EDGE: Two routes to dark matter core formation in ultra-faint dwarfs
Matthew D. A. Orkney, Justin I. Read, Martin P. Rey, Imran Nasim, Andrew Pontzen, Oscar Agertz, Stacy Y. Kim, Maxime Delorme, Walter Dehnen
MMNRAS , 1–14 (2020) Preprint 8 January 2021 Compiled using MNRAS L A TEX style file v3.0
EDGE: Two routes to dark matter core formation in ultra-faint dwarfs
Matthew D. A. Orkney , Justin I. Read , Martin P. Rey , Imran Nasim , Andrew Pontzen , Oscar Agertz ,Stacy Y. Kim , Maxime Delorme , Walter Dehnen , , Department of Physics, University of Surrey, Guildford, GU2 7XH, United Kingdom Department of Physics and Astronomy, University College London, London WC1E 6BT, UK Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden Département d’Astrophysique/AIM, CEA/IRFU, CNRS/INSU, Université Paris-Saclay, 91191 Gif-Sur-Yvette, France Astronomisches Recheninstitut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstraße. 12-14, 69120, Heidelberg, Germany Universitäts-Sternwarte München, Scheinerstraße 1, 81679, München, Germany School for Physics and Astronomy, University of Leicester, University Road, LE1 7RH, UK
Submitted to MNRAS
ABSTRACT
In the standard Lambda cold dark matter paradigm, pure dark matter simulations predict dwarf galaxies should inhabit darkmatter haloes with a centrally diverging density ‘cusp’. This is in conflict with observations that typically favour a constant density‘core’. We investigate this ‘cusp-core problem’ in ‘ultra-faint’ dwarf galaxies simulated as part of the ‘Engineering Dwarfs atGalaxy formation’s Edge’ (EDGE) project. We find, similarly to previous work, that gravitational potential fluctuations withinthe central region of the simulated dwarfs kinematically heat the dark matter particles, lowering the dwarfs’ central dark matterdensity. However, these fluctuations are not exclusively caused by gas inflow/outflow, but also by impulsive heating from minormergers. We use the genetic modification approach on one of our dwarf’s initial conditions to show how a delayed assemblyhistory leads to more late minor mergers and, correspondingly, more dark matter heating. This provides a mechanism by whicheven ultra-faint dwarfs ( 𝑀 ∗ < M (cid:12) ), in which star formation was fully quenched at high redshift, can have their central darkmatter density lowered over time. In contrast, we find that late major mergers can regenerate a central dark matter cusp, if themerging galaxy had sufficiently little star formation. The combination of these effects leads us to predict significant stochasticityin the central dark matter density slopes of the smallest dwarfs, driven by their unique star formation and mass assembly histories. Key words: methods: numerical; galaxies: dwarf; galaxies: evolution; galaxies: formation; galaxies:haloes; dark matter
The Λ CDM paradigm presents us with a Universe that has an en-ergy budget dominated by dark matter and dark energy, and inwhich galaxies are assembled through successive hierarchical merg-ers (White & Rees 1978). It has proven to be extremely successful inpredicting the formation of cosmic structure on large scales (Springelet al. 2006; Clowe et al. 2006; Tegmark et al. 2006; Dawson et al.2013; Oka et al. 2014; Planck Collaboration et al. 2014; Wang et al.2016). However, disagreements between theory and observation en-dure at (sub-)galactic scales that have become collectively known as‘small scale puzzles’ (e.g. Bullock & Boylan-Kolchin 2017).The oldest, and perhaps most challenging, of the small scale puz-zles is the ‘cusp-core problem’ (CC) (e.g. Flores & Primack 1994;Moore 1994; Read et al. 2017). Pure dark matter structure formationsimulations in Λ CDM predict a self-similar radial dark matter den-sity profile – the NFW profile (Navarro et al. 1997). This scales as 𝜌 ∝ 𝑟 − within a scale radius 𝑟 𝑠 , referred to as a ‘cusp’ due to itsdivergence towards the origin. By contrast, observations of rotationcurves in dwarf galaxies appear to favour instead a constant innerdark matter density, referred to as a ‘core’ (e.g. Carignan & Freeman1988; Flores & Primack 1994; Moore 1994; McGaugh et al. 2001;Read et al. 2017).Many solutions to the CC have been proposed in the literature to date. Firstly, observations could have been misinterpreted due toincorrect modelling assumptions. Typical assumptions include spher-ical symmetry, circular orbits and dynamical pseudo-equilibrium, allof which can be reasonably questioned (e.g. Kuzio de Naray & Kauf-mann 2011; Oman et al. 2016; Read et al. 2016b). Secondly, theassumed underlying dark matter model could be incorrect. Alterna-tives such as warm dark matter (e.g. Hogan & Dalcanton 2000; Bodeet al. 2001; Avila-Reese et al. 2001), self-interacting dark matter(e.g. Spergel & Steinhardt 2000; Tulin & Yu 2018) or ultra-light darkmatter (e.g. Schive et al. 2014; Ferreira 2020) all predict a lower darkmatter density at the centres of dwarf galaxies whilst retaining thepredictions of Λ CDM on larger scales. However, in recent years athird class of solution has been gaining traction.The CC problem originates from a comparison of pure dark mattersimulations — that do not model stars or gas (hereafter baryons) —with observations. This opens up the possibility that purely gravita-tional interactions between dark matter particles and baryons couldact to push dark matter out from the centres of dwarf galaxies, trans-forming a cusp into a core. Three main mechanisms have been pro-posed to date:(i) Dynamical friction from infalling dense clumps (e.g. El-Zantet al. 2001; Mo & Mao 2004; Romano-Díaz et al. 2009; Goerdtet al. 2010; Nipoti & Binney 2015). These clumps impart energy © a r X i v : . [ a s t r o - ph . GA ] J a n Matthew D. A. Orkney et al. and angular momentum to the dark matter halo, causing it toexpand.(ii) Dynamical friction from a central stellar or gaseous bar thatacts similarly to infalling clumps, kinematically ‘heating’ thebackground dark matter halo. (e.g. Weinberg & Katz 2007).(iii) A fluctuating gravitational potential driven by gas in-flow/outflow due to cooling, stellar winds and supernovae. Thiscauses dark matter particle orbits to slowly migrate outwards(e.g. Navarro et al. 1996; Read & Gilmore 2005; Mashchenkoet al. 2008; Pontzen & Governato 2012).All three mechanisms owe, ultimately, to a time-varying gravitationalpotential. This allows dark matter particles to exchange orbital energyboth with one another and with the stars and gas in the galaxy.What differs is only the physical mechanism that drives the time-dependent gravitational field. In principle, all three mechanisms canact in tandem as galaxies form and evolve.Despite this diversity of mechanisms in the literature, to datehigh resolution galaxy formation simulations have typically favouredmechanism (iii) at the scale of dwarf galaxies (Pontzen & Governato2012, 2014; Teyssier et al. 2013; Di Cintio et al. 2014; Oñorbe et al.2015; Dutton et al. 2016). Once gas is allowed to cool ( 𝑇 < K)and reach high density ( 𝜌 >
10 atoms/cc; e.g. Pontzen & Gover-nato 2012; Dutton et al. 2016), these simulations find that gas flowsdrive repeated fluctuations in the central galaxy mass of amplitude10 −
20% over a period less than the local dynamical time. Such fluc-tuations gradually lower the inner dark matter density on the scale ofthe stellar half mass radius, 𝑅 / , transforming a dark matter cusp toa core (e.g. Chan et al. 2015; Read et al. 2016a). There is mountingobservational evidence that this process occurs in real dwarf galaxies(e.g. Kauffmann 2014; El-Badry et al. 2016; Sparre et al. 2017; Readet al. 2019; Hirtenstein et al. 2019; but see also Bose et al. 2019;Oman et al. 2019; Genina et al. 2020).While dark matter heating may solve the cusp core problem inisolated gas rich dwarfs, a new puzzle has recently presented itself:there is a growing body of evidence for small dark matter cores evenwithin ‘ultra-faint’ dwarf galaxies, typically defined to have stellarmasses 𝑀 ∗ < M (cid:12) (e.g. Amorisco 2017; Contenta et al. 2018;Sanders et al. 2018; Malhan et al. 2020; Simon et al. 2020). Severalpapers have suggested that galaxies with so few stars have insufficientenergy from stellar feedback to carve out a dark matter core of size0 . − 𝑅 / (Read et al. 2016a), which canbe as small as 30 −
300 pc in ultra-faint dwarfs (e.g. Simon 2019).Such small cores form much more rapidly and require significantlyless energy, raising the possibility that dark matter core formationcould proceed ‘all the way down’ to even the smallest dwarfs (e.g.Read et al. 2016a; Contenta et al. 2018). Furthermore, such smallcores remain dynamically important by construction since they existprecisely where the stars and gas do – i.e. precisely where we canhope to measure the inner dark matter potential.In this paper, we use a suite of high resolution cosmological zoomsimulations from the Engineering Dwarfs at Galaxy formation’s Edge(EDGE) project (Rey et al. 2019; Agertz et al. 2020; Rey et al. 2020;Pontzen et al. 2020) to explore whether dark matter core formationcan proceed even in the very smallest dwarf galaxies. Our simula-tions model galaxies over the mass range 𝑀 ∼ − × M (cid:12) ,consistent with ultra-faint dwarfs, and reach a spatial resolution of ∼ 𝑁 -body code griffin. In Section 4.3, we show how late majormergers can reintroduce a dark matter cusp. In Section 5, we dis-cuss the implications of our results for dark matter cusps and coresin the smallest dwarf galaxies. Finally, in Section 6 we present ourconclusions. The EDGE project is described in detail in Agertz et al. (2020). Here,we briefly summarise the key points. We start with a 512 resolutioncosmological dark matter simulation of a 50 Mpc void region (Figure1). All simulations assume cosmological parameters Ω 𝑚 = . Ω Λ = . Ω 𝑏 = .
