The north-south asymmetry of the ALFALFA H i velocity width function

The number density of extragalactic 21-cm radio sources as a function of their spectral line-widths – the H i width function (H i WF) – is in principle a sensitive tracer of the dark matter halo mass function (HMF). The Λ cold dark matter model predicts that the HMF should be identical everywhere provided it is sampled in suﬃciently large volumes, implying that the same should be true of the H i WF. The ALFALFA 21-cm survey measured the H i WF in two separate, northern (‘spring’) and southern (‘fall’) Galactic ﬁelds and found a systematically higher number density of sources in the spring ﬁeld. Taken at face value, this is in tension with theoretical predictions. We use the Sibelius-DARK N-body simulation and the semi-analytical galaxy formation model GALFORM to create a mock ALFALFA survey. We ﬁnd that the oﬀset in number density likely has two origins: the sensitivity of the survey is diﬀerent in the two survey ﬁelds, which has not been correctly accounted for in previous measurements; and the 1 / 𝑉 eﬀ algorithm used for completeness corrections does not fully account for biases arising from spatial clustering in the galaxy distribution. The latter bias is primarily driven by a foreground overdensity in the spring ﬁeld within a distance of 30 Mpc, but more distant structure also plays a role. We provide an updated measurement of the ALFALFA H i WF (and H i MF) correcting for the variations in survey sensitivity. Only when systematic eﬀects such as these are understood and corrected for can the H i WF fulﬁl its potential as a test of cosmological models.


INTRODUCTION
The standard Lambda Cold Dark Matter (ΛCDM) cosmological model predicts that the number density of self-bound dark matter haloes as a function of mass -the Halo Mass Function (HMF, Frenk et al. 1988) -is well-approximated by a power law with slope of () ∝  −1.9 over almost 20 orders of magnitude below the scale of the largest collapsed structures today (Wang et al. 2020).The lowmass end of the HMF is sensitive to the the power spectrum of density fluctuations in the early universe.For instance, for thermal relic dark matter particles that are lighter than those assumed in ΛCDM models (e.g., warm dark matter; Avila-Reese et al. 2001;Bode et al. 2001), there is a cut-off in the power spectrum at smaller scales.The higher velocities of lighter thermal relics naturally suppress the formation of low-mass haloes because free-streaming effects erase the density perturbations that could seed them.Measuring the HMF therefore offers an opportunity to constrain the particle nature of dark matter.However, the HMF is not directly measurable and hence indirect measurements of the HMF must be made instead.One option is to study the abundances of galaxies as a function of the kinematics of visible tracers orbiting within their dark matter haloes.
There is an apparent tension between the number of low-mass dark matter haloes predicted by ΛCDM dark matter-only simulations in which galaxies are expected to form, and the number of galaxies with kinematics compatible with inhabiting them measured from ob-★ E-mail: richard.brooks.22@ucl.ac.uk servation.This problem was first identified for satellite galaxies of the Milky Way ('too-big-to-fail', Boylan-Kolchin et al. 2011), and subsequently resolved by Sawala et al. (2016) who showed that the inclusion of baryons in the simulations changes the theoretical predictions; the inevitable ejection of baryons from haloes at early times (due to supernova 'feedback') reduces the growth rate of haloes such that, at the present time, a halo in the full-physics simulation is about 10 per cent less massive than its counterpart in the dark matter-only simulation.A similar issue applying to field galaxies was identified by Papastergis et al. (2015), and it is so far unclear whether this issue is similarly resolved.In this instance, dwarf galaxies should be hosted by dark matter haloes that are significantly more massive than those implied by measuring the kinematics within the galaxies.If instead lower mass dark matter halos host dwarf galaxies, observational surveys should measure a much higher number density of galaxies.
The kinematics of observable tracers in a galaxy are linked to the HMF because the maximum circular velocity of a halo,  max , is correlated to its mass (Navarro et al. 1997).Connecting a kinematic tracer to  max often requires additional modelling, e.g. in the case of the 21-cm spectral line width.The 21-cm velocity spectrum is the H mass-weighted line-of-sight velocity distribution.The 21-cm line-width can be parameterised as the full width at half maximum of the spectrum,  50 .If a dark matter halo contains a sufficiently extended H disc, the maximum circular velocity of the dark matter halo is approximately  max ∼  50 /2 sin(), where  is the inclination angle to the line of sight.However, there are considerations which need to be made when computing  max using this approach.In many cases galaxies will have gas discs that are not sufficiently extended to reach the flat part of the velocity profile (Brook & Di Cintio 2015;Ponomareva et al. 2016;Brook & Shankar 2016;Macciò et al. 2016;Brooks et al. 2017).Additionally, the inclination angle is problematic because line-width surveys usually do not spatially resolve the gas structure and therefore rely on optical counterparts of radio sources to get an estimate for  (e.g., Zwaan et al. 2010).There are many other difficulties faced when obtaining  max using line-width measurements, such as: the gas orbits may not be circular (Brook & Shankar 2016); the gas disc may not lie in a single plane (Ponomareva et al. 2016); emission may be mistakenly taken to overlap with a neighbouring source (Jones et al. 2015;Chauhan et al. 2019); the gas disc may be partially supported by turbulent or thermal pressure (Brook & Di Cintio 2015;Ponomareva et al. 2016); etc.Instead of attempting to infer  max from line-width measurements, a potentially more straightforward technique is to predict the number density of extragalactic 21-cm sources as a function of  50 -the H width function (H WF) -and then compare with observational measurements.Comparison of the H WF from theory to an observational equivalent thus provides an alternative avenue to investigate dark matter than to try and infer a HMF from an observed H WF.
For the Arecibo Legacy Fast ALFA 1 (ALFALFA; Giovanelli et al. 2005) survey, the H WF and H mass function (H MF) are measured simultaneously as orthogonal integrations of the twodimensional H mass-width function.Jones et al. (2018) reported that the H MF has significantly different global shapes in the 'spring' (northern Galactic hemisphere) and 'fall' (southern Galactic hemisphere) fields of the survey.The low-mass slope is significantly shallower the in fall field.The large-scale environmental dependence of the low-mass slope in the H MF was tentatively attributed to the presence of the Virgo cluster in the foreground of one half of the survey and a deep void in the other (Jones et al. 2018).The H WF, on the other hand, has been reported to have a similar shape in both survey fields.Oman (2022) tentatively attributed this similarity in shape to possible environmental effects affecting the shape of the H MF, but leaving the shape of the H WF largely unchanged.However, they made no attempt to explain the different overall normalisation of the H MFs and H WFs in the two regions.
The first measurements of the H WF came almost simultaneously from two different 21-cm surveys.Zavala et al. (2009) used an early release of the ALFALFA survey with only 6 per cent of the final data available.Meanwhile, Zwaan et al. (2010) used the H Parkes All-Sky Survey (H PASS).Both of these measurements revealed that the ΛCDM model apparently overpredicts the abundance of sources at the low velocity-width end.The same overabundance problem persists in follow-up work after subsequent ALFALFA data releases (Papastergis et al. 2011;Jones et al. 2018), but can be explained (at least to leading order) by the systematic reduction of the total mass an abundance of structures below  max ∼ 100 km s −1 by gas pressure, reionization, supernova feedback, stripping, and truncated accretion (Sawala et al. 2013, see also Macciò et al. 2016;Papastergis & Shankar 2016;Brooks et al. 2017;Dutton et al. 2019).Zavala et al. (2009) and Zwaan et al. (2010) both measured differences in the normalisation of the H WF in different subsets of the ALFALFA and H PASS surveys, respectively, although their shapes are statistically consistent with being identical.It has been variously speculated that what drives the difference in the normalisation of the H WF between the ALFALFA spring and fall fields is a combination 1 Arecibo L-band Feed Array of: sample variance, distance modelling, the adopted completeness limit, etc. (see, Oman 2022, for further discussion).
In this work we use the Sibelius-DARK N-body simulation (McAlpine et al. 2022) that reproduces the local structure of the Universe on scales larger than ≈ 4 Mpc populated with galaxies using the GALFORM semi-analytical model (Lacey et al. 2016) to create mock surveys similar to the ALFALFA survey.The nature of the Sibelius simulations allows us to investigate the influence of the spatial clustering of galaxies along the line of sight, as well as possible environmental effects on the global shape of the H WF. Previous work has only been able to comment speculatively on the origin of the differences in the H WF between the two fields surveyed by AL-FALFA (Jones et al. 2018;Oman 2022).Our mock surveys provide a suitable footing for an investigation into survey systematics that may be responsible for driving the asymmetry between the spring and fall fields; our approach enables us to provide the first quantitative estimates for the magnitude of these effects and assess whether they can explain the observed asymmetry.
The plan of this paper is as follows.In Sec. 2 we outline the process we use to construct and analyse mock ALFALFA-like 21-cm surveys.In Sec. 3 we present the ALFALFA and Sibelius-DARK plus GAL-FORM H WFs under various conditions and assumptions.In Sec. 4 we comment on possible mitigations for variations in sensitivity for future surveys, and on the possible origins of qualitative differences between our mock H MFs and H WFs and those measured using ALFALFA.We summarise in Sec. 5.

