Estimation of the Mass of Dark Matter Using the Observed Mass Profiles of Late-Type Galaxies
EEstimation of the Mass of Dark Matter Using the Observed Mass Profiles ofLate-Type Galaxies
Ahmad Borzou ∗ EUCOS-CASPER, Department of Physics, Baylor University, Waco, TX 76798, USA (Dated: February 19, 2021)We analyze observations of the mass profiles of 175 late-type galaxies in the Spitzer Photometry& Accurate Rotation Curves (SPARC) database to construct the temperature profile of their darkmatter (DM) halos by assuming that (1) DM in the halos obeys either the Fermi-Dirac or theMaxwell-Boltzmann distribution, and (2) the halos are in the virial state. We derive the dispersionvelocity of DM at the center of the halos and show that its correlation with the halo’s total mass is thesame as the one estimated in N-body simulations. The correlation is also the same as the observedrelation between the two variables for visible matter in galaxies. Taking the latter agreement as avalidation of our analysis, we derive the mass to the temperature of DM at the edge of the halos andshow that it is galaxy independent and is equal to m/T R (cid:39) in natural units. Since the analyzedgalaxies are far away in the sky, we conclude that DM is a thermal relic, and T R in the above ratiocan be expressed in terms of the temperature of the cosmic microwave background (CMB) at thetime of DM decoupling. This result is used to study possible cosmological scenarios. We show thatobservations are at odds with (1) non-thermal DM, (2) hot DM, and (3) collision-less cold DM. Ourfindings are in favor of a warm DM with a mass of ∼ I. INTRODUCTION
Various astronomical and cosmological observationspoint out that a mysterious dark matter (DM) consti-tutes about 85% of the mass of the universe. A reviewof the pieces of evidence for the existence of DM canbe found in [1]. Despite the large stockpile of data thatrefer to the presence of DM, only little is known aboutthe nature of the dark particles, and widely different DMschemes are being investigated at the present.Among a few characteristics of DM that are knownwith high certainty is its galactic mass density, whichis estimated through observations of how visible mattermoves in galaxies. Observations also suggest that DM ha-los in galaxies are stable. Therefore, we can assume withhigh certainty that DM has a statistical distribution inthe galactic halos, which leads to a pressure that con-fronts the attractive gravitational force. See [2] for a re-view. Experiments in particle physics, as well as observa-tions of collisions of galaxies, indicate with high certaintythat DM is either collision-less or its interactions are notstrong. Therefore, depending on its spin, DM distribu-tion in galaxies is either Bose-Einstein or Fermi-Dirac,both of which are equivalent to the Maxwell-Boltzmanndistribution under certain classical conditions. Since thevisible mass in the universe is made of fermions and DMcontributes to the mass of the universe as well, it is morelikely that DM similarly obeys a Fermi-Dirac distribu-tion. Finally, the virial theorem suggests with high cer-tainty that DM halos that have not been involved in mas-sive collisions, i.e. major mergers, are in the virial state.In this paper, we use the above-mentioned propertiesof DM to estimate its temperature profile in galaxies. ∗ ahmad [email protected] We show that the temperature of DM at the edge ofmore than 100 observed late-type galaxies is the same,and derive it in terms of the mass of DM. The analyzedgalaxies are up to 100 mega-parsecs away from each otherand have a diverse range of total masses from 10 to 10 in the unit of the mass of the sun. We conclude that DMis a thermal relic and its universal temperature at theedge of the halos is equal to its temperature when thedark particles decoupled from the rest of the matter inthe early universe and cooled down due to the expansionof the universe. As a validation of our analysis, we esti-mate the temperature to the mass of DM at the centerof the observed halos and show that it is the same as theobserved temperature to the mass of visible matter, andalso consistent with N-body simulations.We investigate the implications of the estimated uni-versal DM temperature within the context of a few pop-ular cosmological models. While a collision-less cold DMis not consistent with our findings, we show that a warmDM with a mass of ∼ II. STABILITY OF HALOS