045 and 𝐻 = .
77 km s − Mpc − , in linewith data from the PLANCK satellite (Planck Collaboration et al.2014).We draw a selection of target haloes from the void volume, cho-sen from a range in halo mass of 10 < 𝑀 / M (cid:12) < × . Thesetarget haloes are resimulated following the zoom simulation tech-nique (Katz & White 1993; Oñorbe et al. 2014) with the AdaptiveMesh Refinement (AMR) code ramses (Teyssier 2002). This grantsus a highly resolved target galaxy within its lower-resolution widercosmological context. The velocity in our initial conditions is thenadjusted to match the velocity of the target halo, which reduces theimpact of numerical diffusion effects (see Pontzen et al. 2020).Key details of our ramses simulations are presented in Table 1.Each simulation is run over the redshift range 99 ≥ 𝑧 ≥
0, with aminimum of 100 outputs spaced linearly with the scalefactor, 𝑎 . Thecontamination fraction, defined as the fraction of lower resolutiondark matter particles within the virial radius, is never greater than2 × − in any of our simulations. This is relevant for the impactof numerical relaxation, which we discuss further in Appendix A.Our simulations are run at a resolution in which the dark matterparticle mass approaches 100 M (cid:12) in the high resolution Lagrangianregion of the target galaxy, with a spatial resolution ∼ (cid:164) 𝜌 ∗ = 𝜖 ff 𝜌 𝑔 𝑡 ff for 𝜌 𝑔 > 𝜌 ★ and 𝑇 𝑔 < 𝑇 ★ , (1)where 𝜌 ★ = 𝑚 proton cm − and 𝑇 ★ =
100 K. Here, (cid:164) 𝜌 ∗ is thestar formation rate density in a gas cell, 𝜌 𝑔 is the density per gascell, 𝑡 ff = √︁ 𝜋 / 𝐺 𝜌 𝑔 is the local free-fall time of the gas, and 𝜖 ff is the star formation efficiency per free-fall time which is set to10% in line with arguments from Grisdale et al. (2019). Each stellarparticle is initialised at 300 M (cid:12) and is representative of a single-age stellar population (SSP) described by a Chabrier initial massfunction (IMF) (Chabrier 2003). Stellar feedback from both Type IIand Ia supernovae are included, and stellar winds from massive and MNRAS000
100 K. Here, (cid:164) 𝜌 ∗ is thestar formation rate density in a gas cell, 𝜌 𝑔 is the density per gascell, 𝑡 ff = √︁ 𝜋 / 𝐺 𝜌 𝑔 is the local free-fall time of the gas, and 𝜖 ff is the star formation efficiency per free-fall time which is set to10% in line with arguments from Grisdale et al. (2019). Each stellarparticle is initialised at 300 M (cid:12) and is representative of a single-age stellar population (SSP) described by a Chabrier initial massfunction (IMF) (Chabrier 2003). Stellar feedback from both Type IIand Ia supernovae are included, and stellar winds from massive and MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs asymptotic giant branch (AGB) stars (see Agertz et al. 2013; Agertz& Kravtsov 2015; Agertz et al. 2020 for details).The epoch of reionisation is modelled as a time-dependent uniformUV background around 𝑧 = .
5, as in the public release of ramses(Haardt & Madau 1996). The exact implementation is discussedfurther in Rey et al. (2020), and is consistent with a late reionisationexpected for a cosmic void (e.g. Keating et al. 2020).In addition, to further test our numerical results, we use the grif-fin code (Dehnen 2014) to run a controlled investigation into theeffects of the merger history on the dark matter halo of one of ourramses simulations. The motivation for these additional simulationswill be made clear in Section 4.2.1. Griffin is a high performance 𝑁 -body integrator that exploits the fast multipole method (FMM),and is an ideal tool to compare against ramses because it is basedon a fundamentally different numerical integration scheme that hascomparable force accuracy to direct summation codes (e.g. Dehnen2014; Gualandris et al. 2017; Nasim et al. 2020a,b). These simula-tions are run at both the equivalent and 10 × better mass resolutioncompared to our ramses runs, with the former being used as a con-vergence study. We employ a force softening length of 10 pc in allof these simulations. Additional tests with a softening length of 5 pcshowed no measurable change in the results. We use the hop halo finder (Eisenstein & Hut 1998) to identify alldistinct bound structures in each simulation output. Hop does notidentify haloes within haloes (subhaloes), and so, where necessary,our analysis is supplemented with the ahf (Amiga Halo Finder)(Knollmann & Knebe 2009). Merger trees and halo properties arecalculated using pynbody (Pontzen et al. 2013) and tangos (Pontzen& Tremmel 2018), respectively. We locate the centre of each boundstructure using the shrinking spheres method of Power et al. (2003),performed exclusively on the dark matter component, as implementedin pynbody. Results from our dark matter only (DMO) simulationsare in all cases corrected for the universal baryon fraction.Throughout this paper, the virial radius, 𝑟 , is defined as thespherical region that is at least 200 times the critical mass density ofthe universe at that redshift. The halo mass 𝑀 , is then the totalmass of all matter contained within that radius. Several dwarfs presented in this paper have been discussed alreadyin previous EDGE collaboration papers , however run at a lower‘fiducial’ resolution ( 𝑚 DM = (cid:12) ). Here, since we are inter-ested in resolving potentially very small dark matter cores, the samedwarfs are resimulated at what we called ‘hires’ resolution in Agertzet al. (2020) ( 𝑚 DM =
117 M (cid:12) ). We show convergence tests betweenthese two resolutions in Appendix B, demonstrating that our resultspresented here do not depend on resolution.Figure 1 shows a surface density plot for the dark matter of thetotal void region from which our initial conditions were selected. Thelocations for each of our haloes in Table 1 are indicated with zoomedsurface density plots, with images taken from the ‘DMO’ simulations.This highlights that our haloes are chosen from a particularly under-dense region, without any major cosmic structure in the near vicinity. Halo1459, Halo1459 GM:Later and Halo1459 GM:Latest appear in Reyet al. (2019). Halo600, Halo605 and Halo624 appear in Rey et al. (2020)
Figure 2 shows a visual representation of each high resolutionbaryonic simulation at 𝑧 =
0. The left-most panels show the cen-tred dark matter surface density out to 𝑟 , which is indicated by adashed circle. The middle panels show the central gas density aver-aged in a 0.2 kpc thick slice through the 𝑧 -axis. The plot is zoomedinto the inner five half light radii, where the 3D half light radiusis marked with a solid circle. Each halo is oriented side-on on theangular momentum vector of the central cold gas ( < K), whereavailable, which represents the central gas disk if it is present. Theright-most panels are a pynbody rendering of the halo stars.For Halo1445 and Halo1459, the gas is extremely under-denseand shows little structure, with the exception of some mild stirringdue to late Type Ia SNe (most apparent in Halo1459). Both Halo600and Halo605 have denser gas, with bubbles forming due to ongoingbursty star formation. Halo624 is the only galaxy to form a structuredgas disc, which is both dense and rotating at 𝑧 =
0. We will presenta detailed study of the observational properties of these simulateddwarfs in forthcoming papers. In this paper, we focus on their darkmatter content and structure.In Figure 3, we present merger trees for each of our haloes usingthe ‘DMO’ simulations. For simplicity, only major mergers onto themain progenitor (coloured line) are shown. The line thickness isrepresentative of each halo mass. The general form of these mergertrees are identical for different resolutions and physics, with the oneexception that the final merger in Halo624 DMO occurs just after 𝑧 = 𝑧 = In Figure 4, we show the evolution of the central dark matter densityfor all of our EDGE simulations. In the upper panels, the inner 3Ddark matter density is plotted at 40 pc. This approaches the inner limitof the region that we consider numerically resolved (see AppendixA). The opaque lines are the results for the baryonic simulations,whereas faint lines are the results for the pure dark matter (DMO)simulations. The corresponding star formation rates of both the mainprogenitor and all progenitors for the baryonic simulation are plottedin the middle panels, averaged over 100 Myr bins. Included in thelower panels are the 3D density profiles for both baryonic and DMOsimulations at 𝑧 = ∼ MNRAS , 1–14 (2020)
Matthew D. A. Orkney et al.
