Galaxy spin as a formation probe: the stellartohalo specific angular momentum relation
Abstract
We derive the stellartohalo specific angular momentum relation (SHSAMR) of galaxies at by combining i) the standard CDM tidal torque theory ii) the observed relation between stellar mass and specific angular momentum (Fall relation) and iii) various determinations of the stellartohalo mass relation (SHMR). We find that the ratio of the specific angular momentum of stars to that of the dark matter i) varies with mass as a double powerlaw, ii) it always has a peak in the mass range explored and iii) it is times larger for spirals than for ellipticals. The results have some dependence on the adopted SHMR and we provide fitting formulae in each case. For any choice of the SHMR, the peak of occurs at the same mass where the stellartohalo mass ratio has a maximum. This is mostly driven by the straightness and tightness of the Fall relation, which requires and to be correlated with each other roughly as , as expected if the outer and more angular momentum rich parts of a halo failed to accrete onto the central galaxy and form stars (biased collapse). We also confirm that the difference in the angular momentum of spirals and ellipticals at a given mass is too large to be ascribed only to different spins of the parent darkmatter haloes (spin bias).
keywords:
galaxies: fundamental parameters  galaxies: formation  galaxies: haloes  galaxies: spiral  galaxies: elliptical and lenticular, cD1 Introduction
Mass and angular momentum are amongst the most important quantities underlying a physically motivated classification of galaxies. Inspired by this belief, Fall (1983) first looked at how galaxies were distributed in the plane of specific angular momentum and stellar mass. This pioneering work (based on data from Rubin et al., 1980; Davies et al., 1983) led to the discovery of an important scaling relation, which we can now call the Fall relation(s): i) for a given morphological type, stellar mass and specific angular momentum of galaxies are related by a powerlaw, ii) spirals and ellipticals follow parallel sequences, with identical slopes but different normalizations, with ellipticals having approximately five times less specific angular momentum than spirals for a given stellar mass.
These findings were more recently confirmed by Romanowsky & Fall (2012, hereafter RF12), who compiled measurements for a sample of nearby galaxies of all Hubble types for over three orders of magnitude in stellar mass, by Obreschkow & Glazebrook (2014), who measured angular momenta of stars, cold atomic and molecular gas in 16 nearby spirals and by Cortese et al. (2016), who extended the sample to nearby galaxies and studied both the stellar and warm gas specific angular momentum (though their measurements reached only one or two effective radii). Perhaps even more noteworthy is that the specific angular momentummass relation was recently measured also for starforming disc galaxies at high redshift () and it was found to be a powerlaw of similar slope and normalization as at with the residuals correlating to galaxy morphology, as observed for nearby galaxies (Burkert et al., 2016; Harrison et al., 2017). Remarkably, the simple nature of the Fall relation (a powerlaw over several decades, with little scatter) is virtually identical to a theoretical relation which is expected to hold for the dark matter haloes from Tidal Torque Theory (TTT, see e.g. Peebles, 1969; Efstathiou & Jones, 1979).
The simple correspondence, in the specific angular momentummass diagram, between stars and dark matter could be interpreted as a major theoretical success, if one assumed that both the mass and the angular momentum of a galaxy are directly proportional to those of their haloes (with some dependence on morphology, but none on the mass). Indeed, some of the earliest theoretical models of galaxy formation (e.g. Fall & Efstathiou, 1980; Dalcanton, Spergel, & Summers, 1997; Mo, Mao, & White, 1998) were based on these assumptions and turned out to be relatively successful in reproducing the observed structural properties and scaling relations of disc and spheroidal galaxies (for more recent similar models see e.g. Somerville et al., 2008; Dutton & van den Bosch, 2012; Kravtsov, 2013).
We now know, however, that at least one of the two assumptions above, i.e. that the stellar masstohalo mass ratio is constant with mass, is not correct. The stellartohalo mass relation (SHMR) is indeed rather complex and far from being linear as it emerged from several recent works, despite the different techniques used to estimate darkmatter halo masses. Consistent results were found in studies that made use of either the socalled abundance matching ansatz (see e.g. Vale & Ostriker, 2004; Moster, Naab, & White, 2013; Behroozi, Wechsler, & Conroy, 2013), sometimes combined with direct estimates via weak lensing and stellar or satellite kinematics (see e.g. Leauthaud et al., 2012; van Uitert et al., 2016), or the spatial clustering of galaxies (see e.g. Yang, Mo, & van den Bosch, 2003; Tinker et al., 2017) or the segregation by colour of galaxy populations (see e.g. Dutton et al., 2010; RodríguezPuebla et al., 2015). In the light of these findings, the simplicity of the observed Fall relation ceases to be a confirmation of expectations and rather starts to be a theoretical challenge. In particular, the following question naturally arises: what fraction of the specific angular momentum of haloes their galaxies need to have to be consistent simultaneously with the Fall relation and the SHMR? Addressing this question is the main aim of this paper.
