MNRAS 000, 1–13 (2020) Preprint 3 November 2021 Compiled using MNRAS LATEX style file v3.0

Satellite mass functions and the faint end of the galaxy mass-halo massrelation in LCDM

Isabel M.E. Santos-Santos1?, Laura V. Sales2, Azadeh Fattahi3 and Julio F. Navarro1

1Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2, Canada2Department of Physics and Astronomy, University of California Riverside, 900 University Avenue, CA 92507, USA3Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

The abundance of the faintest galaxies provides insight into the nature of dark matter and the process of dwarf galaxy

formation. In the LCDM scenario, low mass halos are so numerous that the efficiency of dwarf formation must decline

sharply with decreasing halo mass in order to accommodate the relative scarcity of observed dwarfs and satellites

in the Local Group. The nature of this decline contains important clues to the mechanisms regulating the onset of

galaxy formation in the faintest systems. We explore here two possible models for the stellar mass (M∗)-halo mass

(M200) relation at the faint end, motivated by some of the latest LCDM cosmological hydrodynamical simulations.

One model includes a sharp mass threshold below which no luminous galaxies form, as expected if galaxy formation

proceeds only in systems above the Hydrogen-cooling limit. In the second model, M∗ scales as a steep power-law

of M200 with no explicit cutoff, as suggested by recent semianalytic work. Although both models predict satellite

numbers around Milky Way-like galaxies consistent with current observations, they predict vastly different numbers

of ultra-faint dwarfs and of satellites around isolated dwarf galaxies. Our results illustrate how the satellite mass

function around dwarfs may be used to probe the M∗-M200 relation at the faint end and to elucidate the mechanisms

that determine which low-mass halos “light up” or remain dark in the LCDM scenario.

Key words: galaxies: dwarf – galaxies: haloes – galaxies: luminosity function

1 INTRODUCTION

Ultrafaint dwarfs, defined here as dwarf galaxies with stel-lar masses M∗ < 105 M(Bullock & Boylan-Kolchin 2017),are typically systems whose extremely low surface brightness(µv ≥ 27 mag/arcsec2) hinders their discovery and makesfollow-up studies extremely difficult. Indeed, although recentefforts have led to the discovery of dozens of ultrafaints inthe Milky Way (MW) halo (see Simon 2019, and referencestherein), it remains unclear how many more of them may stilllurk undetected in the vicinity of our Galaxy.

The ultrafaint population also remains largely unexploredin external galaxies, with only loose constraints available onthe massive-end of the ultrafaint regime in M31 (see the dwarfgalaxy catalog compiled and maintained by McConnachie2012). Identifying isolated ultrafaints in the field is even moredifficult, with few, if any, reported so far outside the LocalGroup.

Because of their extreme intrinsic faintness, few bright starsare available for spectroscopic study in ultrafaints, even whenusing some of the largest ground-based telescopes. This im-plies that the characterization of some of their basic prop-

? E-mail: isantos@uvic.ca

erties, such as their metallicity distribution, elemental abun-dances, or velocity dispersion, is subject to large uncertainty.Poorly determined velocity dispersions, in particular, affectour ability to estimate halo masses and to constrain the rela-tion between stellar mass (M∗) and halo virial1 mass (M200)at the very faint end of the galaxy luminosity function.

Indeed, our best constraints on the M∗-M200 relation atthe faint end arguably comes from abundance-matching tech-niques (Conroy et al. 2006; Guo et al. 2010; Moster et al.2013; Behroozi et al. 2013). Because the galaxy stellar massfunction around M∗ ∼ 108 M (the faintest luminosities forwhich it is well constrained; see, e.g., Baldry et al. 2012) issubstantially shallower than the LCDM halo mass function inthat regime (Springel et al. 2008; Boylan-Kolchin et al. 2009),it is clear that galaxy formation must become increasingly in-efficient towards decreasing halo masses. Characterizing thisdecline in galaxy formation efficiency at the low-mass end isdifficult, and there is so far no consensus on how steep the

1 We shall use halo “virial” properties defined within a radius,r200, enclosing a mean density 200 times the critical density forclosure. A subscript ‘200’ identifies quantities defined within or at

that radius.

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decline is, on what the scatter in M∗ at fixed M200 mightbe, and on whether there is a characteristic “threshold” halomass below which no luminous galaxy forms in LCDM.

The lack of consensus concerns not only abundance-matching studies, but also direct cosmological simulationsof the formation of the faintest galaxies. For example, Lo-cal Group simulations from the APOSTLE project (Sawalaet al. 2016; Fattahi et al. 2016) suggest a relation with a fairlysharp cutoff at low halo masses, where few, if any, isolated ha-los with Vmax below ∼ 15 km s−1 host a galaxy (Fattahi et al.2018). At least qualitatively, this is the behaviour expectedin scenarios where luminous galaxy formation only proceedsin halos with masses exceeding the “hydrogen-cooling limit”(HCL) set by the primordial abundance cooling function afteraccounting for the presence of an evolving, ionizing UV back-ground (see; e.g., Gnedin 2000; Okamoto et al. 2008; Benitez-Llambay & Frenk 2020).

On the other hand, some cosmological simulations suggestthat even halos below the HCL may be able to form stars, sothat no clear minimum “threshold mass” for galaxy formationexists. For example, FIRE-2 simulations (Hopkins et al. 2018;Wetzel et al. 2016; Wheeler et al. 2019) or those from theMarvel/JC league collaboration (Munshi et al. 2021) seembetter described by a power-law M∗-M200 relation similar tothat reported by Brook et al. (2014) and which extends wellbelow the HCL mass.

This argument has been strengthened by semi-analyticmodels that attempt to reproduce simultaneously the MWsatellite mass function and its radial distribution. Becausetides may, in principle, disrupt subhalos near the MW disk,accounting for the large number of ultrafaint satellites dis-covered in the inner ∼ 40 kpc of the MW halo has led to thesuggestion that populating subhalos well below the HCL withluminous galaxies may be needed (Kelley et al. 2019; Grauset al. 2019; Nadler et al. 2020).

However, there is still substantial uncertainty aboutwhether Galactic tides are actually able to fully disrupt cuspyLCDM subhalos (van den Bosch et al. 2018; Errani & Navarro2021) and no cosmological simulation has actually reachedthe ultrafaint regime probed by observations. Despite theseuncertainties, it is clear that simulation predictions for thefaintest dwarfs appear to differ, depending on the resolutionand subgrid physics adopted in the simulations. This is prob-lematic, as the steep halo mass function in LCDM impliesthat even small differences in the stellar mass-halo mass rela-tion should result in large differences in the expected numberof faint galaxies.

We explore here how the abundance of ultrafaint satellitesmay be used to place constraints on the behaviour of theM∗-M200 relation at the faint end. Their abundance aroundisolated dwarf primaries is particularly constraining. This isbecause the subhalo mass function is well approximated by apower law (Springel et al. 2008) and, therefore, a power-lawstellar mass-halo mass relation would result in “self-similar”satellite mass functions independent of primary mass (Saleset al. 2013). This is a clear prediction that can be used to gaininsight into the shape of the stellar mass-halo mass relationfor primaries at the faint end.

