THE VELOCITY FUNCTION IN THE LOCAL ENVIRONMENT FROM ΛCDM AND ΛWDM CONSTRAINED SIMULATIONS

Despite the success of the lambda cold dark matter (ΛCDM) paradigm in describing the large-scale structure of the universe, as has been attested by improved measurements of the cosmic microwave background (CMB) and the large-scale clustering of galaxies (Seljak et al. 2006b; Komatsu et al. 2009), some potential problems remain for the model at smaller scales. One of these challenges is related to the prediction of the abundance of low-mass galaxies. Numerical simulations within the ΛCDM model predict many more dark matter subhalos inside galactic-sized host halos than the actual number of observed satellite galaxies around the Milky Way (Klypin et al. 1999; Moore et al. 1999; Diemand et al. 2007; Springel et al. 2008). This discrepancy has been known as the "missing satellite problem." Although some astrophysical processes can be responsible for the suppression of galaxy formation in small-mass halos and lead to the solution of the problem (Somerville 2002; Benson et al. 2002b, 2002a; Gnedin & Kravtsov 2006; Koposov et al. 2009), it is also possible to alleviate the discrepancy by adopting alternative models with a different type of dark matter.

Warm dark matter (WDM) models, for instance, predict significantly less substructure within halos (Colín et al. 2000; Bode et al. 2001). The free-streaming length for dark matter particles depends on their intrinsic properties and establishes a cutoff scale for halo masses below which primordial density perturbations are wiped out. In the case of neutralinos, one of the favorite candidates for CDM models, with m_χ ∼ 100 GeV, the scale is roughly 10⁻⁶ M_☉ (Diemand et al. 2005). For WDM candidates, such as gravitinos with $m_{\tilde{G}}\sim 1$ KeV (see Steffen 2006 for a recent review on gravitino cosmology), this scale is roughly 10¹⁰ M_☉ (see Section 2). Therefore, using the same cosmological parameters but different cutoff scales results in different predictions for the halo mass function (MF). At the low-mass end, the ΛWDM MF is expected to be significantly lower than the ΛCDM MF. Above the cutoff scale, the differences vanish (e.g., Bode et al. 2001; Barkana et al. 2001).

Since for high-density environments both models provide, conceptually different, solutions for the missing satellite problem, it is also important to test their predictions for isolated systems in low-density environments which are not affected by processes such as ram pressure or tidal striping. A comparison between ΛWDM and ΛCDM is then free from these astrophysical phenomena and conclusions drawn from it are more closely related to the nature of dark matter itself. Such an approach has been employed, for example, by Blanton et al. (2008).

Even though low-mass galaxies are difficult to detect, surveys such as the ongoing Arecibo Legacy Fast ALFA (ALFALFA) survey (Giovanelli et al. 2005b) are promising to make such comparison. ALFALFA is exploring the local H i universe with a sky coverage of 7000 deg² and it is designed to detect objects with masses as low as 3 × 10⁷ M_☉ at a Virgo-cluster distance (11.7 h⁻¹ Mpc; Giovanelli et al. 2007). The survey is divided into two regions on the sky. One within the solid angle 07^h30^m< R.A.(J2000.0) <16^h30^m and 0°< decl.(J2000.0) <+36°, which includes the Virgo cluster, and another region within the same declination range and 22^h< R.A.(J2000.0) <03^h. In the reminder of this work, we will loosely refer to these two regions, additionally constrained to distances less than 20 h⁻¹ Mpc, as the "Virgo-direction region (VdR)" and "anti-Virgo-direction region (aVdR)," respectively. We note that the catalogs released by ALFALFA so far in these two regions effectively correspond to a high- and a low-density environment. The H i sources detected by ALFALFA are typically associated with star-forming disk galaxies, spirals in rich clusters are less likely to be detected since these galaxies might be H i deficient (Solanes et al. 2001).

In order to make predictions on the abundance of low-mass galaxies based on the ΛCDM and ΛWDM models for our local environment, it is advantageous to use constrained simulations (CSs) of the local universe (e.g., Bistolas & Hoffman 1998; Mathis et al. 2002; Klypin et al. 2003). These simulations are constructed to reproduce the gross features of the nearby universe, such as the Local Supercluster (LSC) and the Virgo cluster. This can be achieved by setting the initial conditions of the simulations as constrained realizations of Gaussian fields, where actual observational data are used to impose the constraints. By using such simulations, biases which are present because of the particular structure of the local environment, are minimized and a better comparison between simulations and observations can be achieved.

Although the abundance of dark matter halos is usually quantified by the MF, which describes the number density of halos as a function of mass, usually presented either in a differential or integral form (e.g., Reed et al. 2007; Lukić et al. 2007), an alternative approach is to use the velocity function (VF), similarly defined as the number density of halos as a function of maximum circular velocity (e.g., Gonzalez et al. 2000). The advantage of the VF is that it can be compared more directly with observational results since it avoids the more complicated problem of relating dark matter halo masses to galaxy luminosities. Instead, the VF of galaxies can be theoretically estimated using the VF of halos by incorporating a model of baryonic infall; the only processes that affect the VF are those that modify the gravitational potential of galactic systems. In the present work, we show analysis of both the mass and VFs, but concentrate on the latter for comparisons with the ALFALFA survey.

The objective of our paper is to run a ΛCDM and a ΛWDM simulation of the local environment where the same set of constraints have been imposed and to compare the abundance and distribution of dark matter halos for the two cosmologies. In addition, the results are used to predict the abundance of H i sources that will be detected by the ongoing H i blind survey ALFALFA.

The paper is organized as follows. In Section 2, we describe the setting of the simulations. The definition of the coordinate system that we use to impose the field of view of ALFALFA in the CSs is described in Section 3. In Section 4, we present results on the abundance of dark matter halos. In Section 5, we analyze in particular the abundance for the restricted VdR and aVdR cataloged by the ALFALFA survey to date and present predictions on the VF of H i sources. The summary and conclusions of our work are given in Section 6.

We chose the cosmological parameters for our simulations to be consistent with the WMAP 3 year results (Spergel et al. 2007): Ω_m = 0.24, Ω_Λ = 0.76, and H₀ = 100 h km s^-1 Mpc⁻¹ with h = 0.73, n = 0.95, and σ₈ = 0.75. For both cosmologies, the theoretical (unconstrained) linear power spectrum at z = 0 is shown in Figure 1 (blue for CDM and red for WDM). The CDM power spectrum was computed from a Boltzmann code by W. Hu and was kindly provided to us.