From the observational point of view, only very few studies have tried to investigate somewhat similar questions: for instance, Kauffmann et al. (2015) have measured the specific angular momenta of massive gas rich galaxies from rotation curves, they have computed the fraction of stellartohalo specific angular momentum and found it to strongly correlate with galaxy mass. From the theoretical point of view, some models were already built upon the realisation that if the SHMR is not constant with mass, then galaxies should have a nonconstant fraction of the specific angular momentum of their haloes (e.g. Navarro & Steinmetz, 2000). More recently, also other authors have investigated similar questions to ours by means of semianalytical models (e.g. Dutton & van den Bosch, 2012; Stevens, Croton, & Mutch, 2016), as well as of cosmological simulations (Genel et al., 2015; Pedrosa & Tissera, 2015; Teklu et al., 2015; Zavala et al., 2016; Sokołowska et al., 2017). Despite going deep in physicallymotivated prescriptions, all these studies either did not really try or fell short of reproducing the straightness and the small scatter of the Fall relation across its entire mass range and many did not really compare the predictions of their models on the specific angular momentummass diagram and on the SHMR at the same time, hence leaving unanswered questions on the galaxyhalo connection.
In the present contribution, we adopt a complementary approach. Without assuming a specific scenario, we try to reconstruct the stellartohalo specific angular momentum relation (SHSAMR), as a function of mass and morphological type, by requiring the empirical Fall relation to be reproduced. We stress the inherent uncertainties of the process by systematically investigating the dependence of the results on the adopted SHMR. Then, we try to address some interesting related questions, such as i) whether the origin of the specific angular momentummass relation can be interpreted in terms of galaxies forming stars from the innermost and more angular momentumpoor part of the halo and ii) whether the different angular momenta between spirals and ellipticals can be ascribed to differences in the spin parameters of the host dark matter haloes.
The paper is organised as follows. In Section 2, we briefly review some observational and theoretical background concerning the angular momentum of the stellar and dark components of galaxies. In Section 3, we derive our SHSAMR for spirals and ellipticals, adopting different existing SHMR from the literature. In Section 4, we investigate the origin of the Fall relation with two physical models (biased collapse and spin bias). We summarize and conclude in Section 5.
Throughout the paper we use the standard Cold Dark Matter (CDM) model with the cosmological parameters estimated by the Planck Collaboration et al. (2016): are the matter and dark energy density and is the Hubble constant.
2 The angular momentum of galaxies and their haloes
Here we introduce the framework with the equations relevant to our work. We begin by rederiving the specific angular momentum of dark matter haloes in a standard CDM Universe and then we relate this to galaxies by introducing the key parameters of our models.
2.1 The specific angular momentum of dark matter haloes
The mass distribution of a spherically symmetric dark matter halo whose density distribution is described by a Navarro, Frenk, & White (1996, hereafter NFW) profile, is characterized by two parameters: the mass within the virial radius and the concentration .
At , the virial radius of a halo of mass is
(1) 
where is the virial overdensity w.r.t. to the critical density of the Universe (Bryan & Norman, 1998).
The massconcentration relation has been widely studied in the literature, especially by means of cosmological simulations (e.g. Navarro, Frenk, & White, 1997) which have shown that the concentration has a weak dependence on halo mass. In the following we adopt the relation found by Dutton & Macciò (2014)
(2) 
with an intrinsic scatter of about dex.
Dark matter haloes acquire rotation from tidal torques exerted by the surrounding matter (see e.g. Hoyle, 1949) and this is usually quantified with the socalled spin parameter
(3) 
where is the gravitational constant and , and are respectively the angular momentum, mass and energy of the system (see Peebles, 1969). Cosmological simulations have shown that is lognormally distributed and independent on halo mass (e.g. Efstathiou & Jones, 1979; Barnes & Efstathiou, 1987; Porciani, Dekel, & Hoffman, 2002). Here we adopt the results of Macciò et al. (2007), who found lognormal distributions with expectation value and scatter of about dex.