Table 1.
Details of all simulations. The simulation labels denote different reference haloes selected from the initial void simulation. From left to right thecolumns are: the simulation label, the physics scheme employed, the mass resolution, the halo mass ( 𝑀 ) at 𝑧 =
0, total stellar mass within the virial radius( 𝑟 ) at 𝑧 =
0, the virial radius at 𝑧 =
0, the projected half light radius at 𝑧 =
0, and the V-band magnitude at 𝑧 =
0. The simulations are ordered by the 𝑀 mass of the full physics simulations. Name Physics Resolution 𝑀 [M (cid:12) ] 𝑟 [kpc] 𝑀 ∗ [M (cid:12) ] 𝑅 half [pc] 𝑀 𝑉 (mag)[ 𝑚 DM , 𝑚 gas , 𝑚 ∗ ]/ M (cid:12) (projected)Halo1445 DMO DM-only [139, -, -] 1 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . ×
10 Mpc 500 kpc M = × M fl Halo605 M = × M fl Halo1459 M = × M fl Halo1445 M = × M fl Halo624 M = × M fl Halo600
Figure 1.
The location of the EDGE dark matter haloes selected for higher resolution resimulation from a lower resolution void at 𝑧 =
0. Each panel shows asurface density plot of a cube. The zoomed panels for each halo are taken from the ‘DMO’ simulations out to 𝑟 (see Table 1), and show the corresponding 𝑀 mass, as marked. For Halo600, there is a partial-zoom to help illustrate how small the selected haloes are in comparison to the total box size (50 Mpc). the 3D half light radius (black dashed line) at approximately 100 pc.This is also consistent with prior work on the gas flow mechanism(see e.g. Chan et al. 2015; Read et al. 2016a; and Section 1).In Figure 5, we show the gas-to-dark matter central density ratio( 𝜌 gas / 𝜌 DM ) for two representative dwarf galaxies. This is computedat time intervals of Δ 𝑎 = .
01 (the cadence of our simulation outputs),and so the true peak 𝜌 gas / 𝜌 DM may be higher. The upper panel shows Halo1445, which is quenched permanently by 𝑧 = MNRAS000
01 (the cadence of our simulation outputs),and so the true peak 𝜌 gas / 𝜌 DM may be higher. The upper panel shows Halo1445, which is quenched permanently by 𝑧 = MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs H a l o Dark matter
Gas
Stars H a l o H a l o H a l o H a l o ρ DM [M fl kpc − ] 3 4 5 6 7 8 9log ρ gas [M fl kpc − ] Figure 2.
A visual representation of the high resolution baryonic simulations in Table 1.
Left panels : The dark matter surface density out to 𝑟 (dashed circle). Middle panels : The gas density averaged along a 0.2 kpc slice through 𝑧 . The 3D half light radius is indicated by the solid circle. Right panels : The halo starsare rendered in pynbody with the 𝑖 , 𝑣 and 𝑢 -bands over the range 23 ≤ mag arcsec − ≤
28, shown at the same scale as the gas density panels. All images areoriented side-on along the angular momentum vector of the cool gas ( < K) within 1 kpc of the halo centre. The physical size of each frame is indicated by ascale bar in the top left corner. MNRAS , 1–14 (2020)
Matthew D. A. Orkney et al. R e d s h i f t z Halo1445 DMO Halo1459 DMO Halo600 DMO Halo605 DMO Halo624 DMO
Figure 3.
The accretion history of each halo as taken from the ‘DMO’ simulations, where < /
30 mass ratio mergers and sub-mergers are excluded for clarity.The central coloured branch is the main progenitor halo and the line thickness represents the 𝑀 mass of each dark matter halo. ρ D M ( p c ) [ M fl k p c − ] Halo1445
BaryonicDMO − − − S F R [ M fl y r − ] TotalProgenitor − Radius [kpc]10 ρ D M [ M fl k p c − ] z = H a l f l i g h t r a d i u s − Radius [kpc] 10 − Radius [kpc] 10 − Radius [kpc] 10 − Radius [kpc]8.5 3 2 1 0.5 z z z z z Figure 4.
Upper panels:
The evolution of the 3D dark matter density at 40 pc in the main progenitor halo for the high resolution dwarf galaxies. The opaque linesshow simulations run with baryonic physics, whereas the faint lines show pure dark matter (DMO) simulations using the same initial conditions. Our results arequalitatively similar when selecting the density at different inner radii.
Middle panels:
The star formation rate of the baryonic simulations averaged over binsof 100 Myr. The opaque bars show stars formed within 𝑟 of the main progenitor, whereas fainter bars include stars that are brought in with mergers. Lowerpanels:
A comparison of the 3D dark matter density profiles between the baryonic and DMO simulations at 𝑧 =
0. The black dashed lines mark the 3D half lightradii in each case. early times that diminish after the initial quenching. However, as thehalo grows more massive it is able to increase its central gas density(see Rey et al. 2020). Star formation then rejuvenates after 10 Gyr,but the intensity of this late star formation is not sufficient to drivelarge enough fluctuations in 𝜌 gas / 𝜌 DM to further heat the centraldark matter. Our results so far are in line with previous studies in the literature thatsuggest that forming kpc scale dark matter cores becomes inefficientin ultra-faint dwarfs (Peñarrubia et al. 2012; Garrison-Kimmel et al.2013; Di Cintio et al. 2014; Madau et al. 2014; Maxwell et al. 2015;Tollet et al. 2016; Read et al. 2016a). However, the puzzle of apparentdark matter cores in at least some ultra-faints remains (Amorisco2017; Contenta et al. 2018), Section 1.Perhaps the most compelling case for a small dark core in an
MNRAS000
MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs − − − − − ρ g a s ρ D M ( p c ) [ M fl k p c − ] Gas evaporation
Halo14450 2 4 6 8 10 12Time [Gyr]10 − − − − − − ρ g a s ρ D M ( p c ) [ M fl k p c − ] Q u e n c h e d G a s c oo l i n g Rejuvenation
Halo624
Figure 5.
