In the current ΛCDM cosmological scenario, N-body simulations provide us with a universal mass profile, and consequently a universal equilibrium circular velocity of the virialized objects, as galaxies. In this paper we obtain, by combining kinematical data of their inner regions with global observational properties, the universal rotation curve of disc galaxies and the corresponding mass distribution out to their virial radius. This curve extends the results of Paper I, concerning the inner luminous regions of Sb–Im spirals, out to the edge of the galaxy haloes.
Rotation curves (hereafter RCs) of disc galaxies do not show any Keplerian fall-off and do not match the distribution of the stellar (plus gaseous) matter. As a most natural explanation, this implies an additional invisible mass component (Rubin et al. 1980; Bosma 1981; Persic & Salucci 1988) that becomes progressively more conspicuous for the less luminous galaxies (e.g. Persic & Salucci 1988, 1990; Broeils 1992a). Moreover, the kinematical properties of Sb–Im spirals lead to the concept of the universal rotation curve (URC) implicit in Rubin et al. (1985), pioneered in Persic & Salucci (1991) and set in Persic et al. (1996, hereafter PSS, Paper I): RCs can be generally represented out to Rl, the outermost radius where data are available, by VURC(R; P), that is, by a universal function of radius, tuned by some galaxy property P. P can be a global property such as the luminosity and the disc or halo mass or a well-defined local quantity like Vopt. In any case it serves as the galaxy identifier. In PSS individual RCs and a number of co-added RCs proved the URC paradigm being well fitted by an analytical Curve, VURC(r/Ropt, L),1 a function which is the sum in quadrature of two terms: VURCD and VURCH, each representing the disc or halo contribution to the circular velocity:Freeman 1970) of surface density and contributing to the circular velocity V as 2 and In and Kn are the modified Bessel functions computed at 1.6 x. The dark matter component (with ) was described by means of a simple halo velocity profile: (2a) and (2b) well fit all the PSS data with ρ0, a2 specific functions of luminosity (see PSS). Let us remind that disc masses MD of spirals were found in the range 109 M⊙≤MD≤ 2 × 1011 M⊙.
The URC for the purpose of this work matches well the individual RCs of late-type spirals (see also Appendix for a discussion). It is useful to express the URC paradigm in the following way: at any chosen radius, the URC predicts the circular velocity of a (late-type) spiral of known luminosity and disc scalelength, within an error that is one order of magnitude smaller than the variations it shows (i) at different radii and (ii) at any radius, with respect to objects of different luminosity.
Let us remind the reader that the universal curve built in PSS holds out to Rl, and uses the luminosity as the galaxy identifier and the disc scalelength as a unit of measure for the radial coordinate. We will label it as URC0 to indicate it as the first step of a definitive function of the dark radial coordinate, able to reproduce the observed RCs of spirals. URC0 provides fundamental information on the mass distribution (MD) in spirals, while it suffers from three limitations: (1) it strictly holds only in a region extended less than 5 per cent the DM halo size (see below) (2) the velocity profile of the halo component, valid out to Rl, cannot be extrapolated to radii of cosmological interest 3) it identifies objects by their luminosities, rather than by their virial masses. Let us point out that the URC0 has been often and successfully used as an observational benchmark for theories, but this, only for R < Rl and after that a relation between the halo mass and the galaxy luminosity was assumed.
On the other side, high-resolution cosmological N-body simulations have shown that, within the Lambda cold dark matter (ΛCDM) scenario, dark haloes achieve a specific equilibrium density profile characterized by a universal shape and, in turn, a universal halo circular velocity (Navarro, Frenk & White 1996, NFW), VNFW(R, Mvir) in which the virial mass Mvir and virial radius Rvir are the galaxy identifier and radial coordinate.Bullock et al. 2001; Dutton 2006). This leads to Mo, Mao & White 1998). However, it is well known that observations of spiral galaxies favour density concentrations lower than those predicted for CDM by equation (3a): DM haloes detected around spirals do not show the NFW central cusp in favour of a core-like structure (van den Bosch & Swaters 2001; Swaters et al. 2003; Weldrake et al. 2003; Donato et al. 2004; Gentile et al. 2005; Simon et al. 2005; Gentile et al. 2007).