We explore these issues here, and argue that the ultrafaintsatellites of isolated dwarf galaxies is a promising way to elu-cidate how the faintest galaxies form and populate dark halosat the low mass end. This paper is organized as follows. We

begin by motivating in Sec. 3 two particular analytic formsof the faint-end M∗-M200 relation (a power-law and one withan explicit low-mass cutoff) based on results from recent cos-mological hydrodynamical simulations (Sec. 2). We validatethe “cutoff” model in Sec. 4.1 by reproducing results fromthe APOSTLE runs. We then compare the results from bothour models for the ultrafaint satellite population of dwarfprimaries spanning a wide range of stellar mass (Sec. 4.2),and then contrast these results with available data for theLocal Group in Sec. 5. We conclude with predictions for fu-ture satellite surveys of ultrafaint dwarfs around primariessuch as the Large Magellanic Cloud (LMC) in Sec. 5.3 andsummarize our main results in Sec. 6.

2 NUMERICAL METHODS

We shall use results from a number of recent cosmologicalhydrodynamical simulations of dwarf galaxy formation inLCDM. These include simulations of individual galaxies fromthe NIHAO project (Wang et al. 2015; Buck et al. 2019),an ensemble of simulations using the FIRE (Hopkins et al.2018; Wheeler et al. 2019; Garrison-Kimmel et al. 2019) andCHANGA (Munshi et al. 2021) codes, as well as simulationsof constrained Local Group environments from the APOS-TLE project (Sawala et al. 2016; Fattahi et al. 2016, 2018).Since we shall use the latter to calibrate our modeling proce-dure we describe the APOSTLE simulations in some detailbelow. Results from the other runs are taken directly as re-ported in those publications, to which we refer the interestedreader for details.

2.1 The APOSTLE simulations

The APOSTLE project is a set of 12 ‘zoom-in’ cosmologicalvolumes tailored to reproduce the main properties of the Lo-cal Group. Each volume is selected from a large cosmologicalbox to contain a pair of halos with masses, relative radialand tangential velocities, and surrounding Hubble flow, con-sistent with the corresponding values observed for the MilkyWay-Andromeda pair (Fattahi et al. 2016).

The APOSTLE runs used the EAGLE galaxy formationcode (Schaye et al. 2015; Crain et al. 2015), using the so-called‘Reference’ parameters. This code includes subgrid physicsrecipes for radiative cooling, star formation in gass exceedinga metallicity-dependent density threshold, stellar feedbackfrom stellar winds, radiation pressure and supernovae explo-sions, homogeneous X-ray/UV background radiation, super-massive black-hole growth and AGN feedback (note that thelatter has negligible effects on dwarf galaxies and is thereforeunimportant in APOSTLE).

The EAGLE model was calibrated to approximate the ob-served z = 0.1 galaxy stellar mass function in the M∗ = 108-1012 M range. Simulated galaxies thus roughly follow theabundance-matching M∗-M200 relation of Behroozi et al.(2013) and Moster et al. (2013). No extra calibration is madein APOSTLE, and therefore the stellar-halo mass relationthat results for fainter galaxies may be regarded as the ex-trapolation of the same subgrid physics to lower mass halos.

The APOSTLE volumes have been run at three differ-ent levels of resolution. In this paper we use the 5 highest-resolution volumes (labelled ”AP-L1” in Fattahi et al. 2016).

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 3

These runs have initial dark matter and gas particle massesof mDM ∼ 5× 104 M and mgas ∼ 1× 104 M, respectively,and a gravitational softening length of 134 pc at z = 0. TheAPOSTLE volume simulated at highest resolution fully con-tains a sphere of radius r ∼ 3.5 Mpc from the midpoint ofthe MW and M31 analog halos.

The friends-of-friends (FoF) groupfinding algorithm (Daviset al. 1985) (with linking length equal to 0.2 times themean interparticle separation) and the SUBFIND halo finder(Springel et al. 2001; Dolag et al. 2009) were used to identifyhaloes and subhaloes. We shall refer to the galaxies formedin the most massive subhalos of each FoF group as “centrals”,and to the rest of galaxies within the virial radius of each FoFcentral as “satellites”. Throughout the paper we shall use theterm “primary” to refer to a central galaxy that may havesatellites.

APOSTLE assumes a flat LCDM cosmological model fol-lowing WMAP-7 parameters (Komatsu et al. 2011): Ωm =0.272; ΩΛ = 0.728; Ωbar = 0.0455;H0 = 100h km s−1 Mpc−1;σ = 0.81; h = 0.704.

3 MODELING THE SATELLITE STELLAR MASSFUNCTION

The satellite mass function of a primary of given stellar mass,Mpri

∗ , depends mainly on (i) the mass function of subhalospresent in the halo of that system, on (ii) the relation betweenstellar mass and subhalo mass, and on (iii) the possible re-duction of stellar mass due to tidal stripping after infall. Thefirst item depends mainly on the primary halo virial mass,or, equivalently, on V pri

200, and has been extensively studiedthrough cosmological N-body simulations.

For the second item, which, in the case of satellites, appliesbefore first infall into the primary halo, it is customary toexpress the stellar mass not as a function of (sub)halo mass,but rather in terms of its maximum circular velocity, Vmax, aquantity more resilient to tidal effects than virial mass.

The third item is the most difficult to treat analytically,since it depends strongly on the pericentric distance of theorbit, the number of orbits completed, and the radial segrega-tion of stars within each subhalo. Fortunately, as we shall seebelow, the fraction of stellar mass lost to tides is, on average,small, and we shall neglect it in our modeling in the interestof keeping the model as simple as possible.

We describe below the parametrizations we adopt to buildan analytical model for the satellite mass function of a pri-mary of mass Mpri

∗ . These parametrizations are motivatedby the results of cosmological N-body and hydrodynamicalsimulations, as discussed in detail in the remainder of thissection. We note that the two satellite mass function mod-els explored here differ only in the assumptions made for theM∗-Vmax relation.

3.1 Subhalo mass function

The substructure mass function of LCDM halos scales in di-rect proportion to the virial mass of the primary halo and hasbeen shown to be fairly well approximated by a power law.Following Wang et al. (2012), the average number of subhaloswithin the virial radius of an isolated (central) LCDM halo

10−1 100

ν = V satmax/V

pri200

10−2

10−1

100

101

102

103

〈Nsu

b〉(>

ν)

Wang+12 (Vmax, Eq.1)

Tide-corrected Wang+12

APOSTLE Vpri200 > 40 km/s

Aquarius

Phoenix

Figure 1. The subhalo velocity function (i.e., average number of

subhalos with ν = Vmax/V200 above a certain value). The solidblack line shows the function proposed by Wang et al. (2012)

(Eq. 1), which describes well the substructure mass function of

LCDM halos of all masses in the Millenium DM-only cosmologicalsimulation, particularly in the 0.1 < ν < 0.5 range. Extrapola-

tions of this line beyond such range are shown in dotted line style.

The average subhalo velocity function for APOSTLE halos withV pri

200 > 40 km/s is shown with a solid gray line. For comparison,

thin gray dotted and dashed-dotted lines show the average sub-

halo functions found for halos in simulations of the Aquarius andPhoenix projects, respectively. A thick red line shows the ’tide-

corrected’ version of Eq. 1, resulting from statistically convertingVmax values to Vpeak as explained in Sec. 3.

may be expressed as

〈Nsub〉(> ν) = 10.2 (ν/0.15)−3.11 (1)

where ν = Vmax/Vpri200. This function applies to all LCDM

halos regardless of mass, and has been tested well over the0.1 < ν < 0.5 range. The scatter around the average numberat given ν is well approximated by Poisson statistics.

