Galaxy disc scaling relations: A tight linear galaxy -- halo connection challenges abundance matching
Lorenzo Posti, Antonino Marasco, Filippo Fraternali, Benoit Famaey
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2019September 5, 2019
Galaxy disc scaling relations: A tight linear galaxy–haloconnection challenges abundance matching
Lorenzo Posti (cid:63) , Antonino Marasco , , Filippo Fraternali , and Benoit Famaey Université de Strasbourg, CNRS UMR 7550, Observatoire astronomique de Strasbourg, 11 rue de l’Université, 67000 Strasbourg,France. Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, the Netherlands ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD, Dwingeloo, The NetherlandsReceived XXX; accepted YYY
ABSTRACT In Λ CDM cosmology, to first order, galaxies form out of the cooling of baryons within the virial radius of their dark matter halo. Thefractions of mass and angular momentum retained in the baryonic and stellar components of disc galaxies put strong constraints onour understanding of galaxy formation. In this work, we derive the fraction of angular momentum retained in the stellar componentof spirals, f j , the global star formation e ffi ciency f M , and the ratio of the asymptotic circular velocity ( V flat ) to the virial velocity f V ,and their scatter, by fitting simultaneously the observed stellar mass-velocity (Tully-Fisher), size-mass, and mass-angular momentum(Fall) relations. We compare the goodness of fit of three models: (i) where the logarithm of f j , f M , and f V vary linearly with thelogarithm of the observable V flat ; (ii) where these values vary as a double power law; and (iii) where these values also vary as adouble power law but with a prior imposed on f M such that it follows the expectations from widely used abundance matching models.We conclude that the scatter in these fractions is particularly small ( ∼ .
07 dex) and that the “linear” model is by far statisticallypreferred to that with abundance matching priors. This indicates that the fundamental galaxy formation parameters are small-scattersingle-slope monotonic functions of mass, instead of being complicated non-monotonic functions. This incidentally confirms that themost massive spiral galaxies should have turned nearly all the baryons associated with their haloes into stars. We call this the failedfeedback problem.
Key words. galaxies: kinematics and dynamics – galaxies: spiral – galaxies: structure – galaxies: formation
1. Introduction
The current Λ cold dark matter ( Λ CDM) cosmological modelis very successful at reproducing observations of the large-scalestructure of the Universe. However, galactic scales still present tothis day a number of interesting challenges for our understand-ing of structure formation in such a cosmological context (e.g.Bullock & Boylan-Kolchin 2017). These challenges could haveimportant consequences on our understanding of the interplaybetween baryons and dark matter, or even on the roots of thecosmological model itself, including the very nature of dark mat-ter. For instance, the most inner parts of galaxy rotation curvespresent a wide variety of shapes (Oman et al. 2015, 2019), whichmight be indicative of a variety of central dark matter profilesranging from cusps to cores and closely related to the observedcentral surface density of baryons (e.g. Lelli et al. 2013, 2016c;Ghari et al. 2019). In addition to such surprising central correla-tions, the phenomenology of global galactic scaling laws, whichrelate fundamental galactic structural parameters of both baryonsand dark matter, also carries important clues that should informus about the galaxy formation process in a cosmological context.Given the complexity of the baryon physics leading tothe formation of galaxies, which involves for instance gravita-tional instabilities, gas dissipation, mergers and interactions withneighbours, or feedback from strong radiative sources, it is re-markable that many of the most basic structural scaling relationsof disc galaxies are simple, tight power laws (see e.g. van der (cid:63) [email protected]
Kruit & Freeman 2011, for a review); these most basic struc-tural scaling relations, for example, can be between the stellar orbaryonic mass of the galaxy and its rotational velocity (Tully &Fisher 1977; Lelli et al. 2016b), its stellar mass and size (Kor-mendy 1977; Lange et al. 2016), and its stellar mass and stellarspecific angular momentum (Fall 1983; Posti et al. 2018a).The interplay of all the complex phenomena involved in thegalaxy formation process thus conspires to produce a populationof galaxies which is, to first order, simply rescalable. Interest-ingly, in Λ CDM, dark matter haloes also follow simple, tight,power-law scaling relations and their structure is fully rescal-able. Thus, all of this is suggestive of the existence of a sim-ple correspondence between the scaling relations of dark matterhaloes and galaxies (e.g. Posti et al. 2014). In this context, wecan consider to first order a simplified picture in which galax-ies form out of the cooling of baryons within the virial radius oftheir dark matter halo. That is, before any dissipation happens,the fraction of total matter that is baryonic inside newly formedhaloes would not di ff er on average from the current value of thecosmic baryon fraction f b ≡ Ω b / Ω m (cid:39) . Ω b and Ω m are the baryonic and total matter densities of the Universe, re-spectively (Planck Collaboration et al. 2018). In this simplifiedpicture, galaxies are then formed out of those baryons that e ff ec-tively dissipate and sink towards the centre of the potential well,and the final structural properties of galaxies, such as mass, size,and angular momentum, are then directly related to the interplaybetween the (cooling) baryons and (dissipationless) dark matter. Article number, page 1 of 17 a r X i v : . [ a s t r o - ph . GA ] S e p & A proofs: manuscript no. main
Observationally proving that indeed the masses, sizes, andangular momenta of galaxies are simply and directly propor-tional to those of their dark matter haloes, would be a major find-ing. This means that, out of all the complexity of galaxy forma-tion in a cosmological context, a fundamental regularity is stillemerging, which we would then need to understand. Some of theearliest and most influential theoretical models of disc galaxyformation relied on reproducing the observed scaling laws ofdiscs to constrain their free parameters (e.g. Fall & Efstathiou1980; Dalcanton et al. 1997; Mo et al. 1998). These parametersare often chosen to be physically meaningful and fundamentalquantities that synthetically encode galaxy formation, such as aglobal measure of the e ffi ciency at forming stars from the coolingmaterial (e.g. Behroozi et al. 2013; Moster et al. 2013, hereafterM +
13) or a measure of the net gains or losses of the total angu-lar momentum from that initially acquired via tidal torques (Pee-bles 1969; Romanowsky & Fall 2012; Pezzulli et al. 2017). Therich amount of data collected in recent years for spiral galax-ies both in the nearby Universe and at high redshift allows anunprecedented exploitation of the observed scaling laws which,when fitted simultaneously, can yield very strong constraints onsuch fundamental galaxy formation parameters (e.g. Dutton et al.2007; Dutton & van den Bosch 2012; Desmond & Wechsler2015; Lapi et al. 2018).While being the focus of many studies over the past years,the connection between galaxy and halo properties is still nottrivial to measure observationally (see Wechsler & Tinker 2018,for a recent review). However, arguably the most important bitof this connection, the relation between galaxy stellar mass anddark matter halo mass, is very well studied and the results fromdi ff erent groups tend to converge towards a complex, non-linearcorrespondence. As long as galaxies of all types are consideredand stacked together, the same non-linear relation, with a breakat around L ∗ galaxies, is found irrespective of the di ff erent ob-servations used to probe this relation: for instance, the matchof the halo mass function to the observed stellar mass function(the so-called abundance matching ansatz; e.g. Vale & Ostriker2004; Kravtsov et al. 2004; Behroozi et al. 2013; M + ∼
10 times moremassive than the Milky Way. These authors used accurate near-infrared (3.6 µ m) photometry with the Spitzer Space Telescopeand HI interferometry (Lelli et al. 2016a) to determine the stel-lar and dark matter halo masses robustly, by means of fitting theobserved gas rotation curves. Surprisingly, the authors found noindication of a break in the stellar-to-halo mass relation of theirsample of spirals. This finding is in significant tension with ex-pectations of abundance matching models for galaxies with stel-lar masses above 8 × M (cid:12) (see also McGaugh et al. 2010).Since the high-mass slope of the stellar-to-halo mass relation is commonly understood in terms of strong central feedback(e.g. Wechsler & Tinker 2018), we call this observational dis-crepancy the failed feedback problem. This discrepancy mightbe there simply as a result of a morphology-dependent galaxy-halo connection. While the relation found by PFM19 applies todisc galaxies, the stellar-to-halo mass relation from abundancematching instead is an average statistic derived for galaxies ofall types that is heavily dominated by spheroids at the high-massend. This would imply that the galaxy-halo connection for discsand spheroids can be significantly di ff erent, for example it couldbe linear for discs while being highly non-linear for spheroids.If this is the case for disc galaxies in the nearby Universe,then this should leave a measurable imprint on their structuralscaling laws, such as the Tully-Fisher, size-mass, and Fall rela-tions. It is possible to model these three scaling laws (of whichthe last two are dependent) with three (dependent) fundamen-tal galaxy formation parameters: one to determine the stellar-to-halo mass relation, one for the stellar-to-halo specific angu-lar momentum relation, and one for the disc-to-virial rotationvelocity relation. The shape of the observed scaling laws car-ries enough information to constrain these three quantities andtheir scatter together simultaneously, and to disentangle whethera simple, linear galaxy–halo correspondence is preferred for spi-rals or if a more complex, non-linear correspondence is needed(e.g. Lapi et al. 2018).In this paper we use individual, high-quality measurementsof the photometry and gas rotation velocity of a wide sampleof nearby spiral galaxies, from the smallest dwarfs to the mostmassive giant spirals, to fit their observed scaling relations withanalytic galaxy formation models that depend on the three funda-mental parameters mentioned above. We perform fits of modelswith either i) a simple, linear galaxy–halo correspondence, ii) amore complex, non-linear correspondence, and iii) also a com-plex, non-linear correspondence that has an additional prior onthe stellar-to-halo mass relation from popular abundance match-ing models. We then statistically evaluate the goodness of fit inall three cases and, finally, we compare the outcomes of thesethree cases with the halo masses recently measured from the ro-tation curves of the same spirals; thus, we have additional and in-dependent information on which of the models we tried is morerealistic.The paper is organised as follows. In Section 2 we describethe dataset that we use; in Section 3 we introduce the analyticmodels that we adopt to fit the observed scaling relations and ourfitting technique; in Section 4 we present the fitting results, thepredictions of the models, and the a posteriori comparison withthe halo masses measured from the rotation curve decomposi-tions; in Section 5 we summarise and discuss the implications ofour findings.Throughout the paper we use a fixed critical overdensity pa-rameter ∆ =
200 to define dark matter haloes and the standard Λ CDM model, which has the following parameters estimated bythe Planck Collaboration et al. (2018): f b ≡ Ω b / Ω m (cid:39) .
