Dynamical evidence for a morphology-dependent relation between the stellar and halo masses of galaxies
AAstronomy & Astrophysics manuscript no. main © ESO 2021February 24, 2021
Dynamical evidence for a morphology-dependent relation betweenthe stellar and halo masses of galaxies
Lorenzo Posti (cid:63) and S. Michael Fall Université de Strasbourg, CNRS UMR 7550, Observatoire astronomique de Strasbourg, 11 rue de l’Université, 67000 Strasbourg,France. Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA.Received XXX; accepted YYY
ABSTRACT
We derive the stellar-to-halo mass relation (SHMR), namely f (cid:63) ∝ M (cid:63) / M h versus M (cid:63) and M h , for early-type galaxies from theirnear-infrared luminosities (for M (cid:63) ) and the position-velocity distributions of their globular cluster systems (for M h ). Our individualestimates of M h are based on fitting a flexible dynamical model with a distribution function expressed in terms of action-angle variablesand imposing a prior on M h from the correlation between halo concentration and mass in the standard Λ Cold Dark Matter ( Λ CDM)cosmology. We find that the SHMR for early-type galaxies declines with mass beyond a peak at M (cid:63) ∼ × M (cid:12) and M h ∼ × M (cid:12) (near the mass of the Milky Way). This result is consistent with the standard SHMR derived by abundance matching for the generalpopulation of galaxies, and with previous, less robust derivations of the SHMR for early-type galaxies. However, it contrasts sharplywith the monotonically rising SHMR for late-type galaxies derived from extended HI rotation curves and the same Λ CDM prioron M h as we adopt for early-type galaxies. We show that the SHMR for massive galaxies varies more or less continuously, fromrising to falling, with decreasing disc fraction and decreasing Hubble type. We also show that the di ff erent SHMRs for late-type andearly-type galaxies are consistent with the similar scaling relations between their stellar velocities and masses (the Tully-Fisher andthe Faber-Jackson relations). As we demonstrate explicitly, di ff erences in the relations between the stellar and halo virial velocitiesaccount for the similarity of the scaling relations. We argue that all these empirical findings are natural consequences of a picture inwhich galactic discs are built mainly by relatively smooth and gradual inflow, regulated by feedback from young stars, while galacticspheroids are built by a cooperation between merging, black-hole fuelling, and feedback from active galactic nuclei. Key words. galaxies: kinematics and dynamics – galaxies: elliptical – galaxies: spiral – galaxies: structure – galaxies: formation
1. Introduction
Galaxies consist of stars and interstellar gas in relatively com-pact bodies surrounded by more extended halos of dark matterand circumgalactic gas. The composition of the dark matter isunknown, but it is believed to be elementary particles that in-teract only gravitationally with baryons. In the standard Λ ColdDark Matter ( Λ CDM) paradigm, the assembly of galactic halosby gravitational clustering is relatively simple and well under-stood, while the inflow and outflow of gas and the formationof stars by both gravitational and hydrodynamical processes aremuch more complex and are topics of intense current research.One of the most useful empirical constraints in these studies –and the focus of this paper – is the ratio f (cid:63) ≡ M (cid:63) f b M h (1)of the mass in stars M (cid:63) to that in dark matter M h within a galaxynormalised by the cosmic baryon fraction f b . This ratio repre-sents a sort of global star formation e ffi ciency, averaged overspace and time, for that galaxy.The variation of f (cid:63) with M (cid:63) or M h is called the stellar-to-halo mass relation (SHMR). This has now been derived usingseveral di ff erent techniques: abundance matching (Vale & Os-triker 2004; Conroy et al. 2006; Behroozi et al. 2013; Mosteret al. 2013), halo occupation distributions (Peacock & Smith (cid:63) [email protected] f (cid:63) increases with mass to a peak, with f (cid:63) ∼
20% at M (cid:63) ∼ × M (cid:12) and M h ∼ M (cid:12) (near the mass of theMilky Way), and then decreases with mass.The standard explanation for the inverted-U shape of theSHMR is that feedback by young stars is responsible for the low-mass part, while feedback from active galactic nuclei (AGN) isresponsible for the high-mass part. Both types of feedback arepotentially capable of driving outflows from a galaxy and im-peding further inflows, thus quenching star formation. The ef-fect of stellar feedback on the SHMR is fairly well understood: ahigher fraction of gas is driven out of low-mass galaxies becausethey have lower escape speeds (e.g. Dekel & Silk 1986; Veilleuxet al. 2005). Near the peak of the SHMR, much of the gas proba-bly circulates in a self-regulated fountain, without escaping fromthe halo (e.g. Tumlinson et al. 2017, and references therein). Thee ff ect of AGN feedback on the SHMR is less well understood,but it is plausible that it drives energetic outflows that heat someof the circumgalactic gas, thus slowing or reversing its inflows(e.g. Fabian 2012; King & Pounds 2015; Harrison 2017). Merg-ers may also disrupt the inflow and quench star formation, atleast temporarily (e.g. Hopkins et al. 2010b). Both mergers and Article number, page 1 of 16 a r X i v : . [ a s t r o - ph . GA ] F e b & A proofs: manuscript no. main
AGN feedback may cooperate to cause the decline of the SHMRat high mass (e.g. Croton et al. 2006) since mergers can funnelgas to a central black hole, igniting AGN feedback (e.g. Hopkinset al. 2006), while also building galactic spheroids (i.e. classicalbulges, Hopkins et al. 2010a).In practice, the SHMR is usually assumed to be indepen-dent of galactic morphology (Wechsler & Tinker 2018). This as-sumption, however, appears to contradict the reasoning aboveabout the di ff erent roles of stellar and AGN feedback and theobservation that the masses of central black-holes correlate withthe bulge masses of their host galaxies (Kormendy & Ho 2013).Thus, if AGN feedback is important, it should have more ef-fect on the high-mass shape of the SHMR for bulge-dominatedgalaxies than it does for disc-dominated galaxies. More specifi-cally, f (cid:63) should decline with M (cid:63) and M h past the peak in early-type galaxies but rise or level o ff in late-type galaxies. The maingoals of this paper are to confirm this expected dependence ofthe SHMR on galaxy morphology and to explore some of its im-plications for our understanding of galaxy formation.There is already some evidence for secondary correlationsbetween the SHMR and other properties of galaxies. This evi-dence comes from weak lensing (Mandelbaum et al. 2006, 2016;Tinker et al. 2013; Hudson et al. 2015; Taylor et al. 2020), satel-lite kinematics (Conroy et al. 2007; More et al. 2011; Wojtak &Mamon 2013; Lange et al. 2019), empirical models (Rodríguez-Puebla et al. 2015), abundance matching (Hearin et al. 2014;Saito et al. 2016) or a mix of these (Dutton et al. 2010). Theresults of these studies are consistent with the expectation thatearly-type galaxies occupy more massive halos than late-typegalaxies of the same stellar mass. However, in most cases, theresults are based on stacking the observations in large samplesof galaxies to amplify the marginal or undetectable signals fromindividual galaxies, an approach that can sometimes yield spu-rious correlations (and has led to some debate on the topic, seee.g. S6.1 in Wechsler & Tinker 2018). Some recent hydrodynam-ical simulations also display the expected di ff erences betweenthe SHMR of early and late-type galaxies (Grand et al. 2019;Marasco et al. 2020; Correa & Schaye 2020).The most direct approach to deriving the SHMR is to esti-mate the masses of individual halos from the observed kinemat-ics of tracer objects whose space distribution extends well be-yond the luminous parts of galaxies. Since the available tracersalmost never reach the expected outer (virial) radii of the halos,estimates of their total masses require priors such as the corre-lation between concentration and mass found in Λ CDM simu-lations. This is the approach used by Posti et al. (2019a, here-after PFM19) to derive the SHMR of 110 late-type galaxies withextended HI rotation curves in the Spitzer Photometry and Ac-curate Rotation Curves (SPARC) sample (Lelli et al. 2016a) .PFM19 found that the SHMR rises monotonically for all massesand reaches f (cid:63) ∼ . M (cid:63) ∼ × M (cid:12) (dubbed the “failed feedbackproblem”). This result is in stark contrast to the declining high-mass form of the SHMR found in most studies of the generalpopulation of galaxies, which is dominated by early types at thehighest masses (Kelvin et al. 2014).In this paper, we derive the SHMR for early-type galaxiesby methods as similar as possible to those used by PFM19 forlate-type galaxies. In particular, we adopt the same Λ CDM cor-relation between halo concentration and mass. However, insteadof using HI rotation curves to probe the gravitational potential, In this context, HI is a better tracer than H α because it