METHODS
In Sec.2.1 we summarise the defining properties of the ALFALFA survey, detail how extragalactic sources were identified and show how the final survey catalogue is obtained.Next, in Sec.2.2 we provide an overview of the N-body Sibelius-DARK simulation and assess the assumption of identical HMFs in both fields.In Sec.2.3 we give an overview of the GALFORM semi-analytical model.In Sec.2.4 we explain how our GALFORM mock 21-cm survey is created.Finally, in Sec.2.5, we outline the 1/ eff statistical estimator used to correct the H WF for observational incompleteness.

The ALFALFA survey
The ALFALFA survey (Giovanelli et al. 2005) mapped ∼ 7, 000 deg 2 of the sky visible from Arecibo at 21-cm wavelengths out to ∼ 250 Mpc, or  ≤ 18, 000 km s −1 .ALFALFA was specifically designed to investigate the faint end of the H MF in the Local Universe.The survey was completed in 2012, and is composed of two separate fields on the sky; one in the northern Galactic hemisphere, visible during the spring, and the other in the southern Galactic hemisphere, visible during the autumn.By convention, these fields are labeled 'spring' and 'fall', respectively (Jones et al. 2018).
Extragalactic sources in the ALFALFA survey were identified using a matched-filtering technique (Saintonge 2007), supplemented with some sources identified from direct inspection of the raw data cubes.These identified sources were subsequently manually checked to confirm or reject each individual detection, and to assign optical counterparts to detections where possible.The final product is the .100 extragalactic source catalogue described in Haynes et al. (2018).This catalogue lists the coordinates (for the H and associated optical sources), redshifts, 21-cm line flux densities, 21-cm line widths, distances2 , signal-to-noise ratios and H masses of sources, and their uncertainties where relevant.The H mass,  HI , of a galaxy is determined as usual from the flux and distance as: We define a selection of .100 sources from which we measure the H WF in this work in a similar way to Oman (2022).This includes the choice of only 'Code 1' (i.e., S/N > 6.5) sources whose (RA, Dec) coordinates fall in the survey footprint 3 (Jones et al. 2018).Instead of the recessional velocity cut  rec ≤ 15, 000 km s −1 used in Oman (2022), we impose a distance cut d mw ≤ 200 Mpc in order to facilitate comparison with the Sibelius-DARK simulation which is contaminated by low-resolution particles from outside of the zoomin region (McAlpine et al. 2022) beyond this distance.Only sources above the 50 per cent completeness limit (CL) of the survey are selected.The determination of the CL for the ALFALFA survey is described in Haynes et al. (2011).For a flux-limited sample drawn from a uniformly-distributed population of galaxies, number counts as a function of flux are expected to follow a power law with exponent −3/2.Deviation from this form indicates the onset of incompleteness in the survey.There are 20, 857 sources above the 50 per cent CL in the .100 catalogue, of which 13, 006 are in the spring field and 7, 851 are in the fall field (see Table 1).Fig. 1 visualises the .100 sources within the survey footprint on the sky (upper panels), and in a cone diagram over all declinations (lower panel).Oman (2022) used the global 50 per cent CL that they derived throughout their analysis.We adopt the same CL in some contexts, but also make use of the 50 per cent CL derived separately from each of the two survey fields.The fall CL is slightly shallower than the global CL, by 0.011 dex, while the spring CL is slightly deeper, by 0.009 dex, for a net difference of 0.02 dex.Explcitly: where  = log 10 ( 50 /km s −1 ).The CLs at other completeness levels (25 per cent, 90 per cent) are given in Appendix A.