By inverting the definition of (equation 3), the specific angular momentum of a dark matter halo can be conveniently expressed as
(4) 
where
(5) 
is the halo circular velocity at the virial radius and
(6) 
is a dimensionless factor that depends on the structure of the halo. Following Mo, Mao, & White (1998, their equation 23), we write as a function of the concentration :
(7) 
Combining equations (1), (4) and (5) we find that at the specific angular momentum of a halo of mass is
(8) 
which is not very different from , since the dependence of on halo mass is very weak^{1}^{1}1 varies about in the mass range . compared to the powerlaw .
This is similar to the relation adopted by RF12, except for the fact that we use updated cosmological parameters and we take the dependence on the concentration into account.
2.2 The specific angular momentum of the stars
From equation (8), the average specific angular momentum of the stars in a galaxy can be written as
(9) 
where
(10) 
is the ratio of stellar mass to halo mass, while
(11) 
is the ratio of the average specific angular momentum of the stars in the galaxy to that of the dark matter halo^{2}^{2}2Equation (9) is also similar to the relation adopted by RF12, but for the adopted cosmology, the dependence on concentration and the fact that they normalize the stellar fraction by the cosmic baryon fraction..
The ratio of stellar mass to halo mass is sometimes loosely referred to as star formation efficiency, although it encapsulates a variety of very different processes, including e.g. the actual efficiency of the conversion of gas into stars, the fraction of baryons which ever collapses to the centre of the halo (e.g. Kassin et al., 2012), the cumulative effect of feedback from star formation and accretion on the central black hole (e.g. Veilleux, Cecil, & BlandHawthorn, 2005; Harrison, 2017) and, finally, the effect of galactic fountains (which can engage a significant portion of gas in circulating in the circumgalactic medium at various distances from the star forming body of the galaxy, e.g. Fraternali & Binney, 2008). Notice also that, by construction, is bound to be equal or smaller than the cosmic baryon fraction (Planck Collaboration et al., 2016).
Similarly, we will somewhat loosely refer to in equation (11) as the retained fraction of angular momentum. This parameter depends on many phenomena, such as the angular momentum losses due to, e.g., dynamical friction of baryonic massive clumps or induced by galaxy mergers (see e.g. Aarseth & Fall, 1980; Barnes, 1992; Hernquist & Mihos, 1995), but also on the possibility that galaxies form from the inner and less angular momentum rich portion of haloes (biased collapse, e.g. van den Bosch, 1998, RF12), partially counteracted by the opposite effect, namely the selective removal of low angular momentum material as a consequence of galactic winds (e.g. Governato et al., 2007; Brook et al., 2011), and finally, to some extent, angular momentum redistributions due to galactic fountains (e.g. Brook et al., 2012; DeFelippis et al., 2017). Both estimates from recent observations (e.g. Cortese et al., 2016) and expectations from numerical simulations (e.g. Teklu et al., 2015) and semianalytic models (e.g. Dutton & van den Bosch, 2012) agree that this fraction should be of the order of for all galaxies, even if in principle such fraction can be also larger than one.
As discussed in Section 1, observational determinations of the relation find a powerlaw dependence with index close to for each morphological type (the Fall relation). A comparison with equation (9) therefore suggests that the quantity
(12) 
does not vary much with stellar or halo mass, at least for a fixed morphological type. Amongst the ways to have constant there is the possibility that both and are constant (e.g. Mo, Mao, & White, 1998) or that and are correlated (e.g., Navarro & Steinmetz, 2000).
In contrast, any model with varying and constant as a function of galaxy mass are not expected to be in agreement with the data. This is illustrated in Figure 1, where we plot for spirals (left panel) and ellipticals (right panel) the predictions on the specific angular momentummass diagram of a model, given by equation (9), with variable and constant . For this example, we fixed the spin and concentration of the haloes to a typical value ( and ) and we use the widely adopted SHMR of Moster, Naab, & White (2013). Clearly, none of the curves shown provides a satisfactory description of the data. This indicates that is not constant, but rather varies with both mass and morphological type, being close to for lowmass spirals and about for the most massive galaxies of any type. This is true, at least, if both the Fall relation and the SHMR of Moster, Naab, & White (2013) are correct.
In the next Section we investigate the dependence of on galaxy mass and morphological type in detail and for different choices of the SHMR.
3 The retained fraction of angular momentum as a function of galaxy mass and type
3.1 Summary of the model
We now derive the SHSAMR as a function of galaxy mass and morphological type by imposing that both the Fall relation and a given SHMR are satisfied. We proceed as follows:

we draw a sample of galaxies from a uniform distribution^{3}^{3}3 Since we are interested in the behaviour of some scaling relations and not in galaxy/haloes number densities, we do not need to adopt a more realistic galaxy mass function. We work with a uniform mass function for clarity of the presentation of the results. in in the range ;

to each galaxy we assign a dark matter halo whose mass is drawn from a normal distribution in with mean and standard deviation given by the adopted SHMR (see Section 3.2);

we draw a spin parameter from a normal distribution in with mean and standard deviation of (see Section 2.1);

we draw a concentration from a normal distribution in whose mean and standard deviation are given by the as in equation (2);

we compute using equation (8);

we draw the stellar specific angular momentum from a normal distribution in with mean and standard deviation as estimated by RF12. In particular we use the two following estimates of the Fall relation:
(13) for latetype galaxies in the stellar mass range and
(14) for earlytype galaxies in , where is the root mean square scatter. ;

we compute .
Note that this procedure implicitly assumes that the scatters of all the adopted relations are uncorrelated. The implications of this assumption are discussed in Section 3.4.
For simplicity, in what follows we will use the terms latetype and earlytype galaxy as synonyms for spiral and elliptical galaxy. This implies, for instance, that we are assuming that the population of spirals and ellipticals of RF12 traces the population of blue and red galaxies when we adopt coloursegregated SHMR (Dutton et al., 2010; RodríguezPuebla et al., 2015). Also, we do not explicitly treat intermediate galaxy types, e.g. S0s, for which the relation would be intermediate between equations (13) and (14), see RF12. However, in this work we are interested in the main trends of the galaxy populations as a function of morphological type and mass and we expect this rather crude distinction to capture them.
3.2 The SHMR
We employ six different parametrisations of the SHMR amongst those available in the literature. We consider three types of SHMR and two examples per type.
The first two are derived by imposing that the cumulative mass function of haloes matches the observed cumulative galaxy stellar mass function, i.e. the socalled abundance matching technique (see e.g., Vale & Ostriker, 2004). We adopt in particular the two very popular models by Behroozi, Wechsler, & Conroy (2013) and Moster, Naab, & White (2013), which are constrained using the largest compilation of galaxy stellar masses available (including galaxies of all types) at different redshifts. We show the (mean) SHMR and the corresponding in Figure 2 (top panels). The scatter of these two relations is about dex
The second set of SHMR we consider is derived combining constraints from the abundance matching hypothesis together with empirical estimates of and for individual objects, e.g. from stellar, gas or satellite kinematics or from weak galaxygalaxy lensing. This should, in principle, lead to a better estimation of the global SHMR. Note, however, that including individual objects in the fit may bias the results towards a given population of galaxies (e.g. galaxies with wellbehaved gas discs if one uses gas kinematics or galaxies in crowded environments if one uses weak galaxygalaxy lensing). In this work we consider the two parametrizations of Leauthaud et al. (2012) and van Uitert et al. (2016), which we plot in Figure 2 (top panels). The scatter of these two relations is about dex and dex respectively.
Finally, we use two SHMR that have been estimated separately for galaxies with red and blue colours (either with pure abundance matching or by combining different constraints) and in particular those of Dutton et al. (2010) and RodríguezPuebla et al. (2015). The first attempt in this direction was that of Dutton et al. (2010). However, this study has a few limitations: i) the stellar mass range probed is small ( for spirals), ii) measurements of both abundance matching models and stellar/halo masses for individual galaxies have improved since and iii) the bestfitting function itself is found by eye. All these reasons probably lead to a rather flat estimate of for spirals galaxies if compared to those of many other works (see e.g. Figure 2). The more recent work by RodríguezPuebla et al. (2015) improved over these issues as i) they probe a much larger mass range , ii) they use updated data and methods to match the stellar/halo mass functions and iii) they use an automatic routine to find the bestfitting parameters of their model. We will, however, still consider the SHMR by Dutton et al. (2010) in order to be able to compare our results with those of Dutton & van den Bosch (2012), who first inferred the for spiral galaxies. We show these two SHMR segregated by galaxy type in Figure 2 (bottom panels).
3.3 The SHSAMR and the distribution of as a function of mass
Figure 3 shows the mean relation between the specific angular momentum of stars and dark matter (SHSAMR) that we obtain assuming six different parametrisations of the SHMR. This relation is computed separately for spiral and elliptical galaxies and in both cases it is found to be highly nonlinear (in ), irrespectively of the SHMR adopted. Similarly to the relation between stellar mass and halo mass, galaxies on average follow a rather steep SHSAMR at low, which becomes shallower for large galaxies found at higher specific angular momenta.
We now study the full distributions of the retained fractions as a function of halo mass (Figure 4) and stellar mass (Figure 5) for the six different SHMR. For each SHMR we plot the mean relation for late and earlytype galaxies together with a shaded area indicating its uncertainty given the current data, which results from the composition of the scatters in the observed , the SHMR, the and the distribution of . We discuss this point in detail in Section 3.4.
Model  