usually extendsto larger radii (van Albada et al. 1985; Kent 1987). we use the radial velocities of globular clusters (GCs) around25 massive early-type galaxies in the SAGES Legacy UnifyingGlobulars and GalaxieS Survey (SLUGGS, Brodie et al. 2014).We fit a distribution function, expressed in terms of action andangle variables, to the observed kinematics and space distribu-tion of each GC system to estimate its halo mass. This enablesus, for the first time, to make a direct and robust comparisonbetween the SHMR of early types and late types based on indi-vidual estimates of halo masses.The di ff erent SHMRs for early-type and late-type galaxies– one with a prominent bend, the other without – may seempuzzling because both galactic types have similar scaling rela-tions between stellar velocities and masses (the Faber-Jacksonand Tully-Fisher relations). Since the SHMR and the velocityscaling relations both depend on M (cid:63) , one might reasonably ex-pect a bend in the former to impose a bend in the latter. How-ever, since the SHMR also depends on M h , it is possible that thesimilar velocity scaling relations are actually explained by, anddisguised by, di ff erent underlying relations between the stellarvelocities of early-type and late-type galaxies and one or moreproperties of their dark matter halos, thus o ff ering potentiallyimportant clues about the physical mechanisms responsible fordi ff erent galactic morphologies. We explore this issue here forthe first time.The remainder of this paper is organised as follows. In Sec-tion 2, we summarise the data and dynamical models we use toestimate M h for early types and the analogous estimates of M h from PFM19 for late types. Interested readers can find a full de-scription of our models for early-type galaxies in Appendix A.In Section 3, we present our SHMR, showing unambiguouslythat it depends on galaxy morphology and disc fraction. Sec-tion 4 compares our results with previous evidence for di ff erentSHMRs. In Section 5, we reconcile the di ff erent shapes of theSHMRs for late-type and early-type galaxies with their similarvelocity scaling relations in terms of di ff erences between theirstellar and halo velocities, and we interpret this result as a nat-ural consequence of the di ff erent roles of smooth inflow, merg-ing, and AGN feedback in the formation of galactic discs andspheroids. Section 6 summarises our main conclusions.Throughout the paper, we use a fixed critical overdensityparameter ∆ =
200 to define the virial quantities of dark mat-ter halos and a standard Λ CDM model with a Hubble constant H = . − Mpc − (Planck Collaboration et al. 2020). Wedistinguish late-type and early-type galaxies based on publishedmorphological classifications: early types are E and S0 (Hubbletype T < / a, Sa, Sb, Sc, Irr (Hubbletype T ≥
2. Dynamical estimates of halo masses forearly-type and late-type galaxies
The main novelty of this work is our dynamical estimates of halomasses ( M h ) for individual nearby ellipticals and lenticulars. Toderive these, we use observations of the kinematics of the GCsystems around these galaxies and we model explicitly their dis-tribution function. Our method is an adaptation of that of Posti& Helmi (2019), who used it to measure the halo mass of theMilky Way. The method consists of two main ingredients: thedistribution function (DF) of the GC system and the gravitationalpotential. Here we provide an overview of our method with theguidance of Fig. 1, which illustrates the input, fitting, and out-put of our model for a representative galaxy, NGC 4494. This Article number, page 2 of 16. Posti & S. M. Fall: Morphology-dependent stellar-to-halo mass relation r / kpc ° ° ° ° ° ° c o un t s / k p c / r ° / / r °
12 13 log M h / M Ø . . l og c . . log c V c i r c / k m s ° r / kpc ° . . Ø NGC4494_GC.txt
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10 kpc r / kpc æ l o s / k m s ° GCs : SLUGGSPNe : ePN . S (Pulsoni + 18)
10 20 30 40 r / kpc ° ° V l o s / k m s ° b) d) f)c) e)a) Fig. 1.
Illustration of our modelling technique using NGC 4494 as an example. In all panels, GC data are shown as blue points, the DF model isshown as a black solid curve, and the 68% confidence interval of the model is shown as a grey shaded area. a) SDSS colour image of the galaxywith blue crosses marking the spectroscopically confirmed globular clusters by the SLUGGS survey (made with
Aladin ). b) Projected numberdensity profile of the GCs compared to that of the f ( J ) model. c) Marginalised posterior distribution of the halo mass and concentration estimatedwith an MCMC method. Dark and light grey areas encompass respectively 68% and 95% probability, while the black solid lines and cross markthe maximum-likelihood model. d) Line-of-sight velocity as a function of projected radius, V los − r , for the GCs compared with the projectedphase-space density of the maximum-likelihood f ( J ) model. The contours contain 68-95-99% of the projected phase-space density of the model. e) Circular velocity profile, V circ (top), and velocity anisotropy profile, β = − ( σ θ + σ φ ) / σ r (bottom), of the f ( J ) model. f) Line-of-sight velocitydispersion profile of the GCs measured with SLUGGS data compared to that of our f ( J ) model. We also compare to independent measurementsof the σ los profile of planetary nebulae obtained by Pulsoni et al. (2018, red squares). section introduces all the information needed to understand ourresults, while Appendix A provides a detailed description of ourmodel, which the busy reader can skip. Data.
We take the velocity and position data of the glob-ular cluster systems around 27 nearby bright ellipticals andlenticulars from the SLUGGS Survey (Brodie et al. 2014).The centrepiece of this data set is the catalogue of radial ve-locities of the spectroscopically-confirmed GCs obtained withDEIMOS@Keck (Forbes et al. 2017a). Each galaxy has tens orhundreds of GCs, with a significant galaxy-to-galaxy variationand with a typical uncertainty on each radial velocity of about ∼ −
20 km / s. The radial coverage of the GC system is alsoquite varied: the radius containing 90% of the observed GCsranges between 8 and 98 kpc (corresponding to 3 R e and 14 R e )with a median of about 30 kpc (corresponding to 7 . R e ). Thetop panel of Fig. 2 shows the distributions of the median radii( R med , GC ) and the outermost radii ( R last , GC ) of the GC systems,both normalised by the e ff ective radii of the galaxies ( R e ). Tomodel the distribution of baryons, we use the photometric pro-files derived from Spitzer Space Telescope images at 3.6 µ m byForbes et al. (2017b). Out of the 27 galaxies in Brodie et al. (2014), we exclude NGC 4474, since it does not have Spitzerimages, and NGC 4111, since it has fewer than 20 GC veloci-ties. As an example, in Fig. 1a, we show the distribution of theconfirmed SLUGGS GCs around NGC 4494. Distribution function.
We use analytic DFs that depend onaction-angle variables in the form introduced by Posti et al.(2015) specifically to describe spheroidal systems. We refer tothese models as f ( J ), where f is the DF and J are the actionintegrals. The models we use have spherical space distributionsbut anisotropic velocity distributions. The DFs are double powerlaws in the actions such that they generate double power-lawdensity distributions in physical space. Once the two slopes ofthe DF are fitted to the space distributions of GCs (Fig. 1b), thethree remaining parameters specify the velocity distribution ofthe GC system. Gravitational potential.
The total potential in the f ( J ) modelis a superposition of two spherical components: the luminousgalaxy and the dark matter halo. The galaxy is modelled as ade-projected Sersic (1968) profile, based on the the photometryby Forbes et al. (2017a), with an adjustable mass-to-light ratio.The dark matter halo is assumed to have a Navarro et al. (1996, Article number, page 3 of 16 & A proofs: manuscript no. main . . . . log R X / R e N ga l R X = R med , GC R X = R last , GC − log [M ? ( < R X ) / M h ( < R X )] N ga l R X = R last , GC R X = R med , GC R X = R e Fig. 2.
Top panel : distributions of the median radii ( R med , GC , blue) andof the outermost radii ( R last , GC , orange) of the GC systems around theearly types in SLUGGS. Both radii are normalised to the e ff ective radiusof the luminous galaxy ( R e ). Bottom panel : distributions of the ratiosof stellar to halo mass enclosed within the e ff ective radii ( R e , green),within the median radius of the GC system ( R med , GC , blue), and withinthe outermost GC radius ( R last , GC , orange). These are computed from thebest-fit f ( J ) models of each galaxy. The dotted vertical line separatesthe region where the dark matter dominates (left) and where the starsdominate (right). herafter NFW) profile parametrized by its virial mass ( M h ) andconcentration ( c ). For the mass-to-light ratio, we impose a gaus-sian prior with a mean derived by Forbes et al. (2017a), usingstellar population models, and a dispersion σ log( M / L ) = . c and M h found in N-body Λ CDM simulations (Dutton & Macciò2014).
Bayesian parameter estimation.
