GALAXIES IN ΛCDM WITH HALO ABUNDANCE MATCHING: LUMINOSITY–VELOCITY RELATION, BARYONIC MASS–VELOCITY RELATION, VELOCITY FUNCTION, AND CLUSTERING

We use several data sets to construct the LV relation for observed galaxies. Springob et al. (2007) compiled a template I-band TF sample of 807 spiral galaxies of types Sa–Sd in order to calibrate distances to ∼4000 galaxies in the local universe. Template galaxies were chosen to be members of nearby clusters in order to minimize distance errors. Their photometry contains distance uncertainties so the scatter should be taken cautiously and only as an upper limit to the intrinsic TF scatter. Circular velocities were obtained using H i line synthesis observations or optical Hα rotation curves when those were not available. The maximum circular velocity was obtained by using a model fit to the observed profiles. Since the authors correct for the effects of turbulence by subtracting 6.5 km s⁻¹ linearly from the velocity widths, it was necessary to de-correct them by adding this term back in to obtain the true circular velocities.

The Pizagno et al. (2007) sample was selected from the Sloan Digital Sky Survey (SDSS; York et al. 2000). It is one of the most complete and unbiased samples available of Hα rotation curves of disk galaxies and was studied in an attempt to accurately measure the intrinsic scatter in the TF relation. Luminosities were taken from SDSS r-band photometry, yielding the best match with the luminosities assigned to our model galaxies. For this sample we used the asymptotic value of the rotation velocity they obtained using a functional fit to the rotation curves.

In order to test the predictions of the ΛCDM model using abundance matching (ΛCDM + HAM for short) with the largest dynamic range possible, we included in our comparison the latest TF dwarf galaxy sample studied by Geha et al. (2006). Their sample consists of about 110 late-type galaxies with luminosities measured in the r band and rotation velocities measured using H i emission.

The three samples above constitute our major observational data set. We further cut them by selecting galaxies with high inclinations (i > 45° or axis ratio b/a > 0.7) to minimize uncertainties due to projection effects. Additionally, we include only galaxies with better than 10% accuracy in the measurement of the maximum circular velocity. These cuts leave a total of 972 galaxies in the major sample.

For comparison with the data sets mentioned above, we include other smaller samples found in the literature. The sample of Blanton et al. (2008) was also obtained from SDSS and is comprised of only isolated galaxies with high inclinations. The H i galaxy sample used by Sakai et al. (2000) was selected to have small scatter for use as a distance calibrator. It is important to note that while the fit shown here minimizes both the errors in rotation velocity and in luminosity, it may be artificially shallow due to selection effects.

Certain assumptions about galaxy colors had to be made in order to convert the different observational samples to the ^0.1r-band measurements we chose for our model. In order to convert the I-band luminosities measured by Springob et al. (2007) to the r band, we cross-referenced their data with the sample of Pizagno et al. (2007) and used the median (r − I) colors of the galaxies present in both catalogs. To convert from the R-band magnitudes of Sakai et al. (2000) to the SDSS ^0.1r band, we used the transformation equations obtained by Lupton (2005) along with the typical ^0.1(r − i) color of disk galaxies in the SDSS sample studied by Blanton et al. (2003b). In addition, for redshift zero data sets, the k-correction given in Blanton et al. (2003a) was used to convert from z = 0 to z = 0.1 photometric bands.

Lastly, since the obscuring effect of dust extinction as a function of disk inclination is corrected for in TF samples but not in observed LF estimates, we had to de-correct the luminosities of the spiral galaxies in all of the TF samples. To do so, we estimated and added the median extinction in the r band as a function of rotation velocity using the method and sample employed by Pizagno et al. (2007). This correction is ∼0.4 mag for the brightest disks, declining to ∼0.3 mag for V_circ ≈ 100 km s⁻¹. These values are close to those found by Tully et al. (1998) for the extinction in a galaxy with average inclination as a function of H i velocity width. The correction is implemented only when comparing the observations with our model galaxies.

We also include bulge-dominated early-type galaxies (ellipticals and lenticulars) in the LV relation, again measuring the circular velocity at our fiducial 10 kpc radius. The circular velocity in this case is used not as a measure of rotation but merely as a probe of the mass profile, further justifying the use of the term "LV relation" instead of TF relation. Using early-type galaxies allows us to probe closer to the mass regime where the abundance of DM halos drops exponentially (i.e., the knee of the VF), which is more sensitive to the cosmological model. It also allows for the study of halo–galaxy relations without regard to the details of the evolution of the stellar populations within them.

Because of the challenges of both observing and modeling early-type galaxies, so far there exists no comprehensive set of mass measurements for them akin to the spiral galaxy samples. Instead, we compile a set of high-quality LV estimates for individual galaxies from the literature.

To provide the necessary LV data for nearby elliptical and lenticular galaxies, we searched the literature for high-quality mass measurements at ∼10 kpc radii. A variety of different mass tracers were used including hot X-ray gas, a cold gas ring, and kinematics of stars, globular clusters, and planetary nebulae. We required the mass models to incorporate spatially resolved temperature profiles in the case of X-ray studies, and to take some account of orbital anisotropy effects in the case of dynamics. We also used only those cases where V₁₀ was constrained to better than ∼15%. We make no pretense that this is a systematic, unbiased, or especially accurate sample of early-type masses, noting simply that it is preferable to ignoring completely this class of galaxies which dominates the bright end of the luminosity function.

As an aside, we find in comparing to central velocity dispersions σ₀ taken from HyperLeda (Paturel et al. 2003) that the scaling $V_{10}\simeq \sqrt{2} \sigma _0$ works very well on average, suggesting near-isothermal density profiles over a wide range of galaxy masses. It is far easier to measure σ₀ observationally than V₁₀, motivating the use of the former as a proxy for the true V_circ which is more robustly predicted by theory. The ∼15% scatter that we find in the σ₀–V₁₀ relation is relatively small, but it is beyond the scope of this paper to consider the potential systematics of using σ₀ as a proxy. We will use V₁₀ for the LV analysis in this paper.

For the luminosities, we make use of the total B-band apparent magnitudes from the RC3 (de Vaucouleurs et al. 1991), corrected for Galactic extinction. To correct to ^0.1r magnitudes, we use the filter conversions in Blanton & Roweis (2007) together with (B − V) colors obtained from HyperLeda (Paturel et al. 2003). For the distances (required both for absolute magnitudes and for choosing the circular velocity measurement radii in kpc), we use as a first choice the estimates from surface brightness fluctuations (Jensen et al. 2003), and otherwise the recession velocity.

The local data for 55 individual early-type galaxies are presented in Figure 1 (left). Dashed lines show B-band dynamical mass-to-light ratios of M/L_B = 3, 6, 12, and 24; for comparison, early-type galaxies with typical colors (B − V ∼ 0.85–0.95) are expected to have stellar M/L_B ∼ 2.0–2.3 for a Chabrier initial mass function (IMF; e.g., Figure 18 of Blanton & Roweis 2007). A table including the sources of the data is provided in Appendix B. There is no obvious systematic difference between the results from different mass tracers. The galaxies appear to trace a fairly tight LV sequence, except around the L* luminosity (assuming M*_B ≈ −20.6), where there are a few galaxies whose circular velocities appear to be fairly high or low. The low-V₁₀ galaxies include NGC 821 and NGC 4494, which were previously suggested as having a "dearth of DM" (Romanowsky et al. 2003), and as implying a DM "gap" with respect to X-ray bright ellipticals (Napolitano et al. 2009). The present compilation suggests that the galaxy population in the local universe may fill in this gap, although further work will be needed to understand the scatter.

