Constraints on MACHO Dark Matter from Compact Stellar Systems in Ultra-Faint Dwarf Galaxies
aa r X i v : . [ a s t r o - ph . GA ] J un Draft version June 13, 2016
Preprint typeset using L A TEX style emulateapj v. 08/22/09
CONSTRAINTS ON MACHO DARK MATTER FROM COMPACT STELLAR SYSTEMS IN ULTRA-FAINTDWARF GALAXIES
Timothy D. Brandt
Draft version June 13, 2016
ABSTRACTI show that a recently discovered star cluster near the center of the ultra-faint dwarf galaxy Eri-danus II provides strong constraints on massive compact halo objects (MACHOs) of & M ⊙ as themain component of dark matter. MACHO dark matter will dynamically heat the cluster, driving it tolarger sizes and higher velocity dispersions until it dissolves into its host galaxy. The stars in compactultra-faint dwarf galaxies themselves will be subject to the same dynamical heating; the survival of atleast ten such galaxies places independent limits on MACHO dark matter of masses & M ⊙ . BothEri II’s cluster and the compact ultra-faint dwarfs are characterized by stellar masses of just a fewthousand M ⊙ and half-light radii of 13 pc (for the cluster) and ∼
30 pc (for the ultra-faint dwarfs).These systems close the ∼ M ⊙ window of allowed MACHO dark matter and combine withexisting constraints from microlensing, wide binaries, and disk kinematics to rule out dark mattercomposed entirely of MACHOs from ∼ − M ⊙ up to arbitrarily high masses. Subject headings: INTRODUCTIONDozens of ultra-faint dwarf galaxies have recently beendiscovered as satellites of the Galaxy and Andromeda,and as members of the Local Group (McConnachie2012 and references therein; Koposov et al. 2015a;Bechtol et al. 2015; Laevens et al. 2015). These satel-lites have luminosities as low as ∼ L ⊙ , and totalmasses inside the half-light radius that are at least 1–3 orders of magnitude larger than their stellar masses(Simon & Geha 2007; McConnachie 2012 and referencestherein). Many of them host negligible amounts of gas;all are understood to be dominated by their dark mattercontent.Ultra-faint dwarf galaxies are excellent places to learnabout dark matter. They currently provide the bestconstraints on annihilating weakly interacting massiveparticles (WIMPs), ruling out the simplest thermalrelic cross-sections for particle masses of tens of GeV(Ackermann et al. 2015). Dwarf galaxies have also beena source of tension with cosmological simulations: colddark matter simulations have long overpredicted theabundance of massive satellite galaxies (Klypin et al.1999; Moore et al. 1999). This problem may be re-solved through a combination of newly discovered dwarfsand the inclusion of baryonic physics in simulations(Brooks & Zolotov 2014), but has also been suggestedas evidence for exotic forms of dark matter or modifiedgravity (Lovell et al. 2012).While the evidence for dark matter’s existence isoverwhelming (Spergel et al. 2003; Clowe et al. 2006;Planck Collaboration et al. 2015), the identity of thedark matter particles remains mysterious. One intriguingpossibility is that the dark matter consists of black holesformed in the early Universe. Such massive compacthalo objects (MACHOs, Griest 1991) could be detectedin the halo of our Galaxy by gravitational microlensing Institute for Advanced Study, Einstein Dr., Princeton, NJ NASA Sagan Fellow (Paczynski 1986). Microlensing surveys, however, havenow ruled out MACHOs between ∼ − and ∼ M ⊙ asthe dominant component of dark matter in our Galaxy(Alcock et al. 2001; Tisserand et al. 2007). At MACHOmasses & M ⊙ , the existence of fragile, wide halobinaries constrains their abundance (Chanam´e & Gould2004; Yoo et al. 2004), though these limits rely heavilyon just a few systems (Quinn et al. 2009). Quinn et al.(2009) showed that one binary used by Yoo et al. (2004)to claim constraints for MACHOs & M ⊙ is likely spuri-ous, which removes the constraints for masses . M ⊙ .At MACHO masses & M ⊙ , Hern´andez et al.