Dissipative Dark Matter on FIRE: I. Structural and kinematic properties of dwarf galaxies
Xuejian Shen, Philip F. Hopkins, Lina Necib, Fangzhou Jiang, Michael Boylan-Kolchin, Andrew Wetzel
MMNRAS , 1–25 (2021) Preprint 22 February 2021 Compiled using MNRAS L A TEX style file v3.0
Dissipative Dark Matter on FIRE: I. Structural and kinematic propertiesof dwarf galaxies
Xuejian Shen ★ , Philip F. Hopkins , Lina Necib , , , Fangzhou Jiang , , Michael Boylan-Kolchin , AndrewWetzel TAPIR, California Institute of Technology, Pasadena, CA 91125, USA Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA Center for Cosmology, Department of Physics and Astronomy, University of California, Irvine, CA 92697, USA Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA Department of Astronomy, The University of Texas at Austin, 2515 Speedway Stop C1400, Austin, TX 78712, USA Department of Physics & Astronomy, University of California, Davis, CA 95616, USA
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present the first set of cosmological baryonic zoom-in simulations of galaxies including dissipative self-interacting darkmatter (dSIDM). These simulations utilize the Feedback In Realistic Environments (FIRE-2) galaxy formation physics, but allowthe dark matter to have dissipative self-interactions analogous to Standard Model forces, parameterized by the self-interactioncross-section per unit mass, ( 𝜎 / 𝑚 ) , and the dimensionless degree of dissipation, 0 < 𝑓 diss <
1. We survey this parameter space,including constant and velocity-dependent cross-sections, and focus on structural and kinematic properties of dwarf galaxieswith 𝑀 halo ∼ − M (cid:12) and 𝑀 ∗ ∼ − M (cid:12) . Central density profiles (parameterized as 𝜌 ∝ 𝑟 𝛼 ) of simulated dwarfs becomecuspy when ( 𝜎 / 𝑚 ) eff (cid:38) . g − (and 𝑓 diss = . 𝛼 ≈ − . ( 𝜎 / 𝑚 ) eff (cid:28) . g − ,baryonic effects can produce cored density profiles comparable to non-dissipative cold dark matter (CDM) runs but at smallerradii. Simulated galaxies with ( 𝜎 / 𝑚 ) (cid:38)
10 cm g − and the fiducial 𝑓 diss develop significant coherent rotation of dark matter,accompanied by halo deformation, but this is unlike the well-defined thin “dark disks” often attributed to baryon-like dSIDM.The density profiles in this high cross-section model exhibit lower normalizations given the onset of halo deformation. For oursurveyed dSIDM parameters, halo masses and galaxy stellar masses do not show appreciable difference from CDM, but darkmatter kinematics and halo concentrations/shapes can differ. Key words: methods : numerical – galaxies : dwarf – galaxies : haloes – cosmology : dark matter – cosmology : theory
Despite its veiled nature, dark matter is considered the main driverof structure formation in the Universe. The current paradigm — thecosmological constant plus cold dark matter ( Λ CDM) cosmologicalmodel — has been successful in describing the large scale struc-tures in the Universe (Blumenthal et al. 1984; Davis et al. 1985).This model assumes that dark matter is non-relativistic and is effec-tively collisionless, apart from its gravitational interactions with itselfand Standard Model particles. However, in recent decades, evidencefrom astrophysical observations and absence of signal from particlephysics experiments have motivated conjectures on alternative darkmatter models. On the astrophysics side, the Λ CDM model facessignificant challenges in matching observations at small scales (seea recent review Bullock & Boylan-Kolchin 2017). For example, the ★ Contact e-mail: [email protected] core-cusp problem states that the central profiles of dark matter domi-nated systems, e.g. dwarf spheroidal galaxies (dSphs) and low surfacebrightness galaxies (LSBs), are cored (e.g., Flores & Primack 1994;Moore 1994; de Blok et al. 2001; Kuzio de Naray et al. 2006; Gentileet al. 2004; Simon et al. 2005; Spano et al. 2008; Kuzio de Naray &Kaufmann 2011; Kuzio de Naray & Spekkens 2011; Oh et al. 2011;Walker & Peñarrubia 2011; Oh et al. 2015; Chan et al. 2015; Zhuet al. 2016), in contrast to the universal cuspy central density profilefound in dark matter only (DMO) simulations (Navarro et al. 1996a,1997; Moore et al. 1999; Klypin et al. 2001; Navarro et al. 2004; Die-mand et al. 2005). The too-big-to-fail (TBTF) problem states that asubstantial population of massive concentrated subhaloes appears inDMO simulations, which is incompatible with the stellar kinematicsof observed satellite galaxies around the Milky Way or M31 (Boylan-Kolchin et al. 2011; Boylan-Kolchin et al. 2012; Tollerud et al. 2014).This mismatch has been extended to field dwarf galaxies in the Lo-cal Group (Garrison-Kimmel et al. 2014; Kirby et al. 2014) and © a r X i v : . [ a s t r o - ph . GA ] F e b Shen et al. beyond (Papastergis et al. 2015). Although the inclusion of burstystar formation and feedback processes has been shown to allevi-ate the tensions (e.g., Governato et al. 2010; Pontzen & Governato2012; Madau et al. 2014; Brooks & Zolotov 2014; Wetzel et al.2016; Sawala et al. 2016; Garrison-Kimmel et al. 2019a), a popula-tion of compact dwarf galaxies in the local Universe are missing incosmological simulations of CDM (plus baryons) that can producedark matter cores (e.g., Santos-Santos et al. 2018; Jiang et al. 2019;Garrison-Kimmel et al. 2019a). Relate to this, the rotation curves ofdwarf galaxies appear to be more diverse than CDM predictions inthe field (Oman et al. 2015) and Milky Way satellites (Kaplinghatet al. 2019). Therefore, it is important to explore how non-standarddark matter models – in conjunction with baryonic physics – couldhelp solve the small-scale anomalies. On the particle physics side,one of the most popular candidates for CDM (the class of WeaklyInteracting Massive Particles, WIMPs) has not been discovered de-spite decades of efforts and a significant proportion of its parameterspace being ruled out (e.g., Bertone et al. 2005; Bertone 2010; Aprileet al. 2018). The null results in collider production and direct/indirectdetection experiments of classical CDM candidates have motivatedideas about alternative dark matter models (e.g., Hogan & Dalcanton2000; Spergel & Steinhardt 2000; Dalcanton & Hogan 2001; Buck-ley & Peter 2018) and explorations of the rich phenomenology frompotential non-gravitational dark matter interactions. Many of thesealternative dark matter models could behave dramatically differentlyfrom CDM at astrophysical scales and could potentially solve thesmall-scale problems mentioned above.Self-interacting dark matter (SIDM) is an important category ofalternative dark matter models that has been proposed and discussedin the literature for about three decades (e.g., Carlson et al. 1992; deLaix et al. 1995; Firmani et al. 2000; Spergel & Steinhardt 2000). It iswell motivated by hidden dark sectors as extensions to the StandardModel (e.g., Ackerman et al. 2009; Arkani-Hamed et al. 2009; Fenget al. 2009, 2010; Loeb & Weiner 2011; van den Aarssen et al. 2012;Cyr-Racine & Sigurdson 2013; Tulin et al. 2013; Cline et al. 2014).The introduction of SIDM could potentially solve some small-scaleproblems (see the review of Tulin & Yu 2018, and references therein).Dark matter self-interactions enable effective heat conduction andcould result in an isothermal distribution of dark matter with cores athalo centers, which alleviates the core-cusp problem. Meanwhile, itcould also make dark matter haloes (subhaloes) less dense and alle-viate the TBTF problem. Previous DMO simulations have found thata self-interaction cross-section of ∼ g − could solve the core-cusp and TBTF problems in dwarf galaxies simultaneously (e.g.,Vogelsberger et al. 2012; Rocha et al. 2013; Zavala et al. 2013; El-bert et al. 2015). In addition, SIDM with comparble cross-sectionsalso have the potential to explain (e.g., Kamada et al. 2017; Creaseyet al. 2017; Sameie et al. 2020) the diversity of rotation curves ofdwarf galaxies (Oman et al. 2015; Kaplinghat et al. 2019). Follow-ing studies of galaxy clusters in SIDM suggested a cross-sectionof ∼ . g − (e.g., Kaplinghat et al. 2016; Elbert et al. 2018),which motivates the velocity-dependence of self-interaction cross-section. These previous studies on SIDM focused on elastic darkmatter self-interactions. However, in many particle physics realiza-tions of SIDM, dark matter have inelastic (or specifically dissipative)self-interactions (e.g., Kaplan et al. 2010; Cyr-Racine & Sigurdson2013; Cline et al. 2014; Boddy et al. 2014; Foot & Vagnozzi 2015;Schutz & Slatyer 2015; Boddy et al. 2016; Finkbeiner & Weiner2016; Zhang 2017; Blennow et al. 2017; Gresham et al. 2018). Theimpact of dissipative processes of dark matter has not yet been ex-plored in the context of cosmological structure formation.In addition, the focus on purely elastic SIDM (eSIDM) in previ- ous studies has been motivated by solving some small-scale prob-lems (making galaxy centers less dense). Since dissipative darkmatter self-interactions tend to make centers of haloes denser tofirst order consideration, dSIDM was largely omitted in previousstudies of SIDM. However, apart from dark matter physics, somebaryonic physics processes, including bursty star formation and stel-lar/supernovae feedback and tidal disruption, have also been shownto strongly impact the structure of dark matter haloes and help allevi-ate some small-scale problems. Specifically, gas outflows driven bystellar/supernovae feedback could create fluctuations in the centralpotential, which irreversibly transfer energy to CDM particles andgenerate dark matter cores (Governato et al. 2010, 2012; Pontzen& Governato 2012; Madau et al. 2014). Some more recent CDMsimulations could resolve the small-scale problems by more realis-tic modeling of gas cooling, star formation and stellar/supernovaefeedback (e.g., Brooks & Zolotov 2014; Dutton et al. 2016; Fattahiet al. 2016; Sawala et al. 2016; Wetzel et al. 2016; Garrison-Kimmelet al. 2019a; Buck et al. 2019). The interplay between baryons andSIDM in galaxy formation has been more carefully considered insubsequent SIDM simulations that include baryonic physics (e.g.,Vogelsberger et al. 2014; Elbert et al. 2015; Robles et al. 2017;Despali et al. 2019; Fitts et al. 2019; Robles et al. 2019). The inclu-sion of baryons substantially reduces the distinct signatures in dwarfgalaxies caused by elastic dark matter self-interactions, especially inbright dwarfs with 𝑟 / (cid:38)
400 pc (Fitts et al. 2019). This could hidedark matter physics that lead to enhanced central density originally,other than those proposed specifically to lower the central density.The parameter space for dSIDM, as an example of such models,reopens due to these recent developments. The contraction of thehalo driven by dSIDM interactions could help produce the compactdwarf galaxies found in the local Universe that are missing in CDMsimulations plus baryons (e.g., Santos-Santos et al. 2018; Jiang et al.2019; Garrison-Kimmel et al. 2019a) and increase the diversity ofdwarf galaxy rotation curves.A finite self-gravitating system has negative heat capacity and theheat conduction will eventually result in the “gravothermal catastro-phe” of the system (e.g., Lynden-Bell & Wood 1968; Lynden-Bell& Eggleton 1980). In the eSIDM case, effective heat conductionis realized by dark matter self-interactions and the inner cores ofisolated eSIDM haloes will ultimately experience gravothermal col-lapse and cuspy density profiles will reappear (e.g., Burkert 2000;Kochanek & White 2000; Balberg et al. 2002; Colín et al. 2002;Koda & Shapiro 2011; Vogelsberger et al. 2012; Elbert et al. 2015).However, for the most favored elastic self-interaction cross-sections ∼ . g − (assuming velocity-independent), the “gravother-mal catastrophe” would not have enough time to happen in haloeswithin their typical lifetime. In the presence of dissipative self-interactions, the gravothermal evolution of a halo can be acceleratedsignificantly, which affects the structure of dwarf galaxies within aHubble time. Essig et al. (2019) recently used an semi-analytical fluidmodel to investigate the structure of isolated spherically symmetrichaloes in dissipative SIDM (dSIDM) and presented the first con-straint on the energy loss and cross-section of dSIDM. This work wasfollowed by Huo et al. (2019) with non-cosmological N-body simu-lations of isolated dark matter haloes with the NFW profile (Navarroet al. 1996b) initially. Moreover, when dissipation of dark matterself-interaction is strong enough, a patch of dark matter could loseits kinetic energy faster than rebuilding hydrostatic equilibrium withsurrounding matter. Substructures of dissipative dark matter, e.g.dark disks and dark stars, could be generated under this circum-stance. For example, dark matter scenarios with a highly dissipativecomponent (sourced by an 𝑈 ( ) -like hidden sector) have been stud- MNRAS , 1–25 (2021) issipative Dark Matter on FIRE Simulation 𝑀 cdmhalo 𝑅 cdmvir 𝑀 cdm ∗ 𝑟 cdm1 / 𝑟 convdm 𝜎 𝜎 𝜎 𝜎 . 𝜎 ( 𝑣 ) Notesname [ M (cid:12) ] [ kpc ] [ M (cid:12) ] [ kpc ] [ pc ] elastic 𝑓 diss . 𝑓 diss . 𝑓 diss . 𝑓 diss . Ultra faint dwarf m09 2.5e9 35.6 7.0e4 0.46 65 other parameter choices explored
Classical dwarfs m10b 9.4e9 55.2 5.8e5 0.36 77 late-formingm10q 7.5e9 51.1 1.7e6 0.72 73 isolated, early-formingm10v 8.5e9 53.5 1.4e5 0.32 65 isolated, late-forming
Bright dwarfs m11a 3.6e10 86.7 3.7e7 1.2 310 diffuse, coredm11b 4.2e10 90.7 4.2e7 1.7 250 intermediate-formingm11q 1.5e11 138.7 2.9e8 3.1 120 early-forming, cored
Milky Way-mass galaxies m11f 4.5e11 200.2 1.0e10 2.9 280 quiescent late historym12i l.r. 1.1e12 272.3 1.1e11 2.0 290 Milky Way likem12f l.r. 1.5e12 302.8 1.3e11 4.1 310 Milky Way likem12m l.r. 1.5e12 299.3 1.4e11 6.1 360 early-forming, boxy bulge
Table 1. Simulations of the FIRE-2 dSIDM suite.
The simulated galaxies are labelled and grouped by their halo masses. They are classified into fourcategories: ultra faint dwarfs; classical dwarfs, with typical halo mass (cid:46) M (cid:12) ; bright dwarfs, with typical halo mass ∼ − M (cid:12) ; Milky Way-massgalaxies, with typical halo mass ∼ M (cid:12) . These haloes are randomly picked from the standard FIRE-2 simulation suite (Hopkins et al. 2018), samplingvarious star formation and merger histories. All units are physical.( ) Name of the simulation. “l.r.” indicates low-resolution version of the simulation.( ) 𝑀 cdmhalo : Virial mass of the halo (definition given in Section 4.2) in the CDM simulation with baryons at 𝑧 = ) 𝑅 cdmvir : Virial radius of the halo (definition given in Section 4.2) in the CDM simulation with baryons at 𝑧 = ) 𝑀 cdm ∗ : Galaxy Stellar mass (see Section 4.2) in the CDM simulation at 𝑧 = ) 𝑟 cdm1 / : Galaxy stellar half mass radius (see Section 4.2) in the CDM simulation at 𝑧 = ) 𝑟 convdm : Radius of convergence in dark matter properties at 𝑧 = ) Parameters of the dark matter models. 𝜎 (with the number after it) indicates the self-interaction cross-section, 𝜎 / 𝑚 , in unit of cm g − . 𝜎 ( 𝑣 ) denotesthe velocity-dependent cross-section, introduced in Section 2. 𝑓 diss indicates the dimensionless degree of dissipation.( ) Notes: Additional information of each simulation. ied by Fan et al. (2013a,b, 2014); Randall & Scholtz (2015); Foot(2013); Foot & Vagnozzi (2015, 2016). Randall & Scholtz (2015)claimed that a dark disk composed of highly dissipative dark mattercould appear and help explain the exotic mass-to-light ratios of someMilky Way satellites. However, the analytical or semi-analytical stud-ies discussed above were limited to isolated DMO haloes with variousgeometrical simplifications. The influences of baryonic physics, hier-archical halo mergers, deviations from simple fluid approximationsin dark matter haloes were not properly captured in these previ-ous studies. In addition, multi-component dark matter with inelas-tic interactions have been considered in simulations in Todoroki &Medvedev (2019); Vogelsberger et al. (2019), but the dominant pro-cess is exothermic in these studies.In this paper, we perform the first study of dSIDM models us-ing cosmological baryonic (hydrodynamical) zoom-in simulations ofgalaxies. We aim at studying the evolution tracks of dSIDM haloesand looking for properties of dSIDM haloes that distinguish themfrom their CDM counterparts. These simulations have incorporatedthe FIRE-2 model (Hopkins et al. 2018) for hydrodynamics andgalaxy formation physics that could produce galaxies consistent withvarious local and high redshift observables in collisionless CDMsimulations (e.g., Ma et al. 2018; Garrison-Kimmel et al. 2018;Hafen et al. 2019; Garrison-Kimmel et al. 2019a). The setup also en-ables predictions in the regime where hierarchical mergers and strongnon-linear gravitational effects could drive systems away from theidealized analytical solutions. All these factors allow more robust constraints on dSIDM models. The paper is arranged as follows: InSection 2, we discuss the details of the simulations and briefly intro-duce the dSIDM models we study. We derive relevant time scales fordSIDM haloes analytically in Section 3 and study the stellar massesand host halo masses of simulated dwarf galaxies in Section 4.2. Thenwe present the mass density profiles of simulated dwarf galaxies andquantitatively study the impact of dissipation on galaxy structure inSection 4.3. We study the kinematic properties of dark matter andthe shapes of haloes in simulations in Section 4.4 and Section 5.Subsequently, in Section 6, we use analytical methods to explain thephenomena in dSIDM simulations and summarize the evolution pat-tern of dSIDM haloes in different regimes. In Section 7, we explorethe results of simulations with other choices of 𝑓 diss as well as theDMO simulations and compare their differences from the fiducialsimulations. The summary and conclusion of the paper are presentedin Section 8. We present the new FIRE-2 dSIDM simulation suite, which consistsof ∼
40 cosmological hydrodynamical zoom-in simulations of galax-ies chosen at representative mass scales with CDM, eSIDM anddSIDM models. The simulations here are part of the Feedback In
MNRAS000
MNRAS000 , 1–25 (2021)
Shen et al. bright dwarfsclassical dwarfs σ [ km / s ] -4 -3 -2 -1 t X / t H ρ = 2 × M fl / kpc f diss = 0 . dSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM ( / g ) t dyn t relax t diss σ [ km / s ] -4 -3 -2 -1 t X / t H ρ = 2 × M fl / kpc f diss = 0 . − . dSIDM ( v dep . ) dSIDM ( / g ) Figure 1.
Top : Relevant time scales of the physical processes involvedin dSIDM haloes versus one-dimensional velocity dispersion of thesystem.