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The right panel of Figure 1 shows the relation between stellar mass and circular velocity at 10 kpc for the galaxies in the early-type sample along with a few spirals for comparison. Stellar masses were obtained using Equation (8) as explained in Section 6.3. The ellipticals are virtually indistinguishable from the S0s in the regime where they overlap while the spirals seem to contain slightly more stellar mass at the same V_circ. We will discuss this issue in more detail in Section 6.3.

Figure 2 shows the combined LV relation for galaxies with very different morphologies: from Magellanic dwarfs with V_circ ≈ 50 km s⁻¹ to giant ellipticals with V_circ ≈ 500 km s⁻¹. The LV relation is not a simple power law. Dwarf galaxies show a tendency to have lower luminosities as compared with a simple power-law extrapolation from brighter magnitudes. There is a clear sign of bimodality at the bright end of the LV relation with early-type galaxies having ∼20%–40% larger circular velocities as compared with spiral galaxies with the same r-band luminosity (or, conversely, ∼1 mag fainter at fixed V_circ).

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The next step is to separate the DM and baryon components in each halo and allow the baryons to dissipatively sink to the centers of the DM halos. We assume for simplicity that there is a radius at which we could consider most of the cold baryons to be enclosed, with only DM present beyond that point.

This cold baryon component has been observed to comprise only a small fraction of the cosmic abundance of baryons; in other words, the cold baryon fraction f_bar ≡ M_bar/(M_DM + M_bar) in galaxies is much lower (Fukugita et al. 1998; Fukugita & Peebles 2004) than Ω_bar/Ω_matter = 0.17 (Komatsu et al. 2009). We resort to the observations and use the galaxy stellar mass function obtained from the SDSS DR7 by Li & White (2009), who employ estimates of stellar masses by Blanton & Roweis (2007) obtained using five-band SDSS photometry assuming the universal IMF of Chabrier (2003). These masses are consistent with those estimated using single color and spectroscopic techniques (Li & White 2009).

Using the same procedure described above for the luminosity function, we abundance-matched the halos in Bolshoi to the galaxies in the SDSS DR7 starting from the high stellar mass end until reaching our completeness limit at V_circ = 50 km s⁻¹, obtaining stellar masses for each galaxy. Strictly speaking, this procedure results in a one-to-one relation between circular velocity and stellar mass-to-light ratio which should only be interpreted as the average of a population of galaxies with a given V_circ. The scatter (or bimodality) in the mass-to-light ratio as a function of circular velocity could be measured from observations and included in the assignment but would not change our results significantly.

Since dwarfs can have most of their cold baryons in the gas phase instead of in stars (Baldry et al. 2008), we calculated for each model galaxy the total cold gas mass using a parameterization of the observed atomic gas mass fraction as a function of stellar mass from Baldry et al. (2008, their Equation (9), shown as a dashed line in their Figure 11). This includes the total cold atomic gas found in the disk only. The gas-to-stellar mass ratio f_gas depends on stellar mass and it is the largest for dwarfs. For example, f_gas ≈ 4–5 for galaxies with M_* = 10⁸ M_☉. It declines to f_gas ≈ 0.25(0.1) for galaxies with M_* = 10¹⁰(10¹¹) M_☉. It should be even smaller for ellipticals and S0s. It should be noted that, when it comes to dynamical corrections to V_circ, the gas plays a minor role. It only becomes important when we consider the BTF relation.

Lastly, we add the stellar and cold gas masses for each model galaxy and obtain the correction to the circular velocity of the pure DM halo at a radius enclosing the cold baryonic mass. We set this value to 10 kpc for all the halos in our sample. In the case of dwarf galaxies this should be a good approximation to the maximum circular velocity since their rotation curves rise much more slowly and in some cases they peak beyond the optical radius (Courteau 1997). Our assumption allows us to include the peak of the rotation curve for most of these galaxies. In the case that the peak is located well within 10 kpc the correction would be almost negligible since we would still be measuring rotation in the flat regime. Additionally, truncating the cold baryons at a radius of 10 kpc allows us to directly calculate the correction to the circular velocity at that radius without having to resort to more complicated assumptions about the distribution of baryonic matter in galaxies, i.e., exponential lengths and Sérsic indices of disks and bulges as well as extended gas and stellar halos.

To obtain the circular velocity measured at 10 kpc (V₁₀) for the Bolshoi DM halos, we need to use DM profiles and find the DM mass inside a 10 kpc radius. To do this, we could use the individual profile of each halo. However, once we select halos with a given maximum circular velocity, individual halo-to-halo variations are small at 10 kpc (the situation is different if we select halos using virial mass, which results in large deviations in concentration producing large variations in V₁₀). This is why instead of individual profiles we construct average profiles for halos within a narrow range Δlog₁₀V ≈ 0.05 of maximum circular velocity.

We first bin and average the circular velocity profiles of the distinct halos found by the BDM code. These profiles are calculated for each halo (including unbound particles) in logarithmic radial bins in units of R_vir. Using distinct halos is convenient because it gives us density profiles that are less affected by interactions than those of subhalos. For the inner profiles of subhalos the effect is relatively small because the stripping happens preferentially at the outer radii. Using the averaged binned circular velocity profiles we obtain the velocity at 10 kpc. Within about 1.2% of the virial radius, discreteness effects render the profiles unreliable and we use instead extrapolation with the shape of a simple power law in radius. For halos with V_circ < 100 km s⁻¹ the full extent of the profiles suffer from measurement noise which we avoid by extrapolating from the profile of halos with ∼100 km s⁻¹. Figure 3 shows the obtained median relation between the maximum circular velocity V_max and V₁₀ for the Bolshoi DM halos without inclusion of the cold baryons.

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The estimates of the relations obtained when a parametric form of the density profile is used are also shown for the case of the Navarro–Frenk–White (NFW; Navarro et al. 1997)

$$\begin{equation} \rho (r) = \frac{4\rho _s}{(r/r_{\rm s})(1+r/r_{\rm s})^2}, \end{equation} \tag{ 3 }$$

and the Einasto (Einasto 1965; Graham et al. 2006) universal profiles

$$\begin{equation} \rho (r) = \rho _s \exp\lbrace -2n[(r/r_{\rm s})^{1/n} - 1]\rbrace ; \end{equation} \tag{ 4 }$$

where r_s is the radius at which the logarithmic slope of the density profile is −2. Following Graham et al. (2006), we use n = 6.0. The concentration parameter defined for both models as c ≡ R_vir/r_s is given by the relations obtained in Paper I for distinct halos (Klypin et al. 2010; see also Prada et al. 2011):

$$\begin{equation} c = 9.60 \left(\frac{M_{\rm vir}}{10^{12} \mbox{$\, h^{-1}\,M_\odot $}} \right)^{-0.075}, \end{equation} \tag{ 5 }$$

and

$$\begin{equation} M_{\rm vir}= \left(\frac{V_{\rm circ}}{ 2.8 \times 10^{-2} \mbox{ km s$^{-1}$}} \right)^{3.16} \mbox{$\, h^{-1}\,M_\odot $}. \end{equation} \tag{ 6 }$$

Note that in Figure 3 we use total dynamical masses and do not account for the condensation of baryons. For V_circ = 100–450 km s⁻¹, the rotation (or density) profiles of the Bolshoi simulation are extremely well approximated by the Einasto parameterization, whereas NFW underestimates V₁₀ by almost 20% at 450 km s⁻¹. Following the conclusions of Navarro et al. (2004) and Graham et al. (2006), we assume the Einasto profile when extrapolating the inner parts of the largest (V_circ > 450 km s⁻¹) halos.