Evolution of the inner gas-to-dark matter density ratio at 40 pc( 𝜌 gas / 𝜌 DM ) for two representative EDGE dwarfs. Prior to quenching byreionisation (black dashed line), there are large fluctuations in 𝜌 gas / 𝜌 DMcaused by repeated cycles of gas cooling, star formation and stellar feedback.These fluctuations excite ‘dark matter heating’ that lowers the central darkmatter density (Figure 4). Halo1445 experiences no further star formation ordark matter heating after it quenches. By contrast, Halo624 grows in mass,accumulating cold gas and rejuvenating its star formation after ∼
10 Gyr.Subsequent fluctuations are orders of magnitude smaller than at early times,and so there is no late time dark matter heating. ultra-faint dwarf comes from the survival of a lone and extended starcluster, offset from the centre of the Eridanus II galaxy (Amorisco2017; Contenta et al. 2018; Simon et al. 2020). Eridanus II is substan-tially more extended than any of our reference EDGE dwarfs, with 𝑅 half = ±
12 pc. Using the genetic modification approach (Rothet al. 2016; Rey & Pontzen 2018; Stopyra et al. 2020), we can createalternative mass accretion histories for a galaxy. Rey et al. (2019)modified one of the fiducial resolution EDGE dwarfs, Halo1459,such that it assembled later. They found that this leads to a larger,lower surface brightness, and lower metallicity dwarf – more similarto Eridanus II. As such, in this section, we resimulate the geneticallymodified later forming dwarfs from Rey et al. (2019) at higher res-olution to study how late formation impacts the central dark matterdensity.We run two modifications of Halo1459 which we call Halo1459GM:Later and Halo1459GM:Latest (Table 1). The assembly historiesof these modifications were already shown at our fiducial resolutionin Figure 1 of Rey et al. (2019); they are indistinguishable fromthe higher resolution trajectories which we show in Figure 6. Themodified haloes are approximately 3 × (2 × ) less massive at 𝑧 = . H a l o m a ss [ M fl ] Population68 & 95% CI
Halo1459 hiresHalo1459 GM:LaterHalo1459 GM:Latest
Figure 6.
The halo mass growth history of Halo1459 and two modified vari-ants which we call Halo1459 GM:Later and Halo1459 GM:Latest. Includedare grey bands indicating the 68 and 95 per cent scatter for the mass growthhistories of haloes throughout our lower resolution void simulation. The bandsare truncated at ∼ × due to the resolution limit of the void simulation.This shows that our modified haloes are within the expected scatter of assem-bly histories in Λ CDM. for Halo1459 GM:Later (Halo1459 GM:Latest), but grow to the samemass within a 4 per cent margin by 𝑧 = 𝑀 gas / 𝑀 DM ( < 𝑟 halflight ) < − , so any gasflows driven by residual feedback from old stars (as in Rey et al.2020) have a negligible impact on the overall mass distribution.This reduction in the central density is not seen to the same extentin Halo1459 GM:Latest. We will discuss the reasons for this inSection 4.3.The lower panel of Figure 7 shows the radial dark matter den-sity profiles of Halo1459 GM:Later (darker) and Halo1459 DMOGM:Later (lighter) at 𝑧 = 𝑧 = 𝑧 =
0, it still hasa slope of 𝜌 ∝ 𝑟 − . .It is important to rule out the possibility that any apparent darkmatter heating is caused by numerical relaxation. In Appendix A, wecalculate the ‘relaxation radius’, 𝑟 relax , for our EDGE simulations.This is the radius inside which the numerical relaxation time is equalto the simulation run time and so numerical relaxation will becomeimportant. From this calculation, we conclude that our simulationsshould still be well-resolved above 𝑟 relax >
25 pc at 𝑧 =
0, yetthe density clearly evolves on scales larger than this at all times.Therefore, there must be some other mechanism by which the centraldark matter density is lowering. It is well established that dynamical
MNRAS , 1–14 (2020)
Matthew D. A. Orkney et al. × × × ρ D M ( p c ) [ M fl k p c − ] Q u e n c h e d BaryonicDMO − − − S F R [ M fl y r − ] TotalProgenitor − Radius [kpc]10 ρ D M [ M fl k p c − ] H a l f l i g h t r a d i u s Halo1459 GM:Later
Core, ρ ∝ r C u s p , ρ ∝ r − z = z = z Figure 7.
Upper panel:
The evolution of the 3D dark matter density at 40 pc,but for a modified simulation that has a delayed formation history (Halo1459GM:Later). A black dashed line marks the approximate time at which thegalaxy is permanently quenched.
Middle panel:
The star formation rate of thebaryonic simulation, averaged over bins of 100 Myr. Opaque bars are starsformed within 𝑟 of the main progenitor, whereas faint bars include starsthat are brought in with mergers. Lower panel:
A comparison of the 3D darkmatter density profiles between the baryonic and DMO simulations at both 𝑧 = 𝑧 =
0. A black dashed line marks the 3D halflight radius at 𝑧 = heating from dense clumps can contribute to core formation (e.g. El-Zant et al. 2001; Mo & Mao 2004; Romano-Díaz et al. 2009; Goerdtet al. 2010; Nipoti & Binney 2015; and Section 1). In Halo1495GM:Later, the only dense clumps available to drive such a process atlate times are merging dark matter subhaloes.To investigate the veracity of the above late time dark matter heat-ing, and to explore whether merging dark matter subhaloes are indeedthe culprit, in the next section we resimulate the sequence of mergersthat form Halo1459 GM:Later using the non-cosmological 𝑁 -bodyFMM code griffin (Dehnen 2014), as described in Section 2.1. Thisallows us to isolate the heating effects of minor mergers in a non-cosmological setting, and to verify that numerical effects unique toramses are not responsible for the heating. griffin code In this section, we use the griffin 𝑁 -body code to reproduce theassembly history of Halo1459 GM:Later in a controlled manner. Forthis, we simulate a series of halo mergers based upon profile fits tothe reference ramses simulation. We use the coreNFW profile as inRead et al. (2016a), which is a NFW profile (Navarro et al. 1997)adapted to include a parameter 𝑛 that controls the flatness of thecentral density slope.We first fit the spherically symmetric dark matter density profileof the main progenitor halo at 𝑧 =
4. By this time, the central darkmatter halo is well-established and star formation has permanentlyquenched. The total stellar mass is low ( ∼ M (cid:12) ) and the gascontent is negligible, so the system can be safely resimulated usingexclusively its dark matter component.Mergers are defined based on every halo identified with hop at 𝑧 =
4, on the condition that they contain at least 800 dark matterparticles and are destined to merge with the main progenitor. Eachmerging halo is tracked until the snapshot prior to infall (defined asthe point where the merger crosses over 𝑟 of the main progenitor),by which time many of them have coalesced. Despite the large numberof individual haloes at 𝑧 =
4, there are a manageable 38 distinctsubhaloes at the time of merging.For each of these mergers, we perform a spherically symmetriccoreNFW profile fit and generate initial conditions using agama(Vasiliev 2018). We use a multipole potential approximation withone hundred grid nodes, an isotropic velocity distribution function,and model the density out to 3 × 𝑟 for each coreNFW profile fit(which is necessary to ensure there is sufficient dynamical frictionbetween interacting haloes at large radii; Read et al. e.g. 2008). Afurther improvement could include fitting the halo triaxiality, butshould not be necessarily for resolving the leading order effects ofhalo mergers.The initial conditions of the main progenitor are integrated forwardin time with the griffin code, as described in Section 2.1. As thesimulation reaches the time of each merger infall, the correspondinginitial conditions are inserted at the same phase-space location as inthe reference ramses simulation. In this way, the merger history ofthe ramses simulation from 4 ≤ 𝑧 ≤ Λ CDM cosmology increases with decreasing halomass (Stewart et al. 2009; Rodriguez-Gomez et al. 2015). Therefore, asignificant amount of mass accretion is neglected by only consideringmergers above a certain mass threshold. This yields a final 𝑧 = . × less that of the original ramses simulation, althoughmuch of this missing mass is located in the halo outskirts ( ∼ MNRAS000
4, there are a manageable 38 distinctsubhaloes at the time of merging.For each of these mergers, we perform a spherically symmetriccoreNFW profile fit and generate initial conditions using agama(Vasiliev 2018). We use a multipole potential approximation withone hundred grid nodes, an isotropic velocity distribution function,and model the density out to 3 × 𝑟 for each coreNFW profile fit(which is necessary to ensure there is sufficient dynamical frictionbetween interacting haloes at large radii; Read et al. e.g. 2008). Afurther improvement could include fitting the halo triaxiality, butshould not be necessarily for resolving the leading order effects ofhalo mergers.The initial conditions of the main progenitor are integrated forwardin time with the griffin code, as described in Section 2.1. As thesimulation reaches the time of each merger infall, the correspondinginitial conditions are inserted at the same phase-space location as inthe reference ramses simulation. In this way, the merger history ofthe ramses simulation from 4 ≤ 𝑧 ≤ Λ CDM cosmology increases with decreasing halomass (Stewart et al. 2009; Rodriguez-Gomez et al. 2015). Therefore, asignificant amount of mass accretion is neglected by only consideringmergers above a certain mass threshold. This yields a final 𝑧 = . × less that of the original ramses simulation, althoughmuch of this missing mass is located in the halo outskirts ( ∼ MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs Table 2.