Sibelius-DARK
The The simulation assumes a flat CDM cosmology with parameters from the Planck Collaboration et al. (2014): Ω Λ = 0.693, Ω  = 0.307, Ω  = 0.04825,  8 = 0.8288,   = 0.9611, and  0 = 67.77km s −1 Mpc −3 .We note that Haynes et al. (2018) assume a slightly different cosmology that will cause ∼ 10 per cent differences in the normalisation of the ALFALFA and our mock H MFs and H WFs. This is driven by the assumed value of the reduced Hubble constant, ℎ, which when incorporated into the units of the H MF and H WF results in a different normalisation.Below, we restrict ourselves to qualitative comparisons between ALFALFA and GALFORM, so this small quantitative difference does not influence our conclusions.

The halo mass function of Sibelius-DARK
In the top panel of Fig. 2 we show the HMF for the spring and fall fields as well as for the median across eight octants of the Sibelius-DARK sky (the spring and fall regions have volumes equivalent to 80 and 50 per cent of an octant, respectively; all octants and their properties are detailed in Appendix B).The middle panel of Fig. 2 shows the ratio of the N th octant's HMF to that of the median.The scatter in the HMF across the octants on the sky is 20 per cent around the median.
We quantify how closely we should expect the HMFs in regions of these volumes to agree as follows.Hu & Kravtsov (2003) provide a relation for the fractional variance   in the number density of haloes above a given mass threshold  th in their fig.2. We can approximate the low-mass end of their sample variance relation (sample variance dominates over shot noise in our mass range of interest) as   ∝   th 10 13 M 0.15 , where  depends on the survey volume.Assuming an 'effective' survey radius of 111 Mpc -the radius of a sphere enclosing the same volume as surveyed in the entire AL-FALFA survey -we obtain  ∼ 0.116.We therefore expect the mass function between 10 9 and 10 12 M sampled in a volume equivalent to the ALFALFA survey to scatter by about 5 per cent around the cosmic mean HMF in an equivalent volume (or a factor of √︁ (2) more for a volume equivalent to half of the survey).Strictly speaking this calculation applies to cumulative mass functions, but the weak dependence on mass across our range of interest means that it also provides a reasonable order-of-magnitude estimate for the differential mass function.
Given that the fall field has a HMF that is about 8 per cent overdense with respect to the spring field, shown in the lower panel of Fig. 2, and that both fields are overdense with respect to the median (Fig. 2, middle panel), by about 16 and 6 per cent, respectively are therefore consistent with the expectation for a ΛCDM cosmology.
Further reassurance is provided by the fact that the normalisations of the HMFs in the various octants and in the spring and fall survey regions are broadly consistent with the relative galaxy densities of the northern and southern hemispheres reported for Sibelius-DARK by McAlpine et al. (2022, sec 3.1.2 and fig. 8) and similarly for the Simulating the LOcal Web constrained realisation Dolag et al. (2023).The former highlighting that this difference is entirely consistent (within 1 − 2 ) with the expectation for a ΛCDM cosmology.

GALFORM
Since the Sibelius-DARK simulation is a N-body simulation, we model the evolution of the galaxy population using the GALFORM semi-analytical model of galaxy formation.
The GALFORM semi-analytical model calculates the evolution of galaxies in hierarchical theories of structure formation.The processes governing galaxy formation and evolution are modelled as sets of coupled non-linear differential equations (White & Frenk 1991).Since the first GALFORM models (Cole et al. 1994(Cole et al. , 2000)), there have been numerous changes and improvements (e.g., Granato et al. 2000;Benson et al. 2000;Baugh et al. 2005).We model the Sibelius-DARK simulation galaxy population using the Lacey et al. (2016) variant of GALFORM.This is the same variant as used by McAlpine et al. (2022), who present a detailed investigation into the resulting galaxy population (e.g., luminosity function in their fig.5, stellar mass functions in their fig, 6, etc.).This GALFORM variant incorporates different initial mass functions for quiescent star formation versus for starbursts, black hole formation, feedback from supernovae and from active galactic nuclei that suppresses gas cooling in massive haloes, and a new empirical star formation law in galaxy discs based on molecular gas content.A more accurate treatment of dynamical friction acting on satellite galaxies is also introduced, as well as an updated stellar population model.