We have assumed that the local dark matter density is 𝜌 dm = × M (cid:12) / kpc , a typical value at dwarf galaxy centers. We show thecollision time scale ( 𝑡 coll ) and dissipation time scales ( 𝑡 diss ) of all the dSIDMmodels studied in this paper as well as the dynamical time scale ( 𝑡 dyn ). All thetime scales are normalized by the Hubble time scale at 𝑧 = 𝑡 H ≡ / H ).The dissipation time scales are calculated assuming 𝑓 diss = .
5. The shadedregions show the typical one-dimensional velocity dispersions in the classical(e.g. Milky Way satellites) and bright dwarf galaxies (e.g. LSB galaxies). Indwarf galaxies, dissipation and collision time scales are much larger than thedynamical time scale, but can become considerably shorter than the Hubbletime scale. The velocity-dependent model becomes less dissipative ( 𝑡 diss / 𝑡 H becomes larger) in more massive galaxies (with larger velocity dispersion)while models with constant cross-sections become more dissipative. Bottom : Dissipation time scales versus one-dimensional velocity dispersion of thesystem with 𝑓 diss varying from . to . . The symbols are the same as thetop panel. For each model, the upper boundary of the shaded region corre-sponds to the case 𝑓 diss = . 𝑓 diss = . Realistic Environments project (FIRE, Hopkins et al. 2014), specif-ically the “FIRE-2” version of the code with details described inHopkins et al. (2018). The simulations adopt the code Gizmo (Hop-kins 2015), with hydrodynamics solved using the mesh-free La-grangian Godunov “MFM” method. The simulations include heatingand cooling from a meta-galactic radiation background and stellarsources in the galaxies, star formation in self-gravitating molecular,Jeans-unstable gas and stellar/supernovae/radiation feedback. TheFIRE physics, source code, and numerical parameters are identi-cal to those described in Hopkins et al. (2018); Garrison-Kimmel et al. (2019b). For dwarf galaxies, the baryonic particle masses ofsimulations are 𝑚 b (cid:39)
250 - 2000 M (cid:12) . For Milky Way-mass galax-ies, the high-resolution ‘latte’ runs have 𝑚 b = (cid:12) while thelow-resolution runs have 𝑚 b = (cid:12) . In all simulations, thedark matter particle masses are roughly five times larger, accordingto the universal baryon fraction. For dwarf galaxies, the minimumgravitational force softening length reached by gas in the simula-tions is ℎ b (cid:39) . ℎ b (cid:39) . . . 𝜖 dm =
40 pc (30 pc). Force softeningfor gas uses the fully conservative adaptive algorithm from Price& Monaghan (2007), meaning that the gravitational force assumesthe identical mass distribution as the hydrodynamic equations (re-sulting in identical hydrodynamic and gravitational resolution). Thesimulations are identified with the main “target” halo around whichthe high-resolution zoom-in region is centered. In post-processing,we identify subhaloes (of the main “target” halo) with the Rock-star (Behroozi et al. 2013a) halo finder and create merger trees ofhaloes (subhaloes) with the code Consistent Trees (Behroozi et al.2012, 2013b). As shown in Table 1, the simulation suite consists ofone ultra faint dwarf ( m09 ), three classical dwarf galaxies ( m10q,m10b, m10v ), three bright dwarf galaxies ( m11a, m11b, m11q )and four Milky Way-mass galaxies ( m11f, m12i, m12f, m12m ). Theanalysis in this paper will primarily focus on the classical and brightdwarf galaxies and we defer analysis on Milky Way-mass galaxies toa follow-up work.
Dark matter self-interactions are simulated in a Monte-Carlo fashionfollowing the implementation in Rocha et al. (2013). In this paper, westudy a simplified empirical dSIDM model: two dark matter particleslose a constant fraction, 𝑓 diss , of their kinetic energy in the center ofmomentum frame when they collide with each other. This empiricalmodel is motivated by particle physics models of composite stronglyinteracting dark matter in a hidden non-Abelian sector (e.g., Clineet al. 2014; Boddy et al. 2014) or asymmetric dark matter boundstates (e.g., Wise & Zhang 2014; Gresham et al. 2018), the kineticenergy of which can be absorbed by internal degrees of freedomduring self-interactions. In some models, the energy could be even-tually carried away by some low mass or massless dark carriers.For each galaxy, we run simulations with a default dissipation frac-tion 𝑓 diss = . and with constant self-interaction cross-sections ( 𝜎 / 𝑚 ) = . / /
10 cm g − or a velocity-dependent cross-sectionmodel: 𝜎 ( 𝑣 ) 𝑚 = ( 𝜎 / 𝑚 ) + ( 𝑣 / 𝑣 ) , (1)where the fiducial choice of parameters is ( 𝜎 / 𝑚 ) =
10 cm g − and 𝑣 =
10 km s − . The velocity dependence of the self-interactioncross-section is empirically motivated by the relatively tight con-straints on SIDM at galaxy cluster scale (e.g., Markevitch et al.2004; Randall et al. 2008; Kaplinghat et al. 2016) and the relativelyhigh cross-section needed to solve some small-scale problems (e.g.,Vogelsberger et al. 2012; Rocha et al. 2013; Zavala et al. 2013; Elbertet al. 2015; Kaplinghat et al. 2016). Meanwhile, the velocity depen-dence is a generic feature of many particle physics realizations ofdark matter. The asymptotic ( 𝑣 / 𝑣 ) − velocity dependence we adopt Other choices of 𝑓 diss are explored with m09 in Section 7.1.MNRAS , 1–25 (2021) issipative Dark Matter on FIRE is motivated by particle physics models featuring dark matter self-interactions mediated by light gauge bosons (e.g., Feng et al. 2009;Kaplan et al. 2010; Cyr-Racine & Sigurdson 2013; Boddy et al. 2016;Zhang 2017). The sharp decline in cross-section could also appearin some models of strongly interacting composites. In these mod-els, when the de Broglie wavelength of the particle become smallerthan the characteristic length scale of the interaction, ∼ / Λ dm , theself-interaction cross-section is expected to drop significantly (e.g.,Boddy et al. 2014; Cline et al. 2014; Tulin & Yu 2018). In this section, we derive analytical formulae for relevant time scalesin dSIDM haloes, including the dynamical time scale, the collisiontime scale and the dissipation time scale. These analytical formulaecan be used to understand the influence of dissipation on galaxy struc-tures in different circumstances. We will present results for modelswith constant and velocity-dependent cross-sections, respectively.
The local dynamical time scale in a system is defined as 𝑡 dyn ≡ √︄ 𝜋𝐺 𝜌 = . (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − / , (2)where 𝐺 is the gravitational constant and 𝜌 is the local matter density.At the centers of dwarf galaxies, the mass density is dominated bydark matter, so 𝜌 is simply the local dark matter mass density. The collision time scale of dark matter self-interaction is 𝑡 coll ≡ (cid:104) 𝜌𝑣 rel 𝜎𝑚 (cid:105) , (3)where 𝜌 is local dark matter mass density, 𝑣 rel is the relative veloc-ity between dark matter particles and (cid:104) ... (cid:105) denotes the average overall possible encounters. This measures the time scale that one darkmatter particle is expected to have one self-interaction with any otherdark matter particles. For simplicity, we assume that the velocitiesof dark matter particles locally obey the Maxwell-Boltzmann distri-bution. Therefore, the average can be treated as a thermal average (cid:104) 𝑋 (cid:105) = √ 𝜋𝜎 ∫ ∞ d 𝑣 rel 𝑣 𝑒 − 𝑣 / 𝜎 𝑋, (4) where 𝜎 is the local one-dimensional velocity dispersion of darkmatter. After taking the thermal average, the collision time scale is 𝑡 coll = .
206 Gyr (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 ) g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − [ constant cross-section ] ; 𝑡 coll = .
661 Gyr (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 )
10 cm g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − (cid:16) 𝜎 𝑣 (cid:17) (cid:34) − (cid:16) 𝑣 𝜎 (cid:17) cos (cid:16) 𝑣 𝜎 (cid:17) + sin (cid:16) 𝑣 𝜎 (cid:17) (cid:16) 𝜋 − (cid:16) 𝑣 𝜎 (cid:17)(cid:17)(cid:35) − (cid:39) .
165 Gyr (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 )
10 cm g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − (cid:16) 𝜎 𝑣 (cid:17) ln (cid:16) 𝜎 𝑣 (cid:17) − [ 𝜎 (cid:29) 𝑣 ][ velocity-dependent cross-section ] , (5)where Si ( 𝑥 ) = ∫ 𝑥 d 𝑡 sin ( 𝑡 )/ 𝑡 and Ci ( 𝑥 ) = − ∫ ∞ 𝑥 d 𝑡 cos ( 𝑡 )/ 𝑡 are sineand cosine integrals, ( 𝜎 / 𝑚 ) and 𝑣 are parameters of the velocity-dependent cross-section. For our fiducial choice of 𝑣 =
10 km s − ,galaxies of masses (cid:38) M (cid:12) (massive dwarfs/Milky Way-massgalaxies) will have velocity dispersions in the limit 𝜎 (cid:29) 𝑣 . Wecan see that the collision time scale of the velocity-dependent modelis usually much larger than the constant cross-section model after thethermal average. This is due to the velocity suppression of collisionsbetween particles with high relative velocities, which contribute moreto the total interaction rate. In addition, the collision time scale indifferent models scales with velocity dispersion in opposite ways.For the models with constant cross-sections, the collision time scaleis shorter in systems with higher densities or higher velocity disper-sions, which indicates that self-interaction has stronger impact inmore massive systems . On the other hand, for the velocity-dependent model, the collision time scale sharply increases in systems withhigher velocity dispersions, which indicates that self-interaction hasweaker impact in more massive systems . The dissipation time scale here is defined as the time scale for an orderunity fraction of local dark matter kinetic energy to be dissipatedaway through dark matter self-interactions 𝑡 diss ≡ 𝜌𝜎 / 𝐶, (6)where 𝜎 is the one-dimensional velocity dispersion and 𝐶 is theeffective cooling rate defined as 𝐶 ≡ (cid:68) 𝑛 ( 𝜌𝑣 rel 𝜎𝑚 ) 𝐸 loss (cid:69) = (cid:68) 𝜌 𝜎𝑚 𝑣 rel 𝐸 loss 𝑚 (cid:69) , (7)where 𝑛 is the local number density of dark matter particles, 𝐸 loss is the kinetic energy loss per collision in the center of momentumframe and (cid:104) ... (cid:105) again denotes the thermal average. For the fractionaldissipation model we study in this paper, 𝐸 loss / 𝑚 = ( / ) 𝑓 diss 𝑣 .The dissipation time scale measures how fast the kinetic energy isdissipated away from the system and, after order one dissipation timescale, the local dark matter structure is expected to be dramaticallyaffected. MNRAS000
10 km s − ,galaxies of masses (cid:38) M (cid:12) (massive dwarfs/Milky Way-massgalaxies) will have velocity dispersions in the limit 𝜎 (cid:29) 𝑣 . Wecan see that the collision time scale of the velocity-dependent modelis usually much larger than the constant cross-section model after thethermal average. This is due to the velocity suppression of collisionsbetween particles with high relative velocities, which contribute moreto the total interaction rate. In addition, the collision time scale indifferent models scales with velocity dispersion in opposite ways.For the models with constant cross-sections, the collision time scaleis shorter in systems with higher densities or higher velocity disper-sions, which indicates that self-interaction has stronger impact inmore massive systems . On the other hand, for the velocity-dependent model, the collision time scale sharply increases in systems withhigher velocity dispersions, which indicates that self-interaction hasweaker impact in more massive systems . The dissipation time scale here is defined as the time scale for an orderunity fraction of local dark matter kinetic energy to be dissipatedaway through dark matter self-interactions 𝑡 diss ≡ 𝜌𝜎 / 𝐶, (6)where 𝜎 is the one-dimensional velocity dispersion and 𝐶 is theeffective cooling rate defined as 𝐶 ≡ (cid:68) 𝑛 ( 𝜌𝑣 rel 𝜎𝑚 ) 𝐸 loss (cid:69) = (cid:68) 𝜌 𝜎𝑚 𝑣 rel 𝐸 loss 𝑚 (cid:69) , (7)where 𝑛 is the local number density of dark matter particles, 𝐸 loss is the kinetic energy loss per collision in the center of momentumframe and (cid:104) ... (cid:105) again denotes the thermal average. For the fractionaldissipation model we study in this paper, 𝐸 loss / 𝑚 = ( / ) 𝑓 diss 𝑣 .The dissipation time scale measures how fast the kinetic energy isdissipated away from the system and, after order one dissipation timescale, the local dark matter structure is expected to be dramaticallyaffected. MNRAS000 , 1–25 (2021)
Shen et al.
After taking the thermal average, the dissipation time scale is 𝑡 diss = 𝑓 diss 𝑡 coll = .
310 Gyr (cid:16) 𝑓 diss . (cid:17) − (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 ) g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − [ constant cross-section ] ; 𝑡 diss = .
926 Gyr (cid:16) 𝑓 diss . (cid:17) − (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 )
10 cm g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − (cid:16) 𝜎 𝑣 (cid:17) (cid:34) (cid:16) 𝜎 𝑣 (cid:17) − (cid:16) 𝑣 𝜎 (cid:17) sin (cid:16) 𝑣 𝜎 (cid:17) − cos (cid:16) 𝑣 𝜎 (cid:17) (cid:16) 𝜋 − (cid:16) 𝑣 𝜎 (cid:17)(cid:17)(cid:35) − (cid:39) .
991 Gyr (cid:16) 𝑓 diss . (cid:17) − (cid:16) 𝜌 M (cid:12) / kpc (cid:17) − (cid:16) ( 𝜎 / 𝑚 )
10 cm g − (cid:17) − (cid:16) 𝜎
10 km s − (cid:17) − (cid:16) 𝜎 𝑣 (cid:17) , [ 𝜎 (cid:29) 𝑣 ][ velocity dependent model ] . (8)In the model with a constant cross-section, the dissipation time scalehas the same scaling behaviour as the collision time scale defined inEquation 5 and differs only by a factor of 0 . / 𝑓 diss . In the velocity-dependent model, the scaling behaviours of the dissipation and col-lision time scales are also quite similar when 𝜎 (cid:29) 𝑣 . The dissi-pation time scale of the velocity-dependent model is usually muchlarger than the constant cross-section model after thermal average.This again can be attributed to the velocity suppression of collisionsbetween particles with high relative velocities, which not only con-tribute more to the total collision rate but also induce higher energyloss per collision. Similar to what has been found for the collisiontime scale, dissipation is more significant in more massive systems in the models with constant cross-sections. Dissipation, however, is less significant in more massive systems in the velocity-dependent model.In Figure 1, we show the relevant time scales discussed aboveas a function of the one-dimensional velocity dispersion of the sys-tem; in particular, we show the collision and dissipation time scalesof the dSIDM models studied in this paper as well as the dynami-cal time scale, assuming that the local dark matter mass density is 𝜌 = × M (cid:12) / kpc , which is a typical value at dwarf galaxycenters. The time scales are all normalized by the Hubble time scaleat 𝑧 =
0, roughly representing the lifetime of the system. In the toppanel, the dissipation time scales are calculated assuming 𝑓 diss = . 𝑡 diss with 𝑓 diss = . .
9. With the vertical shaded regions inboth panels, we show the typical ranges of one-dimensional velocitydispersions of the classical (e.g., Milky Way satellites) and brightdwarf galaxies (e.g., LSB galaxies). For the dSIDM models withconstant cross-sections, the collision time scales are always propor-tional to the dissipation time scales and, they are order of magnitudecomparable to each other. Both of them are shorter than the Hubbletime scale but larger than the dynamical time scale in dwarf galaxies.The dissipation time scale decreases in systems with higher veloc-ity dispersions, so we expect these constant cross-section models tobecome more dissipative in more massive dwarfs. For the velocity-dependent dSIDM model, the collision and dissipation time scales are no longer proportional to each other, and they both increase asthe velocity dispersion increases, opposite to the behaviour of mod-els with constant cross-sections. The dissipation time scale of thevelocity-dependent model is comparable to the Hubble time scalein the classical dwarfs but becomes at least an order of magnitudelarger than the Hubble time scale in the bright dwarfs, suggestingnegligible effects of dissipation in this case.
The cooling induced by dissipative dark matter self-interactions canbe compared to the cooling of baryons, which is usually described bythe cooling function Λ . For dSIDM, the effective cooling function is Λ eff ∼ 𝑇𝑛𝑡 diss ∼ ( 𝜎 / 𝑚 ) 𝑓 diss 𝜎 ∼ (cid:40) 𝜎 ∼ 𝑇 / [ constant cross-section ] 𝜎 − ∼ 𝑇 − / [ velocity dependent model ] (9)where 𝑇 is 𝑚𝜎 / 𝑘 B for weakly collisional dark matter. The coolingfunction in the constant cross-section model is similar to the coolingcurve of gas below ∼ K while the cooling function in the velocitydependent model is similar to the 10 − K gas cooling curve.Other behaviours are possible if a velocity-dependence of 𝑓 diss isintroduced, e.g. Λ eff would be a constant if 𝑓 diss ∼ 𝑇 / with thesame velocity-dependent cross-section. However, the most importantqualitative difference between the dSIDM studied here and baryonsis not the behaviour of the cooling curve but the fact that baryons(gas) are effectively in the 𝑓 diss → ( 𝜎 / 𝑚 ) → ∞ regime.The effective interaction cross-section of gas is enormous comparedto favored SIDM interaction cross-sections and the energy loss per“collision” is small. Gas cooling is the result of a large amount ofparticle interactions in a locally thermalized region. On contrary,dSIDM with 𝑡 coll order of magnitude comparable to 𝑡 diss cannotachieve local thermalization effectively when cools down. It is useful to define an "effective cross-section" for the velocity-dependent dSIDM model (cid:16) 𝜎𝑚 (cid:17) eff = (cid:68) 𝜎𝑚 𝑣 rel (cid:69) /(cid:104) 𝑣 rel (cid:105) , (10)where 𝑣 rel is the relative velocity between encountering particles and (cid:104) ... (cid:105) is a thermal average as discussed in Section 3.2. This definitionensures that a dSIDM model with a constant cross-section taking thevalue of this "effective cross-section" will result in the identical rateof dark matter self-interaction, assuming that dark matter particles arein thermal equilibrium. This definition allows a proper comparisonbetween velocity-dependent and independent SIDM models. UsingEquation 4, we find (cid:16) 𝜎𝑚 (cid:17) eff = ( 𝜎 / 𝑚 ) (cid:16) 𝑣 𝜎 (cid:17) (cid:34) − (cid:16) 𝑣 𝜎 (cid:17) cos (cid:16) 𝑣 𝜎 (cid:17) + sin (cid:16) 𝑣 𝜎 (cid:17) (cid:16) 𝜋 − (cid:16) 𝑣 𝜎 (cid:17)(cid:17)(cid:35) , (11)where the notation is the same as Equation 5. The asymptotic be-haviour of ( 𝜎 / 𝑚 ) eff is dominated by the 𝜎 − term, which is similarto the velocity-dependent cross-section defined in Equation 1. Thefactor 32 in the denominator comes from the thermal average andindicates that dSIDM models with velocity-dependent cross-section MNRAS , 1–25 (2021) issipative Dark Matter on FIRE Figure 2. Visualizations of four dark matter haloes in simulations with CDM versus dSIDM.