The griffin simulations used to investigate the late-time densityreduction in Halo1459 GM:Later. From left to right, the columns give: thesimulation names, whether mergers were included or the main progenitorwas isolated, the dark matter particle mass resolution, and the force softeninglength.
Name Mergers Resolution Force softening [pc][ 𝑚 DM /M (cid:12) ] Isolated low (cid:55)
117 10Mergers low (cid:51)
117 10Isolated (cid:55) (cid:51) × × ρ D M ( p c ) [ M fl k p c − ] DM compressionIsolatedMergers z Figure 8.
The evolution of the 3D dark matter density at 40 pc in the griffinsimulations. We start the 𝑥 -axis at redshift 𝑧 = 𝑧 = the orbits of merging haloes begin to diverge from the referencesimulation after two pericentre passages, and it is already establishedthat reproducing exact orbital behaviour of mergers is challenging(e.g. Lux et al. 2010). Lastly, any mergers already within the virialradius of the main progenitor by 𝑧 = ≈ . × M (cid:12) kpc − over 12 Gyr. However, this contrasts with a muchmore substantial drop in the central density of the simulation withmergers of ≈ . × M (cid:12) kpc − . The small heating present in theisolated simulation is not immediately obvious in any of our non-GMramses simulations, and the differences in numerical setup and lackof cosmological growth may be contributing to this. We do not runany ramses simulations without cosmological accretion, so a strictcomparison is difficult.Finally, notice the three prominent ‘spikes’ in the inner dark matterdensity at ∼ . ∼ . ∼ . Along with mechanisms that flatten the central dark matter density,there are mechanisms that can rebuild it. Laporte & Penarrubia (2015)investigate a scenario where dense mergers can reintroduce dynami-cally cold dark matter into a cored parent halo, thereby rebuilding thecentral density cusp. These events require that the merger is able tofall into the centre of the parent halo intact, which would demand themerging structures are resistant to the tidal disruption of the parentgalaxy.In Figure 9, we show an example of such a merger in Halo1459GM:Latest. The left panel shows the central dark matter densities ofthe merging system at several key times. The merging halo is shownin red at 𝑧 = .
79, by which time it has permanently crossed over the 𝑟 radius of the parent halo. The parent halo is shown at the sametime in black, and has already begun to depart from a primordialdensity cusp (black dotted line), primarily due to the action of minormergers as in Section 4.2. By 𝑧 = .
51, the dark grey line shows thatthe central density of the parent halo has continued to decline to itslowest point. Finally, the light grey line shows the parent and merginghalo combined at 𝑧 = .
47, with the central density returning to asteep primordial cusp (which is coincidentally well described by thecuspy profile fit made at 𝑧 = . 𝑧 = .
79 and 𝑧 = .
51, but the central density is retained.The final panel conveys how the increase in central density at 𝑧 = . We have established that gas flows driven by star formation are able todrive sufficiently large potential fluctuations to erode central densitycusps in our EDGE simulations (Section 4.1). However, this is seenonly at early times when star formation rates exceed 1 × − M (cid:12) yr − and fluctuate on a timescale of order the local dynamical time. De-spite the late-time rejuvenation of star formation in several of oursimulations, this second phase of star formation is not sufficient todrive significant gas flows and the central dark matter density isunaffected.However, bursty star formation is not the only means by which thecentral gravitational force can be varied. We have also shown that MNRAS , 1–14 (2020) Matthew D. A. Orkney et al. − Radius [kpc]10 ρ D M [ M fl k p c − ] Merger [ z = z = z = z = z = z = z = M fl kpc − M fl kpc − M fl kpc − Figure 9.
This plot illustrates how the reintroduction of denser dark matter material to the Halo1459 GM:Latest simulation erases the effect of earlier dark matterheating.
Leftmost panel:
The central 3D dark matter density profiles of a cuspy subhalo (red line) and of the parent halo evolution (black, dark grey and lightgrey lines). The density profile of the host at 𝑧 = .
51 is centred on only the host particles to avoid any bias due to the merger. Removing the merger particlesfrom the density profile calculation does not qualitatively change the results. A black dashed line represents a NFW profile fit to the parent halo at 𝑧 = . Right panels:
The evolution of the merger system shown in three panels, where the parent halo is in grey-scale and the merging halo isred. The merging halo has permanently fallen into the central 100 pc of the parent halo by 𝑧 = .
47. In all cases, the merger system has been oriented such thatthe centres of both haloes are in the 𝑥𝑦 -plane. − − R a d i u s [ k p c ] t d y n Inner region
Figure 10.