(2004) showed that some dwarf galaxy cores wouldbe dynamically relaxed, in tension with the rela-tively constant-density cores that have been inferred(S´anchez-Salcedo et al. 2006; Goerdt et al. 2006). MA-CHOs of very high mass ( & M ⊙ ) are also ruled outby the kinematics of the Galactic disk (Lacey & Ostriker1985). The only constraints on a population of MA-CHOs between ∼ M ⊙ and ∼ M ⊙ , however, cur-rently come from limits on spectral distortions of the cos-mic microwave background (CMB): black holes may haveaccreted during the early Universe, leaving an imprint onthe CMB (Ricotti et al. 2008). However, other authorshave argued that these constraints may not be definitive(Bird et al. 2016). Mu˜noz et al. (2016) have shown thatMACHOs of these masses may also be probed by lensedfast radio bursts (FRBs).In this paper, I derive MACHO constraints fromthe compact stellar distributions of ultra-faint dwarfgalaxies and, in particular, from the survival of a starcluster in Eridanus II. Eridanus II was discovered aspart of the Dark Energy Survey (Koposov et al. 2015a;Bechtol et al. 2015). It has an absolute magnitude of M V = − . ±
17 kpc (Crnojevi´c et al. 2016). Eri II hosts asingle star cluster of absolute magnitude M V = − . ∼ L ⊙ ,V ) and half-light radius r h = 13 pc. Thestar cluster appears to be nearly coincident with thegalaxy’s center. Several other ultra-faint dwarfs havestellar masses of a few thousand M ⊙ and half-light radiiof ∼
30 pc (Bechtol et al. 2015; Koposov et al. 2015a,b;Laevens et al. 2015); these galaxies provide independentMACHO constraints.Eri II is one of the few dwarf galaxies with a star clus-ter, but it is not unique in this respect. The Fornax dwarfspheroidal galaxy has long been known to host globularclusters (Baade & Hubble 1939; Hodge 1961). Its fiveknown globular clusters range from 240 pc to 1.6 kpcin projected separation from the galaxy center, and from ∼ × M ⊙ to ∼ × M ⊙ in mass (Mackey & Gilmore2003). Dynamical friction should cause these clusters tospiral in towards the center of Fornax (Tremaine 1976);their current existence outside of its core places interest-ing constraints on the properties of Fornax’s dark halo(S´anchez-Salcedo et al. 2006; Goerdt et al. 2006; Inoue2011; Cole et al. 2012). Other dwarf galaxies do host nu-clear star clusters (Georgiev et al. 2009); these are invari-ably much more massive and more tightly bound thanthe cluster in Eri II.In this paper I show that the star cluster in Eri II hasimportant implications for MACHO dark matter, andthat the population of compact ultra-faint dwarfs pro-vides similar, independent limits. The paper is organizedas follows. In Section 2, I apply the theory of collisionalstellar systems to dynamical heating of the cluster byMACHOs. Section 3 presents the constraints on MA-CHO dark matter from the cluster in Eri II and from thepopulation of compact ultra-faint dwarfs. I discuss andconclude with Section 4. HEATING OF A STAR CLUSTER BY MACHOSA star cluster is a dynamic environment where gravi-tational interactions lead to the exchange of energy be-tween stars. These interactions cause the system to dy-namically relax; they may be approximated as diffusionterms using the Fokker-Planck equation (Chandrasekhar1943). This approximation has enabled modeling ofcluster density and velocity distributions (King 1966;Meylan & Heggie 1997 and references therein). Whena range of masses is present, stellar interactions lead tomass segregation, in which the most massive bodies havethe most spatially compact distribution (Spitzer 1969).Two-body interactions tend to equalize the mean ki-netic energies of different mass groups at a given location.In a system consisting of > M ⊙ MACHOs and stars,the stars will gain energy from the MACHOs; a compactstellar system will puff up. This can be treated as a diffu-sion problem, with weak scatterings gradually changingeach star’s velocity. The sum of the diffusion coefficientsfor the parallel and perpendicular components of the ve-locity describes the evolution of a star’s kinetic energy.Assuming an isotropic Maxwellian velocity distributionfor the dark matter particles and a locally uniform darkmatter density, the relevant diffusion coefficient is givenby D h (∆ v ) i = 4 √ πG f DM ρm a ln Λ σ (cid:20) erf( X ) X (cid:21) , (1)where ln Λ is the Coulomb logarithm, m a and σ are theMACHO mass and velocity dispersion, ρ is the total dark matter density, and f DM is the fraction of dark matter inMACHOs of mass m a (Binney & Tremaine 2008). Thevariable X is the ratio of the stellar velocity to that of theMACHOs, X ≡ v ∗ / ( √ σ ). I will assume that the starsare relatively cold compared to the dark matter, v ∗ . σ ,which implies that erf( X ) /X ∼