The images are dark matter surface density maps, projectedalong the z-direction of simulation coordinates, at 𝑧 = . × 𝑅 vir of the CDM run. In the first row, we show the haloes in the CDM. In the second row,we show the haloes in the velocity-dependent dSIDM model. In the third row, we show haloes in the dSIDM model with constant cross-section 1 cm g − . Thehaloes are ordered from left to right by their virial masses. In each image, the outer dotted circle indicates the radius 𝑅 (the density enclosed is 500 timesthe critical density at 𝑧 =
0) which represents the overall size of the halo. The inner dashed circle indicates the radius 𝑅 core ≡ × 𝑅 . (the mass enclosedin a sphere of radius 𝑅 . is 0 .
1% the virial mass of the halo) which represents the core size of the halo. Comparing the core sizes, the haloes in the dSIDMmodel are visibly more concentrated than their CDM counterparts. For the velocity-dependent dSIDM model, since the self-interaction cross-section decreasesin more massive haloes, the increased concentration of halo is less apparent in more massive haloes. For the dSIDM with constant cross-section, haloes of allmasses are consistently more concentrated than their CDM counterparts. are not as efficient as those with constant cross-sections, owing againto the velocity suppression.
In this section, we present the structural and kinematic properties ofsimulated dwarf galaxies in different dark matter models and studythe impact of dissipation on galaxy structures.
In Figure 2, we show images of four dark matter haloes in our sim-ulation suite at 𝑧 =
0. Each image is a two-dimensional surfacedensity map of dark matter, projected along the z-direction of simu-lation coordinates, with a logarithmic stretch. The dynamical ranges are adjusted based on the maximum and median intensities of pix-els. The haloes are ordered from left to right by their halo masses(see Section 4.2 for the definition). We show the images in CDM,the dSIDM with constant cross-section ( 𝜎 / 𝑚 ) = g − and thevelocity-dependent dSIDM model for comparison. The haloes indSIDM models are visibly more concentrated than their CDM coun-terparts when comparing their core sizes (dashed circles). For thevelocity-dependent dSIDM model, since the self-interaction cross-section decreases in more massive haloes which typically have highervelocity dispersions, the increased concentration of the halo becomesless apparent. On contrary, in dSIDM models with constant cross-sections, haloes of all masses are consistently more concentrated thantheir CDM counterparts. Meanwhile, the substructures also appearto be more abundant and concentrated in dSIDM models, but wewill focus on the main halo in this paper and defer the analysis onsubstructures to follow-up work. MNRAS000
0. Each image is a two-dimensional surfacedensity map of dark matter, projected along the z-direction of simu-lation coordinates, with a logarithmic stretch. The dynamical ranges are adjusted based on the maximum and median intensities of pix-els. The haloes are ordered from left to right by their halo masses(see Section 4.2 for the definition). We show the images in CDM,the dSIDM with constant cross-section ( 𝜎 / 𝑚 ) = g − and thevelocity-dependent dSIDM model for comparison. The haloes indSIDM models are visibly more concentrated than their CDM coun-terparts when comparing their core sizes (dashed circles). For thevelocity-dependent dSIDM model, since the self-interaction cross-section decreases in more massive haloes which typically have highervelocity dispersions, the increased concentration of the halo becomesless apparent. On contrary, in dSIDM models with constant cross-sections, haloes of all masses are consistently more concentrated thantheir CDM counterparts. Meanwhile, the substructures also appearto be more abundant and concentrated in dSIDM models, but wewill focus on the main halo in this paper and defer the analysis onsubstructures to follow-up work. MNRAS000 , 1–25 (2021)
Shen et al. d e n s i t y p r o fi l e -1 r [kpc] ρ t o t [ M fl / k p c ] m10q CDMdSIDM ( v dep . ) dSIDM ( / g ) dSIDM ( / g ) eSIDM ( / g ) -1 r [kpc] ρ t o t [ M fl / k p c ] m11a CDMdSIDM ( v dep . ) dSIDM ( / g ) dSIDM ( / g ) eSIDM ( / g ) r [kpc] ρ t o t [ M fl / k p c ] m11q CDMdSIDM ( v dep . ) dSIDM ( / g ) eSIDM ( / g ) c i rc u l a r v e l o c i t y r [kpc] V c i r c [ k m / s ] m10q 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r [kpc] V c i r c [ k m / s ] m11a 2 4 6 8 r [kpc] V c i r c [ k m / s ] m11q v e l o c i t y d i s p er s i o n r [kpc] σ d [ k m / s ] m10q 10 r [kpc] σ d [ k m / s ] m11a 10 r [kpc] σ d [ k m / s ] m11q v e l o c i t ya n i s o t r o p y r [kpc] β m10q 10 r [kpc] β m11a 10 r [kpc] β m11q r o t a t i o n v er s u s d i s p er s i o n r [kpc] -2 -1 V r o t / σ d m10q 10 r [kpc] -2 -1 V r o t / σ d m11a 10 r [kpc] -2 -1 V r o t / σ d m11q Figure 3. A gallery view of the structural and kinematic properties of dwarf galaxies in simulations.
From top to bottom, in each row, we show thethree-dimensional total mass density ( 𝜌 tot = 𝜌 dm + 𝜌 star + 𝜌 gas ), circular velocity ( 𝑉 circ ≡ √︁ 𝐺𝑀 totenc ( 𝑟 )/ 𝑟 ), three-dimensional velocity dispersion of dark matter( 𝜎 ≡ √︃ 𝜎 + 𝜎 𝜃 + 𝜎 𝜙 ), velocity anisotropy of dark matter ( 𝛽 ≡ − ( 𝜎 𝜃 + 𝜎 𝜙 )/ 𝜎 ) and rotation velocity versus velocity dispersion of dark matter( 𝑉 rot / 𝜎 ) averaged in spherical shells as a function of galactocentric distance for three simulated galaxies. We compare three categories of dark matter models:CDM; eSIDM (elastic SIDM model with a constant cross-section 1 cm g − ); dSIDM (dissipative SIDM models with various cross-sections, as defined inTable 1). The gray shaded regions in the first row of plots indicate 0 . − . 𝑅 cdmvir , which is the aperture we will later use to measure the slopes of the densityprofiles (see Section 4.3 and Figure 5-7). The gray dashed horizontal line in the fourth row is a reference line, indicating isotropic velocity dispersion ( 𝛽 =0). Ingeneral, dSIDM models produce cuspy central density profiles in the simulated dwarf galaxies, opposed to the cored central density profile in CDM and eSIDMmodels. As a consequence, the circular velocities at the center of the galaxies increase. In dSIDM models with ( 𝜎 / 𝑚 ) ≥ g − , coherent rotation of darkmatter becomes prominent and random velocity dispersion is suppressed.MNRAS , 1–25 (2021) issipative Dark Matter on FIRE log( M halo [M fl ]) l og ( M ∗ [ M fl ] ) M ∗ = f b a r y o n M h a l o Moster+
CDMdSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM ( / g ) eSIDM ( / g ) Figure 4. Stellar mass versus halo mass relation of galaxies in simulations.
The stellar masses and halo masses of simulated dwarf galaxies are presentedwith open markers (as labelled). We compare them with the observationalresults derived through abundance matching from Moster et al. (2014); Brooks& Zolotov (2014); Garrison-Kimmel et al. (2017). The black dashed linesshow ∼
95% inclusion contour assuming the scatter of the relation estimatedin Garrison-Kimmel et al. (2017). Regardless of the dark matter model, thesimulated galaxies are consistent with the observational relation.
In Figure 3, we present a gallery view of the total mass density, cir-cular velocity, three-dimensional velocity dispersion of dark matter,velocity anisotropy of dark matter, rotation velocity versus velocitydispersion of dark matter, averaged in spherical shells as a functiongalactocentric distance for three simulated galaxies. Details of themeasurements of the kinematic properties and relevant definitionsare introduced in Section 4.4. Under the influence of baryonic feed-back, the density profiles in CDM are generally shallower than thecuspy NFW profiles at galaxy centers, which is expected for thesegalaxies for their 𝑀 ∗ / 𝑀 halo values (e.g., Di Cintio et al. 2014; Chanet al. 2015; Oñorbe et al. 2015; Tollet et al. 2016; Lazar et al. 2020).In the eSIDM model, due to effective heat conduction, the profilesare even flatter at galaxy centers compared to the CDM case, but thedifference becomes less apparent in the bright dwarf ( m11q ) wherethermal conduction through self-interactions is subdominant com-pared to baryonic feedback. In dSIDM models, when the effectiveself-interaction cross-section is large (and equivalently dissipation isefficient assuming a fixed 𝑓 diss ), the central density profiles are cuspyand power-law like. For the velocity-dependent dSIDM model, in theclassical dwarf galaxies like m10q , the velocity-dependent cross-section is high and a cuspy central profile emerges. In more massivegalaxies like m11a and m11q , the velocity-dependent cross-sectionthere becomes much smaller, accompanied by stronger baryonic feed-back. As a consequence, the profiles in these systems become coredagain though the central mass density is still higher than the CDMcase. An interesting outlier here is the dSIDM model with constant ( 𝜎 / 𝑚 ) =
10 cm g − , exhibiting cuspy central density profile butwith lower normalization, which is likely due to the deformed shapeof the halo (see Section 5). A more detailed discussion on the massdensity profiles will be presented in Section 4.3.In addition to the density profile, the kinematic properties of haloesare also quite different in different dark matter models. Despite somevariations, there are some important features shared by the simula-tions of different haloes. When the cross-section is high, the rotationcurves of dwarf galaxies in dSIDM models are significantly higher atsmall radii compared to their CDM counterparts. The differences are consistent with the findings in density profiles. Again, an outlier isthe dSIDM model with ( 𝜎 / 𝑚 ) =
10 cm g − , with the normalizationof rotation velocities lower than other models. For the velocity dis-persion profile, the ones in eSIDM are flat at halo centers indicatingan isothermal distribution of dark matter particles. The velocity dis-persions in dSIDM models in general decreases towards halo centers.Particularly, the dSIDM model with ( 𝜎 / 𝑚 ) =
10 cm g − shows dra-matic decrease in velocity dispersion at 𝑟 (cid:46)
10 kpc. This indicatesmore coherent motion of dark matter particles and a decreasing sup-port from random velocity dispersion. For the velocity anisotropyprofile, the dSIDM models with ( 𝜎 / 𝑚 ) ≥ g − have lowervelocity anisotropies than their CDM counterparts at halo centers,indicating that the velocity dispersions are more dominated by thetangential component. At the same time, the coherent rotation is alsostronger in these dSIDM models. An extreme case is the dSIDMmodel with ( 𝜎 / 𝑚 ) =
10 cm g − where the sub-kpc structure isclearly in transition from dispersion supported to coherent rotationsupported. The ratio between coherent circular velocity and veloc-ity dispersion is significantly higher than others. In Section 4.4, thekinematic properties of simulated galaxies will be investigated indetail. We measure the bulk properties of the dark matter haloes and galaxiesin simulations following what has been done for the standard FIRE-2 simulations as described in Hopkins et al. (2018). We define thehalo mass 𝑀 halo and the halo virial radius 𝑅 vir using the overdensitycriterion introduced in Bryan & Norman (1998). We define the stellarmass 𝑀 ∗ as the total mass of all the stellar particles within an apertureof 0 . 𝑅 vir and correspondingly define the stellar half-mass radius 𝑟 / as the radius that encloses half of the total stellar mass. Forthe isolated dwarf galaxies in simulations, these definitions on thestellar mass and the stellar half-mass radius give similar results towhat derived using the iterative approach described in Hopkins et al.(2018).In Figure 4, we compare the stellar mass versus halo mass of sim-ulated dwarf galaxies with the scaling relations derived based onobservations (Moster et al. 2014; Brooks & Zolotov 2014; Garrison-Kimmel et al. 2017). The black dashed lines show 95% inclu-sion contour assuming the scatter estimated in Garrison-Kimmelet al. (2017). The simulated dwarfs are consistent with observa-tions in the stellar mass versus halo mass relation and the galax-ies we sampled in the simulation suite well represent the "median"galaxies in the real Universe. With mild dark matter self-interaction( ( 𝜎 / 𝑚 ) (cid:46) g − ), the halo and stellar masses of galaxies arenot significantly affected compared to their CDM counterparts, inagreement with previous studies of eSIDM (e.g., Vogelsberger et al.2014; Robles et al. 2017; Fitts et al. 2019). However, in the dSIDMmodel with ( 𝜎 / 𝑚 ) =
10 cm g − , both the halo masses and the stel-lar masses decrease for about 0 . . 𝑀 halo (cid:46) M (cid:12) . Although this level of dif-ferences is still minor compared to the scatter of the relation, it isworth to note that the model with ( 𝜎 / 𝑚 ) =
10 cm g − behavesqualitatively different from other models explored. This aspect willbe discussed in Section 4.3 and Section 4.4 in the following. In this section, we present the total mass density profiles (includ-ing the contribution from dark matter, stars and gas) of simulated
MNRAS000
MNRAS000 , 1–25 (2021) Shen et al. ρ t o t [ M fl / k p c ] Classical dwarfsm10q m10b m10v
CDMdSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM (
10 cm / g ) eSIDM ( / g ) r [kpc] ρ t o t / ρ n f w t o t α r [kpc] -10 α − α n f w Figure 5.
Left : Total mass density profiles of the classical dwarf galaxies in simulations.
The three classical dwarfs presented here are m10q , m10b and m10v . The total mass density profiles in different dark matter models are shown (as labelled). They can be compared to the NFW profiles derived by fitting thedensity profiles at large radii of the haloes (0 . 𝑟 cdm1 / < 𝑟 < 𝑟 cdm1 / ), and the ratios of the density profiles to the NFW fits are shown in the lower sub-panel. Thegray shaded region denotes the range of radii where we measure the slopes of the density profiles below. The purple dotted vertical line indicates the averageconvergence radius ( ∼
70 pc) of the classical dwarfs (see Table 1).
Right : Local power-law slopes of density profiles of the classical dwarf galaxies.
Theslopes are derived via fitting the nearby density profile with power-law. In these classical dwarfs, the CDM model predicts cored central density profiles due tobaryonic feedback. The eSIDM model produces cores of slightly bigger sizes and shallower slopes. The dSIDM model with ( 𝜎 / 𝑚 ) = . g − still producescored profiles but with higher central densities and steeper slopes than their CDM counterparts. The dSIDM models with effective cross-section > . g − all produce cuspy central density profiles with power-law slopes centering around − .
5. These profiles are even steeper than the NFW profiles. dwarf galaxies in dSIDM models with different parameters and com-pare them with the CDM predictions. We note that, for the dwarfgalaxies in simulations, the mass density profiles are dominated bydark matter. We divide the simulated dwarf galaxies into two cate-gories: ( i ) classical dwarfs, e.g. the m10’s , with typical halo mass of (cid:46) M (cid:12) and sub-kpc stellar half-mass radius; ( ii ) bright dwarfs,e.g. the m11’s , with typical halo mass of (cid:38) M (cid:12) and stellar half-mass radius of several kpc. We will investigate the extent at which thedissipative dark matter self-interactions affect the structure of thesedwarfs.In the left panel of Figure 5, we show the total mass densityprofiles of the classical dwarf galaxies in simulations with CDM,eSIDM and dSIDM models at 𝑧 = . The effective cross-section ( 𝜎 / 𝑚 ) eff of the velocity-dependent dSIDM model in these classicaldwarfs is ∼ . g − calculated using Equation 11, plugging inthe density and one-dimensional velocity dispersion of dark matterparticles enclosed in a sphere of radius 1 / 𝑟 cdm1 / , where 𝑟 cdm1 / is thestellar half-mass radius in the CDM model. We fit the density pro-files at large radii of the haloes (0 . 𝑟 cdm1 / < 𝑟 < 𝑟 cdm1 / ) with theNFW profile. In the lower sub-panel, we show the ratios betweenthe density profiles in different models and the NFW fits. In theright panel of Figure 5, we show the local power-law slopes of the The bursty star formation history in dwarf galaxies could create fluctuationsin density profiles, which leads to uncertainties in the profile measured at the 𝑧 = 𝑧 = density profiles. In the lower sub-panel, we show the differences inthe slopes versus the NFW fits. In the classical dwarfs, the centraldensity profiles are cored in the CDM case due to baryonic feed-back. The eSIDM model produces profiles with much larger coresand shallower slopes than CDM. However, the dSIDM models allpredict cuspy and power-law like central density profiles at sub-kpc scale, except for the one with low self-interaction cross-section0 . g − . These profiles are even steeper than the NFW profiles,with power-law slopes ∼ − . − . g − still produces cored centralprofiles in two galaxies, but the central densities are higher, and thecore sizes are smaller than their CDM counterparts. The profiles inthe velocity-dependent dSIDM model lie between the profiles in thedSIDM models with ( 𝜎 / 𝑚 ) = . g − , which is consistentwith the estimate of ( 𝜎 / 𝑚 ) eff in these systems. Surprisingly, increas-ing the self-interaction cross-section to 10 cm g − does not lead tofurther contraction of the haloes. Instead, the density profiles in themodel have lower normalization out to ∼
10 kpc, although the profilesstill have cuspy shapes at galaxy centers. The classical dwarf galaxythat exhibits the strongest decrease in density profile normalizationin this model is m10q . This decreased normalization of density pro-files measured spherical shells is likely related to the deformation ofhaloes (e.g. with the same energy budget, a disk-like structure willhave lower spherically averaged density than a spherical structure).Assuming that the radial contraction is adiabatic which preservesspecific angular momentum, the radial contraction of dSIDM haloes
MNRAS , 1–25 (2021) issipative Dark Matter on FIRE ρ t o t [ M fl / k p c ] Bright dwarfsm11a m11b m11q
CDMdSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM (
10 cm / g ) eSIDM ( / g ) r [kpc] ρ t o t / ρ n f w t o t α r [kpc] -10 α − α n f w Figure 6.
Left : Total mass density profiles of the bright dwarf galaxies in simulations.
The three bright dwarfs presented here are m11a , m11b and m11q .The notation is the same as Figure 5. The purple dotted vertical line here indicates the average convergence radius ( ∼
200 pc) of the bright dwarfs (see Table 1).
Right : Local power-law slopes of the density profiles of the bright dwarf galaxies.
In these bright dwarfs, the CDM model again predicts cored centraldensity profiles with even larger cores ( ∼ kpc) than the classical dwarfs due to stronger baryonic feedback. The eSIDM model produces cores of similar sizesand slopes. The velocity-dependent dSIDM model has relatively low effective cross-sections ( ∼ .
01 cm g − ) in these dwarfs. This model still produce coresbut with slightly higher central densities than their CDM counterparts. The dSIDM models with relatively high effective cross-sections ( (cid:29) .
