Connecting the Galactic and Cosmological Scales: Dark Energy and the Cuspy-Core Problem
aa r X i v : . [ a s t r o - ph ] D ec Connecting the Galactic and Cosmological Scales: Dark Energyand the Cuspy-Core Problem
A. D. Speliotopoulos ∗ Department of Mathematics, Golden Gate University, San Francisco, CA 94105, andDepartment of Physics, Ohlone College, Fremont, CA 94539-0390 (Dated: November 30, 2007)
Abstract
We propose a solution to the ‘cuspy-core’ problem by extending the geodesic equations of motionusing the Dark Energy length scale λ DE = c/ (Λ DE G ) / . This extension does not affect the motionof photons; gravitational lensing is unchanged. A cosmological check of the theory is made, and σ is calculated to be 0 . ± . , compared to 0 . +0 . − . for WMAP. We estimate the fractional densityof matter that cannot be determined through gravity at 0 . ± . , compared to 0 . +0 . − . , thefractional density of nonbaryonic matter. The fractional density of matter that can be determinedthrough gravity is estimated at 0 . +0 . − . , compared to 0 . +0 . − . for Ω B . ∗ Electronic address: [email protected] . INTRODUCTION The recent discovery of Dark Energy [1, 2] has not only broadened our knowledge ofthe universe, it has brought into sharp relief the degree of our understanding of it. Only asmall fraction of the mass-energy density of the universe is made up of matter that we havecharacterized; the rest consists of Dark Matter and Dark Energy, both of which have notbeen experimentally detected, and both of whose precise properties are not known. Bothare needed to explain what is seen on an extremely wide range of length scales. On thegalactic ( ∼
100 kpc parsec), galactic cluster ( ∼
10 Mpc), and supercluster ( ∼
100 Mpc)scales, Dark Matter is used to explain phenomena ranging from the formation of galaxiesand rotation curves, to the dynamics of galaxies and the formation of galactic clusters andsuperclusters. On the cosmological scale, both Dark Matter and Dark Energy are needed toexplain the evolution of the universe.While the need for Dark Matter is ubiquitous on a wide range of length scales, ourunderstanding of how matter determines dynamics on the galactic scale is lacking. Recentmeasurements by WMAP [3] have validated the ΛCDM model to an unprecedented precision;such is not the case on the galactic scale, however. Current understanding of structureformation is based on [4], and both analytical solutions [5] and numerical simulations [6, 7,8, 9, 10] of galaxy formation have been done since then. These simulations have consistentlyfound a density profile that has a cusp-like profile [6, 8, 10], instead of the pseudoisothermalprofile commonly observed. Indeed, De Blok and coworkers [11] has explicitly shown that thedensity profile from [6] attained through simulation does not fit the density profile observedfor Low Surface Brightness galaxies; the pseudoisothermal profile is the better fit.This is the cuspy-core problem. There have been a number of attempts to solve itwithin ΛCDM [9, 10], with varying degrees of success. While the problem does not existfor MOND [12], there are other hurdles MOND must overcome. Our approach to thisproblem, and to structure formation in general, is more radical; therefore, its consequencesare correspondingly broader. It is based on the observation that with the discovery of DarkEnergy, Λ DE , there is a universal length scale, λ DE = c/ (Λ DE G ) / , associated with theuniverse. Extensions of the geodesic equations of motion (GEOM) can now be made thatwill satisfy the equivalence principal, while not introducing an observable fifth force. Whileaffecting