Therefore, the reconstruction of the MD of DM haloes from observations in parallel with that emerging from N-body simulations is required not only as a normal scientific routine, but also in view of a likely theory-versus-observations disagreement.
As an alternative to the simulation method, we will support the URC paradigm by means of a set of proper observational data and we will derive an analytical form for this curve, valid from the galaxy centre out to its virial radius and characterized by the halo mass as the galaxy identifier. In detail, we extend/improve the URC0 in PSS (i) by adopting a different halo profile, proper to describe the halo distribution out to the virial radius, (ii) by using a number of RCs substantially more extended than those in PSS and (iii) by exploiting the relationship between the disc mass MD, and the virial galaxy mass Mvir, recently obtained by Shankar et al. (2006). This will allow to build an ‘observational’ universal curve, VURC(R; Mvir), extended out to Rvir and having the virial mass as the galaxy identifier. This curve is the observational counterpart of the universal ΛCDM NFW N-body generated profile.
Although the concept behind the URC may be valid also for galaxies of different Hubble types (see Salucci & Persic 1997), a number of issues are still open and will be dealt elsewhere:
Sa galaxies amount, by number, to less than 10 per cent of the whole spiral population, and are important objects in view of the dual nature of their stellar distribution. They show RC profiles with a clear systematics with luminosity (Rubin et al. 1985), but, not unexpectedly, with some difference from those of the URC0 (Noordermeer 2007).
Dwarf spirals with Vopt < 50 km s−1 are not well studied and included in the URC yet, also because in these objects the RCs do not coincide with the circular velocity, being significant the complex asymmetric drift correction.
The kinematical properties of spirals of very high stellar disc mass are not presently investigated with a suitably large sample.
A possible additional URC physical parameter (e.g. the surface stellar density) to take care of the (small) variance of the RCs profiles that seems to be unaccounted by the luminosity.
Finally, let us remind the reader that, in a flat cosmology with matter density parameter ΩM= 0.27 and Hubble constant H0= 71 km s−1 Mpc−1, at the present time, the halo virial radius Rvir, that is, the size of the virialized cosmological perturbation of mass Mvir scales with the latter asEke et al. 1996).
2 THE UNIVERSAL HALO VELOCITY PROFILEwith M0= 1.6 ρ0r30, so that (2b) in the URC0 (Salucci & Burkert 2000; Gentile et al. 2004). At larger radii, the mass diverges only logarithmically with radius and converges to the NFW velocity profile, provided that r0≪Rvir.
We fit the set of individual and co-added RCs of PSS with VURC(R; MD, ρ0, r0) and derive the model parameters MD, ρ0, r0 (see Salucci & Burkert 2000):
Equations (1), (2a), (5b), (6a) and (6b) define the URC out to Rl, VURC(R, MD, r0), from the ‘baryonic perspective’. Let us notice that (as a result of modelling the RC mass) r0, differently from MD and ρ0, has quite large fitting uncertainties, namely, δr0/r0= 0.3–0.5. Following our empirical approach, we do not extrapolate the URCH determined inside Rl out to Rvir≫Rl, because this will be uncertain as well as of unknown validity. This quantity will be derived in the next section.
3 THE URC OUT TO THE VIRIAL RADIUS
We overcome the two main limitations of the URC0, its problematic extrapolation between Rl and Rvir and the uncertainty in the estimate of the core radius, by determining the latter by means of a new outer observational quantity, the halo virial velocity Vvir≡[G Mvir/Rvir(Mvir)]1/2, related to the virial mass through equation (4). In detail, we obtain Vvir from the disc mass, suitably measured from inner kinematics through its relationship to the virial mass found by Shankar et al. (2006):Bell et al. 2003; Baldry et al. 2004) and of (ii) the cosmological halo mass function as indicated by N-body ΛCDM simulations.3 Let us notice that this relationship is obtained without assuming any halo density profile, so that it can be combined with the mass modelling of the inner kinematics.
Let us first derive RD(Mvir), the disc scalelength as a function of the halo mass, by inserting equation (7) in the relationshipequation (8).
It is worth to compute the radial extrapolation needed to reach Rvir from Rl= 6RD, a quantity that can also be associated with the baryonic collapse factor F=Rvir/RD; we find , that is, about 25 times the disc size ∼3 RD.