Mock 21-cm survey
Four steps are required to construct a mock 21-cm survey from the Sibelius-DARK + GALFORM simulation: calculation of the galactic circular velocity curve, determination of the amount of H gas as a function of line-of-sight velocity to produce a 21-cm spectrum, convolution with a kernel to model the thermal broadening of the 21-cm line, and application of the selection criteria consistent with the chosen 21-cm survey.
To define the radial mass profiles of galaxies, we follow Lacey et al. (2016).The relevant mass-bearing components of a galaxy are the dark matter halo, stellar bulge, stellar disc, hot gas and cold gas.Dark matter haloes are assumed to follow an NFW profile (Navarro et al. 1996(Navarro et al. , 1997) ) described by a virial4 mass and concentration.Stellar bulges follow a 'de Vaucouleurs' ( 1/4 ) law for their surface mass density profile.This profile would require numerical integration to obtain an enclosed mass profile.We follow Lacey et al. (2016) and assume the simpler, though very similar in shape, Hernquist (1990) mass profile for bulges instead.The stellar disc is modelled as an infinitesimally thin disc with an exponentially decaying density profile with a half-mass radius  disc .The hot gas is assumed to settle into a spherically symmetric distribution with density profile: with gas core radius   = 0.1  vir , where  vir is the virial radius of the dark matter halo.The cold gas is modelled as an infinitesimally thin disc with exponentially decaying surface density profile with the same half-mass radius as the stellar disc,  disc .This is partitioned into atomic and molecular gas components, expressed as the fraction of cold gas that is molecular  mol = Σ mol /(Σ mol + Σ atom ), calculated assuming vertical hydrostatic pressure equilibrium and a gas velocity dispersion of  gas = 10 ± 2 km s −1 (for a detailed description, see Lagos et al. 2011).The atomic hydrogen surface density is assumed to be  H = 0.74 times the atomic gas surface density.The galaxy components are assumed to be azimuthally symmetric.Circular velocities for spherically symmetric components are calculated as  2  () =   (< )/, while for thin exponential discs we use: where  disc is the total mass of the disc,  = /2 disc , and   and   are the modified Bessel functions.The total circular velocity at any radius is then the sum in quadrature over all components.The resulting circular velocity curve, with its decomposition into different components, is shown in the left panels of Fig. 3 for two example galaxies.
We define the 21-cm velocity spectrum as the H mass-weighted distribution of line-of-sight velocities.To determine the line-of-sight velocities, the inclination  of the galaxy to the observer must be set.This is done by drawing a uniformly distributed random value for cos() between 0 and 1.We have repeated our analysis using 10 unique random seeds to assign galaxy inclinations.These cause only small differences of about 8 per cent or less in the number of galaxies detected in our mock survey at fixed  50 .
To calculate how much H gas contributes at each line-of-sight velocity, the model H disc is discretized in the radial (50 bins from  = 0 to  = 4  disc ) and azimuthal angular (45 bins from  = 0 to  = 2) coordinates.The line-of-sight velocity of each radial-angular element is  circular () sin  cos .The radial extent is sufficient to enclose ≥ 90 per cent of the cold gas mass, enough to obtain a converged value for  50 .We always calculate the total flux of galaxies    using the total atomic hydrogen mass, integrated to infinity.We match the effective ALFALFA spectral resolution of 10 km s −1 .
We also account for the velocity dispersion of the H gas. The 21-cm line is thermally broadened, hence influencing the measured value of the 21-cm line width.Following Lagos et al. (2011), we assume an empirically determined amplitude for the H velocity dispersion of   = 10 ± 2 km s −1 .GALFORM also assumes the gas velocity dispersion to be 10 ± 2 km s −1 (see, Lacey et al. 2016, and references therein).We implement this by convolving the spectrum with a Gaussian kernel of standard deviation,   .The right-hand panels in Fig. 3 show 21-cm velocity spectra for the same two example galaxies as shown in the left panels.The example in the upper panels is a galaxy where the velocity dispersion is comparable to the maximum circular velocity.The example in the lower panels is a galaxy where the velocity dispersion is much smaller than the maximum circular velocity.In both cases, the black curves show the spectra before convolution with the thermal broadening kernel, while the red bins show the spectra including thermal broadening.We measure the 21-cm line width,  50 , as the full width at half maximum of the spectra.We choose  50 over alternatives such as the full width at 20 per cent maximum,  20 , because  50 is less sensitive to noise in the 21-cm spectra, enabling more sources to be used (Zwaan et al. 2010).
To replicate the ALFALFA survey in order to study the observational and statistical effects on the measurements of the H WF, we first need to create a mock catalogues of Sibelius-DARK + GAL-FORM sources that would have been detected by an ALFALFA-like survey.We draw our initial set of galaxies from the  = 0 output of the GALFORM model. 5e apply two criteria to determine which galaxies are included in our mock survey catalogue.First, only sources in the ALFALFA survey footprints should be included (Jones et al. 2018, tables D1-D4).The Sibelius-DARK halo catalogues provide (RA, Dec) coordinates which we use to determine whether a source is included within the footprints.Second, the ALFALFA CL must be applied.We implement a continuous CL by linearly interpolating between the 25, 50, and 90 per cent spring and fall CLs given in equations (A1-A6), and linearly extrapolating to 0 and 100 per cent completeness.We draw a uniformly distributed random number between 0 and 1 for each GALFORM galaxy.If this number exceeds the survey completeness for the given 21-cm line-width and flux of the galaxy, the galaxy is discarded from the catalogue.The result is a mock GALFORM survey containing 21, 031 sources, of which 12, 408 are in the spring field and 8, 623 are in the fall field (see Table 1, which additionally includes the total number of GALFORM sources before the selection criteria are applied).The distribution of the mock survey sources in (RA, Dec) and distance is overlaid on the ALFALFA survey source distributions in Fig. 1.
For a survey like ALFALFA, there can be instances whereby two or more galaxies fall inside the beam at the same time and overlap in frequency, an effect termed 'beam confusion'.Our method to construct a mock survey does not account for beam confusion.This could plausibly be a limitation when comparing our mock survey to observations.Obreschkow et al. (2013) found that 'confused' galaxies typically have high H mass and  50 , with  HI > 10 10 M and  50 > 300 km s −1 , albeit for the HIPASS survey, which has a larger beam than ALFALFA.Subsequently, Chauhan et al. (2019) found lower levels of confusion in these ranges for their mock ALFALFA surveys and that any confusion only reduced the total number of galaxies in their sample by less than 1 per cent.Jones et al. (2015) also found that beam confusion can only slightly change the shape of the H MF, by no more than would already be allowed by the random errors on the measurements.It therefore seems unlikely that beam confusion is one of the main drivers of the systematic difference between the spring and fall H WFs, so we omit further discussion of this effect from our analysis below.
Fig. 4 shows the counts in the catalogues per unit volume as a function of H mass (left panel) and  50 (right panel) for the ALFALFA .100 catalogue (dashes) and Sibelius-DARK + GALFORM (crosses) for the spring (green) and fall (purple) fields individually.The GAL-FORM H mass distribution overall appears similar in shape to that of ALFALFA.Noticeable differences include the overdensity of GALour analysis using the  = 0.05 output of the same GALFORM model and find only small differences of about 5 per cent or less in the number of galaxies detected in our mock survey at fixed  50 .
FORM sources at intermediate masses, 10 8  HI /M 10 9 , and the underdensity of sources at the highest H masses.The GALFORM  50 distribution, on the other hand, has a starkly different shape to that measured in ALFALFA.There is an overdensity of sources for line widths below ∼ 150 km s −1 , and an underdensity above.We comment further on these differences in Sec. 4.
The distribution of sources in the  50 −  HI plane for the 'true GALFORM' data (detected and undetected galaxies in the simulation), the 'mock GALFORM' data (only those simulation sources within the ALFALFA selection criteria) and the ALFALFA data are shown in Fig. 5.The true GALFORM data displays a bi-modal distribution of sources which can be attributed to the relative contribution from satellite and central galaxies, respectively.There are obvious differences in the distribution between the true GALFORM and the mock, or real, ALFALFA survey.Given that the simulation will produce many galaxies that are either low in flux, and/or possess large values for  50 , these sources will not satisfy the ALFALFA selection criteria and hence 'drop out' of the final survey catalogue.Alongside these  50 −  HI distributions, we show the relative abundance of sources between the ALFALFA and mock GALFORM surveys in the bottom-right panel of Fig. 5. Generally, the ALFALFA survey contains more sources at the edges of the  50 −  HI distribution.In particular, the smallest bin in  50 ( 50 ∼ 20 − 30km s −1 ) contains no sources for our GALFORM mock.This comes from a limitation in our method when accounting for thermal broadening of the 21cm emission line; see Sec 2.4 for details.On the other hand, the mock GALFORM survey has an abundance of sources relative to ALFALFA for ( HI ,  50 ) ∼ (10 8 − 10 10 M , 25 − 150 km s −1 ).This is consistent with the corresponding 1D  HI and  50 distributions shown in Fig. 4. We comment further on differences in the  50 −  HI plane between our mock and the real ALFALFA survey in Sec.3.4.