the motion of massive test particles, photons will still travel along null geodesics,2nd gravitational lensing is not changed. For a model galaxy, the extend GEOM results ina nonlinear evolution equation for the density of the galaxy. This equation is the minimumof a functional of the density, which is interpreted as an effective free energy for the system.We conjecture that like Landau-Ginzberg theories in condensed matter physics, the systemprefers to be in a state that minimizes this free energy. Showing that the pseudoisothermalprofile is preferred over cusp-like profiles reduces to showing that it has a lower free energy.Here, phenomena on the galactic scale are inexorably connected to phenomena on thecosmological scale, and a cosmological check of our theory is made. The Hubble lengthscale λ H = c/hH naturally appears in our approach, even though a cosmological model isnot mentioned either in its construction, or in its analysis . Using the average rotationalvelocity and core sizes of 1393 galaxies obtained through four different sets of observations[11, 13, 14, 15] spanning 25 years, we calculate σ to be 0 . ± . , in excellent agreement with0 . +0 . − . from [3]. We also calculate Ω asymp , the fractional density of matter that cannot bedetermined through gravity, to be 0 . ± . , which is nearly equal to the fractional densityof nonbaryonic matter Ω m − Ω B = 0 . +0 . − . [3]. We then find the fractional density ofmatter in the universe that can be determined through gravity, Ω Dyn , to be 0 . +0 . − . , whichis nearly equal to Ω B = 0 . +0 . − . . Details of our calculations and theory is in [16]. II. EXTENDING THE GEOM AND GALACTIC STRUCTURE
Any extension of the geodesic action requires a dimensionless, scalar function of someproperty of the spacetime folded in with some physical property of matter. While beforeno such properties existed, with the discovery of Dark Energy there is now λ DE and theseextensions can be made. As we work in the nonrelativistic, linearized gravity limit, weconsider the simplest extension: L Ext = mc (cid:16) D (cid:2) Rc / Λ DE G (cid:3) (cid:17) (cid:18) g µν dx µ dt dx ν dt (cid:19) ≡ mc R [ Rc / Λ DE G ] (cid:18) g µν dx µ dt dx ν dt (cid:19) (1)with the constraint v = c for massive test particles. Here, D ( x ) is a function functiongiven below, and R is the Ricci scalar. For massive test particles, the extended GEOMis v ν ∇ ν v µ = c ( g µν − v µ v ν /c ) ∇ ν log R [4 + 8 πT / Λ DE c ], where v µ is the four-velocity ofa test particle, T µν is the energy-momentum tensor, T = T µµ , and we take Λ DE to bethe cosmological constant. As the action for gravity+matter is a linear combination of3he Hilbert action and the action for matter, any changes to the equation of motion for testparticles can be accounted for in T µν , and we still have R = 4Λ DE G/c + 8 πGT /c in Eq. (1).For massless particles, v ν ∇ ν ( R [4 + 8 πT / Λ DE c ] v µ ) = 0 instead. With the reparametization dt → R dt , the extended GEOM for massless test particles reduces to the GEOM. Ourextended GEOM does not affect the motion of photons.Because the geodesic Lagrangian is extended covariantly, Eq. (1) explicitly satisfies thestrong equivalence principal. For T µν , we may still take T µν = ( ρ + p/c ) v µ v ν − pg µν for aninviscid fluid with density ρ and pressure p [16]. While for the GEOM T geo-Dust µν = ρv µ v ν for dust, for the extended GEOM the pressure does not vanish [16]; it is a functional of ρ and R . Nevertheless, in the nonrelativistic limit p << ρc , and T Ext-Dust µν ≈ ρv µ v ν still[16]. Moreover, because v µ v µ = c for the extended GEOM, the