01 cm g − ) stillproduce cuspy and power-law like central density profiles. The power-law slopes center around − . − − will eventually be halted by the growing centrifugal force from coher-ent dark matter rotation. This will also make dSIDM haloes deformfrom spherical to oblate in shape and the density profiles will ap-pear with lower normalization. In subsequent sections, we will seemore evidence for this phenomenon from the analysis of kinematicproperties (Section 4.4) and shapes (Section 5) of dark matter haloes.In the left panel of Figure 6, we show the total mass density pro-files of the bright dwarf galaxies in simulations with CDM, eSIDMand dSIDM models. The ( 𝜎 / 𝑚 ) eff of the velocity-dependent dSIDMmodel in these bright dwarfs is ∼ .
01 cm g − . In the right panel ofFigure 6, we show the local power-law slopes of the density profilesof the bright dwarfs. The phenomena in the bright dwarfs are qualita-tively consistent with those in the classical dwarfs shown above. In thebright dwarfs, the central density profiles are cored in the CDM case.The decrease of the central density compared to the NFW profile isstronger than that in the classical dwarfs, due to stronger baryonicfeedback in the bright dwarfs. The eSIDM model again produceslarger cores and shallower slopes in these galaxies compared to theCDM case. In dSIDM models, the shapes of the density profiles varywith the self-interaction cross-section (or equivalently the efficiencyof dissipation, assuming fixed 𝑓 diss ). The velocity-dependent dSIDMmodel has relatively low effective cross-section in the bright dwarfsand thus the central density profiles are still cored, similar to the CDMcase. However, in the dSIDM model with ( 𝜎 / 𝑚 ) = . g − ,cuspy and power-law like central profiles show up in two out of thethree bright dwarfs and the only cored one shows enhanced centraldensities at 𝑟 (cid:46) kpc. In the dSIDM model with ( 𝜎 / 𝑚 ) = g − ,the central profiles of all three bright dwarfs are cuspy with power-lawslopes centering around − . ( 𝜎 / 𝑚 ) =
10 cm g − , the density profiles have lower normal-ization although they are still cuspy, similar to the phenomenon wefound in the classical dwarfs. Here, the bright dwarf galaxy that ex-hibits the strongest decrease in density profile normalization in thismodel is m11b .Comparing the density profiles of the classical dwarfs and brightdwarfs, we find that the dSIDM model with the same constant cross-section can behave qualitatively differently in galaxies of differentmasses. For example, the model with ( 𝜎 / 𝑚 ) = . g − pro-duces cored central profiles in two of the classical dwarfs but pro-duces cuspy central profiles in two of the bright dwarfs. As discussedin Section 3, the dissipation time scale of models with constant cross-section inversely depends on density and velocity dispersion of thesystem. The bright dwarfs typically have much higher velocity dis-persion at their centers than the classical dwarfs while the centraldensities are comparable to the classical dwarfs. As expected, dissi-pation has stronger impact in the bright dwarfs. On the other hand, thevelocity-dependent dSIDM model produces cuspy central profiles inthe classical dwarfs but produces cored central profiles in the brightdwarfs. The dissipation time scale of the velocity-dependent modelinversely depends on density but exhibits a 𝑣 asymptotic depen-dence on velocity dispersion. The opposite dependence on velocitydispersion makes the impact of dissipation stronger in the classicaldwarfs.To quantify the impact of dissipation on galaxy structures, wemeasure the slopes of the total mass density profiles at galaxy centers.The aperture we choose for this measurement is 0 . − . 𝑅 cdmvir (as indicated by the gray bands in Figure 5 and 6), where 𝑅 cdmvir is MNRAS000
10 cm g − , the density profiles have lower normal-ization although they are still cuspy, similar to the phenomenon wefound in the classical dwarfs. Here, the bright dwarf galaxy that ex-hibits the strongest decrease in density profile normalization in thismodel is m11b .Comparing the density profiles of the classical dwarfs and brightdwarfs, we find that the dSIDM model with the same constant cross-section can behave qualitatively differently in galaxies of differentmasses. For example, the model with ( 𝜎 / 𝑚 ) = . g − pro-duces cored central profiles in two of the classical dwarfs but pro-duces cuspy central profiles in two of the bright dwarfs. As discussedin Section 3, the dissipation time scale of models with constant cross-section inversely depends on density and velocity dispersion of thesystem. The bright dwarfs typically have much higher velocity dis-persion at their centers than the classical dwarfs while the centraldensities are comparable to the classical dwarfs. As expected, dissi-pation has stronger impact in the bright dwarfs. On the other hand, thevelocity-dependent dSIDM model produces cuspy central profiles inthe classical dwarfs but produces cored central profiles in the brightdwarfs. The dissipation time scale of the velocity-dependent modelinversely depends on density but exhibits a 𝑣 asymptotic depen-dence on velocity dispersion. The opposite dependence on velocitydispersion makes the impact of dissipation stronger in the classicaldwarfs.To quantify the impact of dissipation on galaxy structures, wemeasure the slopes of the total mass density profiles at galaxy centers.The aperture we choose for this measurement is 0 . − . 𝑅 cdmvir (as indicated by the gray bands in Figure 5 and 6), where 𝑅 cdmvir is MNRAS000 , 1–25 (2021) Shen et al. dSIDM cuspsNFWbaryonic feedbackeSIDM cores -4 -3 -2 log( M ∗ /M halo ) α [ . − . % R v i r ] Classicaldwarfs Brightdwarfs
CDMdSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM (
10 cm / g ) eSIDM ( / g ) Figure 7. Slopes of the central density profiles of dwarf galaxies in the simulation suite.
The slopes are measured at 0 . − . 𝑅 cdmvir . The slopes measuredin simulations with different dark matter models are shown in open markers (as labelled). Galaxies are ordered from left to right based on their stellar-to-halomass ratios ( 𝑀 ∗ / 𝑀 halo ), and are classified as classical dwarfs and bright dwarfs. (The ultra-faint dwarf m09 in the suite also has its 𝑀 ∗ / 𝑀 halo value lying inthe classical dwarf regime.) The asymptotic behaviours of the slopes at the low mass end are clearly different between different dark matter models. In low-massdwarf galaxies, the density profiles in dSIDM models with ( 𝜎 / 𝑚 ) ≥ g − and the velocity-dependent model converge to a slope of ∼ − . − ∼ − . ( 𝜎 / 𝑚 ) = . g − can still produce small cores in some dwarf galaxies with relativelystrong baryonic feedback, with 𝛼 ∼ − 𝛼 ∼ − .
5. The dSIDM models with constant cross-sections stillproduce cuspy density profiles with slopes centering around − . − −
1. Unlike dSIDM models, density profiles in CDM are shallowerthan the NFW profile and are shallower in more massive dwarf galaxies, due to stronger baryonic feedback there (indicated by the thick cyan line). The eSIDMmodel consistently produces cored density profiles with slope ∼ − . the virial radius of the halo in the CDM model. This has beenchosen since it is an appropriate aperture to illustrate the impact ofdissipation at small radii while remaining larger than the convergenceradii of dark matter profiles in these runs (see Table 1). In Figure 7,we show the power-law slopes of the density profiles (measured at0 . − . 𝑅 cdmvir ) of simulated dwarf galaxies versus their stellar-to-halo mass ratios ( 𝑀 ∗ / 𝑀 halo ). The slopes of the density profiles indifferent models show four different “tracks”: • The NFW profile has an asymptotic − power-law slope at galaxycenters. The virial radius does not vary much in simulations with different darkmatter models. Using the virial radius in the CDM run is simply to ensurethat the aperture is identical for different dark matter models. The slope of the NFW profile varies with radius. At the radii we measurethe slopes, the NFW profile has a slope of ∼ − . • In CDM, baryonic feedback drives gas outflow and creates fluctua-tions in the central gravitational potential which significantly affectsthe distribution of dark matter. Dwarf galaxies have shallower den-sity profiles than the NFW profile. The difference in slope peaks inmost massive bright dwarfs where baryonic feedback is most effi-cient in perturbing galaxy structures, as has been found in previousstudies (e.g., Di Cintio et al. 2014; Chan et al. 2015; Oñorbe et al.2015; Tollet et al. 2016; Lazar et al. 2020). • In eSIDM, elastic dark matter self-interaction drives the halo tothermal equilibrium and produces an isothermal density profile witha core at the center. The power-law slopes of the central profiles areclose to zero in most of the simulated dwarf galaxies, regardless oftheir mass. • In dSIDM, dissipative dark matter self-interaction is a competingfactor against baryonic feedback in shaping the central density profile.When ( 𝜎 / 𝑚 ) eff > . g − , dark matter dissipation becomesdominant and the central density profiles in dwarf galaxies are steeper MNRAS , 1–25 (2021) issipative Dark Matter on FIRE log(( σ/m ) eff [cm / g]) ∆ α [ . − . % R v i r ] dSIDM ( v dep . ) dSIDM ( . / g ) dSIDM ( / g ) dSIDM ( / g ) log(( t cdiss /t H ) ∆ α [ . − . % R v i r ] Figure 8.
Top : Slope change versus effective self-interaction cross-sectionof dwarf galaxies in simulations. Δ 𝛼 is defined as the difference in slopesmeasured at 0 . − . 𝑅 cdmvir between galaxies in dSIDM and CDM. The reddashed line labels the qualitative trend (not rigorous fitting). In the regimewhere ( 𝜎 / 𝑚 ) eff < g − , the steepening of central profiles inducedby dissipative dark matter self-interactions becomes progressively stronger insystems with higher effective cross-sections. In the regime where ( 𝜎 / 𝑚 ) eff > g − , the steepening of central profiles saturates. Bottom : Slope changeversus dissipation time scale at halo center.
When log ( 𝑡 cdiss / 𝑡 H ) > − 𝑡 cdiss decreases while the steepeningsaturates when log ( 𝑡 cdiss / 𝑡 H ) < − than the ones in the CDM model. In the classical dwarfs, the power-law slopes are steeper than the − ∼ − .
5. In the bright dwarfs, the power-law slopes have largerscatter, ranging from − −
1. When the ( 𝜎 / 𝑚 ) eff is relatively low(e.g. the model with ( 𝜎 / 𝑚 ) = . g − in the classical dwarfsand the velocity-dependent model in the bright dwarfs), the centraldensity profiles are affected by a mixture of dark matter dissipationand baryonic feedback, which compete with each other. In somedwarfs with relatively strong feedback effects, the slopes becomeshallower than the ∼ − . 𝛼 (cid:38) − .
5) at smaller radii as shown inthe right panels of Figure 5 and 6.To demonstrate the net impact of dissipation, in the top panel ofFigure 8, we show the slope change Δ 𝛼 versus the effective self- We verify that the impact of baryonic feedback becomes negligible in thisregime through the comparison with DMO simulations in Section 7.2. interaction cross-section ( 𝜎 / 𝑚 ) eff . Δ 𝛼 is defined as the difference inslopes measured at 0 . − . 𝑅 cdmvir between galaxies in dSIDM andCDM, Δ 𝛼 = 𝛼 dsidm − 𝛼 cdm . More negative Δ 𝛼 indicates strongerimpact of dissipation on the steepness of the density profile. The ef-fective self-interaction cross-section is calculated using Equation 11,plugging in the density and one-dimensional velocity dispersion ofdark matter particles enclosed in a sphere of radius 1 / 𝑟 cdm1 / . The reddashed line shows the qualitative trend (not rigorous fitting) of Δ 𝛼 versus ( 𝜎 / 𝑚 ) eff . When ( 𝜎 / 𝑚 ) eff (cid:46) g − , the steepening of thecentral density profiles induced by dissipation becomes progressivelystronger in systems with higher effective cross-sections. The changeof the power-law slope scales roughly linearly as the logarithm ofthe effective cross-section. When ( 𝜎 / 𝑚 ) eff is larger than 1 cm g − ,the steepening of the central density profiles saturates. The Δ 𝛼 when ( 𝜎 / 𝑚 ) eff (cid:39)
10 cm g − is comparable to the ( 𝜎 / 𝑚 ) eff (cid:39) . g − case. In the bottom panel of Figure 8, we show the slope change Δ 𝛼 versus the dissipation time scale at halo center 𝑡 cdiss , calculated us-ing Equation 8. The steepening of the central density profiles occurswhen 𝑡 cdiss becomes comparable to 𝑡 H . The slope difference becomeslarger as 𝑡 cdiss decreases when 𝑡 cdiss (cid:38) . 𝑡 H . When 𝑡 cdiss (cid:46) . 𝑡 H , thesteepening of the central profile saturates, similar to the trend in thetop panel. This is likely related to the increasing rotation support ofdark matter when ( 𝜎 / 𝑚 ) eff (cid:38) g − , which will be shown in thefollowing section. In this section, we will explore the kinematic properties of dark matterparticles in the simulated dwarf galaxies. These properties includevelocity dispersion, coherent rotation velocity, velocity anisotropyand the velocity distribution function of dark matter.To evaluate these properties, we first divide a simulated halo intospherical shells with respect to the halo center. In each shell, wemeasure the total angular momentum of dark matter particles andalign the z-axis of the coordinate system with the direction of theangular momentum. This helps us define the azimuthal and zenithdirections (note that different shells could have different directions ofangular momentum and thus different definitions of the z-axis). Thevelocities of dark matter particles are decomposed to the radial, zenithand azimuthal components ( 𝑣 r , 𝑣 𝜃 and 𝑣 𝜙 ) in spherical galactocentriccoordinates. The coherent rotation velocity 𝑉 rot of particles in theshell is calculated as 𝑉 rot = 𝐽 dm 𝐼 shell 𝑅 shell ,𝐼 shell = 𝑀 dm 𝑟 − 𝑟 𝑟 − 𝑟 , 𝑅 shell = 𝑟 o + 𝑟 i 𝐽 dm is the total angular momentum of dark matter particles inthe shell, 𝑀 dm is the total mass of dark matter in the shell, 𝐼 shell is themoment of inertia of the shell, 𝑟 o and 𝑟 i are the outer and inner radiiof the shell, 𝑅 shell is the median radius of the shell. Here, we have as-sumed that the mass is uniformly distributed in the shell in the calcu-lation of moment of inertia. We also measure the mean inflow/outflowvelocity ( 𝑣 r ) of dark matter particles in the shell. We subtract both thecoherent rotation velocity and the mean inflow/outflow velocity be-fore measuring the velocity dispersion 𝜎 r , 𝜎 𝜃 and 𝜎 𝜙 correspondingto the radial direction, and the azimuthal and zenith angles, respec-tively. Finally, the three-dimensional velocity dispersion is calculatedas: 𝜎 = √︃ 𝜎 + 𝜎 𝜃 + 𝜎 𝜙 . The one-dimensional velocity dispersionis estimated as: 𝜎 = √︃ ( 𝜎 + 𝜎 𝜃 + 𝜎 𝜙 )/
3. The degree of velocity
MNRAS000
MNRAS000 , 1–25 (2021) Shen et al. -2 -1 r/R vir -2 -1 V r o t / σ d CDMdSIDM ( / g ) dSIDM ( / g ) Figure 9. Coherent rotation velocity relative to velocity dispersion ofdark matter in simulations.
The coherent rotation velocities and the velocitydispersions are measured in spherical shells as discussed in the main text. Wepresent the results in CDM and dSIDM with ( 𝜎 / 𝑚 ) = g − .For each model, we show the results of five dwarf galaxies: m10q , m10b , m10v , m11a and m11b . The coherent rotation becomes more prominentinside ∼ 𝑅 vir as the self-interaction cross section increases, but not inevery galaxy. The two galaxies that have rotation velocities comparable tovelocity dispersions are m10q and m11b . isotropic10 -2 -1 r/R vir β radialtangential CDMdSIDM ( / g ) dSIDM ( / g ) Figure 10. Velocity anisotropy profiles of dark matter in simulated dwarfgalaxies.
The velocity anisotropies are calculated using Equation 13. Wepresent the results in CDM and dSIDM with ( 𝜎 / 𝑚 ) = g − . Foreach model, we show the results of the same five galaxies as in Figure 9. Thevelocity anisotropy decreases as the self-interaction cross-section increasesand eventually becomes negative, suggesting that the velocity dispersion ismore dominated by the tangential component. This is consistent with morecoherent rotation found in Figure 9. anisotropy is calculated as 𝛽 = − 𝜎 𝜙 + 𝜎 𝜃 𝜎 , (13)Under this definition, 𝛽 = 𝛽 = 𝛽 to a velocity dispersion dominatedby the tangential component. Coherent rotation:
A natural consequence of dissipative interac- tions is that particles tend to move in a more coherent fashion, ratherthan in random dispersion. If the energy dissipation is faster thanthe relaxation processes (either through dark matter self-interactionsor gravitational interactions), the coherent rotation would graduallybecome prominent in the system if angular momentum is conserved.In Figure 9, we show the ratio between coherent rotation veloc-ity and three-dimensional velocity dispersion of dark matter mea-sured in spherical shells in CDM and dSIDM with ( 𝜎 / 𝑚 ) = g − . For each model, each line corresponds to one of the sim-ulated dwarf galaxies: m10q , m10b , m10v , m11a and m11b . Quali-tatively, the coherent rotation velocity at small galactocenric radii be-comes progressively more prominent as self-interaction cross-sectionbecomes higher (and dissipation becomes more efficient). At largeradii, the systematic difference becomes negligible. Quantitatively,there are apparent galaxy-to-galaxy variations. The ratio can reach ∼ . ∼ 𝑅 vir (roughly sub-kpc scale in dwarfs) in m10q and m11b in dSIDM with ( 𝜎 / 𝑚 ) =
10 cm g − , while in m11a and m10b , the ratio remains (cid:46) . ∼ 𝑅 vir in any models. Theseevidences suggest that, at the centers of galaxies, some dSIDM real-izations are in a transition from a pure dispersion supported systemto a system supported by a mixture of random velocity dispersionand coherent rotation. The radial scale for this transition to take placeis a few percent of the virial radius. Such scale is quite consistentwith the centrifugal barrier ∼ 𝑠𝑅 vir ( 𝑠 is the halo spin parameterwith typical value ∼ .
01 - 0 .
1) found for dissipative gas in CDMhaloes (e.g., Mo et al. 1998).