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2.1. ΛWDM Simulation Settings

2.2. Constrained Simulations

The initial conditions for the CSs were set up using the (Hoffman & Ribak 1991) algorithm of constrained realizations of Gaussian random fields. Two types of data sets are used as input for the algorithm. The first data set is made of radial velocities of galaxies drawn from the catalogs: MARK III (Willick et al. 1997), surface brightness fluctuations (Tonry et al. 2001), and the Catalog of Nearby Galaxies (Karachentsev et al. 2004). Present epoch peculiar velocities are less affected by nonlinear effects and are therefore imposed as linear constraints on the primordial perturbation field (Zaroubi et al. 1999). This approach follows the CSs performed by Kravtsov et al. (2002) and Klypin et al. (2003). The other data set is obtained from the catalog of nearby X-ray selected clusters of galaxies (Reiprich & Böhringer 2002). Assuming the spherical top-hat model and using the virial parameters of a cluster, the linear overdensity of the cluster is derived. The estimated linear overdensity is imposed on the virial mass scale of the cluster as a constraint. The density and velocity fields on scales larger than 5 h⁻¹ Mpc are strongly constrained by the imposed data.

Using the above initial conditions, we carried out the simulations using the code GADGET-2 (Springel 2005). Both simulations follow the evolution of 1024³ dark matter particles from z = 50 to z = 0 in a box of size L = 64 h⁻¹ Mpc. The associated Nyquist frequency and fundamental mode are represented with vertical lines in Figure 1. The dark matter particle mass is m_DM = 1.63 × 10⁷ h⁻¹ M_☉. The simulations were started with a fixed comoving softening length (Plummer equivalent) of = 1.6 h⁻¹ kpc. Once the corresponding physical comoving softening reached = 0.8 h⁻¹ kpc, it was kept constant at this value.

Figure 2 displays the projected dark matter distribution of the CSs at z = 0 for the ΛCDM and the ΛWDM models, respectively. The highest and lowest concentrations are colored red and black, respectively. Both panels show the matter distribution within a slice that is 8 h⁻¹ Mpc thick projected onto the X–Y plane and centered on Z = 24 h⁻¹ Mpc. The figure captures the LSC which is the filamentary structure crossing the image plane horizontally. It is the most prominent feature. The locations of the virtual Local Group (LG)⁶ and the Virgo cluster are marked as well. A description of the identification of these objects will follow below. A visual inspection of the two figures already reveals a deficit of small-scale structure in the ΛWDM simulation, as expected.

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Halos in the simulations were identified using AHF,⁷ AMIGA's halo finder (Knollmann & Knebe 2009). AHF identifies halos as local density maxima in an adaptively smoothed density field using a hierarchy of grids and a refinement criterion. The latter was chosen so that a grid cell is subdivided until each subsection contains less than five particles. In this way, the size of the smallest cells is comparable to the force resolution of our simulations. Halos, and/or subhalos, are formed by groups of particles that are gravitationally bound to a given density peak. For each halo in the final catalog, AHF provides a list of internal properties, the most important quantities for the current study are the viral radius r_vir, defined as the radius that contains a mean overdensity $\bar{\rho }(r_{\rm vir})=\Delta \rho _{\rm crit}$, where ρ_crit is the critical density and we have chosen Δ ∼ 94 at z = 0, a value computed according to the spherical collapse model for our cosmological parameters;⁸ the corresponding virial mass M_vir; the maximum rotational velocity V_max; and the spin parameter $\lambda =J\sqrt{\vert E\vert } / GM_{\rm vir}^{5/2}$ (Peebles 1969), where J and E are the total angular momentum and energy of the halo. Subhalos identified within each halo were removed from the catalog; for the reminder of our analysis we will use main halos only.

The significance of our study relies on a proper simulation of the local environment. However, the constraints imposed on the initial conditions of the simulations leave some freedom for the evolution, mainly on small scales (<5 h⁻¹ Mpc). This can partly be accounted for by a careful adjustment of the coordinate system. We have done so using the following steps. First, we identified an appropriate LG within the CSs. To that purpose, we followed the criteria described in Macciò et al. (2005; see also Table 2 of Martinez-Vaquero et al. 2007) to identify LG candidates within the CSs. These are (1) match in mass, proximity, and kinematics of an MW–M31-halo-like binary system, i.e., we look for a pair of halos with a maximum rotational velocity, V_max, between 125 km s^-1 and 270 km s^-1, with a distance between the pair ≲1 h⁻¹ Mpc and with a negative relative velocity; (2) absence of a massive nearby halo, i.e., no halos with masses larger than the members of the pair within a radius of 2 h⁻¹ Mpc; and (3) presence of a Virgo-like halo, 500 < V_max < 1500  km s^-1, at the appropriate distance, 5–12 h⁻¹ Mpc.

In addition, we favor LG candidates which are located close to the center of the simulation box since the constrained initial conditions place the LG progenitor exactly at the center. Subsequent dynamical evolution may displace the entire environment by some Mpcs.

Slices containing the best LG candidates for the CDM and WDM run are shown in Figure 2 where we have marked the location of the LG and the Virgo cluster. If not stated otherwise, "LG" and "Virgo" denote these best possible candidates. Table 1 contains some properties of the main objects in our simulations: the LG, and the clusters Virgo and Fornax (see below). Observed estimates for these properties are also given in the table.

Based on the locations of the LG, Virgo, and the local environment, we now aim to define a supergalactic coordinate system (de Vaucouleurs et al. 1991, Chapter XII; see also Lahav et al. 2000). We will use such coordinate system as the basis to identify the ALFALFA regions in the (simulated) sky. To define it, we first assume that the equatorial plane of the supergalactic coordinate system lies in the supergalactic plane (SGP), which is spanned by the LG and the LSC. Thus, besides Virgo, we need to find another cluster belonging to the LSC to mathematically define the LSC. As revealed by Figure 2, there is a prominent cluster on the right-hand side of Virgo. Its location and mass closely resemble those of the observed Ursa Major cluster (1.6 × 10¹³ h⁻¹ M_☉ at a distance of ∼11.1 h⁻¹ Mpc from the LG). Albeit this choice seems natural, we have also tested other clusters within the simulated LSC to define alternative coordinate systems. The final results turned out to be almost independent of this choice. Using this SGP, we construct an orthonormal vector basis with the origin placed at the MW and rotate it until the simulated Virgo cluster is located at the same longitude of the observed one, i.e., at a supergalactic longitude (SGL) of 10245. Finally, we tilt the plane to achieve a supergalactic latitude (SGB) of 284 to match the one that is observed. With this procedure, the simulated Virgo is located at the same angular position as the observed one. Below we will introduce an alternative coordinate system. To avoid confusion, we refer to this first one as SG_zero.