Dissipation allows the baryons to condense into galaxies in the central regions of DM halos dragging the surrounding DM into a new more concentrated equilibrium configuration. If the density of the DM halo increases considerably within the extent of the disk, the peak circular velocity could be much larger than our previous estimates. There are different approximations for the adiabatic compression of the DM. Blumenthal et al. (1986) provide a simple analytical expression, which is known to overpredict the effect. The approximation proposed by Gnedin et al. (2004) predicts a significantly smaller increase in the density of the DM. More recent simulations indicate an even smaller compression (Tissera et al. 2010; Duffy et al. 2010). However, at a 10 kpc radius the DM contributes a relatively large fraction of mass even for large galaxies. As a result, the difference between the strong compression model of Blumenthal et al. (1986) and no-compression is only 10%–20% in velocity.

We use the standard AC model of Blumenthal et al. (1986) to bracket the possible effect. We thus assume that following the condensation of the baryons, the DM particles adjust the radius of their orbits while conserving angular momentum in the process. We solve the equation

$$\begin{equation} M_{\rm tot}(r_{i})r_{i} = [M_{\rm DM}(r_{i})(1-f_{\rm bar}) + M_{\rm bar}(r_{f})]r_{f}, \end{equation} \tag{ 7 }$$

where r_f = 10 kpc, M_bar(r_f) is the total baryonic mass assigned to each halo and f_bar = Ω_bar/Ω_matter is the universal fraction of baryons. We solve Equation (7) for r_i and then add the DM mass M_DM(r_i)(1 − f_bar) to the mass of cold baryons to find the circular velocity. Note that this implies that only cold baryons (i.e., stars and cold gas) are left in the central regions of the galaxy, while the remaining hot baryons are at larger radii. As expected, only the halos that are dominated by baryons at their centers suffer a significant increase in their measured circular velocities due to the increase in concentration of DM as a result of AC.

Figure 4 shows the predicted LV relation for galaxies in the ΛCDM model obtained using HAM. We binned together the major observational samples described in Section 2 and include them for comparison. The internal extinction de-correction for late-type galaxies described in Section 2.1 was implemented for a fair comparison with the models. The full curve in this plot shows results obtained using the Montero-Dorta & Prada (2009) LF of galaxies in the SDSS DR6 sample and uses the AC model of Blumenthal et al. (1986). The 1σ and 2σ width of the distribution of model galaxies in bins of luminosity is represented by the shaded regions (details about the model used to add scatter can be found in Section 6.2.4). Predictions without AC (with the cold baryon contribution added in quadrature) are shown as the dot-dashed curve. The dashed curve shows the effect of a steeper slope at the faint end of the LF that accounts for potential surface brightness incompleteness. (For details see Section 5.) Although dwarf galaxies with V_circ < 80 km s⁻¹ seem to agree better with a model using the Montero-Dorta & Prada (2009) luminosity function, the scatter of the observed dwarf LV relation is so large that both LFs used in conjunction with the abundance of DM halos produce results that are consistent with the available data.

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One may note that the AC model misses late-type points with V_circ = 150–250 km s⁻¹ and the no-AC model practically fits most of the late-type measurements. This should not be interpreted as either an advantage for the no-AC model or a disadvantage for the AC model. Our predictions apply to the average population across all types of galaxies. Because of the dichotomy of the LV diagram, a model that goes through either early-type galaxies or through late-type galaxies is an incorrect model. The correct answer should be a model which tends to be close to spirals at low luminosities (where spirals dominate the statistics) and gradually slides toward the ellipticals at the high-luminosity tail where they dominate. It seems that the AC model does exactly that, but it may overpredict the circular velocities at the very bright tail of the LV relation.⁸

As demonstrated in Section 6.2.4, our stochastic HAM model accounts for galaxies that reside in halos with both smaller and larger V_circ than the average. For example, since the most massive spiral galaxies comprise a very small percentage of the galaxy population with V_circ > 250 km s⁻¹ (less than 10%), in our model they are assigned to the low-V_circ wing of the distribution shown in Figure 4. Hence, even though the model makes predictions for the average galaxy population, it also accounts for the morphological bimodality observed in the LV relation.

One also should not overestimate the quality of the observations. The fact that in Figure 4 the brightest spirals with M_r − 5log₁₀ ≲ −21 are more than 2σ away from the mean of the models is not a problem because of the size of the uncertainties in the observational data. For instance, there is a systematic ∼10% velocity offset between the measurements of Pizagno et al. (2007) and Springob et al. (2007), which seems to point to the current uncertainties in the LV relation.

Considering the current level of the uncertainties, our model galaxies show remarkable agreement with observations spanning an order of magnitude in circular velocity (or, equivalently, three orders of magnitude in halo mass) and more than three orders of magnitude in luminosity. For galaxies above 200 km s⁻¹, our model produces results that agree extremely well with the observed luminosities of early types (Es and S0s). Given how simple the prescription is, it is perhaps surprising how closely we can reproduce the properties of observed galaxies. For galaxies with V_circ = 100–200 km s⁻¹, DM halos without any corrections already occupy the region expected for galaxies. The dynamical corrections improve the fits, but it is important to emphasize that the abundance matching method yields the correct normalization of the LV relation regardless of the details of the corrections we implement. Another feature of the relation, its steepening below 100 km s⁻¹, is caused by the shallow faint-end slope of the luminosity function. Although our model galaxies in this regime are slightly underluminous as compared to a simple power-law extrapolation from brighter galaxies, observed dwarfs seem to predict a deviation from a power-law TF in the same general direction.

In the way it was constructed, our model galaxy sample does not include uncertainties in either the halo VF or in the galaxy luminosity function. This produces an LV relation with no scatter. Section 6.2.4 examines the effects of including scatter in the halo matching procedure.

We now discuss in greater detail the results of the individual steps explained in Section 5.

6.2.1. Measuring Circular Velocity at 10 kpc

Observations do not always provide measurements of the circular velocity at 10 kpc. This is especially true for dwarf galaxies where the last measured point of the rotation curve can be at 3–5 kpc. What are the errors associated with using measurements at different radii? Figure 5 presents three typical examples of the circular velocities expected for galaxies with vastly different masses.

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The top panel shows a model of a giant elliptical galaxy with 1.5 × 10¹¹ M_☉ of stellar mass distributed according to an R^1/4 law with a half-light radius of 5.5 kpc. The stellar component is embedded in a DM halo with virial mass M_vir = 10¹³ M_☉ and median concentration c = 7. The maximum circular velocity (310 km s⁻¹) of the halo is reached at 160 kpc. The middle panel shows an MW size model with maximum circular velocity 190 km s⁻¹, virial mass M_vir = 1.7 × 10¹² M_☉, and median concentration c = 9 for its mass. The cold baryonic component consists of a Hernquist bulge (M_bulge = 10¹⁰ M_☉, half-mass radius R_bulge = 1 kpc) and an exponential disk (M_disk = 5 × 10¹⁰ M_☉, scale radius R_disk = 3.5 kpc). The bottom panel shows a dwarf galaxy model with M_vir = 7 × 10¹⁰ M_☉, c = 12, V_max = 65 km s⁻¹. Its cold baryons have two exponential disks: one for stars and one for cold gas with a mass ratio of 1:4 and radii R_* = 1.5 kpc and R_gas = 3.0 kpc. The total mass in cold baryons is M_bar = 3 × 10⁸ M_☉. When we include baryons, we assume that most of them were blown out from the models and the only baryons left are in the form of stars and cold gas. As in Bolshoi, the "DM" circular velocities in Figure 5 include a cosmological amount of baryons that traces the distribution of the DM. This contribution is removed from the mass profiles before adding the cold baryons. In all three cases we use the Einasto DM profiles (Equation (4)) with n = 6. For the models labeled "DM+Baryons" at each radius we simply add the mass of cold baryons and the mass of DM. The models termed "DM+Baryons+AC" include adiabatic compression of DM according to the prescription of Blumenthal et al. (1986).