157 and H = . − Mpc − .
2. Data
Our primary data catalogue comes from the sample of 175nearby disc galaxies with near-infrared photometry and HI ro- We call the relation between stellar mass and stellar specific angularmomentum the “Fall relation” hereafter, due to the pioneering work byFall (1983) who realised the importance of this law in galaxy formation.Article number, page 2 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations tation curves (SPARC) collected by Lelli et al. (2016a, hereafterLMS16). These galaxies span more than 4 orders of magnitudein luminosity at 3.6 µ m and all morphological types, from irreg-ulars to lenticulars. The sample was primarily collected for stud-ies of high-quality, regular, and extended rotation curves; thusgalaxies have been primarily selected on the basis of interfer-ometric radio data. Moreover, the catalogue selection has beenrefined to include only galaxies with near-infrared photometryfrom the Spitzer Space Telescope. Hence, even though it is notvolume limited, this sample provides a fair representation of thefull population of nearby (regularly rotating) spirals. Samples ofspirals with a much higher completeness and with high-qualityHI kinematics will soon be available with the Square KilometreArray precursors and pathfinders, such as MeerKAT or APER-ture Tile In Focus (APERTIF).In what follows we consider only galaxies with inclinationslarger than 30 ◦ , since for nearly face-on spirals the rotationcurves are highly uncertain. This introduces no biases, sincediscs are randomly orientated with respect to the line of sight.We used the gas rotation velocity along the flat part of the ro-tation curve as a representative velocity for the system because itis known to minimise the scatter of the (baryonic) Tully-Fisherrelation (e.g. Verheijen 2001; Lelli et al. 2019). We used thesame estimate of V flat as in Lelli et al. (2016b), which is basi-cally an average of the three last measured points of the rotationcurve, with the condition that the curve is flat within ∼
5% overthese last three points. When fitting the models in the followingsections, we only consider the sample of galaxies that satisfiesthis condition; this includes 125 galaxies. We nonetheless showthe locations on the scaling relations of the other 33 galaxies(with inclinations larger than 30 ◦ ) that do not satisfy that crite-rion (white filled circles); also for these objects we adopted thedefinition of V flat and its uncertainty from Lelli et al. (2016b).The disc scale lengths R d have also been derived by Lelliet al. (2016a) with exponential fits to the outer parts of the mea-sured surface brightness at 3.6 µ m with Spitzer. These authorsdid this to exclude the contamination from the bulge (if present)in the central regions of the galaxy. We computed the stellarmasses M (cid:63) by integrating the observed surface brightness pro-files, which are decomposed into a disc and bulge component asin Lelli et al. (2016a), and by assuming a constant mass-to-lightratio for the two components of ( M / L [3 / , M / L [3 / ) = (0 . , . j (cid:63) − M (cid:63) relation, aka the Fall relation, is now verywell established observationally. Several independent measure-ments now agree perfectly both on the slope and normalisationof this relation at least for spirals (Romanowsky & Fall 2012;Obreschkow & Glazebrook 2014; Posti et al. 2018a; Fall & Ro-manowsky 2013, 2018). The total specific angular momentumof the stellar disc is, instead, measured as in Posti et al. (2018a).Given the stellar rotation curve V (cid:63) , estimated from the HI ro-tation curve , and the stellar surface density Σ (cid:63) , we calculated After accounting for the support from the stellar velocity disper-sion, or the so-called asymmetric drift correction, following the mea-surements from Martinsson et al. (2013). This correction is found tobe negligible for the determination of j (cid:63) for most systems (Posti et al.2018a). j (cid:63) = (cid:82) d R R Σ (cid:63) ( R ) V (cid:63) ( R ) (cid:82) d R R Σ (cid:63) ( R ) . (1)We used this measurement (and associated uncertainty as givenby Eq. 3 in Posti et al. 2018a) for the 92 SPARC galaxies with“converged” cumulative j (cid:63) profiles, meaning that they flatten inthe outskirts to within ∼
10% (following the definition by Postiet al. 2018a). For the other 33 galaxies with flat rotation curves,but with non-converged cumulative j (cid:63) profiles, we adopted themuch simpler estimator (see e.g. Romanowsky & Fall 2012) j (cid:63) = R d V flat , (2)which comes from Eq. (1) under the assumption of an exponen-tial stellar surface density profile with a flat rotation curve. Inthis equation, we are implicitly assuming that the gas rotation, V flat , is a reasonable proxy for the rotation velocity of stars, atleast in the outer regions of discs. Stars are indeed found on al-most circular orbits in the regularly rotating discs analysed inthis work (Iorio et al. 2017; Posti et al. 2018a). The simple j (cid:63) estimator in Eq. (2) is widely used and known to be reasonablyaccurate for spirals, provided that the measurements of R d and V flat are sound (e.g. Fall & Romanowsky 2018). In particular,Posti et al. (2018a), studying the sample of 92 SPARC galaxieswith converged profiles, determined that the estimator (2) is un-biased and yields a typical uncertainty of 30 −
40% on the true j (cid:63) . Thus, in what follows, we also consider j (cid:63) measurementsobtained with Eq. (2) and with an uncertainty of 40% for the33 SPARC galaxies with flat rotation curves, but non-convergedcumulative j (cid:63) profiles. We added a sub-sample of galaxies drawn from the Local Irreg-ulars That Trace Luminosity Extremes, The HI Nearby GalaxySurvey Survey (LITTLE THINGS, Hunter et al. 2012) to the cat-alogue described above. These are 17 dwarf irregulars that havefairly regular HI kinematics and are seen at inclinations largerthan 30 ◦ .This sample has been recently analysed by Iorio et al. (2017)who determined the rotation curve of each system from the de-tailed 3D modelling of the HI data. We used their results and ap-plied the same criterion on the rotation curve flatness as for theSPARC sample. We found that 4 out of 17 galaxies (CVnIdwA,DDO53, DDO210, UGC8508) have rotation curves which do notflatten to within ∼
5% over the last three data points, and thuswe excluded these galaxies from the fits but we still show thesein the plots (as white filled diamonds).We determined the size of these galaxies from their opti-cal R-band or V-band images using publicly available data from1-2 meter class telescopes at the Kitt Peak National Observa-tory (KPNO; Cook et al. 2014). In the cases where no KPNOdata were available, we used Sloan Digital Sky Survey (SDSS)data (CVnIdwA, DDO 101, DDO 47, and DDO 52; Baillard et al.2011) or Vatican Advanced Technology Telescope (VATT) data(UGC 8508; Taylor et al. 2005) instead. While a number of LIT-TLE THINGS systems come with IRAC Spitzer images, theseare vastly contaminated by bright point-like sources that wefound di ffi cult to treat properly. Also considering the superiorquality of the optical data, we decided to use the latter for oursize measurements.Using these images, we derived the surface brightness pro-files for all 17 systems following the procedure fully described Article number, page 3 of 17 & A proofs: manuscript no. main
RA [J2000] D E C [ J ] ddo52 r-band image * [ a r b i t r a r y un i t s ] Surface brightness profile
Fig. 1.