1. This assumption issatisfied for all of the cluster and dark matter param-eters used in the following section. For the Coulomblogarithm,ln Λ ≈ ln (cid:18) b max h v i G ( m + m a ) (cid:19) ≈ ln (cid:18) r h σ G ( m + m a ) (cid:19) , (2)where m and m a are the masses of the cluster stars andMACHOs, respectively, h v i ≈ σ is the typical rela-tive velocity, and b max is the maximum impact parameter(Binney & Tremaine 2008), which I take to be the half-light radius r h . For 10 M ⊙ MACHOs with σ = 10 km s − and r h = 13 pc, ln Λ ≈
10. As usual, the Coulomb loga-rithm is very insensitive to the assumed halo properties.If dark matter is a mixture of MACHOs and low-massparticles like WIMPs, dynamical heating will competewith dynamical cooling. The cooling rate from WIMPsis given by D [∆ E ] = vD [∆ v || ] = − πG ρv ∗ m ∗ (1 − f DM ) ln Λ σ G ( X ) , (3)with X ≡ v ∗ / ( √ σ ) as before, m ∗ being the mass of atypical star, and G ( X ) = 12 X (cid:20) erf( X ) − X √ π exp − X (cid:21) . (4)Dynamical heating will dominate over cooling by a factorheatingcooling ∼ m a σ √ m ∗ v ∗ (cid:18) erf( X ) XG ( X ) (cid:19) (cid:18) f DM − f DM (cid:19) , (5)or a factor of ∼ f DM / (1 − f DM ) for 10 M ⊙ MACHOs.Equation (5) is always much larger than unity for thelimiting f DM derived in the following section; I neglectWIMP cooling for the rest of this paper.Heating by MACHOs will add energy to the cluster,causing it to expand. If the cluster is embedded in a darkmatter core of constant density ρ , its potential energy perunit mass is given by UM = constant − α GM ∗ r h + βGρr h , (6)where M ∗ is the cluster’s stellar mass, r h is its projectedhalf-mass radius, and α and β are proportionality con-stants that depend on the mass distribution. For a coredS´ersic profile, α ∼ . β ∼
10; I will adopt thesevalues throughout the paper and assume them to remainconstant as the cluster expands. The measured profile ofthe cluster in the dwarf galaxy Eridanus II is a S´ersic pro-file with n ≈ . sersic (Novak et al.2012) gives values of α ≈ .
36 and β ≈ .
2, which wouldmake this cluster expand slightly faster than with myfiducial α and β .Using the virial theorem, E tot = − U , Equations (6)and (1) combine to give an implicit equation for the evo- Fig. 1.—
Dynamical heating of a 6000 M ⊙ star cluster by 30 M ⊙ MACHOs at three fiducial densities, neglecting mass loss from thecluster. The cluster expands slowly until its mean density equalsthat of the MACHOs, and then expands as r h ∼ √ t . lution of the half-light radius, dr h dt = 4 √ πGf DM m a σ ln Λ (cid:18) α M ∗ ρr h + 2 βr h (cid:19) − . (7)As long as the star cluster is dark-matter dominated,Equation (7) is independent of the dark matter density.A compact stellar system will expand slowly until it be-comes dominated by its dark matter content, and thenexpand with r h ∼ √ t . Figure 1 demonstrates this behav-ior for a 6000 M ⊙ cluster with an initial half-light radiusof 1 pc for 30 M ⊙ MACHOs at three fiducial dark matterdensities, taking α = 0 . β = 10.Motivated by Equation (7), I define two characteristiclifetimes for a stellar system. The first is the time for itto puff up to its observed size from the ∼ √ CONSTRAINTS FROM THE ULTRA-FAINTDWARFSI now combine Equation (7) with the observed survivalof compact ultra-faint dwarf galaxies and of the star clus-ter in the core of Eridanus II to constrain MACHO darkmatter. As described in the previous section, I define twocharacteristic lifetimes: (1) the time for the cluster to ex-pand to its current size from the ∼ The Cluster in Eridanus II
The star cluster in Eridanus II is believed to be atleast ∼ ∼