The 1/𝑉 eff maximum likelihood estimator
The 1/ eff estimator is used to estimate the abundance of undetected galaxies with a given  HI and  50 from the abundance of those galaxies that were detected.Galaxies can be undetected in the survey due to being low-mass, and hence low-flux sources and/or due to having wider line widths which spread their emission over more channels in the detector.The original SWML estimator (Efstathiou et al. 1988) is applicable to galaxy samples that are integrated-flux limited.The SWML can be extended to become the bivariate stepwise maximum likelihood (2DSWML) estimator for surveys that have a selection function that depends on two observable quantities, such as ALFALFA.The only difference between the 1/ eff and 2DSWML estimators is that in the 1/ eff case, the effective volumes are iteratively calculated for each individual galaxy, instead of calculated per 2D bin in  HI - 50 space as in the 2DSWML approach (Zwaan et al. 2005;Martin et al. 2010).The effective volumes found for each galaxy are maximum likelihood counterparts of the classical 1/ max volumes (Schmidt 1968) with the important difference that the 1/ eff method in principle corrects for spatial non-uniformity in the source distribution.
The 1/ eff estimator is 2D in the sense that the CL of the AL-FALFA survey depends both on the integrated 21-cm flux,  21 and line-width,  50 , of the source.The implementation of the 1/ eff estimator is the same as in Oman (2022) with the only difference being that we adopt separate CLs for the two survey fields.The 1/ eff method requires knowledge of the survey CL in order to produce the 2D H mass-width function.Instead of using the global 50 per cent CL as in Oman (2022, equation A5), we adopt the 50 per cent CL appropriate to each survey field (Equations 2 & 3).From the summation of the values of the 1/ eff weights in 2D bins in  HI and  50 we compute the 2D H mass-width function.The sum along the  50 axis gives the H MF, while the same along the mass axis gives the H WF.

ALFALFA AND SIBELIUS-DARK + GALFORM H I WIDTH FUNCTIONS
Understanding the origins of spatial variations of the normalisation of the ALFALFA H WF is crucially important in the context of using it as a constraint on cosmology.The ΛCDM cosmological model predicts that the dark matter HMF should be universal (in shape and normalisation, e.g.Frenk et al. 1988;Mo & White 1996;Sheth & Tormen 2002;Crain et al. 2009) and therefore similar in the two fields surveyed in ALFALFA, because the volumes sampled are sufficiently large.We have checked this explicitly in the Sibelius-DARK simulation: the HMFs in the spring and fall volumes differ by no more than 8 per cent (within their uncertainties) at any halo mass 10 8 <  vir /M < 10 14 (see Sec. 2.2).
The most straightforward prediction for the H WF is that it should also have the same shape and normalisation (within about 8 per cent) in the two fields (see Sec. 5.3.2,Oman 2022, for a detailed ac-count of the connection between the HMF and the H WF). Indeed, the Sibelius-DARK + GALFORM galaxy catalogue confirms this.In the right panels of Fig. 6 we show the H WF of all galaxies with  HI > 10 6 M in the spring and fall survey fields (regardless of whether they would be detected) with the green and purple solid lines, respectively.Analogously to the HMFs in the two regions, the two H WF curves differ by no more than 8 per cent at any line width.If the approximately factor of 2 difference in normalisation of the H WF for the spring and fall ALFALFA fields cannot otherwise be explained, then the ΛCDM model could be called into question (Zwaan et al. 2010;Papastergis et al. 2011;Oman 2022, where the first entry highlights the issue in the HIPASS survey and those after for ALFALFA).We have already outlined above (Sec.2.4) that random scatter in the HIWF driven by random inclination angles and the systematic effects due to the redshift evolution of galaxies within the survey volume are much too small to explain the large observed difference in normalisation, so we now turn our attention to other potential sources of error.
In Sec.3.1 we measure the H WF of our mock Sibelius-DARK + GALFORM survey using exactly the same method as Oman (2022), and compare to their measurement for the ALFALFA survey.Next, in Sec.3.2, we repeat the measurement assuming the separately derived CLs for the spring and fall fields in the calculation of the 1/ eff weights, and compare to the same approach applied to the .100 catalogue.In Sec.3.3 we investigate effect of the clustering of sources along the line of sight upon the H WFs. Finally, in Sec.3.4 we investigate the effect of the galaxy distribution in the H massspectral line width plane.