Photometry for DDO 52 as a representative example for the LIT-TLE THINGS galaxies. Top panel: r-band image with the concentric el-lipses showing the annuli where the surface brightness is computed. Thegreen regions are foreground sources that we mask during the derivationof the profile. The blue ellipse is drawn at the disc scale length. Bottompanel: surface brightness profile, normalised to the total light within theoutermost ring. The thick dashed black line indicates the exponentialfit, while the blue arrow indicates the exponential scale length. in Marasco et al. (2019), adopting as galaxy centres, inclinationsand position angles the values determined by Iorio et al. (2017).We then fit these profiles with exponential functions to deter-mine the galaxy scale lengths, which we found to be in excellentagreement with those inferred by Hunter & Elmegreen (2006). InFigure 1 we illustrate the procedure we use for the representativecase of DDO 52. Finally, for the LITTLE THINGS galaxies weused the estimator (2) for the stellar specific angular momentumwith a conservative error bar of 40%.
3. Model
We started with dark matter haloes, which are described by theirstructural properties – mass ( M h ), radius ( R h ), velocity ( V h ), andspecific angular momentum ( j h ) – defined in an overdensity of ∆ times the critical density of the Universe. Haloes, then, adhereto the following scaling laws (e.g. Mo et al. 2010): M h = GH (cid:114) ∆ V ; (3) R h = H (cid:114) ∆ V h ; (4) j h = λ H √ ∆ V , (5)where G is the gravitational constant and λ = j h / √ R h V h isthe halo spin parameter, as in the definition by Bullock et al.(2001, which is conceptually equivalent to the classic definitionin Peebles 1969). The distribution of λ for Λ CDM haloes is verywell studied and it is known to have a nearly log-normal shape– with mean log λ ≈ − .
456 and scatter σ log λ ≈ .
22 dex –irrespective of halo mass. Henceforth, since λ is not a functionof V h , Eq. (5) is a simple power law j h ∝ V , while also Eq. (3)-(4) are obviously similar power laws. We very simply parametrise the intrinsically complex processesof galaxy formation, by considering that, to first order, galaxiesform out of the cooling of baryons within the virial radius oftheir halo. The fundamental parameters we consider are then thefollowing fractions: f M ≡ M (cid:63) M h ; f j ≡ j (cid:63) j h ; f V ≡ V flat V h ; f R ≡ R d R h . (6)The aim of this work is to unveil the galaxy–halo connectionconstraining and characterising the four galaxy formation frac-tions above using the observed global properties of disc galaxies. – The first, and arguably most important, of these parameters isthe stellar mass fraction f M , which is also sometimes looselyreferred to as global star formation e ffi ciency parameter (e.g.Behroozi et al. 2013; M + ffi ciency of gas-to-stars conversion integrated over time. Onaverage, this parameter has an obvious strict upper limit setby the cosmic baryon fraction f b (cid:39) . – The second is the specific angular momentum fraction f j ,also known as the retained fraction of angular momentum(e.g. Romanowsky & Fall 2012). After the halo collapsed,but before galaxy formation started, tidal torques suppliedbaryons and dark matter with nearly equal amounts of an-gular momentum, so j baryon / j h = f j that can easily deviate from unity. Article number, page 4 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations – The third is the velocity fraction f V , which is the ratio of thecircular velocity at the edge of the galactic disc to that at thevirial radius. While this factor in principle can take any valuedepending on the galaxy and halo mass distribution, but alsodepending on the extension of the measured rotation curve, itis typically expected to be f V (cid:38) M (cid:63) / M (cid:12) >
9, see e.g. Papastergis et al. 2011). – The last parameter is the size fraction f R , i.e. the ratio of thedisc exponential scale length ( R d ) to the halo virial radius( R h ). However, if we assume that the size of the galaxy isregulated by its angular momentum (Fall & Efstathiou 1980;Mo et al. 1998; Kravtsov 2013), then f j and f R are not inde-pendent. It is easy to work out their relation as a function ofthe dark matter halo profile, which turns out to be analytic inthe case of an exponential disc with a flat rotation curve (seeAppendix A), i.e., f R = λ √ f j f V . (7)An analogous result was already derived analytically by Fall(1983). For more realistic haloes, for example a Navarroet al. (1996, NFW) halo, a similar proportionality still ex-ists, and can be worked out with an iterative procedure (seee.g. Mo et al. 1998).With these definitions we can rewrite the dark matter rela-tions of Eqs. (3)-(5) now for the stellar discs as M (cid:63) = f M GH (cid:114) ∆ (cid:32) V flat f V (cid:33) ; (8) R d = λ f j H √ ∆ V flat f V ; (9) j (cid:63) = λ f j H √ ∆ (cid:32) V flat f V (cid:33) . (10)In this form, the above equations involve all observable quan-tities ( V flat , M (cid:63) , R d , j (cid:63) ) and the three fundamental fractions( f M , f j , f V ). In what follows, we use observations on the R d − V flat and the j (cid:63) − V flat diagrams, together with the usual stellar massTully-Fisher M (cid:63) − V flat , instead of the more canonical size-massand Fall relations. The main reason for this is that when high-quality HI interferometric data are available, V flat is a very well-measured quantity (typically within ∼ M (cid:63) su ff ersfrom many systematic uncertainties (e.g. on the stellar initialmass function). Thus, we use the observed scaling relations (8)-(10) to constrain the behaviour of the three fundamental frac-tions as a function of V flat . However, we show in Appendix B theresult of fitting the more canonical Tully-Fisher, size-mass, andFall relations, hence deriving the fractions (6) as a function of M (cid:63) . We note that, as might be expected, we find similar resultsfor the fractions f M , f j , and f V when having either V flat or M (cid:63) asthe independent variable for the scaling laws. f M , f j , and f V The three scaling laws (8)-(10) provide us with constraints on thethree fundamental galaxy formation parameters f M , f j , and f V .In particular, these are generally not constant (e.g. M +
13 for f M ;Posti et al. 2018b for f j ; Papastergis et al. 2011 for f V ) and their variation from dwarf to massive galaxies is encoded in the scal-ing laws. We use parametric functions to describe the behaviourof f M , f j , and f V as a function of V flat and then we look for theparameters that yield the best match to the observed scaling re-lations. The ansatz on the functional form of f = f ( V flat ), where f is any of the three fractions, and the prior knowledge imposedon some of the free parameters, define the three models that wetest in this paper. (i) In the first and simplest model that we consider, the threefractions log f to vary linearly as a function of log V flat asfollows:log f = α log V flat / km s − + log f . (11)Thus, we have a slope ( α ) and a normalisation ( f ) for eachof the three fractions f M , f j , and f V . In this case, we adoptuninformative priors for all the free parameters. (ii) The second model assumes a more complicated doublepower-law dependence of f on V flat , f = f (cid:32) V flat V (cid:33) α (cid:32) + V flat V (cid:33) β − α . (12)We have two slopes ( α , β ) and a normalisation ( f ) that aredi ff erent for each of the three f ; while the scale velocity( V ), which defines the transition between the two power-lawregimes, is the same for the three fractions for computationalsimplicity. Also in this case, we use uninformative priors forall the free parameters. (iii) The last model has the same functional form as model (ii),i.e. Eq. (12), with uninformative priors for f j and f V ; whilewe impose normal priors on the slopes ( α , β ), normalisation( f ) and scale velocity ( V ) such that the global star forma-tion e ffi ciency follows the results of the abundance matchingmodel by M +
13. In order to properly account for the sharpmaximum of f M at M h ≈ × M (cid:12) , we slightly modify thefunctional form of f M = f M ( V flat ) as f M = f (cid:32) V flat V (cid:33) α (cid:34) + (cid:32) V flat V (cid:33) γ (cid:35) β − α , (13)where γ = M h ∝ V .While the ansatz (i) was chosen because it is the simplestpossible, with the smallest number of free parameters, the func-tional form and priors adopted in cases (ii) and (iii) were inspiredby many results obtained using di ff erent methods on the stellar-to-halo mass relation (see Wechsler & Tinker 2018, and refer-ences therein). Thus, in case (ii) we allow f M , but also f j and f V ,to follow the double power-law functional form, which is typi-cally used to parametrise how f M varies for galaxies of di ff erentmasses; while in case (iii) we additionally impose priors on the f M parameters, following the results of one of the most popularstellar-to-halo mass relations (M + V is the onlyparameter that we assume to be the same for f M , f j , and f V .The reason is mainly statistical, as the data are not informa-tive enough to disentangle between breaks occurring at di ff erent V for di ff erent fractions. The observed scaling relations carryenough statistical information to distinguish only basic trends(for instance, whether or not there is a peak in f M , f j , and / or f V ) Article number, page 5 of 17 & A proofs: manuscript no. main and cannot really discriminate between detailed, degenerate be-haviours. Moreover, both f j and f V are thought to be physically,closely related to f M (e.g. Navarro & Steinmetz 2000; Cattaneoet al. 2014; Posti et al. 2018b), so it makes sense to investigatea scenario in which they have a transition at the same physicalgalaxy mass scale. In what follows, we dub the models (i)-(ii)-(iii) as linear, double power law and M +
13 prior, respectively.Finally, we note that we also tried letting free the parametergoverning the sharpness of the transition of the two power-lawregimes; i.e. γ in Eq. (13). Again, we find that the data do nothave enough information to constrain this variable, thus we de-cided to fix it to γ = γ =
3, as in the M +
13 prior model)yields similar results to those presented below.