first law of thermodynamicsstill holds for the fluid, and the standard thermodynamical analysis of the evolution of theuniverse under the extended GEOM follows much in the same way as before. All dynamical effects of extension can be interpreted as the rest energy gained or lostby the test particle due to variations in the local curvature. For these effects not to havealready been seen, D (4 + 8 πT / Λ DE c ) must change very slowly at current experimentallimits. As such, we take D ( x ) = χ ( α Λ ) R ∞ x (1 + s α Λ ) − ds , where α Λ ≥ χ ( α Λ ) isset by D (0) = 1. This D ( x ) was chosen for three reasons. First, there is only one freeparameter, α Λ , to determine. Second, it ensures that the effects of the additional termsin the extended GEOM will not already have been observed; Λ DE = (7 . +0 . − . ) × − g/cm , and ρ ≫ Λ DE / π in all current experimental environments so that D ≈
0. Alower experimental bound of 1.35 for α Λ can be found [16]. Third, D ′ ( x ) is negative, andwill contribute an effective repulsive potential to the extended GEOM that mitigates theNewtonian 1 /r potential.While definitive, a first principles calculation of the galactic rotation curves using theextended GEOM would be analytically intractable. Instead, we show that given a model,stationary galaxy with a specific rotation velocity curve v ( r ), we can derive the mass densityprofile of the galaxy. We use a spherical model for the galaxy that has three regions. RegionI = { r | r ≤ r H , and ρ ≫ Λ DE / π } , where r H is the galactic core radius. Region II= { r | r > r H , r ≤ r II , and ρ ≫ Λ DE / π } is the region outside the core containingstars undergoing rotations with constant rotational velocity; it extends out to r II , which isdetermined by the theory. A Region III = { r | r > r II , and ρ ≪ Λ DE / π } also appears in4he theory.As all the stars in the model galaxy undergo circular motion, the acceleration of a star, a ≡ ¨ x , is a function of is location, x , only. Taking the divergence of the extended GEOM, f ( x ) = ρ − κ ( ρ ) (cid:26) ∇ ρ − α Λ πρ/ Λ DE (cid:18) π Λ DE (cid:19) |∇ ρ | (cid:27) , (2)where κ ( ρ ) ≡ (cid:8) πρ/ Λ DE ) α Λ (cid:9) /χλ DE , and f ( x ) ≡ −∇ · a / πG . We do notdifferentiate between baryonic matter and Dark Matter in ρ . Near the galactic core 1 /κ ( ρ ) ∼ λ DE [Λ DE / πρ H ] (1+ α Λ ) / , where ρ H is the core density. Even though λ DE = 14010 +800 − Mpc,because ρ H ≫ Λ DE / π , α Λ can be chosen so that 1 /κ ( r ) is comparable to typical r H . Doingso sets α Λ ≈ / v ( r ), a ( r ) can be found and f ( r ) determined. We idealize the observed velocitycurves as v ideal ( r ) = v H r/r H for r ≤ r H , while v ideal ( r ) = v H for r > r H , where v H is theobserved asymptotic velocity. This v ideal ( r ) is more tractable than the pseudoisothermalvelocity curve, v p-iso ( r ), used in [11]. As it has the same limiting forms in both the r ≪ r H and r ≫ r H limits, v ideal ( r ) is also an idealization of v p-iso ( r ).For cusp-like density profiles [10], it is the density profile that is given. While it is possibleto integrate the general density profile to find the corresponding curves v cusp ( r ), both themaximum value of v cusp ( r ) and the size of the core are different depending on the profile.These core sizes would thus have to be scaled appropriately to compare one profile withanother. Doing so is possible in principle, but would be analytically intractable in practice.We instead take f ( r ) = ρ H ( r H /r ) γ if r ≤ r H , and f ( r ) = ρ H ( r H /r ) β / r > r H for thedensity profiles. Here, γ < β ≥ r H . The γ = 0 , β = 2 case corresponds to