THE VELOCITY WIDTH FUNCTION OF GALAXIES FROM THE 40% ALFALFA SURVEY: SHEDDING LIGHT ON THE COLD DARK MATTER OVERABUNDANCE PROBLEM

The ongoing ALFALFA survey is a wide-area, blind 21 cm emission-line survey that takes advantage of the increased survey speed offered by the seven-feed Arecibo L-band Feed Array (ALFA) receiver at the Arecibo Observatory. The ALFALFA data are acquired in a minimally invasive drift-scan mode in two passes, ideally separated by several months in order to enable the discrimination between narrowband radio frequency interference (RFI) and small spectral width cosmic signals. When completed (i.e., when all data will have been processed), the survey will have detected >30,000 galaxies over an area of ∼7000 deg² of sky out to cz ≈ 18,000 km s⁻¹. The ALFALFA survey is more sensitive than the previous generation HIPASS survey (Meyer et al. 2004; Zwaan et al. 2004), with a 5σ detection limit of 0.72 Jy km s⁻¹ for a source with a profile width of 200 km s⁻¹ as compared to a 5σ sensitivity of 5.6 Jy km s⁻¹ for the same source in HIPASS (Giovanelli et al. 2005). In addition to greater sensitivity, ALFALFA has a finer velocity resolution (11.2 km s⁻¹ versus 26.4 km s⁻¹ for smoothed data) and better angular resolution (36 versus 13' FWHM), resulting in a more accurate identification of optical counterparts.

ALFALFA catalogs have so far been extracted (Giovanelli et al. 2007; Saintonge et al. 2008; Kent et al. 2008; Stierwalt et al. 2009; Martin et al. 2009; M. P. Haynes et al. 2011, in preparation) for a total area of 2934 deg². The current ALFALFA footprint consists of four distinct regions: two in the northern Galactic hemisphere, hereafter referred to as the Virgo direction region (VdR: 07^h30^m < α < 16^h30^m, 4° < δ < 16°, and 24° < δ < 28°), and two in the southern Galactic hemisphere, hereafter referred to as the anti-Virgo direction region (aVdR: 22^h < α < 03^h, 14° < δ < 16°, and 24° < δ < 32°). From this primary data set we only select extragalactic objects detected at high significance (signal-to-noise ratio (S/N) >6.5, designated Code 1), and we further restrict ourselves to the redshift range cz ⩽ 15, 000 km s⁻¹, beyond which radar interference from the nearby San Juan airport causes a significant drop of the ALFALFA detection efficiency. This final sample, corresponding to ∼40% of the ALFALFA survey area (hereafter the α.40 sample), contains a total of 11,086 galaxies.

Figure 1 shows the spatial distribution of the α.40 sources in the Virgo and anti-Virgo directions, respectively, and puts in evidence the complex large-scale structure present in both volumes. Density fluctuations in the survey volume can be the dominant source of statistical uncertainty in surveys like ALFALFA, where the sample size ensures small counting errors. Our statistical estimator, described in Section 3.1, is chosen to minimize this structure-induced bias.

Zoom In Zoom Out Reset image size

Figure 2 displays some statistical properties of the α.40 sample. Histograms (a) and (b) represent the distribution of heliocentric velocity, v_☉, and of signal profile width, w₅₀, which are both directly measured quantities (Giovanelli et al. 2007; Saintonge 2007). The signal profile width is measured at the 50% flux level of each of the two peaks of the typical double-horned HI profile (or at 50% of the single peak flux, for single-peaked profiles). The value of w₅₀ reported in the ALFALFA catalogs is further corrected for instrumental broadening. Histogram (c) displays the distribution of galaxy HI mass, M_HI, which is a distance dependent (and hence derived) quantity. Unlike previous HI surveys, we assign distances to nearby galaxies through a peculiar velocity flow model (Masters 2005) and use Hubble flow distances only for galaxies with cz > 6000 km s⁻¹ (see Section 4.2 for a detailed discussion on the impact of distance uncertainties on ALFALFA results).