The result of our coordinate definition is visualized in the left panel of Figure 3. It shows a sky map with the angular distribution of halos in equatorial coordinates (R.A., decl.) in a Mollweide projection for the ΛCDM simulation. Only halos within 20 h⁻¹ Mpc from the origin (simulated MW) are included. The size of the sphere is limited by the size of the simulation box. Larger radii are likely to contain spurious structures caused by periodic boundary conditions. The map was created using the HEALPIX software⁹ using N_pix = 12(64)² pixels. The value of each pixel is given by a mass-weighted count of all halos located in that pixel. Afterward, a smoothing of the map was done using a Gaussian beam with an FWHM of 7°; the different color scale is a measure of the values in the map, from red-to-blue for high-to-low values. The angular positions of all halos with masses larger than 5 × 10⁹ h⁻¹ M_☉ also appear in the map as black points. The voids and high-density regions within the local simulated volume can be clearly appreciated in the sky map. The prominent high-density region crossing the whole map almost vertically at the center is the simulated LSC with Virgo roughly in the middle (black circle). By construction, the location of the real Virgo (black square) is identical with the simulated one. The boxes in the center and on the sides of the map give the boundaries of the VdR and aVdR, respectively, accessible to the ALFALFA survey.

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Since the LSC is roughly in place in our CSs and because the simulated Virgo and the real one are at the same angular position in the sky (and almost at the same distance from the LG), we believe that the procedure described in the last paragraph led to an appropriate coordinate system allowing us to use our CSs to simulate the VdR of the sky surveyed by ALFALFA. However, a visual impression of the aVdR (enclosed by the boxes on the sides of the sky map in the left panel of Figure 3) indicates tentative problems with this simulated area of the sky. It contains a significant part of a filamentary structure with a high density of halos, stretching from around (0°, 0°) to (45°, +20°). Such structure is associated with the simulated Fornax cluster. This cluster was imposed as a part of the simulation constraints, see Table 1, it appears as a black circle in the right corner of the sky map. Its angular position is, however, out of place, the real angular position of the Fornax cluster is marked as the lower right black square in Figure 3.

Such deviation is within the expected variations of CSs due to intrinsic uncertainties. For instance, the velocity constraints used to produce the CSs have still large errors, making the random component of the CSs more significant. Also, the present day positions of the massive clusters obtained from observational data are imposed on the initial conditions of the CSs. Both of these conditions imply that the dynamical evolution of the simulations shifts the positions of the clusters and modify the structures finally obtained in the CSs. Furthermore, since scales smaller than ∼5 h⁻¹ Mpc are unconstrained, the position of the LG is not imposed directly by the constraints. Finally, to be able to explore low-mass halos we have used a small box size for the CSs; since the LSC stretches from one side of the box to the other, periodic boundary conditions distort the shape of the LSC. Despite all these difficulties, we are confident that the CSs and the choice we have made for the location of the LG are reliable enough to make the comparison we intend in this work.

The result to keep in mind is that Fornax appears inside the aVdR whereas in reality it does not. Then, by using SG_zero we would overpredict the abundance of halos in this region since by mistake our simulated survey probes a region of higher density than the one expected from observations.

To ameliorate this problem, we carried out an additional adjustment of the coordinate system. We relax the requirement that the angular position of the simulated Virgo has to coincide exactly with the angular position of the real Virgo. Instead, we look for a coordinate system with the same origin, and that minimizes the quadratic sum of the distances between the simulated and real clusters Virgo and Fornax. We refer to this coordinate system as SG_min. The distribution of halos after this rotation can be seen in the sky map at the right panel of Figure 3. In the coordinate system SG_min, the distances between the real and simulated Virgo and Fornax clusters are 4.5 h⁻¹ Mpc and 5 h⁻¹ Mpc, respectively. The main effect of this rotation is that the (ALFALFA) aVdR region no longer includes significant parts of the filamentary structure associated with the Fornax cluster.

We are confident that the adoption of the new coordinate system, SG_min, is consistent with the freedom inherent to the CSs. In Section 5.1, we investigate the impact of the rotation in a more general sense. There we can show that minor rotations, as the one adopted for the final adjustment of the coordinate system, change the halo abundance in the VdR by less than 30%. Thus, the adjustment of the coordinate system introduces only a small uncertainty that we will, however, keep in mind for the interpretation of our results. Here, we have only discussed the adjustment of the ΛCDM coordinate system, we repeated the same procedure for the ΛWDM simulation.

4.1. Global Mass Function

The blue and red solid lines in the upper panel of Figure 4 display the differential MF for the ΛCDM and ΛWDM CSs, respectively. Analytical predictions for both cosmologies are shown as dashed lines following the Sheth and Tormen formalism (S–T formalism; Sheth & Tormen 1999, 2002; Sheth et al. 2001). For their computation, we used the public code described in Reed et al. (2007). In this prediction, the only difference between both cosmologies is the suppression of the power spectrum at small scales in the ΛWDM case. The statistical error bars presented in the figure are Poisson errors, employing the definition given in Lukić et al. (2007): $\sigma _{\pm }=\sqrt{N+1/4}\pm 1/2$, where N is the number of halos per bin. The value of the filtering mass for the WDM simulation, 1.1 × 10¹⁰ h⁻¹ M_☉, is marked in the figure with a vertical solid line.

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At the high-mass end, there is an excess of massive halos in our simulations. This is caused by the constraints which enforce the growth of a very massive structure, the LSC, within the relatively small volume of the simulation box. At the low-mass end, we see discreteness features as described in Wang & White (2007). According to their study, the limiting mass that can be trusted is given by the formula: $M_{\rm lim}=10.1\bar{\rho } d k_{\rm peak}^{-2}$, where d is the interparticle separation, k_peak is the wavenumber for which k³P(k) reaches its maximum, and $\bar{\rho }=\Omega _{DM}\rho _{\rm crit}$. For the case of our ΛWDM simulation: M_lim = 3 × 10⁹ h⁻¹ M_☉. As can be seen in Figure 4, this value (indicated by the dotted vertical line) marks the mass limit below which there is an artificial rise in the WDM MF, an indication for the onset of discreteness effects.

