Scaling Relations of Halo Cores for Self-Interacting Dark Matter
SScaling Relations of Halo Cores for Self-Interacting Dark Matter
Henry W. Lin and Abraham Loeb Institute for Theory & Computation, Harvard-Smithsonian Centerfor Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA (Dated: September 11, 2018)Using a simple analytic formalism, we demonstrate that significant dark matter self-interactionsproduce halo cores that obey scaling relations nearly independent of the underlying particle physicsparameters such as the annihilation cross section and the mass of the dark matter particle. For dwarfgalaxies, we predict that the core density ρ c and the core radius r c should obey ρ c r c ≈
41 M (cid:12) pc − with a weak mass dependence ∼ M . . Remarkably, such a scaling relation has recently beenempirically inferred. Scaling relations involving core mass, core radius, and core velocity dispersionare predicted and agree well with observational data. By calibrating against numerical simulations,we predict the scatter in these relations and find them to be in excellent agreement with existingdata. Future observations can test our predictions for different halo masses and redshifts. Introduction.
The standard cosmological model witha cosmological constant Λ and cold dark matter (CDM)has been confirmed on large scales by a wealth of suc-cessful predictions. Yet despite its simplicity and exper-imental success, a variety of challenges to the ΛCDMparadigm persist. On the observational side, the CDMparadigm suffers from the well-known cusp-core problem(see [1] for a review); numerical simulations [2, 3] predictthat the density of dark matter halos should rise sharplytowards the center of the halo forming a “cusp”, whereasobservations (see e.g. [4–7] and references therein) in-dicate that the dark matter in the center of halos risesmore gently, resembling a “core” even in dwarf galax-ies where the baryonic content is negligible [8]. Moti-vated by observations, various models of interacting darkmatter have been proposed to alleviate this tension in-cluding scattering [9, 10] and annihilation [11], althoughsome authors, e.g [12], argue that dark matter cores donot appear upon a more careful analysis. Many inter-esting experiments (for recent progress see e.g. [13, 14]and references therein) have been devised to constrainsome combination of mass and interaction cross section.Here, we focus on the model-independent observationalsignatures of annihilation. Remarkably, we find that scal-ing relations exist that are generic predictions of self-interactions where any dependence on the cross section σ and the particle mass cancel. We can thus make predic-tions about annihilation signature without introducingany additional parameters beyond what is required in aminimal ΛCDM cosmology. Throughout this Letter, weassume h ≡ H / (100 km s − Mpc − ) = 0 .
7, Ω m = 0 . Λ = 0 .
73 [15].
Core scaling relations
To derive the core scaling re-lations, we need two simple ingredients. The first ingre-dient is the initial condition of the dark matter densityprofile, before enough cosmic time has passed for annihi-lation to significantly alter the profile. We take the darkmatter profiles to be of the Navarro-Frenk-White (NFW)form [2]: ρ ( r, t = t ) = ρ ( r/r s )(1 + r/r s ) . (1) Here t is the age of the universe when the dark matterhalo virialized. We will comment shortly on the modifi-cation of our results if the inner profile of the dark matterhalo is more accurately described by the Einasto profile[16], which is likely a better fit to simulations [17]. Todetermine ρ and r s for a halo of mass M and redshift z , we adopt an expression for the concentration param-eter c ( M, z ) ≡ r s /r , where r is defined as theradius interior to which the average density is 200 timesthe critical density of the universe ρ crit = (3 H / πG ),where H is the Hubble parameter.We adopt the fitting formulae of Dutton and Macci`o[18], their equations 7, 10, and 11, calibrated from theresults of their N-body simulations:log c = a + b log( M/ h − M (cid:12) ) a = 0 .
520 + 0 .
385 exp ( − . z . ) b = − .