We compare in Fig. 1 the results of three sets of cosmo-logical simulations with the predictions from Eq. 1 (thickblack line). The simulations include the average of all MilkyWay-sized halos of the Aquarius project (dotted black line;Springel et al. 2008), that of the cluster-sized halos of thePhoenix project (dot-dashed black line; Gao et al. 2012), aswell as that of all halos with V200 > 40 km/s in the APOS-TLE project (solid grey line). As is clear from this figure,Eq. 1 reproduces quite well the subhalo mass function of ha-los spanning a wide range of virial mass.

However, this function is expressed in terms of the present-day subhalo maximum circular velocity, Vmax, which mayhave been affected by tidal stripping after infall. Since thestellar content of a subhalo is more closely tied to Vpeak,the maximum circular velocity prior to infall, Eq. 1 musttherefore be corrected to yield the distribution of Vpeak val-ues needed in the modeling.

MNRAS 000, 1–13 (2020)

4 Santos-Santos et al.

10−1 100

Vmax/V200

10−1

100

Vp

eak/V

200

APOSTLE

0.2 0.4 0.6 0.8 1.0

Vmax/Vpeak

0.0

0.2

0.4

0.6

0.8

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mal

ized

N(<

)

All subhalos

0.05 ≥ Vmax/V200 < 0.1

0.1 ≥ Vmax/V200 < 0.2

0.2 ≥ Vmax/V200 < 0.3

0.3 ≥ Vmax/V200 < 1

Figure 2. Vmax/V200 vs. Vpeak/V200 (left) and normalized cumulative distribution of Vmax/Vpeak (right) for APOSTLE subhalos. Lines

of different colors show results for subhalos in different Vmax/V200 bins, as indicated in the legend.

To this end, we explore the relation between Vmax and Vpeak

in APOSTLE halos. This is shown in the left-hand panel ofFig. 2, scaled by the virial velocity of the primary, V200, atz = 0. As expected, APOSTLE subhalos had Vpeak valuessystematically larger than Vmax. The distribution of the ratioVpeak/Vmax is shown (in cumulative form) in the right-handpanel of Fig. 2, for various bins in Vmax/V200. This panelshows that, on average, the reduction in subhalo maximumcircular velocity that result from tides is fairly modest andlargely independent of subhalo mass.

Only the most massive subhalos (i.e., Vmax/V200 > 0.3) de-viate from this trend, and appear substantially less affectedby tides than less massive subhalos. The median Vmax/Vpeak

is ∼ 0.82 for low mass subhalos, but climbs to ∼ 0.93 atthe massive end. Why do tides seem to affect more massivehalos less? This is most likely a result of the rapid dynami-cal friction-driven evolution of massive halos, which tend tomerge with the primary halo quickly after accretion. In otherwords, the (few) very massive halos present at any given timeresult from recent accretion events where tides have not hadany substantive effect yet.

The results shown in Fig. 2 can be used to statistically cor-rect the distribution of Vmax/V200 measured in cosmologicalsimulations and to estimate the Vpeak subhalo distribution ofa given halo. The result is illustrated by the thick red linein Fig. 1, which shows the tide-corrected form of Eq. 1. Weshall hereafter adopt the tide-corrected version of Eq. 1 (withPoisson scatter) to model the Vpeak distribution of a halo ofgiven V200.

3.2 The stellar mass-halo mass relation

What is the stellar mass expected for a subhalo of givenVpeak? Fig. 3 motivates our choice of models for the stel-lar mass-halo mass relation. This figure shows the M∗-

Vmax relation reported for central galaxies (i.e., isolated andnot tidally-stripped systems) selected from recent cosmolog-ical hydrodynamical simulations, as indicated in the legend.When necessary, we have transformed quoted halo massesinto Vmax assuming they follow a Navarro-Frenk-White den-sity profile (hereafter, NFW, Navarro et al. 1996, 1997) with amass-concentration relation as given by Ludlow et al. (2016).

These simulations suggest two different behaviours for theM∗-Vmax relation. On the left panel of Fig. 3 we have groupedsimulations where M∗ and Vmax seem better described by asimple power law that extends down from Vmax ∼ 200 km/sto less than ∼ 10 km/s, deep into the ultrafaint regime (M∗ ∼10-102M).

Interestingly, the power-law follows closely the extrapo-lated M∗-Vmax relation from Moster et al. (2013) (dashedred line),

M∗ =0.0702

[(m1)−β + (m1)γ ]M200, (2)

where m1 = M200/M1, M1 = 1011.59 M, β = 1.376, andγ = 0.608. As above, M200 in this relation can be easily tran-formed into Vmax assuming an NFW density profile and amass-concentration relation.

On the other hand, the panel on the right in Fig. 3 groupssimulations whose results seem better described by a rapidlysteepening relation between M∗ and Vmax towards decreas-ing Vmax, suggesting the presence of a cutoff in the relation.Following Fattahi et al. (2018), this “cutoff” relation may beparametrized as:

M∗ = ηα exp(−ηµ)M0, (3)

with η = Vmax/50 km/s, and (M0, α, µ) = (3 ×108 M, 3.36, −2.4), shown by the solid black line in Fig.3.

Although those authors fitted only results for central galax-ies, an indistinguishable fit is obtained when adding the M∗-Vpeak data for APOSTLE satellites, which justifies the use of

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 5

101 102

Vmax [km/s]

101

103

105

107

109

1011

M∗

[M

]

NIHAO (Wang+15)

FIRE HR (Wheeler+19)

FIRE (Hopkins+17)

Fattahi+18

Moster+13

MW (Cautun+20)

101 102

Vmax [km/s]

101

103

105

107

109

1011

fbarx M200

FIRE (GarrisonKimmel+19)

APOSTLE (Fattahi+18)

CHANGA (Munshi+21)

NIHAO-UHD (Buck+19)

Figure 3. M∗-Vmax relations for simulated ’central’ galaxies at z = 0 from recent cosmological hydrodynamical simulations (see legends).These have been divided into two groups, depending on whether they either roughly follow a “power-law”-like relation (left panel), or a

relation with a sharp “cutoff” in M∗ at low Vmax (right panel). The power-law results are well described by the extrapolated abundance-matching relation from Moster et al. (2013) (red dashed line); the cuttoff results may be approximated by the fit to APOSTLE data

from Fattahi et al. (2018) (black line). Open red star symbols indicate FIRE-HR galaxies with 1 and 15 stellar particles. Backsplash

galaxies in APOSTLE are shown as smaller black points. In the right-hand panel, we mark all simulated galaxies from the other samplesshowing Vmax < 20 km/s with smaller symbols as they are likely to have been affected by tides as well. Indeed, the NIHAO-UHD sample

avoids most backsplash galaxies by selecting central dwarfs outside 2.5 × r200 of a massive primary, and shows in general Vmax > 20

km/s. For reference, a dotted gray line marks the maximum total amount of baryons inside M200 as expected from the cosmic mix, wherefbar = Ωbar/Ωm.