theidealized psuodoisothermal profile.Since ρ ≫ Λ DE / π in Regions I and II, Eq. (2) minimizes F [ ρ ] = Λ DE c π (cid:0) χ / λ DE (cid:1) Z d u ( α Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∇ (cid:18) Λ DE πρ (cid:19) α Λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − α Λ α Λ − (cid:18) Λ DE πρ (cid:19) α Λ − + (cid:18) Λ DE πρ (cid:19) α Λ πf ( u )Λ DE ) , (3)which we identify as a free energy functional; here, u = r/χ / λ DE . For γ = 0,Eq. (2) gives ρ ( r ) = ρ H in Region I; the free energy for this solution is I F γ =0 = − Λ DE r H (Λ DE / πρ H ) α Λ − / α Λ − γ > I F γ greater than I F γ =0 [16]. This results because ∼ |∇ ρ | ≥ |∇ ρ | only vanishes for the constant densitysolution.For Region II, the density, ρ II , is first found asymptotically in the large r limit. Withthe anzatz f ( r ) ≪ ρ ( r ) for large r , Eq. (2) reduces to a homogeneous equation [16] with thesolution ρ asymp ( u ) = Λ DE Σ( α Λ ) / πu α Λ , where Σ( α Λ ) = [2(1 + 3 α Λ ) / (1 + α Λ ) ] α Λ . Toinclude the galaxy’s structural details, we take ρ II ( r ) = ρ asymp ( r ) + ρ II ( r ) and to first orderin ρ II , ρ II ( r ) = ρ asymp ( r ) + 13 A β ρ H (cid:16) r H r (cid:17) β + (cid:16) r H r (cid:17) / ( C cos cos [ ν log r/r H ] + C sin sin [ ν log r/r H ]) . (4)where ν = [2(1 + 3 α Λ ) / (1 + α Λ ) − / / , C cos and C sin are determined by boundaryconditions, and A β = 1 for β = 2 ,
3. The first part, ρ asymp ( r ), of ρ II ( r ) corresponds toa background density. It is universal, and has the same form irrespective of the detailedstructure of the galaxy.
The second part, ρ II ( r ), gives the structural details.The free energy, II F , for Region II separates into the sum of three parts. The first partdepends only on ρ asymp ; it is positive, and is independent of β . The second part is II F asymp − β ( χ / λ DE ) = c Z D II d u f ( u ) (cid:18) Λ DE πρ asymp (cid:19) α Λ + 8 πα Λ c Λ DE Σ α Λ ) Z ∂D II u ρ II ( u ) ∇ ρ asymp · d S , (5)where D II is Region II. It is negative because the minimum ρ must be positive. Indeed, wefind that II F asymp − β ∼ − ( r H /r II ) β for β < / II F asymp − β ∼ − ( r H /r II ) / for 5 / ≤ β < − / (1 + α Λ ); and II F asymp − β ∼ ± ( r H /χ / λ DE ) − / (1+ α Λ ) for 5 − / (1 + α Λ ) < β . Clearly,free energy is lowest for β = 2. The third part depends on ( ρ II ( r )) , and is negligibly small.The total free energy in this region is thus smaller for β = 2 than for β >
2. Combined withthe calculation for I F , we conclude that the pseudoisothermal density profile has the lowestfree energy, and is the preferred state of the system. We thus take γ = 0 and β = 2 in thefollowing.In Region III, ρ ≪ Λ DE / π , and κ ( ρ ) ≈ (1 + 4 α λ ) /χλ DE ; Eq. (2) reduces to theundriven, modified Bessel equation. As such, the density vanishes exponentially fast in thisregion on the scale 1 /κ ( r ). This sets r II = [ χ/ (1 + 4 α λ )] / λ DE .The extended GEOM can be written as ¨ x = −∇ V . The dynamics of test particles isgoverned by an effective potential V ( x ) = Φ( x ) + c log ( R [4 + 8 πρ Λ DE ]), and not by thegravitational potential Φ( x ). For Φ( r ) in Region I, we obtain the Newtonian gravity result6( r ) = v H r / r H + constant. In Region II, Φ( r ) is dominated by four terms. The first isthe usual 1 /r term. The second is a log( r/r H ) term due to f ( r ). This term is long ranged,and in addition to galactic rotation curves, could explain the interaction observed betweengalaxies and galactic clusters. The third is a ρ II ( r ) r term, and contains terms ∼ /r / .The fourth term is a r α Λ / (1+ α Λ ) term due to ρ asymp , and is proportional to c .This last term grows as r / for α Λ = 3 /