Zoom In Zoom Out Reset image size

Figure 3 displays the distribution of α.40 sources in the velocity width (w₅₀) versus integrated-flux (S_int) plane. As expected, the detection limit of the survey is a function of signal profile width and correctly scales as S_{int, lim} ∼ w^1/2₅₀. Due to the large density of sources near the detection limit, we evaluate the completeness limit of the survey (red dashed line in Figure 3) based on the actual data rather than on simulations using synthetic sources.

Zoom In Zoom Out Reset image size

In Figure 4 we present the ALFALFA WF, based on 10,744 galaxies drawn from the α.40 sample. For the calculation of the WF we restrict ourselves to α.40 galaxies which are positioned in the portion of the flux-width plane where the ALFALFA survey is complete (i.e., above the red dashed line in Figure 3) and have profile widths broader than w₅₀ ≳ 18 km s⁻¹. This cut results in the elimination of ≈330 galaxies from the calculation. An additional 13 very nearby sources are eliminated, for which the flow model assigned distances are subject to large uncertainty.

Zoom In Zoom Out Reset image size

The WF is calculated in logarithmic width bins, according to the Σ 1/V_eff method (Zwaan et al. 2005). The Σ 1/V_eff method is a non-parametric maximum likelihood method and, as such, it is insensitive to the presence of large-scale structure in the survey volume. As its name suggests, it closely resembles the traditional Σ 1/V_max method (Schmidt 1968) and consists of summing the number of detections in each width bin, weighted by the inverse of the "effective" volume available to each source. More precisely, the space density of galaxies belonging to width bin k (k = 1, 2, ..., N_w) is

$$\begin{eqnarray} &&\phi _k = \sum _i \frac{1}{V_{{\rm eff},i}} \end{eqnarray} \tag{ 1 }$$

for all galaxies i in width bin k.

In the case of a spatially homogeneous survey volume, V_{eff, i} would coincide with V_{max, i}, the latter defined as the volume within which galaxy i could be placed and still be detectable by the survey. On the other hand, if the survey volume displays significant density variations, V_{eff, i} takes into account the relative density of the volume available to galaxy i with respect to the mean density of the total survey volume. As with all density-independent estimators, the overall normalization is lost and has to be calculated afterward. The normalization is fixed by matching the integral of the distribution to the average number density of galaxies in the survey volume (see Martin et al. 2010; see their Appendix B.1 for more details).

Due to its spectral resolution and sensitivity, ALFALFA can push the low-width limit of the WF to w ≈ 20 km s⁻¹, a factor of two lower than the HIPASS survey. Over the full measured range (20 km s⁻¹<w < 800 km s⁻¹) the ALFALFA WF is very well described by a modified Schechter function of the form⁵

$$\begin{equation} \phi (w) = \frac{dn}{d\log w} = \ln (10) \: \phi _\ast \left(\frac{w}{w_\ast }\right)^\alpha e^{-({w}/{w_\ast })^\beta }. \end{equation} \tag{ 2 }$$

The least-squares parameters⁶ are ϕ_* = 0.011 ± 0.002h³₇₀ Mpc⁻³ dex⁻¹, log w_* = 2.58 ± 0.03, α = −0.85 ± 0.10, and β = 2.7 ± 0.3 (uncertainties are statistical 1σ errors due to Poisson errors on the individual bin values). Note, however, that the final sample contains 163 sources that lack a confidently identified optical counterpart. Some of these sources correspond to tidal debris from nearby interacting galaxies and may not be hosted by individual DM halos. Excluding these galaxies from the WF calculation leads to a somewhat shallower narrow-end slope of α = −0.68 ± 0.11.