After adding the cold baryons the circular velocity profiles become rather flat in the inner 5–10 kpc regions of the galaxies implying that measurements of circular velocities anywhere in this region are accurate enough to provide the value of the circular velocity at 10 kpc.

There are some caveats in choosing 10 kpc as a fiducial radius for either extremely massive spirals or giant ellipticals. Considering the correlation between the central surface brightness and disk scale length found by Courteau et al. (2007), the most luminous disks appear to have scale lengths as large as 15 kpc, whereas according to Courteau (1997) their rotation curves peak at about 1 scale length. Hence, we may be underestimating the maximum observed rotation velocity of these galaxies in our sample. We may also overestimate circular velocities when we assume that most of the cold baryon mass is inside 10 kpc radius. In principle, some corrections can be applied to compensate for this effect. However, our estimates show that at most this is a ∼20% effect for spirals and somewhat smaller for ellipticals because they are more compact for the same luminosity. Considering existing uncertainties and complexities of implementing the correction, we decided not to use them.

6.2.2. Dark Matter Profiles

To illustrate the effect of tidal stripping, Figure 6 shows the results (dashed curve) obtained when the luminosity assignment is performed using the peak historical value of each halo's circular velocity (V_acc) as compared with the circular velocity at z = 0 (dotted curve). The reason why the dashed curve is rightward of the dotted one is that for subhalos the circular velocity estimated at z = 0 is smaller than its value at accretion V_acc. If we compare luminosities at the same circular velocity, the differences can be substantial: almost 1 mag for galaxies with V_circ = 50–60 km s⁻¹ due to the steep slope of the LV relation for dwarfs. In terms of velocities, the differences are much smaller. Neglecting the effects of stripping in the assignment scheme affects mostly dwarf galaxies, overestimating their circular velocities by a maximum of ∼20%. For larger galaxies the effect decreases to less than 5%. This is due to the fact that stripping only affects subhalos and they comprise only a minority (about 20%) of the total halo population. In addition, only the small number of subhalos which orbit close to their host halo's center get significantly stripped and experience a substantial decline in their circular velocity.

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Comparison of LV relations constructed using V_acc, one with the maximum circular velocity (the dashed curve in Figure 6) and another with velocities V_acc estimated at 10 kpc (the full curve), indicates that this affects the largest halos the most. For example, the V₁₀ velocity is almost a factor of two smaller than V_max for the group-size halos presented in the plot. Taking the circular velocity at 10 kpc also makes the LV relation much less curved as compared with the maximum velocity. Below ∼80 km s⁻¹ the maximum circular velocity of the DM halo happens near or within 10 kpc, which explains why the curve does not shift in this regime.

6.2.3. The Effects of Cold Baryons and Adiabatic Compression

Figure 7 shows how cold baryons change the circular velocity at a 10 kpc radius. Here, we use two extreme approximations that bracket the effect. The first approximation assumes that there is no change in the distribution of the DM. All simulations so far indicate that there is some compression. Hence, the no-compression approximation definitely underpredicts the circular velocity V₁₀. The second approximation uses the adiabatic compression model of Blumenthal et al. (1986) which produces the largest increase in the density of the DM. (The full and dashed curves were already shown in Figure 4.) There are some differences between the LV relations predicted by those approximations. However, the largest effect is just adding the cold baryons in quadrature to the circular velocity of the DM. Adiabatic compression increases the circular velocity even further, but the effect is relatively minor because the fraction of cold baryons gets progressively smaller for larger galaxies. The amount of cold baryons used for the models is crucial for this test. As we discuss in Section 6.3, the abundance matching predicts relatively small cold baryon masses for dwarfs and giants, and this is why the adiabatic compression in Figures 5 and 7 is 10%–20% at the most. Again, dwarfs below ∼80 km s⁻¹ are insensitive to the cold baryon presence. Here, the full curve (DM+baryons) is slightly below the V_circ of the DM curve because the latter includes a cosmological fraction of baryons which trace the DM, most of which are assumed to be blown out from galaxies (see Section 5).

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Figure 8 shows the LV relation that we would obtain by assuming instead that half of all baryons within the virial radius or equivalently 8% of the virial mass are retained and are used to build the central galaxy (while luminosity is not affected). Both the shape and the normalization of the LV relation are incorrect, with the circular velocities systematically larger than observations by up to 50%. Clearly, it is difficult to obtain the observed LV relation assuming a constant cold baryon fraction in the framework of the ΛCDM cosmology.

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6.2.4. The Effects of Including Scatter in Luminosity at Fixed Halo Circular Velocity

So far, the abundance matching procedure we have used assumed a monotonic one-to-one relation between halo circular velocity and galaxy luminosity or stellar mass. This assumption produces average relations that can be compared with the medians of the observations. As shown by previous studies, a more detailed treatment of the scatter between halo and galaxy properties may yield average relations of the brightest galaxies that deviate significantly from the case with no scatter. For instance, Tasitsiomi et al. (2004) showed that iteratively introducing a log-normal scatter of width 1.5 mag in the assignment of luminosities to DM halos produces an average TF relation with massive galaxies that are brighter by about 1 mag compared to the monotonic assignment. By treating the scatter analytically, Behroozi et al. (2010) found that performing HAM using their preferred value of 0.16 dex of log-normal scatter reduces the average stellar mass assigned to massive halos with total masses >10¹³ M_☉ by up to 70% (when binning using virial mass) but does not affect less massive galaxies below the knee of the stellar mass function.

Appendix A gives a detailed description of the method we employ to introduce scatter in our model. In short, we obtain luminosities for each of the galaxies in our sample by stochastically scattering the values obtained in the monotonic assignment while forcing the preservation of the observed luminosity function. When scattering the values of luminosity we do not constrain the shape of the probability distribution (e.g., log-normal) or require its width to be constant for all circular velocities. This is well justified since the shape of the intrinsic scatter is more difficult to constrain observationally (e.g., due to observational systematics).

One parameter that our model does not currently predict is the width of the probability distribution of luminosity at a fixed halo circular velocity. This scatter originates from three main sources. The first is the observational error in the determination of the true luminosities of galaxies. Since we use the LF from the SDSS spectroscopic sample, these are the sum of the errors in photometry plus the errors in the distances obtained from spectroscopy. At the mean redshift of the sample these combined errors are expected to be typically much less than 0.1 mag. The second source of scatter is the one present in the intrinsic relation between halo circular velocity and galaxy luminosity due to variation in the physical processes of galaxy formation. Verheijen (2001) studied the nature of the intrinsic scatter in the TF relation from H i observations and obtained a value of $\sigma _{M_r}^{\rm int} = 0.38$ in the R band. This value is consistent with the distribution in the observational samples used in this work. Lastly, since we assign luminosities that are uncorrected for the inclination of disk galaxies, we also need to include the scatter that results from the distribution of dust extinction corrections observed in SDSS. Maller et al. (2009) find a fit to this distribution as a weak function of r-band luminosity and disk scale length. We adopt the value $\sigma _{M_r}^{\rm ext} = 0.28$ they use for a galaxy with M_K = −20. We neglect the errors due to photometry and distances and add the remaining two contributions in quadrature to obtain $\sigma _{M_r} \approx 0.5$, which we use to introduce scatter to the model galaxies below the knee of the LF. Above this luminosity, where early types dominate, we assume that the lack of significant internal extinction slightly reduces the scatter to ∼0.3.