Eq. 4 to model the stellar content of a satellite of given Vpeak.(For centrals Vpeak ≈ Vmax at z = 0, by construction.)

The“power-law”and“cutoff”relations between stellar massand peak velocity are the sole difference between the twomodels we explore in this paper. These two relations are plot-ted in both panels of Fig. 3 for ease of comparison (red dashedcurve for “power-law” and solid black for “cutoff”). Becausetheir predictions differ for systems like the Milky Way, whichwe shall use to calibrate our models, we adopt a single M∗-Vmax relation for Vmax > 84 km/s (where the Fattahi et al.2018 and the Moster et al. 2013 lines cross each other). Thisis shown by the power-law solid gray line depicted in Fig. 3and may be expressed as

M∗/M = 3.29× 109 (Vmax/84 km/s)4.52, (4)

applicable only for Vmax > 84 km/s.

In what follows, we shall express the stellar mass-halo massrelation in terms of Vpeak, defined as the maximum circularvelocity of a satellite before infall, or, for centrals, as Vmax atz = 0.

3.2.1 Scatter

As is clear from Fig. 3, the M∗-Vpeak relation has substan-tial scatter. We account for this assuming that M∗ follows,at given Vpeak, a log-normal distribution with a dispersion,σM∗ , that increases toward decreasing halo masses. Follow-ing Garrison-Kimmel et al. (2017) and Munshi et al. (2021),we parametrize σM∗ as a broken power-law of Vpeak, as il-lustrated by the solid black line in the right-hand panel ofFig. 4,

σM∗ =

σ0 Vpeak > 57 km/s

κ log10(Vpeak/V0) Vpeak < 57 km/s(5)

with σ0 = 0.24 dex, κ = −1.26, and V0 = 88.6 km/s. Forcomparison, we show the scatter measured for APOSTLEgalaxies (see open black circles in the right panel of Fig. 4).

The left and middle panels of Fig. 4 show the assumed“cut-off” and “power-law” relations, respectively, where the cutoffrelation is compared to the APOSTLE results used to moti-vate Eq. 5. The shaded bands delineate the 10-90 percentilescatter in M∗ at given Vpeak. As expected, the assumed cut-off relation reproduces the APOSTLE results well. Assumingthe same scatter, the “power-law” model also reproduces the

MNRAS 000, 1–13 (2020)

6 Santos-Santos et al.

101 102

Vpeak/km s−1

102

104

106

108

1010

M∗/

M

Cutoff

APOSTLE

101 102

Vpeak/km s−1

102

104

106

108

1010Power-law

FIRE

1.5 2.0

log(Vpeak/km s−1)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

σM∗

APOSTLE

This work

Munshi+21

Figure 4. M∗-Vpeak relations for the two models studied in this paper. Left: “cutoff”; Middle: “power-law”. The “cutoff” relation has been

calibrated to match the APOSTLE simulation data for subhalo centrals at z = 0 and for subhalo satellites of MW/M31 analog primaries

before infall, when their circular velocities peak (black circles). We show our “power-law” model in comparison to FIRE simulation data(black symbols in middle panel; Hopkins et al. 2018; Wheeler et al. 2019). The shaded bands show the 10 − 90 percentile range in M∗ at

given Vpeak. The scatter is assumed to be a function of Vpeak, following Eq. 5. The black line in the rightmost panel shows our assumed

scatter, compared with that proposed by Munshi et al. (2021).

results of the FIRE simulations well, as shown in the mid-dle panel of Fig. 4. For simplicity, we assume that both thecutoff and the “power-law” model have the same M∗ scatterdependence on Vpeak given by Eq. 5.

3.3 Stellar mass loss

As mentioned above, our simple modeling shall neglect tidallyinduced stellar mass loss in subhalos. This is clearly a simpli-fication, but finds support in the results for APOSTLE satel-lites, which show that the effects of stellar mass loss are quitemodest. This is shown in Fig.5, where we plot the stellar massof APOSTLE satellite galaxies at z = 0 versus that at zpeak,the redshift when its maximum circular velocity peaked. Un-like Vmax, M∗ changes, on average, very little after infall intothe main halo. Half of APOSTLE satellites have lost less than∼ 22% of their peak mass, and only 10% have lost more than∼ 58% since infall. In the interest of simplicity, we have de-cided not to include any corrections for stellar mass loss, buthave checked that none of our main conclusions are altered ifa correction of the magnitude suggested by Fig. 5, is imple-mented.

3.4 The cutoff and “power-law” models

The assumptions discussed above allow us to compute the ex-pected satellite stellar mass function for a system of arbitraryvirial mass. To summarize, for a halo of given V200 we firstdraw a realization of the subhalo Vpeak function consistentwith the tide-corrected Eq. 1, assuming Poisson scatter. Foreach subhalo, we then draw a stellar mass using either the“cutoff” or the “power-law” models described in Sec. 3.2, withscatter as given by Eq. 5. Unless otherwise specified, we shallalways show median results obtained by combining at least

104 105 106 107 108 109 1010

M sat∗ /M (z = 0)

103

104

105

106

107

108

109

1010

Msa

t∗/M

(zp

eak)

APOSTLE satellites

Figure 5. Stellar mass of APOSTLE satellite galaxies measuredat z = 0 versus that measured at the time when the maximum

circular velocity of a subhalo peaks, zpeak. The 1:1 correspondenceis marked with a dashed gray line.

∼ 100 independent realizations of each primary, together withthe 10-90 percentile range.

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 7

102 105 108

M sat∗ /M

100

101

N(>

)

V200[km/s]=154.9±18.0

Npri =10

APOSTLE

Cutoff

102 105 108

M sat∗ /M

V200[km/s]=109.3±8.8

Npri =5

102 105 108

M sat∗ /M

V200[km/s]=62.1±10.5

Npri =31

102 105 108

M sat∗ /M

V200[km/s]=26.8±8.0

Npri =306

Figure 6. Satellite stellar mass functions of APOSTLE primaries (grey) compared to those predicted by the “cutoff” model (blue). Eachpanel corresponds to results for different bins in virial velocity of the primaries, V200, as given in the legend. The models assume the same

V200s as those of APOSTLE primaries in each bin. The number of APOSTLE primaries in each bin, Npri, is also indicated in the legend.

Solid lines show the median results and shaded areas the 10-90 percentile range. For APOSTLE, results below M∗ < 105 M are shownwith a dashed linestyle to indicate that objects below this mass are likely not well resolved.

102 105 108

M sat∗ /M

100

101

102

N(>

)

Mpri∗ /M=3× 1010

Cutoff

Power-law

102 105 108

M sat∗ /M

Mpri∗ /M=3× 109

102 105 108

M sat∗ /M

Mpri∗ /M=3× 108

102 105 108

M sat∗ /M

Mpri∗ /M=3× 107

Figure 7. Comparison of the satellite stellar mass functions predicted by the “cutoff” (blue) and “power-law” (red) models. Each panel

shows results assuming a different fixed value of the stellar mass of the primary, Mpri∗ (see legend). Solid lines show median results and

shaded bands the 10-90 percentile range.