ALFALFA finds significantly more high-width galaxies than HIPASS (a factor of 3 at w ≈ 400 km s⁻¹, growing to a factor of ∼10 at w ≈ 800 km s⁻¹), which is also evident from the marked difference in the value of the position of the "knee" of the WF for the two surveys (log w_* = 2.58 ± 0.03 for ALFALFA versus log w_* = 2.21 ± 0.10 for HIPASS,⁷ in disagreement at the >3σ level). Despite the fact that the nominal HIPASS volume is a factor of ∼5 larger than the α.40 volume, ALFALFA is able to find more high-width galaxies thanks to its better sensitivity (see Figure 5). The same effect can be seen in the HIMFs published by the two surveys, with ALFALFA (Martin et al. 2010) finding a factor of a few more of the highest HI-mass galaxies compared to HIPASS (Zwaan et al. 2005).

Zoom In Zoom Out Reset image size

On the low-width end, ALFALFA finds a rising slope (α < 0) which is, however, by no means steep enough to match the CDM prediction (see Section 5). Despite the vastly different value for the narrow-end slope reported by the two surveys (α = 0.10 ± 0.39 for HIPASS versus α = −0.85 ± 0.10 for ALFALFA) the HIPASS and ALFALFA data points are consistent in the width range 40 km s⁻¹ ≲ w ≲ 200 km s⁻¹. The HIPASS α parameter is not well constrained, as their WF does not extend to low enough widths and suffers from considerable counting error in the low-width bins.

Measurement errors on w₅₀ can shift galaxies among width bins, altering the bin counts and therefore the inferred space density. The w₅₀ value for ALFALFA sources is subject to two separate sources of error: one is statistical in nature and present for all sources, while the other is systematic and concerns only a fraction of the α.40 sample. The former is due to the distortion of the signal profile shape by noise; the latter results from the fact that the measurement of the spectral width of a signal relies on the accurate visual identification of its spectral boundaries, which is non-trivial for a number of sources (especially those found in the vicinity of RFI). The final width error reported in the ALFALFA catalogs, Δw₅₀, is the sum in quadrature of the random and systematic error terms described above. Owing to the fact that all α.40 galaxies are detected with high S/N and have a clean spectral profile in the vast majority of cases, the typical α.40 width error is relatively small and its distribution well behaved. The median error is Δw_{50, median} ≈ 8 km s⁻¹ and ∼70% of the sources have a fractional error of Δw₅₀/w₅₀ ⩽ 10%.

In order to assess the effect of Δw₅₀ on the WF, we create 50 mock galaxy samples by reassigning random widths to every galaxy i in the primary ALFALFA data set according to their individual measured width (w_{50, i}) and error (Δw_{50, i}). Each mock sample is subject to the same cuts as the α.40 sample and a new realization of the WF is calculated ("1×" set). In order to illustrate the systematic trends introduced, we also perform an additional set of WF realizations with artificially inflated width errors (twice the reported ALFALFA width errors, "2×" set).

The results are shown in Figure 6: overplotted to the original ALFALFA WF (data points and solid black line) are a modified Schechter fit to the mean WF corresponding to the 1× (red solid line) and 2× (dashed red line) realizations. Width errors at the ALFALFA error levels seem to only slightly affect the high-width end of the WF. As evidenced by the 2× run, width errors generally lead to a rise of the high-width end, due to a net "diffusion" of galaxies from intermediate-width bins with large number counts to high-width bins with lower number counts.

Zoom In Zoom Out Reset image size

Since velocity width is a distance-independent quantity, galaxy counts in width bins are not altered by distance errors. However, the weights (1/V_{eff, i}) that each galaxy contributes to its bin depend on HI-mass (see Equation (1) and discussion in Section 3.1), and therefore on the assumed distance. Masters et al. (2004) have shown that ignoring the local peculiar velocity field can lead to biased estimates of galaxy statistical distributions, especially for surveys drawing a large fraction of their sample from the Virgo direction (VdR). To avoid this bias ALFALFA uses redshift distances only for distant (cz > 6000 km s⁻¹) galaxies and assigns distances to nearby galaxies through a parametric flow model developed by Masters (2005). The model includes two attractors (Virgo Cluster & Great Attractor), a dipole component (Local Group peculiar velocity), a quadrupole component (Local Group asymmetric expansion), and a random thermal residual of σ_local ≈ 160 km s⁻¹. Here we assume that most of the coherent motion of nearby galaxies is correctly described by the flow model, and no significant bias results from this systematic component of galaxy peculiar velocities. Contrary to intuition however, even the random component σ_local can induce a systematic bias through the "Eddington effect" (see for example Figure 6 in Zwaan et al. 2003).