101 + 0 . z. (2)These formulae are in agreement with a recently proposeduniversal model of [19].The second ingredient is the evolution of the dark mat-ter profiles with time. We will only consider the simplestcase of s-wave annihilation where (cid:104) σv (cid:105) = const. Thisis a reasonable approximation for most annihilation pro-cesses unless a quantum selection rule prevents s-waveannihilation or the cross section acquires a sharp veloc-ity dependence due to Sommerfeld enhancement. Ignor-ing the gravitational back reaction, the time derivativeof the density is given by˙ ρ ( x, t ) = − Γ ρ ( x, t ) , (3)where Γ ρ has units of inverse time. The solution to thisequation is ρ ( r, t ) = ρ ( r, t )1 + f Γ tρ ( r, t ) , (4)where we have inserted a fudge factor f > f = 1). a r X i v : . [ a s t r o - ph . GA ] F e b Plugging in our initial condition yields ρ ( r, t ) = ρ c ( r/r c )(1 + r/r s ) + 1 , (5)where we define a core density and core radius ρ c ≡ ( f Γ t ) − and r c ≡ ( f Γ t ρ ) r s , respectively. For the mo-ment, we simply define r c the core radius, we will showshortly that it corresponds approximately to the notionof core radius defined by other authors.Since different authors define these quantities differ-ently, it is worth noting what these quantities represent,so that they can be sensibly compared to observationalwork. It is straightforward to verify that both quantitiescan be expressed in terms of properties of the central den-sity profile: the central density is simply ρ c = ρ ( r = 0),and Taylor expanding the logarithmic slope − d log ρd log r = (cid:18) rr c (cid:19) + O (cid:0) r (cid:1) (6)shows that r c is the length scale over which the logarith-mic slope significantly deviates from 0. A full compari-son of the behavior of our density profiles with existingwork is given in Figure 1, which shows that the numericalvalue of r c (obtained by fitting to data) may differ fromthe values of ( r c ) Burkert or ( r c ) Zavala , depending on theprecise value of r s . However, the difference is unlikely tobe greater than ∼ . . r s , our def-inition of r c is consistent with ( r c ) Burkert or ( r c ) Zavala upto a factor of order unity.These non-parametric properties of the central den-sity profile allow us to connect our definitions with otherparametric profiles. For example, the above considera-tions lead us to identify ( ρ ) Burkert = ρ c and ( r c ) Burkert ≈ r c where the subscripts denote definitions found in Burk-ert [7], since the core radii in both our profile and Burk-ert’s profile are turnover radii, e.g., the length scale overwhich the logarithmic slope becomes non-negligible.Note that we have not made any assumptions aboutthe constancy of f with particle physics or halo parame-ters. Kaplinghat et al. [11] have argued that 3 (cid:46) f (cid:46) rM ( r ) isconserved; numerical simulations of annihilating [11, 20]and self-interacting (scattering only) [21] dark matterconfirm the accuracy of this approximation. These simu-lations also show that our results are qualitatively unaf-fected by the choice of NFW or Einasto profiles, at leastwhen the dark matter interaction is strong enough so thatthe core radius r c (cid:38) − r s and the differences betweenNFW and Einasto are significant.More quantitatively, Zavala et al. [22] showed that self-interacting dark matter profiles with σ/m = 0 . g − are well fit by a profile of the form (using our notation) ρ Zavala ( r ) = ρ c ( r/r c )(1 + r/r s ) + (1 + r/r s ) . (7) For small radii r (cid:28) r s , the last term (1 + r/r s ) ≈ r (cid:29) r s ,the r term in the denominator will dominate, and thequadratic term (1 + r/r s ) can be neglected. In eitherregime, our model is in agreement with the results ofZavala et al. [22]. The typical volume averaged differencebetween our models ( ρ − ρ Zavala ) /ρ is not more than afew percent within 2 r s , and is entirely negligible for muchlarger volumes.Although we have motivated equation (3) by consid-ering annihilating dark matter, we note that scatteringalso has a similar effect, as scattering will kick parti-cles from the high density inner regions to larger radii,where their contribution to the mass budget is negligible.Since the scattering rate is also proportional to ρ , Γ inequation (3) should also include contributions from