Figure 9 shows the luminosity-binned distribution of model galaxies in the LV relation obtained from the stochastic HAM scheme and compares it to the monotonic assignment discussed in Section 6.1. Since we are left with a choice regarding which quantity to average over, we choose to bin in the r-band magnitude to be consistent with the binning of the observations. The mean relation is almost identical to the case with no scatter for galaxies below 200 km s⁻¹, while it becomes brighter by up to 0.3 mag for more massive galaxies. Galaxies below L* show a distribution of luminosities that is close to Gaussian as far as 2σ away from the mean but has a slightly longer bright tail. Galaxies brighter than L* show a trend of narrowing of the distribution as well as a skewness that reduces the number of upscattered galaxies with increasing luminosity.

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To check the consistency of our approach we also calculated the average LV relation of our model galaxies obtained using the deconvolution method described in Behroozi et al. (2010) and log-normal scatter. This procedure yields a deviation of the mean relation for V_circ > 200 km s⁻¹ toward higher luminosities that depends on the assumed width of the scatter. Figure 17 in Appendix A shows this effect for constant $\sigma _{M_r} = 0.5$ (left panel) and $\sigma _{M_r}$ decreasing from 0.5 to 0.3 past L* (right panel). Even with a variable width that mimics our approach, the luminosity-binned spread obtained with the method of Behroozi et al. (2010) is unrealistically large at the bright end.

The differences between the results of the two methods actually reside in the assumptions about the shape of the spread. Since our stochastic assignment scheme does not constrain the scatter distribution to be log-normal and centered on the monotonic relation, the resulting skewness beyond L* allows it to preserve the median LV relation of the scatterless sample. In addition, without a skewed distribution it is extremely difficult to obtain a narrower distribution of galaxies at the bright end of the LV relation.

From this analysis we conclude that the introduction of scatter in luminosity at a given halo circular velocity yields a median relation at the bright end that is sensitive to the shape and width of the probability distribution function used. The median LV relation is thus robust to uncertainties in the nature of the scatter below the shoulder of the LF, allowing for a direct comparison with observations. We prefer our scatter method for two reasons. First, it exactly preserves the luminosity function while the deconvolution method only does so approximately. Second, it naturally produces an observed luminosity-binned distribution that becomes narrower for the brightest galaxies, in agreement with that expected from observations.

For the LV relation, the cold baryons played an ancillary role: they provided a correction to the circular velocity at 10 kpc. The correction is small for galaxies below 100 km s⁻¹. For large galaxies the cold baryon contribution increases and typically is about half of the mass within 10 kpc. Regardless of their role in the LV relation, baryons are one of the prime subjects for the theory of galaxy formation. Unfortunately, accurate measurements of baryonic masses are also prone to some uncertainties. Dynamical measurements of the baryonic component are difficult because of DM-baryon degeneracies (e.g., Dehnen & Binney 1998). In other words, the baryon mass depends on what is assumed about the DM. Population synthesis provides an independent estimate of the stellar mass, but it has its share of complexities including the uncertainty in the IMF. In addition to the stellar mass, most galaxies have an important (if not dominant) fraction of their cold baryons in the form of neutral hydrogen gas. For consistency, in this paper we make use of stellar population synthesis estimates of stellar masses whenever possible.

The BTF relation is one way of displaying the amount of cold baryons in galaxies. The BTF relation has been investigated over the years (McGaugh et al. 2000, 2010; Bell & de Jong 2001; Verheijen 2001; McGaugh 2005; Stark et al. 2009). Here, we use the recent observational samples of Stark et al. (2009) and Leroy et al. (2008), along with Verheijen (2001) and the Geha et al. (2006) sample used for the LV relation. Stark et al. (2009) include gas-dominated spiral galaxies, which makes the results much less sensitive to the uncertainties in the IMF. For consistency, we calculate stellar masses for the Stark et al. (2009) and Geha et al. (2006) samples using a simple linear fit to the distribution of V-band mass-to-light ratios versus (B − V) color shown in Figure 18 of Blanton & Roweis (2007):

$$\begin{equation} M/L_V = 3.0(B-V) - 0.6. \end{equation} \tag{ 8 }$$

Unlike the Bell et al. (2003) models, this relation fits well the measured mass-to-light ratios of both blue and red galaxies in the SDSS. These estimates are fully compatible with the stellar masses used in the stellar mass function of Li & White (2009) as part of our model. Leroy et al. (2008) present results based on the H i Nearby Galaxy Survey (THINGS): Walter et al. (2008). The measurement of luminosities in the infrared using Spitzer results in reliable estimates of the stellar masses that are consistent with the Blanton & Roweis (2007) results. In this case we adjust their stellar masses from the Kroupa to the Chabrier (2003) IMF used by Blanton & Roweis (2007) by subtracting 0.05 dex. Selecting only galaxies with high inclinations (i > 45° or b/a > 0.7) and better than 15% accuracy in the circular velocity data leaves a total of 161 galaxies. We also include the results of mass modeling of the MW and M31 (Klypin et al. 2002; Widrow & Dubinski 2005).

We also employ Equation (8) to obtain stellar masses for the early-type galaxies. This ensures a fair comparison between the baryonic masses of early- and late-type galaxies.

Figure 10 shows the cold baryon fraction relative to the universal value as a function of stellar mass in our model. The cold baryon fraction f_b peaks at ≈0.2 for the stellar masses typical of MW type galaxies and sharply falls on both sides of the mass spectrum. Our results are broadly consistent with Guo et al. (2010). We note that even the peak of f_b ≈ 0.2 is almost a factor of two smaller than what a few years ago was considered a fiducial value (Mo et al. 1998).

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The BTF relation is shown in Figure 11. Theoretical estimates from abundance matching provide a good fit to observational results for galaxies ranging from dwarfs with V_circ ≈ 60 km s⁻¹ to giants with V_circ ≈ 500 km s⁻¹. In a remarkable agreement with the LV relation result, the model with AC seems to also provide a better fit to the BTF compared to the model with no contraction. There is a hint that observations show more baryonic mass for dwarfs below V_circ = 40 km s⁻¹ as compared to an extrapolation of the model. It is not clear whether this is a real problem because of the uncertainties involved in the observations. First, the small sample size could produce biased results. Second, there is an uncertainty at the faint end of the luminosity function. The results of abundance matching are sensitive to the number density of galaxies with absolute magnitudes M_r > −14, which is poorly constrained.

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As in the case of the LV relation, the model BTF relation agrees very well with the average population of galaxies in each morphological regime. Below ≃ 200 km s⁻¹ it follows late-type disks while it accurately describes massive early types above this threshold. The observations show no preference for a model with no halo contraction versus one with maximum contraction. Both cases fit well within the systematic and statistical uncertainties in the observations.

Although S0 and elliptical galaxies seem to contain slightly less mass in cold baryons than massive spirals, there is also a hint that the bimodality observed in the observed LV relation in Figures 2 and 4 between early- and late-type galaxies is not merely the result of a variation in the mass-to-light ratio. Other authors have come to the same conclusions (e.g., Williams et al. 2010; Dutton et al. 2010, 2011). This would imply that early types are not just the result of passive fading of late-type disks but are fundamentally different. It also requires that they inhabit deeper potential wells which may be the result of different formation or environmental processes. These results have deep implications for galaxy formation but in order to draw conclusions we would need consistent stellar and gas mass estimates for a larger sample of galaxies, which are not currently available.