In all models we allow the three fractions f M , f j , and f V to havea non-null intrinsic scatter σ . This parameter has an importantphysical meaning, as it encapsulates all the physical variationsof the complex processes that lead to the formation of galaxies.The information on this parameter comes from the intrinsic ver-tical scatter (at fixed V flat ) observed in the three di ff erent scalingrelations considered in this work. All of the measured scatters σ log M (cid:63) , σ log R d , and σ log j (cid:63) are given by the combination of theintrinsic scatter of f V with that of f M or f j . This combination isclearly degenerate and the information encoded in the data is notenough to distinguish the two of them . Henceforth, for simplic-ity we assume that the intrinsic scatter is the same for all threefractions.With this simplifying assumption, the scatter on log f ( σ log f ) is related to the observed intrinsic vertical scatter of thethree scaling relations as σ log M (cid:63) = √ σ log f , (14) σ log R d = σ log j (cid:63) = (cid:113) σ f + σ λ , (15)where σ log λ ≈ .
22 dex is the known scatter on the halo spinparameter. These formulae come from standard propagation ofuncertainties in Eqs. (8)-(10), where only the non-null intrinsicscatters of the fractions f and the halo spin parameter λ are con-sidered. An additional free parameter in every model we triedis σ log f ; thus, all in all, model (i) has 7 free parameters, whilemodels (ii)-(iii) have 11 free parameters. We use Bayesian inference to derive posterior probabilities ofthe free parameters ( θ ) in our three sets of models, i.e. P ( θ | V flat , M (cid:63) , R d , j (cid:63) ) ∝ P ( V flat , M (cid:63) , R d , j (cid:63) | θ ) P ( θ ) , (16)where ( V flat , M (cid:63) , R d , j (cid:63) ) are the data, P ( θ ) is the prior, and P ( V flat , M (cid:63) , R d , j (cid:63) | θ ) is the likelihood. The prior is uninforma-tive (flat) for all free parameters, except in model (iii) where it isnormal for the four parameters describing f M where means and Indeed, we tried letting free both the scatter of f V and that of f M or f j finding a non-flat posterior in only one of the two, which happens tobe compatible with the value we quote in Tab. 1 standard deviations have been taken from the abundance match-ing model of M +
13. The likelihood is defined as a sum of stan-dard χ , i.e.ln P ( V flat , M (cid:63) , R d , j (cid:63) | θ ) = ln P M + ln P R + ln P j , (17)whereln P M = − N (cid:88) i = (cid:104) M (cid:63) − M (cid:63) ( V flat ) Eq . (8) (cid:105) σ M (cid:63) + δ M (cid:63) −
12 log (cid:104) π (cid:16) σ M (cid:63) + δ M (cid:63) (cid:17)(cid:105) , (18)ln P R = − N (cid:88) i = (cid:104) R d − R d ( V flat ) Eq . (9) (cid:105) σ R d + δ R d −
12 log (cid:104) π (cid:16) σ R d + δ R d (cid:17)(cid:105) , (19)ln P j = − N (cid:88) i = (cid:104) j (cid:63) − j (cid:63) ( V flat ) Eq . (10) (cid:105) σ j (cid:63) + δ j (cid:63) −
12 log (cid:104) π (cid:16) σ j (cid:63) + δ j (cid:63) (cid:17)(cid:105) , (20)and δ M (cid:63) , δ R d , and δ j (cid:63) are the measurement uncertainties on therespective quantities. We note that this likelihood does not ac-count for the observational uncertainties on V flat , which are muchsmaller than those on the other observable quantities. This im-plies that the intrinsic scatter σ log f that we fit is vertical and thatit is greater or equal to the intrinsic perpendicular scatter.Given these definitions, we construct the posterior P ( θ | V flat , M (cid:63) , R d , j (cid:63) ) with a Monte Carlo Markov Chainmethod (MCMC; and in particular with the python implemen-tation by Foreman-Mackey et al. 2013). In each of the threecases (i)-(ii)-(iii), we define the “best model” to be the modelthat maximises the log-likelihood.Finally, we assess which one between the three best mod-els is preferred by the data using standard statistical informationcriteria: the Akaike information criterion (AIC) and Bayesian in-formation criterion (BIC). These are meant to find the best statis-tical compromise between goodness of fit (high ln P ) and modelcomplexity (less free parameters), in such a way that any gainin having a larger likelihood is penalised by the amount of newfree parameters introduced. The preferred model is then chosenas that with the smallest AIC and BIC amongst those explored.
4. Results
We modelled the observed M (cid:63) − V flat , R d − V flat and j (cid:63) − V flat relations with the three models described in Sec. 3.3. We havefound the best model, defined as the maximum a posteriori, in thethree cases (i)-(ii)-(iii) and we show how they compare with theobservations in Figure 2. In each row of this Figure we show oneof the three scaling relations considered; while in each columnwe present the comparison of the data with the three models.Table 1 summarises the posterior distributions that we derive forthe parameters of the three models (with their 16th-50th-84thpercentiles).We only fitted the data for galaxies which have a flat rotationcurve according to the definition in Sec. 2 (i.e. black-, grey- and Article number, page 6 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations
Table 1.
Posterior distributions of the parameters of the three mod-els considered in this study. The three columns are for the linear(Eq. 11), double power law (Eq. 12) and M +
13 prior models, respec-tively (Eq. 13). The four row blocks, instead, refer to the retained frac-tion of angular momentum f j , the star formation e ffi ciency f M , the ratioof asymptotic-to-virial velocity f V , and their intrinsic scatter σ log f . Theposteriors of the parameters are all summarised with their 16th-50th-84th percentiles. linear double power-law M +
13 priorlog f , j − . + . − . . + . − . . + . − . V / km s − – 63000 + − + − α j . + . − . . + . − . . + . − . β j – 1 + − − . + . − . log f , M − . + . − . . + . − . − . + . − . α M . + . − . − . + . − . . + . − . β M – 0 + − . + . − . log f , V . + . − . . + . − . . + . − . α V . + . − . . + . − . . + . − . β V – − + − − . + . − . σ log f . + . − . . + . − . . + . − . Table 2.
Goodness of fit of the three best models.
Model ln P max ∆ AIC ∆ BIClinear − . − . . . +
13 prior − . . . j (cid:63) − V flat planewhere the best M +
13 prior model seems to favour slightly higherangular momentum dwarfs than observed. We report in Table 2the values of the maximum-likelihood models in the three cases.While the general trend of stellar mass, size, and specific an-gular momentum as a function of V flat is well captured by thethree best models, the inferred intrinsic vertical scatters of thethree scaling relations are also well reproduced. While we mea-sure a vertical scatter of 0 .
21, 0 . , and 0 .
23 dex for the observed M (cid:63) − V flat , R d − V flat and j (cid:63) − V flat relations, respectively (with atypical uncertainty of 0.03 dex); the vertical intrinsic scatters ofthe three scaling laws predicted by the three best models (withEqs. 14-15) are written as( σ log M (cid:63) , σ log R d , σ log j (cid:63) ) = (0 . , . , . (linear) ( σ log M (cid:63) , σ log R d , σ log j (cid:63) ) = (0 . , . , . (double power law) ( σ log M (cid:63) , σ log R d , σ log j (cid:63) ) = (0 . , . , . (M +
13 prior) with a typical uncertainty of 0.02 dex. The scatter of M (cid:63) in themodels perfectly matches the observed scatter, while it is slightlylarger for R d and j (cid:63) albeit being consistent within the uncertain-ties. It is well known that the observed scatter on j (cid:63) for galac-tic discs is smaller than that expected only from the distributionof halo spin parameters (Romanowsky & Fall 2012), which al-ready suggests that the intrinsic scatter of f j has to be particu-larly small and also that the scatter of λ likely correlates withthat of other properties of the galaxy–halo connection (e.g. f j or f V ; see Posti et al. 2018b). Interestingly despite the di ff erencesin the three models of Sec. 3.3, we find consistently in all casesthat the preferred value of the intrinsic scatter on the three fun-damental fractions f M , f j , and f V is σ log f = . ± .