The difference between the MF of the different simulations becomes even more clear in the lower panel of Figure 4, where we plot the ratio of the measured MFs to the value of the S–T MF for the ΛCDM cosmology. Clearly, the abundance of low-mass halos in the ΛWDM case (for masses larger than M_lim) is considerably lower than the predictions using the S–T formalism. This discrepancy has been found before (e.g., Bode et al. 2001), so one should not expect good agreement between the S–T approach in the range of masses close to and below the ΛWDM filtering mass. In the following, we will use the ΛCDM S–T MF as a reference to present some of our results.

4.2. Global Velocity Function

As was mentioned in Section 1, a more direct comparison for the abundance of structure between our CSs and observations of the local universe can be achieved by constructing the VF of dark matter halos. The maximum rotational velocities are used instead of the masses to calculate the abundance of halos per logarithmic V_max bin. Provided that a physical connection between V_max and the measured maximum rotational velocity of spiral galaxies can be established, our VF can be directly compared with observations of galactic discs. Further below, in Section 5.2, we describe a simplified model which is designed to accomplish this goal.

The upper panel of Figure 5 shows the differential VF for halos in both simulations (the line styles and colors are the same as in Figure 4). Also shown is a prediction for the VF in the ΛCDM case (dashed line) obtained using the procedure outlined in Sigad et al. (2000). In brief, the procedure is the following: (1) a "virtual" sample of halos is generated in such a way that its MF mimics the one given by the S–T formalism; (2) a concentration value taken from a log-normal distribution (Jing 2000) is assigned to each halo; mean and standard deviation values for the log-normal distribution were taken from the analysis by Macciò et al. (2008) for the set of simulations with WMAP3 cosmology: 〈logc〉 = 1.775 − 0.088 log M₂₀₀ and σ_logc = 0.132; (3) assuming a Navarro–Frenk–White (NFW) profile, we compute V_max for each halo using the concentration and virial velocity corresponding to that halo (e.g., see Equation (7) of Sigad et al. 2000). A prediction in the case of the ΛWDM simulation is not given since the previously described steps are all uncertain in this case.

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The vertical solid and dotted lines in Figure 5 are the estimated values for the maximum velocities corresponding to the filtering and limiting masses for the ΛWDM simulation. For the computation of the corresponding virial velocities, we used the relation between the virial mass of the halo and its virial radius: $M_{\rm vir}=\frac{4}{3}\pi \Delta \rho _{\rm crit}r_{\rm vir}^3$. Then, the virial velocity is simply given by V²_vir = GM_vir/r_vir. Using the same mass–concentration relation described in the paragraph above, we compute the maximum circular velocities for the filtering and limiting mass and find the values of 36 km s^-1 and 24 km s^-1.

The lower panel of Figure 5 shows the ratio of the measured VFs to the analytical one displayed as dashed line in the upper panel.¹⁰ This figure is analogous to the ratios of the MFs as shown in the lower panel of Figure 4. The analytical estimate for the VF follows closely the result of the ΛCDM CS for most of the velocity range. The difference at the high-mass end is due to the constraints imposed in the simulation.

The downward bend for the ΛCDM VF at the low-velocity end is caused by the mass cutoff for halos (∼4 × 10⁸ h⁻¹ M_☉). The reason why we do not see an abrupt cutoff is due to the spread in halo concentrations which causes a similar spread in V_max. The behavior of the ΛWDM VF is analogous to the behavior of the ΛWDM MF. Discreteness effects can be seen for velocities lower than the limiting V_max (dotted line). For velocities just above this limiting value, the difference between both simulations is approximately an order of magnitude.

4.3. Abundance of Halos in the Local Environment

We now turn our attention from the global to the local abundance of dark matter halos. In Figure 6, we show the ratio of the differential MFs to the ΛCDM S–T prediction for spheres of different sizes centered on the LG. The upper and lower panels show the results for the ΛCDM and ΛWDM simulations. In both panels, the ratios decrease with increasing radius of the spheres. The smallest sphere has a radius of 15 h⁻¹ Mpc resulting in the highest abundance (uppermost curve in each panel). The following sequence of curves corresponds to radii increased by intervals of 5 h⁻¹ Mpc up to a maximum radius of 40 h⁻¹ Mpc. The curve corresponding to a sphere with radius 20 h⁻¹ Mpc is highlighted with a thicker dash-dotted line. For reference, the result for the MF in the whole cubic box is presented as a thick solid line (see Figure 4). For spheres growing beyond the border of the simulation box, we use periodic boundary conditions to fill in the volume of the sphere.

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According to Figure 6, the local environment shows an overabundance of halos compared to the mean abundance in the whole box. Within 20 h⁻¹ Mpc (which was the radius used to produce Figure 3), the halo abundance is about two times larger than that in the entire box. In what follows all results are restricted to a sphere with radius of 20 h⁻¹ Mpc, which is close to the maximum radius that a sphere centered at the LG can have and still lie completely inside the simulation box.

5.1. Simulated Field of View of ALFALFA

Figure 7 displays the distributions of halos within the ALFALFA field of view projected onto the plane of the supergalactic coordinate system, SG_min, which was introduced at the end of Section 3. The panels on the left are for the ΛCDM simulation and on the right for the ΛWDM simulation. The red and blue dots give the position of halos within the VdR and the aVdR, respectively. Only halos with a distance to the LG less than 20 h⁻¹ Mpc and with masses larger than 5 × 10⁹ h⁻¹ M_☉ are shown. The difference in halo abundance between both regions is clearly visible.

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Figure 8 shows the radial dependence of the number density of halos, n_h, normalized to the total number density of halos, n_sph, in the 20 h⁻¹ Mpc sphere. The red and blue lines show the result for the VdR and aVdR, and the black line for the whole sphere. The upper and lower panels are for the ΛCDM and ΛWDM simulations, respectively. The peak at a distance of ∼11 h⁻¹ Mpc in the VdR (red line) is caused by halos associated with the LSC in the vicinity of the Virgo cluster. For radii larger than 10 h⁻¹ Mpc, the VdR is significantly overdense, compared to the density in the sphere, whereas the aVdR is underdense at all radii.

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The difference between VdR and aVdR is also clear in Figure 9 where we show the differential VF for both simulations (solid lines for ΛCDM and dotted lines for ΛWDM) in different regions around the LG: the whole 20 h⁻¹ Mpc sphere (black solid and dotted lines), the VdR (red lines), and the aVdR (blue lines). The values for the filtering (vertical solid line) and limiting (vertical dotted line) velocities for the ΛWDM simulation are also shown for reference.