thescattering cross section. These results are in agreementwith numerical simulations (e.g. the results of Elbert et al. [21]) which predict a flat slope interior to a coreradius ∼ r c and a profile which converges to the orig-inal NFW form at large radii. Note, however, that ifa self-interacting dark matter halo is in the gravother-mal collapse regime, our results do not apply. Indeednumerical simulations [23, 24] show that during collapse,the inner density profile becomes significantly more cuspythan a constant density core. The failure of our formal-ism in this regime is not surprising, as processes suchas gravothermal collapse obviously do not preserve adia-batic invariants.Although r c and ρ c will have a large spread due to thedifferent formation times of halos, there exists an overallscaling relation ρ c r c = ρ r s . (8)We emphasize that the above equation was not derived bymaking any assumptions about whether or not r c (cid:28) r s .Note that the formation time t , the fudge factor f , andthe annihilation rate Γ no longer appear in this result forthe core surface density. (However if the cross section isnegligible, r c → ρ r s = N c ( ρ crit ) / M / log( c + 1) − c / (1 + c ) . (9)where the normalization N = 10(10 /π ) / / (3 / ) ≈ . c ( M, z ) is a very weak function of mass, we ex-pect that the right hand side to scale roughly with M / .A more detailed calculation shows that the mass depen-dence is even weaker ρ r s ∝ M . at low redshifts andeven at a relatively high redshift z = 3, ρ r s ∝ M . .Thus, if we consider a subclass of halos such as dwarfgalaxies, ρ c r c will be approximately constant. Thisis precisely the scaling relation inferred empirically in ���� � � � � � � � � � ��� - - - - - - - - - ��� ( � / � � ) FIG. 1: Comparison of our density profile with Burkert [7] and Zavala et al. [22]. On the y -axis is the logarithmicslope of the density profile, dρ/d log r , on the x -axis is the logarithmic radius in units of r c . The gray lines show ourprofile for r s = r c to r s = 16 r c in log increments. Similarly, the pink dashed lines show different values of r s usingthe parameterization found in Zavala et. al . The thick black line is from [7]. Note that observations provide the bestconstraints in the region r/r c (cid:46)
1. This graph illustrates that our definition of r c is approximately consistent withother definitions of r c found in the literature. For example, the value of r c inferred by fitting a density profile tosome given data may be somewhat larger or smaller compared to the value of ( r c ) Burkert or ( r c ) Zavala . ρ � � � [ � / � � � ] - � ��� [ � ⊙ ] z = z = ρ � � � [ � / � � � ] - - �������� � l og ( M / M ⊙ ) = l o g ( M / M ⊙ ) = FIG. 2: Predicted ρ c r c as a function of mass M and redshift z . As sample points, we display the Milky Way [25]in black, a typical dwarf galaxy in gray, and the Phoenix galaxy cluster [26] in red. On the right panel, differentcurves display different halo masses M , which increase in unit log-increments. On the left panel, the differentcurves represent different redshifts which increase in increments of 0.5.Spano et al. [4], Donato et al. [5], Kormendy and Free-man [6] and most recently Burkert [7].Ultimately, the near-constancy of ρ r c is a consequenceof the fact that the initial conditions provide by (non-interacting) LCDM simulations have a density profilethat scales approximately ∝ r − . (The resulting profileswith interaction taken into account do not need ρ ∝ r − ).For example, if dark matter halos were more accuratelydescribed by older secondary infall models [27], the innerprofiles would be isothermal: ρ ∝ r − , which would notlead to the near-constancy of ρ c r c . Comparison with observations.
Although it is obser-vationally difficult to measure the corresponding M ofa dwarf galaxy, numerical simulations [28] suggest that atypical value is M ∼ M (cid:12) . Using this as a fiducialvalue, we predict ρ c r c = 41 M (cid:12) pc − × (cid:18) M M (cid:12) (cid:19) . , (10) where the logarithmic slope ( M b with b = 0 .
18) has beenobtained by appropriately (log-log) linearizing the massdependence at the fiducial value. Although the slope iscalibrated at the fiducial value, it remains approximatelyconstant over a large range of masses; even for a galaxycluster b = 0 .