In order to assess the effect of σ_local on the WF, we proceed as in Section 4.1 and create 50 mock samples by adding Gaussian noise on the cataloged distance of each α.40 galaxy. We calculate the WF corresponding to each sample realization and use the obtained average distribution to investigate the effect of distance uncertainties on the WF. We adopt the Masters (2005) value of σ_local ≈ 160 km s⁻¹, but we also perform simulations with double the fiducial dispersion (σ_local ≈ 320 km s⁻¹).

The results are displayed graphically in Figure 7, where the data points and solid line correspond to the original ALFALFA WF, and the blue solid and dotted lines correspond respectively to the results of the σ_local = 160 km s⁻¹ and σ_local = 320 km s⁻¹ simulation sets. The largest effect is an overall increase in the amplitude of the WF at intermediate widths, which is probably due to the net transport of sources toward lower HI masses and therefore larger values of 1/V_eff. Unlike in the case of the HIMF, the low-end slope α does not seem to be affected in any systematic way. We conclude that, apart from a mild increase in amplitude at intermediate widths, the WF is relatively insensitive to distance uncertainties due to galaxy peculiar motions.

Zoom In Zoom Out Reset image size

The WF presented in Figure 4 aspires to represent the distribution in a cosmologically representative volume. The sensitivity of the ALFALFA survey allows ∼w_* and broader galaxies to be detected throughout the full α.40 volume (V_α.40 ≈ 3.1 · 10⁶ h³₇₀ Mpc³), which ensures a cosmologically fair sampling of the MW-sized galaxy population. On the other hand, low-width galaxies tend to be faint systems that can only be detected in smaller volumes. As a result, the low-width bins of the WF are subject to increased uncertainty caused by the deviation of the galaxy distribution from homogeneity on small scales, which is referred to as cosmic variance (see Figure 8 for a graphical illustration).

Zoom In Zoom Out Reset image size

In order to quantitatively assess the effects of cosmic variance on the ALFALFA WF, we jackknife resample the α.40 survey volume, by splitting it into 14 parts equally spaced in R.A. Then, we reevaluate the WF excluding each part in turn. The resulting scatter for each parameter, x, is given by

$$\begin{equation} \sigma _x^2 = \frac{N-1}{N} \: \sum (x-\bar{x})^2 \;, \;\;\; N=14. \end{equation} \tag{ 3 }$$

The scatter calculated by Equation (3) would be equal to the purely statistical error if the survey volume were homogeneous, and so any excess noise results from the presence of inhomogeneities. The method described above provides a measurement of cosmic variance on linear scales smaller than those probed by the full survey and hence yields a conservative estimate of the true uncertainty (cosmic variance generally increases with decreasing scale).

The full uncertainties on the fit parameters (including cosmic variance) are ϕ_* = 0.011 ± 0.003 (0.002) h³₇₀ Mpc⁻³ dex⁻¹, log w_* = 2.58 ± 0.04 (0.03), α = −0.85 ± 0.19 (0.10), and β = 2.7 ± 0.3 (0.3), where the term in parentheses represents the purely Poisson error reported in Section 3.1. Indeed, parameters w_* and β, which dictate the shape of the WF at high widths, show a very modest increase in their uncertainty due to cosmic variance. On the other hand, the narrow-end slope α is significantly affected, with cosmic variance contributing a large fraction of the full error.