We use a Markov ChainMonte Carlo (MCMC) method to derive posterior probabilitiesof the free parameters of our model (Fig. 1c). The likelihood isgiven by the product of the DF convolved by the error distri-bution for each cluster. Since we need 3 positions and 3 veloc-ities to evaluate the DF, we sample the missing position fromthe observed density distribution of the clusters and the twomissing velocities uniformly in the range allowed by the escapespeed of the model (Fig. 1d). We then use our f ( J ) models toderive the intrinsic properties of the potential (the circular ve-locity V circ ) and of the GC system (the anisotropy parameter β = − ( σ θ + σ φ ) / σ r ) shown in Fig. 1e. In the bottom panelof Fig. 2, we use the f ( J ) models to compute the ratios of stel-lar to halo mass enclosed within progressively larger radii: theluminous R e , the median GC radius R med , GC , and the outermostGC radius R last , GC . Evidently, dark matter is negligible relative tostars near R e , is comparable near R med , GC ( ∼ − R e ), and is dom- inant near R last , GC ( ∼ − R e ). We can also compute a posteriorithe model line-of-sight velocity dispersion profile and check thatit is consistent with the observed profile for GCs in the SLUGGSSurvey and planetary nebulae in the ePN.S Survey (Pulsoni et al.2018). Fig. 1f shows this consistency for NGC4494. PFM19 determined halo masses for a sample of nearby spiralsby fitting galaxy plus halo models to their extended HI rotationcurves. Here we just summarise their analysis and results andrefer the reader to their paper for full details.
Data.
The sample consists of 110 nearby spirals with 3.6 µ mSpitzer images and HI rotation curves drawn from the SPARCdatabase compiled by Lelli et al. (2016a, see also the original ref-erences therein). The rotation curves, taken from various sourcesin the literature, were derived from interferometric HI observa-tions that extended well beyond the optical discs of the galaxies.The sample spans a large range in stellar masses, from dwarfs( M (cid:63) ∼ M (cid:12) ) to giants ( M (cid:63) ∼ M (cid:12) ). Rotation curve decomposition.
The observed rotation curvesare decomposed into gas, stars, and dark matter components.The gas contribution is derived directly from the HI flux, whilethe stellar contribution is computed from the 3.6 µ m photome-try with an adjustable mass-to-light ratio. The dark matter halois modelled as a standard NFW profile with variable virial mass( M h ) and concentration ( c ) following the c − M h relation from Λ CDM simulations (Dutton & Macciò 2014). In massive spi-rals, which are the focus of this work, models with cuspy darkmatter halos (such as NFW) match the observed rotation curves,yielding fits that are statistically indistinguishable from those ob-tained with other halo models (e.g. pseudo-isothermal or coredhalos, see de Blok et al. 2008; Martinsson et al. 2013; Katz et al.2017; PFM19; Li et al. 2020). This contrasts with the situationfor dwarf galaxies, whose rotation curves are often matched bet-ter with cored halo models (e.g. de Blok et al. 2001; Oh et al.2011).
Bayesian parameter estimation.
The MCMC approach isused to fit the rotation curve and to estimate the posterior distri-bution of the three free parameters of the gravitational potential:the stellar mass-to-light ratio, the halo mass, and concentration.It is important to note here that the key assumptions of thedynamical models for late types and early types are the same: anadjustable mass-to-light ratio at 3.6 µ m, a spherical NFW halo,a prior following the Λ CDM c − M h relation and an MCMCapproach to sample the posterior. This makes the results fromour f ( J ) models for early-type galaxies directly comparable withthose of PFM19 for late-type galaxies.
3. The SHMR for different galaxy types
In Fig. 3, we plot our estimates of f (cid:63) versus stellar mass M (cid:63) (leftpanel) and halo mass M h (right panel) for the 25 SLUGGS early-type galaxies. We compare these with the estimates of f (cid:63) fromPFM19 for SPARC late-type galaxies and with the abundancematching model of Moster et al. (2013). We find that at a fixedstellar mass, above ∼ × M (cid:12) , early types have systematicallylower f (cid:63) than late types of similar stellar mass, by a factor of ∼ M (cid:63) ∼ M (cid:12) .In order to guard against the possibility that the trends visiblein Fig. 3 are induced by correlated errors in the plotted variables, M (cid:63) / M h versus M (cid:63) or M h , we also show in Fig. 4 the SHMR Article number, page 4 of 16. Posti & S. M. Fall: Morphology-dependent stellar-to-halo mass relation M ? / M (cid:12) − − − − f ? ≡ M ? / f b M h abundance matchinglate types (PFM19)early types (this work) M h / M (cid:12) Fig. 3.
SHMR in the form of the ratio f (cid:63) ≡ M (cid:63) / f b M h as a function of stellar mass (left) or halo mass (right) for the sample of spiral galaxies inSPARC (blue diamonds, PFM19) and for the sample of ellipticals and lenticulars in SLUGGS (red circles, this work). The halo masses of late typesare estimated from HI rotation curves, those of early types from the kinematics of the GC system. We compare to the SHMR from the abundancematching model by Moster et al. (2013, grey band). directly in the form M (cid:63) versus M h . In particular, we zoom in onthe high-mass regime of the SHMR ( M (cid:63) > M (cid:12) ), which isof most interest here. Fig. 4 confirms that late types and earlytypes are separated from each other in the same way as indicatedin Fig. 3; massive late types occupy systematically less massivehalos than early types of the same stellar mass.The grey bands in Fig. 3-4, representing the conventionalSHMR derived by abundance matching, are displayed only forcomparison purposes. We stress that all the main results of thispaper come from dynamical analyses of late-type and early-typegalaxies and do not depend in any way on abundance matching.In these figures, we show the SHMR from Moster et al. (2013)because it represents a consensus in the field (see Fig. 2 in Wech-sler & Tinker 2018). In the SHMR derived by Kravtsov et al.(2018), massive galaxies tend to occupy slightly less massivehalos with respect to the Moster et al. (2013) SHMR. However,even in this case, the qualitative picture presented in Fig. 3-4remains valid.For early types with M (cid:63) (cid:38) M (cid:12) , we measure a scatter of ≈ . f (cid:63) at a fixed M (cid:63) . This scatter reflects a combinationof several e ff ects, which we assume to be independent to firstorder: i) the observational errors in the GC velocities and theuncertainty in the velocity dispersion due to sparse sampling,ii) the uncertainty in the mass-to-light ratio, iii) the scatter inthe c − M h relation, and iv) the intrinsic scatter in f (cid:63) at fixed M (cid:63) . The first term varies substantially from galaxy to galaxy, asit is related to the signal-to-noise of the GC spectra and to thenumber of GCs observed, but for a typical galaxy this is of theorder of 25%, i.e. 0.1 dex. The second term is of the order of0.1-0.2 dex (Forbes et al. 2017b). The third term is an output ofcosmological simulations and is ≈ .
11 dex (Dutton & Macciò2014). The fourth term can be estimated from the conventionalSHMR. We generate a population of halos from a standard halomass function (Tinker et al. 2008) and we assign an M (cid:63) to eachhalo following the Moster et al. (2013) SHMR. They estimatethe scatter of f (cid:63) to be 0.15 dex at a fixed halo mass M h , whichcorresponds to the grey bands in f (cid:63) versus M (cid:63) and M h in Fig. 3, M h / M (cid:12) M ? / M (cid:12) abundance matchinglate types (PFM19)early types (this work) Fig. 4.
SHMR in the form of stellar mass ( M (cid:63) ) as a function of halomass ( M h ). Symbols are as in Fig. 3, however here we zoom in on thehigh-mass regime of the SHMR. and in M (cid:63) versus M h in Fig. 4. The resulting scatter at a fixedstellar mass M (cid:63) is about 0.08 dex below the turnover at M (cid:63) ∼ × M (cid:12) , but then increases substantially, reaching about 0.34dex at M (cid:63) ∼ M (cid:12) . Combining these four e ff ects, we can This happens as a result of the combination of the SHMR with thesteeply declining halo mass function. Above the peak, low-mass halosthat are high- f (cid:63) outliers of the f (cid:63) − M h relation are about as common ashigh-mass halos with a typical f (cid:63) . This not only increases the scatter at aArticle number, page 5 of 16 & A proofs: manuscript no. main − − − f ? fiducial M h / M (cid:12) c M h / M (cid:12) large scatter M h / M (cid:12) M h / M (cid:12) steep c − M h M h / M (cid:12) M h / M (cid:12) shallow c − M h M h / M (cid:12) M ? / M (cid:12) − − − f ? M ? / M (cid:12) M ? / M (cid:12) M ? / M (cid:12) Fig. 5. E ff ect of varying the prior on the c − M h correlation in our dynamical determination of the SHMR. The top (bottom) panels show theresulting f (cid:63) − M h ( f (cid:63) − M (cid:63) ) relation, where the halo masses of both late types (blue diamonds) and early types (red circles) are calculated with aprior on the c − M h relation that follows the purple shaded area (1 σ ) in the middle panel. From left to right, the four columns are for the followingpriors: i) the fiducial c − M h from cosmological simulations by Dutton & Macciò (2014, with slope = − .