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In both simulations, the difference between the VdR and aVdR is clear and qualitatively as expected, the former being an overdense region and the latter an underdense region, relative to the whole region contained within the sphere of 20 h⁻¹ Mpc radius. We note, however, that the aVdR has actually a similar, although smaller, density than the mean cosmological density, especially for the ΛCDM simulation. This becomes apparent from the comparison between the blue solid and black dashed lines representing the aVdR and the S–T prediction, respectively, in the ΛCDM case (see also Figure 5). This fact is partially related to the limitations of our CSs (already mentioned in detail in Section 3). For halo velocities lower than 100 km s^-1 and larger than the WDM limiting velocity (24 km s^-1), the difference between the VdR and aVdR is approximately constant in both cosmologies, the ratio of their differential VFs in this range is roughly 3.

To conclude this section, we investigate the robustness of our results in relation to the choice of the coordinate system. For that purpose, Figure 10 displays a comparison of the VFs for moderate rotations of the supergalactic coordinate system defined in Section 3. The comparison is done using the ratio of the VFs to the VF in the whole sphere with 20 h⁻¹ Mpc radius (VF/VF₂₀). The black and blue dashed lines show the ratios of the VFs for the VdR and aVdR based on the initial coordinate system, SG_zero. This coordinate system was set up to optimize the agreement between the real and simulated Virgo location, but failed to get the Fornax cluster in place. In this system, the simulated aVdR contains a significant part of the filamentary structure associated with the Fornax cluster (see left panel of Figure 3), which is not being surveyed in the observed field of view. Therefore, the abundance of halos in the simulated aVdR is unexpectedly high. The solid black and blue lines show the same quantities using SG_min. With it, the simulated aVdR does not contain Fornax any longer and the halo abundance decreases substantially. The results based on the systems SG_zero and SG_min differ by ∼15%–20% for the VdR and by ∼40%–50% for the aVdR. The red region shown in Figure 10 displays the ratio of the VFs functions in the VdR for rotations of system SG_zero up to 20° in both SGB and longitude. This figure indicates that the volume density of halos in the VdR stays within 20%–30% of its value in the coordinate system SG_zero for moderate rotations around it and halo velocities below 100 km s^-1 (which is the range we are ultimately interested in). The VF in the VdR is therefore robust against moderate rotations of the coordinate system due to the appearance of adjacent high- and low-density regions. For the aVdR, the results are more sensitive to rotations of the original coordinate system. These general results for the choice of coordinate system apply in the ΛWDM case.

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5.2. Velocity Function of Disk Galaxies

In order to directly compare our simulations with observations, we need to populate the dark matter halos with galaxies. Essentially, we need to use a method to connect the maximum circular velocity measured for disk galaxies with the properties of the hosting halo. Since a full semianalytical treatment goes beyond the scope of the current study, we use a simplified scheme that, nevertheless, allows us to make predictions on the VF of galaxies in the local environment. First, appropriate halos need to be selected. The high-velocity (large-mass) halos are associated with groups and clusters of galaxies. In our scheme, we exclude halos with masses larger than 10¹³ h⁻¹ M_☉ since we are interested in low-mass-isolated galaxies. Next, we assume each of the remaining halos to contain only one disk galaxy (for isolated galaxies in the local universe, the fraction of spiral galaxies lies between 80% and 90%, e.g., Sulentic et al. 2006; Hernández-Toledo et al. 2008). Although many of these halos contain significant substructures, which in principle can be populated with satellite galaxies, we restrict the present study to main halos and the central galaxies within them. This can be justified because the fraction of satellite galaxies lies between 10% and 40% (Zheng et al. 2007) and approximately 50% of them are unlikely to be detected in surveys like ALFALFA. The latter estimate is roughly correct because, for optically selected samples, a large fraction of satellite galaxies have red colors (Weinmann et al. 2006; Wang et al. 2007; Font et al. 2008) and have probably already lost most of their gas. H i studies of nearby groups (e.g., Kilborn et al. 2005) similarly show that about half of the satellites would lay below the sensitivity threshold for H i detection. Therefore, our results may underestimate the abundance of H i sources by 5%–20%, and thus, the lack of satellites in our modeling is not a major source of uncertainty.

We compute the circular velocities of the disks lying at the center of each halo using the analytical model of Mo et al. (1998). The principal hypotheses of this model are the conservation of specific angular momentum of both the dark matter and the gaseous components, and the equality of disk and halo specific angular momenta (J_disk/M_disk = J_h/M_h) during the process of disk formation. These assumptions are key conditions for the formation of realistic disks in hydrodynamical simulations (Zavala et al. 2008). The rotation curve of the galactic system (disk+halo) is given by the combined gravitational effects of the disk (which ends up with an exponential surface density profile) and the adiabatically contracted halo. The maximum rotational velocity (V_max: disk+halo) of the disk and the maximum rotational velocity of the pure dark matter halo (V_max,h) are related by V_max = G(λ, f_disk)V_max,h, the function G(λ, f_disk) depends on the spin parameter of the halo, λ, and the fraction of baryonic mass that is used to assemble the disk, f_disk = M_d/M_vir. The V_max/V_max,h ratio increases with f_disk and decreases with λ. We note that this ratio is nearly independent of the halo concentration; for fixed values of f_disk and λ, variations of c within the typical scatter in the mass–concentration relation produce a change of less than 2%. The function G(λ, f_disk) has been approximated in Zavala (2003) with a fitting function that has an accuracy larger than 96% for λ ∈ [0.02, 0.1]:

$$\begin{equation} G(\lambda,f_{\rm disk})=1.04\left(1-\frac{0.11f_{\rm disk}+5\times 10^{-4}}{\lambda }\right)^{-1}. \end{equation} \tag{ 1 }$$

The parameter f_disk actually depends on the processes occurring during galaxy formation, such as gas cooling and feedback. For simplicity. we take a constant value f_disk = 0.03 for all galaxies. However, at the end of the following section we address the effects of supernova (SN) feedback. For the constant value of f_disk, we take λ_lim = 0.02 as a limiting value, below which the disk would be unstable (Mo et al. 1998). We note, however, that the large majority of the halos in our CSs have λ>λ_lim, halos with lower values are not considered in the analysis.

Applying the scheme described above, we obtain the VF of the modeled disk galaxies. The result is shown in Figure 11, colors and line styles are the same as those used in Figure 9. We have included in the figure the values of the filtering and limiting velocities for halos related to the ΛWDM (vertical solid and dotted lines). These values mark lower limits for the corresponding quantities in the case of modeled disks in the ΛWDM scenario.