22. In particular, for the Phoenix galaxycluster [26], ρ c r c ≈ . × M (cid:12) / pc − . We show the fulldependence on mass and redshift in Figure 2.Remarkably, such a scaling relation has been empiri-cally inferred by Burkert [7], who reports that (in our no-tation) ρ c r c = 75 +85 − M (cid:12) pc − over 18 magnitudes in bluemagnitude, covering a sample that ranges from dwarfgalaxies to giant galaxies. Here we have identified thecentral density ρ ( r = 0) = ρ c and the core radius r c asthe length scale associated with a turnover in logarithmicslope. The reported uncertainties are not 1 σ uncertain-ties but encompass all but 1 or 2 of the 48 data points. Al-though the median dark matter halo mass of their sampleis not easily measured and therefore not reported, a rea-sonable value is M = 10 . M (cid:12) which lies between dwarfgalaxies ( M ∼ M (cid:12) ) and giant galaxies ( M ∼ ).Taking M = 10 . M (cid:12) and z = 0, we have from equation(7) that ρ c r c = 78 M (cid:12) pc − , which fully consistent withthe value that is empirically inferred.The predicted scaling of ρ c r c with mass can also betested. Since the luminosity L ∗ of a galaxy is propor-tional to the number (and thus the mass M ∗ of stars, L ∗ ∝ M ∗ . For low mass galaxies M (cid:46) . M (cid:12) , ob-servations [29] indicate M ∝ M β ∗ with β = 0 . +0 . − . ,with the slope gradually steepening to ∝ M . ∗ for verymassive galaxies. Hence, for low mass M (cid:46) . M (cid:12) (intermediate mass M ∼ . M (cid:12) ) galaxies, we pre-dict that ρ c r c ∝ M . ∝ M . ∗ ∝ L γ ∗ with γ = 0 . γ = 0 . ρ c r c ∝ L γ ∗ , where γ = 0 . ± . M ∗ – L ∗ relation is steeper, could further con-strain γ in order to provide a more stringent test of ourpredictions. We also note that our model predicts virtu-ally no redshift evolution for z (cid:46)
1. Hence a detection ofredshift dependence in ρ c r c would falsify our model.It is worth mentioning that our derivation of the scal-ing relations allows us to compute the expected scatterin the observed relations. The two sources of scatter inour model are the scatter in c that exists even fora fixed mass and redshift, and the scatter in M if theobserved population of dwarf galaxies contains galaxiesof different virial masses. Note that from equation (7), ρ c r c ∝ c for large values of c , whereas the explicitmass dependence is much weaker. A typical amount ofscatter (at the 1 σ level) associated with c is ∼ . ∼ . ∼ c will dominate. To compare to empirical results,we will assume that the population of dwarf spheroidalgalaxies observed in [7] does not contain such a diversityof masses. Certainly if the scatter in mass were muchlarger than an order of magnitude, the fact that all oftheir halos have log M . = 7 . ± . M . is themass enclosed in the inner 0.3 kpc would seem peculiar,since the total mass is expected to be a strong functionof M . . In particular, for an NFW profile gives leadsto a mass dependence [30] M ∝ ( M . ) . , though de-viations from ρ ∝ r − in the inner regions will changethis dependency. Assuming that c has a 1 σ scatterof 0.15 dex, we predict using Monte Carlo methods that ρ c r c = 78 +33 − M (cid:12) pc − with a 68% confidence interval.Considering that the uncertainties reported in Burkert[7] contain ∼
96% of the data and are thus close to 2 σ bounds, there is good agreement between the predictedand observed scatter.Our scaling laws are also in agreement with the olderresults of Spano et al. [4] and Donato et al. [5], which find in their sample of galaxies and dwarf galaxies ρ c r c =10 . ± . M (cid:12) pc − [5]. The somewhat higher value (by afactor of ∼
2) of ρ c r c compared to the results of Burkert[7] could be due to the fact that their sample is dominatedby more massive halos.As a consistency check, we may derive additional scal-ing relations from our model. For example, if we assumethat r c (cid:28) r s (an assumption that we so far have not yetmade use of), the density in the region r (cid:46) r c is approx-imately constant, and M c r c ≈ π ρ c r c ≈ const , (11)and the core velocity dispersion σ c obeys σ c r − c ∼ GM ( r < r c ) /r c ∼ Gρ r c . Hence, σ r − c ≈ const. (12)The last two scaling laws have also been empirically in-ferred in [7] and have been shown to be consistent withavailable data. However, we caution that these additionalresults can only be derived in our formalism if r (cid:46) r c ,which is not the general case. Conclusion.