Fiducial analysis of the mock surveys
We make the measurement of the H MF and H WF for the spring and fall mock Sibelius-DARK + GALFORM catalogues separately, following the procedure outlined in Oman (2022) where the globally derived CL for the ALFALFA survey (Oman 2022, equation A5) is assumed in the 1/ eff estimator.The top-left and right panels of Fig. 6 respectively show the H MFs and H WFs measured using this approach.The measurements of Oman (2022), now adapted to retain only sources within 200 Mpc (Sec.2.1), are also shown in these panels for comparison.
We restrict ourselves to a qualitative comparison of the .100ALFALFA (dash marker with shaded box for the spring and fall fields in green and purple, respectively) and 'mock GALFORM' (crosses of corresponding colours) H WFs. We find that the ALFALFA and mock GALFORM H WFs have similar, almost constant low linewidth slopes for  50 100 km s −1 .For larger  50 , GALFORM underpredicts the number density of sources significantly.
Despite the 'true GALFORM' curves for the spring and fall fields (green and purple solid lines) differing by less than 8 per cent across the entire range in  50 , we find that our mock 21-cm survey has a qualitatively similar offset between the spring and fall H WFs as is observed in ALFALFA.The number density in the spring field exceeds that in the fall field throughout the line-width range.The median ratio between the spring to fall H WFs is log 10 (  /   ) = 0.36 ± 0.09 (16 th -84 th percentile scatter about the median) for ALFALFA, and 0.17 ± 0.13 for mock GALFORM.Our mock GALFORM H WFs capture the shapes of their true counterparts reasonably well, although they do underestimate them by up to a factor of 3 at  50 100 km s −1 , and up to a factor of 10 at higher line widths.The reason for this is that there are combinations of H masses and line widths where galaxies exist, but none are ob-served (because the survey is not sensitive to detect them); this issue is discussed in detail by Oman (2022, sec. 5.2.6).
We focus our attention on the puzzling result that the mock GAL-FORM H WFs are systematically offset from each other in the same sense as ALFALFA.We first consider the influence of the choice of CL for each survey field.

Influence of the survey completeness limit
We repeat the measurement of the H MF and H WF for the spring and fall .100 and mock Sibelius-DARK + GALFORM catalogues as in Sec.3.1, but this time assuming CLs derived separately for the spring and fall ALFALFA fields (equations 2 & 3) in the calculation of the 1/ eff weights.The bottom-right panel of Fig. 6 shows the measured H WF using this approach.
The .100 ALFALFA and mock GALFORM H WFs have qualitatively similar shapes to those discussed in Sec.3.1.However, the spring and fall mock survey H WFs are no longer systematically offset from each other.Therefore, when the same CLs used to construct the mock catalogues (Sec.2.5) are used to correct them for incompleteness, the 1/ eff algorithm correctly recovers the fact that the H WFs in the two fields have indistinguishable shapes and amplitudes (although the true number densities are still underestimated).Quantifying this, the median ratio between the spring to fall H WFs is log 10 (  /   ) = 0.05 ± 0.14.
For ALFALFA, the systematic offset between the spring and fall H WFs is reduced when separate spring and fall CLs are used in the calculation of the 1/ eff weights, but does not disappear: the median ratio between the spring to fall H WFs is log 10 (  /   ) = 0.25 ± 0.09.Because there is empirical evidence that the ALFALFA survey has different CLs in the spring and fall fields (Haynes et al. 2011;Oman 2022), our view is that these measurements should be taken to supersede those of Oman (2022) when taking the same cut in recessional velocity -we tabulate them in Appendix C for both the H MFs and H WFs in the spring and fall regions.We also include a combined measurement for the entire survey in which the 1/ eff weights have been derived separately for galaxies in the two regions accounting for the different survey sensitivity in each.
The fact that the spring and fall ALFALFA H WFs are still offset from each other after correcting for the different CLs in the two fields suggests that this may not be the only systematic effect influencing the measurement.There are a limited number of other factors that can influence the outcome of the 1/ eff measurement.We have identified only two: differences in the clustering along the line of sight in the two fields and/or differences in the properties of galaxies in the two fields as reflected by their distribution in the 2D space of  HI −  50 .
In order to focus on these other effects, for our analysis in the remainder of this section we remove the influence of the CL by imposing a very conservative completeness cut on both the ALFALFA survey and on our mock surveys: we impose a CL offset in  21 by 0.10 dex above that empirically derived for the fall field (equation 3).We also assume this CL in the calculations of 1/ eff weights below.