Equation (7) in combination with equation (4) allows us to add, to the PSS set of kinematical data leading to the URC0, a new observational quantity: Vvir(MD) =GMvir/Rvir, relative to the virial radius. Then, we determine the core radius not from the inner kinematics, but as the value of r0 for which the velocity model described by equations (1), (2a), (5b) and (6a), matches (at Rvir) the virial velocity Vvir given by equations (7) and (4). Let us write this asequation (7) inserted in equation (6a). From the PSS inner kinematics we get the values of ρ0 and MD according to equation (6a), so that equation (9a) becomes an implicit relation between r0 and Mvir (c1, c2 are known numerical constants):
Let us stress that the present derivation of r0 is very solid with respect to observational uncertainties: errors up to a factor 2 in Mvir in equation (7) trigger errors in r0 lower than 40 per cent, and errors in the outer halo velocity slope (0.1 ≤R/Rvir≤ 1) lower than 0.1. This is certainly smaller than the scatter of values with which this quantity is found by N-body simulations and by smoothed particle hydrodynamics/semi-analytical studies of galaxy formation including the baryonic components.
It is worth investigating a number of recently published superextended (SE) RCs (Salucci et al. 2003; Gentile et al. 2004; Donato et al. 2004). They reach a radius larger than 5 per cent (and up to 15 per cent) of the virial radius, that is, a radius at least twice as extended as those of the synthetic curves in PSS. The mass modelling of these SE RCs (made in the original papers) shows an r0 versus Mvir relationship that is in good agreement with equation (10) (see Fig. 1). Relation (10) and the above individual values differ by 20–40 per cent from those determined from the inner kinematics alone and given by means of equation (6b). Since in this paper (also because Rl∼r0), equations (6a) and (6b) are considered a prediction of the inner mass modelling rather than an actual measurement, such good agreement indicates the soundness of the PSS mass modelling.
Let us notice that only for a range of values of the crucial quantity V(Rl)–Vvir, with the first term obtained by the inner kinematics and the second one via equations (4) and (7), there is a solution for equation (9b), therefore, the existence of equation (10) and the agreement of the values of the Burkert core radii, measured independently at 0.05Rvir, 0.1Rvir and Rvir are important tests passed by this profile.
Then, by means of equations (1), (2a), (5b), (6a) and (10), we construct the full URC, extended out to the virial radius and with the virial mass as the galaxy indicator. It is useful to show the relationships we use (see Fig. 2). The mass model includes a Burkert DM halo of central density ρ0, of core radius of size r0 and a Freeman disc of mass MD. The URC fits nicely the available velocity data out to Rl and it is valid out to the virial radius, where it exactly matches Vvir. Moreover, since Mvir is the quantity that in theoretical studies identifies a galaxy, we overcome the main limitation of URC0.
We consider all of the three coordinate systems r, r/RD, r/Rvir equivalent to represent the main structural properties of the MD in spirals, but each of them showing some particular aspects. More specifically, it is then possible and useful to build several ‘URCs’, that is, VURC(r/Rcoo; P), where P is a galaxy identifier (MD, Mvir, L) and Rcoo a radial coordinate (r, r/RD, r/Rvir). Although not all these URCs are independent in a statistical sense, they all are relevant in that they all well reproduce the individual RCs and each of them highlights particular properties of the MD.
In Fig. 3 we show VURC(r; Mvir), the URC in physical units with the objects identified by the halo virial mass; each line refers to a given halo mass in the range 1011 M⊙≲Mvir≲ 1013 M⊙; the halo mass determines both the amplitude and the shape of the curve. Note the contribution of the baryonic component, negligible for small masses but increasingly important in the larger structures, mirrors the behaviour of the Mvir–MD relation. The general existence of an inner peak is evident but, especially at low masses, it is due to both dark and stellar components. Remarkably, the maximum value of the circular velocity occurs at about 15 ± 3 kpc, independent of the galaxy mass: this seems to be a main kinematic imprint of the DM – luminous mass interaction occurring in spirals. Furthermore, Fig. 3 shows that the ‘cosmic conspiracy’ paradigm has no observational support: there is no fine tuning between the dark and the stellar structural parameters to produce the same particular RC profile in all objects (e.g. a flat one). Conversely, a number of relationships between the various structural parameters produce a variety of RC profiles. Moreover, the peak velocity of the stellar component Vpeakdisc=VD(2.2 RD) =G MD/RDk, with k= constant, is not a constant fraction of the virial velocity as is found in ellipticals (i.e. σ∝Vvir), but it ranges between the values 1 and 2 depending on the halo mass.