Beam confusion arises from the fact that the ALFA 33 × 38 beam occasionally produces blends of small galactic groups at moderate distances, especially when individual galaxies are poorly separated in redshift space. The qualitative effect of such blends is to transform two or more independent sources into a single HI profile of larger w₅₀ than each of its constituents. We do not attempt to quantify the effect of confusion bias, but we anticipate it to be more pronounced at the high-width end of the WF. This is because galaxies with w ≳ 550 km s⁻¹ are preferentially found at large distances, where beam confusion is more severe. It is worth noting that this bias, even though present, cannot account for the discrepancy between the ALFALFA and HIPASS WFs at high widths, since the latter suffers from more confusion due to its larger beam size (13' FWHM).

CDM predictions start diverging from the observational results at low widths, and so the behavior of the theoretical WF for w < 200 km s⁻¹ is of great importance. Unfortunately, the very interesting work of O09 is only reliable for w ≳ 100 km s⁻¹ due to the limitations in the mass resolution of the Millennium Simulation. We employ instead two recent high-resolution CDM simulations that lack however modeling of the HI component of their virtual galaxy samples.

Figure 10 compares the ALFALFA measurement with the WF of the galaxy population corresponding to the Bolshoi simulation (Klypin et al. 2010), as modeled by Trujillo-Gomez et al. (2010, hereafter TG10). Each Bolshoi halo was assigned realistic stellar and cold gas masses, based on empirical relations. Subsequently, two models were considered, one where the gravitational potential of the baryons is simply superimposed on the DM potential (solid green line) and one where the DM halo adiabatically contracts in response to the presence of the baryons (dash-dotted green line). Note that TG10 define v_rot as the value of the simulated rotation curve at a radius of 10 kpc. We argue that their modeling scheme and use of v_10 kpc provide a good approximation of the measured velocity for galaxies with both flat and rising rotation curves.

Zoom In Zoom Out Reset image size

Also plotted in Figure 10 is the WF of simulated galaxies based on the Zavala et al. (2009, hereafter Za09) constrained N-body simulation (blue solid line). Za09 perform a modest volume (64 h⁻¹ Mpc on a side) but very high-resolution (v_lim = 24 km s⁻¹) constrained simulation, designed to reproduce the large-scale structure of the local universe. Virtual galaxies are modeled according to the analytical results of Mo et al. (1998), assuming a disk-to-virial mass ratio of f_disk ≡ M_disk/M_vir = 0.03 independent of halo size. Lastly, the maximum amplitude of the rotation curve (v_{rot, max}) for each galaxy is calculated by combining the disk and DM halo contributions.

Since neither model considers the distribution of the velocity field tracer (i.e., HI) in simulated galaxies, we convert rotational velocities into HI velocity widths by assuming the relationship

$$\begin{equation} w = 2 v_{{\rm rot}} \sin i + w_{{\rm eff}}. \end{equation} \tag{ 4 }$$

Galaxies are assumed to be randomly oriented with respect to the line of sight (cos i is uniformly distributed in the [0, 1] interval), while w_eff is a small "effective" term used to reproduce the broadening effect of turbulence and non-circular motions on HI linewidths. The use of Equation (4) is only justified if the HI disk is extended enough to sample the value of v_rot adopted by the model under consideration (e.g., v_10 kpc for TG10 and v_{rot, max} for Za09). We adopt the value w_eff = 5 km s⁻¹ for the broadening term,⁸ which is added linearly for galaxies with v_rot > 50 km s⁻¹ and in quadrature for lower velocity galaxies.

Figure 10 puts in evidence the marked departure of the theoretical distributions from the ALFALFA measurement at w < 200 km s⁻¹, which becomes more dramatic with decreasing width. According to the TG10 WF, the difference is approximately a factor of ∼4 at w = 100 km s⁻¹, exhibiting an increasing trend. The Za09 WF,⁹ implies a difference of a factor of ∼8 at the lowest width where the simulation is complete (w ≈ 50 km s⁻¹) and displays a much steeper low-width slope than the ALFALFA measurement. An extrapolation of the Za09 WF to the ALFALFA width limit (w = 20 km s⁻¹) would result in a discrepancy of a factor of ∼100.