The orbit of one example merger from our ‘Mergers’ griffin sim-ulation. The orbit between each simulation output has been reconstructed witha two-body integration in agama, assuming a spherically symmetric back-ground potential and using a multipole fit to each simulation snapshot. Whilstthese orbit reconstructions are imperfect due to perturbations from other sub-haloes and triaxiality, they provide a reasonable estimate for our purposes.A red bar shows the width of one dynamical time 𝑡 dyn = 𝜋 √︁ / 𝜋𝐺𝜌 ( 𝑟 ) over a range of orbital radii, and a black star marks the time at which themerger dissolves. The approximate inner region is indicated at 0.1 kpc with ahorizontal dashed line. passing subhaloes act to fluctuate the central density driving darkmatter heating (Section 4.2). In Figure 10, we show an example orbitof one merging subhalo taken from our griffin simulation. Noticethat this merger repeatedly punctures the inner region of its hostgalaxy on a timescale shorter than the local dynamical time (red). The first close passage of this subhalo corresponds to the dark matterdensity spike at 6.5 Gyr shown in Figure 8. The other density spikesin that Figure correspond to close passages from different mergingsubhaloes. Taken together, this indicates that the late time dark matterheating is being driven by tidal shocks from the merging subhaloeson their host.The above late time dark matter heating due to minor mergersoccurs in both our baryonic and DMO ramses simulations (Figure7). However, in the DMO simulations the inner dark matter density,while lower, remains cuspy (Figure 7, bottom panel). By contrast, inthe baryonic simulation – and in our griffin replica of this simulation– these minor mergers flatten the cusp. This occurs because in thesesimulations, the cusp is already weakened at early times by darkmatter heating due to star formation (Figure 7, upper panel).Note that this minor merger induced heating has been discussedpreviously in the literature albeit in different contexts. Naab et al.(2009) propose a mechanism by which the central concentrationsof massive elliptical galaxies are reduced through repeated minormergers, with similar effects also seen in Bédorf & Portegies Zwart(2013). And, Leung et al. (2020) propose that mergers could expandthe orbits of globular clusters in the Fornax dwarf spheroidal galaxy,solving a long-standing puzzle as to why they have not sunk to thecentre of Fornax via dynamical friction. This same mechanism wouldalso expand the orbits of the dark matter particles too. We have presented a suite of cosmological zoom simulations of theultra-faint dwarf galaxies performed with the adaptive mesh refine-ment code ramses as part of the EDGE project. These simulationshave a spatial and mass resolution of 3 pc and 120 M (cid:12) , respectively,sufficient to resolve the formation of very small dark matter cores.Our key result is that we uncover two distinct pathways to darkmatter core formation at sub-kpc scales in the 10 < 𝑀 / M (cid:12) < × halo mass regime. These are able to drive reductions in thecentral (40 pc) dark matter density of up to approximately a factor oftwo as compared to pure dark matter simulations. The first pathway is MNRAS000
The orbit of one example merger from our ‘Mergers’ griffin sim-ulation. The orbit between each simulation output has been reconstructed witha two-body integration in agama, assuming a spherically symmetric back-ground potential and using a multipole fit to each simulation snapshot. Whilstthese orbit reconstructions are imperfect due to perturbations from other sub-haloes and triaxiality, they provide a reasonable estimate for our purposes.A red bar shows the width of one dynamical time 𝑡 dyn = 𝜋 √︁ / 𝜋𝐺𝜌 ( 𝑟 ) over a range of orbital radii, and a black star marks the time at which themerger dissolves. The approximate inner region is indicated at 0.1 kpc with ahorizontal dashed line. passing subhaloes act to fluctuate the central density driving darkmatter heating (Section 4.2). In Figure 10, we show an example orbitof one merging subhalo taken from our griffin simulation. Noticethat this merger repeatedly punctures the inner region of its hostgalaxy on a timescale shorter than the local dynamical time (red). The first close passage of this subhalo corresponds to the dark matterdensity spike at 6.5 Gyr shown in Figure 8. The other density spikesin that Figure correspond to close passages from different mergingsubhaloes. Taken together, this indicates that the late time dark matterheating is being driven by tidal shocks from the merging subhaloeson their host.The above late time dark matter heating due to minor mergersoccurs in both our baryonic and DMO ramses simulations (Figure7). However, in the DMO simulations the inner dark matter density,while lower, remains cuspy (Figure 7, bottom panel). By contrast, inthe baryonic simulation – and in our griffin replica of this simulation– these minor mergers flatten the cusp. This occurs because in thesesimulations, the cusp is already weakened at early times by darkmatter heating due to star formation (Figure 7, upper panel).Note that this minor merger induced heating has been discussedpreviously in the literature albeit in different contexts. Naab et al.(2009) propose a mechanism by which the central concentrationsof massive elliptical galaxies are reduced through repeated minormergers, with similar effects also seen in Bédorf & Portegies Zwart(2013). And, Leung et al. (2020) propose that mergers could expandthe orbits of globular clusters in the Fornax dwarf spheroidal galaxy,solving a long-standing puzzle as to why they have not sunk to thecentre of Fornax via dynamical friction. This same mechanism wouldalso expand the orbits of the dark matter particles too. We have presented a suite of cosmological zoom simulations of theultra-faint dwarf galaxies performed with the adaptive mesh refine-ment code ramses as part of the EDGE project. These simulationshave a spatial and mass resolution of 3 pc and 120 M (cid:12) , respectively,sufficient to resolve the formation of very small dark matter cores.Our key result is that we uncover two distinct pathways to darkmatter core formation at sub-kpc scales in the 10 < 𝑀 / M (cid:12) < × halo mass regime. These are able to drive reductions in thecentral (40 pc) dark matter density of up to approximately a factor oftwo as compared to pure dark matter simulations. The first pathway is MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs stellar feedback, in agreement with previous literature. This requiresa sufficiently high star-formation rate over an extended period of time,which in our EDGE simulations only occurs at high redshift prior toreionisation. At these early times, we found that the star formationrate fluctuated on the order of the local dynamical time with anaverage amplitude of ∼ × − M (cid:12) yr − . This caused the orbits ofdark matter particles to migrate outwards, lowering the dwarf’s innerdark matter density.However, even after quenching by reionisation, we found that asecond mechanism can cause dark matter cores to continue to grow:impulsive heating from minor mergers. To demonstrate this, we ‘ge-netically modified’ the initial conditions for one dwarf such that itassembled later from many minor mergers. We found that, in thiscase, the dwarf’s inner dark matter density continued to drop longafter star formation ceased. We tested the veracity of this result byrunning an independent ‘replica’ simulation using the griffin 𝑁 -body code, finding excellent agreement between the griffin andramses calculations.While all of our dwarfs experienced some dark matter heating priorto reionisation, we showed that dense major mergers can replenishkinematically cold dark matter, thereby reintroducing a density cuspat late times. This demonstrates that the central density of the small-est dwarf galaxies at 𝑧 = ACKNOWLEDGEMENTS