101 and scatter = c − M h , but with about twice the scatter (0.25 dex), iii) a steeper one (slope = − . = − . c − M h relation for comparison. In the top and bottom rows the grey shaded area shows the SHMR fromMoster et al. (2013) based on abundance matching. nicely explain the observed scatter of 0.4 dex in our estimates of f (cid:63) . Plotting f (cid:63) as a function of M h demonstrates clearly thatearly types occupy halos of a wide range of masses (10 M (cid:12) (cid:46) M h (cid:46) M (cid:12) ). In contrast, late types of similar stellar massare all found in halos of mass M h ∼ M (cid:12) and virtually noneoccupies halos more massive than M h ∼ × M (cid:12) . This is po-tentially a very important finding since it hints at the existenceof an upper limit to the masses of halos within which discs canform (e.g. Dekel & Birnboim 2006).Of all the assumptions in our modelling technique, we foundthat the prior on the c − M h correlation has the largest e ff ect onestimates of M h . In Fig. 5, we show the results of some teststo assess the robustness of our findings. We re-fitted our f ( J )models to the SLUGGS data with di ff erent priors for the haloconcentration-mass relation: in particular, we doubled the scat-ter and we increased and decreased the slopes of the c − M h relation so as to span the 1 σ range of the the standard Λ CDM fixed M (cid:63) , but it also increases the average f (cid:63) at a fixed M (cid:63) with respectto that obtained by inverting f (cid:63) ( M h ) (see e.g. Moster et al. 2020). relation over the range of halo masses probed here . These pri-ors are shown in the middle row of panels in Fig. 5. Each col-umn of Fig. 5 shows the f (cid:63) − M h (top) and f (cid:63) − M (cid:63) (bottom)relation that we obtain when assuming these di ff erent priors. Tocompare early types (red) and late types (blue) consistently, were-computed the rotation curve decompositions of the late typesin SPARC with each prior. From the results plotted in Fig. 5, wenote that, while there can be significant di ff erences for individ-ual galaxies, the general trends for the populations of late typesand early types remains robust.These tests give us confidence that the systematic di ff erenceof the SHMR of massive late-type and early-type galaxies is real.An important point that we need to emphasise here is that the c − M h priors that we have used for these tests are deliberatelyextreme. In fact, such a large scatter (0.25 dex) or such steepor shallow slopes are outside of the range of published c − M h relations for the standard Λ CDM cosmogony (e.g. Diemer &Kravtsov 2015). This allows us to exclude, with high confidence, We also repeated this test with systematically larger and smaller con-centrations, i.e. with c following the fiducial c − M h relation ± σ , findingsimilar results to those shown in Fig. 5Article number, page 6 of 16. Posti & S. M. Fall: Morphology-dependent stellar-to-halo mass relation . . . . . . D / T − . − . . . . . l og ( f ? / f ? , A M ) − − − Hubble Type T
Fig. 6.
Residuals of our dynamical estimate of f (cid:63) relative to the abundance-matching value f (cid:63), AM at the same stellar mass from Moster et al. (2013)versus the disc-to-total ratio D / T (left) and Hubble type T (right), for massive late types ( M (cid:63) > × M (cid:12) , blue diamonds) and early types (redcircles). The grey area shows the scatter of the Moster et al. (2013) SHMR around M (cid:63) ∼ M (cid:12) . Several early types pile up at D / T = that the two branches of the SHMR revealed in Fig. 3 are theresult of late types and early types occupying halos with system-atically di ff erent concentrations. In Sect. 3.1, we investigated the di ff erence in f (cid:63) between twobroadly defined galaxy samples, late types and early types, whilehere we look in more detail at how f (cid:63) depends continuously ondisc fraction and morphological type for massive galaxies.For 21 of the 25 early types in SLUGGS, we rely on thephotometric bulge / disc decompositions in the r -band performedby Krajnovi´c et al. (2013), who fitted a Sérsic plus exponentialfunctions to the observed 1D photometric profiles. We note that,for half of these, Krajnovi´c et al. (2013) found no significantcontribution from an exponential disc component. For the fourremaining galaxies, we fitted the 1D R -band profiles from theCarnegie-Irvine Galaxy Survey (Li et al. 2011) with Sérsic plusexponential functions, in order to perform a similar analysis tothat of Krajnovi´c et al. (2013).We include only the 20 late types with the largest stellar massfor this comparison ( M (cid:63) > × M (cid:12) ). We take the disc-to-totalratios D / T from either: i) the Spitzer Survey of Stellar Structurein Galaxies (Sheth et al. 2010), based on 2D bulge / disc decom-positions of the 3.6 µ m Spitzer images with the code galfit (Peng et al. 2002, 2010) when available; ii) otherwise, fromthe kinematic decompositions reported by Fall & Romanowsky(2013, 2018, and references therein). For a few galaxies that donot have D / T from i) or ii), we used the decomposition of the1D surface brightness profiles at 3.6 µ m performed by Lelli et al.(2016a).For all of these massive galaxies, we calculate the residualof our estimate of f (cid:63) relative to the abundance-matching value f (cid:63), AM at the same mass from Moster et al. (2013) and expressit in the form log( f (cid:63) / f (cid:63), AM ). We plot this quantity in Fig. 6 asa function of D / T (left) and as a function of the Hubble type T (right), both for late types (blue diamonds) and early types(red circles). We find that bulge-dominated galaxies ( D / T < . T < −
3) have small residuals with respect to abundance match-ing, as they lie within its scatter (grey area). In contrast, disc-dominated galaxies ( D / T > . T >
4) are found to have sys-tematically larger f (cid:63) . The transition between these two regimesoccurs at around D / T ∼ . T ∼
2. The spirals with lowestdisc fractions in this sample ( D / T ∼ . T ∼
2) are indeed inbetter agreement with abundance matching, although the scatterof the points is substantial. The dependence of f (cid:63) on D / T and on T that we observe in Fig. 6 indicates that the location of a galaxyin the f (cid:63) − M (cid:63) diagram depends fairly continuously on its discfraction.From this new perspective, we can now see that derivationsof the SHMR that include galaxies of all types are likely to over-estimate the scatter at the high-mass end. This is because thetrend of f (cid:63) with D / T (Fig. 6), if not recognised, will simply becounted as scatter. However, this e ff ect is likely to be small sincedisc-dominated galaxies are relatively rare at the high-mass end( ∼
10% at M (cid:63) (cid:38) M (cid:12) , see e.g. Kelvin et al. 2014; Ogle et al.2019). To gain some intuition on the magnitude of this e ff ect,we performed a simple calculation in an extreme case of a bi-nary population of galaxies: pure discs and pure spheroids. At afixed M (cid:63) = M (cid:12) , we have 90% spheroids on the abundancematching relation, i.e. with f (cid:63), AM ( M (cid:63) ), and we have 10% discswith f (cid:63) systematically o ff set from this by a factor 0.8 dex. In or-der to match the scatter of the Moster et al. (2013) SHMR at thatstellar mass, which is ≈ ≈ .
27 dex. Thisexercise suggests that calculations that ignore the dependence ofthe SHMR on morphology will overestimate the intrinsic scatterby about 0 .
08 dex.
Article number, page 7 of 16 & A proofs: manuscript no. main M ? / M (cid:12) − − − − f ? ≡ M ? / f b M h AM : Rodriguez − Puebla + 15WL : Mandelbaum + 16SK : More + 11Mix : Dutton + 10 M h / M (cid:12) − Lapi + 18 − Fig. 7.
Same as Fig. 3, but we compare our SHMR with others that are representative of studies based on di ff erent techniques: abundance matching(Rodríguez-Puebla et al. 2015), weak-lensing (Mandelbaum et al. 2016), satellite kinematics (More et al. 2011) and a combination of the above(Dutton et al. 2010). Blue colour is used for late types, red colour is used for early types. The insets show the comparison to the estimates of f (cid:63) for late-type galaxies by Lapi et al. (2018). We omit the errorbars for clarity.