01 dex.This small scatter indicates that the galaxy–halo connection isextremely tight in disc galaxies, independently of their complexformation process. The connection with baryons is likely to beeven tighter than with stars, as hinted by the very small scatter ofthe baryonic Tully-Fisher relation. This means that studying theobserved baryonic fractions instead of stellar fractions should beparticularly illuminating in the future.Of the three best models that we have found, the doublepower law model has the highest likelihood. This is not surpris-ing, as this model has the most freedom to adapt to the observeddata. Employing the statistical criteria of both AIC and BIC, itturns out that the gain in a larger value of the likelihood does notstatistically justify the inclusion of four more free parameterswith respect to the linear model. On the other hand, the M + ff erencewith respect to the linear model. Thus, we have to conclude thatto fit the current observations of the scaling laws of nearby discs,any model more complex than a single power law statistically re-sults in an overfit. These results are summarised in Tab. 2. In Figure 3 we show the predictions of the three best models (oneach column) of the three fundamental fractions, respectively f j , f M , and f V , as a function of V flat (on each row). The most im-portant and most striking result to notice is that the predictionsof the three fractions behave similarly in the linear and doublepower-law models. For the vast majority of galaxies the predic-tions of these two models, which are by far statistically preferredto the M +
13 prior model, are in remarkable agreement, consid-ering that they have very di ff erent functional forms and degreesof freedom. The fact that the agreement is so detailed in f M , f j ,and f V ensures that the result is robust and confirms that the datahave enough information to infer these fractions. This can, thus,be regarded as a major success of the modelling approach pre-sented in this work.Along the same lines, another interesting result is that evenwhen allowing the behaviour of the three fractions to changeslope at a characteristic velocity (log V ), i.e. the parameters pre-ferred by the data, f M and f V do not have a significant break atthe scale of Milky Way-sized galaxies. This is a key predictionof abundance matching models. Considering that the best M + V ≈
125 km / s is statistically dis-favoured, we conclude that the observed scaling laws of nearbydiscs do not provide clear indications of any break in the be- Article number, page 7 of 17 & A proofs: manuscript no. main M ? / M (cid:12) linear model double power-law model M ? − V fl a t r e l a t i o n M+13 prior model R d / k p c − R d − V fl a t r e l a t i o n V flat / km s − j ? / k p c k m s − V flat / km s − V flat / km s − j ? − V fl a t r e l a t i o n Fig. 2.
Comparison of the three best models obtained with di ff erent assumptions on f j , f M , and f V with the data from SPARC galaxies (circles)and LITTLE THINGS galaxies (diamonds). Each column shows the fits for a given model, following the assumptions in Sec. 3.3. The top row isfor the stellar Tully-Fisher relation, the middle row is for the size-velocity relation, while the bottom row is for the angular momentum-velocityrelation. The white filled points in the plots are the galaxies which do not satisfy the Lelli et al. (2016b) criterion on the flatness of their rotationcurves. The yellow filled points in the j (cid:63) − V flat relation are the 92 SPARC galaxies with ‘onverged j (cid:63) profiles, following Posti et al. (2018a). haviour of the fundamental fractions at the scale of L ∗ galaxies(e.g. McGaugh et al. 2010).Both the best linear and double power-law models have aglobal star formation e ffi ciency which grows monotonically withgalaxy mass, approximately as M / (cid:63) . Henceforth, the most ef-ficient galaxies at forming stars are the most massive spirals( M (cid:63) (cid:38) M (cid:12) , V flat (cid:38)
250 km / s), qualitatively confirming pre-vious results on detailed rotation curve decomposition (PFM19,see Sec. 4.3 for a more in-depth comparison). We also note thatthe most massive spirals in both models have f M ∼ f b , which implies that these systems have virtually no missing baryons(PFM19).The retained fraction of angular momentum is, on the otherhand, remarkably constant ( f j ≈ .
6) over the entire rangeprobed by our galaxy sample ( ∼ ∼ f M and f j ,we are now able to cast new light on why disc galaxies todayhave comparable angular momenta to those of their dark haloes.Since the slopes of the power-law j − M relations for galaxiesand haloes are nearly the same within the uncertainties ( ∼ / j (cid:63) ∝ f j f − / M M / (cid:63) it follows that the factor f j f − / M Article number, page 8 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations − f j ≡ j ? / j h linear model − double power-law model − M+13 prior model − − − − − f M ≡ M ? / M h − − − − − − − − − − V flat / km s − − f V ≡ V fl a t / V h V flat / km s − − V flat / km s − − . . . . . . . . . . Fig. 3.
Model predictions. We show how f j (top row), f M (middle row), and f V (bottom rows) vary as a function of V flat for the three best models(columns). In the middle row the dot-dashed line shows the value of the cosmic baryon fraction f b = . f V =
1. The insets show a zoom-in of the plots in linear scale. has to be nearly constant with mass (e.g. Romanowsky & Fall2012, their Eq.s 15-16). This implies that the retained fractionof angular momentum has to correlate with the global star for-mation e ffi ciency (log f j ∝ log f M ) to reproduce the observedscalings. Most of the earlier investigations on f j found that itwas nearly constant with mass (Dutton & van den Bosch 2012;Romanowsky & Fall 2012; Fall & Romanowsky 2013, 2018),since they all adopted a monotonic f M (from Dutton et al. 2010).Posti et al. (2018b), instead, used di ff erent models for the stellar-to-halo mass relation to derive f j as a function of mass suchthat the observed Fall relation was reproduced. Since most ofthe contemporary and popular models for f M = f M ( M (cid:63) ) have a bell shape, the constraint log f j ∝ log f M led these authors toconclude that a bell-shaped f j = f j ( M (cid:63) ) was also favoured. This,in turn, implies for instance that dwarfs should have significantlysmaller f j than L ∗ galaxies (e.g. El-Badry et al. 2018; Marshallet al. 2019). However, the recent halo mass estimates by PFM19indicated that spirals are following a simpler stellar-to-halo massrelation, roughly f M ∝ M / (cid:63) . This, together with the constraint f j f − / M ≈ const . implies a very weak dependence of f j on stellarmass, roughly f j ∝ M / (cid:63) . The comprehensive analysis presentedin this paper confirms this and points towards an even weakerdependence of f j on mass, which is consistent with this valuebeing constant ( f j ≈ .
6) within the scatter.
Article number, page 9 of 17 & A proofs: manuscript no. main M ? / M (cid:12) − − − − f M ≡ M ? / M h cosmic baryon fraction Fig. 4.
Global star formation e ffi ciency f M ≡ M (cid:63) / M h as a function of M (cid:63) for the SPARC and LITTLE THINGS galaxies. The measurementsof the halo masses come from PFM19 and Read et al. (2017), respec-tively. Symbols are the same as in Fig. 2. The red line indicates the f M − M (cid:63) relation derived in the linear model for guidance. The velocity fraction f V is found to be always compatiblewith unity in the linear and double power-law models. The ex-pression f V ≈ f j , f V also turns outto depend substantially on f M . If f M is monotonic then f V is alsomonotonic and close to unity; while, if f M has a non-monotonicbell shape, then f V also follows a similar behaviour, rapidlyplunging below unity for both dwarfs and high-mass spirals.Considering a Tully-Fisher relation of the type V flat ∝ M δ(cid:63) andwriting V flat ∝ f V f − / M M / (cid:63) (see e.g. Appendix B), then it fol-lows that f V ∝ f / M M δ − / (cid:63) , which means that roughly f V ∝ f / M since the extra dependence on M (cid:63) is very weak ( δ (cid:39) . − . f M has a bell shape as expectedfrom abundance matching models, then also f V will have a simi-lar shape (e.g. Cattaneo et al. 2014; Ferrero et al. 2017). With ourcomprehensive analysis we find that such models are statisticallydisfavoured by the data, which instead favour a monotonicallyincreasing f M and a roughly constant f V ≈
1. We can now con-clude that our results provide a simple and appealing explanationto why the observed scaling laws are single, unbroken powerlaws: the galaxy–halo connection is linear and the fractions (6)are single-slope functions of velocity (or mass), instead of beingcomplicated non-monotonic functions which, when combined asin Eqs. (8)-(10), conspire to yield power-law scaling relations.To make sure that these results are not valid only for theSPARC + LITTLE THINGS sample we considered, we repeatedthe same analysis on the much larger galaxy sample fromCourteau et al. (2007). This sample contains about 1300 spiralgalaxies found in di ff erent environments and it has a higher com-pleteness than SPARC. However, the mass range is more limited(8 (cid:46) log M (cid:63) / M (cid:12) (cid:46) .