Velocity anisotropy:
In Figure 10, we show the velocity anisotropyof dark matter measured in spherical shells in CDM and dSIDMwith ( 𝜎 / 𝑚 ) = g − . The velocity anisotropies are cal-culated using Equation 13. The measured anisotropy is not sensitiveto the bulk motion of dark matter in the shell since we have sub-tracted the mean rotation/inflow/outflow velocities. For each model,we show the results of the same five galaxies as in Figure 9. CDMhaloes are almost isotropic at the centers with mild radial velocitydispersion anisotropy at the outskirt, which is consistent with previ-ous studies (e.g, Lemze et al. 2012; Sparre & Hansen 2012; Wojtaket al. 2013). In dSIDM models, it is similar to the CDM case thatthe velocity anisotropy increases towards larger galactocentric radii.However, as dissipation becomes more efficient, the normalizationof the velocity anisotropy decreases and eventually becomes nega-tive at small radii. In the dSIDM model with ( 𝜎 / 𝑚 ) =
10 cm g − ,the velocity anisotropy drops to ∼ − . 𝑟 ∼ 𝑅 vir , suggestingthat the tangential component of the velocity dispersion is relativelystronger there. This phenomenon is inline with the more prominentcoherent rotation developed in dSIDM haloes. Phase space distribution:
In Figure 11, we present the densitydistribution function of dark matter in the 𝑣 𝜙 − 𝑣 r phase space,d 𝜌 dm / d 𝑣 r d 𝑣 𝜙 , of m10q and m10v . We compare the results in CDMand dSIDM with ( 𝜎 / 𝑚 ) =
10 cm g − to better illustrate the contrast.The phase space distributions are measured in three radial bins: cen-tral, 𝑟 < 𝑟 cdm1 / / ∼
100 - 200 pc); intermediate, 𝑟 cdm1 / / < 𝑟 < 𝑟 cdm1 / ( ∼ kpc) and “outskirt”, 3 𝑟 cdm1 / < 𝑟 < . 𝑅 cdmvir ( (cid:38)
10 kpc). The az-imuthal and zenith directions are defined based on the direction ofthe total angular momentum of dark matter in each radial bin re-spectively. From inside out, each contour is determined such that itencloses a certain percentile (as labelled on the contour line) of darkmatter particles in the bin. We note that, different from the measure-ment of velocity dispersions, the coherent rotation or inflow/outflowvelocity has not been subtracted when determining 𝑣 r and 𝑣 𝜙 . Darkmatter at small and intermediate radii in the dSIDM model with MNRAS , 1–25 (2021) issipative Dark Matter on FIRE central ( ∼ − ) intermediate ( ∼ kpc ) outskirt ( (cid:38)
10 kpc )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r < r / / m10q % % % % % % CDMdSIDM ( / g )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r / / < r < r / m10q % % % % % % CDMdSIDM ( / g )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r / < r < R vir / m10q % % % % % % % CDMdSIDM ( / g )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r < r / / m10v % % % % % CDMdSIDM ( / g )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r / / < r < r / m10v % % % % % CDMdSIDM ( / g )
30 20 10 0 10 20 30 v r [km / s] v φ [ k m / s ] r / < r < R vir / m10v % % % % % % % CDMdSIDM ( / g ) Figure 11. Phase space distribution function of dark matter in simulated classical dwarfs.
We present the two-dimensional density distribution of darkmatter in the 𝑣 𝜙 − 𝑣 r phase space, d 𝜌 dm / d 𝑣 r d 𝑣 𝜙 . In the three columns, we show the distribution in three radial bins: central, 𝑟 < 𝑟 cdm1 / /
3, intermediate, 𝑟 cdm1 / / < 𝑟 < 𝑟 cdm1 / , and “outskirt”, 3 𝑟 cdm1 / < 𝑟 < . 𝑅 cdmvir , respectively. From inside out, each contour is determined such that it encloses a certain percentileof dark matter particles in the bin. The percentiles range from 10% to 90% with 20% as interval, as labeled on the contours. The dots represent the locationswhere the velocity distribution function peaks. Dark matter in dSIDM models exhibit positive median 𝑣 𝜙 while the phase space distribution is almost isotropicin CDM. The differences consistently show up in the three radial bins and suggest a coherent rotation built up in dSIDM haloes. The phase space distribution inthe dSIDM model is also more peaky than the CDM case, at least for the central and intermediate radial bins. ( 𝜎 / 𝑚 ) =
10 cm g − exhibits a median 𝑣 𝜙 (cid:39) −
10 km s − contraryto the almost zero median 𝑣 𝜙 in the CDM case. The distribution inthe dSIDM model is also more peaky than the CDM case. The dif-ferences here is consistent with the coherent rotation of dark matterin dSIDM found above. At the outskirt of the galaxy, the increase inthe median of 𝑣 𝜙 is still visible but the scatter in the phase space alsobecomes larger.In Figure 12, we show the velocity ( | 𝑣 | ) distribution functionsof dark matter in the classical dwarfs in CDM and dSIDM with ( 𝜎 / 𝑚 ) = g − . We present the results at small ( 𝑟 <𝑟 cdm1 / /
3) and intermediate galactocentric radii ( 𝑟 cdm1 / / < 𝑟 < 𝑟 cdm1 / ),respectively. We also show the distribution function in log-log scaleto emphasize the low velocity tail. Compared to the CDM case,the velocity distributions in dSIDM models show apparent suppres-sion at the high velocity tail and bumps at lower velocities, due torelatively high interaction rates of particles with high absolute ve-locities. The low velocity tail is less affected by dissipation due torelatively low interaction rates there. The peak velocity decreasesas self-interaction cross-section becomes larger. The phenomenon isactually opposite to the prediction of the “gravothermal collapse” inSIDM haloes (e.g., Balberg et al. 2002; Essig et al. 2019). The differ-ence reflects the deviation of dSIDM haloes from both dynamical andthermal equilibrium in the phase of radial contraction, as well as thefact that one cannot assume velocity distributions as purely isotropicin relaxed dSIDM haloes. Compared with the Maxwell-Boltzmanndistribution, the velocity distributions in CDM have extended tailsat both the low and high velocity tail, since CDM particles are col-lisionless and are not locally thermalized. The distributions in thedSIDM models are suppressed the high velocity tail. At small galac-tocentric radii, the asymptotic behaviour of the velocity distributionfunction in CDM and dSIDM are quite different from the Maxwell- Boltzmann distribution, decreasing slower towards lower velocities.However, at intermediate radii, both CDM and dSIDM have distribu-tions that resemble the Maxwell-Boltzmann distribution at the lowvelocity tail. The change in halo shape is another important signature for alterna-tive dark matter physics. This aspect has been explored in detail forthe eSIDM case (e.g., Zemp et al. 2011; Peter et al. 2013; Robleset al. 2017; Brinckmann et al. 2018; Sameie et al. 2018). In dSIDMhaloes, morphological changes in response to the energy dissipationare also expected, inline with the steepening of the density profileand the increased rotation support found in previous sections.To measure the shape of dark matter haloes, we determine theorientation and magnitude of the principal axes of dark matter distri-bution by computing the eigenvectors and eigenvalues of the shapetensor of dark matter mass distribution, defined as S = ∫ 𝑉 𝜌 ( r ) r r T d 𝑉 ∫ 𝑉 𝜌 ( r ) d 𝑉 , (14)where 𝜌 ( r ) is the dark matter mass density at position r with respectto halo center. In terms of discrete dark matter particles, each elementof the tensor is calculated as 𝑆 ij = (cid:205) k 𝑚 k ( 𝑟 k ) i ( 𝑟 k ) j (cid:205) k 𝑚 k . (15)where 𝑚 k is the mass of the k-th dark matter particle and ( 𝑟 k ) i is thespatial coordinate of the k-th particle. The three eigenvectors of theshape tensor give the three axes of the mass distribution. Specifically, MNRAS000
3) and intermediate galactocentric radii ( 𝑟 cdm1 / / < 𝑟 < 𝑟 cdm1 / ),respectively. We also show the distribution function in log-log scaleto emphasize the low velocity tail. Compared to the CDM case,the velocity distributions in dSIDM models show apparent suppres-sion at the high velocity tail and bumps at lower velocities, due torelatively high interaction rates of particles with high absolute ve-locities. The low velocity tail is less affected by dissipation due torelatively low interaction rates there. The peak velocity decreasesas self-interaction cross-section becomes larger. The phenomenon isactually opposite to the prediction of the “gravothermal collapse” inSIDM haloes (e.g., Balberg et al. 2002; Essig et al. 2019). The differ-ence reflects the deviation of dSIDM haloes from both dynamical andthermal equilibrium in the phase of radial contraction, as well as thefact that one cannot assume velocity distributions as purely isotropicin relaxed dSIDM haloes. Compared with the Maxwell-Boltzmanndistribution, the velocity distributions in CDM have extended tailsat both the low and high velocity tail, since CDM particles are col-lisionless and are not locally thermalized. The distributions in thedSIDM models are suppressed the high velocity tail. At small galac-tocentric radii, the asymptotic behaviour of the velocity distributionfunction in CDM and dSIDM are quite different from the Maxwell- Boltzmann distribution, decreasing slower towards lower velocities.However, at intermediate radii, both CDM and dSIDM have distribu-tions that resemble the Maxwell-Boltzmann distribution at the lowvelocity tail. The change in halo shape is another important signature for alterna-tive dark matter physics. This aspect has been explored in detail forthe eSIDM case (e.g., Zemp et al. 2011; Peter et al. 2013; Robleset al. 2017; Brinckmann et al. 2018; Sameie et al. 2018). In dSIDMhaloes, morphological changes in response to the energy dissipationare also expected, inline with the steepening of the density profileand the increased rotation support found in previous sections.To measure the shape of dark matter haloes, we determine theorientation and magnitude of the principal axes of dark matter distri-bution by computing the eigenvectors and eigenvalues of the shapetensor of dark matter mass distribution, defined as S = ∫ 𝑉 𝜌 ( r ) r r T d 𝑉 ∫ 𝑉 𝜌 ( r ) d 𝑉 , (14)where 𝜌 ( r ) is the dark matter mass density at position r with respectto halo center. In terms of discrete dark matter particles, each elementof the tensor is calculated as 𝑆 ij = (cid:205) k 𝑚 k ( 𝑟 k ) i ( 𝑟 k ) j (cid:205) k 𝑚 k . (15)where 𝑚 k is the mass of the k-th dark matter particle and ( 𝑟 k ) i is thespatial coordinate of the k-th particle. The three eigenvectors of theshape tensor give the three axes of the mass distribution. Specifically, MNRAS000 , 1–25 (2021) Shen et al. | v | [km / s] N o r m a li ze d P r o b a b ili t y d ρ d m / d | v | r < r / / CDMdSIDM ( / g ) dSIDM ( / g ) Maxwell − Boltzmann | v | = 20km / s | v | [km / s] -5 -4 -3 -2 -1 l og ( d ρ d m / d | v | ) r < r / / | v | [km / s] N o r m a li ze d P r o b a b ili t y d ρ d m / d | v | r / / < r < r / CDMdSIDM ( / g ) dSIDM ( / g ) Maxwell − Boltzmann | v | = 25km / s | v | [km / s] -5 -4 -3 -2 -1 l og ( d ρ d m / d | v | ) r / / < r < r / Figure 12. Velocity distribution functions of dark matter in the classical dwarfs.
Top left : Velocity distribution function at small galactocentric radii( 𝑟 < 𝑟 cdm1 / / ( 𝜎 / 𝑚 ) = g − (as labelled). As a reference, a Maxwell-Boltzmanndistribution is shown with the thick gray line. Compared to CDM, the velocity distribution functions in dSIDM models are more suppressed at the high velocitytail as the cross-section increases and the peaks of the distributions also decrease systematically. Top right : Same velocity distribution functions as the top leftpanel but in log-log scale to highlight the asymptotic behaviour at the low velocity tail. Both CDM and dSIDM models have velocity distribution functions thatdecreases slower than the Maxwell-Boltzmann distribution at the low velocity tail. Dissipation has limited impact at low velocities due to small interaction ratesthere.
Bottom left : Velocity distribution function at intermediate galactocentric radii ( 𝑟 cdm1 / / < 𝑟 < 𝑟 cdm1 / ). Similar differences in the velocity distribution ofCDM and dSIDM are found compared to the one at small radii. Bottom right : The same velocity distribution function as the bottom left panel but in log-logscale. Both CDM and dSIDM models have velocity distributions that overall resemble the Maxwell-Boltzmann distribution at the low velocity tail. the major, intermediate and minor axes will be denoted as a , b and c ,respectively. The ratios between the eigenvalues of the shape tensorgive the axis ratios of the mass distribution.For the simulated dark matter haloes, we perform this measure-ment in a fixed volume of 𝑉 = 𝜋𝑟 /
3, where 𝑟 lim is chosen to be1 kpc. The volume is an ellipsoid with its major, intermediate andminor axes ( a , b and c are set to 𝑟 lim initially) updated iterativelyuntil convergence is reached. This gives an estimation of the shapeof the dark matter halo at kpc scale. In the top panel of Figure 13,we show the minor/intermediate axis ratio ( 𝑐 / 𝑏 ) versus the inter-mediate/major axis ratio ( 𝑏 / 𝑎 ) of dark matter mass distribution at 𝑧 = ( 𝜎 / 𝑚 ) = g − , haloes are quite spherical with 𝑏 / 𝑎 (cid:38) . 𝑐 / 𝑏 (cid:38) .
8. The radial contraction washes the initialtriaxiality of the haloes and the increased central force makes haloesmore spherical. However, in the model with ( 𝜎 / 𝑚 ) =
10 cm g − ,two of the haloes become oblate in shape, with 𝑐 / 𝑏 drops to around0 . .
7, while the other three are still quite spherical in the end.In the bottom panel of Figure 13, we show the evolution of theaxis ratios of m10q from 𝑧 (cid:39) . 𝑧 = m10q as an example, since it has dramatic changes in itsshape in dSIDM models. The markers with darker colors representmeasurements at lower redshifts. The CDM halo stays triaxial since 𝑧 (cid:39) . ( 𝜎 / 𝑚 ) = g − is already more spherical than CDM andeSIDM counterparts at 𝑧 (cid:39) . 𝑐 / 𝑏, 𝑏 / 𝑎 > .
95) at 𝑧 =
0. However, the halo in dSIDM with ( 𝜎 / 𝑚 ) =
10 cm g − initially follows the track of becoming morespherical but then turns oblate in shape. We note that, though notshown explicitly here, the other halo ( m11b ) which ends up oblate MNRAS , 1–25 (2021) issipative Dark Matter on FIRE Intermediate / Major axis ratio b/a M i n o r / I n t e r m e d i a t e a x i s r a t i o c / b m o r e p r o l a t e m o r e o b l a t e m o r e s p h e r i c a l z = 0 CDMeSIDM ( / g ) dSIDM ( / g ) dSIDM ( / g ) Intermediate / Major axis ratio b/a M i n o r / I n t e r m e d i a t e a x i s r a t i o c / b m10q ( z = 2 . to z = 0 ) m o r e p r o l a t e m o r e o b l a t e m o r e s p h e r i c a l CDMeSIDM ( / g ) dSIDM ( / g ) dSIDM ( / g ) Figure 13.
Top : Axis ratios of dark matter haloes at central kpc in simu-lations at 𝑧 = . We show the minor/intermediate axis ratio ( 𝑐 / 𝑏 ) versus theintermediate/major axis ratio ( 𝑐 / 𝑎 ) of dark matter mass distribution in differ-ent simulations. The axes are measured iteratively while fixing the volume ofan ellipsoid as 4 𝜋 / 𝑟 , where 𝑟 lim is chosen to be 1 kpc. When 𝑐 / 𝑏 ( 𝑏 / 𝑎 )is close to unity, the system is a prolate (oblate) spheroid. When both 𝑐 / 𝑏 and 𝑏 / 𝑎 are close to unity, the system is spherically symmetric. In CDM, darkmatter haloes are triaxial ellipsoids with a clear hierarchy of minor, interme-diate and major axes. The CDM haloes lean towards prolate shapes, drivenby mild radial dispersion anisotropy. In the dSIDM model with ( 𝜎 / 𝑚 ) = g − , dark matter haloes behave as oblate spheroids, driven by thecoherent rotation of dark matter. In the extreme cases (e.g., m10q in dSIDMwith ( 𝜎 / 𝑚 ) =
10 cm g − ), 𝑐 / 𝑏 drops to as low as ∼ . 𝑏 / 𝑎 staysaround unity. At larger radii ( 𝑟 (cid:29) kpc), the qualitative trends are similar butthe differences between dark matter models become rapidly smaller. Bottom : Evolution of the axis ratios of m10q at central kpc from 𝑧 (cid:39) . to 𝑧 = . The markers with darker colors represent measurements at lower redshifts.The CDM halo stays triaxial since 𝑧 (cid:39) . ( 𝜎 / 𝑚 ) = g − is already more spherical than CDM and eSIDM counterparts at 𝑧 (cid:39) . 𝑧 =
0. However, the halo in dSIDM with ( 𝜎 / 𝑚 ) =
10 cm g − initially follows the track of becoming more sphericalbut then turns oblate in shape. ( 𝑐 / 𝑏 ∼ . 𝑧 =
0) in the model with ( 𝜎 / 𝑚 ) =
10 cm g − hassimilar evolutionary track in the axis ratio plane. However, the threehaloes ( m10b , m10v , m11a ) that end up spherical ( 𝑐 / 𝑏, 𝑏 / 𝑎 (cid:38) . 𝑧 =
0) are still in the phase of turning spherical.The morphological differences found here are consistent with ourfindings in the previous sections that coherent rotation develops indSIDM haloes with ( 𝜎 / 𝑚 ) =
10 cm g − and could also result in the lower normalization of the density profiles (measured in sphericalshells) found in Section 4.3. In the model with ( 𝜎 / 𝑚 ) =
10 cm g − ,the two haloes that become oblate in shape at 𝑧 = m10q and m11b ) are the haloes with the most significant coherent rotation(as presented in Section 4.4) and also with the most significant de-crease in density profile normalization (as presented in Section 4.4).When the coherent rotation velocity becomes comparable to thevelocity dispersion, a self-gravitating spheroidal system consistingof collisionless particles flattens. This is a well-known behaviourin the stellar distribution of elliptical galaxies (e.g., Davies et al.1983; Cappellari et al. 2007) and models of isotropic oblate rotatingspheroids (Binney 1978; Binney & Tremaine 1987, 2008). Similar tothese previous studies, the response of the ellipticity of the spheroidto 𝑉 rot / 𝜎 is weak. In the simulated dwarfs m10q and m11b , sig-nificant coherent rotation of 𝑉 rot / 𝜎 ∼ . 𝑐 / 𝑏, 𝑐 / 𝑎 ∼ . − . 𝑟 (cid:46) kpc). However,the coherent rotation and halo deformation are weaker in other sim-ulated dwarfs and this is likely related to the differences in the massassembly history of the dwarfs.We note that, for the oblate spheroids we found here, the mi-nor and major axes are still comparable to each other. The shapeis qualitatively different from the thin "dark disk" discussed in theliterature (albeit for Milky Way-sized galaxies) regarding dissipativedark matter (Fan et al. 2013a,b, 2014; Randall & Scholtz 2015; Foot2013; Foot & Vagnozzi 2015, 2016). The dissipation time scale inthe model studied here is still orders of magnitude longer than thedynamical time scale of the system, which prevents fragmentationof the dark matter into e.g., “dark stars” and other compact struc-tures (e.g., Hoyle 1953; Rees 1976; Gammie 2001). This is quali-tatively different from baryon-like dissipative dark matter models.In addition, unlike those models that assume dissipative dark matteris a sub-component of all the dark matter, the model studied hereassumes that all the dark matter are dissipative. In our case, therewould be no external gravitational force that can suppress the growthof secular gravitational instabilities (e.g., Ostriker & Peebles 1973;Christodoulou et al. 1995), which prevents the formation of a coldand thin "dark disk" completely supported by rotation. In previous sections, we have presented several signatures of dSIDMmodels in dwarf galaxies that differ from their CDM counterparts. Inthis section, we discuss these phenomena in more detail and providesome physical explanations to the behaviours using simple analyticalarguments.