4 RESULTS

4.1 The cutoff model and APOSTLE

We start by comparing the results of the “cutoff” modelwith satellite mass functions from the APOSTLE simula-tions. This is shown in Fig. 6, where each panel shows, in grey,the APOSTLE satellite mass functions for central galaxies,binned by halo virial velocity. The average V200 and standarddeviation in each bin is given in the legend of each panel. Solidlines show the median satellite mass function in the bin, whilethe shaded area represents the 10-90 percentile distribution.

Although we show mass functions down to stellar massesas low as M∗ > 102 M we note that objects with M∗ <105 M in APOSTLE are resolved with fewer than 10 starparticles. Therefore, below that mass APOSTLE results are

best regarded as lower limits rather than actual simulationpredictions.

By construction, the first bin (leftmost panel) includesthe 10 primaries that are considered MW and M31 analogsin the APOSTLE volumes. For these APOSTLE primaries(V200 ≈ 150 km/s, or, equivalently, M200 ∼ 1.2 × 1012 M),the median number of satellites with M∗ > 105 M is∼ 24.1+18.5

−6.7 , where the uncertainties represent the 10-90 per-centile range. For dwarf primaries with V200 ≈ 62 km/s (thirdpanel from the left) the number of satellites in APOSTLE isdrastically reduced by the cutoff in the M∗-Vpeak relation,with a median of only ∼ 2.0+1.7

−1.0 satellites with M∗ > 105

M. Finally, the last panel shows that no luminous satellitesare found in APOSTLE around primaries with V200 . 35km/s.

MNRAS 000, 1–13 (2020)

8 Santos-Santos et al.

10−9 10−7 10−5 10−3 10−1

M sat∗ /Mpri

∗

100

101

102

N(>

)

Cutoff

10−9 10−7 10−5 10−3 10−1

M sat∗ /Mpri

∗

Power-law

Mpri∗ /M=3× 1010

Mpri∗ /M=7× 109

Mpri∗ /M=109

Mpri∗ /M=3× 108

Figure 8. Scaled satellite stellar mass functions (i.e., Msat∗ /Mpri

∗ ) predicted by the “cutoff” (blue) and “power-law” (red) models. Different

linestyles show results assuming a different fixed value of the stellar mass of the primary, Mpri∗ (see legend). Lines show median results

and shaded bands the 10-90 percentile range. Note that for Mpri∗ . 5 × 109 M the “power-law” mass function becomes independent of

primary mass, as discussed by Sales et al. (2013).

The blue bands in Fig. 6 show the results of the “cutoff”model, applied to a sample of primaries whose number andV200 distribution matches that in each APOSTLE bin. Weuse 10 independent realizations of the satellite mass functionof each primary to obtain robust results.

There is in general good agreement between the analyti-cal “cutoff” model and the simulation results, especially forsatellites with M∗ > 105 M. Even the number of massive(M∗ > 108 M) satellites is well reproduced, with a medianof ∼ 1-2 LMC or SMC-mass satellites expected around MW-mass primaries (leftmost panel).

This is not unexpected, given that we have motivated themodel on APOSTLE results, but it provides validation for ourapproach. It also allows us to predict the population of dwarfsfainter than currently resolved by APOSTLE and other sim-ulations. Importantly, the “cutoff” model predicts a steadydecline in the number of satellites surrounding dwarfs of de-creasing mass, approaching zero as the mass of the primaryapproaches the threshold mass (rightmost bin in Fig. 6).

4.2 Cutoff vs power-law model satellite mass functions

We now compare the satellite stellar mass functions predictedby each model, as a function of the stellar mass of the pri-mary. This is shown in Fig. 7, where each panel correspondsto a different Mpri

∗ , given in the panel legends (cutoff in blue,power-law in red). The most obvious difference is the largedifference in the number of faint satellites predicted by eachmodel. Hundreds of ultrafaints with M∗ > 102 M are ex-pected in the“power-law”model, even for primaries as faint asthe Magellanic Clouds, whereas ultrafaint numbers are muchless numerous in the case of the “cutoff” model.

The difference between models is more clearly appreciatedwhen comparing the normalized satellite mass functions; i.e.,

the satellite mass function expressed in terms of M sat∗ /Mpri

∗ .This is shown in Fig. 8 for all primary stellar mass bins inthe “cutoff” model (left) and “power-law” model (right). Forthe “power-law” model the normalized satellite mass func-tion changes little with primary stellar mass. In particular,primaries with Mpri

∗ < 109 M would be expected to sharethe same normalized satellite mass function, as shown by theoverlap of the red dotted and long-dash-dotted lines in theright-hand panel of Fig. 8.

As discussed by Sales et al. (2013), this near “self similar-ity” arises because the subhalo mass function and the stel-lar mass-halo mass relation in this model are both close topower-laws, and thus scale-free. This is particularly true atM∗ < 109 M (see middle panel Fig. 4), which explains whylower mass bins overlap in their normalized satellite massfunction. On the other hand, if the stellar-halo mass relationis not scale-free, as is the case for the “cutoff” model, thenormalized satellite mass function declines with decreasingprimary mass (see blue curves on the left panel of Fig. 8).The large differences between models suggest that the satel-lite mass function around isolated primaries spanning a widerange of mass (and, in particular, including dwarfs) may beused to infer the shape of the stellar mass-halo mass relationat the faint end.

Another, perhaps more intuitive contrast between modelsmay be obtained comparing the expected total number ofsatellites more massive than a given stellar mass as a func-tion of primary halo virial mass. We show this in Fig. 9,where different line-styles indicate the cumulative number ofsatellites above a given M sat

∗ , as labeled on the left panel, asa function of either the virial velocity of the host (a proxyfor the primary halo mass; lower x-axis) or the correspond-ing stellar mass of the primary according to each of the twomodels (upper x-axis).

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 9

20 30 40 60 80 100 200

V200[km/s]

100

101

102

N(>

)

Cutoff

M sat∗ /M > 103

M sat∗ /M > 104

M sat∗ /M > 105

M sat∗ /M > 106

20 30 40 60 80 100 200

V200[km/s]

100

101

102

Power-law

6 7 8 9 10

log Mpri∗ /M

6 7 8 9 10

log Mpri∗ /M

Figure 9. The cumulative number of satellites with Msat∗ above certain values (see legend) predicted by the “cutoff” and “power-law”

models for primaries with fixed values of V200, as indicated in the x-axis. The corresponding Mpri∗ values, according to each model, can

be read from the top x-axis. Lines show median results and shaded bands show the 10-90 percentile range.

For massive satellites (i.e., M sat∗ > 106 M, solid line) the

predictions of the two models are rather similar, with ∼ 2.5satellites on average in hosts with V200 = 75 km/s and 10-30satellites for hosts in our most massive bin, V200 ≥ 150 km/s.However, the predictions of the two models differ appreciablywhen considering fainter satellites and, in particular, in theregime of ultrafaint dwarfs.

For example, in the “power-law” model, a dwarf primarywith Mpri

∗ ∼ 109 M (like the LMC) is expected to host∼ 70 satellites with M sat

∗ ≥ 103 M, ∼ 50 of which would beultrafaint (M sat

∗ < 105 M). On the other hand, in the “cut-off” model only 8 satellites are expected with M sat

∗ ≥ 103 Mfor the same primary. As we have seen before, the populationof ultrafaint satellite dwarfs is heavily suppressed in mod-els with a sharp cutoff in the stellar mass-halo mass relationlike the one explored here. Deep imaging and spectroscopicsurveys of the surroundings of isolated dwarfs designed toconstrain the satellite population within their virial radiusshould thus yield key insights into the stellar mass-halo massrelation at the faint end.