7) and the data quality is poorer, sinceit relies on optical (H α ) rotation curves, the disc scale lengthsare typically more uncertain and we have to use estimator (2) to compute j (cid:63) for all galaxies. Nevertheless, when we built thethree scaling relations and we fitted the three models, we arrivedat basically the same main conclusions as above: the linear anddouble power-law models have very similar predictions for f j , f M , and f V and they are statistically preferred to the M +
13 priormodel. Thus, from this test we conclude that the results we in-ferred on the fundamental fractions using the SPARC + LITTLETHINGS sample are generally applicable for all regularly rotat-ing disc galaxies.Finally, we note that assuming a linear or double power-lawfunctional form for the behaviour of the three fractions as a func-tion of V flat does not bias our results. We tested this by fitting anon-parametric model, where we do not assume any functionalform for the behaviour of f j , f M , and f V as a function of V flat .Instead, we bin the range in V flat spanned by the data with fivebins of di ff erent sizes, such that the number of galaxies per binis roughly equal. We, thus, constrained the five values of f j , f M ,and f V , together with the intrinsic scatter σ log f , for a grand to-tal of 16 degrees of freedom. The resulting predictions on thethree fractions are very well compatible with those of the lin-ear or double power-law models; we show these predictions inAppendix C. f M with detailed rotationcurve decomposition The three best models that we fitted to the stellar Tully-Fisher,size-mass, and Fall relations, directly predict the virial massesof the dark matter haloes hosting these spirals. Luckily all thesegalaxies have good enough photometric and kinematic data to al-low for an accurate decomposition of their observed HI rotationcurve, which can be used to get a robust measurement of theirhalo masses. In particular, PFM19 and Read et al. (2017) havecarefully performed fits to the observed rotation curves for theSPARC and LITTLE THINGS samples, respectively, and haveprovided measurements of V h . We show in Figure 4 the mea-surements of f M for these galaxies. Since these measurementsrely on fits of the dark matter halo profile and since they havenot been used in the model fit carried out in this paper, we cannow check a posteriori if the predictions of our three best modelsagree with the global shape of the halo profile inferred from theHI rotation curves.We show this comparison in Figure 5, in which we plot theobserved V flat against the V h measured from the rotation curvedecomposition; predictions from the three best models are su-perimposed. The predictions of the linear model are by far inbest agreement with the measurements.The double power law is in a similar remarkable agreementfor all galaxies. From this check we conclude that these twomodels both provide a good description of the observed discgalaxy population, but with a preference for the linear modelfrom a statistical point of view, i.e. from the standard statisti-cal criteria AIC and BIC. On the other hand, the M +
13 priormodel manifestly fails at reproducing the measured distributionof galaxies in the V flat − V h plane, both at low masses and, possi-bly, at high masses. According to the predictions of this model,both dwarfs and very massive spirals should inhabit much moremassive dark matter haloes than what it is suggested from theirHI rotation curves. This has already been noted and dubbed the“too big to fail" problem in the field (Papastergis et al. 2015).Thus, we conclude that a simple empirical model (of the typeEqs. 8-10), in which all disc galaxies follow a stellar-to-halomass relation which has a peak at M (cid:63) ∼ × M (cid:12) , predictsgalaxy formation fundamental parameters that are discrepant Article number, page 10 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations V flat / km s − V h / k m s − linear model V flat / km s − double power-law model V flat / km s − M+13 prior model
Fig. 5.
Comparison of the predictions of the three models in the V flat − V h plane, with data for the SPARC (circles) and LITTLE THINGS galaxies(diamonds). The halo virial velocities have been obtained with a careful rotation curve decomposition by Read et al. (2017) for the LITTLETHINGS galaxies and by PFM19 for the SPARC galaxies. In all panels, the grey dot-dashed line is the 1:1 and the symbols are the same as inFig. 2. with measurements of the kinematics of cold gas in spirals. Asimple tight and linear galaxy–halo connection, in disagreementwith abundance matching, however fully cures this too big to failproblem.
5. Conclusions
In this paper we used the observed stellar Tully-Fisher, size-mass, and Fall relations of a sample of ∼
150 nearby disc galax-ies, from dwarfs to massive spirals, to empirically derive threefundamental parameters of galaxy formation: the global star for-mation e ffi ciency ( f M ), the retained fraction of angular momen-tum ( f j ), and the ratio of the asymptotic rotation velocity to thehalo virial velocity ( f V ).Under the usual assumption that the galaxy size is relatedto its specific angular momentum, we used an analytic modelto predict the distribution of discs in the mass-velocity, size-velocity, and angular momentum-velocity planes. We definedthree models with di ff erent parametrisations of how the threefundamental parameters vary as a function of asymptotic ve-locity (or galaxy mass): we thus tested a linear model, a dou-ble power-law model, and another with a double power-law be-haviour, but with prior imposed such that the model follows theexpectations from widely used abundance matching stellar-to-halo mass relations for the global star formation e ffi ciency (theM +
13 prior model).We find the best-fitting parameters in each of these modelsand their posterior probabilities performing a Bayesian analysis.We briefly summarise our main findings: – We find reasonably good fits of the observed scaling relationsin all three cases that we have tested. – By assuming that the intrinsic scatter is the same for allthree fundamental fractions (for computational simplicity),we find that this scatter has to be particularly small ( σ log f (cid:39) . ± .
01 dex) to account for the intrinsic scatters of thethree observed scaling relations. – We determined that the statistically preferred model is thatwhere the fundamental galaxy formation parameters varylinearly with galaxy velocity (or mass) using standard statis-tical criteria (AIC & BIC). On the other hand, the model with standard abundance matching priors (from M +
13) is largelydisfavoured by the data. We conclude that models where thegalaxy–halo connection is complex and non-monotonic sta-tistically provide an overfit to the structural scaling relationsof discs. – We empirically derive and show how the three fundamen-tal parameters vary as a function of galaxy rotation veloc-ity. We find that in the best-fitting linear and double power-law models the three fractions have a remarkable similarbehaviour, despite having completely di ff erent functionalforms. This ensures that the observed scaling laws really pro-vide a strong, data-driven inference on the galaxy–halo con-nection. – We confirm previous indications that the retained fraction ofangular momentum and the ratio of the asymptotic-to-virialvelocity strongly depend on the global star formation e ffi -ciency (e.g. Navarro & Steinmetz 2000; Cattaneo et al. 2014;Posti et al. 2018b); in particular, they are non-monotoniconly if the latter is non-monotonic. – In the statistically preferred models, the retained fraction ofangular momentum is relatively constant across the entiremass range ( f j ∼ .