When 𝜎 / 𝑚 becomes large enough such that the dissipation timescale is comparable or lower than the Hubble time scale (1 / 𝐻 ), allthe dSIDM haloes in simulations first undergo radial contraction,accompanied by the steepening of the central density profiles. It issurprising that, during this phase, the asymptotic power-law slopesof the central density profiles of dwarf galaxies converge to ∼ − . ∼ . MNRAS000
When 𝜎 / 𝑚 becomes large enough such that the dissipation timescale is comparable or lower than the Hubble time scale (1 / 𝐻 ), allthe dSIDM haloes in simulations first undergo radial contraction,accompanied by the steepening of the central density profiles. It issurprising that, during this phase, the asymptotic power-law slopesof the central density profiles of dwarf galaxies converge to ∼ − . ∼ . MNRAS000 , 1–25 (2021) Shen et al. result in different evolution patterns. Gas clouds exhibit much higherparticle scattering rates and less energy dissipation per scattering, sothe collisional relaxation time scale is orders of magnitude shorterthan the cooling time scale, which means that global thermal equi-librium is easier to be established in gas clouds. During the earlycontraction of gas clouds, it is often assumed that the compressionalheating will offset the radiative loss of thermal energy and keep thecloud nearly isothermal (e.g., Gaustad 1963; Shu 1977). However, indSIDM haloes, since the dissipation time scale is comparable to thecollision time scale (see Section 3), the dSIDM fluid cannot adjustitself to global thermal equilibrium during the contraction of the sys-tem, which is qualitatively different from the isothermal contractionof gas clouds. This is supported by the fact that the velocity disper-sion profiles (shown in Figure 3) at the centers of simulated dwarfsin are never flat in dSIDM models, contrary to the isothermal profilesin eSIDM cases.For gas clouds, the isothermal contraction will gradually increasethe imbalance of gravitational forces over thermal pressure forces,which eventually results in the free-fall collapse of the central partof the cloud (e.g., Bodenheimer & Sweigart 1968; Penston 1969a,b;Larson 1969; Shu 1977; Hunter 1977; Foster & Chevalier 1993).In terms of time scales, the free-fall collapse will happen when thecooling time scale becomes shorter than the dynamical time scale ofthe cloud. However, in dSIDM haloes, this is also prohibited, sincethe dissipation time scale (in the surveyed parameter space) is ordersof magnitude larger than the dynamical time scale of the system. Asthe dissipation of thermal/kinetic energy drives the contraction ofthe halo on the dynamical time scale, dark matter particles could begravitationally accelerated again, which would effectively increasethe thermal pressure and slow down the collapse. Moreover, on thedynamical time scale, dark matter particles from different radii can“mix” because they are only weakly collisional as oppose to gas. Asa consequence, even though the global thermal equilibrium of thesystem is broken, the contraction would still be much slower than thefree-fall collapse of gas clouds (as found in Figure 11).We find the behaviour of our systems can be reasonably describedby the solution for a “slow” quasi-equilibrium cooling flow (with neg-ligible thermal conduction) rather than rapid isothermal or gravother-mal “collapse”. Following Stern et al. (2019), the continuity equationof an adiabatic slow-cooling halo, that is spherically symmetric andisotropic Jeans pressure supported, can be written asd ln 𝜌 d ln 𝑟 + d ln 𝑣 r d ln 𝑟 = − , (16)where 𝜌 is the density of the fluid and 𝑣 r is the radial inflow velocity.The momentum equation and the entropy equation of the system canbe reduced to (Stern et al. 2019)d ln 𝑣 r d ln 𝑟 (cid:16) 𝑣 𝑐 − (cid:17) = − 𝑣 𝑐 − 𝑟 / 𝑣 r 𝛾𝑡 cool , (17)where 𝑣 c is the circular velocity, 𝑐 s is the adiabatic sound speed, 𝛾 is the adiabatic index and 𝑡 cool is the cooling time scale of the fluid.Applying the solution to the cooling flow of dark matter, we replacethe sound speed 𝑐 s with the one-dimensional velocity dispersion ofdark matter 𝜎 and the cooling time scale 𝑡 cool with the dissipationtime scale 𝑡 diss of dark matter self-interactions. In the "subsonic"limit ( 𝑣 r (cid:28) 𝜎 ), the second equation becomes − d ln 𝑣 r d ln 𝑟 = − 𝑣 𝜎 − 𝑟 / 𝑣 r 𝛾𝑡 diss . (18)A simple self-similar solution exists by requiring that all the logarith-mic derivatives of dark matter properties are constants. Then 𝑣 / 𝜎 and ( 𝑟 / 𝑣 r )/ 𝑡 diss also need to be constants. If we assume 𝜌 ∼ 𝑟 𝛼 , weobtain the scaling of the one-dimensional velocity dispersion as 𝜎 ∼ 𝑣 c ∼ √︁ 𝐺 𝑀 enc ( 𝑟 )/ 𝑟 ∼ 𝑟 + 𝛼 / . (19)In the meantime, Equation 16 implies that 𝑣 r ∼ 𝑟 − 𝛼 − . Accordingto Equation 8, the dissipation time scale 𝑡 diss scales with densityand velocity dispersion as 𝜌 − 𝜎 − ∼ 𝑟 −( + 𝛼 / ) . If we plug in thescaling of 𝑣 r and 𝑡 diss to the term ( 𝑟 / 𝑣 r )/ 𝑡 diss , we obtain 𝑟 / 𝑣 r 𝑡 diss ∼ 𝑟 𝑟 𝛼 + 𝑟 −( + 𝛼 / ) ∼ 𝑟 + 𝛼 / . (20)So the power-law solution (which requires the term to be a constantat all radii) has 𝛼 = − /
5. Quantitatively, the slope of the densityprofile given by this “dark cooling flow” solution is consistent withthe finding in dSIDM simulations that the asymptotic slopes of thedensity profiles converge to around − .
5. It also predicts 𝜎 ∼ 𝑟 . ,which is consistent with the central velocity dispersions of simulateddwarfs that mildly increase with radii. In general, “thermal conduction” and dissipation are the two mainmechanisms in SIDM haloes to transfer kinetic energy of dark matter.“Thermal conduction” is dark matter collisional energy transfer. Thedetailed form of the heat conductivity depends on the nature of theheat conduction. In the theory of thermal conductivity of an idealfluid, the heat flux is the averaged one-way flux of particles across animaginary surface multiplied by the difference in energy per particlebetween the starting and ending points. Up to order unity corrections,this gives 𝜅 (cid:39) 𝑘 B 𝑚 𝜌 𝑙 𝜏 (21)where 𝑘 B is the Boltzmann constant, 𝑙 is the characteristic distancebetween the starting and ending points and 𝜏 is the time between col-lisions. In SIDM haloes, the collision (or close encounters) betweenparticles is governed by dark matter self-interactions since the colli-sion time scale of dark matter self-interaction is significantly lowerthan the two-body gravitational relaxation time scale. Thus, we have 𝜏 = 𝑡 coll . If the mean free path between collisions is significantlyshorter than the physical size of the system (referred to as the ShortMean Free Path (SMFP) regime), dark matter will behave like a fluidand the heat conductivity is fully regulated by the mean free path ofdark matter particles ( 𝑙 = 𝜆 = /( 𝜌𝜎 / 𝑚 ) ). Therefore, in this regime,the thermal conductivity is: 𝜅 = 𝑘 B 𝑚 𝐶 𝜌 𝜆 𝑡 coll , (22)where 𝐶 is an order unity constant and has been found to be ( √ 𝜋 / )/( /√ 𝜋 ) in the Chapman-Enskog theory (e.g., Chapman& Cowling 1970; Lifshitz & Pitaevskii 1981) and 0 . /( /√ 𝜋 ) innumerical simulations (Koda & Shapiro 2011).On the other hand, this picture is not valid when the mean free pathbetween collisions is much larger than the gravitational scale height 𝐻 of the system (referred to as the Long Mean Free Path (LMFP)regime), defined as: 𝐻 = √︄ 𝜎 𝜋𝐺 𝜌 . (23)In this regime, particles can travel several orbits before experiencinga collision. Lynden-Bell & Eggleton (1980) found that the character-istic distance between encounters in this limit (for weakly collisional MNRAS , 1–25 (2021) issipative Dark Matter on FIRE r [kpc] l og ( | ˙ E c o ll / ˙ E d i ss | ) m10q˙ E coll = ˙ E diss ( σ/m ) = 1 .
00 cm / g , f diss = 0 . collisional energy gaincollisional energy loss r [kpc] l og ( | ˙ E c o ll / ˙ E d i ss | ) m11a˙ E coll = ˙ E diss ( σ/m ) = 1 .
00 cm / g , f diss = 0 . collisional energy gaincollisional energy loss Figure 14. Dark matter energy transfer rates via “thermal conduction”(dark matter collisional energy transfer) versus dissipation energy lossrates, measured in spherical shells, as a function of galactocentric radii.
We show the heat gain or loss of dark matter via collisions ( (cid:164) 𝐸 coll , Equation 26)versus the energy dissipation rate ( (cid:164) 𝐸 diss , Equation 27) in circles (red for (cid:164) 𝐸 coll >
0, blue for (cid:164) 𝐸 coll < m10q and in one of the bright dwarfs m11a . In both galaxies, with 𝑓 diss = .
5, the collisional energy transfer rate is always roughly an order ofmagnitude lower than the energy dissipation rate. fluid) can be roughly described by the gravitational scale height( 𝑙 = 𝐻 ). In this case, the thermal conductivity is: 𝜅 = 𝑘 B 𝑚 𝐶 𝜌 𝐻 𝑡 coll , (24)where 𝐶 is an order unity constant and has been found to be 0 .
75 innumerical simulations (Koda & Shapiro 2011). For the fiducial modelstudied in the paper, the mean free path of dark matter self-interactionis always orders of magnitudes larger than the gravitational scaleheight of the systems (or translated to time scale, the collision timescale of dark matter self-interaction is orders of magnitudes largerthan the dynamical time scale of the system). So, these haloes all stayin the LMFP regime.The flux of thermal energy transferred outward through a sphereof radius 𝑟 can be calculated as: 𝑗 coll ( 𝑟 ) = − 𝜅 𝜕𝑇 ( 𝑟 ) 𝜕𝑟 = − 𝜅 𝑚𝑘 B 𝜕𝜎 ( 𝑟 ) 𝜕𝑟 , (25)where 𝜅 takes the conductivity in the LMFP regime defined in Equa-tion 24. The net collisional energy gain per unit volume in a spherical shell can be calculated as: (cid:164) 𝐸 coll ( 𝑟 ) = − 𝜋𝑟 𝜕 ( 𝜋𝑟 𝑗 coll ( 𝑟 )) 𝜕𝑟 . (26)The second mechanism of energy transfer is energy dissipation dueto dark matter self-interactions. Different from “thermal conduction”,the dissipation we modelled here is not regulated by any characteristiclength scale, since the dissipated energy will not be reabsorbed andeffectively has an infinite mean free path. The dissipation energy lossper unit volume in a spherical shell is the volumetric cooling rate: (cid:164) 𝐸 diss ( 𝑟 ) = 𝐶 ( 𝑟 ) = 𝜌 ( 𝑟 ) 𝜎 ( 𝑟 )/ 𝑡 diss ( 𝑟 ) . (27)The relative importance of collisional energy transfer and dissipationis determined by the comparison between 𝑡 coll and 𝑡 diss . For thedSIDM model studied in this paper, 𝑡 coll and 𝑡 diss always have similardependence on density and velocity dispersion. Thus, their ratio isalmost a constant over the evolution of the halo and only dependson 𝑓 diss . For the fiducial model with 𝑓 diss = . 𝑡 diss is of the sameorder of magnitude as 𝑡 coll (e.g., 𝑡 diss = . 𝑡 coll / 𝑓 diss for the modelswith constant cross-sections). In this regime, dissipation is alwaysthe dominant mechanism for energy transfer and is responsible fortriggering the contraction of the halo. Collisional energy transfer isnegligible. Therefore, the evolution pattern of dSIDM haloes in thisregime will be qualitatively different from the canonical gravothermalcollapse of eSIDM haloes.In Figure 14, we demonstrate the dominance of dissipation overcollisional energy transfer in simulations. We show the collisionalenergy transfer rate, (cid:164) 𝐸 coll , relative to the energy loss rate due todissipation, (cid:164) 𝐸 diss , of spherical shells as a function of galactocentricradii. In the classical and bright dwarfs, assuming the fiducial choiceof 𝑓 diss , the rate of energy transfer via collisions is always roughlyan order of magnitude lower than the energy dissipation rate. When dissipation dominates over collisional energy transfer of darkmatter, the evolution track of an isolated dSIDM halo can be dividedinto four regimes, depending on the dissipation time scale 𝑡 diss • Regime A ( 𝑡 diss (cid:29) 𝑡 H ): The halo evolves in the same way as analo-gous CDM halo since both 𝑡 diss and 𝑡 coll are significantly longer thanthe lifetime of the system. • Regime B ( 𝑡 H (cid:38) 𝑡 diss (cid:38) . 𝑡 H ): The halo undergoes radial contrac-tion. The density profile within the radius where 𝑡 H (cid:38) 𝑡 diss steepensand becomes cuspy with power-law slopes asymptoting to ∼ − . • Regime C (0 . 𝑡 H (cid:38) 𝑡 diss (cid:29) 𝑡 dyn at the halo center): At a certainstage of the radial contraction, prominent coherent rotation of darkmatter will develop in the system. The system is in a transitionfrom purely dispersion supported to being supported by a mixtureof random velocity dispersion and coherent rotation. During thistransition, the radial contraction of the halo and the steepening of thedensity profile are stopped by centrifugal forces. The halo becomesoblate in shape during this phase and the normalization of the densityprofile measured in spherical shells decreases. • Regime D ( 𝑡 dyn (cid:38) 𝑡 diss ): Local instability starts to build up and re-sults in fragmentation of the halo. Numbers of dark "clumps" wouldstart to form within the local free-fall time scale. None of our simula-tions has reached this regime and it would require order-of-magnitudelarger self-interaction cross-sections to test. MNRAS000
75 innumerical simulations (Koda & Shapiro 2011). For the fiducial modelstudied in the paper, the mean free path of dark matter self-interactionis always orders of magnitudes larger than the gravitational scaleheight of the systems (or translated to time scale, the collision timescale of dark matter self-interaction is orders of magnitudes largerthan the dynamical time scale of the system). So, these haloes all stayin the LMFP regime.The flux of thermal energy transferred outward through a sphereof radius 𝑟 can be calculated as: 𝑗 coll ( 𝑟 ) = − 𝜅 𝜕𝑇 ( 𝑟 ) 𝜕𝑟 = − 𝜅 𝑚𝑘 B 𝜕𝜎 ( 𝑟 ) 𝜕𝑟 , (25)where 𝜅 takes the conductivity in the LMFP regime defined in Equa-tion 24. The net collisional energy gain per unit volume in a spherical shell can be calculated as: (cid:164) 𝐸 coll ( 𝑟 ) = − 𝜋𝑟 𝜕 ( 𝜋𝑟 𝑗 coll ( 𝑟 )) 𝜕𝑟 . (26)The second mechanism of energy transfer is energy dissipation dueto dark matter self-interactions. Different from “thermal conduction”,the dissipation we modelled here is not regulated by any characteristiclength scale, since the dissipated energy will not be reabsorbed andeffectively has an infinite mean free path. The dissipation energy lossper unit volume in a spherical shell is the volumetric cooling rate: (cid:164) 𝐸 diss ( 𝑟 ) = 𝐶 ( 𝑟 ) = 𝜌 ( 𝑟 ) 𝜎 ( 𝑟 )/ 𝑡 diss ( 𝑟 ) . (27)The relative importance of collisional energy transfer and dissipationis determined by the comparison between 𝑡 coll and 𝑡 diss . For thedSIDM model studied in this paper, 𝑡 coll and 𝑡 diss always have similardependence on density and velocity dispersion. Thus, their ratio isalmost a constant over the evolution of the halo and only dependson 𝑓 diss . For the fiducial model with 𝑓 diss = . 𝑡 diss is of the sameorder of magnitude as 𝑡 coll (e.g., 𝑡 diss = . 𝑡 coll / 𝑓 diss for the modelswith constant cross-sections). In this regime, dissipation is alwaysthe dominant mechanism for energy transfer and is responsible fortriggering the contraction of the halo. Collisional energy transfer isnegligible. Therefore, the evolution pattern of dSIDM haloes in thisregime will be qualitatively different from the canonical gravothermalcollapse of eSIDM haloes.In Figure 14, we demonstrate the dominance of dissipation overcollisional energy transfer in simulations. We show the collisionalenergy transfer rate, (cid:164) 𝐸 coll , relative to the energy loss rate due todissipation, (cid:164) 𝐸 diss , of spherical shells as a function of galactocentricradii. In the classical and bright dwarfs, assuming the fiducial choiceof 𝑓 diss , the rate of energy transfer via collisions is always roughlyan order of magnitude lower than the energy dissipation rate. When dissipation dominates over collisional energy transfer of darkmatter, the evolution track of an isolated dSIDM halo can be dividedinto four regimes, depending on the dissipation time scale 𝑡 diss • Regime A ( 𝑡 diss (cid:29) 𝑡 H ): The halo evolves in the same way as analo-gous CDM halo since both 𝑡 diss and 𝑡 coll are significantly longer thanthe lifetime of the system. • Regime B ( 𝑡 H (cid:38) 𝑡 diss (cid:38) . 𝑡 H ): The halo undergoes radial contrac-tion. The density profile within the radius where 𝑡 H (cid:38) 𝑡 diss steepensand becomes cuspy with power-law slopes asymptoting to ∼ − . • Regime C (0 . 𝑡 H (cid:38) 𝑡 diss (cid:29) 𝑡 dyn at the halo center): At a certainstage of the radial contraction, prominent coherent rotation of darkmatter will develop in the system. The system is in a transitionfrom purely dispersion supported to being supported by a mixtureof random velocity dispersion and coherent rotation. During thistransition, the radial contraction of the halo and the steepening of thedensity profile are stopped by centrifugal forces. The halo becomesoblate in shape during this phase and the normalization of the densityprofile measured in spherical shells decreases. • Regime D ( 𝑡 dyn (cid:38) 𝑡 diss ): Local instability starts to build up and re-sults in fragmentation of the halo. Numbers of dark "clumps" wouldstart to form within the local free-fall time scale. None of our simula-tions has reached this regime and it would require order-of-magnitudelarger self-interaction cross-sections to test. MNRAS000 , 1–25 (2021) Shen et al. -1 r [kpc] ρ t o t [ M fl / k p c ] m09 CDMNFW fitdSIDM ( . / g , f diss = 0 . ) dSIDM ( / g , f diss = 0 . ) dSIDM ( / g , f diss = 0 . ) -1 r [kpc] l og ( | ˙ E c o ll / ˙ E d i ss | ) m09˙ E coll = ˙ E diss ( σ/m ) = 0 .
56 cm / g , f diss = 0 . collisional energy gaincollisional energy loss -1 r [kpc] l og ( | ˙ E c o ll / ˙ E d i ss | ) m09˙ E coll = ˙ E diss ( σ/m ) = 1 .
00 cm / g , f diss = 0 . collisional energy gaincollisional energy loss -1 r [kpc] l og ( | ˙ E c o ll / ˙ E d i ss | ) m09˙ E coll = ˙ E diss ( σ/m ) = 5 .