2, and would dominate the motion of test particlesin the galaxy if the extended GEOM depended on Φ( x ) instead of V ( x ). We instead findthat V ( x ) ≈ Φ( x ) − [ u χc (1 + α Λ ) / α Λ (1 + 3 α Λ )] (Λ DE / π )( ρ asymp − α Λ ρ II ). The last twoterms in this expression cancel both the ρ II ( r ) r and the r α Λ / (1+ α Λ ) terms in Φ( r ); theresultant V ( r ) increases as log r/r H , agreeing with observation.The r α Λ / (1+ α Λ ) term in Φ( x ) comes from the background density ρ asymp . Thus, a goodfraction of the mass in the observable galaxy does not contribute to the motion of testparticles in the galaxy . It is rather the near-core density ρ II ( r ) that contributes to V ( x ).As inferring the mass of structures through observations of the dynamics under gravity oftheir constituents is one of the main ways of estimating mass, the motion of stars in galaxiescan only be used to estimate ρ II ; the matter in ρ asymp ( r ) is present, but cannot be “seen”in this way. Moreover, as ρ asymp ( r ) ≫ ρ II ( r ) when r ≫ r H , the majority of the mass in theuniverse cannot be seen using these methods . III. A COSMOLOGICAL CHECK
We have extrapolated our results for a single galaxy to the cosmological scale. This ispossible because recent measurements from WMAP, the Supernova Legacy Survey, and theHST key project show that the universe is essentially flat; h = 0 . +0 . − . and of the ageof the universe t = 13 . +0 . − . Gyr were determined using this assumption. The largestdistance between galaxies is thus ct ≡ K (Ω) λ H , where K (Ω) = 1 . ± . .Next, the density of matter of our model galaxy dies off exponentially fast at r II ; theextent of matter in the galaxy is fundamentally limited to 2 r II . This size does not depend onthe detailed structure of the galaxy; it is inherent to the theory. Given a Ω Λ = 0 . ± . , wecan express r II = [8 πχ/ Λ (1 + 4 α Λ )] / λ H [16] as well [3], and numerically r II = 0 . λ H for α Λ = 3 /
2. Although α Λ was set to 3 / ρ ( r )naturally cuts off at λ H /
2. 7o accomplish the extrapolation, we consider our model galaxy to be the representativegalaxy for the observed universe. This representative galaxy could, in principal, be found bysectioning the observed universe into three-dimensional, non-overlapping cells of differentsizes centered on each galaxy. By surveying these cells, a representative galaxy, with anaverage v ∗ H and r ∗ H , can be found, and used as inputs for the model galaxy. Even thoughsuch a survey has not yet been done, a large repository of galactic rotation curves and coreradii [11, 14, 15] is present in the literature. Taken as a whole, these 1393 galaxies arereasonably random, and are likely representative of the observed universe at large.While we were able to estimate of α Λ = 3 / require that r II = K (Ω) λ H /
2, which in turn gives α Λ as the solution of K (Ω) (1 +4 α Λ ) = 32 πχ ( α Λ ) / Λ ; this sets α Λ = 1 . ± . .A calculation of σ has been done [16] using Eq. (4). The resultant σ is dominated bytwo terms. The first is due to the background density ρ asymp . It depends only on α Λ , andcontributes a set amount of 0.141 to σ . The second is the larger one, and is due primarilyto the 1 /r term in Eq. (4). It depends explicitly on the rotation curves through the term( v ∗ H /c ) (8 h − Mpc /r ∗ H ).Although there have been a many studies of galactic rotation curves in the literature,both v H and r H are needed here. This requires fitting the observed velocity curve to somemodel. To our knowledge, both values are available from four places in the literature: Thede Blok