Late type galaxies  
Behroozi, Wechsler, & Conroy (2013)  
Moster, Naab, & White (2013)  
Leauthaud et al. (2012)  
van Uitert et al. (2016)  
Dutton et al. (2010)  
RodríguezPuebla et al. (2015)  
Early type galaxies  
Behroozi, Wechsler, & Conroy (2013)  
Moster, Naab, & White (2013)  
Leauthaud et al. (2012)  
van Uitert et al. (2016)  
Dutton et al. (2010)  
RodríguezPuebla et al. (2015) 
At a fixed halo mass, the retained fraction of angular momentum is dependent on galaxy’s morphology, with latetype galaxies having systematically larger than that of earlytype galaxies by roughly a factor , consistent with the findings of RF12. For a fixed morphological type, generally has a maximum at the halo mass corresponding to the peak of (at about for all the SHMR adopted here, see Figure 2). For all the six SHMR the mean tends to mildly decrease at masses smaller than , by roughly a factor of at . At masses much larger than all models (but that with the SHMR by Dutton et al., 2010, which does not extend to that mass range) exhibit a sharp decrease of the mean , by roughly a factor at . Given the scatter that we estimate for these distributions (see Section 3.4 for details) the variation of with halo mass is significant in all models, except those with the SHMR by Dutton et al. (2010) which are consistent with a constant for spirals and for ellipticals. However, by comparing the top right panel of Figure 4 with the other panels it is clear that this result is mainly driven by the small mass range probed by that specific SHMR. From this we conclude that the finding of Dutton & van den Bosch (2012) that is approximately constant with galaxy mass is mainly due to their choice of the SHMR. Finally, we notice that in the mass range observationally probed by the Fall relation of RF12 (darker shaded areas and curves in Figures 45) also the distribution of for spiral galaxies with the SHMR of RodríguezPuebla et al. (2015) appears to be marginally consistent with a constant even if the mean sharply decreases at large masses. This is because the scatter we have conservatively estimated is rather large (see Section 3.4). Similar trends are present in the distribution of as a function of stellar mass , thus similar conclusions can be drawn from Figure 5.
One of the main results of our work is that the detailed shape of the distribution of as a function of halo mass significantly depends on the SHMR. As discussed in Section 2.2, if the Fall relation is a powerlaw for each morphological type, then must decrease for masses larger (and smaller) than by an amount that balances the decrease of . In particular, and a massindependent spin parameter for the dark matter yield
(15) 
In particular, if varies as a powerlaw in a given stellar mass range, then in the same range also behaves as a powerlaw
(16) 
or, equivalently, if varies as in a given halo mass range, then goes as
(17) 
In the stellar mass range probed by RF12 the Fall relations for spirals and ellipticals have and , respectively. For spiral galaxies, for instance, this implies that for a Dutton et al. (2010) SHMR the retained fraction roughly behaves as and at low and highmasses respectively; while for a Moster, Naab, & White (2013) SHMR equations (16)(17) imply the behaviours and at low and highmasses respectively.
Our results are a consequence of the fact that the empirical Fall relation is well represented by a single powerlaw of index close to . All the observations available at the moment are consistent with a powerlaw relation, whose normalization (and maybe logarithmic slope) vary with galaxy morphology. However, it is not definitively clear whether an upward bend of such relation for very massive () discs can be completely ruled out (see e.g. Fall & Romanowsky, 2013) and the determination of for the most massive nearby disc galaxies known would be extremely helpful in shedding some light on this issue. At stellar masses smaller than observational constraints on both the Fall relation and the SHMR are either very uncertain or missing, hence our conclusions can not reliably extend much further down this mass. However, if we were to extrapolate the behaviour of both the Fall relation and the SHMR to the dwarf galaxies regime, we would find that decreases with decreasing galaxy mass. At the moment there are only some indications that the Fall relation may be extended through the dwarf regime (Chowdhury & Chengalur, 2017) and some studies which estimate the SHMR for dwarfs and compare it to that of more massive galaxies (e.g. Miller et al., 2014), so we are not able to draw any significant conclusion here. However, we notice that our expectation of a decreasing retained fraction with decreasing stellar mass below is in agreement with the recent hydrodynamical simulations of ElBadry et al. (2017), whose model galaxies closely follow the SHMR of Moster, Naab, & White (2013), at least for .
3.3.1 A simple description of the relation
We find that a reasonable description of the mean as a function of is provided by the following double powerlaw model
(18) 
where , , and are constants representing respectively a normalization, a mass scale and the low and highmass slopes. In Table 1 we report the bestfit values of the above parameters to the twelve distributions of as a function of halo mass, in the range of stellar masses probed by the RF12 Fall relations, presented in Figure 4 (found by minimising a likelihood) together with their uncertainties (computed by sampling the posterior distribution using a Monte Carlo Markov Chain method, assuming uninformative priors). The values of the bestfitting lowmass and highmass slopes and can be compared to the scaling given in equation (17) for a powerlaw Fall relation and in the regimes where the SHMR is approximated by a powerlaw.
3.4 Scatter of the SHSAMR
The large shaded areas in Figure 4 derive from a combination of the intrinsic and observational scatters in the plane and SHMR, as well as from the intrinsic scatter of the theoretical relation and distribution. Each of these scatters sum up in quadrature to yield the final width
(19) 
While encompassing the current uncertainties, this should not be considered as an estimate of the intrinsic scatter of the SHSAMR , unless one makes the assumption that all the terms in equation (19) are uncorrelated, which is not very realistic, considering that at least the observed scatter will most likely depend on , and . Thus, the width of the shaded areas should be regarded as a very conservative upper limit on the intrinsic scatter of the relation.
The observed scatter of the relation is found to be much smaller than the sum in quadrature of , and (see Table 2). This suggests that the scatters of these quantities are likely to be correlated in such a way that when combined as in equation (9) they yield a scatter of dex for the of spirals and dex for that of ellipticals. Note that these conclusions are mostly driven by the scatter in (see Table 2), which is theoretically well established.
Galaxies  