12 Gyr(Crnojevi´c et al. 2016). At an age of 3 Gyr, the V -band mass-to-light ratio for a metal-poor stellar system is ∼ M ⊙ /L ⊙ , while this ratio is ∼ M ⊙ /L ⊙ for an old sys-tem (Maraston 2005). The cluster’s observed M V = − . ∼ M ⊙ at an age of3 Gyr, or a mass of ∼ M ⊙ at an age of 12 Gyr. Thesystem has an observed half-light radius r h = 13 pc. Iassume the system to have resided within the core of thedark matter halo for its entire life, and derive MACHOlimits by requiring the timescales for dynamical heatingto be longer than the cluster’s age.Figure 2 shows the constraints for a range of plausi-ble dark matter halo properties, with three-dimensionalvelocity dispersions of 5–10 km s − and dark matter den-sities of 0.02–1 M ⊙ pc − . These values span the rangeof parameters characteristic of ultra-faint dwarf galaxies(Simon & Geha 2007; McConnachie 2012 and referencestherein). At an age of 3 Gyr (left panel), MACHOs . M ⊙ are excluded from making up all of the darkmatter unless the Eri II cluster was initially compact andremains embedded in a low-density, high-dispersion halo.In this case, a cluster of the observed size is a transientphenomenon; similar objects should be rarer than com-pact low-mass clusters. If the cluster has spent ∼
12 Gyrnear the center of its halo (right panel), the constraintsstrengthen.The preceding discussion assumed a roughly constantdark matter density profile (a core larger than the clus-ter). Assuming a cuspy dark matter profile with the clus-ter at the dynamical center would strengthen the con-clusions. Such an assumption would make the clusterdominated by dark matter at a smaller half-light radius;it would quickly begin to evolve with r h ∼ √ t inde-pendently of dark matter density (Equation (7)). Fur-ther, the velocity dispersion of the dark matter parti-cles is expected to fall toward the center of an NFWhalo (Ferrer & Hunter 2013). Lower velocity dispersionswould make MACHOs even more effective at dynamicalheating, improving constraints on their abundance. If,on the other hand, the cluster were slightly offset fromthe dynamical center of a strong dark matter cusp, itwould be tidally shredded in a dynamical time.3.2. Constraints from Other Ultra-Faint Dwarfs
The entire stellar population of a dwarf galaxy willalso be dynamically heated by MACHOs. Many com-pact ultra-faint dwarf galaxies are now known, with stel-lar masses . M ⊙ (assuming a mass-to-light ratio M/L V = 3 M ⊙ /L ⊙ ,V ), half-light radii .
30 pc, and cen-tral densities ∼ M ⊙ pc − . Table 1 lists some basic prop-erties of ten compact ultra-faint dwarfs (plus the starcluster in Eri II); all but three were discovered since 2015.Where measured, the ages of the stars are consistent with ∼
10 Gyr (Bechtol et al. 2015; Laevens et al. 2015). Themean densities listed are vulnerable to different defini-tions of the half-light or half-mass radius, and should betreated as uncertain to at least a factor of ∼
2. The com-pact ultra-faint dwarfs constrain MACHO dark matterin the same way as the star cluster in Eri II: I use thesame two heating timescales and require one or the otherto be longer than 10 Gyr.Figure 3 shows the limits on MACHO dark matterimplied by a fiducial compact ultra-faint dwarf, with r h = 30 pc, M ∗ = 3000 M ⊙ , and a central dark matterdensity ρ = 1 M ⊙ pc − , for three-dimensional velocity Fig. 2.—
MACHO constraints from the survival of the star cluster near the core of Eridanus II, assuming a cluster age of 3 Gyr (leftpanel) and 12 Gyr (right panel). The units for the dark matter density ρ and velocity dispersion σ , are M ⊙ pc − and km s − , respectively.The limits come from requiring that the timescale to grow from r h, = 2 pc to the observed r h = 13 pc is longer than the cluster age (redlines), or from requiring that the timescale to double in area (increase by √ r h ) is longer than the cluster age (blue lines). TABLE 1Properties of Compact Ultra-Faint Dwarf Galaxies
Name r h † L V ρ / σ ∗ Ref. †† pc L ⊙ M ⊙ pc − km s − Wil I 25 ± . +2 . − . +8 −
300 3 3 . ± . ± . +2 . − . +2 − /55 +5 − * . +1 . − . +4 − /60 +76 − * . +2 . − . +9 − /43 +153 − * +6 − /33 +20 − * +13 − /12 +2 − * +13 − /11 +8 − * +8 − ** ± † Where two values are given, the first is from Koposov et al.(2015a) and the second from Bechtol et al. (2015). †† References abbreviated as: 1 (Martin et al. 2007); 2(Martin et al. 2008); 3 (Simon et al. 2011); 4 (Belokurov et al.2009); 5 (Bechtol et al. 2015); 6 (Koposov et al. 2015a); 7(Koposov et al. 2015b); 8 (Laevens et al. 2015); 9 (Crnojevi´c et al.2016) * Geometric means of Koposov et al. (2015a) and Bechtol et al.(2015), rounded to 500 L ⊙ . ** Values are for the central star cluster only. dispersions of 5 and 10 km s − . The observed ultra-faintdwarfs lie within this range; with one-dimensional veloc-ity dispersions between 3 and 6 km s − (Table 1). Thesurvival of the compact ultra-faint dwarfs listed in Table1 rules out dark matter consisting entirely of MACHOsof mass & M ⊙ . DISCUSSION AND CONCLUSIONSThe star cluster in the core of the newly discovereddwarf galaxy Eridanus II provides strong constraints ona region of MACHO parameter space difficult to probewith either microlensing or wide Galactic binaries; thepopulation of compact, ultra-faint dwarfs provides sim-ilar, independent limits. Figure 4 compares the con-straints derived in Section 3 using conservative assump-