5 MODELS VS. OBSERVATIONS

5.1 Milky Way and M31 satellites

The most complete available census of faint satellites is in theLocal Group, which provides therefore a good testbed for theideas explored above. We compare in Fig. 10 the predictionsof the theoretical models with data for MW and M31 satellites(left and middle panels)2. Black symbols connected by a solid

2 Stellar masses for observed Local group satellites have been esti-mated using luminosities from McConnachie (2012) and assuming

appropriate mass-to-light ratios according to Woo et al. (2008).

curve show the observational data, taken from McConnachie(2012)’s updated compilation of Local Group dwarfs whereobjects within 300 kpc of the MW/M31 are considered satel-lites.

For the models, we choose a virial velocity of V200 = 150km/s for the MW, and a somewhat larger V200 = 165 km/sfor M31, in agreement with current available mass constraints(see; e.g., Cautun et al. 2020; Sofue 2015; Fardal et al. 2013).Each virial velocity is sampled 100 times; the resulting me-dian and 10th-90th percentiles are shown in blue for the “cut-off” model and in red for the “power-law” model.

The number of MW satellites with M sat∗ ≥ 106 M is

in reasonable agreement with both models (see left panel inFig. 10), as well as with data from the SAGA survey, whichtargeted the bright end of the satellite population within 300kpc of MW-like primaries (Mao et al. 2020). We note thatour models refer to satellites within the virial radius of theassumed halo (r200 ∼ 215 kpc for our choice of V200 = 150km/s) rather than the 300 kpc used in the observational data.The thin black line in the left-hand panel of Fig. 10 shows theknown MW satellites inside that smaller radius; the differenceis quite small.

Both the “power-law” and the “cutoff” model predict thesame number of satellites with M∗ > 106 M (roughly ∼ 18),interestingly well in excess of the known number of such sys-tems orbiting the Milky Way. The discrepancy worsens be-tween 104 and 106 M, where the MW satellite mass functionappears to have a sizable “gap”. It is unclear what the sig-nificance of such gap may be, but it is tempting to associateit with increasing incompleteness in observational detections(see; e.g., the discussion in Fattahi et al. 2020). The numbersclimb rapidly in the 102-104 M range, to almost match thepredictions of the “cutoff” model.

As discussed in Sec. 4.2, it is in the ultrafaint regime wherethe “power-law” and “cutoff” models can be best differen-

MNRAS 000, 1–13 (2020)

10 Santos-Santos et al.

102 104 106 108 1010

M sat∗ /M

100

101

102

N(>

)

V200 = 150 km/s

MW

SAGA

Cutoff

Power-law

102 104 106 108 1010

M sat∗ /M

100

101

102

V200 = 165 km/s

M31

102 104 106 108 1010

M sat∗ /M

100

101

102

V200 = 50, 100 km/s

LMC, possible

LMC, most-likely(Santos-Santos+21)

Figure 10. Predictions of the “cutoff” (blue) and “power-law” (red) models for the satellite stellar mass functions of observed host galaxies.Left: Milky Way; Middle: M31; Right: LMC (note the different y-axis limits). For each case, we assume a fixed value of V200 of the primary

motivated by literature estimates (see text for details). Solid lines show median results and shaded bands the 10-90 percentile range. Black

points show the observed satellite stellar mass function of each host. In the case of MW and M31, thicker lines show results for satelliteswithin 300 kpc, while thinner lines correspond to satellites within r200 (214 kpc for V200 = 150 km/s, and 236 kpc for V200 = 165 km/s,

assuming h=0.7). The first panel additionally shows the satellite stellar mass function of MW-mass analogs observed as part of the SAGAsurvey (Mao et al. 2020). For the case of the LMC, we show model results assuming V200 = 50 (colored, solid) as well as 100 (colored,

long-dashed) km/s, compared to likely Magellanic satellites according to Santos-Santos et al. (2021).

tiated. For faint dwarfs with M sat∗ < 105 M, the “cut-

off” model predicts substantially fewer ultrafaints than the“power-law” model. Interestingly, this comparison suggeststhat if the stellar mass-halo mass relation does indeed have alow-mass cutoff, the majority of ultrafaint dwarfs in the MWmight have already been discovered, leaving little room toaccommodate a large missing population of ultrafaints. Onthe other hand, the “power-law” model suggests the presenceof a numerous, yet undetected population of ultrafaints inthe MW halo. Upcoming surveys of the MW satellite pop-ulation, especially those which account for satellites hiddenbehind the disk, or missing due to the incomplete spatial andsurface brightness coverage of existing surveys, should be ableto distinguish clearly between the two models proposed here.

The middle panel of Fig. 10 compares model predictionswith current estimates of the M31 satellite population. Al-though the surveyed population in M31 does not go as deep asin the MW, the total number of satellites with M∗ ≥ 105 Mseems to fall below the“power-law”model predictions, at leastfor the virial mass explored here. There is a hint that theobserved satellite mass function compares more favourablywith the “cutoff” model, which predicts roughly half as manysatellites in that mass range as the “power-law” model.

The “cutoff” model predicts at least ∼ 50 new ultrafaintM31 satellites in the range 102 < M∗/M < 104.6 (the massof And XX, the least massive M31 satellite known), bringingthe total population to ∼ 90-110 total dwarfs above a stellarmass 100 M. By contrast, the “power-law” model predicts atotal of ∼ 680-740 satellites with M∗ > 102 M. We note thatthese numbers are quite sensitive to the choice of virial massfor the M31 halo; doubling the mass (i.e., increasing V200

to 208 km/s) would yield roughly twice as many satellitesfor either model, although the relative differences in massfunction shape would be preserved.

5.2 Satellites of isolated LMC-like dwarfs

Finally, the right-hand panel of Fig. 10 shows the predic-tions for the satellite population of isolated dwarf galaxieswith stellar mass comparable to that of the LMC (M∗ ∼3 × 109M) , or, more precisely, dwarf primaries inhabitinghalos with V200 in the range 50 to 100 km/s. This is consis-tent with the virial mass range (2.5 < M200/1010 M < 45)of galaxies with comparable stellar mass in the APOSTLEsimulations (see; e.g., Santos-Santos et al. 2021). These au-thors use kinematic information to identify LMC-associateddwarfs; their list of most likely LMC satellites include 7 satel-lites: the SMC, Hydrus 1, Horologium 1, Carina 3, Tucana4, Reticulum 2, and Phoenix 2 (dashed black line). A lesslikely, but still plausible association is also ascribed to Ca-rina, Horologium 2, Grus 2, and Fornax, bringing the totalto 11 (solid black line).