4. Comparison with other estimates of the SHMR
In this section, we compare our derivation of the f (cid:63) − M (cid:63) relationfor individual late-type and early-type galaxies with other esti-mates from the literature. Taken together, these results provideadditional evidence that galaxies of di ff erent types occupy halosof di ff erent masses. In Fig. 7, we show how our SHMR compares to those derivedby di ff erent techniques, for late (blue) and early types (red) sep-arately. We notice that there is general agreement among thesestudies, with our SHMR showing perhaps the largest di ff erencesbetween late and early types.Satellite kinematics (e.g. Conroy et al. 2007; More et al.2011; Wojtak & Mamon 2013; Lange et al. 2019) and especiallyweak lensing (e.g. Mandelbaum et al. 2006, 2016; Tinker et al.2013; Hudson et al. 2015; Taylor et al. 2020) are, in principle,reliable tracers of halo masses out to very large radii. However,these methods rely heavily on stacking (hundreds or thousandsof) galaxies that are usually grouped into late types and earlytypes via a hard cut in colour. These analyses have the advantageof including large numbers of galaxies but the disadvantage thatcolour is an imperfect proxy for morphology, since it depends ona combination of other factors, including star formation rate andhistory, dust reddening, and metallicity. Hence, the di ff erencesbetween late types and early types will be artificially attenuated.Similar considerations apply also to SHMRs based on empiricalmodels, which are mostly constrained by observed stellar massfunctions, with only indirect estimates of halo masses (e.g. Dut-ton et al. 2010; Rodríguez-Puebla et al. 2015; Behroozi et al.2019; Moster et al. 2020),Our work reinforces these results since it is based on carefulestimates of the halo masses of individual galaxies in samplesof late types and early types specifically selected for dynamicalstudies. Our derivation of the SHMR has opposite strengths andweaknesses with respect to the statistical estimates above. It is therefore not surprising that we find a somewhat larger di ff erencein f (cid:63) between late-type and early-type galaxies. The SLUGGS globular cluster data have already been used byAlabi et al. (2016, 2017) to estimate the total halo masses of thegalaxies by a simpler method: the so-called tracer mass estimator(TME, Watkins et al. 2010, with original formulation by Bahcall& Tremaine 1981). In this case, the total mass is taken to be M TME = C ˆ R ˆ V / G where G is the gravitational constant, ˆ R andˆ V are a characteristic radius and velocity of the system and C isa dimensionless constant of order unity that is calibrated a prioriwith some simple assumptions. We compare in Fig. 8 our halomasses with those of Alabi et al. (2017) on a galaxy-by-galaxybasis, finding overall consistency within a factor of a few.One of the shortcomings of the TME method is that it doesnot clearly partition between luminous and dark components.This is of particular importance in the context of the SLUGGSearly types, since the TME is sensitive to the dynamical massnear the median radius of the tracer population ( R med , GC ), whichhappens to be where the masses of stars and dark matter are com-parable (see Fig. 2). Therefore, it is not surprising that the M h estimates of this method are consistent within a factor of a fewwith those of our f ( J ) models.Bílek et al. (2019) recently estimated the halo masses of theearly-type galaxies in SLUGGS, by analysing the kinematics oftheir GC systems. In particular, they used the GC velocity dis-persion profiles together with the Jeans equations to constrainthe gravitational potential. They modelled the dark matter halowith a spherical NFW profile and they imposed a prior on the c − M h relation from cosmological simulations, as in our ap-proach. However, rather than allowing the data to constrain thevelocity anisotropy of the GC system, as we do here, they assumea β profile as an input to their model. We compare their estimatesof halo masses with ours in Fig. 8. Despite the di ff erences and Article number, page 8 of 16. Posti & S. M. Fall: Morphology-dependent stellar-to-halo mass relation N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C N G C l og M h / M (cid:12) Alabi + 17Bilek + 19this work
Fig. 8.
Comparison of our estimates of halo masses for early-type galaxies in SLUGGS based on f ( J ) models (red circles) with those of Alabiet al. (2017), using the TME (green squares), and those of Bílek et al. (2019), based on a Jeans analysis (grey pentagons). limitations of their approach, we find that their results are consis-tent with ours within the uncertainties. This comparison adds toour confidence that our estimates of M h are not biased by model-dependent systematic e ff ects. Several other studies have derived the SHMR of spiral galax-ies from their rotation curves, with results consistent with thosefrom PFM19 shown here in Fig. 3. Lapi et al. (2018) found asimilar trend from stacked H α rotation curves in an independentsample of spirals (see Fig. 7). However, H α rotation curves typi-cally do not extend as far as HI rotation curves and thus, at best,provide only weak constraints on dark matter halo masses (vanAlbada et al. 1985; Kent 1987).Katz et al. (2017) also estimated the halo masses of spi-ral galaxies in the SPARC sample from HI rotation curve de-composition. However, they either used an unconstrained fit (i.e.with no prior on the c − M h relation), leaving M h undetermined,or they imposed a prior on M h from the Moster et al. (2013)SHMR. While the SHMR they derive naturally follows closelythe Moster et al. SHMR by construction, their estimates of M h are nevertheless in fair agreement with the ones we obtain with-out imposing a prior on the SHMR. Li et al. (2020) recently re-vised and expanded their approach to several di ff erent types ofhalo profiles and reached similar conclusions.It is also interesting to notice that a systematic di ff erence be-tween late types and early types was also reported by Tortoraet al. (2019) when looking at the dark matter fraction f DM withinone e ff ective radius R e as a function of stellar mass. They no-ticed that late types lie on a decreasing f DM − M (cid:63) relation, whilefor massive early types this relation inverts, analogous to our re-sults for 1 / f (cid:63) versus M (cid:63) based on the total masses within thevirial radii of the halos. This suggests that, at a fixed M (cid:63) , mas-sive discs are less dominated by dark matter than spheroids, bothglobally and locally (see also Marasco et al. 2020). Another novel method for estimating dark matter halo massesof galaxies is based on the total mass of their GC system, M GCS . Several recent studies have demonstrated the existence of a convincing linear relation between M GCS and M h for bothlate types and early types (e.g. Blakeslee et al. 1997; Spitler& Forbes 2009; Georgiev et al. 2010; Harris et al. 2013, 2017;Hudson et al. 2014; Burkert & Forbes 2020). The physical ori-gin of this relation is not well understood (e.g. Kravtsov &Gnedin 2005; Boylan-Kolchin 2017; El-Badry et al. 2019), butwe can exploit it as a semi-independent method for deriving theSHMR nonetheless . Thus, we may regard the relation between L K / M GCS and L K as a direct analogue of the f (cid:63) − M (cid:63) relation,where L K is the K -band luminosity. The L K / M GCS − L K relationrequires only photometry, rather than spectroscopy of the GCs,making it relatively easy to derive. Harris et al. (2013, see alsoreferences therein) have assembled a catalogue of M GCS for 422nearby galaxies of all morphological types covering a wide rangein luminosities.We plot in Fig. 9 the ratio L K / M GCS as a function of L K forthe galaxies in the Harris et al. (2013) sample, separating theminto late types (blue crosses) and early types (orange squares).The similarity between the L K / M GCS − L K relation in Fig. 9 andthe f (cid:63) − M (cid:63) relation in Fig. 3 is striking; early types turnover atabout L K ∼ × L (cid:12) , while bright late types seem to lie ona separate rising branch (see also Kim et al. 2019, for a similaranalysis of early types).The Harris et al. (2013) sample contains 24 out of the 25SLUGGS early types that we analysed in this work, but onlythree of the 20 massive spirals in SPARC (NGC 891, NGC 5907and NGC 7331). The galaxies in common with our detailed anal-ysis are indicated in Fig. 9 by symbols with darker and thickeredges. We notice that the three spirals have systematically higher L K / M GCS than any of the early types of similar L K , in quali-tative agreement with what we find on the f (cid:63) − M (cid:63) diagram.From this analysis we conclude that, despite the large scatter,the L K / M GCS − L K relation is quite consistent with our more ro-bust derivation of the f (cid:63) − M (cid:63) relation from GC kinematics andHI rotation curves. A caveat we note here is that studies that determined the M GCS − M h relation often assumed halo masses from a standard SHMR, typicallynot taking into account the morphology dependence that we highlightin Fig. 3. Irregulars are excluded from this analysisArticle number, page 9 of 16 & A proofs: manuscript no. main L K / L (cid:12) L K / M G C S ( L (cid:12) / M (cid:12) ) Harris et al . (2013) catalogue E , S0 Sa , Sb , Sc Fig. 9.
Ratio of K -band luminosity to the mass of the GC system( L K / M GCS ) versus K -band luminosity for galaxies in the Harris et al.(2013) catalogue. We plot spirals with turquoise crosses, and ellipti-cals and lenticulars with orange squares. We highlight the galaxies incommon between the Harris et al. (2013) catalogue and our work withdarker edge colours.