et. al. data set [11]; the CF data set [14]; the Mathewson et. al. data set [15, 17]analysed in [14]; and the Rubin et. al. data set [13]. Except the last set, the observed velocitycurves is fitted to either v p-iso ( r ), or to a functionally similar velocity curve [14]. The last setgives only the galactic rotation curves, and they have been fitted to v p-iso ( r ) in [16]. Whilethe URC of [18] has a constant asymptotic velocity, it has a r . behavior for r small. Thisbehavior is different from v ideal , and was not considered here [16].While v H is easily identified for all four data sets, determining r H is more complicated;this is determination is done in [16]. The resultant values are used to obtain v ∗ H and r ∗ H foreach set, which are then used to calculate the σ and ∆ σ for it. Results of these calculationsare in Table I. Four of the five data sets give a σ that agrees with the WMAP value atthe 95% CL. The Rubin et. al. set does not, but it is known that these galaxies were notrandomly selected [13]. 8 ata Set v ∗ H ∆ v ∗ H r ∗ H ∆ r ∗ H σ ∆ σ t-test deBlok et. al. (53) 119.0 6.8 3.62 0.33 0.613 0.097 1.36CF (348) 179.1 2.9 7.43 0.35 0.84 0.18 0.43Mathewson et. al. (935) 169.5 1.9 15.19 0.42 0.625 0.089 1.34Rubin et. al. (57) 223.3 7.6 1.24 0.14 2.79 0.82 2.46Combined (1393) 172.1 1.6 11.82 0.30 0.68 0.11 0.70TABLE I: The v ∗ H (km/s), r ∗ H (kps), and resultant σ , ∆ σ , and t-test comparison with the WMAPvalue of σ . We have estimated Ω asymp by averaging ρ asymp ( r ) over a sphere of radius r II , and foundΩ asymp = 0 . ± . . In calculating this average, we assumed that there is only a singlegalaxy within the sphere, however. While this is a gross under counting of the numberof galaxies in the universe, ρ asymp is an asymptotic solution, and ρ II → r .Additional galaxies may change the form of ρ asymp , but these changes are expected to beequally short ranged; we expect that our calculation is an adequate estimate of Ω asymp .Such is not the case for Ω Dyn , however. Direct calculation of Ω
Dyn would require knowingboth the detailed structure of galaxies, and the distribution of galaxies in the universe.Instead, we note that Ω m = Ω asymp + Ω Dyn , and using Ω m = 0 . +0 . − . from WMAP, findΩ Dyn = 0 . +0 . − . . IV. CONCLUDING REMARKS
Given how sensitive σ is to v ∗ H , r ∗ H , and α Λ , that our predicted values of σ is withinexperimental error of the WMAP value is surprising. Even in the absence of a direct exper-imental search for α Λ , this agreement provides a compelling argument for the validity of ourextension of the GEOM. It also supports our free energy conjecture; our calculation of σ would be very different if β = 3, say, was used instead of β = 2. With α Λ = 1 .
51 so close tothe experimental lower bound for α Λ of 1 .
35, direct measurement of α Λ may also be possiblein the near future.Interestingly, Ω m − Ω B = 0 . +0 . − . is nearly equal to Ω asymp in value. Correspondingly,Ω B [3] is nearly equal to Ω Dyn . It would be tempting to identify Ω asymp with Ω m − Ω B , espe-cially since matter in ρ asymp ( r ) is not “visible” to inferred-mass measurements. That Ω Dyn B is consistent with the fact that most of the mass inferredthrough gravitational dynamics are indeed made up of baryons. We did not differentiatebetween normal and dark matter in our theory, however. Without a specific mechanismfunneling nonbaryonic matter into ρ asymp and baryonic matter into ρ − ρ asymp , we cannot atthis point rule out the possibility that Ω m − Ω B = Ω asymp and Ω B ≈ Ω Dyn is a numericalaccident.
Acknowledgments
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