DATA AVAILABILITY
Data available upon request.
REFERENCES
Agertz O., Kravtsov A. V., 2015, ApJ, 804, 18Agertz O., Kravtsov A. V., Leitner S. N., Gnedin N. Y., 2013, ApJ, 770, 25Agertz O., et al., 2020, MNRAS, 491, 1656Amorisco N. C., 2017, ApJ, 844, 64 Avila-Reese V., Colín P., Valenzuela O., D’Onghia E., Firmani C., 2001, ApJ,559, 516Bédorf J., Portegies Zwart S., 2013, MNRAS, 431, 767Binney J., Tremaine S., 1987, Galactic dynamicsBode P., Ostriker J. P., Turok N., 2001, ApJ, 556, 93Bose S., et al., 2019, MNRAS, 486, 4790Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343Carignan C., Freeman K. C., 1988, ApJ, 332, L33Chabrier G., 2003, PASP, 115, 763Chan T. K., Kereš D., Oñorbe J., Hopkins P. F., Muratov A. L., Faucher-Giguère C. A., Quataert E., 2015, MNRAS, 454, 2981Clowe D., Bradač M., Gonzalez A. H., Markevitch M., Randall S. W., JonesC., Zaritsky D., 2006, ApJ, 648, L109Contenta F., et al., 2018, MNRAS, 476, 3124Dawson K. S., et al., 2013, AJ, 145, 10Dehnen W., 2014, Computational Astrophysics and Cosmology, 1, 1Dehnen W., Read J. I., 2011, European Physical Journal Plus, 126, 55Di Cintio A., Brook C. B., Dutton A. A., Macciò A. V., Stinson G. S., KnebeA., 2014, MNRAS, 441, 2986Diemand J., Moore B., Stadel J., Kazantzidis S., 2004a, MNRAS, 348, 977Diemand J., Moore B., Stadel J., 2004b, MNRAS, 353, 624Dutton A. A., et al., 2016, MNRAS, 461, 2658Eisenstein D. J., Hut P., 1998, ApJ, 498, 137El-Badry K., Wetzel A., Geha M., Hopkins P. F., Kereš D., Chan T. K.,Faucher-Giguère C.-A., 2016, ApJ, 820, 131El-Zant A., Shlosman I., Hoffman Y., 2001, ApJ, 560, 636Ferreira E. G. M., 2020, arXiv e-prints, p. arXiv:2005.03254Flores R. A., Primack J. R., 1994, ApJ, 427, L1Garrison-Kimmel S., Rocha M., Boylan-Kolchin M., Bullock J. S., Lally J.,2013, MNRAS, 433, 3539Genina A., Read J. I., Fattahi A., Frenk C. S., 2020, arXiv e-prints, p.arXiv:2011.09482Goerdt T., Moore B., Read J. I., Stadel J., 2010, ApJ, 725, 1707Grisdale K., Agertz O., Renaud F., Romeo A. B., Devriendt J., Slyz A., 2019,MNRAS, 486, 5482Gualandris A., Read J. I., Dehnen W., Bortolas E., 2017, MNRAS, 464, 2301Haardt F., Madau P., 1996, ApJ, 461, 20Hirtenstein J., et al., 2019, ApJ, 880, 54Hogan C. J., Dalcanton J. J., 2000, Phys. Rev. D, 62, 063511Katz N., White S. D. M., 1993, ApJ, 412, 455Kauffmann G., 2014, MNRAS, 441, 2717Keating L. C., Weinberger L. H., Kulkarni G., Haehnelt M. G., Chardin J.,Aubert D., 2020, MNRAS, 491, 1736Kennicutt Robert C. J., 1998, ApJ, 498, 541Kimm T., Cen R., Devriendt J., Dubois Y., Slyz A., 2015, MNRAS, 451, 2900Knollmann S. R., Knebe A., 2009, ApJS, 182, 608Kuzio de Naray R., Kaufmann T., 2011, MNRAS, 414, 3617Laporte C. F. P., Penarrubia J., 2015, MNRAS, 449, L90Leung G. Y. C., Leaman R., van de Ven G., Battaglia G., 2020, MNRAS,493, 320Lux H., Read J. I., Lake G., 2010, MNRAS, 406, 2312Madau P., Shen S., Governato F., 2014, ApJ, 789, L17Malhan K., Valluri M., Freese K., 2020, MNRAS,Mashchenko S., Wadsley J., Couchman H. M. P., 2008, Science, 319, 174Maxwell A. J., Wadsley J., Couchman H. M. P., 2015, ApJ, 806, 229McGaugh S. S., Rubin V. C., de Blok W. J. G., 2001, AJ, 122, 2381Mo H. J., Mao S., 2004, MNRAS, 353, 829Moore B., 1994, Nature, 370, 629Munshi F., et al., 2013, ApJ, 766, 56Naab T., Johansson P. H., Ostriker J. P., 2009, ApJ, 699, L178Nasim I., Gualandris A., Read J., Dehnen W., Delorme M., Antonini F.,2020a, MNRAS,Nasim I., Gualandris A., Read J. I., Antonini F., Dehnen W., Delorme M.,2020b, arXiv e-prints, p. arXiv:2011.04663Navarro J. F., Eke V. R., Frenk C. S., 1996, MNRAS, 283, L72Navarro J. F., Frenk C. S., White S. D. M., 1997, ApJ, 490, 493Nipoti C., Binney J., 2015, MNRAS, 446, 1820 MNRAS , 1–14 (2020) Matthew D. A. Orkney et al.
Oñorbe J., Garrison-Kimmel S., Maller A. H., Bullock J. S., Rocha M., HahnO., 2014, MNRAS, 437, 1894Oñorbe J., Boylan-Kolchin M., Bullock J. S., Hopkins P. F., Kereš D., Faucher-Giguère C.-A., Quataert E., Murray N., 2015, MNRAS, 454, 2092Oka A., Saito S., Nishimichi T., Taruya A., Yamamoto K., 2014, MNRAS,439, 2515Oman K. A., Navarro J. F., Sales L. V., Fattahi A., Frenk C. S., Sawala T.,Schaller M., White S. D. M., 2016, MNRAS, 460, 3610Oman K. A., Marasco A., Navarro J. F., Frenk C. S., Schaye J., Benítez-Llambay A. r., 2019, MNRAS, 482, 821Peñarrubia J., Pontzen A., Walker M. G., Koposov S. E., 2012, ApJ, 759, L42Planck Collaboration et al., 2014, A&A, 571, A16Pontzen A., Governato F., 2012, MNRAS, 421, 3464Pontzen A., Governato F., 2014, Nature, 506, 171Pontzen A., Tremmel M., 2018, ApJS, 237, 23Pontzen A., Roškar R., Stinson G. S., Woods R., Reed D. M., Coles J., QuinnT. R., 2013, pynbody: Astrophysics Simulation Analysis for PythonPontzen A., Rey M. P., Cadiou C., Agertz O., Teyssier R., Read J., Orkney M.D. A., 2020, MNRAS,Power C., Navarro J. F., Jenkins A., Frenk C. S., White S. D. M., Springel V.,Stadel J., Quinn T., 2003, MNRAS, 338, 14Read J. I., Gilmore G., 2005, MNRAS, 356, 107Read J. I., Lake G., Agertz O., Debattista V. P., 2008, MNRAS, 389, 1041Read J. I., Agertz O., Collins M. L. M., 2016a, MNRAS, 459, 2573Read J. I., Iorio G., Agertz O., Fraternali F., 2016b, MNRAS, 462, 3628Read J. I., Iorio G., Agertz O., Fraternali F., 2017, MNRAS, 467, 2019Read J. I., Walker M. G., Steger P., 2019, MNRAS, 484, 1401Rey M. P., Pontzen A., 2018, MNRAS, 474, 45Rey M. P., Pontzen A., Agertz O., Orkney M. D. A., Read J. I., Saintonge A.,Pedersen C., 2019, ApJ, 886, L3Rey M. P., Pontzen A., Agertz O., Orkney M. D. A., Read J. I., Rosdahl J.,2020, MNRAS, 497, 1508Rodriguez-Gomez V., et al., 2015, MNRAS, 449, 49Romano-Díaz E., Shlosman I., Heller C., Hoffman Y., 2009, ApJ, 702, 1250Roth N., Pontzen A., Peiris H. V., 2016, MNRAS, 455, 974Sanders J. L., Evans N. W., Dehnen W., 2018, MNRAS, 478, 3879Schive H.-Y., Chiueh T., Broadhurst T., 2014, Nature Physics, 10, 496Schmidt M., 1959, ApJ, 129, 243Simon J. D., 2019, ARA&A, 57, 375Simon J. D., et al., 2020, arXiv e-prints, p. arXiv:2012.00043Sparre M., Hayward C. C., Feldmann R., Faucher-Giguère C.-A., MuratovA. L., Kereš D., Hopkins P. F., 2017, MNRAS, 466, 88Spergel D. N., Steinhardt P. J., 2000, Phys. Rev. Lett., 84, 3760Springel V., Frenk C. S., White S. D. M., 2006, Nature, 440, 1137Stewart K. R., Bullock J. S., Barton E. J., Wechsler R. H., 2009, ApJ, 702,1005Stopyra S., Pontzen A., Peiris H., Roth N., Rey M., 2020, arXiv e-prints, p.arXiv:2006.01841Tegmark M., et al., 2006, Phys. Rev. D, 74, 123507Teyssier R., 2002, A&A, 385, 337Teyssier R., Pontzen A., Dubois Y., Read J. I., 2013, MNRAS, 429, 3068Tollet E., et al., 2016, MNRAS, 456, 3542Tulin S., Yu H.-B., 2018, Phys. Rep., 730, 1Vasiliev E., 2018, AGAMA: Action-based galaxy modeling framework(ascl:1805.008)Wang W., White S. D. M., Mandelbaum R., Henriques B., Anderson M. E.,Han J., 2016, MNRAS, 456, 2301Weinberg M. D., Katz N., 2007, MNRAS, 375, 460White S. D. M., Rees M. J., 1978, MNRAS, 183, 341
APPENDIX A: DERIVING THE RELAXATION RADIUSFOR OUR EDGE SIMULATIONS