In the context of the “cutoff” model, these numbers seemto rule out a virial velocity as low as 50 km/s (bottom bluecurve), and suggest a virial velocity a little below 100 km/s(top blue curve). In contrast, a virial velocity near the lowerbound would be favoured in the case of the “power-law”model. An LMC halo as massive as 100 km/s seem quiteinconsistent with the data in this case. Note that the pre-dictions of the two different models differ substantially evenfor satellites with M∗ ∼ 104 M. This limit seems withinreach of what may be achievable in future surveys of LMC-

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 11

102 103 104 105 106 107 108

M sat∗ /M

100

101

102

N(>

)

DELVEM∗/M : (0.5− 3)× 109

V200 = 75 km/s

Power-law

Cutoff

Power-law No scatter

Cutoff No scatter

Figure 11. Satellite stellar mass functions predicted by the “cut-

off” (blue) and “power-law” (red) models, assuming a fixed pri-mary halo virial velocity of V200 = 75 km/s. A solid line shows the

median result, while the shaded band shows the 10-90 percentile

range. In addition, the dashed lines show the results obtained as-suming no scatter in the M∗-Vpeak relation. This has a strong

effect on the power-law results (in orange), but affects very lit-tle the cutoff ones. These results summarize the predictions of our

two models for the satellite mass functions of the 4 DELVE Magel-

lanic analogs NGC300, NGC55, IC5152 and SextansB, with stellarmasses in the [0.5, 3] × 109 M range (Drlica-Wagner et al. 2021).

Thin individual lines show the results for 4 individual random re-

alizations of the model (with scatter), to illustrate the expectedobject-to-object variation.

like primaries, turning them into strong constraints of thestellar mass-halo mass relation of faint galaxies, a subject weaddress in more detail below.

5.3 Predictions for future surveys

Beyond the Local Group, several ongoing (and future) ob-servational efforts have the potential to measure the satel-lite population of isolated LMC-like galaxies, and thus de-liver strong constraints on the stellar mass-halo mass rela-tion at the faint end. To reduce fluctuations due to object-to-object scatter, it is desirable to survey several primaries ofsimilar stellar mass while simultaneously reaching the ultra-faint satellite regime. This is why dwarf galaxies are the mostpromising primaries: within the Local Volume (i.e., within 10Mpc from the Milky Way) there are only 8 MW-like galaxies(M∗ ≥ 1010.5 M) outside the Local Group but there are 112known dwarfs with 108 < M∗/M < 109.5 (Tully et al. 2009,2016).

As an example, we provide in Fig. 11 expectations fromthe “cutoff” and “power-law” models for the satellite massfunction of 4 LMC-like dwarfs expected to be surveyed as partof the DES DELVE campaign (Drlica-Wagner et al. 2021).

This includes NGC 300, NGC 55, IC 5152 and Sextans B,which span a stellar mass range M∗ = [0.5, 3]× 109 M. Themodel predictions are based on 100 realizations of dwarfs withfixed virial velocity, V200 = 75 km/s, and are shown by the topred curves and the bottom blue curves. (Thin lines correspondto 4 individual realizations, to illustrate the expected object-to-object scatter.)

As in our earlier discussion, this figure makes clear thatreaching satellites with M∗ ∼ 104 M should be enough todifferentiate between models, since the “power-law” modelpredicts almost 3 times more such satellites than the “cutoff”model. The difference is most striking when reaching ultra-faints with M∗ ∼ 102 M, where only 10 satellite dwarfs areexpected around LMC-analogs in the case of a cutoff whereasmore than ∼ 160 are predicted for the “power-law” model.Should future surveys fail to discover a large number of ul-trafaint dwarfs around isolated LMC-analogs, this would bestrong evidence in favor of some kind of cutoff in the stellarmass-halo mass relation.

5.4 Comparison with previous work on satellites of LMC-likehosts

It is interesting to compare our results with previous work inthe literature on the satellite population of LMC-like hosts.For instance, assuming a power-law relation between stellarand halo mass, Nadler et al. (2020) predict 48 ± 8 satellites3 with M∗ ≥ 102 M, about a factor of three lower than the∼ 160 dwarf satellites predicted by our “power-law” model.This is not due to differences in our assumptions about theprimary virial mass nor about the subhalo abundance: wehave explicitly checked that the number of subhalos in ourLMC-like primaries is consistent with Nadler et al. (2020).Indeed, we find 48-58 subhalos (10th-90th percentiles) withVpeak > 10 km/s within the virial radius of primaries withV200 = 75 km/s, in good agreement with the 52 ± 8 quotedby those authors. The difference must therefore be due to theway each model populates those subhalos with galaxies.

The slope in the low-mass end of the M∗-Vpeak relation as-sumed in Nadler et al. (2020) is somewhat shallower than theone adopted here (∼ 5.1 compared to ∼ 7.4 in our “power-law” model), but we have identified two main factors con-tributing to the smaller number of dwarf satellites predictedby Nadler et al. (2020) compared to our work. One is thatwe model the scatter in the M∗ − Vpeak relation as velocity-dependent, increasing from ∼ 0.22 dex for MW-like objectsto ∼ 1 dex in halos with Vpeak ∼ 10 km/s. On the otherhand, Fig. 6 (in combination with their Table 1) in Nadleret al. (2020) suggests that they assume a roughly constant∼ 0.2 dex scatter in the dwarf regime.

The effect of the larger assumed scatter in our model isappreciable. Indeed, assuming zero scatter in the stellar mass- velocity relation, the “power-law” model would decreasethe predicted numbers from ∼ 160 to ∼ 67 satellites withM∗ ≥ 102 M (see middle dashed orange curve in Fig. 11),in better agreement with the 48±8 predicted in Nadler et al.(2020). This is also in agreement with the ∼ 70 satellites with

3 Nadler et al. (2020) quote numbers above an absolute V-

magnitude MV = 0, corresponding to a M∗ ∼ 90M assuming

a mass-to-light ratio M/L = 1.

MNRAS 000, 1–13 (2020)

12 Santos-Santos et al.

−7 < MV < −1 predicted by Jethwa et al. (2016) via dynam-ical modelling of the Magellanic Cloud satellite population.We refer the reader to Garrison-Kimmel et al. (2017) for adetailed discussion of the degeneracies in the slope/scatterof abundance matching models and the expected number ofdwarfs.

A second factor affecting the number of ultrafaints inNadler et al. (2020) is the assumption of an occupationfraction. Indeed, Nadler et al. (2020) assumes that belowVpeak ∼ 9 km/s an increasing fraction of halos with decreas-ing Vpeak remain dark and never host a galaxy (modeled ac-cording to their Eq. 3), while our “power-law” model assumesan occupation fraction equal to 1 at all Vpeak. We note thatadding an occupation fraction to a power-law M∗-Vpeak rela-tion effectively makes it steeper and more comparable to the“cutoff” model, lowering the total number of predicted faintsatellites.

Our predictions may also be compared with the work ofDooley et al. (2017), who explored the satellite population ofLMC-like hosts using (power-law) extrapolations of severalabundance matching models, including that of Moster et al.(2013). The main difference with our own “power-law” modelis that they also include an occupation fraction to model theeffects of reionization. As such, their predictions are moresimilar to our “cutoff” model, with ∼ 10-15 (median, depend-ing on which particular abundance matching relation) dwarfsatellites with M∗ ≥ 102 M within a 50 kpc radius of theirhosts. These results are bracketed by the predictions of our“cutoff” model, with 6-13 (10th-90th percentiles), and our“power-law” model, with 145 − 178 satellites, although ournumbers are within a larger volume of r200 ∼ 107 kpc (cor-responding to V200 = 75 km/s).