Moreover, as in the NFW (and Burkert) RC profiles, the URC profiles are found (moderately) decreasing over most of the halo radial extent. The paradigm of flat RCs is obviously incorrect even/especially intended as an asymptotic behaviour at large radii. In fact, we find that both V(0.05 Rvir), the velocity at the farthest radius with available kinematical in PSS and V(3RD), a main reference velocity of the luminous regions of spirals, are significantly (10–30 per cent) higher than the (observational) value of Vvir. This rules out a V= constant extrapolation of the inner RCs out to regions non-mapped by the kinematics and DM dominated regions. We note that this result is independent of the adopted halo density profile and is far from being granted on theoretical grounds.
In Fig. 4 we frame the URC from a full DM perspective by plotting VURC(R/Rvir; Mvir). We set the virial mass Mvir as the galaxy identifier and R/Rvir as the radial ‘dark’ coordinate, thus normalizing the amplitudes by Vvir∝M1/3vir. This ensemble of curves, a main goal of the present work, is parallel to those emerging in N-body simulations and aims to represent the actual velocity profiles of spirals. In these variables the DM haloes are self-similar; the whole system is self-similar in the outer regions, while in the innermost 30 per cent of the halo size the baryons have influenced the dynamics and broken the self-similarity. From these coordinates it easily emerges that the maximum of the RC occurs at very different radii, namely, at ≃2 RD for the most massive objects and at ∼10 RD for the least massive ones. Then, no reference circular velocities, which should be considered as the actual physical counterparts of the empirical velocities of the Tully–Fisher relationship, exist in actual galaxy RCs.
In Fig. 5 we zoom into the URC to look for the inner (luminous) regions of spirals from a baryonic perspective: the URC is so expressed as a function of the ‘baryonic’ radial coordinate r/RD. This figure corresponds to Fig. 4 of PSS, with the important difference that here the virial mass, rather than the galaxy luminosity, is the galaxy identifier. Plainly, an inverse correlation between the average steepness of the RC slope and the halo mass holds, similar to the slope–luminosity relationship found by Persic & Salucci (1988). In this coordinate the stellar matter is closely self-similar, and the different shapes of the RCs are mainly due to the Mvir–MD relation. In the space defined by normalized circular velocity–dark radius–halo mass, spirals do not occupy random positions, but a well-defined plane of very small thickness. We clearly see that, by filling only less than 10−3 per cent of the available volume, the available kinematics of spiral galaxies defines the URC. Let us notice that, in principle, theories of the formation of spirals do not trivially imply the existence of such a surface that underlies the occurrence of a strong dark-luminous coupling.
We now show the URC DM density distribution. In Fig. 6 we show it as a function of x and Mvir. For x < 10−1 the well known core–cusp discrepancy emerges, that is, the DM density of actual haloes around spirals is about one order of magnitude smaller and radially much more constant than the NFW predictions. At x > 0.4, for a concentration parameter c= 13 (Mvir/1012 M⊙)−0.13, the observed halo densities are consistent with the NFW predictions for haloes of the same virial mass. Note that this is a direct test: for haloes with density profiles at x > 0.5 very different from the Burkert or the NFW profiles, equation (9b) does not have solution.equation (11), that is, ε≠ 0. Then in Fig. 7 we show that, even assuming large uncertainties in Vvir, in order to match both V(Rl) and Vvir, we must have ε < 0.1. This is a first direct support for the Burkert and the NFW density law to be able to represent the outer regions (0.3 ≤r/Rvir≤ 1) of DM galaxy haloes.
Notice that weak-lensing shear fields, at several hundreds kpc from the galaxy centres, are found compatible with the predictions of the NFW density profile, but cannot exclude non-NFW profiles (Kleinheinrich et al. 2006, and references therein).