Projecting the distribution of galaxies in the LV plane onto the luminosity axis produces the luminosity function, while projecting onto the circular velocity axis yields the circular VF of galaxies: the number–density of galaxies with given circular velocity. From a theoretical cosmology point of view, the VF is an ideal characterization because it does not include uncertain predictions for the luminosity and requires relatively modest corrections for the baryonic masses. Unfortunately, it is more difficult to obtain it from observations and so far, there have been only a few attempts to do so (Gonzalez et al. 2000; Kochanek & White 2001; Zavala et al. 2009; Chae 2010; Zwaan et al. 2010).

For the theory the starting point is the VF of DM halos (e.g., Klypin et al. 2010). For halos with V_circ < 500 km s⁻¹ it is well approximated by a power law n(> V_circ)∝V^−α_circ, where α ≈ 3. This only applies to velocities taken at the maximum of the circular velocity curves of DM halos. For galaxies, the results must be adjusted to V₁₀ and corrected for the dynamical effects of cold baryons.

The most recent measurement of the VF of nearby late-type galaxies was obtained by Zwaan et al. (2010). Their result is based on the blind H i sample of the HIPASS survey, which is complete down to $M_{\rm H\,\mathsc{i}} = 5.5\times 10^7 \mbox{ $M_\odot $}$ at a distance of 5 Mpc (Zwaan et al. 2010). Since gas-rich galaxies are thought to dominate at the low-mass end, their sample should provide an accurate measurement of the abundance of dwarfs if these galaxies contain enough neutral gas to be detected. To obtain a galaxy VF for all morphological types we also include the determination of the early-type VF done by Chae (2010), using the conversion between velocity dispersion and circular velocity found in Zwaan et al. (2010). Even though their VF was obtained indirectly using the observed relation between luminosity and stellar velocity dispersion, it agrees with previous direct measurements.

Figure 12 shows the results, as well as the modified Schechter fit to the VF of late-type galaxies (Zwaan et al. 2010) and the fit for early types found in Chae (2010). At intermediate to large masses (80 km s⁻¹ < V_circ < 400 km s⁻¹), where the completeness of the surveys is hard to question, the VF of our model sample reproduces the observed abundances reasonably well. The abundance of MW type galaxies is predicted to within 50% when AC is taken into account, and within a few percent when no contraction takes place. From our earlier analysis of the LV relation in Section 6, we are led to believe that halo contraction is needed to obtain the correct position of elliptical and S0 galaxies in the plot. A more detailed treatment of AC might be necessary in order to better match the abundance of galaxies larger than the MW. Our model galaxy VF overestimates the abundance of the most massive and rarest galaxies with V_circ > 400 km s⁻¹ regardless of whether or not we implement the correction for contraction of the halos. Most of these extremely bright galaxies inhabit the centers of clusters, where it is very likely that the simplistic observational estimate of V_circ is breaking down. At small velocities (V_circ < 80 km s⁻¹) the theory significantly overpredicts the number of dwarfs. This "missing dwarfs" problem remains unresolved in ΛCDM (Tikhonov & Klypin 2009; Zavala et al. 2009; Zwaan et al. 2010). It should be noted that the variance of the VF of the model galaxies below 60 km s⁻¹ in regions of radius 5 Mpc can be as large as one order of magnitude. This shows that environmental bias may be an important factor in explaining the underabundance of dwarfs in our model compared to HIPASS.

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To illustrate the effect that each of the steps in our procedure has on the VF, we show in Figure 13 the VF of DM halos only. It also shows that when the stripping due to the merger history of each halo is considered, the halo VF does a slightly better job at matching the abundance of galaxies.

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As we previously noted, the corrections due to the presence of the cold baryonic component affect dwarfs (V_circ < 100 km s⁻¹) very little, resulting in a negligible shift in their abundance compared to that of their host DM halos at the low-mass end of the VF in Figure 12. One interpretation of this is that the dwarf overabundance problem cannot be resolved if both the LV relation and the VF of dwarf galaxies are to be reproduced simultaneously. In other words, these galaxies must undergo a process that limits their abundance without changing their dynamical mass. The first possible origin for the large discrepancy between our model galaxies and the HIPASS VF could be observational bias. HIPASS is a blind H i survey and does not detect gas-poor galaxies. Only if gas-poor dwarf spheroidals dominate the galaxy population below ∼100 km s⁻¹ would it be possible to reconcile our results with the survey. This is highly unlikely since these types of galaxies are only a small fraction of the total dwarf population. On the other hand, if the HIPASS H i mass detection limit (5.5 × 10⁷ M_☉ at 5 Mpc) is relatively high at the distances where most of their sample is found, incompleteness effects might explain the discrepancy.

Assuming that the surveys are complete, a possible solution to the problem is a mapping of all the dwarf galaxies below 50 km s⁻¹ to DM halos in the range of 50–100 km s⁻¹. This in turn implies that the measured rotation curves of a large fraction of dwarfs must severely underestimate the true maximum circular velocities of these galaxies. The only possible explanation for this bias would be that the optical and H i disk is truncated well inside the radius where the rotation curve flattens out. Another solution to the missing dwarf problem requires most of these galaxies to have a low enough surface brightness in H i to be undetectable in current surveys. This would imply the existence of a large number of small halos containing little or no neutral gas.

The most important success of the HAM technique is considered to be reproducing the observed galaxy clustering measured in the form of the galaxy correlation function in its various forms, both in the local universe and at high redshift (Tasitsiomi et al. 2004; Conroy et al. 2006; Guo et al. 2010; Wetzel & White 2010). Most of these works claimed to match the observed clustering although they relied on simulations with either very low resolution or outdated cosmological parameters. The high resolution and large volume of the Bolshoi simulation allow us to calculate the galaxy two-point correlation function at a range of scales comparable to the latest results from the final data release of the SDSS (Zehavi et al. 2011). In addition, its up-to-date set of cosmological parameters allows for a direct comparison between observations and the predictions of ΛCDM+HAM. The comparisons in this section do not make use of the dynamical corrections that were necessary to obtain the LV relation and the VF. Instead, the calculation of the galaxy correlation function only requires the position and velocity information of the halos in the simulation along with their luminosities obtained from HAM. This makes the correlation function an even more robust prediction of the model.

In order to compare our model with observations, we use the most recent measurement of the SDSS galaxy projected autocorrelation function done by Zehavi et al. (2011). To make the best comparison possible we use projected galaxy separations (a direct observable) and the same luminosity and projected radii bins as Zehavi et al. (2011). We also integrate along the line of sight using the same distance bins while including the peculiar velocities of the model galaxies in the redshift calculation. The integration is traditionally performed to wash out the effects of redshift distortions. We limit the calculation of the correlation function to distances below 30 h⁻¹ Mpc to avoid scales at which much of the power comes from long waves that are absent in the simulation due to the finite box size. The small-scale correlation function of DM halos is extremely sensitive to the abundance of satellite halos near the centers of hosts. As a result of this, the clustering in N-body simulations could be underestimated due to artificial disruption of just a few satellites. Since it is beyond the scope of this paper to perform a comprehensive study of this effect, we choose to compare our model to galaxies in the range −19 > M_r − 5log h > −22. In this and all following sections we refer to the ^0.1r band in shorthand as simply the r band.

Figure 14 shows the projected two-point correlation function of the model galaxies in Bolshoi and compares it to the full SDSS sample results. The clustering amplitude of the model galaxies is in excellent agreement with observations for galaxies with luminosities around L* in the range of −20 > M_r − 5log₁₀h > −21. Model galaxies with luminosities −19 > M_r − 5log₁₀h > −21 agree very well with the observations at scales beyond 1–2 h⁻¹ Mpc where the clustering is dominated by halos of different hosts (the so-called two-halo term). Below 1 h⁻¹ Mpc there is a marked decline in the number of pairs as the separation decreases. For bright galaxies with −21 > M_r − 5log₁₀h > −22 the situation is different; ΛCDM + HAM slightly overpredicts the clustering over all scales, with the disagreement increasing to ∼30% beyond 10 h⁻¹ Mpc.