Our predictions in Fig. 11 might also inform other satellitesearches around LMC-analogs in the field such as the LBT-SONG survey or MADCASH (Carlin et al. 2021). At leasttwo faint satellites have been identified around the Magellanicdwarf NGC 628 (Davis et al. 2021) surveying only a fractionof its inferred virial extension with the Large Binocular Tele-scope as well as the confirmation of DDO113 as (interacting)satellite of NGC 4214. Additionally, two dwarfs have beenconfirmed as satellites of the Magellanic analogs NGC 2403and NGC 4214 with HST data for the Hyper Suprime Camsurvey MADCASH. Muller & Jerjen (2020) report, in addi-tion, two candidate faint dwarfs possibly associated to NGC24 in the Sculptor group using the Dark Energy Camera. Asdata continues to accumulate around dwarf primaries, thecensus of their satellite population is starting to emerge asthe most promising and effective way to constrain the galaxy-halo connection at the low mass end.

6 SUMMARY AND CONCLUSIONS

We have studied the effects of different stellar mass-halomass relations on the predicted population of faint and ul-trafaint dwarf satellites around primaries spanning a widerange of stellar mass. The models are motivated by re-sults of recent state-of-the-art cosmological hydrodynamicalsimulations, extrapolated to the ultrafaint regime, down toM∗ ∼ 102 M.

Two faint-end stellar mass-halo mass model relations areexplored: one is a “power-law” motivated by recent semian-

alytic results about the abundance of satellites in the LocalGroup, and by recent high resolution simulations from theFIRE project, which follow closely a power-law extrapola-tion to the faint regime of the abundance-matching resultsfrom Moster et al. (2013).

A second is a “cutoff” model where the stellar mass-halomass relation gradually steepens towards decreasing mass sothat no luminous dwarf exists beyond a minimum thresholdhalo virial mass of order M200 ∼ 109 M. The “cutoff” modelis motivated by results from the APOSTLE simulations, andby analytic considerations that disfavour the formation ofgalaxies in halos below the“hydrogen-cooling” limit (see; e.g.,Benitez-Llambay & Frenk 2020, and references therein). Weassume the same subhalo mass function and the same mass-dependent scatter in the M∗-Vpeak relation in both models.

Our main finding is that satellite mass functions of primarygalaxies spanning a wide range of mass is an excellent probeof the shape of the stellar mass-halo mass relation at thefaint end. Satellite mass functions are particularly constrain-ing if they reach deep into the ultrafaint regime. For exam-ple, the “cutoff” model predicts ∼ 9(19) times fewer dwarfswith M∗ ≥ 102 M than the“power-law”model for primarieswith M∗ ∼ 3×1010(3×107)M. The difference becomes moremarked as the stellar mass of the primary decreases, implyingthat the satellites of dwarf primaries, in particular, provideparticularly strong constraints on the stellar mass-halo massrelation at the faintest end.

The models also predict different normalized satellite massfunctions, i.e., the number of satellites expressed as a func-tion of M sat

∗ /Mpri∗ rather than M sat

∗ . While the normalizedfunction declines with decreasing primary stellar mass in the“cutoff” model, it is nearly independent of primary mass inthe “power-law” model. This self-similar behavior results be-cause the subhalo mass function is also a power-law in LCDM,as discussed by Sales et al. (2013).

Our findings have important implications when applied tonearby galaxies, where the surveying of ultrafaint dwarfs isor will become feasible in the near future. For a MW-massprimary (i.e., V200 ∼ 150 km/s), the “power-law” and “cut-off” models predict ∼ 612+35

−25 vs. ∼ 77+10−11 satellites above the

nominal M∗ = 102 M mass cut, respectively. Interestingly,in the MW itself the number of already discovered satellitedwarfs is quite close to the “cutoff” model prediction, leavingonly little room for the detection of large numbers of newultrafaint dwarfs. This is a prediction that should also betestable by ongoing and future surveys of the satellite popu-lation around nearby galaxies.

LCDM predicts that dwarf galaxies should also host a num-ber of fainter companions. The models described above maybe used to compute the number of dwarf satellites expectedaround LMC-mass systems in the field. We find, on aver-age, that the “cutoff” model predicts ∼ 3-22 satellites withM∗ > 100M; the number, on the other hand, grows to ∼ 65-300 for the power-law case, where the range corresponds toassuming a virial velocity range between 50 and 100 km/sfor the LMC halo. This highlights the potential for ultrafaintdiscoveries in regions surrounding Magellanic-like dwarfs inthe field, a particularly exciting prospect in light of ongoingefforts such as DELVE (Drlica-Wagner et al. 2021), MAD-CASH (Carlin et al. 2021) or LBT-SONG (Davis et al. 2021),which target the surroundings of isolated LMC-like dwarfs.

We conclude that the satellite population of dwarf galaxies

MNRAS 000, 1–13 (2020)

Satellites and the faint galaxy mass-halo mass relation 13

in the field offer a powerful way to constrain the faint end ofthe stellar mass-halo mass relation (see also Wheeler et al.2015). Only if the relation extends well below the hydrogen-cooling limit (as envisioned in the “power-law” model) thenone would expect dwarfs to have numerous ultrafaint com-panions. In the cutoff case, as the mass of the primary ap-proaches the cutoff the number of satellites should declinerapidly. For the particular cutoff we explore in this paper,dwarfs with M∗ ∼ 3 × 108 M or less should have virtuallyno luminous satellites, regardless of luminosity.

Dwarf primaries are also good probes because the galaxymass is, in relative terms, much smaller that that of their sur-rounding halo. The galaxy’s effect on the subhalo populationis therefore proportionally reduced compared to galaxies likethe Milky Way, where the disk is massive enough to affect no-ticeably the evolution and survival of subhalos in its vicinity(Jahn et al. 2019). Finally, dwarf primaries are more abun-dant in the Local Volume than giant spirals like the MW orM31, so surveying a statistically meaningful sample becomes,in principle, more feasible.

We conclude that the satellite mass function of dwarf galax-ies in the field represents an efficient and attractive approachfor constraining the mapping between stars and dark matterhalos in the low mass end with deliverables expected in theforeseeable future.

ACKNOWLEDGEMENTS

ISS is supported by the Arthur B. McDonald Canadian As-troparticle Physics Research Institute. LVS is grateful for fi-nancial support from the NSF-CAREER-1945310 and NASA-ATP-80NSSC20K0566 grants. JFN is a Fellow of the Cana-dian Institute for Advanced Research. AF is supported by aUKRI Future Leaders Fellowship (grant no MR/T042362/1).This work used the DiRAC@Durham facility managed bythe Institute for Computational Cosmology on behalf of theSTFC DiRAC HPC Facility (www.dirac.ac.uk). The equip-ment was funded by BEIS capital funding via STFC capi-tal grants ST/K00042X/1, ST/P002293/1, ST/R002371/1,and ST/S002502/1, Durham University and STFC opera-tions’ grant ST/R000832/1. DiRAC is part of the Nationale-Infrastructure.

DATA AVAILABILITY

The APOSTLE simulation data and model results un-derlying this article can be shared on reasonable requestto the corresponding author. For data from other simu-lations we refer the interested reader to the correspond-ing references cited in each case. The observational datafor Local group satellites used in this article comes fromMcConnachie (2012, see http://www.astro.uvic.ca/~alan/Nearby_Dwarf_Database_files/NearbyGalaxies.dat, andreferences therein).

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