To maintain the stability of the halos, the forces ofgravity and pressure should be equal at any distance from a r X i v : . [ a s t r o - ph . C O ] F e b the center − ρ ( r ) dPdr = G M ( r ) r , (1)where ρ ( r ) is the mass density of DM, P ( r ) is the pressureof DM, G is the Newton gravitational constant, and M ( r )is the total mass enclosed in the radius r . On the otherhand, the pressure of fermionic DM reads P ( r ) = km ρ ( r ) T ( r ) h ( r ) , (2)where k is the Boltzmann constant, m is the mass of DM, T ( r ) is the temperature of DM, and h ( r ) ≡ f /f is theratio of the Fermi-Dirac integrals. When the quantumnature of the fermionic DM is negligible, the distributionreduces to the Maxwell-Boltzmann and the ratio wouldbe equal to one.In this paper, we assume that ρ ( r ) is known fromobservations. Therefore, using equations (1) and (2), y ( r ) ≡ T ( r ) h ( r ) /T h is given in terms of ρ ( r ) by y ( r ) = ρ ρ ( r ) (cid:18) − Gρ σ (cid:90) r ρ ( r (cid:48) ) M ( r (cid:48) ) r (cid:48) dr (cid:48) (cid:19) , (3)where the subscript naught refers to the value of thequantities at the center, and the dispersion velocity atthe center is σ = kT h /m . It should be noted thatthe equation above is valid regardless of the degeneracylevel of fermionic DM. Even when the temperature tendsto zero, corresponding with the full degeneracy level, theratio of the Fermi-Dirac integrals tends to infinity suchthat the right-hand side remains finite and non-zero. Theequation is also valid for bosonic DM provided that it isnot in the Bose-Einstein condensate.In general, ρ ( r ) and σ in the equation above are in-dependent and should be separately derived from obser-vations. Nevertheless, by observing the mass profile, wecan estimate a lower bound on the dispersion velocity.Since the left-hand side of equation (3) is positive by def-inition, the central dispersion velocity has to satisfy thefollowing inequality [3] σ > Gρ Max (cid:18)(cid:90) r ρ ( r (cid:48) ) M ( r (cid:48) ) r (cid:48) dr (cid:48) (cid:19) , (4)where ’Max’ refers to the value of the integral at a dis-tance r where the integral reaches its maximum. III. VIRIAL STATE
In this section, we use the virial theorem to derive thedispersion velocity of DM at the center of the halos interms of ρ ( r ), such that y ( r ) in equation (3) is entirelyknown if the mass density is constructed out of observa-tions.According to the virial theorem, the total kinetic en-ergy U of DM halo is equal to minus half of its total gravitational potential energy W if the galaxy has notbeen participating in a major merger recently.The gravitational potential energy of the halo is givenby W = 12 (cid:90) R πr ρ ( r ) φ ( r ) dr, (5)where the gravitational potential is φ ( r ) = − πG (cid:32) r (cid:90) r ρ ( r (cid:48) ) r (cid:48) dr (cid:48) + (cid:90) R ρ ( r (cid:48) ) r (cid:48) dr (cid:48) (cid:33) . (6)Since the density of the kinetic energy of DM is equalto P , we use equations (2) and (3) to calculate the totalkinetic energy of the halos U = 6 πρ σ (cid:90) R r (cid:18) − Gρ σ (cid:90) r ρ ( r (cid:48) ) M ( r (cid:48) ) r (cid:48) dr (cid:48) (cid:19) dr. (7)Using the virial theorem, and equations (5), and (7),the dispersion velocity of DM at the center of the halosread σ = − πρ R (cid:18) W + U (cid:19) , (8)where U refers to the second term on the right hand sidein equation (7).Therefore, the ratio of the mass of DM m over its tem-perature at the edge of the halo T ( R ) reads mT ( R ) = kσ y ( R ) , (9)where the right-hand side is known in terms of observed ρ ( r ). Moreover, we have safely assumed that at the edgeof the large halos, h ( R ) (cid:39)
1. It should be noted that thelatter approximation remains valid up to high degrees ofpartial degeneracy. The approximation fails if DM is inthe full degeneracy level at the edge of the halos, wherethe mass density has fallen by orders of magnitude andquantum effects are less important.