6) as is the ratio of the asymptotic-to-virial velocity ( f V ∼ ffi ciency is found to be a monotonically in-creasing function of mass, implying that the most e ffi cientgalaxies at forming stars are the most massive spirals (with f M ∼ f b ), whose star formation e ffi ciency has not beenquenched by strong feedback (the failed feedback problem). – Finally, we compared a posteriori the predictions of the threemodels with the dark matter halo masses found by Readet al. (2017) and PFM19 from the detailed analysis of ro-tation curves in the LITTLE THINGS and SPARC galaxysamples. We found that the M +
13 prior model is stronglyrejected since it significantly overpredicts the halo massesespecially at low V flat , but also at high V flat . This too big tofail problem (Papastergis et al. 2015) is fully solved in thelinear model, which best describes the measurements.Our analysis leads us to conclude that the statisticallyfavoured explanation to why the observed scaling laws of discsare single, unbroken power laws is the simplest possible: the Article number, page 11 of 17 & A proofs: manuscript no. main fundamental galaxy formation parameters for spiral galaxies aretight single-slope monotonic functions of mass, instead of beingcomplicated non-monotonic functions.The present study and the associated failed feedback prob-lem concern only disc galaxies. It is known that when includ-ing also spheroids, which dominate the galaxy population atthe high-mass end, the inferred galaxy-halo connection becomeshighly non-linear. In particular, it appears that there is a clear dif-ference in the stellar-to-halo mass relations for spirals and ellip-ticals at least at the high-mass end, as probed statistically usingmany observables (e.g. Conroy et al. 2007; Dutton et al. 2010;More et al. 2011; Wojtak & Mamon 2013; Mandelbaum et al.2016; Lapi et al. 2018). Thus, the results found in this work andthose of PFM19 could in principle be reconciled with conven-tional abundance matching if the galaxy-halo connection is madedependent on galaxy type. This can be achieved for instance ifdiscs and spheroids have significantly di ff erent formation path-ways, i.e. in accretion history, environment etc., which are stillencoded today in their di ff erent structural properties (e.g. alsoTortora et al. 2019). Whether this is the case in current simula-tions of galaxy formation, and whether the failed feedback prob-lem in massive discs can be addressed within those simulationsis the next big question to be asked.The model preferred by the SPARC and LITTLE THINGSdata has a monotonic f M approximately proportional to M / (cid:63) .With this global star formation e ffi ciency, it turns out that the re-tained fraction of angular momentum f j needs to be high and rel-atively constant for discs of all mass ( f j ≈ . − . f j ≈ const , a simple correspondence j (cid:63) ∝ j h is in remarkableagreement with the observations (Romanowsky & Fall 2012;Posti et al. 2018b). This implies that the current measurementsare compatible with a model in which discs have overall retainedabout all the angular momentum that they gained initially fromtidal torques. This is to be intended in an integrated sense in theentire galaxy: it can not simply have happened with every gaselement having conserved their angular momentum (sometimesreferred to as “detailed” or “strong” angular momentum conser-vation) because dark matter haloes and discs have completelydi ff erent angular momentum distributions today (Bullock et al.2001; van den Bosch et al. 2001). Thus, even if stars and darkmatter appear to have a simple correspondence j (cid:63) ∝ j h , it re-mains unclear and unexplained how the angular momentum ofgas and stars has redistributed during galaxy formation and whythe total galaxy’s specific angular momentum is still proportionalto that of its halo.Some disc galaxies, especially dwarfs, have huge cold gasreservoirs which sometimes dominate over their stellar budget.These systems are typically outliers of the (stellar) Tully-Fisher,but they instead lie on the baryonic Tully-Fisher relation, whichis obtained by replacing the stellar mass with the baryonic mass( M baryons = M (cid:63) + M HI , see e.g. McGaugh et al. 2000; Verheijen2001). More galaxies adhere to this relation, which is also tighterthan the stellar Tully-Fisher, suggesting that it is a more fun-damental law (e.g. Lelli et al. 2016b; Ponomareva et al. 2018).Thus, considering baryonic fractions instead of stellar fractions(Eq. 6) in a model such as that of Sec. 3 would presumably giveus deeper and more fundamental insight into how baryons cooledand formed galaxies. To do this, it is thus imperative to havea baryonic counterpart of the size-mass and Fall relations (e.g.Obreschkow & Glazebrook 2014; Kurapati et al. 2018), for asample of spirals su ffi ciently large in mass. We plan to report onthe latter soon, establishing first whether the observed baryonic Fall relation is tighter and more fundamental than the stellar Fallrelation. Acknowledgements.
We thank the anonymous referee for an especially care-ful and constructive report. LP acknowledges support from the Centre Nationald’Etudes Spatiales (CNES). BF acknowledges support from the ANR projectANR-18-CE31-0006.
References
Baillard, A., Bertin, E., de Lapparent, V., et al. 2011, A&A, 532, A74Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013, ApJ, 770, 57Bullock, J. S. & Boylan-Kolchin, M. 2017, ARA&A, 55, 343Bullock, J. S., Dekel, A., Kolatt, T. S., et al. 2001, ApJ, 555, 240Cattaneo, A., Salucci, P., & Papastergis, E. 2014, ApJ, 783, 66Conroy, C., Prada, F., Newman, J. A., et al. 2007, ApJ, 654, 153Cook, D. O., Dale, D. A., Johnson, B. D., et al. 2014, MNRAS, 445, 881Courteau, S., Dutton, A. A., van den Bosch, F. C., et al. 2007, ApJ, 671, 203Dalcanton, J. J., Spergel, D. N., & Summers, F. J. 1997, ApJ, 482, 659Desmond, H. & Wechsler, R. H. 2015, MNRAS, 454, 322Dutton, A. A., Conroy, C., van den Bosch, F. C., Prada, F., & More, S. 2010,MNRAS, 407, 2Dutton, A. A. & van den Bosch, F. C. 2012, MNRAS, 421, 608Dutton, A. A., van den Bosch, F. C., Dekel, A., & Courteau, S. 2007, ApJ, 654,27El-Badry, K., Quataert, E., Wetzel, A., et al. 2018, MNRAS, 473, 1930Fall, S. M. 1983, in IAU Symposium, Vol. 100, Internal Kinematics and Dynam-ics of Galaxies, ed. E. Athanassoula, 391–398Fall, S. M. & Efstathiou, G. 1980, MNRAS, 193, 189Fall, S. M. & Romanowsky, A. J. 2013, ApJ, 769, L26Fall, S. M. & Romanowsky, A. J. 2018, ApJ, 868, 133Ferrero, I., Navarro, J. F., Abadi, M. G., et al. 2017, MNRAS, 464, 4736Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125,306Ghari, A., Famaey, B., Laporte, C., & Haghi, H. 2019, A&A, 623, A123Hunter, D. A. & Elmegreen, B. G. 2006, ApJS, 162, 49Hunter, D. A., Ficut-Vicas, D., Ashley, T., et al. 2012, AJ, 144, 134Iorio, G., Fraternali, F., Nipoti, C., et al. 2017, MNRAS, 466, 4159Katz, H., Lelli, F., McGaugh, S. S., et al. 2017, MNRAS, 466, 1648Kormendy, J. 1977, ApJ, 218, 333Kravtsov, A. V. 2013, ApJ, 764, L31Kravtsov, A. V., Berlind, A. A., Wechsler, R. H., et al. 2004, ApJ, 609, 35Kurapati, S., Chengalur, J. N., Pustilnik, S., & Kamphuis, P. 2018, MNRAS, 479,228Lange, R., Mo ff ett, A. J., Driver, S. P., et al. 2016, MNRAS, 462, 1470Lapi, A., Salucci, P., & Danese, L. 2018, ApJ, 859, 2Leauthaud, A., Tinker, J., Bundy, K., et al. 2012, ApJ, 744, 159Lelli, F., Fraternali, F., & Verheijen, M. 2013, MNRAS, 433, L30Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016a, AJ, 152, 157Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016b, ApJ, 816, L14Lelli, F., McGaugh, S. S., Schombert, J. M., Desmond, H., & Katz, H. 2019,MNRAS, 484, 3267Lelli, F., McGaugh, S. S., Schombert, J. M., & Pawlowski, M. S. 2016c, ApJ,827, L19Mandelbaum, R., Seljak, U., Kau ff mann, G., Hirata, C. M., & Brinkmann, J.2006, MNRAS, 368, 715Mandelbaum, R., Wang, W., Zu, Y., et al. 2016, MNRAS, 457, 3200Marasco, A., Fraternali, F., Posti, L., et al. 2019, A&A, 621, L6Marshall, M. A., Mutch, S. J., Qin, Y., Poole, G. B., & Wyithe, J. S. B. 2019,arXiv e-prints [ arXiv:1904.01619 ]Martinsson, T. P. K., Verheijen, M. A. W., Westfall, K. B., et al. 2013, A&A, 557,A131McGaugh, S. S., Schombert, J. M., Bothun, G. D., & de Blok, W. J. G. 2000,ApJ, 533, L99McGaugh, S. S., Schombert, J. M., de Blok, W. J. G., & Zagursky, M. J. 2010,ApJ, 708, L14Mo, H., van den Bosch, F. C., & White, S. 2010, Galaxy Formation and Evolution(Cambridge University Press)Mo, H. J., Mao, S., & White, S. D. M. 1998, MNRAS, 295, 319More, S., van den Bosch, F. C., Cacciato, M., et al. 2011, MNRAS, 410, 210Moster, B. P., Naab, T., & White, S. D. M. 2013, MNRAS, 428, 3121Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563Navarro, J. F. & Steinmetz, M. 2000, ApJ, 538, 477Obreschkow, D. & Glazebrook, K. 2014, ApJ, 784, 26Oman, K. A., Marasco, A., Navarro, J. F., et al. 2019, MNRAS, 482, 821Oman, K. A., Navarro, J. F., Fattahi, A., et al. 2015, MNRAS, 452, 3650Papastergis, E., Giovanelli, R., Haynes, M. P., & Shankar, F. 2015, A&A, 574,A113 Article number, page 12 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations
Papastergis, E., Martin, A. M., Giovanelli, R., & Haynes, M. P. 2011, ApJ, 739,38Peebles, P. J. E. 1969, ApJ, 155, 393Pezzulli, G., Fraternali, F., & Binney, J. 2017, MNRAS, 467, 311Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018, ArXiv e-prints[ arXiv:1807.06209 ]Ponomareva, A. A., Verheijen, M. A. W., Papastergis, E., Bosma, A., & Peletier,R. F. 2018, MNRAS, 474, 4366Posti, L., Fraternali, F., Di Teodoro, E. M., & Pezzulli, G. 2018a, A&A, 612, L6Posti, L., Fraternali, F., & Marasco, A. 2019, A&A, 626, A56Posti, L., Nipoti, C., Stiavelli, M., & Ciotti, L. 2014, MNRAS, 440, 610Posti, L., Pezzulli, G., Fraternali, F., & Di Teodoro, E. M. 2018b, MNRAS, 475,232Read, J. I., Iorio, G., Agertz, O., & Fraternali, F. 2017, MNRAS, 467, 2019Romanowsky, A. J. & Fall, S. M. 2012, ApJS, 203, 17Schombert, J. & McGaugh, S. 2014, PASA, 31, 36Shi, J., Lapi, A., Mancuso, C., Wang, H., & Danese, L. 2017, ApJ, 843, 105Taylor, V. A., Jansen, R. A., Windhorst, R. A., Odewahn, S. C., & Hibbard, J. E.2005, ApJ, 630, 784Tortora, C., Posti, L., Koopmans, L. V. E., & Napolitano, N. R. 2019, arXiv e-prints [ arXiv:1902.10158 ]Tully, R. B. & Fisher, J. R. 1977, A&A, 54, 661Vale, A. & Ostriker, J. P. 2004, MNRAS, 353, 189van den Bosch, F. C., Burkert, A., & Swaters, R. A. 2001, MNRAS, 326, 1205van den Bosch, F. C., Norberg, P., Mo, H. J., & Yang, X. 2004, MNRAS, 352,1302van der Kruit, P. C. & Freeman, K. C. 2011, ARA&A, 49, 301van Uitert, E., Cacciato, M., Hoekstra, H., et al. 2016, MNRAS, 459, 3251Verheijen, M. A. W. 2001, ApJ, 563, 694Wechsler, R. H. & Tinker, J. L. 2018, ARA&A, 56, 435Wojtak, R. & Mamon, G. A. 2013, MNRAS, 428, 2407Yang, X., Mo, H. J., & van den Bosch, F. C. 2008, ApJ, 676, 248Zheng, Z., Coil, A. L., & Zehavi, I. 2007, ApJ, 667, 760
Appendix A: Size and angular momentum fractionsin disc galaxies
The specific angular momentum of an exponential disc with aflat rotation curve V flat = f V V h is j (cid:63) = (cid:82) d R R exp ( − R / R d ) V flat (cid:82) d R R exp ( − R / R d ) = R d f V V h . (A.1)From Eqs. (4)-(5) and introducing f j and f R as in Eq. (6), wehave R d = f R H (cid:114) ∆ V h ; (A.2) j (cid:63) = λ f j H √ ∆ V . (A.3)Plugging Eq. A.1 into Eq. A.3 we obtain R d = f j λ f V H √ ∆ V h , (A.4)which, using Eq. A.2, can be rearranged as f R = λ √ f j f V . (A.5)A very similar relation to this was already derived by Fall & Ef-stathiou (1980) and Mo et al. (1998), who started by assumingthat j (cid:63) = R d V h to replace Eq. (A.1) and got f R = λ f j / √ f j , f M , and f V , thus we prefer to use the formulationin Eq. (A.5), which allows the flat asymptotic circular velocity V flat to di ff er from the halo virial velocity V h . We have, however,checked that using f R = λ f j / √ Appendix B: Fitting f j , f M , and f V as a function of M (cid:63) In this Appendix we demonstrate that considering M (cid:63) as theindependent observable, and thus fitting the canonical Tully-Fisher, size-mass, and Fall relations, yields similar predictionsfor the fractions f j , f M , and f V to what we obtained above.We start from the equations for dark matter, i.e. V h = (cid:114) ∆ GHM h / ; (B.1) R h = (cid:32) GM h ∆ H (cid:33) / ; (B.2) j h = λ ( ∆ H ) / (2 GM h ) / . (B.3)After introducing the three fractions, we have V flat = f V (cid:114) ∆ GHM (cid:63) f M / ; (B.4) Article number, page 13 of 17 & A proofs: manuscript no. main R d = λ f j f V GM (cid:63) √ ∆ H f M / ; (B.5) j (cid:63) = λ f j ( ∆ H ) / (cid:32) GM (cid:63) f M (cid:33) / . (B.6)The three equations above are used to fit the observations in the V flat − M (cid:63) , R d − M (cid:63) and j (cid:63) − M (cid:63) diagrams.Similar to Sec. 3.3, we try three di ff erent models as follows: (i) the linear model, wherelog f = α log M (cid:63) / M (cid:12) + log f . (B.7) (ii) the double power-law model, where f = f (cid:32) M (cid:63) M (cid:33) α (cid:32) + M (cid:63) M (cid:33) β − α . (B.8) (iii) the M +
13 prior” model, which is the same as case (ii), butwith priors on f M from the abundance matching model ofM + / model comparisons and the constraints on the fun-damental fractions, respectively. The behaviour of the modelsis largely identical to those in the main text; the only notice-able di ff erence is that the double power-law model now has aslight break at low masses (at ∼ M (cid:12) ). However the statisti-cal significance of this break is very low since it is driven byjust a few data points in the dwarf regime, where the uncer-tainties are higher. On the other hand, for this model we stillfind no indications of a break at around L ∗ galaxies. Finally, wenote that also in this case we find that the statistically preferredmodel is the linear model, according to the AIC & BIC criteria:( ∆ AIC , ∆ BIC) = (1 . , .
3) with respect to the double power-law model and ( ∆ AIC , ∆ BIC) = (28 . , .
1) with respect to theM +
13 prior model.
Appendix C: Non-parametric model
In this Appendix, we describe a model in which the variation of f j , f M , and f V as a function of V flat has a completely free form.This model is aimed to test whether the functional forms that wehave chosen in Sec. 3.3 are too restrictive for the data that weconsidered, and if the data themselves are informative enough toconstrain a di ff erent behaviour.We binned the range in V flat spanned by the data, [15,320]km / s, into five bins of di ff erent sizes, such that the number ofgalaxies in each bin is roughly equal. We, then, constrained thefive (constant) values of f j , f M , and f V in each bin maximisingthe same likelihood as in Sec. 3.5. Together with the intrinsicscatter σ log f , this model has a total of 16 degrees of freedom.Figure C.1 shows the three fractions in this model as a func-tion of V flat . A part from a small di ff erence in the lowest V flat bin, where dwarf galaxies with the highest uncertainties domi-nate, the predictions of this model are in remarkable agreementwith those of the linear and double power-law models. Also theintrinsic scatter that we fit with this model is very well compara-ble with that of the other cases, where σ log f = . ± .
05. Thisensures that the linear or double power-law functional forms thatwe adopted for our fiducial models are not too restrictive for thedata that we have at hand.
Table B.1.
Posterior distributions for the three models with M (cid:63) as themain independent observable quantity. This Table is analogous to Tab. 1and the models and the equations fitted described in App. B. linear double power law M +
13 priorlog f , j − . + . − . − . + . − . . + . − . log M / M (cid:12) – 3 . + . − . . + . − . α j . + . − . + − . + . − . β j – 0 . + . − . − . + . − . log f , M − . + . − . − . + . − . − . + . − . α M . + . − . + − . + . − . β M – 0 . + . − . . + . − . log f , V . + . − . . + . − . . + . − . α V . + . − . − + − . + . − . β V – 0 . + . − . − . + . − . σ log f . + . − . . + . − . . + . − . Article number, page 14 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations V fl a t / k m s − linear model double power-law model T u ll y − F i s h e rr e l a t i o n M+13 prior model R d / k p c S i ze − m a ss r e l a t i o n M ? / M (cid:12) j ? / k p c k m s − M ? / M (cid:12) M ? / M (cid:12) F a ll r e l a t i o n Fig. B.1.
Comparison of the three models described in Appendix B with the data on the Tully-Fisher, size-mass, and Fall diagrams. Curves andsymbols are analogous to Fig. 2. Article number, page 15 of 17 & A proofs: manuscript no. main − f j ≡ j ? / j h linear model − double power-law model − M+13 prior model − − − − f M ≡ M ? / M h − − − − − − − − M ? / M (cid:12) × − × − × − × − × f V ≡ V fl a t / V h M ? / M (cid:12) × − × − × − × − × M ? / M (cid:12) × − × − × − × − × . . . . . . . . Fig. B.2.
Behaviour of the three fractions as a function of M (cid:63) in the three models of Appendix B. Curves and symbols are analogous to Fig. 3.Article number, page 16 of 17. Posti, A. Marasco, F. Fraternali & B. Famaey: Galaxy disc scaling relations V flat / km s − − f j ≡ j ? / j h V flat / km s − − − − − − f M ≡ M ? / M h V flat / km s − − f V ≡ V fl a t / V h . . . . . Fig. C.1.
Fundamental fractions as a function of the asymptotic rotation velocity for the non-parametric model (gold dot-dashed lines, with theband encompassing the intrinsic scatter σ log f ). The three fractions are binned in 5 bins in V flatflat