Late types  0.25  0.11  0.11  0.19 
Early types  0.25  0.11  0.14  0.24 
The considerations above are critical also to interpret the results on the massdependence of the retained fraction. For instance, the distribution of spirals in the case of the SHMR of RodríguezPuebla et al. (2015) is consistent with a constant only if the intrinsic scatter of the SHSAMR is as large as the displayed shaded area, which we argued above to be rather unlikely.
3.5 The distribution of as a function of
In Figure 6 we show the retained fraction of angular momentum as a function of the stellar fraction, or star formation efficiency, . These two quantities are clearly correlated such that galaxies with smaller stellar fraction also have smaller retained fraction of angular momentum. The correlation roughly follows , so that the quantity defined in equation (12) is about constant with galaxy mass. The deviations of the observed Fall relation from can be interpreted as a consequence of the deviations of the relation from (see equation 15).
Note that similar conclusions to ours can be reached by modelling galaxies that are on the massrotational velocity relation, i.e. the Tully & Fisher (1977) relation, within a CDM framework. Combining a relation with the scalings for dark matter haloes summarized in Section 2.1 one finds where , which also immediately yields (see Navarro & Steinmetz, 2000). Currently this type of models are able to reasonably represent the observed TullyFisher, masssize and SHMR simultaneously (see Ferrero et al., 2017). However, even if the TullyFisher relation is much tighter and much better known w.r.t. the Fall relation, unlike the angular momentum retention fraction which is directly related to the main phenomena driving galaxy evolution (e.g. the accretion history and feedback), the parameter has no clear physical meaning and its link with galaxy evolution is much more obscure.
The relation indicates that galaxies which are globally efficient at forming stars are also efficient at “retaining” angular momentum. Hence the formation history of a galaxy which has very efficiently turned its available gas into stars is such that its spin is comparable to that of the dark halo (and viceversa), as cosmological hydrodynamical simulations also suggest. Recent studies have found that such galaxy/halo mass and spinconnection is regulated either by the effect of stellar feedback, which preferentially removes the low material (see Pedrosa & Tissera, 2015; DeFelippis et al., 2017), together with the halo assembly history, since mergers tend to lower the angular momentum of stars (see Sokołowska et al., 2017), and/or by the accretion history of gas onto the galaxy, which proceeds from the low material first and then continues to the high one for systems which are still efficient at forming stars at later times (see ElBadry et al., 2017). In Section 4 we expand the discussion of this latter kind of models, which we call biased collapse models following RF12.
All these factors potentially play key roles in shaping galaxies as we see them today and they must be considered when attempting to explain the current morphology and structural properties of galaxies. Any galaxy formation model, which aims at explaining the formation and evolution of galaxies in terms of their history of accretion and star formation, has necessarily to deal with the fact that the global star formation efficiency is intimately linked to the global efficiency at angular momentum retention.
4 Physical models for the angular momentum of galaxies
In this Section we describe two simple physicallybased models for the angular momentum content of galaxies and we compare their predictions on the specific angular momentummass diagram with observations from RF12. This approach is complementary to that adopted in Section 3 and may be useful to test the predictions of some specific scenarios. In particular, we first ask whether a powerlaw Fall relation naturally emerges in a biased collapse scenario, in which the global efficiency of star formation correlates with the efficiency at angular momentum retention (Section 4.1); then we ask whether the different normalization of the Fall relations for spirals and ellipticals can be ascribed to a spinbias scenario and not to a variable with galaxy mass (Section 4.2). For simplicity, here we use only the SHMR of RodríguezPuebla et al. (2015) and we are implicitly assuming that their blue and red galaxy populations trace the spirals and ellipticals of RF12.
4.1 Biased collapse models: a powerlaw relation
We first test a simple model in which varies with galaxy mass. and are strongly correlated both in an empirical CDM model based on the Fall relation (Section 3 and Figure 6) and in one based on the TullyFisher relation (see Navarro & Steinmetz, 2000, and Section 3.5). The same happens also in a model in which the global star formation efficiency depends on the gas cooling efficiency: if the gas has a cumulative angular momentum distribution and it is accreted (and cools) onto the galaxy insideout, then (see van den Bosch, 1998; Dutton & van den Bosch, 2012)