00 cm / g , f diss = 0 . collisional energy gaincollisional energy loss Figure 15.
Top left:
Total mass density profiles of m09 in dSIDM models with other combinations of 𝑓 diss and 𝜎 / 𝑚 . We choose three combinations of 𝑓 diss and 𝜎 / 𝑚 that give the same dissipation time scale: 𝑓 diss = . , 𝜎 / 𝑚 = g − ; 𝑓 diss = . , 𝜎 / 𝑚 = g − ; 𝑓 diss = . , 𝜎 / 𝑚 = .
56 cm g − . Otherpanels:
Collisional energy transfer rates versus energy dissipation rate of dark matter (as Figure 14) . The energy transfer rate via collisions is subdominantcompare to dissipation in the model with 𝑓 diss = . .
9. In the model with 𝑓 diss = .
1, collisional heating overtakes dissipation at the center of the galaxy.This model actually produces denser and cuspier central density profile, as the halo experiences the gravothermal collapse and a dense core in the SMFP regimeemerges at the center. In all models, at large radii ( ∼
10 kpc), collisional energy transfer rates become comparable to the dissipation rate, but the absolute valueof both terms at these radii are too small to make a difference.
We note that the specific simulations studied in this paper have as-sumed that the dimensionless degree of dissipation is 𝑓 diss = . 𝑓 diss and 𝜎 / 𝑚 . Therefore, when dissipation is the dominantmechanism for energy transfer, different combinations of 𝑓 diss and 𝜎 / 𝑚 should give rise to similar predictions as long as the dissipationtime scale is the same. In this section, we vary the dissipation fraction 𝑓 diss and test how the results are affected in explicit simulations.We use the ultra faint dwarf m09 as the test halo. The halo is idealfor the test since the density profile is dark matter dominated andbaryonic feedback is weak considering its 𝑀 ∗ / 𝑀 halo (cid:46) × − . Wechoose three combinations of 𝑓 diss and 𝜎 / 𝑚 that give the same dissi-pation time scale: 𝑓 diss = . , 𝜎 / 𝑚 = g − ; 𝑓 diss = . , 𝜎 / 𝑚 = g − ; 𝑓 diss = . , 𝜎 / 𝑚 = .
56 cm g − . In Figure 15, we showthe total mass density profile of m09 in these three models comparedwith the CDM counterpart and the NFW profile. The models with 𝑓 diss = . 𝑓 diss = . 𝑓 diss = . 𝜎 / 𝑚 = g − ) produces a qualitatively different profile from theother two models. The density follows the NFW profile at (cid:38)
100 pcwhile gets enhanced by about two orders of magnitude at the scale (cid:46)
100 pc compared to the extrapolation of the NFW profile, andis even denser than the cuspy profile in the other two models. It iscounterintuitive that the model with a lower degree of dissipationgives rise to higher central densities. The phenomenon can be ex-plained by the increased importance of collisional energy transfer inthis model. When 𝑓 diss = .
1, the collision time scale becomes anorder of magnitude lower than the dissipation time scale and the halois no longer purely dominated by dissipation. Under the influence ofcollisional energy transfer, the evolution track of the halo resemblesthe “gravothermal catastrophe” of eSIDM haloes, where “thermalconduction” is responsible for energy transfer. The analytical modelof the “gravothermal catastrophe” of SIDM haloes (e.g., Balberget al. 2002) predicts that a halo initially in the LMFP regime willcontract while maintaining a cored, self-similar density profile untilthe central part of the halo reaches the SMFP regime. Subsequently,a dense, optical thick core (in the SMFP regime) will form while theoutskirt of the halo stays in the LMFP regime. In the simulation with
MNRAS , 1–25 (2021) issipative Dark Matter on FIRE Figure 16. A cartoon of the dSIDM parameter space.
The dSIDM model isparameterized with 𝜎 / 𝑚 and 𝑓 diss . When 𝜎 / 𝑚 is small enough, both elasticand dissipative SIDM models become analogous to CDM in the lifetime ofthe Universe. When 𝑓 diss becomes small enough, dSIDM becomes essentiallyeSIDM-like since collisional energy transfer dominates over dissipation inthis regime. When the product of 𝜎 / 𝑚 and 𝑓 diss becomes large enough, thedissipation time scale could drop below the local dynamical time scale ofthe system and results in fragmentation of dSIDM into compact dark objects.Effectively, baryon-like models are located at the low 𝑓 diss , high 𝜎 / 𝑚 cornerof the plot. The dSIDM models studied in this paper live in the parameterspace, which is not immediately ruled out but can still give rise to uniquephenomena different from CDM or eSIDM models. 𝑓 diss = .
1, at the center of m09 , the density reaches 10 M (cid:12) / kpc and the collision time scale there is comparable to the dynamicaltime scale (assuming a typical one-dimensional velocity dispersion ∼
10 km s − ) which indicates that the center of the halo is indeed inthe SMFP regime. It is striking that the enhanced central density dueto the gravothermal evolution is even higher than that produced bymodels with higher degree of dissipation.We verify that the phenomenon discussed above is indeed causedby increased importance of “thermal conduction” by showing the col-lisional energy transfer rates versus dissipation rates in simulationsin Figure 15. In the model with 𝑓 diss = . , 𝜎 / 𝑚 = .
56 cm g − or 𝑓 diss = . , 𝜎 / 𝑚 = g − , the collisional energy transferrate is always subdominant compared to dissipation. However, in themodel with 𝑓 diss = . , 𝜎 / 𝑚 = g − , the collisional energytransfer rate overtakes dissipation at small radii ( (cid:46) . | (cid:164) 𝐸 coll | (cid:29) | (cid:164) 𝐸 diss | at halo centers, which occurs for 𝑓 diss (cid:46) .
1, thehalo behaves more like an eSIDM halo and the higher central densityis primarily due to the gravothermal evolution driven by collisionalenergy transfer (but potentially accelerated by dissipation).To better illustrate the parameter space of dSIDM (including thespace that haven’t been explored in this paper), we create a cartoonimage (Figure 16) which qualitatively divides the dSIDM parame-ter space into several regions. The dSIDM models are parameter-ized with 𝜎 / 𝑚 and 𝑓 diss . Both eSIDM and dSIDM models becomeCDM-like when 𝜎 / 𝑚 is small enough such that the collision timescale becomes much longer than the lifetime of the Universe. dSIDMbecomes essentially eSIDM-like when 𝑓 diss becomes small enough,since collisional energy transfer dominates over dissipation in thisregime. When the product of 𝜎 / 𝑚 and 𝑓 diss becomes large enough,the dissipation time scale could drop below the local dynamical time -1 r [kpc] ρ t o t [ M fl / k p c ] m10q CDMdSIDM ( / g ) dark matter only(DMO) -1 r [kpc] ρ t o t [ M fl / k p c ] m11q Figure 17. Total mass density profiles of galaxies in DMO simulationsand full physics simulations.
We present the density profiles of m10q and m11q in CDM and dSIDM with ( 𝜎 / 𝑚 ) = g − . The results of fullphysics simulations are shown in solid lines while the results of DMO simu-lations are shown in dashed lines. The purple dotted vertical line indicates theconvergence radius in DMO runs (see Table 1). In CDM, the central densityprofiles in DMO simulations are similar to the NFW profile before reachingthe convergence radii. The full physics simulation of m11q produces a kpcsize core at the center due to strong baryonic feedback there. However, in thedSIDM model with ( 𝜎 / 𝑚 ) = g − , the DMO and full physics simula-tions produce almost identical results, indicating that dissipative interactionsof dark matter completely determine the evolution of the dark matter haloand the impact of baryonic feedback becomes negligible. This is generallytrue when the dissipation time scale becomes significantly shorter than theHubble time scale. scale of the system and results in fragmentation of dSIDM into com-pact dark objects. For higher value of 𝑓 diss and 𝜎 / 𝑚 , the scenariothat all dark is dissipative would be ruled out by observations (e.g.,constraints from merger clusters (Markevitch et al. 2004; Randallet al. 2008); lensing constraints on compact dark matter substruc-tures). If we put baryons (and baryon-copy dSIDM models) in thisspace effectively, they will be located at the low 𝑓 diss , high 𝜎 / 𝑚 corner of the plot. Thus the interesting dSIDM parameter space thatgives unique phenomena but is not immediately ruled out is roughlyaround 𝑓 diss (cid:39) . − ( 𝜎 / 𝑚 ) (cid:39) . −
100 cm g − . The analysis and discussion in the main paper revolve around theimpact of dissipative dark matter interactions on galaxy structures.
MNRAS000
MNRAS000 , 1–25 (2021) Shen et al.
However, baryonic physics could also impact galaxy structures in var-ious ways. For instance, the gas outflow driven by stellar/supernovaefeedback could irreversibly transfer energy to dark matter and in-duce cores at galaxy centers (e.g., Governato et al. 2010; Pontzen &Governato 2012; Madau et al. 2014); the gravitational influence ofbaryons condensed at galaxy centers could induce adiabatic contrac-tion of dark matter haloes (e.g., Blumenthal et al. 1986; Gnedin et al.2004). The contamination of baryonic physics processes is an im-portant factor when studying the influence of alternative dark matterphysics.We explore this aspect by performing dark matter only (DMO)simulations of the same haloes in the simulation suite and compar-ing the results. In Figure 17, we compare the total mass densityprofiles of dwarf galaxies m10q and m11q in DMO simulations andfull physics simulations. It is not surprising that, in the CDM case,the density profiles produced by DMO simulations are cuspy andNFW-like before reaching the convergence radii. In full physics sim-ulations, m11q exhibits a kpc size core while m10q still exhibits acuspy profile like its DMO counterpart. The difference results fromthe different level of baryonic feedback in the two galaxies. However,in the dSIDM model with ( 𝜎 / 𝑚 ) = g − , the DMO and fullphysics simulations produce almost the same density profiles, indi-cating that baryonic physics no longer affect the density profiles ofdwarf galaxies once dissipation is strong enough . This check alsovalidates the results presented in this paper against uncertainties inmodelling the baryonic physics processes in simulations. In this paper, we present the first suite of cosmological baryonic (hy-drodynamical) zoom-in simulations of galaxies in dSIDM. We adopta dSIDM model where a constant fraction 𝑓 diss of the kinetic energy islost during dark matter self-interaction. We sample models with dif-ferent constant self-interaction cross-sections as well as a model withvelocity-dependent cross-section. The dSIDM models explored hereare weakly collisional ( 𝜎 / 𝑚 (cid:46)
10 cm g − ) but strongly dissipative( 𝑓 diss (cid:38) .
1) and are qualitatively different from some previouslyproposed baryon-like dSIDM models (e.g., Fan et al. 2013a; Foot2013; Randall & Scholtz 2015), which are limited to explain a subsetof all dark matter in the Universe. The simulations utilize the FIRE-2 model for hydrodynamics and galaxy formation physics, whichallows for realistic predictions on the structural and kinematic prop-erties of galaxies. This simulation suite consists of various galaxies,from ultra faint dwarfs to Milky Way-mass galaxies. In this paper,we primarily focus on the analysis of dwarf galaxies in dSIDM andexplore galaxy/halo’s response to dissipative self-interactions of darkmatter. The following signatures of dSIDM models in dwarf galaxiesare identified and explored: • The dark matter halo masses and galaxy stellar masses are notsignificantly affected in dSIDM models with ( 𝜎 / 𝑚 ) (cid:46) g − compared to the CDM case (see Figure 4). The dwarf galaxies inthe dSIDM model with ( 𝜎 / 𝑚 ) =
10 cm g − have slightly lower(0 . . • Energy dissipation due to dark matter self-interactions induces ra-dial contraction of dark matter halo. This mechanism competes withbaryonic feedback in shaping the central profiles of dwarf galaxies(see Figure 5 and 6). When the effective self-interaction cross-sectionis low, the central profiles are still cored despite higher densities and smaller core sizes. When the effective self-interaction cross-sectionis larger than ∼ . g − , assuming 𝑓 diss = .
5, the central densityprofiles of dwarf galaxies become cuspy and power-law like. The re-sulting asymptotic power-law profile is steeper than the NFW profile.The power-law slopes asymptote to ∼ − . − − − . • Interestingly, further increasing the effective cross-section to10 cm g − does not lead to further contraction of the halo or steep-ening of the density profile. Instead, the normalization of the densityprofiles drops. A likely explanation is that the centrifugal force in-creases faster than the gravitational attraction as the halo contractswith specific angular momentum conserved. This eventually haltsthe contraction, increases the rotation support of the halo and drivesthe halo deformation (to oblate), which makes the density measuredin spherical shell decreased. • Through time scale analysis (Section 3), we show that the dSIDMmodels with constant cross-sections will have stronger impact inmore massive galaxies while the velocity-dependent model has theopposite dependence. This is demonstrated by the simulations ofclassical dwarfs and bright dwarfs with the same dark matter model(see Figure 5 and 6). The dSIDM model with ( 𝜎 / 𝑚 ) = . g − produces small cores in two of the classical dwarfs but producescuspy profiles in two of the bright dwarfs. The velocity-dependentdSIDM model produces cuspy profiles in all the classical dwarfswhile producing cored profiles in the bright dwarfs that are almostidentical to the CDM case. • The kinematic properties of the dark matter change in parallel tothe contraction of dark matter halo (see Section 4.4). As the self-interaction cross-section of dSIDM increases, the coherent rotationbecomes more prominent compared to random velocity dispersion.In the meantime, the velocity dispersions are more dominated bythe tangential component than the radial component, reflected by thenegative velocity anisotropies in dSIDM haloes. The central partsof the galaxies are in transition from dispersion supported to ro-tation supported. Meanwhile, the velocity distribution function issuppressed at high velocities while it increases at low velocities indSIDM models. As the cross-section increases, the median velocityis also shifted lower. • The shape of the halo is affected by dissipation (see Figure 13). Inthe dSIDM model with ( 𝜎 / 𝑚 ) = g − , the halo becomes morespherical towards lower redshifts, contrary to the triaxial shape ofCDM haloes. The spherical “dark cooling flow” washes out the initialtriaxiality of the halo and makes the halo compact and spherical in theend. However, in the dSIDM model with ( 𝜎 / 𝑚 ) =
10 cm g − , thehalo shape shows a response to the more prominent coherent rotationof dark matter. Haloes are initially on the track of becoming morespherical, but later turn oblate in shape due to the halt of sphericalcontraction and increased rotation support. • As shown in Section 6.2, the energy transfer in dSIDM haloes (withthe degree of dissipation 𝑓 diss = .
5) is dominated by dissipationrather than “thermal conduction” (collisional energy transfer). Whenwe vary 𝑓 diss to (cid:46) .
1, collisional energy transfer becomes importantand the density at small radii ( 𝑟 (cid:46)
100 pc) is significantly enhanced(see Figure 15), which resembles the gravothermal collapse of eS-IDM haloes. This gives the counterintuitive prediction that a modelwith a lower degree of dissipation (but higher cross-section to makethe dissipation time scale invariant) can produce even denser haloesthan models with higher degrees of dissipation. • The density profiles in full physics simulations of CDM are more
MNRAS , 1–25 (2021) issipative Dark Matter on FIRE cored than the ones in DMO simulations, caused by the inclusion ofbaryonic physics. However, the DMO simulations of dSIDM mod-els show little difference from the full physics simulations (see Fig-ure 17), likely due to the dominance of dark matter energy dissipationover perturbations from baryonic feedback. This shows that the struc-tural properties of dSIDM haloes is insensitive to baryonic physicsin this regime and demonstrates the robustness of our results againstvarious uncertainties in the baryonic sector in simulations.In this paper, we present the first study of dwarf galaxies in dSIDMmodels using cosmological hydrodynamical simulations. We findseveral observable signatures of dSIDM models in dwarf galaxies andsystematically study the evolution patterns of dSIDM haloes, whichdiffers from canonical astrophysical systems. Analytical explanationsare provided to explain the phenomena found in simulations. Thefindings in this paper could serve as effective channels to constraindSIDM models when compared to observations. This aspect will beconsidered in follow-up work in this series. ACKNOWLEDGEMENTS
Support for PFH was provided by NSF Research Grants 1911233& 20009234, NSF CAREER grant 1455342, NASA grants80NSSC18K0562, HST-AR-15800.001-A. Numerical calculationswere run on the Caltech compute cluster “Wheeler”, allocationsFTA-Hopkins/AST20016 supported by the NSF and TACC, andNASA HEC SMD-16-7592. LN is supported by the DOE underAward Number DESC0011632, the Sherman Fairchild fellowship,the University of California Presidential fellowship, and the fel-lowship of theoretical astrophysics at Carnegie Observatories. FJis supported by the Troesh scholarship. MBK acknowledges sup-port from NSF CAREER award AST-1752913, NSF grant AST-1910346, NASA grant NNX17AG29G, and HST-AR-15006, HST-AR-15809, HST-GO-15658, HST-GO-15901, HST-GO-15902, HST-AR-16159, and HST-GO-16226 from the Space Telescope ScienceInstitute, which is operated by AURA, Inc., under NASA contractNAS5-26555. AW received support from NASA through ATP grants80NSSC18K1097 and 80NSSC20K0513; HST grants GO-14734,AR-15057, AR-15809, and GO-15902 from STScI; a Scialog Awardfrom the Heising-Simons Foundation; and a Hellman Fellowship.