The ALFALFA measurement of the WF confirms the results of the HIPASS survey (Zwaan et al. 2010), which obtained its WF at lower sensitivity and velocity resolution. This fact excludes the possibility that the CDM overabundance problem is an artifact of the limited performance characteristics of past blind 21 cm surveys. The reason for the observed discrepancy can be therefore most likely attributed to one of the two following factors.

• 1.  
  The inaccuracy of standard CDM simulations, presumably due to the inadequacy of the assumed DM model.
• 2.  
  The improper comparison of simulated halos with observed galaxies. This could be due either to

  • (a)  
    the inadequate modeling of the baryonic counterparts hosted by DM halos, which leads to wrong predictions for maximum rotational velocities, or
  • (b)  
    the incorrect interpretation of inclination-corrected HI linewidths as maximum rotational velocities.

In what follows, we will consider these possibilities in more detail and argue about their prospects as solutions of the CDM overabundance problem.

Most large-scale simulations of cosmic structure conform to the standard ΛCDM cosmological model. In particular, they assume that all DM is cold (i.e., has negligible free streaming length), non-self-interacting, and stable (i.e., non-decaying). These properties are appropriate for a universe where DM consists of stable weakly interacting massive particles (WIMPs). WIMPs are currently the favored DM particle candidate and are expected to have masses in the GeV–TeV range and weak scale self-interaction cross sections, justifying the DM attributes most commonly assumed in cosmological N-body simulations.

However, the picture changes considerably if DM is composed of relatively light (∼keV) particles, in which case it is referred to as warm dark matter (WDM). Structure on large scales would be the same as in a CDM universe, but on small scales halo formation would be heavily suppressed due to the non-negligible free-streaming length of the light WDM particle. Za09 have considered this alternative scenario and carried out a second run of their very high-resolution simulation assuming a 1 keV WDM particle. They subsequently populate their halos with synthetic galaxies, employing the same modeling scheme as in their CDM run (Section 5.1). The result is shown by the red solid line in Figure 11, superposed on the ALFALFA WF (data points with error bars and black solid line) and the result of their CDM run (blue solid line).

Zoom In Zoom Out Reset image size

Strikingly, the synthetic WF in the WDM case exhibits a shallow slope at the low-width end, in good agreement with the slope measured by ALFALFA. Such a shallow slope results from the suppressed production of low-mass halos in a WDM universe, which directly translates into a lower abundance of low-width visible galaxies. WDM could therefore provide a simple and elegant solution of the overabundance problem.

Despite its appeal in this specific context, the general prospects of WDM also depend on its overall viability as the dominant constituent of non-baryonic matter in the universe. A number of theoretical microscopic models for WDM have been proposed, most commonly involving sterile neutrinos (Dodelson & Widrow 1994; Fuller et al. 2003; Asaka et al. 2005; Kusenko 2009). Constraints on the particle's mass can be placed by astrophysical and cosmological considerations. In particular, Lyα forest data places lower limits on the neutrino mass (a lighter particle generally results in suppression of power at larger scales), while X-ray observations can place upper mass limits (radiative decay into X-ray photons generally becomes more efficient at higher masses). The limits on the neutrino mass imposed by these observational constraints depend on the assumed neutrino production mechanism. Abazajian & Koushiappas (2006) find that non-resonantly produced neutrinos are ruled out, using a compilation of Lyα forest and X-ray data (see references therein). Boyarsky et al. (2009) have considered sterile neutrino production in the context of the νMSM (Minimal Standard Model + 3 sterile neutrinos) and argue that neutrinos with m_sn > 2 keV are viable.

The second class of potential solutions attribute the disagreement between theory and observation to the process used to translate the output of simulations into actual galaxies. In particular, a number of important effects need to be taken into account (identified as items 1–4 in Section 5) to ensure a successful comparison of simulated halos with observed galactic samples.