Influence of the clustering of sources along the line of sight
Our use of the 1/ eff estimator has so far treated the spring and fall fields as fully independent: when calculating the H WF and H MF for the spring field, no information is available about the large-scale structure in the fall field, and vice versa.We now wish to investigate whether the different clustering along the line of sight in each field causes a spurious systematic offset between the H WFs. We proceed as follows.We calculate 1/ eff weights for the full .100catalogue, thereby obtaining effective volumes for each source without any knowledge of the different clustering of sources along the line of sight in each field.The 1/ eff algorithm makes no use of the (RA, Dec) coordinates of sources, so this approach treats the survey as a single contiguous volume.We then separately compute the H WF for each field using the effective volumes for sources in that field.This exercise should erase the offset between the spring and fall fields if the distribution of sources along the line of sight is its direct cause.
The resulting H WFs are shown in the top panel of Fig. 7.We label the case for which the survey fields are treated independently as 'independent  eff ', and the case for which the survey fields are analysed jointly as 'full survey  eff '.We find that there is almost no change in the systematic offset between the spring and fall H WFs; the median ratio between the spring to fall H WFs derived using the independent  eff weights is log 10 (  /   ) = 0.22±0.11.Therefore, the different distribution of sources along the line of sight in the two fields is not an important driver of the systematic offset.This leaves the distribution of sources in the 2D  HI −  50 parameter space to be investigated.

Influence of the galaxy distribution in 𝑀 HI − 𝑤 50
The distribution of detected sources in the  HI −  50 plane is the fundamental input into the 1/ eff estimator.Even if the dark matter HMF is identical in the spring and fall fields, should galaxy formation and evolution proceeded differently in the two fields this could lead to differences between their respective H WFs. We therefore check for measurable differences in the  HI −  50 distribution of sources between the two survey fields, which could be a symptom of such differential galaxy evolution.We show ratios of the  HI −  50 distributions in the two fields in Fig. 8.The intrinsic distribution of GALFORM galaxies ('true GALFORM', left panel) is slightly overdense in the fall field.The amplitude of the fall overdensity in the  HI −  50 plane is consistent with that in the HMF (see Sec 2.2).We interpret this to imply that galaxy formation in this GALFORM model proceeds essentially identically in the two fields.There is also some noise due to low number counts at high  HI and/or  50 .
In the centre panel, we show the ratio of the distributions of GAL-FORM galaxies detected in our mock survey, weighted by their 1/ eff weights (calculated separately for the spring and fall fields).Large differences would arise due to the different CLs in the two fields, so we make this comparison after imposing the aforementioned more conservative completeness cut.We repeat the same process for the ALFALFA survey and show the result in the right panel of the figure.In the mock survey, the two distributions agree over most of the parameter space, except around ( HI ,  50 ) ∼ (10 8.5 M , 120 km s −1 ).
The ALFALFA survey has a similar feature at the same location, but also has log( s / f ) > 0 over most of the space.This overall overdensity in the spring field is the same one reflected in the systematically higher number density in the same field in the H MF and H WF: these functions are the integrals of the H mass-width function along the width and mass axes, respectively.We note that this does not necessarily imply that the bias need be near-uniform across the  HI −  50 plane.Fig. 8 therefore suggests that the difference in number density seems to arise across all line widths and H masses and is not tied to any particular region in this parameter space.The local overdensity near ( HI ,  50 ) ∼ (10 8.5 M , 120 km s −1 ) bears further investigation.
We have traced the origin of this feature to the difference in the number of detected sources as a function of distance (see Fig. 9) The spring (green) and fall (purple) .100 AL-FALFA H WF as derived using the 1/V eff weights assigned independently (described in Sec.3.3; dashes with shaded 1 uncertainties) and from the full survey catalogue (points with error bars showing 1 uncertainties).The completeness limit adopted has been raised by 0.10 dex above that empirically derived for the fall field.Sources detected at all distances from the Milky Way are included.The lower sub panel shows the spring-to-fall ratio for the independent (red points) and full survey (black points) 1/V eff weights.Middle panel: The spring (green) and fall (purple) H WF derived using the 1/V eff (dashes) and 1/V max (open circles) weights assigned independently but including only those sources detected at distances  mw > 30 Mpc.The lower sub panel again shows the spring-to-fall ratio for the independent 1/V eff (red points) and independent 1/V max (open black circles) 1/V eff weights.Lower panel: The median H WF ratio for spring-to-fall regions (derived using the 1/V eff weights) as a function of the minimum distance from the Milky Way at which sources are retained, d mw .
in the foreground of the survey ( mw 30 Mpc).In Sec.3.3 we confirmed that the systematic offset between the H WFs in the two survey fields is not due to a global difference in the clustering of sources in distance.However, local features could still play a role.Essentially, the 1/V eff estimator seems to be incorrectly extrapolating the foreground overdensity in the spring field through the entire survey volume, biasing the spring field to higher number densities.This effect is similar to what would occur if we used the 1/ max estimator (Schmidt 1968), which assumes that galaxies are uniformly distributed in space, but less severe -the 1/ eff algorithm is intended to compensate for non-uniformity in the galaxy distribution, but does so imperfectly.The spring field in our mock GALFORM survey also has slightly more sources than the fall field at small distances, but the effect is much less pronounced, explaining why this particular bias is less pronounced in the mock survey.This can be ascribed to the Sibelius-DARK simulation being an imperfect match to the actual local cosmic web.We note an increased density of GALFORM sources at ∼ 15 Mpc in the spring field, associated with the Virgo cluster.We suggest that this is the cause of the localised overdense spring feature at ( HI ,  50 ) ∼ (10 8.5 M , 120 km s −1 ) in Fig. 8.
We confirm this interpretation of the origin of the offset between the ALFALFA spring and fall H WFs by removing all sources from the ALFALFA catalogue with distances  mw < 30 Mpc.This choice of distance cut is further motivated in the lower panel of Fig. 7. Here, we show the median ratio of the spring and fall H WFs as a function of the minimum distance from the Milky Way at which sources are retained.For all distances beyond about 30 Mpc the median ratio stays about the same, indicating that the foreground galaxies have the most influence within about this distance.In the middle panel of Fig. 7, we show the H WFs in the two fields measured with both (i) the conservative CL described above imposed and (ii) sources within 30 Mpc removed.In this case the H WFs in the two fields are very close to agreement; the median ratio between the spring to fall H WFs derived using the independent  eff weights is log 10 (  /   ) = 0.07 ± 0.12.We attribute the small remaining differences to the 1/ eff estimator slightly over-compensating for clustering of sources along the line of sight beyond 30 Mpc.For comparison, we also show the H WFs in the two fields measured using the 1/ max algorithm (i.e.assuming a spatially uniform galaxy distribution) in the middle sub-panel Fig. 7.In this case, the sign of the offset between spring and fall is reversed (due to an overdensity of sources in the fall field at distances of ∼ 50 − 80 Mpc) with a median ratio between the spring to fall H WFs derived using the independent  max weights being log 10 (  /   ) = −0.01 ± 0.12.
In summary, we attribute the differences between the spring and fall ALFALFA H WFs (e.g. as measured by Oman 2022) to: (i) the adopted completeness limit for the survey; and (ii) the 1/ eff estimator incorrectly extrapolating the foreground overdensity in the spring field through the entire survey volume (in that field).Accounting for these systematic effects leads to H WFs in the spring and fall fields that are consistent with being identical, which is in turn consistent with the two fields having identical HMFs (within a few per cent) as expected in a ΛCDM cosmology.