4 DISCUSSION AND CONCLUSIONS
In this paper, we have built the URC of spiral galaxies by means of kinematical and photometric data. We physically extended the URC, established for the inner region of galaxies in PSS out to Rvir and have been able to employ the virial mass Mvir as the parameter that characterizes spiral galaxies, and the virial radius Rvir as a unit of measure for the radial coordinate. This URC is meant to be the observational counterpart of the NFW RC, emerging from cosmological simulations performed in the CDM scenario. The URC yields the gravitational potential at any radius and it allows us to link the local properties in the inner luminous regions with the global properties of the DM haloes.
DM haloes have one (and likely just one) characteristic length-scale, r0∝M0.6vir, which it is not naturally present in current scenarios of galaxy formation. Thus, they do not show any sign of an inner cuspy region of size rs∝M0.4vir. The halo velocity contribution VURCH rises with radius like a solid body at r∼ 0, decelerates to reach a maximum at 3.24 r0 from where it start to slowly decrease out to Rvir with a slope that it is consistent with that of the NFW haloes. The main significance of the URC concerns the full MD. First, it is possible to immediately exclude the following scenarios (and combinations of them): (i) individual behaviour, every object has its own MD; (ii) unique behaviour, every object has almost the same MD. Instead, the MD in spirals shows a remarkable mass-dependent systematics: both the dark and the stellar matter are distributed according to profiles that are functions of the total mass Mvir (see Fig. 8). Finally, the DM halo becomes the dominant mass component in galaxies at different radii, according to the galaxy mass: from ∼10−2Rvir for the lowest masses, to ∼10−1Rvir for the highest ones.
A mathematica code for the figures in this paper is available at http://www.novicosmo.org/salucci.asp.
We thank L. Danese for helpful discussions, and the referee for useful comments.
In this appendix we discuss the observational evidence for the URC claim, the nature and the implications of which it is worth to clarify. The paradigm states that, when binned by luminosity, the RCs form a set of smooth, low-scatter synthetic curves, whose profiles and amplitudes are strong functions of the luminosity bin.4 Furthermore, the URC paradigm implies: (i) rotation velocity slopes versus rotation velocity amplitudes relationships (see figs 2 and 3 of PSS) and (ii) a set of relations (radial Tully–Fisher relationship) holding at different radii x, defined asPersic & Salucci 1991).
Evidence for the URC claim and/or its above implications comes from: (i) detailed analyses of independent samples: Catinella et al. (2006), (2200 RCs, see their fig. 12), Swaters (1999), (60 extended RCs, see chapter 4); (ii) independent analyses of the PSS sample: Rhee (1996), Roscoe (1999); (iii) the finding of a very tight RTF in PSS and other three different samples Willick (1999, see below), Yegorova et al. (2006).
The claim has been also tested by comparing the RCs of two samples of spirals (Courteau 1998, 131 objects; Verheijen 1997, 30 objects) with the circular velocities predicted by the URC0, once that the values of galaxy luminosity and disc length-scale are inserted in it. The face-value result of the test: 2/3 of the RCs are in pretty good agreement with the universal curve, while 1/3 show some disagreement, indicates that the URC is a useful tool to investigate the systematics of the MD in spirals, but also it questions about its universality. However, while some of this disagreement may reflect an inefficiency of the URC0 to reproduce the RCs, the actual performance of the URC is better than it is claimed. In fact, spurious data versus predictions disagreements are created in performing this test and precisely when they insert in URC0 the values of LB and RD, affected by (occasionally large) observational errors. By taking into account this effect the URC0 success rate reaches 80 per cent and more.
Willick (1999) found, by studying a large sample of RCs, a radial variation of the scatter of the inverse RTF defined above and he interpreted it as an evidence against the URC. Let us show that this argument is incorrect and that au contraire the properties of the RTF support the URC paradigm. The increase/decrease of the scatter found is very small (Willick 1998): the scatter ranges from 0.065 dex (at 2RD) to 0.080 dex (at 0.5RD and at 3RD) and it implies, if totally intrinsic, a prediction error in log V(x) of (0.082–0.0652)0.5= 0.04 dex. Moreover, some of the scatter increase/decrease is due to the larger random observational errors present in the outermost measurements; in fact, a refined analysis of the issue (Yegorova et al. 2006) finds a smaller predicting error for three large sample of spirals. Therefore, from the RTF we have that, in the region considered, the luminosity statistically predicts the circular velocity at any radius and in any galaxy within an error of 5–10 per cent, a quantity much smaller than the variations of the latter in each galaxy and among galaxies.