Starting from an NFW profile and a sim-ple treatment of annihilation, we have derived a universalscaling relation ρ c r c ≈ M (cid:12) pc − × (cid:0) M / M (cid:12) (cid:1) . at z = 0. Said differently, it has been remarked thatanomalies with ΛCDM are associated with an accelera-tion scale a ∼ − cm s − [8] where new physics be-comes relevant. In units where c = G = 1, a character-istic acceleration is equivalent to a characteristic surfacedensity. In this work, we have derived the surface densityscale (see equation 9), assuming that the new physics isdark matter self-interactions.Remarkably, the derivation did not involve any addi-tional parameters beyond what is needed in LCDM. Thisscaling relation holds independently of the dark mattermass and annihilation or scattering cross sections as wellas the amount of adiabatic expansion f experienced bythe core. Our results are also relatively insensitive toslow motions of baryonic matter in the dark matter halo,provided that the effect of the baryons is to adiabaticallyexpand or contract the core. This scaling relation is thusa robust signature of self-interaction, independent of thedetailed properties of the dark matter particles. Special-izing to the case of dwarf galaxies, we have shown thatboth the magnitude of ρ c r c and the scatter are in ex-cellent agreement with the data. We have also checkedthe model by deriving additional scaling relations whichalso agree with experiment. The predicted evolution ofhalo properties with redshift and mass can be tested withfuture observations. Acknowledgements.
This work was supported in partby NSF grant AST-1312034. [1] W. J. G. de Blok, Advances in Astronomy , 789293(2010), arXiv:0910.3538 [astro-ph.CO]. [2] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro-phys. J. , 493 (1997), astro-ph/9611107. [3] B. Moore, T. Quinn, F. Governato, J. Stadel, andG. Lake, MNRAS , 1147 (1999), astro-ph/9903164.[4] M. Spano, M. Marcelin, P. Amram, C. Carignan,B. Epinat, and O. Hernandez, MNRAS , 297 (2008),arXiv:0710.1345.[5] F. Donato, G. Gentile, P. Salucci, C. Frigerio Martins,M. I. Wilkinson, G. Gilmore, E. K. Grebel, A. Koch, andR. Wyse, MNRAS , 1169 (2009), arXiv:0904.4054.[6] J. Kormendy and K. C. Freeman, ArXiv e-prints (2014),arXiv:1411.2170.[7] A. Burkert, ArXiv e-prints (2015), arXiv:1501.06604.[8] M. G. Walker and A. Loeb, Contemporary Physics ,198 (2014), arXiv:1401.1146.[9] D. N. Spergel and P. J. Steinhardt, Physical Review Let-ters , 3760 (2000).[10] A. Loeb and N. Weiner, Physical Review Letters ,171302 (2011), arXiv:1011.6374 [astro-ph.CO].[11] M. Kaplinghat, L. Knox, and M. S. Turner, PhysicalReview Letters , 3335 (2000), astro-ph/0005210.[12] L. E. Strigari, C. S. Frenk, and S. D. M. White, ArXive-prints (2014), arXiv:1406.6079.