DATA AVAILABILITY
The simulation data of this work was generated and stored on thesupercomputing system Frontera at the Texas Advanced Comput-ing Center (TACC), under the allocations FTA-Hopkins/AST20016supported by the NSF and TACC, and NASA HEC SMD-16-7592.The data underlying this article cannot be shared publicly immedi-ately, since the series of paper is still in development. The data willbe shared on reasonable request to the corresponding author. REFERENCES
Ackerman L., Buckley M. R., Carroll S. M., Kamionkowski M., 2009, Phys.Rev. D, 79, 023519Aprile E., et al., 2018, Phys. Rev. Lett., 121, 111302Arkani-Hamed N., Finkbeiner D. P., Slatyer T. R., Weiner N., 2009, Phys.Rev. D, 79, 015014 https://frontera-portal.tacc.utexas.edu/about/ Balberg S., Shapiro S. L., Inagaki S., 2002, ApJ, 568, 475Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T., Klypin A. A., PrimackJ. R., 2012, Consistent Trees: Gravitationally Consistent Halo Catalogsand Merger Trees for Precision Cosmology (ascl:1210.011)Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013a, ApJ, 762, 109Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T., Klypin A. A., PrimackJ. R., 2013b, ApJ, 763, 18Bertone G., 2010, Nature, 468, 389Bertone G., Hooper D., Silk J., 2005, Phys. Rep., 405, 279Bett P., Eke V., Frenk C. S., Jenkins A., Helly J., Navarro J., 2007, MNRAS,376, 215Binney J., 1978, MNRAS, 183, 501Binney J., Tremaine S., 1987, Galactic dynamicsBinney J., Tremaine S., 2008, Galactic Dynamics: Second EditionBlennow M., Clementz S., Herrero-Garcia J., 2017, J. Cosmol. Astropart.Phys., 2017, 048Blumenthal G. R., Faber S. M., Primack J. R., Rees M. J., 1984, Nature, 311,517Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27Boddy K. K., Feng J. L., Kaplinghat M., Tait T. M. P., 2014, Phys. Rev. D,89, 115017Boddy K. K., Kaplinghat M., Kwa A., Peter A. H. G., 2016, Phys. Rev. D,94, 123017Bodenheimer P., Sweigart A., 1968, ApJ, 152, 515Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2011, MNRAS, 415, L40Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2012, MNRAS, 422, 1203Brinckmann T., Zavala J., Rapetti D., Hansen S. H., Vogelsberger M., 2018,MNRAS, 474, 746Brooks A. M., Zolotov A., 2014, ApJ, 786, 87Bryan G. L., Norman M. L., 1998, ApJ, 495, 80Buck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS,483, 1314Buckley M. R., Peter A. H. G., 2018, Phys. Rep., 761, 1Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343Burkert A., 2000, ApJ, 534, L143Cappellari M., et al., 2007, MNRAS, 379, 418Carlson E. D., Machacek M. E., Hall L. J., 1992, ApJ, 398, 43Chan T. K., Kereš D., Oñorbe J., Hopkins P. F., Muratov A. L., Faucher-Giguère C. A., Quataert E., 2015, MNRAS, 454, 2981Chapman S., Cowling T. G., 1970, The mathematical theory of non-uniformgases. an account of the kinetic theory of viscosity, thermal conductionand diffusion in gasesChristodoulou D. M., Shlosman I., Tohline J. E., 1995, ApJ, 443, 551Cline J. M., Liu Z., Moore G. D., Xue W., 2014, Phys. Rev. D, 90, 015023Colín P., Avila-Reese V., Valenzuela O., Firmani C., 2002, ApJ, 581, 777Creasey P., Sameie O., Sales L. V., Yu H.-B., Vogelsberger M., Zavala J.,2017, MNRAS, 468, 2283Cyr-Racine F.-Y., Sigurdson K., 2013, Phys. Rev. D, 87, 103515Dalcanton J. J., Hogan C. J., 2001, ApJ, 561, 35Davies R. L., Efstathiou G., Fall S. M., Illingworth G., Schechter P. L., 1983,ApJ, 266, 41Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985, ApJ, 292, 371Despali G., Sparre M., Vegetti S., Vogelsberger M., Zavala J., Marinacci F.,2019, MNRAS, 484, 4563Di Cintio A., Brook C. B., Macciò A. V., Stinson G. S., Knebe A., DuttonA. A., Wadsley J., 2014, MNRAS, 437, 415Diemand J., Zemp M., Moore B., Stadel J., Carollo C. M., 2005, MNRAS,364, 665Dutton A. A., Macciò A. V., Frings J., Wang L., Stinson G. S., Penzo C.,Kang X., 2016, MNRAS, 457, L74Elbert O. D., Bullock J. S., Garrison-Kimmel S., Rocha M., Oñorbe J., PeterA. H. G., 2015, MNRAS, 453, 29Elbert O. D., Bullock J. S., Kaplinghat M., Garrison-Kimmel S., Graus A. S.,Rocha M., 2018, ApJ, 853, 109Essig R., McDermott S. D., Yu H.-B., Zhong Y.-M., 2019, Phys. Rev. Lett.,123, 121102Fan J., Katz A., Randall L., Reece M., 2013a, Physics of the Dark Universe,2, 139 MNRAS000
Ackerman L., Buckley M. R., Carroll S. M., Kamionkowski M., 2009, Phys.Rev. D, 79, 023519Aprile E., et al., 2018, Phys. Rev. Lett., 121, 111302Arkani-Hamed N., Finkbeiner D. P., Slatyer T. R., Weiner N., 2009, Phys.Rev. D, 79, 015014 https://frontera-portal.tacc.utexas.edu/about/ Balberg S., Shapiro S. L., Inagaki S., 2002, ApJ, 568, 475Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T., Klypin A. A., PrimackJ. R., 2012, Consistent Trees: Gravitationally Consistent Halo Catalogsand Merger Trees for Precision Cosmology (ascl:1210.011)Behroozi P. S., Wechsler R. H., Wu H.-Y., 2013a, ApJ, 762, 109Behroozi P. S., Wechsler R. H., Wu H.-Y., Busha M. T., Klypin A. A., PrimackJ. R., 2013b, ApJ, 763, 18Bertone G., 2010, Nature, 468, 389Bertone G., Hooper D., Silk J., 2005, Phys. Rep., 405, 279Bett P., Eke V., Frenk C. S., Jenkins A., Helly J., Navarro J., 2007, MNRAS,376, 215Binney J., 1978, MNRAS, 183, 501Binney J., Tremaine S., 1987, Galactic dynamicsBinney J., Tremaine S., 2008, Galactic Dynamics: Second EditionBlennow M., Clementz S., Herrero-Garcia J., 2017, J. Cosmol. Astropart.Phys., 2017, 048Blumenthal G. R., Faber S. M., Primack J. R., Rees M. J., 1984, Nature, 311,517Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27Boddy K. K., Feng J. L., Kaplinghat M., Tait T. M. P., 2014, Phys. Rev. D,89, 115017Boddy K. K., Kaplinghat M., Kwa A., Peter A. H. G., 2016, Phys. Rev. D,94, 123017Bodenheimer P., Sweigart A., 1968, ApJ, 152, 515Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2011, MNRAS, 415, L40Boylan-Kolchin M., Bullock J. S., Kaplinghat M., 2012, MNRAS, 422, 1203Brinckmann T., Zavala J., Rapetti D., Hansen S. H., Vogelsberger M., 2018,MNRAS, 474, 746Brooks A. M., Zolotov A., 2014, ApJ, 786, 87Bryan G. L., Norman M. L., 1998, ApJ, 495, 80Buck T., Macciò A. V., Dutton A. A., Obreja A., Frings J., 2019, MNRAS,483, 1314Buckley M. R., Peter A. H. G., 2018, Phys. Rep., 761, 1Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343Burkert A., 2000, ApJ, 534, L143Cappellari M., et al., 2007, MNRAS, 379, 418Carlson E. D., Machacek M. E., Hall L. J., 1992, ApJ, 398, 43Chan T. K., Kereš D., Oñorbe J., Hopkins P. F., Muratov A. L., Faucher-Giguère C. A., Quataert E., 2015, MNRAS, 454, 2981Chapman S., Cowling T. G., 1970, The mathematical theory of non-uniformgases. an account of the kinetic theory of viscosity, thermal conductionand diffusion in gasesChristodoulou D. M., Shlosman I., Tohline J. E., 1995, ApJ, 443, 551Cline J. M., Liu Z., Moore G. D., Xue W., 2014, Phys. Rev. D, 90, 015023Colín P., Avila-Reese V., Valenzuela O., Firmani C., 2002, ApJ, 581, 777Creasey P., Sameie O., Sales L. V., Yu H.-B., Vogelsberger M., Zavala J.,2017, MNRAS, 468, 2283Cyr-Racine F.-Y., Sigurdson K., 2013, Phys. Rev. D, 87, 103515Dalcanton J. J., Hogan C. J., 2001, ApJ, 561, 35Davies R. L., Efstathiou G., Fall S. M., Illingworth G., Schechter P. L., 1983,ApJ, 266, 41Davis M., Efstathiou G., Frenk C. S., White S. D. M., 1985, ApJ, 292, 371Despali G., Sparre M., Vegetti S., Vogelsberger M., Zavala J., Marinacci F.,2019, MNRAS, 484, 4563Di Cintio A., Brook C. B., Macciò A. V., Stinson G. S., Knebe A., DuttonA. A., Wadsley J., 2014, MNRAS, 437, 415Diemand J., Zemp M., Moore B., Stadel J., Carollo C. M., 2005, MNRAS,364, 665Dutton A. A., Macciò A. V., Frings J., Wang L., Stinson G. S., Penzo C.,Kang X., 2016, MNRAS, 457, L74Elbert O. D., Bullock J. S., Garrison-Kimmel S., Rocha M., Oñorbe J., PeterA. H. G., 2015, MNRAS, 453, 29Elbert O. D., Bullock J. S., Kaplinghat M., Garrison-Kimmel S., Graus A. S.,Rocha M., 2018, ApJ, 853, 109Essig R., McDermott S. D., Yu H.-B., Zhong Y.-M., 2019, Phys. Rev. Lett.,123, 121102Fan J., Katz A., Randall L., Reece M., 2013a, Physics of the Dark Universe,2, 139 MNRAS000 , 1–25 (2021) Shen et al.
Fan J., Katz A., Randall L., Reece M., 2013b, Phys. Rev. Lett., 110, 211302Fan J., Katz A., Shelton J., 2014, J. Cosmol. Astropart. Phys., 2014, 059Fattahi A., et al., 2016, MNRAS, 457, 844Feng J. L., Kaplinghat M., Tu H., Yu H.-B., 2009, J. Cosmol. Astropart. Phys.,2009, 004Feng J. L., Kaplinghat M., Yu H.-B., 2010, Phys. Rev. Lett., 104, 151301Finkbeiner D. P., Weiner N., 2016, Phys. Rev. D, 94, 083002Firmani C., D’Onghia E., Avila-Reese V., Chincarini G., Hernández X., 2000,MNRAS, 315, L29Fitts A., et al., 2019, MNRAS, 490, 962Flores R. A., Primack J. R., 1994, ApJ, 427, L1Foot R., 2013, Phys. Rev. D, 88, 023520Foot R., Vagnozzi S., 2015, Phys. Rev. D, 91, 023512Foot R., Vagnozzi S., 2016, J. Cosmol. Astropart. Phys., 2016, 013Foster P. N., Chevalier R. A., 1993, ApJ, 416, 303Gammie C. F., 2001, ApJ, 553, 174Garrison-Kimmel S., Boylan-Kolchin M., Bullock J. S., Kirby E. N., 2014,MNRAS, 444, 222Garrison-Kimmel S., Bullock J. S., Boylan-Kolchin M., Bardwell E., 2017,MNRAS, 464, 3108Garrison-Kimmel S., et al., 2018, MNRAS, 481, 4133Garrison-Kimmel S., et al., 2019a, MNRAS, 487, 1380Garrison-Kimmel S., et al., 2019b, MNRAS, 489, 4574Gaustad J. E., 1963, ApJ, 138, 1050Gentile G., Salucci P., Klein U., Vergani D., Kalberla P., 2004, MNRAS, 351,903Gnedin O. Y., Kravtsov A. V., Klypin A. A., Nagai D., 2004, ApJ, 616, 16Governato F., et al., 2010, Nature, 463, 203Governato F., et al., 2012, MNRAS, 422, 1231Gresham M. I., Lou H. K., Zurek K. M., 2018, Phys. Rev. D, 98, 096001Hafen Z., et al., 2019, MNRAS, 488, 1248Hayashi E., Navarro J. F., Springel V., 2007, MNRAS, 377, 50Hogan C. J., Dalcanton J. J., 2000, Phys. Rev. D, 62, 063511Hopkins P. F., 2015, MNRAS, 450, 53Hopkins P. F., Kereš D., Oñorbe J., Faucher-Giguère C.-A., Quataert E.,Murray N., Bullock J. S., 2014, MNRAS, 445, 581Hopkins P. F., et al., 2018, MNRAS, 480, 800Hoyle F., 1953, ApJ, 118, 513Hunter C., 1977, ApJ, 218, 834Huo R., Yu H.-B., Zhong Y.-M., 2019, arXiv e-prints, p. arXiv:1912.06757Jiang F., Dekel A., Freundlich J., Romanowsky A. J., Dutton A. A., MacciòA. V., Di Cintio A., 2019, MNRAS, 487, 5272Kamada A., Kaplinghat M., Pace A. B., Yu H.-B., 2017, Phys. Rev. Lett.,119, 111102Kaplan D. E., Krnjaic G. Z., Rehermann K. R., Wells C. M., 2010, J. Cosmol.Astropart. Phys., 2010, 021Kaplinghat M., Tulin S., Yu H.-B., 2016, Phys. Rev. Lett., 116, 041302Kaplinghat M., Valli M., Yu H.-B., 2019, MNRAS, 490, 231Kirby E. N., Bullock J. S., Boylan-Kolchin M., Kaplinghat M., Cohen J. G.,2014, MNRAS, 439, 1015Klypin A., Kravtsov A. V., Bullock J. S., Primack J. R., 2001, ApJ, 554, 903Kochanek C. S., White M., 2000, ApJ, 543, 514Koda J., Shapiro P. R., 2011, MNRAS, 415, 1125Kuzio de Naray R., Kaufmann T., 2011, MNRAS, 414, 3617Kuzio de Naray R., Spekkens K., 2011, ApJ, 741, L29Kuzio de Naray R., McGaugh S. S., de Blok W. J. G., Bosma A., 2006, ApJS,165, 461Larson R. B., 1969, MNRAS, 145, 271Lazar A., et al., 2020, MNRAS, 497, 2393Lemze D., et al., 2012, ApJ, 752, 141Lifshitz E. M., Pitaevskii L. P., 1981, Physical kineticsLoeb A., Weiner N., 2011, Phys. Rev. Lett., 106, 171302Lynden-Bell D., Eggleton P. P., 1980, MNRAS, 191, 483Lynden-Bell D., Wood R., 1968, MNRAS, 138, 495Ma X., et al., 2018, MNRAS, 478, 1694Madau P., Shen S., Governato F., 2014, ApJ, 789, L17Markevitch M., Gonzalez A. H., Clowe D., Vikhlinin A., Forman W., JonesC., Murray S., Tucker W., 2004, ApJ, 606, 819 Mo H. J., Mao S., White S. D. M., 1998, MNRAS, 295, 319Moore B., 1994, Nature, 370, 629Moore B., Ghigna S., Governato F., Lake G., Quinn T., Stadel J., Tozzi P.,1999, ApJ, 524, L19Moster B. P., Macciò A. V., Somerville R. S., 2014, MNRAS, 437, 1027Navarro J. F., Frenk C. S., White S. D. M., 1996a, ApJ, 462, 563Navarro J. F., Frenk C. S., White S. D. M., 1996b, ApJ, 462, 563Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493Navarro J. F., et al., 2004, MNRAS, 349, 1039Oñorbe J., Boylan-Kolchin M., Bullock J. S., Hopkins P. F., Kereš D., Faucher-Giguère C.-A., Quataert E., Murray N., 2015, MNRAS, 454, 2092Oh S.-H., Brook C., Governato F., Brinks E., Mayer L., de Blok W. J. G.,Brooks A., Walter F., 2011, AJ, 142, 24Oh S.-H., et al., 2015, AJ, 149, 180Oman K. A., et al., 2015, MNRAS, 452, 3650Ostriker J. P., Peebles P. J. E., 1973, ApJ, 186, 467Papastergis E., Giovanelli R., Haynes M. P., Shankar F., 2015, A&A, 574,A113Penston M. V., 1969a, MNRAS, 144, 425Penston M. V., 1969b, MNRAS, 145, 457Peter A. H. G., Rocha M., Bullock J. S., Kaplinghat M., 2013, MNRAS, 430,105Pontzen A., Governato F., 2012, MNRAS, 421, 3464Power C., Navarro J. F., Jenkins A., Frenk C. S., White S. D. M., Springel V.,Stadel J., Quinn T., 2003, MNRAS, 338, 14Price D. J., Monaghan J. J., 2007, MNRAS, 374, 1347Randall L., Scholtz J., 2015, J. Cosmol. Astropart. Phys., 2015, 057Randall S. W., Markevitch M., Clowe D., Gonzalez A. H., Bradač M., 2008,ApJ, 679, 1173Rees M. J., 1976, MNRAS, 176, 483Robles V. H., et al., 2017, MNRAS, 472, 2945Robles V. H., Bullock J. S., Boylan-Kolchin M., 2019, MNRAS, 483, 289Rocha M., Peter A. H. G., Bullock J. S., Kaplinghat M., Garrison-KimmelS., Oñorbe J., Moustakas L. A., 2013, MNRAS, 430, 81Sameie O., Creasey P., Yu H.-B., Sales L. V., Vogelsberger M., Zavala J.,2018, MNRAS, 479, 359Sameie O., Yu H.-B., Sales L. V., Vogelsberger M., Zavala J., 2020, Phys.Rev. Lett., 124, 141102Santos-Santos I. M., Di Cintio A., Brook C. B., Macciò A., Dutton A.,Domínguez-Tenreiro R., 2018, MNRAS, 473, 4392Sawala T., et al., 2016, MNRAS, 457, 1931Schutz K., Slatyer T. R., 2015, J. Cosmol. Astropart. Phys., 2015, 021Shu F. H., 1977, ApJ, 214, 488Simon J. D., Bolatto A. D., Leroy A., Blitz L., Gates E. L., 2005, ApJ, 621,757Spano M., Marcelin M., Amram P., Carignan C., Epinat B., Hernandez O.,2008, MNRAS, 383, 297Sparre M., Hansen S. H., 2012, J. Cosmol. Astropart. Phys., 2012, 049Spergel D. N., Steinhardt P. J., 2000, Phys. Rev. Lett., 84, 3760Stern J., Fielding D., Faucher-Giguère C.-A., Quataert E., 2019, MNRAS,488, 2549Todoroki K., Medvedev M. V., 2019, MNRAS, 483, 3983Tollerud E. J., Boylan-Kolchin M., Bullock J. S., 2014, MNRAS, 440, 3511Tollet E., et al., 2016, MNRAS, 456, 3542Tulin S., Yu H.-B., 2018, Phys. Rep., 730, 1Tulin S., Yu H.-B., Zurek K. M., 2013, Phys. Rev. D, 87, 115007Vogelsberger M., Zavala J., Loeb A., 2012, MNRAS, 423, 3740Vogelsberger M., Zavala J., Simpson C., Jenkins A., 2014, MNRAS, 444,3684Vogelsberger M., Zavala J., Schutz K., Slatyer T. R., 2019, MNRAS, 484,5437Walker M. G., Peñarrubia J., 2011, ApJ, 742, 20Warren M. S., Quinn P. J., Salmon J. K., Zurek W. H., 1992, ApJ, 399, 405Wetzel A. R., Hopkins P. F., Kim J.-h., Faucher-Giguère C.-A., Kereš D.,Quataert E., 2016, ApJ, 827, L23Wise M. B., Zhang Y., 2014, Phys. Rev. D, 90, 055030Wojtak R., Gottlöber S., Klypin A., 2013, MNRAS, 434, 1576Zavala J., Vogelsberger M., Walker M. G., 2013, MNRAS, 431, L20MNRAS , 1–25 (2021) issipative Dark Matter on FIRE Zemp M., Gnedin O. Y., Gnedin N. Y., Kravtsov A. V., 2011, ApJS, 197, 30Zhang Y., 2017, Physics of the Dark Universe, 15, 82Zhu Q., Marinacci F., Maji M., Li Y., Springel V., Hernquist L., 2016, MN-RAS, 458, 1559de Blok W. J. G., McGaugh S. S., Rubin V. C., 2001, AJ, 122, 2396de Laix A. A., Scherrer R. J., Schaefer R. K., 1995, ApJ, 452, 495van den Aarssen L. G., Bringmann T., Pfrommer C., 2012, Phys. Rev. Lett.,109, 231301This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000