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The discrepancy in the clustering of the faintest bin at small separations may be a result of numerical effects such as artificial disruption or halo misidentification in dense environments. A small deviation at the closest separations (r_p < 400 h⁻¹ kpc) is likely to have the same origin. Since the fraction of halos that are satellites decreases sharply at large halo masses (see Klypin et al. 2010), the model correlation functions of the brightest galaxies do not suffer from these effects. Further scrutiny is necessary to understand the origin of the effect and make more robust comparisons with observations.

6.5.1. Effect of Scatter on the Correlation Function

Since only the brightest galaxies in the LV relation are affected by scatter, the obtained two-point correlation function of galaxies brighter than M_r ≈ −22 will be sensitive to the choice of scatter distribution. In the past few years, some studies of the correlation function of DM halos have suggested the scatter to be an essential ingredient in reproducing the observations (e.g., Tasitsiomi et al. 2004; Wetzel & White 2010; Behroozi et al. 2010).

Figure 15 shows the galaxy autocorrelation function obtained for our model galaxies using the stochastic method described in Appendix A to perform HAM. Here, we assume the same distribution described in Section 6.2.4, with $\sigma _{M_r} \approx 0.5$ below L* and $\sigma _{M_r} \approx 0.3$ above. The correlation amplitude of model galaxies fainter than M_r − 5log₁₀h = −21 is essentially unchanged compared to the monotonic assignment result shown in Figure 14. The clustering of the bright galaxies with −22 < M_r − 5log₁₀h < −21 shows a slight decrease at all separations except the smallest ones, and thus better agreement with the SDSS data. The amplitude decreased due to the fact that the same galaxies now get assigned to less massive halos on average, and these halos are less clustered. This is consistent with the upward shift in the average luminosity of the brightest galaxies in the LV relation (Figure 9). The correlation functions of galaxies in fainter bins are indistinguishable from those without scatter.

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In summary, applying our physically motivated scatter model maintains (and even improves) the excellent agreement of the ΛCDM+HAM model with the observed galaxy clustering. The clustering of the most massive and rare galaxies (those above M_r ≈ −22) will be more sensitive to the addition of scatter and the model used to implement it. Once other dominant sources of uncertainty in the simulations and observations are better understood, a robust test of the cosmological model could be done using these objects.

To perform HAM while including scatter in luminosity as a function of circular velocity we perform the following procedure. For brevity, wherever we use M_r we refer to $M_{^{0.1}r} - 5\log _{10} h$.

• 1.  
  Start with the monotonic assignment described in Section 5: order the list of DM halos from largest to smallest V_circ. Using the integral LF, solve for the unique luminosities of the galaxies that have the same number densities as the DM halos in Bolshoi. This matching gives the monotonic relation M^mono_r(V_circ).
• 2.  
  For the halo with the largest V_circ, draw a luminosity value M_r at random from a Gaussian distribution of width σ centered at a point 1σ brighter than the value of M^mono_r for that halo. Mathematically, this is equivalent to $M_r = M_r^{\rm mono}(V_{\rm circ}) - \sigma + \mathcal {G}(0,\sigma)$, where $\mathcal {G}(0,\sigma)$ is a random realization of a Gaussian probability distribution with standard deviation σ centered at zero.
• 3.  
  If the randomly drawn M_r is brighter than the brightest galaxy in the LF, another draw is performed and the process is repeated until a suitable value is found.
• 4.  
  The random M_r is compared to the list of available M_r values in the list. The closest available value becomes the luminosity of that halo. If the value is already taken, another random draw is performed until an available one is found. This luminosity value is flagged to prevent it from being used again for another halo. This step ensures that the observed luminosity function is preserved by only assigning each luminosity once.
• 5.  
  In order to avoid having unassigned values of M_r in the list on the bright tail of the distribution, we check whether there is any unassigned luminosity which is more than 3σ brighter than M^mono_r. If such unassigned value exists, the next halo in the list is assigned to it. Since we are stepping along the list of luminosities as we assign them, the process is intrinsically asymmetric and there will always be some leftover M_r values that get assigned in this step. The offset used in step 1 ensures that these make up only a few percent of the sample.
• 6.  
  Repeat steps 2–5 for each halo in the ordered list until the 3σ faint tail of the Gaussian for a given halo reaches the completeness limit of the sample as defined in Section 3. The procedure is stopped at this point to prevent placing a hard constraint on the faint end of the LF where it is most uncertain. The faintest halos (up to 0.5 mag brighter than the cutoff) are removed from the sample to insure the preservation of the LF throughout.

In spite of the fact that a constant Gaussian distribution is used in the algorithm, the final distribution of M_r for a given circular velocity V_circ is not a Gaussian and the width of the M_r–V_circ relation is not constant. This happens because of the asymmetry in the distribution of the galaxies: there are always more galaxies with smaller luminosities than with larger ones. Thus, the distribution and the spread of the final M_r–V_circ relation are the result of a convolution of a Gaussian distribution with the luminosity function.

The effect of the asymmetry of the LF is more pronounced for the brightest galaxies in the sample and accounts for the small shift in the median compared to the monotonic assignment. The algorithm also gives a natural narrowing of the distribution of luminosities as V_circ increases. This occurs because in the exponential tail of the LF the number of available M_r values changes rapidly across the width of the Gaussian. Increasingly fewer available values on the bright side of the median force the selection of most values to take place in a narrower interval on the faint side.

Since our assignment method reduces the width of the obtained distribution of luminosities, we choose the input value σ = 0.7. This yields a distribution with $\sigma _{M_r} \approx 0.5$ below ∼200 km s⁻¹ and gradually decreasing to $\sigma _{M_r} \approx 0.3$ above ∼300 km s⁻¹.

Figure 16 shows the distribution of luminosities in four bins of circular velocity, V₁₀. Below 200 km s⁻¹, galaxies show a near-normal distribution that is centered very close to the values obtained from the monotonic assignment without scatter. For V_circ > 250 km s⁻¹ the distributions get progressively more skewed as galaxies move from the bright to the faint tail. The spread of the distribution of luminosities increases with V_circ: the widths are $\sigma _{M_r} \approx 0.50$, 0.45, 0.43, and 0.35 for the bins centered at 102.5, 205.0, 307.5, and 520.0 km s⁻¹, respectively.

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Figure 17 shows the LV relation obtained with the scatter model of Behroozi et al. (2010) in the case of constant scatter (left panel) as well as assuming the same variable width used in our model: $\sigma _{M_r} \approx 0.5$ for V_circ < 250 km s⁻¹ declining to ∼0.3 for larger V_circ. Evidently, the median of the distribution in luminosity stays relatively unchanged only when forcing a small scatter width at the bright end. Even when we impose the same variable width used in the stochastic HAM scheme, the resulting spread is too large for the brightest galaxies.

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Table 1 gives some of the properties of our early-type sample along with the source of data for each galaxy.