IV. ANALYSIS OF OBSERVED LATE-TYPEGALAXIES
In this section, we use observations of more than ahundred late-type galaxies in the SPARC database [4],together with the theoretical framework presented in thepreceding sections, to investigate the temperature of DMin those galaxies.The SPARC database contains both H I / H α rotationcurves and near-infrared surface photometry. The latterhelps with the construction of the DM mass profile closeto the center while the former can be used to learn thehalos’ outer mass. The observations are subsequentlyused to set the free parameters of a few popular massmodels for each of the galaxies. Log(M/M )
Log ( T h / m ) [ K e V ] EinastoLuckypIsoBurkert
Log(M/M )
Log ( T R h R / m ) [ K e V ] EinastoLuckypIsoBurkert
FIG. 1. The temperature to the mass of DM versus the total mass of the halos of the SPARC galaxies. The left panel refersto the DM temperature at the center of the halos and the right panel refers to the temperature of DM at the edge of the haloswhere the mass density of DM is only 200 time the critical mass density in the universe. It can be seen that the temperatureof DM at the center depends on the total mass of the halo. However, the DM temperature at the edge of the halos is almostuniversal. The figure also shows that in light halos with the mass of ∼ M (cid:12) , DM temperature at the center is almost thesame as the universal outer temperature. The four mass models that we use in this paper are ρ ( r ) = ρ (cid:20)(cid:16) rr (cid:17) (cid:18) (cid:16) rr (cid:17) (cid:19)(cid:21) − Burkert ρ (cid:20) (cid:16) rr (cid:17) (cid:21) − pIso ρ exp (cid:16) − α (cid:15) (cid:104)(cid:16) rr (cid:17) α (cid:15) − (cid:105)(cid:17) Einasto ρ (cid:20) (cid:16) rr (cid:17) (cid:21) − Lucky13 , (10)where, for each of the 175 galaxies, the scale radius r and the characteristic mass density ρ , as well as α (cid:15) inthe Einasto model are estimated using the observationsand are provided in [5].We use the above four mass models as ρ ( r ) in equa-tions (8) and (9) to derive σ and T ( R ) /m of DM in thecorresponding halos. We show that the two quantities donot strongly depend on the mass models of ρ ( r ) as far asthey are not singular at the center.Figure 1 shows the temperature of DM divided by itsmass. The left panel shows that the temperature at thecenter depends on the total mass of the halos. A moreconventional variant of this plot is shown in the appendixin figure 6, which shows that the logarithm of the dis-persion velocity of DM at the center of halos is linearlyrelated to the logarithm of the total mass of the halos.The slope in this plot coincides with the same slope esti-mated in the N-body simulations of DM as well as withthe observed slope for the dispersion velocity of visiblematter [6]. This agreement is in favor of our assumptionthat the analyzed halos of the SPARC dataset are in thevirial state. We report that the estimated dispersion ve-locities in this figure are approximately equal to the lowerbound derived in equation (4). Distance [Mpc]
Log ( T R / m ) [ K e V ] EinastoLuckypIsoBurkert
FIG. 2. The distance of the SPARC galaxies from us in thex-axis and the mass over the edge temperature of DM inthe y-axis. The plot indicates that the halos are far enoughfrom each other that cannot be in thermal equilibrium at thepresent epoch.
The right panel of figure 1 shows that the DM temper-ature at the edge of the halos is universal, and at 95%confidence T ( R ) m = (1 . ± . × − (K · eV − ) , (11)where R is conventionally defined as the edge of thehalos, where the mass density of DM is 200 times thecritical mass density.Figure 1 also shows that in light halos of mass ∼ M (cid:12) , the DM temperature at the center is not differ- r (kpc) ( r ) ( W / + U ) ( k m s ) EinastoLuckypIsoBurkert
FIG. 3. The y-axis refers to equation (8) with a varying haloradius. The four curves are cut on the x-axis where the den-sity reaches as low as 200 times the critical density of theuniverse. As can be seen the y-axis reaches a flat plateauat small distances from the center, and does not depend onwhere R is defined. The figure also indicates that σ has aslight dependence on the mass model. However, wight theorder of magnitude precision, all of the mass models refer to ∼