5. The SHMR and galaxy scaling laws
In the previous sections, we established that the SHMR of latetypes and early types follows two distinct branches: one where f (cid:63) increases with mass for spirals, and another one for ellip-ticals and lenticulars where f (cid:63) decreases beyond a peak near M (cid:63) ∼ × M (cid:12) . Yet, late-type and early-type galaxies areknown to obey very similar scaling relations between their stel-lar masses and velocities; the Tully & Fisher (1977) and Faber& Jackson (1976) relations are observed to be pure power lawswith no significant features. These two facts may appear to beat odds with each other, since velocity is a proxy for dynamicalmass and one might therefore expect the shape of the SHMR toimpact the mass-velocity scaling laws (e.g. Ferrero et al. 2017;Posti et al. 2019b). How can we reconcile these seemingly con-tradictory facts?The resolution of this apparent paradox logically must in-volve the relations between the characteristic velocities of galax-ies and those of their dark matter halos, which need to be dif-ferent for late types and early types on the two branches of theSHMR. In this section, we demonstrate this di ff erence and dis-cuss its physical implications. We take measurements of the flat parts of HI rotation curvesfor spirals in SPARC ( V flat , Lelli et al. 2016b) and stellar ve-locity dispersions measured within a fixed radius of 1 kpc forthe ellipticals and lenticulars in SLUGGS ( σ (cid:63), , Brodie et al.2014). These are the two characteristic velocities that we use todefine the Tully-Fisher and Faber-Jackson relations, which we compactly write as M (cid:63) ∝ V a gal , (2)where V gal is V flat for late types and σ (cid:63), for early types. Alongeither the rising branch of the SHMR for late types or along thedeclining one for massive early types, the SHMR can also beapproximated by a power law f (cid:63) ∝ M b h . (3)Given the definition of f (cid:63) (Eq. 1) and the fact that, in Λ CDMcosmogonies, halo masses and virial velocities are related by M h ∝ V , we can rearrange the two equations above into a re-lation between V gal and V h , which then becomes the power law V gal ∝ V c h , (4)with c = b + / a . (5)Eq. (4) is key here since it relates the familiar scaling laws (Eq. 2)with the two branches of the SHMR (Eq. 3). With our set of mea-surements – M (cid:63) from 3.6 µ m photometry, V gal from observed HI( V flat ) or stellar kinematics ( σ (cid:63), ), M h from dynamical modelsof rotation curves and GC kinematics – we perform power-lawfits to Eq. (2)-(4), finding a = . ± . , b = . ± . , c = . ± . , (late types) (6) a = . ± . b = − . ± . , c = . ± .
17 (early types) . (7)In Fig. 10, we show (on the top row) the data and the power-lawfits of the three relations ( f (cid:63) − M h , M (cid:63) − V gal , and V gal − V h )for the late types (blue) and early types (red). While the mass-velocity scaling laws have a similar slope ( a ), the SHMR and the V gal − V h relations have significantly di ff erent slopes ( b and c ) onthe two branches. At a fixed stellar mass, the large di ff erence inhalo mass between discs and spheroids is hidden in the similarTully-Fisher and Faber-Jackson relations by the di ff erent V gal − V h relations.The slope c ≈ V flat / V h is nearly the same (andabout equal to unity) for discs of all masses (Posti et al. 2019b).This is yet another manifestation of the so-called disc-halo con-spiracy, i.e. rotation curves are observed to be flat from the in-ner, baryon-dominated parts of galactic discs to the outer, darkmatter-dominated parts (e.g. van Albada et al. 1985; Kent 1987).On the other hand, for early types along the falling branch ofthe SHMR, we find a very di ff erent result: c ≈ .
4, which im-plies that the ratio σ (cid:63), / V h decreases with both stellar andhalo mass.While V flat for late types is measured at large radii and tracesthe potential of the dark halo, σ (cid:63), for early types is mea-sured in the inner regions where the potential is dominated bystars. One might then wonder whether this is responsible for thedi ff erence in the V flat / V h and σ (cid:63), / V h ratios. We check forthis in the bottom panels of Fig. 10, where we replace the ob-served V gal with the circular velocity evaluated at a fixed radiusof 2 R e for both late-type and early-type galaxies in our sample.For late types, we obtain the circular velocities directly from theobserved rotation curves, while for early types, they are an out-put of our f ( J ) dynamical models. Fig. 10 shows that the di ff er-ence persists in the relation V circ (2 R e ) ∝ V c (cid:48) h . Now we find c (cid:48) ≈ c (cid:48) ≈ .
57 for early types.
Article number, page 10 of 16. Posti & S. M. Fall: Morphology-dependent stellar-to-halo mass relation M h / M (cid:12) − − − f ? SHMR V ga l / k m s − V gal = { V flat σ ?, Tully - Fisher & Faber - Jackson V ga l / k m s − V gal − V h relation M ? / M (cid:12) V c i r c ( R e ) / k m s − V h / km s − V c i r c ( R e ) / k m s − Fig. 10.
SHMR ( f (cid:63) − M h , left), stellar mass-velocity scaling law ( M (cid:63) − V gal , middle) and relation between galaxy velocity and halo velocity( V gal − V h , right) for the population of late types (blue) and early types (red). In the top-middle and top-right panels, the characteristic velocityof galaxies V gal is the velocity along the flat parts of HI rotation curves for late types ( V flat , Lelli et al. 2016b) and the velocity dispersion of starswithin 1 kpc for early types ( σ (cid:63), , Brodie et al. 2014). In the bottom-middle and bottom-right panels, we substitute the observed V gal with thecircular velocity evaluated at the same radius (2 R e ) for both galaxy types. In all panels, we show power-law fits to the data of late types (bluedashed lines) and early types (red dotted lines). The observed scaling relations between the rotation velocities,sizes, and stellar masses of spiral galaxies indicate that they rep-resent a self-similar population of objects, homologous to theirdark matter halos (e.g. Posti et al. 2019b). In particular, galacticdiscs have almost as much specific angular momentum as theirdark halos, as expected from simple conservation arguments(e.g. Fall & Efstathiou 1980; Dalcanton et al. 1997; Mo et al.1998). Stellar feedback modifies this behaviour, making gas re-tention mass dependent, and thus creating the rising branch ofthe SHMR. Gravitational clustering and accretion moves galax-ies along the scaling laws and up the rising branch of the SHMR.This introduces no features in either the scaling laws or the f (cid:63) − M (cid:63) relation.However, galaxies in crowded environments, such as groupsand clusters, often collide and merge with each other. Depend-ing on the mass ratio of the galaxies and on whether they aregas rich or gas poor, mergers have a couple of important impli-cations for the evolution of massive galaxies. First, the stellarbody is dynamically heated, causing the spheroidal componentto grow at the expense of the disc component, thus leading tomorphological transformation (e.g. Quinn et al. 1993; Hopkinset al. 2010a; Martin et al. 2018). Second, some of the gas in themerging galaxies may be funnelled into the central black hole, thus triggering AGN feedback (e.g. Hopkins et al. 2006). Out-flows and radiation from the AGN may then impede further in-flows and star formation, hence reducing f (cid:63) . As a consequenceof merging and AGN feedback, massive discs are driven o ff therising branch of the SHMR, becoming passive spheroids on thefalling branch.In the previous subsection, we showed that late types followa V gal − V h relation with a slope of c ≈
1, implying that theratio of binding energy per unit mass of the luminous galaxy tothat of its dark halo ( ∝ V / V ) is approximately independent ofmass. In contrast, early types have c ≈ .
4, indicating that theratio of galaxy-to-halo binding energy per unit mass decreasesas mass increases. Both merging and AGN feedback may reducethe binding energy per unit mass, leading to c <
1, as observed.To understand how mergers can lower the ratio V gal / V h weconsider a typical collision between two galaxies on a weaklybound orbit (seee.g. Khochfar & Burkert 2006). While the halomass M h and the virial velocity V h increase during merging, ide-alised simulations have shown that the internal velocity disper-sion of the stars typically remains constant or decreases, depend-ing on the mass ratio, gas fraction and orbital parameters (e.g.Nipoti et al. 2003; Cox et al. 2006; Naab et al. 2009; Hilz et al.2012; Posti et al. 2014). After some Gyrs of evolution in a denseenvironment, a massive early type that experiences several merg-ers will therefore lower its V gal / V h ratio as its mass increases. Article number, page 11 of 16 & A proofs: manuscript no. main
This e ff ect is also observed in cosmological simulations, wherefrequent minor mergers deposit stars primarily in the outskirtsof massive galaxies, thus lowering their binding energy per unitmass (e.g. Oser et al. 2012; Gabor & Davé 2012).At the same time, outflows launched by the AGN will inter-act with the circumgalactic medium, pushing some of it outward,depending on the opening angle of the outflow. If this gas is everable to condense and form stars, this would also tend to lowerthe binding energy per unit mass of the host galaxy.Mergers and AGN feedback may thus combine to trans-form star-forming discs on the rising branch of the SHMR intoquenched spheroids on the falling branch. These two processesare contemporaneous but episodic. Both mergers and AGN feed-back tend to disrupt inflow onto galactic discs, thus suppressingdisc growth while promoting spheroid growth. Between theseepisodes, relatively smooth inflow can resume, thus promot-ing the regrowth of discs. This reasoning suggests that massivegalaxies may evolve along complicated, essentially stochastic,paths in the region of the f (cid:63) − M (cid:63) plane bounded by the risingpure-disc branch and the declining pure-spheroid branch.
