limit on the dark matter density in the so-lar system, ρ upper (cid:12) ∼ . × M (cid:12) kpc − , found by completely differentmethods and ideas [142].There are limitations with almost all well known density profiles in thesense that they fit the data so well in one sector, but fail in the other. Forinstance, it has been argued [143] that the NFW profile does not alwaysfollow from the gas rotation curves of large samples. For a constant velocityanisotropy, the PI profile is ruled out, while a truncated flat (TF) model[144] and NFW model are consistent with the data. Incidentally, it mightbe noted that the TF model expands up to r like both in the PI and BECprofiles, and further, like the MO model, TF is described solely by twoparameters, mass and the scale length. Nesti and Salucci [132] argue thatNFW and/or PI halos are not supported by present day observations inexternal galaxies due to recent improvement of simulation techniques. URCprofile for velocity distribution seems to fit the data incredibly well up to ∼
30 kpc [97]. The present model based on EiBI Eq.(3), which is akin tothe quantum BEC model, is probably no better or worse than the others.Nevertheless, the foregoing study hopefully provides some new definitiveinformation in an analytic way using a metric solution (7) of EiBI theory.22 able I. Lower bound on average density [eq. (36) ].Galaxy R WR R term κ (cid:104) ρ (cid:105) lower γ β = αγα = 12 . ( M (cid:12) kpc − )ESO1870510 39 .
66 52 .
96 318 .
83 2 . × .
62 20 . .
77 60 .
15 406 .
11 1 . × .
07 13 . .
45 52 .
65 315 .
37 2 . × .
32 04 . .
25 56 .
62 361 .
70 2 . × .
07 38 . .
02 52 .
03 308 .
53 2 . × .
97 12 . .
75 55 .
91 353 .
21 2 . × .
20 02 . .
18 142 .
57 2284 .
84 1 . × .
40 05 . .
80 98 .
08 1073 .
95 3 . × .
24 28 . .
08 52 .
11 309 .
47 2 . × .
70 08 . .
10 67 .
53 508 .
76 1 . × . . .
50 55 .
57 349 .
11 2 . × . . able II. Central densities of dark matter and upper bounds [eq.(31)]. All densities are in units of M (cid:12) kpc − , α = 12 . ρ lower0 ρ BEC0 ρ PI0 ρ NFW0 ρ upper0 ESO 8 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × U11557 6 . × . × . × . × . × U11748 4 . × . × . × . × . × U11819 1 . × . × . × . × . × U11583 8 . × . × . × . × . × F568-3 4 . × . × . × . × . × F583-1 7 . × . × . × . × . × igure captions Figure 1: Plot of v vs R (cm) using Eq.(15) for ESO 1200211. The lu-minous mass is M lum = 5 . × M (cid:12) so that N ∗ = 5 . × . The otherconstants are in (16) and R = 2 kpc [44]. One finds the velocity terminatingradius R term = 52 .
04 kpc. 25igure 2: Plot of V (cid:48)(cid:48) vs R (cm) using Eq.(21). The crossing shows the haloradius R WR = 39 .
033 kpc ( ≡ . × cm), which is the maximallyallowed radius supporting stable circular orbits in the halo of ESO 1200211,for which R = 2 kpc. The plot is made for the radii R > R . Figure 3: Plot of V (cid:48)(cid:48) vs R ∈ [0 . π, π ] using Eq.(30) for ESO 1200211. Thechosen parameters are: r = 0 . v ∞ = 0 . r opt = 8 kpc ( ≡ . × cm). Here ρ = 8 . × M (cid:12) kpc − , which corresponds to ρ =0 . π . The orbits are unstable in the chosen entire radial range R ∈ [0 . π, π ]because V (cid:48)(cid:48) > V (cid:48)(cid:48) vs R ∈ [0 . π, π ] using Eq.(30) for ESO 1200211. Thechosen parameters are: r = 0 . v ∞ = 0 . r opt = 8 kpc. Here centraldensity is further lowered to ρ = 5 . × M (cid:12) kpc − , which correspondsto ρ = 0 . π . The orbit are unstable in some intermediate radii as V (cid:48)(cid:48) ispartly positive and partly negative.Figure 5: Plot of V (cid:48)(cid:48) vs R ∈ [0 . π, π ] using Eq.(30) for ESO 1200211. Thechosen parameters are: r = 0 . v ∞ = 0 . r opt = 8 kpc. Here centraldensity is further lowered to ρ = 5 . × M (cid:12) kpc − , which correspondsto ρ upper0 = 0 .
94 = β upper π , where β upper = 0 . stable inthe entire chosen range for R . The corresponding ρ is the upper limit oncentral density ρ upper0 specific to the sample.27igure 6: ρ ( r ) vs r for three profiles for the Milky Way. Bottom brown isBEC profile ( ρ BEC = 1 . × M (cid:12) kpc − , ξ = πR DM = 0 .
028 kpc − ), Bluecurve is Burkert profile ( ρ BP = 4 . × M (cid:12) kpc − , r = 9 .
26 kpc) and thegreen one is NFW profile ( ρ NFW = 1 . × M (cid:12) kpc − , r S = 16 . cknowledgments One of us (Ramil Izmailov) was supported by the Ministry of Educationand Science of Russian Federation. This work was supported in part byan internal grant of M. Akmullah Bashkir State Pedagogical University inthe field of natural sciences. The authors are thankful to Guzel Kutdusova,Regina Lukmanova and Almir Yanbekov for technical assistance.