10 (km · s − ). We report the same behavior in all of thegalaxies although this plot refers particularly to DDO170. ent from the outer universal temperature. This observa-tion implies that, in such halos, the gravitational energythat is converted to the kinetic energy of dark particlesis negligible.Figure 2 shows that the temperature of DM at the edgeof the halos is the same even though the galaxies are in awide range of distances from us and cannot communicateas rapidly as needed to maintain a thermal equilibriumat the present.The universality of DM temperature at the edge of thehalos suggests that it has been in thermal contact in theearly universe. The decoupled DM keeps the distribu-tion but cools down via the expansion of the universeuntil it is trapped in the halos, where the conversion ofthe gravitational potential warm it up. Although othermeans also can warm-up DM, the acquired energy hasbeen lost due to the gravitational contraction accordingto the virial theorem. Consequently, T R can be expressedin terms of the temperature of DM at the decoupling inthe early universe, T freeze .Since the exact location where DM halo ends is notknown with certainty and R is only an estimation,in the following, we investigate which one of our resultsdepend on the definition of the edge of the halo, andquantify such possible dependencies. For the sake ofease in presentation, we only show the results for galaxyDDO170.To show that the dispersion velocity of DM at the cen-ter of halos is independent of the edge, we invoke equa-tion (8) and plot σ in terms of R . The result can be seen r (kpc) G r d r M r EinastoLuckypIsoBurkert
FIG. 4. The second term in the parentheses in equation (3) isplotted versus the distance from the center. The figure showsthat this term does not vary with distance in the outer regionof halos. We report the same behavior in all of the galaxiesalthough this plot refers particularly to DDO170. r (kpc) / c EinastoLuckypIsoBurkert r (kpc) T h / m ( K e V ) FIG. 5. The mass profile, the temperature profile and theenclosed mass of the halo of DDO170 galaxy. in figure 3 and indicates that after a small distance fromthe center, the right hand side of equation (8) reaches aconstant value.To investigate how T R varies with respect to our defi-nition of the edge of the halo, we inspect the componentsof equation (3). As can be seen in figure 4, the secondterm in the parentheses reaches a plateau in the outerregions. Therefore, at large distances from the center, y ( r ) ∝ ρ ( r ) − , and T R l = 200 l − T R , where R l is a ra-dius at which the mass density of the halo is l times thecritical density of the universe. Consequently, the massof DM in terms of its temperature at the edge reads m (cid:39) l × . × T ( R l )(eV · K − ) . (12)Figure 5 shows the mass and temperature profiles ofthe halo of DDO170. It should be noted that the formeris the observed mass model reported in [5]. The latter isthe combination of the observations and the theoreticalframework presented in this paper. As can be seen fromthe figure, the temperature increases with the distancefrom the center up to a point before it starts to decrease.This temperature profile can lead to an additional at-tractive force in the inner region of the halo. The originof the extra attractive force can be understood by not-ing that the left hand side of equation (1) is the forcedue to the pressure. Since both temperature and massdensity are functions of r , the force of the pressure hastwo components. In figure 7, we show the two compo-nents of the force of pressure relative to gravity, where dP dr is due to the temperature derivative. The figure il-lustrates that the extra attractive force can be orders ofmagnitude stronger than gravity at the center leadingto higher compression of DM and lowering the so-calledphase-space lower bounds on the mass of DM [3]. V. COSMOLOGICAL SCENARIOS
In the preceding section, we expressed the mass of DMin terms of its universal temperature in the outer part ofhalos. We discussed that the temperature should havebeen set in the early universe. Hence, we should be ableto find the mass of DM in terms of the temperature ofCMB. In the following, we investigate the relation be-tween the two within the context of a few cosmologicalschemes.
A. Non-Relativistic Decoupling
If DM is cold, its decoupling temperature and the uni-versal temperature in the outer regions of halos are re-lated by the scale-factor through T R l (cid:39) T freeze a freeze . Onthe other hand, the freeze-out temperature of DM interms of the present temperature of CMB reads T freeze (cid:39) T CMB0 a − freeze . Therefore, using equation (12), the mass ofDM reads m = l × T CMB0 T freeze (eV) , (13)where we have used T CMB0 (cid:39) . (cid:114) . l
200 10 eV < m, (14)where for l ∼ ∼
10 eV.On the other hand, the decoupling of DM from CMBhas to be before the photon last scattering at ∼ . m < l
200 500 eV (15)which for l ∼
200 reads m <
500 eV. Using equation 13, in this model, DM decouples fromCMB when the temperature of the radiation is between0 . < T freeze <
10 eV. This scenario is not acceptablesince otherwise, we should have seen the signature of DMin the particle physics experiments. Therefore, collision-less cold DM is at odds with the observations and is nota viable scheme.