(20) 
where is the cosmic baryon fraction. Hence is expected to correlate with if stars form preferentially from the gas at highest densities, i.e. closest to the centre where the gas cooling was more effective (e.g. Kassin et al., 2012, see also RF12, Section 6.3.2). We refer to this scenario as biased collapse.
In the model presented in this Section we interpret as the fraction of the halo’s specific angular momentum present in the gas actually able to cool and form stars in the galaxy. In practice, we are considering galaxies formed out of gas at angular momentum between , where is the maximum specific angular momentum of the material that collapsed to form the central galaxy. A correlation between and is expected also in the presence of feedback (as it is seen in cosmological simulations, e.g. Pedrosa & Tissera, 2015), though it may alter the value of the exponent in equation (20) due to angular momentum mixing (see below).
Motivated by these results we make the ansatz:
(21) 
We repeat the steps (i) to (iv) as in Section 3.1, with the SHMR of RodríguezPuebla et al. (2015); then, we assign to each galaxy as in equation (21), with variable and and no scatter; finally, we use equation (9) to compare with the observations of RF12.
We determine and by minimising a likelihood and we also allow for the presence of outliers in the observed data. Then, we compute the posterior probabilities of the two parameters by assuming flat priors (using a Monte Carlo Markov Chain method). We obtain
(22)  
(23) 
Our model estimates a fraction of of possible outliers in the Fall relation for spirals at , while it is consistent with zero for ellipticals. The slope of the relation (21) is unsurprisingly in agreement with the value for both galaxy types: in fact, this is the only value of that generates a powerlaw Fall relation (also with slope ). For reference, the case would correspond to in equation 21 (e.g. Mo, Mao, & White, 1998, RF12) and we find it to be disfavoured by observations. On the other hand, our best fit value is significantly smaller than the value suggested by (Dutton & van den Bosch, 2012), under the assumption that the baryons and the dark matter have exactly the same angular momentum distribution. Note that equation (20) is consistent with an angular momentum distribution of baryons with . The best fit would imply , which is interestingly in between the values and expected for a dark matter halo and a pure disc, respectively (e.g. van den Bosch, Burkert & Swaters, 2001). This may be an indication of mixing processes, possibly related to feedback, having redistributed the angular momentum of baryons, resulting in a shallower angular momentum distrbution (e.g. Brook et al., 2012).
Remarkably, the value of the normalization for spirals is consistent with , as expected from equation (20) with . For ellipticals, dynamical friction induced by mergers may have contributed to net losses of angular momentum resulting in an overall smaller normalisation.
We show in the lefthand panel of Figure 7 the predictions of this model compared with the galaxies observed by RF12. A remarkable agreement can be appreciated. Though, the scatter of the predicted relation is still too large compared to the observed scatter: the model predicts dex w.r.t. the observed dex and dex for late and early types. This is because, as discussed in Section 3.4, we are overestimating the scatter of the Fall relation since we are assuming the scatters in the SHMR, distribution, and relations to be uncorrelated. Moreover, even with this simplified assumption of nointrinsic scatter in the relation we are still overpredicting the scatter in the relation.
4.2 Spinbias models: lowrotating galaxies in lowspin haloes
Another hypothesis one could envisage is that the retention fraction is actually constant with mass and galaxy type and, instead, the scatter of the halo spin distribution correlates with galaxy type and specific angular momentum. In this picture, at a fixed halo mass, haloes with larger spins host preferentially galaxies of later morphological types and with larger specific angular momenta (see e.g. RF12, Section 6.2). We refer to this scenario as spinbias.
To test this hypothesis, we consider a model in which, for a given halo mass, the haloes with the largest are occupied by spirals and those with the smallest are occupied by ellipticals. In our simplified picture of galaxies of two morphological types well traced by their colours, we can define for each halo mass a critical value which divides haloes populated by galaxies of the two types. This must be a function of mass since haloes of different masses have different probabilities to contain galaxies of a given type (i.e. morphology depends on galaxy mass, e.g. Kelvin et al., 2014). To find the function we use an iterative procedure. Starting from a given SHMR and an observed distribution of late/early types fraction as a function of stellar mass , taken from Kelvin et al. (2014), we proceed as follows:

we generate a uniform population of in and we divide it into 12 equallyspaced bins;

starting from a flat guess for the function , in a given halo mass bin we assign to haloes with the type “spiral” and the type “elliptical” to haloes with ;

we compute stellar masses for each halo using the SHMR of RodríguezPuebla et al. (2015) for the two galaxy populations;

we bin again the entire population, but this time in stellar mass; in each bin we compute the fraction of spirals and ellipticals as determined by our guess function and we compare the result to the observed distribution from Kelvin et al. (2014);

we update the guess , iterating the steps above, until a satisfactory match to the target of Kelvin et al. (2014) is found.
With this procedure we have defined a model galaxy population for which the morphological type distribution is matched to the halo spin distribution consistently with the given SHMR. Now we use equation (9), with an arbitrary constant for both galaxy type and mass, to predict the Fall relation.
We also tried with a different approach, that is, starting from a stellar mass distribution, in each stellar mass bin we assumed a lognormal distribution (for the dark matter) and we set to be the th percentile of the distribution, where is the Kelvin et al. (2014) type fraction in that stellar mass bin. The results that we now discuss are basically identical in these two procedures.
The righthand panel of Figure 7 shows the comparison of this model with the observations of RF12. In this case the model provides a poor representation of the data in two respects. First, a pure spinbias scenario, in which ellipticals/spirals form in low/high haloes at a given mass, can not account for the difference in the average at a fixed stellar mass as a function of galaxy type. This is not surprising and can be understood if one considers that the scatter of the distribution ( dex) is smaller than the systematic shift of the Fall relations of discs and ellipticals ( dex, RF12, ). Second, the shape of the predicted relation is nonlinear and it steepens at about the peak mass of : this results in an inconsistency with the most massive discs and ellipticals (). Hence we confirm previous claims (RF12) that a pure spinbias scenario is not able to explain the systematic difference of the specific angular momentum of spirals and ellipticals of a given stellar mass, even when coupling it with stateoftheart SHMR.
Even if we also take into account the fact that elliptical galaxies have typically stopped forming stars several Gyrs ago, when the specific angular momentum of the halo was smaller (, see equation 4), we are still not able to explain all the discrepancy between the relation for spirals and ellipticals in a spinbias scenario. This is illustrated by the black arrow in the righthand panel of Figure 7, which shows that even an extreme scenario, in which all ellipticals were already quenched at , would be at most marginally consistent with the observed shift in normalization with respect to the spirals^{4}^{4}4 Only modest deviations from the SHMR at are expected for that at in the stellar mass range relevant for ellipticals (Behroozi, Wechsler, & Conroy, 2013, e.g.). Hence, for simplicity, in this calculation we have used the SHMR at . .
Note, also, that our model is extreme, for the transition between spirals and ellipticals is sharp and unlikely to be realistic. In the case of a smoother transition the two distributions would move closer and they would be even more inconsistent with the large systematic separation observed for the Fall relations of spirals and ellipticals.
A pure spinbias scenario has been recently invoked to explain the formation of the socalled Ultra Diffuse Galaxies (UDGs, see van Dokkum et al., 2015), which are very faint and extended galaxies that are extreme outliers of the masssize relation. At a given stellar mass, these systems have on average higher angular momentum than the bulk of the galaxy distribution (e.g. Leisman et al., 2017) and this has been related to them being formed in the high tail of the halo distribution (Amorisco & Loeb, 2016). However, this is not necessarily the case since for a given stellar mass, a galaxy with an higherthanaverage retained fraction would also have an high specific angular momentum for its mass even if the dark matter halo has an average spin (see equation 9). This is also in agreement with results from recent cosmological simulations, in which the analogous of these galaxies tend to live in nonpeculiar dark matter haloes (see e.g. Di Cintio et al., 2017). In these simulations stellar feedback prevents the collapse of low gas in the central regions and induces energy and angular momentum exchanges between the dark matter halo and the galaxy, which ends up having larger angular momentum with respect to other galaxies of similar mass.
5 Summary and Conclusions
In this contribution we have used analytic models in a standard CDM framework to study the average retention of angular momentum of galaxies as a function of their mass and type. We did this by comparing the stellar specific angular momenta of galaxies on the empirical Fall relation with that of their dark matter haloes, hence constructing a stellartohalo angular momentum relation. The main improvements with respect to previous works are that we derive the retained fraction of angular momentum by imposing consistency with both the empirical Fall relation and the SHMR and that we explore the dependence of the results on the adopted SHMR. The main results are the following:

we confirm that is smaller than unity for all galaxies and that, on average, latetype galaxies have larger fractions with respect to earlytypes (by about a factor ) at any given mass;

we find a significant dependence of on stellar or halo mass for both morphological types and almost all SHMR, with the only exception of the Dutton et al. (2010) prescription for spirals, mostly due to the limited mass range considered there;

we infer a very conservative upper limit on the scatter of the relation of about dex;

we find that the ratio of stellar to halo angular momentum depends on the star formation efficiency roughly as ;

we confirm that the spinbias scenario (in which galaxies with different morphologies inhabit haloes with different spin) is not sufficient to explain the segregation of spirals and ellipticals in the plane;

our results are consistent with a biased collapse scenario, in which . The relatively low value required by the Fall relation is suggestive of mixing processes having altered the angular momentum distribution of baryons in the halo before or while a fraction of them collapsed to form the central galaxy.
During the completion of this work, Shi et al. (2017) published a model of biased collapse to reproduce the Fall relation and the star formation efficiency. The main differences with our work are i) their best model predicts a significant flattening of the Fall relation for spirals at low masses, ii) they do their calculations for only one choice of the SHMR (relatively similar to the one by Dutton et al., 2010), iii) they try to disentangle a fraction of accreted material from a fraction of retained angular momentum, based on metallicity evolution arguments. We refrained from (iii) due to the inherent uncertainties and degeneracy in the two parameters and, in fact, our includes both processes and should be compared with their .
Acknowledgements
LP thanks Emmanouil Papastergis and Crescenzo Tortora for many useful discussions. LP acknowledges financial support from a Vici grant from NWO. GP acknowledges support from the Swiss National Foundation grant PP00P2_163824. EDT acknowledges the support of the Australian Research Council (ARC) through grant DP160100723.
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