Both theoretical works presented in Section 5.1 address issues 1 and 2. For example, Za09 set an explicit limit on the mass of halos hosting individual galaxies at M_vir = 10¹³ h⁻¹ M_☉. The influence of baryons on the shape of galactic rotation curves is also taken into account by both works, albeit using slightly different prescriptions and definitions of galaxy rotational velocity. Despite the use of numerous simplifying assumptions by TG10 (e.g., all baryons within 10 kpc) and Za09 (e.g., fixed disk-to-virial mass ratio for all galaxies), their theoretical WFs are in fair agreement with the ALFALFA measurement at intermediate widths (200 km s⁻¹<w < 500 km s⁻¹).

The last two issues are related specifically to the atomic hydrogen content of galaxies, which is not modeled by either TG10 or Za09. Specifically, issue 3 concerns the detectability of a galaxy in a 21 cm survey. In principle, there exists the possibility that most of the low-mass halos predicted by CDM cosmology correspond to HI-devoid, dwarf spheroidal galaxies. In reality, a solution involving a multitude of isolated early-type dwarf systems seems rather unlikely. Direct observations (Garnett 2002; Swaters & Balcells 2002; Noordermeer et al. 2005), as well as other empirical arguments, suggest that the HI-to-stellar mass ratio grows with decreasing mass for galaxies in the field. HI surveys should thus have an advantage, rather than a disadvantage, in detecting the baryonic counterparts hosted by low-mass DM halos. In addition, optical surveys suggest that isolated early-type dwarfs in medium/low density environments are relatively rare (Karachentsev et al. 2004). A second issue relates to the fact that satellite galaxies may be underrepresented in the α.40 sample, since they are generally redder (and have presumably lower gas fractions) than central galaxies of the same luminosity (e.g., Font et al. 2008). This bias could result in a ≲ 30% underestimate of the abundance of low-width galaxies by ALFALFA (e.g., Yang et al. 2008; Klypin et al. 2010)—not nearly enough to explain the observed discrepancies.

Issue 4 regards the size and detailed spatial distribution of the atomic hydrogen component in galaxies, which determines the way in which its rotation curve is converted into an HI velocity width. In particular, w_HI is a fair tracer of the maximum rotational velocity, only if the HI disk is extended enough to reach the flat (or decreasing) part of the galactic rotation curve. The use of Equation (4) in the derivation of the theoretical WFs implicitly assumes this situation to be true; observationally however, this is often times not the case. For example, the Catinella et al. (2006) set of template rotation curves provides evidence of the fact that lower rotational velocity galaxies tend to have steeper outer rotation curves (see their Figures 1 and 4). The dwarf galaxy samples of Spekkens et al. (2005) and Swaters et al. (2009) suggest that the effect becomes more dramatic at the lowest velocities (see Figures 3 and 4 in the respective references).

This systematic trend for lower velocity galaxies to host less extended HI disks can be understood in terms of the expected baryon depletion of low-mass halos. Results from N body + hydrodynamics simulations (e.g., Hoeft et al. 2006; Ricotti 2010) indicate that halos with mass below some critical value lose a significant fraction of their cosmic share of baryonic matter, due to environmental and internal feedback processes. In particular, UV heating of the intergalactic medium after reionization is believed to lead to substantial gas removal from low-mass halos (v_rot ≲ 20–30 km s⁻¹, corresponding to M_vir ≲ 10⁹–10^9.5 h⁻¹ M_☉). Internal feedback processes such as supernova winds may also be important, but their efficacy is strongly model dependent.

The above considerations could lead to a solution of the overabundance problem that would not require a modification of the extremely successful ΛCDM paradigm. In simple terms, the overabundance problem would be the result of the inability of HI to trace the maximum halo rotational velocity of low-mass systems, which leads to a severe underestimate of their true mass. The same argument has been identified as a possible solution of the "mini-void size" problem by Tikhonov & Klypin (2009), while a similar effect has been proposed by Peñarrubia et al. (2008) as a solution to the "missing satellites" problem.