B. Relativistic Decoupling
If DM decouples from the visible matter when it isstill relativistic, the temperature at the edge of galaxies isapproximately equal to the present temperature of CMB.Hence, using equation (12), the mass of DM reads m (cid:39) l × (eV) . (16)Since we expect that l/ ∼ O (1), the equation aboveimplies that the mass of DM is equal to ∼ ∼ ∼ VI. CONCLUSIONS
We have analyzed observations of more than 100 late-type galaxies by assuming that (i) DM obeys either theFermi-Dirac or the Maxwell-Boltzmann distribution, and(ii) the halos are in the virial state. Hence, using thestability of the halos, the temperature of DM has beenexpressed in terms of the mass density that is estimatedfrom observations.The dispersion velocities of DM at the center of the ha-los have been estimated. We have shown that the depen-dence of the latter on the total mass of the correspondinghalos is the same as in N-body simulations, suggestingthat the halos are in the virial state as assumed.The temperature of DM at the center and the edgeof the halos have been estimated. At the center, DMtemperature increases with the total mass of the halo.However, at the edge, the temperature is independent ofthe halo properties and its ratio with respect to the massof DM is universally equal to (1 . ± . × − (eV · K − )at 95% confidence. This result indicates that DM at theedge of the halos is an unperturbed thermal relic whosetemperature is equal to its decoupling temperature in theearly universe, times a power of the scale factor due tothe expansion of the universe.We have studied the implications of the observed uni-versal temperature within the context of two cosmologi-cal models. It has been shown that hot, and cold DM arenot consistent with the observations, while a warm DMscenario with a mass of ∼ Appendix A: Additional figures
Log(M/M )
Log ( ) [ k m s ] EinastoLuckypIsoBurkert
FIG. 6. The central dispersion velocity of DM as a function ofthe total mass of the halo. It should be noted that this figureis nearly insensitive to where the edge of the halo is defined.[1] P. J. E. Peebles, Growth of the nonbaryonic darkmatter theory, Nature Astronomy , 0057 (2017),arXiv:1701.05837 [astro-ph.CO].[2] P. Salucci, The distribution of dark matter in galaxies,Astron Astrophys Rev , 2 (2019), arXiv:1811.08843[astro-ph.GA].[3] A. Borzou, On the stability of fermionic non-isothermaldark matter halos, European Physical Journal C , 1076(2020), arXiv:2003.04532 [astro-ph.CO].[4] F. Lelli, S. S. McGaugh, and J. M. Schombert, SPARC:Mass Models for 175 Disk Galaxies with Spitzer Photom-etry and Accurate Rotation Curves, AJ , 157 (2016),arXiv:1606.09251 [astro-ph.GA].[5] P. Li, F. Lelli, S. McGaugh, and J. Schombert, A Com-prehensive Catalog of Dark Matter Halo Models forSPARC Galaxies, The Astrophysical Journal SupplementSeries , 31 (2020), arXiv:2001.10538 [astro-ph.GA].[6] H. J. Zahid, M. J. Geller, D. G. Fabricant, and H. S.Hwang, The Scaling of Stellar Mass and Central Stel-lar Velocity Dispersion for Quiescent Galaxies at z¡0.7,Astrophys. J. , 203 (2016), arXiv:1607.04275 [astro-ph.GA]. [7] A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada,Where Are the Missing Galactic Satellites?, Astrophys.J. , 82 (1999), arXiv:astro-ph/9901240 [astro-ph].[8] B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn,J. Stadel, and P. Tozzi, Dark Matter Substructure withinGalactic Halos, The Astrophysical Journal , L19(1999), arXiv:astro-ph/9907411 [astro-ph].[9] J. Dubinski and R. G. Carlberg, The Structure of ColdDark Matter Halos, Astrophys. J. , 496 (1991).[10] J. F. Navarro, C. S. Frenk, and S. D. M. White, TheStructure of Cold Dark Matter Halos, Astrophys. J. ,563 (1996), arXiv:astro-ph/9508025 [astro-ph].[11] M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat,Too big to fail? The puzzling darkness of massive MilkyWay subhaloes, Monthly Notices of the Royal Astro-nomical Society , L40 (2011), arXiv:1103.0007 [astro-ph.CO].[12] M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat,The Milky Way’s bright satellites as an apparent failureof ΛCDM, Monthly Notices of the Royal AstronomicalSociety , 1203 (2012), arXiv:1111.2048 [astro-ph.CO].[13] M. R. Lovell, V. Eke, C. S. Frenk, L. Gao, A. Jenkins, r (kpc) f o r c e s o f p r e ss u r e Einasto g dP dr g dP dr sum r (kpc)Lucky g dP dr g dP dr sum r (kpc)pIso g dP dr g dP dr sum r (kpc)Burkert g dP dr g dP dr sum FIG. 7. Each of the two components of the force of the pressure is divided by the gravitational acceleration. The componentcontaining the temperature derivative is indicated by dP dr . As can be seen, the forces of the pressure are significantly largerthan the gravitational force at close to the center of the halo. Since the sum of the two ratios is equal to one at all distances,the gravitational force is strong enough to balance the repulsive force and maintain the stability.T. Theuns, J. Wang, S. D. M. White, A. Boyarsky, andO. Ruchayskiy, The haloes of bright satellite galaxies in awarm dark matter universe, Monthly Notices of the Royal Astronomical Society420