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5.3. Comparison with the Early ALFALFA Catalog Release

5.3.1. Sample Selection and Corrections to the Line Width W₅₀

By the end of the writing of this paper, only three catalogs have been publicly released by the ALFALFA collaboration, two in the VdR (Giovanelli et al. 2007; Kent et al. 2008), and the other one in the aVdR (Saintonge et al. 2008). These catalogs comprise only ∼6% of the final volume. The first two cover an area from 11^h30^m ≲ R.A. ≲ 14^h and +08° < decl. < +16° and the last one from 22^h < R.A. < 03^h and +26° < decl. < +28°. Despite the limited volume, we make an attempt to compare our results to the observational data released so far.

For such comparison, we take a sample of the ALFALFA sources according to the following criteria: (1) distances lower than 20 h⁻¹ Mpc, (2) exclusion of high-velocity clouds (HVCs) and sources with no measurement of H i mass, (3) removal of sources with no inclination measurement or with i ⩽ 30°,¹¹ and finally, (4) removal of sources showing clear signs of interaction within a projected angular distance smaller than the beam size of the Arecibo antenna (∼35).

Criterion (1) is the most stringent of all, only 14% of the sources in the three original catalogs fulfill it. Of the remaining galaxies, 81% satisfy criteria (2)–(4). The final sample consists of 186 galaxies in the VdR and 15 in the aVdR.

It is necessary to obtain measurements for the inclination of the sources since the 21 cm line width W₅₀ (measured at the 50% peak level) can be associated with V_max only after appropriate corrections, one of them, deprojection to an edge-on view. We discuss further below how we corrected the values of W₅₀.

To get the inclinations, we extracted the minor-to-major axis ratios (b/a) by cross-checking the sources with the GOLD Mine Database¹² (Gavazzi et al. 2003), the Cornell H i Archive of pointed sources¹³ (Springob et al. 2005), and the NASA/IPAC Extragalactic Database (NED).¹⁴ Using the axis ratios, we compute the inclinations with the formula

$$\begin{equation} \cos ^2(i)=\frac{(b/a)^2-q_0^2}{1-q_0^2}, \end{equation} \tag{ 2 }$$

where q₀ is the intrinsic axial ratio of a galaxy seen edge-on, we adopt q₀ = 0.2 as a fiducial value for all galaxies in our sample, independent of morphological type (e.g., Tully et al. 2009). For criterion (4) we checked, whenever it was possible, the optical counterpart of the sources using the Sloan Digital Sky Survey (SDSS) Web site.¹⁵

The value of W₅₀ given in the raw catalogs has already been corrected for instrumental broadening. We further corrected these values for turbulent motions and for inclination effects following the procedure given in Verheijen & Sancisi (2001):

$$\begin{eqnarray} W_{50}^2 &=& \frac{1}{{\rm sin}(i)}\bigg[W_{50,R}^2+W_{t,50}^2\left(1-2e^{-\left(\frac{W_{50,R}}{W_{c,50}}\right)^2}\right)\nonumber\\ && -2W_{50,R}W_{t,50}\left(1-e^{-\left(\frac{W_{50,R}}{W_{c,50}}\right)^2}\right)\bigg], \end{eqnarray} \tag{ 3 }$$

where W_50,R is the raw value corrected for instrumental broadening, W_t,50 = 5 km s^-1 and W_c,50 = 100 km s^-1. After these corrections, the maximum rotational velocity of the galaxies can be estimated with reasonable accuracy as W₅₀/2. We make, however, the following comment on the estimation of V_max using W₅₀: although the H i line width provides no information on the radial rotation profile of the galaxy, most of the H i gas is located in the outer part of disk galaxies (typically beyond three scale lengths).¹⁶ For large disks, W₅₀ provides a measure of rotational velocity in the regions where the rotation curve is already flat or rising slowly (e.g., Catinella et al. 2006, 2007). In the case of galaxies with lower velocities (<75 km s^-1), their rotation curves are typically thought to be still rising to the last measured point. However, Swaters et al. (2009) have recently analyzed in detail a sample of dwarf galaxies and found that the shapes of their rotation curves are similar to those of more massive galaxies; in particular, the rotation curves typically start to flatten at two disk scale lengths. If these results hold for the galaxies in our sample, then we can be confident to use W₅₀ to get V_max without an important systematic underestimation.

5.3.2. The H i Velocity Function

The angular position of the final sample of galaxies is shown in Figure 12 using equatorial coordinates (upper and middle panels for the VdR and aVdR, respectively). As a reference, the position of M87 is marked in the figure with a red star. The lower panel shows the number density of sources as a function of their distance to the MW. The prominent overdensity around 11 h⁻¹ Mpc is caused by galaxies in the vicinity of the Virgo cluster. A similar feature is present in our CSs (see Figure 8). We note that distances in the ALFALFA catalogs take into account the peculiar velocity field according to the model of Tonry et al. (2000).

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The H i VF of the sample of galaxies in the VdR is shown in Figure 13 with red square symbols. The values were computed using the $\Sigma (1/\mathbb {V}_{\rm max})$ weighting method proposed by Schmidt (1968). $\mathbb {V}_{\rm max}$ is the volume given by the maximum distance, D_max, where a given source could be placed and still be detected by the survey. This distance is determined by the sensitivity limit of the survey, given in the ALFALFA survey design for a signal-to-noise threshold of 6 (see Equation (5) of Giovanelli et al. 2005a)

$$\begin{equation} \left(\frac{D_{\rm max}}{\rm Mpc}\right)^2=\frac{1}{0.49}f_{\beta }t_s^{1/2}\left(\frac{\rm {{\it M}_{H\,{\scriptscriptstyle I}}}}{10^6{\:M_\odot }}\right) \left(\frac{W_{50}}{200\:{\rm km\,s^{-1}}}\right)^{\gamma }, \end{equation} \tag{ 4 }$$

where $M_{\rm H\,{\scriptscriptstyle I}}$ is the H i mass associated with the source, γ = −1/2 for W₅₀ < 200 km s^-1 and γ = −1 for W₅₀ ⩾ 200 km s^-1. The parameter f_β quantifies the fraction of the source flux detected by the telescope's beam, for simplicity, we treat all sources as point sources and take f_β = 1. Since the beam size of the Arecibo antenna is 35, the majority of the sources can be treated in this way. We take a fiducial value of 48 s for the integration time t_s. For comparison with our simulations, we are restricting the analysis to distances lower than 20 h⁻¹ Mpc, thus, effectively: $\mathbb {V}_{\rm max}=\hbox{\rm min}(\mathbb {V}(D_{\rm max}),\mathbb {V}(20\,h^{-1}\hbox{Mpc}))$. For the majority of the sources, D_max>20 h⁻¹Mpc, therefore, the volume–weights have only a minor impact in the number count. However, three sources in the lower velocity bins have a very low value of V_max and their weights deviate strongly from the average in their respective bins; we removed them since they are not statistically representative and can lead to a strong overestimation of the VF.