The dark matter ‘particles’ in collisionless cosmological simulationslike ramses are really ‘super-particles’ that represent unresolvedpatches of the dark matter fluid. This approximation leads to overly large two-body relaxation that causes an artificial reduction in theinner dark matter density of a halo over time (e.g. Power et al. 2003;Diemand et al. 2004a,b; Dehnen & Read 2011).The two-body relaxation timescale is given by Binney & Tremaine(1987): 𝑡 relax = 𝑁𝑡 orb 𝜋 log Λ , (A1)where 𝑁 is the number of particles within some radius 𝑟 , 𝑡 orb is theorbital time of the system given by 𝑡 orb = 𝜋 √︁ 𝑅 / 𝐺 𝑀 and Λ isthe ‘Coulomb logarithm’ given by Λ = 𝑏 max / 𝑏 min . The maximumimpact parameter 𝑏 max and the minimum impact parameter 𝑏 min de-fine the largest and smallest scales at which particles are expectedto interact with each other. The relaxation time gives the time takenfor the particle velocities to change by 90 ° , and can be considered asthe time taken for a system to lose dynamical memory of its initialconditions. A large relaxation time is preferred because this impliestwo-body relaxation has a minimal influence on the particle kinemat-ics. For standard 𝑁 -body methods, the simplest way to increase therelaxation time is by increasing the number of dark matter particles, 𝑁 , that sample the system (e.g. Dehnen & Read 2011).There is some debate in the literature over the best choices forthe impact parameters 𝑏 max and 𝑏 min . Here, we define 𝑏 max to bethe total virial size of the system 𝑟 , and 𝑏 min to be the sidelength of the highest resolution grid cell in ramses. This is theapproximate distance above which Newtonian gravity is recovered.Although multiple resolutions of grid cell are used throughout oursimulated haloes, the central regions that are of interest here arepredominantly at the highest resolution.We now perform a brief resolution study in order to determine atwhat radius numerical relaxation effects become important. In Fig.A1, we plot dark matter density profiles from a DM-only simulationat three different resolutions. These three resolutions each exhibita different amount of central density flattening due to numericalrelaxation, with the flattening becoming stronger with increasingparticle mass and with time. At 𝑧 =
0, all three resolutions areconverged beyond ≈
300 pc, but begin to deviate from the expectedNFW form within some critical radius.We define the ‘relaxation radius’, 𝑟 relax , to be the radius at whichthe relaxation time (equation A1) for the enclosed dark matter par-ticles is equal to the simulation age for any particular simulationoutput: (cid:18) 𝑟 relax kpc (cid:19) = 𝜂 (cid:18) 𝑡 sim Gyr (cid:19) 𝛼 (cid:18) 𝑀 ( 𝑟 < relax ) M (cid:12) (cid:19) − (cid:18) (cid:104) 𝑚 (cid:105) M (cid:12) (cid:19) (A2)where we have substituted 𝑡 relax for the total run time of thesimulation, (cid:104) 𝑚 (cid:105) is the mean particle mass and 𝜂 and 𝛼 are fit-ting parameters. From the data in Figure A1, we find 𝛼 = / 𝜂 = 𝐺 log Λ / . The mass within the relaxation radius, 𝑀 ( 𝑟 relax ) , is calculated directly from our simulation data and soequation A2 can be solved numerically to find 𝑟 relax . This is thenused to predict the resolution limit for our EDGE simulations inthis paper. The relaxation radii, calculated in this way, are markedon Figure A1 by the vertical lines. Notice that the dark matter den-sity profiles are shallower leftwards of 𝑟 relax in the lower resolutionsimulations as compared to the higher resolution simulations. APPENDIX B: SIMULATION CONVERGENCE
The dark matter particle mass resolution used in this paper is anorder of magnitude smaller than used in the fiducial EDGE simula-tions (Rey et al. 2019; Agertz et al. 2020; Rey et al. 2020; Pontzen
MNRAS000
MNRAS000 , 1–14 (2020) wo routes to dark matter core formation in ultra-faint dwarfs − − Radius [kpc]10 ρ D M [ M fl k p c − ] z = m DM =
139 M fl m DM = fl m DM = fl − Radius [kpc] z = − Radius [kpc] z = Figure A1.
The evolution of the 3D density profiles for Halo1459 DMO at three dark matter particle resolutions, as stated in the legend. The two lowest particlemasses (green and cyan lines) correspond to the high resolution and fiducial resolutions, respectively (see Table 1). The fainter lines indicate the radii withinwhich at least one relaxation time (estimated using equation A1) has passed, with the transition marked using an additional vertical line for clarity. In somecases, the density profiles are not plotted at smaller radii because there are insufficient particles. M ∗ / M = − M ∗ / M = − M [M fl ]10 M ∗ [ M fl ] Halo1445Halo1459Halo600Halo605Halo624FiducialHires
Figure B1.
The stellar mass-halo mass relation through time for our mainsimulation suite, where we compare our fiducial resolution simulations (faintdashed lines) to our high resolution simulations (solid lines). The black circleson the high resolution lines mark intervals of 1 Gyr in time. The grey diagonallines show constant ratios of 𝑀 ∗/ 𝑀 in powers of ten, as marked et al. 2020). In this Appendix, we perform a convergence study todetermine how our results are impacted by dark matter resolution.As previously in this paper, we distinguish between the lower andhigher resolution simulations by appending the labels ‘fiducial’ and‘hires’, respectively, to their name.In Figure B1, we plot the total stellar mass within 𝑟 versus 𝑀 as a function of time for the fiducial (faint dashed) and hires(opaque solid) EDGE simulations. We find good convergence inshape of the general evolution of each simulated dwarf. The higher resolution simulations form systematically more stars (as first notedin Agertz et al. 2020). However, the final stellar masses typicallyagree within ∼
30% and at worst differ by a factor of ∼ two (forHalo600). This is within the expected uncertainties due to modellinggalaxy formation (Agertz et al. 2020).In Figure B2, we compare the 3D dark matter density profiles forour main simulation suite at two redshifts, 𝑧 = 𝑧 = 𝑟 > 𝑟 relax (vertical lines), and there is good agreement also for the baryonicsimulations Halo1445, Halo1459 and Halo624, at both 𝑧 = 𝑧 =
0. However, two of our EDGE simulations show poor convergence.Halo600 fiducial substantially rejuvenates its star formation after 𝑧 =
1, forming 17% of its final stellar mass after this time. Thisextended period of star formation is replaced by a singular burst inHalo600 hires. As a result, Halo600 fiducial undergoes significantlymore dark matter heating that causes its final inner density profile tobe substantially lower than Halo600 hires. Similarly, Halo605 fiducialforms a larger dark matter core than Halo605 hires at early times.However, this disparity is largely resolved before 𝑧 = 𝑀 ∗ / 𝑀 (e.g.Peñarrubia et al. 2012; Di Cintio et al. 2014; Read et al. 2019). Thisis true to leading order. However, it also matter how those stars form,as illustrated by Halo600 fiducial versus Halo600 hires. The latteractually forms more stars, but because at late time they form in asingle burst, this leads to less dark matter heating. And, at least atthe very edge of galaxy formation, it also matters what the mergerhistory is. Halo605 fiducial goes on to partially lose its dark mattercore due to a late cuspy merger. MNRAS , 1–14 (2020) Matthew D. A. Orkney et al. ρ D M [ M fl k p c − ] Halo1445 z = z = z = z = z = − Radius [kpc]10 ρ D M [ M fl k p c − ] z = DMO hiresDMO fiducialHiresFiducial − Radius [kpc] z = − Radius [kpc] z = − Radius [kpc] z = − Radius [kpc] z = Figure B2.
The 3D dark matter density profiles of our main simulation suite, where we compare our DM-only simulations (faint lines) and baryonic simulations(opaque lines) at fiducial (dashed lines) and high (solid lines) resolution. The upper panels show this comparison at 𝑧 =
4, by which time all haloes are quencheddue to reionisation (some will later reignite their star formation). The lower panels show this comparison at 𝑧 =
0. The numerical relaxation radius, as definedin Appendix A, is indicated by a short vertical line for the baryonic simulations. This limit is similar for the corresponding DMO simulations.This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000