6. Summary and Conclusions
In this paper, we have derived the SHMR for a sample of 25 mas-sive early-type galaxies from estimates of their individual halomasses. We accomplished this by comparing a dynamical modelwith a flexible analytical distribution function with position andvelocity data for the globular cluster systems around these galax-ies. Combining our new results for early types with those fromPFM19 for late types based on extended HI rotation curves, wederived, for the first time, the f (cid:63) − M (cid:63) relation for galaxies of dif-ferent morphologies with identical assumptions about their haloproperties. Our main findings can be summarised as follows.(i) At the high-mass end of the SHMR ( M (cid:63) > × M (cid:12) ), latetypes are found to have significantly higher f (cid:63) than earlytypes of the same stellar mass (by about a factor ∼ M (cid:63) ∼ M (cid:12) ). While f (cid:63) increases with M (cid:63) for late types(PFM19), it decreases for early types, in broad agreementwith expectations from abundance matching (e.g. Mosteret al. 2013). Our determinations show unequivocally that theSHMR has a secondary correlation with galaxy type at thehigh-mass end.(ii) For massive galaxies ( M (cid:63) > × M (cid:12) ), we studied how f (cid:63) deviates from the expectations of abundance matching( f (cid:63), AM ) as a function of disc fraction and Hubble type. Wefind a fairly continuous transition between close agreement,log( f (cid:63) / f (cid:63), AM ) ∼
0, for pure spheroids, and an order of mag-nitude discrepancy, log( f (cid:63) / f (cid:63), AM ) ∼
1, for pure discs. Thistransition occurs at about D / T ∼ .
6, or T ∼
2, suggestiveof scenarios involving merging and AGN feedback.(iii) We have tested the sensitivity of our M h estimates with re-spect to our adopted priors on the c − M h correlation. We findthat the secondary correlation of the SHMR with galaxy typeis robust relative to any reasonable adjustments to this prior.We have also compared our results both with other statisticalderivations of the SHMR (e.g. using weak-lensing or satel-lite kinematics) and with other individual estimates of halomasses based on di ff erent data and / or techniques. We findthese estimates to be compatible within the uncertainties, al-lowing us to conclude that the issue of whether the SHMRhas a secondary correlation with galaxy type is now settled.(iv) We investigated the apparent paradox between the two sep-arate branches of the SHMR – a rising one for discs and a falling one for massive spheroids – and the similar power-law relations between stellar masses and velocities for latetypes and early types, the Tully-Fisher and Faber-Jackson re-lations. We demonstrated that this happens because the rela-tions between galaxy velocity and halo velocity are di ff erentfor galaxies of di ff erent types. Discs have a constant ratio V gal / V h ≈ V gal / V h with mass. We suggest that this is asignature of the combined e ff ects of merging and AGN feed-back.As suggested above, both merging and AGN feedback arelikely responsible for splitting the SHMR and the V gal − V h re-lation into di ff erent branches for discs and spheroids, but theirexact roles remain to be determined. The growth of discs andspheroids in massive galaxies may be intermittent, with discgrowth during periods of relatively smooth inflow, interruptedby spheroid growth during episodes of merging and AGN feed-back. Hydrodynamical simulations may shed light on the under-lying physical processes, so long as they are relatively insensitiveto numerical resolution and subgrid recipes for stellar and AGNfeedback. A careful census of black holes in a large sample ofhost galaxies of di ff erent morphologies and masses likely wouldalso be instructive. Acknowledgements.
We thank Michal Bílek, Benoit Famaey, and Filippo Fra-ternali for encouragement in the early stages of this project and Romeel Davé,Ken Freeman, and Andrey Kravtsov for helpful comments in the later stages.LP acknowledges support from the Centre National d’Etudes Spatiales (CNES).This research has made use of "Aladin sky atlas" developed at CDS, StrasbourgObservatory, France (Bonnarel et al. 2000)
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Appendix A: Action-based dynamical models ofearly-type galaxies
Here, we describe the dynamical models that we use to repre-sent the distribution function of globular cluster systems around25 ellipticals and lenticulars and their dark matter halos. We firstsummarise the basic principles of models based on action-anglevariables, and then we describe our application to the study ofearly types. For a more complete introduction to action-anglevariables, we refer the reader to the monographs by Born (1927)and Arnold (1978). We use the code
AGAMA (Vasiliev 2019) toevaluate actions, potentials, and distribution functions, and togenerate the dynamical models in this work.
Appendix A.1: Preliminaries
We begin with the distribution function (DF) for a globular clus-ter system f , defined such that f ( x , v )d x d v is the probability offinding a cluster in the infinitesimal volume element d x d v at theposition-velocity point ( x , v ) in phase-space. According to thestrong form of the Jeans (1915) theorem, in a steady state, f is afunction of the integrals of motion (see also Lynden-Bell 1962).Without loss of generality, we may choose these to be the threeaction integrals J i = π (cid:73) p i d q i for i = , , , (A.1)where p i and q i are canonically conjugate momenta and coordi-nates, and write the DF as f ( J ).Actions J and their canonically conjugate angles θ , are the“natural” coordinates of galactic dynamics since i) the descrip-tion of orbits becomes mathematically simplest, ii) they describesystems both in and out of equilibrium and iii) actions are adia-batic invariants, i.e. they are constant under slow changes of thepotential (see e.g. Binney & Tremaine 2008). f ( J ) models havebeen somewhat underused in galactic dynamics, mainly becauseactions generally cannot be expressed with algebraic functionsof positions and velocities and need to be computed numerically.In recent years, several crucial advances have made it feasible tocalculate J e ffi ciently in arbitrary potentials (see e.g. Sanders &Binney 2016, and references therein). This, in turn, has led to theintroduction of several analytic f ( J ) DFs tailored to model dif-ferent galaxy components (e.g. Binney 2010; Posti et al. 2015;Sanders & Evans 2015; Pascale et al. 2018; Vasiliev 2019).In this work, we deal mostly with spherical potentials, whichgreatly simplifies the numerical calculations. In the case of aspherical system the motion of particles is confined to a planeand all orbits can be characterised by two actions. One of theseis the total angular momentum L = | L | , and the other is the radialaction J r = π (cid:90) r peri r apo p r d r = π (cid:90) r peri r apo (cid:16) E − Φ − L / r (cid:17) / d r , (A.2)where p r and r are the radial momentum and position, E = p r / + L / r + Φ is the energy, Φ is the gravitational poten-tial and r apo and r peri are the apocentre and pericentre of theorbit. Thus, in the spherical case, we have J = ( J r , L ) and f ( J ) = f ( J r , L ). Appendix A.2: Distribution function
To describe the phase-space distribution of globular clusters weuse the DF introduced by Posti et al. (2015, see also Williams & Evans 2015, Vasiliev 2019). This is f ( J ) = M (2 π J ) + (cid:32) J h ( J ) (cid:33) A Γ / A + (cid:32) g ( J ) J (cid:33) A (B − Γ ) / A (A.3)where h ( J ) = ν h J r + − ν h L , g ( J ) = ν g J r + − ν g L . (A.4)Here M is a parameter proportional to the mass of the systemdescribed by the DF; since we are treating the globular clustersas tracers of the potential, M is unimportant in this context. TheDF of Eq. (A.3) has 6 free parameters, all with specific physicalmeanings. Γ and B control the asymptotic slopes of the densityprofile in the inner ( r →
0) and outer parts ( r → ∞ ) respec-tively, while the parameter A controls the sharpness of the tran-sition between these regimes. In the case A =
1, the two slopes Γ and B have a direct correspondence to the asymptotic slopes ofthe density distribution . J is a characteristic action that definesthe radial scale at which the transition between the two regimesoccurs. The last two parameters, ν h and ν g , control the velocityanisotropy of the model in the inner and outer parts, respectively.An important advantage of the double power-law f ( J ) inEq. (A.3), over models that depend on ( E , L ), is that in the formercase the density distribution e ff ectively decouples from the ve-locity distribution. This allows us to fix at the outset the param-eters of the DF that regulate the density profile of a GC system(A, B, Γ ) and then to fit only for those that determine the veloc-ity anisotropy of the system ( J , ν h , ν g ). Such decoupling is pos-sible because, for double power-law models, the homogeneousfunctions h ( J ) and g ( J ) are designed to approximate surfaces ofconstant energy in action space (Williams et al. 2014; Posti et al.2015). Thus, h and g largely determine the di ff erential energydistribution d N / d E , and hence the density profile of the model(see S4.3 in Binney & Tremaine 2008). Starting from a quasi-ergodic model, where J r and L appear on an equal footing in h and g , one can easily make the model anisotropic by varying ν h and ν g without altering the radial density profile (Binney 2014;Posti et al. 2015).As a first step in our modelling procedure, we fix the twoslopes Γ and B and the sharpness A by matching to the ob-served number density profile of GCs. In Fig. 1b, we show thisfit for the GC system of the galaxy NGC 4494, with (A, B, Γ ) = (2 . , . , . J , ν h and ν g , are instead allowed tovary, but, for internal consistency of the DF, we need to require0 < ν h , ν g < Appendix A.3: Gravitational potential