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The results from our CSs for the corresponding field of view are also shown in Figure 13. The red dashed and dotted areas encompass the 1σ regions, using Poisson statistics, for the predictions in the ΛCDM and ΛWDM cases, respectively. The VFs were constructed taking into account the sensitivity limit of the survey, using the same $\Sigma (1/\mathbb {V}_{\rm max})$ weighting method as for the observations and renormalizing the result according to the fraction of galaxies that were excluded from the final observational sample due to the 30° inclination cutoff. To properly apply the sensitivity limit, we would need to give an estimate of the gas fraction, f_gas = M_H I/M_disk, in the modeled disks, that depends on the efficiency of gas transformation into stars and the gas infall history. For simplicity we take f_gas = 1, but we note that the value of f_gas is irrelevant for our particular analysis, see discussion by the end of the section. The sensitivity limit has a relevant effect only for the low-velocity disks.

For velocities larger than ∼80 km s^-1, the VF of both CSs match reasonably well the observational data. This result is not trivial, for example, at V_max = 100 km s^-1, the value of the VF for halos in the whole simulated box is approximately an order of magnitude lower than the value associated with the observed sample in the VdR. Therefore, we confirm that the CSs are able to simulate properly the overdense VdR. For velocities in the range 35–80 km s^-1, the ΛCDM simulation overpredicts the value of the VF, increasingly for lower velocities. On the other hand, the ΛWDM simulation is in good agreement with the observed data, with values slightly higher. The low-velocity end, V_max < 35 km s^-1, is not suitable for comparison since both simulations and observations are not complete for the lower velocities. The minimum halo mass in our simulations that is reliable, according to a comparison of the MF with theoretical expectations in the low-mass end (see Figure 4), sets a lower limit of completeness for V_max. For the ΛCDM simulation, this mass is ∼10⁹ h⁻¹ M_☉, corresponding to V_max ∼ 24 km s^-1; in the ΛWDM case is given by the limiting mass 3 × 10⁹ h⁻¹ M_☉ related to V_max ∼ 29 km s^-1. The typical sensitivity limit for the ALFALFA survey goes down to $M_{{\rm H\,{\scriptscriptstyle I}}}=10^7\:h^{-1}\, M_\odot$ for distances up to 20 h⁻¹ Mpc, see Equation (4), a source with this mass can have a V_max value within a broad range, due to the natural scatter on the V_max–M_H i relation. An examination of our galaxy sample shows that this range is ∼13–35 km s^-1, which puts a lower limit of ∼35 km s^-1 for the completeness of the sample. Taking the theoretical and observational limits into account, a consistent comparison can only be done for values of V_max above ∼35 km s^-1.

So far we have used a constant disk baryon fraction f_disk = 0.03 to populate the dark matter halos with galaxy disks. A more realistic approach would be to include the effects of SN feedback in f_disk. This inclusion would only have an important effect for low-velocity halos if the gas outflow produced by SN is sufficient to deplete the forming galaxy of gas and push it below the sensitivity limit of ALFALFA. We now incorporate this effect using the disk galaxy evolutionary model of Dutton & van den Bosch (2009) for the case where the outflow of gas is energy driven: the kinetic energy of the wind is a fraction _EFB = 0.25 of the kinetic energy produced by SN. In this case, f_disk becomes a strong function of halo mass (see upper panel of Figure 8 in Dutton & van den Bosch 2009). We adopt the fitting function obtained by the authors for the energy driven feedback model (see their Equation (43) and Table 3) and apply it to our model. The results appear in Figure 13 as dashed and dotted lines for the ΛCDM and ΛWDM CSs. The net impact is a marginal reduction of the VF at the low-velocity end. A more drastic reduction would be possible with a model that significantly lowers the value of f_gas, which we have set to one for simplicity. For instance, the lowest value we have set for comparison, V_max = 35 km s^-1, is related on average to a halo of 6 × 10⁹ h⁻¹ M_☉, which in the case of the SN feedback model corresponds to f_disk = 0.02, thus the modeled disk would lie below the sensitivity limit of the survey only if f_gas is lower than 0.08. In our scheme, the gas fraction only determines if a given halo should be counted or not, thus, the previous result indicates that the specific value of f_gas has no impact unless the galaxies we are aiming to compare with have very low gas fractions. H i deficient low-mass galaxies are typically present inside galaxy clusters like Virgo and are unlikely to be detected by ALFALFA; they also have no counterpart in our simulations since we have dealt exclusively with halos, excluding the subhalos within them. Galaxies in the field are less likely to have such low gas fractions. Nevertheless, we explicitly checked that this is indeed the case for the sample of galaxies in the VdR. For that purpose, we identified the SDSS optical counterparts of the H i sources and used the method described in Blanton & Roweis (2007) to estimate the stellar masses of 73% of the galaxies in the final VdR sample. We found that the majority of these galaxies have large gas fractions, approximately 90% of the ones with V_max < 100 km s^-1 have f_gas>0.1.

In Figure 14, we show analogous results for the aVdR. Although in this case the statistical significance of the observed sample is much lower (only 15 sources), the main results observed in the case of the VdR are reproduced, indicating that simulations are able to probe properly this region as well.

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N-body simulations with constrained initial conditions, set up to resemble the spatial distribution of dark matter in the local universe, are a powerful tool to make detailed comparisons between predictions of the dark matter paradigm and observational evidence in our local environment.

Using the algorithm of constrained realizations developed by Hoffman & Ribak (1991) and the approach laid out in Kravtsov et al. (2002) and Klypin et al. (2003), we have run a pair of CSs that incorporate nearby observational data sets as input for their initial conditions. One simulation follows the evolution of structure in a CDM model and the other one is based on a thermal WDM particle with a mass of 1 keV resulting in an effective filtering mass of ∼10¹⁰ h⁻¹ M_☉.

After an appropriate choice of the coordinate system, the simulations are able to reproduce the overall spatial distribution of the most significant structures within 20 h⁻¹ Mpc of the LG, namely, the LSC, including the Virgo cluster, as well as the Fornax cluster laying in the opposite direction (see Section 3).