[13] K. R. Dienes”, Physical Review Letters (2015),10.1103/PhysRevLett.114.051301.[14] M. Kaplinghat”, Physical Review Letters (2015),10.1103/PhysRevLett.114.211303.[15] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Ar-naud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J.Banday, R. B. Barreiro, J. G. Bartlett, and et al., ArXive-prints (2015), arXiv:1502.01589.[16] J. Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata , 87 (1965).[17] D. Merritt, A. W. Graham, B. Moore, J. Diemand, andB. Terzi´c, Astronomical Journal , 2685 (2006), astro-ph/0509417.[18] A. A. Dutton and A. V. Macci`o, MNRAS , 3359(2014), arXiv:1402.7073.[19] B. Diemer and A. V. Kravtsov, Astrophys. J. , 108(2015), arXiv:1407.4730.[20] R. Dav´e, D. N. Spergel, P. J. Steinhardt, and B. D. Wan-delt, Astrophys. J. , 574 (2001), astro-ph/0006218.[21] O. D. Elbert, J. S. Bullock, S. Garrison-Kimmel,M. Rocha, J. O˜norbe, and A. H. G. Peter, ArXiv e-prints (2014), arXiv:1412.1477. [22] J. Zavala, M. Vogelsberger, and M. G. Walker, MNRAS , L20 (2013), arXiv:1211.6426 [astro-ph.CO].[23] C. Kochanek and M. White, The Astrophysical Journal , 514 (2000).[24] J. Koda and P. R. Shapiro, MNRAS , 1125 (2011),arXiv:1101.3097.[25] P. J. McMillan, MNRAS , 2446 (2011),arXiv:1102.4340.[26] M. McDonald, M. Bayliss, B. A. Benson, R. J. Foley,J. Ruel, P. Sullivan, S. Veilleux, K. A. Aird, M. L. N.Ashby, M. Bautz, G. Bazin, L. E. Bleem, M. Brodwin,J. E. Carlstrom, C. L. Chang, H. M. Cho, A. Cloc-chiatti, T. M. Crawford, A. T. Crites, T. de Haan,S. Desai, M. A. Dobbs, J. P. Dudley, E. Egami, W. R.Forman, G. P. Garmire, E. M. George, M. D. Glad-ders, A. H. Gonzalez, N. W. Halverson, N. L. Har-rington, F. W. High, G. P. Holder, W. L. Holzapfel,S. Hoover, J. D. Hrubes, C. Jones, M. Joy, R. Keisler,L. Knox, A. T. Lee, E. M. Leitch, J. Liu, M. Lueker,D. Luong-van, A. Mantz, D. P. Marrone, J. J. McMa-hon, J. Mehl, S. S. Meyer, E. D. Miller, L. Mocanu,J. J. Mohr, T. E. Montroy, S. S. Murray, T. Natoli,S. Padin, T. Plagge, C. Pryke, T. D. Rawle, C. L. Re-ichardt, A. Rest, M. Rex, J. E. Ruhl, B. R. Saliwanchik,A. Saro, J. T. Sayre, K. K. Schaffer, L. Shaw, E. Shi-rokoff, R. Simcoe, J. Song, H. G. Spieler, B. Stalder,Z. Staniszewski, A. A. Stark, K. Story, C. W. Stubbs,R. ˇSuhada, A. van Engelen, K. Vanderlinde, J. D. Vieira,A. Vikhlinin, R. Williamson, O. Zahn, and A. Zen-teno, Nature (London) , 349 (2012), arXiv:1208.2962[astro-ph.CO].[27] J. E. Gunn and J. R. Gott, III, Astrophys. J. , 1(1972).[28] J. O˜norbe, M. Boylan-Kolchin, J. S. Bullock, P. F. Hop-kins, D. Kerˇes, C.-A. Faucher-Gigu`ere, E. Quataert, andN. Murray, ArXiv e-prints (2015), arXiv:1502.02036.[29] Y. Zu and R. Mandelbaum, MNRAS , 1161 (2015),arXiv:1505.02781.[30] L. E. Strigari, J. S. Bullock, M. Kaplinghat, J. D. Si-mon, M. Geha, B. Willman, and M. G. Walker, Nature(London)454