We model the mass distribution of each galaxy with two spheri-cal components; the stellar body of the galaxy and its dark mat-ter halo – the globular cluster system is then regarded as a tracerwith negligible mass. The stellar distribution is described by a Posti et al. (2015) showed that a self-consistent model with DF as inEq. (A.3) and A = αβγ -model (Zhao 1996) with α = Γ = (6 − γ ) / (4 − γ ) andB = β − numerically deprojected Sersic (1968) profile, whose parame-ters are taken from the photometry of 3.6 µ m Spitzer images byForbes et al. (2017b). We fix all the parameters of the stellarcomponent, except its mass-to-light ratio at 3.6 µ m, which weallow to vary with a log-normal prior with a central value esti-mated by Forbes et al. (2017b) from stellar population models,and a dispersion of 0.2 dex.The dark matter halo in our model is described by a standardNFW profile. This has two free parameters: the virial mass ( M h )and concentration ( c ), which we allow to vary. While we adopta flat (uninformative) prior on the halo mass, we use a prior forthe concentration that follows the mean c − M h correlation from Λ CDM simulations, with a scatter of 0.11 dex (Dutton & Macciò2014). Thus, overall our models have six free parameters: threefor the potential and three for the DF.Several of our galaxies appear flattened on the sky, so it is im-portant to evaluate whether the assumption of spherical symme-try for the stellar component of the potential significantly a ff ectsour results. To check this, we have re-run all of our models withan axisymmetric galaxy mass distribution that has the same 3Dflattening as the 2D image (Forbes et al. 2017b), while the darkmatter halo is still spherical. With respect a spherical galaxy withthe same mass, the deviations in the resulting halo masses are al-ways well within the uncertainties . We are therefore confidentthat the assumption of spherical symmetry in the galactic massdistribution does not significantly bias our results. Appendix A.4: Parameters estimation
We estimate the posterior distributions of the model parameters( (cid:36)(cid:36)(cid:36) ) with standard Bayesian inference: P ( (cid:36)(cid:36)(cid:36) | d ) ∝ P ( d | (cid:36)(cid:36)(cid:36) ) P ( (cid:36)(cid:36)(cid:36) ),where d are the data, P ( d | (cid:36)(cid:36)(cid:36) ) is the likelihood, and P ( (cid:36)(cid:36)(cid:36) ) is theprior. We adopt a prior that is flat (uninformative) for four pa-rameters (log M h , log J , ν h and ν g ), gaussian for log M / L . ,with a mean estimated for each galaxy by Forbes et al. (2017b)with stellar population models and a dispersion of 0.2 dex, andgaussian for log c , with a mean given by the Λ CDM relation anda dispersion of 0.11 dex.The DF in Eq. (A.3) itself is a probability distribution thatcan serve as the likelihood in our framework. Specifically, for aset of N particles with position-velocity coordinates ( x i , v i ) or-biting in a given potential Φ , the likelihood, given the model f ( J ), is simply (cid:81) Ni = f [ J ( x i , v i )]. In reality, when dealing withdata, one does not know the positions and velocities with infiniteprecision; thus, a convolution of the DF with the observed errordistribution is needed (see Binney & Wong 2017; Posti & Helmi2019).In our case, we also lack information about the two trans-verse velocities and the precise positions of the GCs along theline-of-sight (LOS). To take this into account, we marginalisethe likelihood over all of the realistically possible transverse ve-locities and LOS positions of the clusters. For the two unknowntransverse velocities ( v x , v y ), we adopt uniform distributions inthe range [ − V esc , V esc ], where V esc is the escape velocity of thepotential. For the unknown LOS position z , we adopt the depro-jected density distribution of the GC system ρ ( s ), where s is thespherical radius, s = x + y + z , evaluated at a fixed po- We also recall that the potential is always more spherical than themass distribution that it generates (Binney & Tremaine 2008). sition on the sky ( x GC , y GC ). Thus, we have the following errordistribution E ( u | d ) = δ ( x − x GC ) δ ( y − y GC ) G ( v z | V los , (cid:15) V los ) ρ ( z ) U ( v x | − V esc , V esc ) U ( v y | − V esc , V esc ) , (A.5)where u = ( x , y , z , v x , v y , v z ) is a point in phase space in a Carte-sian frame centred on the galaxy, and d = ( x GC , y GC , V los , (cid:15) V los )are the observations. Here ρ is the deprojected density distri-bution derived from the observed GC number counts profile(Fig. 1b), G ( v z | V los , (cid:15) V los ) is a gaussian distribution with mean V los and dispersion (cid:15) V los , U ( v | − V esc , V esc ) is a uniform distribution inthe range [ − V esc , V esc ], and δ ( x − x GC ) is a Dirac delta distributioncentred on x GC . We use a δ distribution because the uncertaintyin the sky positions of the clusters is negligible. Finally, the like-lihood of our model is given by the convolution of the DF withthe E distribution of each cluster, i.e. P ( d | (cid:36)(cid:36)(cid:36) ) = N (cid:89) i = (cid:90) d u E ( u | d i ) f [ J ( u )] . (A.6)In practice, we evaluate Eq. (A.6) with a Monte Carlomethod, sampling the integral and the E distribution of eachcluster with 1000 realisations. Fortunately, the likelihood inEq. (A.6) turns out to be quite insensitive to the specific form ofboth the density distribution ρ and the distributions of the miss-ing velocities; in fact, we verified that using a gaussian instead ofa uniform distribution in v x and v y does not alter significantly ourresults on the halo masses. As an example, in Fig. 1d, we showthe distribution of clusters around NGC 4494 on the observableprojection of the phase-space, the V los − r plane, compared tothe prediction of the maximum-likelihood f ( J ) model for thisgalaxy. The V los − r plane is e ff ectively the sub-space where weare fitting our models to the data.With the prior and likelihood defined as above, we evalu-ate the posterior distribution of the six free parameters of themodel with a Markov Chain Monte Carlo (MCMC) method; inparticular, we use the a ffi ne-invariant sampler implemented inthe code emcee by Foreman-Mackey et al. (2013). For all 25SLUGGS early-type galaxies, we find that the chains convergequite rapidly around a well-defined peak in the posterior aftera short burn-in phase. As an example, in Fig. 1c, we show themarginalised posterior distributions for the halo mass and con-centration for the galaxy NGC 4494. Clearly, both M h and c are well constrained by our analysis, despite having an unavoid-able degeneracy. For each parameter, we take the median of themarginalised posterior as the best-fit value and the interval be-tween the 16th and 84th percentiles as a measure of its uncer-tainty. Appendix A.5: Derived quantities
From our model, with parameters optimised for each GC systemin the SLUGGS sample, we can now derive several other prop-erties of interest. In Fig. 1e, we show, as examples, the circularvelocity curve of the mass distribution, V circ , and the (spheri-cally averaged) velocity anisotropy profile of the GC system, β = − ( σ θ + σ φ ) / σ r . While V circ depends only on the three freeparameters of the gravitational potential ( M h , c , M / L . ), β de-pends mostly on the three free parameters of the DF ( ν h , ν g , J ).This means, incidentally, that the uncertainty on V circ , which we Article number, page 15 of 16 & A proofs: manuscript no. main estimate with random realisations of the model from the poste-rior (1 σ grey band in Fig. 1e), is fully determined by the widthof the posterior on the parameters of the potential (Fig. 1c).We can also compute the profile of the LOS velocity disper-sion of the GC systems ( σ los ), which depends on both the po-tential and the DF. We show this model profile for NGC 4494 inFig. 1f, where we compare it with the observed profile (see Fos-ter et al. 2016). Such a comparison is meaningful since we do notinput directly the σ los profile to our fitting routine, although wedo, of course, input the same individual velocities that determine σ los . The agreement that we observe for NGC 4494 (Fig. 1f), andalso for the other galaxies in our sample (not shown), thus servesas a useful consistency check on our procedure.For 18 of the 25 early types in our sample, Pulsoni et al.(2018) measured the velocity dispersion profile of the popula-tion of planetary nebulae orbiting around the host galaxy andfound that in most cases it agrees quite well with the σ los profileof the GC system from SLUGGS. Fig. 1f shows this agreementfor NGC 4494. In a few cases, however, the σ los for the planetarynebulae is ∼ −
40% lower than for the globular clusters. Thisdi ff erence in the σ los likely reflects the di ff erent density profilesof the two types of tracers orbiting in the same gravitational po-tential. Even among GCs, there are di ff erences in σ los when thesystem is subdivided by colour. Red GCs have lower σ los thanblue GCs, and are in better agreement with both the velocity dis-persion of planetary nebulae and the stellar bodies of galaxies(e.g. Pota et al. 2015).thanblue GCs, and are in better agreement with both the velocity dis-persion of planetary nebulae and the stellar bodies of galaxies(e.g. Pota et al. 2015).