Table 1. Luminosity and Circular Velocity Data for Nearby Early-type Galaxies

Name	Type	MB	±	$\log _{10}\left(\frac{M_*}{M_\odot }\right)$	V10	+	−	V15	+	−	V20	+	−	Probe	Ref.
NGC 4889	E	−22.61	0.16	11.87	520	60	50	...	...	...	...	...	...	stars	T+07
NGC 4874	E	−22.51	0.16	11.65	500	30	50	510	60	70	...	...	...	stars	T+07
NGC 0315	E	−22.27	0.25	11.66	520	10	5	530	30	10	550	35	20	stars	K+00
NGC 1316	S0	−22.20	0.19	11.46	381	29	29	466	52	52	361	49	49	X-ray	NM09
NGC 4839	E	−22.14	0.15	11.52	385	65	30	400	80	50	425	105	70	stars	T+07
NGC 0057	E	−22.06	0.19	11.56	491	20	49	494	19	52	495	22	52	X-ray	O+07
NGC 4555	E	−22.05	0.21	11.50	614	58	62	606	61	66	598	63	74	X-ray	OP04
NGC 4952	E	−21.75	0.19	11.37	405	10	15	435	20	30	455	25	45	stars	T+07
NGC 6407	E/S0	−21.73	0.26	11.51	445	25	35	450	50	40	460	60	60	stars	MB01
NGC 4472	E	−21.64	0.12	11.41	415	40	40	...	...	...	...	...	...	stars	MB01
NGC 7626	E	−21.56	0.20	11.41	420	9	10	410	15	15	390	20	15	stars	K+00
NGC 5044	E	−21.49	0.27	11.31	273	17	17	328	11	11	365	13	13	X-ray	NM09
NGC 4486	E	−21.40	0.16	11.31	503	47	34	...	...	...	...	...	...	GCs	M+11
NGC 3923	E	−21.35	0.43	11.26	365	28	28	324	40	40	324	40	40	X-ray	NM09
NGC 4816	E/S0	−21.35	0.23	11.19	300	45	40	300	50	40	310	50	40	stars	T+07
NGC 1395	E	−21.32	0.17	11.26	374	17	17	...	...	...	...	...	...	X-ray	NM09
NGC 4944	S0	−21.31	0.32	11.19	275	2	2	280	2	2	280	2	2	stars	T+07
NGC 4382	S0/a	−21.31	0.16	11.12	260	19	19	260	19	19	198	51	51	X-ray	NM09
NGC 4649	E	−21.29	0.16	11.27	425	10	10	436	21	22	444	35	37	X-ray	HB10
NGC 4374	E	−21.25	0.12	11.26	382	20	20	386	20	20	393	20	20	PNe	N+11
NGC 4827	E/S0	−21.25	0.23	11.20	350	50	30	...	...	...	...	...	...	stars	T+07
NGC 0128	S0	−21.20	0.20	11.18	370	14	14	361	15	15	...	...	...	stars	W+09
IC 1459	E	−21.17	0.21	11.26	338	53	53	258	53	53	282	45	45	X-ray	NM09
NGC 4957	E	−21.14	0.19	11.21	325	2	2	310	10	2	295	15	2	stars	T+07
NGC 4261	E	−21.04	0.20	11.20	362	27	29	338	27	29	332	30	33	X-ray	HB10
NGC 6703	E/S0	−21.07	0.20	11.11	220	20	15	...	...	...	...	...	...	stars	K+00
NGC 3665	S0	−21.06	0.23	11.13	423	52	52	404	32	32	404	32	32	X-ray	NM09
NGC 5846	E	−21.04	0.24	11.19	340	5	5	...	...	...	...	...	...	stars	K+00
NGC 4908	E	−21.00	0.15	11.19	320	40	40	...	...	...	...	...	...	stars	T+07
NGC 0720	E	−20.99	0.18	11.18	317	12	13	323	13	13	330	11	12	X-ray	HB10
NGC 7796	E	−20.99	0.20	11.15	308	46	26	305	37	28	297	35	26	X-ray	O+07
NGC 4365	E	−20.98	0.18	11.15	333	26	26	333	26	26	...	...	...	X-ray	NM09
NGC 3607	E/S0	−20.91	0.20	11.07	278	28	28	278	28	28	265	28	28	X-ray	NM09
NGC 1399	E	−20.88	0.19	11.11	430	25	30	...	...	...	...	...	...	stars	K+00
NGC 3585	E	−20.77	0.22	10.99	295	46	46	...	...	...	...	...	...	X-ray	NM09
NGC 2974	E	−20.76	0.20	11.06	304	10	10	...	...	...	...	...	...	Gas	W+08
NGC 5084	S0	−20.73	0.21	11.05	282	8	8	...	...	...	...	...	...	stars	W+09
NGC 6771	S0/a	−20.72	0.17	10.92	340	18	18	331	18	18	320	23	23	stars	W+09
NGC 4807	E/S0	−20.64	0.23	10.99	295	30	15	...	...	...	...	...	...	stars	T+07
NGC 4931	S0	−20.62	0.23	11.02	280	5	15	275	10	25	270	15	30	stars	T+07
NGC 0821	E	−20.58	0.21	10.88	182	13	13	...	...	...	...	...	...	stars	FG10
NGC 1332	E/S0	−20.56	0.22	10.94	291	9	10	291	9	10	291	9	10	X-ray	HB10
IC 0843	S0	−20.55	0.19	10.83	380	10	5	340	20	10	320	25	15	stars	T+07
ESO151-G004	S0	−20.48	0.26	11.10	291	19	19	308	19	19	295	25	25	stars	W+09
NGC 4494	E	−20.40	0.17	10.77	198	10	10	188	14	14	184	18	18	PNe	N+09
NGC 4869	E	−20.38	0.15	10.91	280	50	30	...	...	...	...	...	...	stars	T+07
NGC 4636	E	−20.38	0.17	10.85	430	16	16	491	22	22	538	29	29	X-ray	NM09
NGC 1032	S0/a	−20.33	0.21	10.82	270	20	20	...	...	...	...	...	...	stars	W+09
NGC 4697	E	−20.20	0.18	10.73	235	15	10	234	17	19	231	23	26	PNe	dL+08
IC 4045	E	−20.19	0.15	10.85	390	40	30	...	...	...	...	...	...	stars	T+07
NGC 3203	S0/a	−19.89	0.25	10.47	229	7	7	...	...	...	...	...	...	stars	W+09
NGC 3379	E	−19.84	0.11	10.69	206	31	13	192	42	19	181	48	22	PNe	dL+09
NGC 3957	S0/a	−19.24	0.20	10.38	199	13	13	...	...	...	...	...	...	stars	W+09
NGC 4710	S0/a	−19.10	0.21	10.12	182	10	10	...	...	...	...	...	...	stars	W+09
NGC 4469	S0/a	−18.77	0.17	10.22	182	13	13	...	...	...	...	...	...	stars	W+09

Notes. The galaxies are sorted by absolute magnitude M_B, which is corrected for Galactic extinction; the quoted errors include the statistical uncertainties in distance and photometry. The stellar masses are based on B−V colors, and correspond to a Chabrier IMF (see the text). Circular velocities V₁₀, V₁₅, and V₂₀ are measured at 10, 15, and 20 kpc, respectively, and are in units of km s⁻¹. dL+08: de Lorenzi et al. 2008; dL+09: de Lorenzi et al. 2009; FG10: Forestell & Gebhardt 2010; HB10: Humphrey & Buote 2010; K+00: Kronawitter et al. 2000; M+11: Murphy et al. 2011; MB01: Magorrian & Ballantyne 2001; N+09: Napolitano et al. 2009; N+11: Napolitano et al. 2011; NM09: Nagino & Matsushita 2009; OP04: O'sullivan & Ponman 2004; O+07: O'sullivan et al. 2007; T+07: Thomas et al. 2007; W+08: Weijmans et al. 2008; W+09: Williams et al. 2009.

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