Fig. 3.—
MACHO constraints from the observed sizes of compactultra-faint dwarf galaxies, assuming a stellar mass of 3000 M ⊙ , acurrent half-light radius r h = 30 pc, and an age of 10 Gyr. Theunits for the dark matter density ρ and velocity dispersion σ , are M ⊙ pc − and km s − , respectively. The limits come from requiringthat the timescale to grow from r h, = 2 pc to r h = 30 pc is longerthan 10 Gyr (red lines), or from requiring that the timescale todouble in area (increase by √ r h ) is longer than 10 Gyr (bluelines). tions about the dark matter halos to constraints frommicrolensing (Alcock et al. 2001; Tisserand et al. 2007)and wide Galactic halo binaries (Quinn et al. 2009). Thekinematics of the Galactic disk provide an independentlimit on the abundance of very massive ( & M ⊙ ) MA-CHOs (Lacey & Ostriker 1985). For dark matter ha-los consistent with measured dwarf properties (Table 1),MACHO dark matter is ruled out over the entire openregion of masses. If Eri II’s cluster is old and was ini-tially puffier than Galactic clusters, it provides especiallystrong limits.While Eri II’s cluster likely provides the best limits onMACHOs from ∼ M ⊙ up to thousands of M ⊙ , thereare ways to evade its constraints. The cluster, for exam-ple, could have recently spiraled into the center of Eri IIdue to dynamical friction, having spent most of its lifeas a compact cluster in a low-density MACHO environ-ment. However, the inspiral timescale is inversely pro- Fig. 4.—
Constraints on MACHO dark matter from microlens-ing (blue and purple, Alcock et al. 2001; Tisserand et al. 2007) andwide Galactic binaries (green, Quinn et al. 2009), shown togetherwith the constraints from the survival of compact ultra-faint dwarfgalaxies and the star cluster in Eridanus II. I conservatively adopt adark matter density of 0 . M ⊙ pc − in Eri II and 0 . M ⊙ pc − inthe ultra-faint dwarfs, assume a three-dimensional velocity disper-sion σ = 8 km s − , and use two definitions of the heating timescale.A low-density halo and initially compact cluster weaken the con-straints from Eri II. Even in this case, assuming dark matter halosto have the properties that are currently inferred, MACHO darkmatter is excluded for all MACHO masses & − M ⊙ . portional to the cluster mass (Binney & Tremaine 2008),and the cluster in Eri II is 1.5–2 orders of magnitude lessmassive than Fornax 4 (Mackey & Gilmore 2003), theFornax globular cluster nearest the center of that dwarf(at 240 pc in projected separation). This scenario there-fore requires very different dark matter halos in the twogalaxies or severe mass loss during Eri II’s inspiral, andalso luck to catch the cluster on the point of disruption.This problem of coincidence is generic to any scenario inwhich Eri II’s cluster was initially compact. The proba-bility of observing the system in such a transient state issignificantly higher if the cluster’s age is ∼ ∼
12 Gyr.Other possibilities to evade the constraints includean intermediate-mass black hole ( & M ⊙ ) to provide binding energy, or a chance alignment such that the clus-ter only appears to reside in the center of Eri II. Bothwould be surprising. Such a black hole would have a masscomparable to the total stellar mass of its host galaxy. Amassive black hole would also be expected to host a re-laxed MACHO cluster of comparable mass, in which caseit may not avoid the problem of dynamical heating at all.A chance alignment of a cluster physically located at thegalaxy’s half-light radius is possible; the most na¨ıve esti-mate, the fraction of solid angle lying within a few r h inprojection, gives a chance alignment probability of ∼ ∼