The mass and VFs of halos in the whole simulated boxes are consistent with theoretical expectations. The ΛCDM case follows closely the estimates from the S–T formalism, except in the highest mass end due to the influence of the LSC. For the ΛWDM cosmogony, the results lie close to the ΛCDM case for halos with masses higher than the filtering mass; for lower masses, the mass and VFs flatten and then rise due to spurious numerical fragmentation for masses ≲3 × 10⁹ h⁻¹ M_☉ confirming the limiting mass formula given by Wang & White (2007). The mass resolution of the CSs, m_DM = 1.63 × 10⁷ h⁻¹ M_☉, allows us to derive robust results for halos with masses larger than this limiting mass, corresponding to maximum rotation velocities of 24 km s^-1.

For the local environment, the general prediction of the CSs is that the region within 20 h⁻¹ Mpc of the LG is overdense by a factor of ∼2 compared to the MF of the whole simulated boxes (see Figure 6). Such result goes in the same direction as a recent result reported by Tikhonov & Klypin (2009), showing that the luminosity function in a sample of galaxies within 8 h⁻¹ Mpc is larger than the universal luminosity function by a factor of 1.4.

Since we have shown that our CSs are capable of reproducing the abundance of halos in the local environment, we have obtained predictions for the VF of halos in the field of view, within 20 h⁻¹ Mpc, that is being surveyed by ALFALFA. The VF has the important advantage over the MF that it can be compared more directly with observational data since it avoids the problem of relating halo masses to galaxy luminosities.

After completion, the ALFALFA survey will detect H i sources covering 7000 deg² of the sky in two different regions. In this work, we have referred to these regions, additionally constrained to 20 h⁻¹ Mpc, as the VdR and the aVdR. Our CSs predict that the VF of halos in the VdR exceeds the universal VF by a factor of ∼3. The VF in the aVdR is only slightly underdense (∼10%) than the universal value (see Figure 9).

We have used a simplified model to populate our halos with disk galaxies. It only incorporates the dynamical effects of the disks. With this model, we are able to predict the VF of disk galaxies in the two regions explored by ALFALFA. Although the survey is not complete yet we have compared our predictions and the results from a sample of galaxies taken from the catalogs released so far, which cover only 6% of the total planned volume.

For velocities in the range between 80 km s^-1 and 300 km s^-1, the VFs predicted for the ΛCDM and ΛWDM simulations agree quite well with the VF of the sample of galaxies. For the VdR, this result is particularly encouraging and reassures the confidence in our CSs to properly simulate the local environment: despite the small volume used for comparison, the simulations are able to predict the shape and normalization of the VF in the high-velocity regime. In the VdR, the normalization is an order of magnitude larger than for the universal VF.

For velocities larger than the minimum mass we can trust for comparison, ∼35 km s^-1, and lower than 80 km s^-1, the predictions agree well for the ΛWDM cosmogony (contrary to recent claims; see Section 9 of Blanton et al. 2008), with the VF being approximately flat in this regime. The ΛCDM model, however, predicts a steep rise in the VF toward low velocities; for V_max ∼ 35 km s^-1, it predicts ∼10 times more sources than the ones observed. Using the same set of simulations, Tikhonov et al. (2009) found that the observed spectrum of mini-voids in the local volume is in good agreement with the ΛWDM model but can hardly be explained within the ΛCDM scenario.

Although we have only explored a simplified model to populate our halos with disk galaxies, our results indicate a potential problem of the ΛCDM paradigm in the low-velocity regime of dwarf galaxies. Nevertheless, there are several issues that need to be addressed before reaching a strong conclusion.

On the observational side, the sample of galaxies we have analyzed comprises only 186 galaxies in the VdR. It is of key importance to obtain the VF for the complete volume of ALFALFA and check if the flattening at low velocities is reproduced. Another important issue is to determine to which extent the value of W₅₀ is representative of the maximum rotational velocity for the least massive galaxies. A recent analysis by Swaters et al. (2009) indicates that the rotation curves for late-type dwarf galaxies are similar to those of more massive galaxies, starting to flatten at two disk scale lengths, thus, W₅₀ would not underestimate strongly the value of V_max for these galaxies since the H i gas typically extends beyond three disk scale lengths.

From a theoretical perspective, astrophysical phenomena such as SN feedback and UV photoionization play an important role to deplete gas from low-mass halos. The relevant influence of these effects on the VF comes from the typical sensitivity limit of the survey, allowing to detect H i masses larger than 10⁷ h⁻¹ M_☉ at distances D ⩽ 20 h⁻¹ Mpc. We have explored the former of these effects by using the model presented in Dutton & van den Bosch (2009) and found little impact on the VF. Regarding UV background heating, Hoeft et al. (2006) found that this mechanism is not very efficient in evaporating all baryons from dwarf sized halos below a characteristic mass scale of 6 × 10⁹ h⁻¹ M_☉ (V_max ∼ 35 km s^-1). Since according to our findings, a strong suppression of gas is needed for halos with characteristic velocities V_max < 60 km s^-1 to flatten the VF in the ΛCDM case, it is unlikely that UV photoionization could account for it.

The simulations used in this work were performed at the Barcelona Supercomputing Centre (BSC), the Leibniz Rechenzentrum Munich (LRZ), and the Shanghai Supercomputer Center. The cpu time used at BSC and LRZ was partly granted by the DEISA Extreme Computing Project (DECI) SIMU-LU. J.Z. thanks Volker Springel for helpful comments in running the simulations, and Alexander Knebe and Steffen Knollman for help with AHF. J.Z. and A.F. are supported by the Joint Postdoctoral Program in Astrophysical Cosmology of the Max Planck Institute for Astrophysics and the Shanghai Astronomical Observatory. J.Z. was partially supported by the CAS Research Fellowship for International Young Researchers. Y.P.J. is supported by NSFC (10533030, 10821302, 10878001), by the Knowledge Innovation Program of CAS (No. KJCX2-YW-T05), and by 973 Program (No. 2007CB815402). Our collaboration was supported by the ASTROSIM network of the ESF. This research has been supported by the Israel Science Foundation (13/08 at the HU). G.Y. acknowledges support of the Spanish Ministry of Education through research grants FPA2006-01105 and AYA2006-15492-C03. Some of the results in this paper have been derived using the HEALPiX (Górski et al. 2005) package. This research has made use of the GOLD Mine Database and the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made use of the SDSS archive. Its full acknowledgment can be found at http://www.sdss.org.