DISCUSSION
In Sec.4.1, we comment further on the influence of the choice of CL on the H WF and the importance for future surveys of mitigating variations in the CL.Then in Sec.4.2 we discuss the possible origins of the qualitative differences between the ALFALFA H MF and H WF and those that we obtain from simulations.

Survey completeness limit
For ALFALFA, the CL is determined empirically.Oman (2022) found that the CLs in the ALFALFA spring and fall fields are different by about 0.02 dex in  21 when derived separately.This small quantitative difference was assessed to be insufficient to have a strong influence on the shape (low-velocity slope, and location and sharpness of exponential cutoff at high velocities -i.e.excluding the overall normalisation) of the H WF. We agree with this assessment, but emphasise that the overall normalisation of the H WF is extremely sensitive to the chosen CL.Given this, we have investigated whether differences between the spring and fall CLs could plausibly fully explain the overall difference in number density between the spring and fall H WFs. Beginning from the global 50 per cent CL (Oman 2022, eq.A5), we gradually 'raised' the fall and 'lowered' the spring CLs by equal increments until the overall difference in number density in the two fields vanished.We found that the minimum offset between the CLs in order to fully account for the variation would have to be at least 0.1 dex, or five times greater than the empirically determined difference.Such a large difference between the CLs is completely incompatible with the .100 catalogue.The difference between the CLs in the two fields can therefore only explain part of the overall difference in amplitude between the spring and fall ALFALFA H WFs.
Since a difference in sensitivity between the spring and fall fields is apparent, it is worth considering whether there may also be variations in sensitivity internally within each field.Indeed, this has previously been measured: using a catalogue covering 40 per cent of the final surveyed area, Haynes et al. (2011) measured a CL that is 0.02 dex 'deeper' than that measured by Oman (2022) using the entire surveyed area in the spring region 6 .
We suggest that it is worth considering possible measures to mitigate variations in sensitivity across the sky in the design of future 21-cm surveys intending to measure the H WF (or H MF). As an illustrative example, a survey built up by covering a wide area to shallow depth and repeating this process to build up sensitivity might be expected to have a more spatially uniform CL in the presence of timevarying radio frequency interference than one where a small field is observed to a final target depth before moving on to a subsequent field.
We also comment briefly on the assumed form of the CL for ALFALFA: a two-segment broken power law.There is what we would characterise as tentative evidence in the ALFALFA catalogue data for departures from this form.For instance, there is a hint of an upturn at low line widths, and the sharp transition between the two power law segments may not accurately capture what is in reality likely to be a more gradual transition in slope.The relatively small number of sources observed in ALFALFA makes it challenging to make stronger statements, but the order of magnitude increase in source counts expected from the next generation of H surveys should allow for much more tightly constrained determinations of their CLs.

The H MF
The true GALFORM H MF does not reproduce the almost constant low-mass slope seen for ALFALFA (left panels, Fig. 6).Instead, there is a fall and rise in the number density of sources over the mass range 10 6 <  HI /M < 10 8 .This is caused by the relative contributions of central and satellite galaxies to the H MF, which in GALFORM are easily separable.In GALFORM, all galaxies are assumed to originate as central galaxies, but when the dark matter halo of a galaxy merges 7 with another more massive halo, that galaxy is assumed to become a satellite in the new larger halo.The evolution of galaxy properties is modelled separately for centrals and satellites.
For instance, as soon as a galaxy becomes a satellite, its hot gas halo is instantly stripped away by the ram pressure of the central galaxy, and added to the central's hot gas halo.Consequently, no further gas is allowed to cool and accrete onto the satellite.This aggressive stripping influences the cold gas content of a galaxy, causing the dip in the distribution in H mass, seen in our Fig.6 to occur for  HI 10 8 M .The exact location of this transition is affected by the halo mass resolution of the simulation (Lacey et al. 2016, Sec. 4.2.2).However, the dominance of the H MF by central and satellite galaxies in the high-and low-mass ends, respectively, is independent of the model adopted (Lagos et al. 2011, Sec. 4.1.1).Smoothing these two peaks into a constant low-mass slope for the H MF may be possible if satellites' hot gas is not instantaneously ram pressure stripped and/or gas is allowed to cool onto satellites.

The H WF
The true GALFORM H WF when compared to ALFALFA shows a distinct lack of larger line width sources, particularly for  50 > 200 km s −1 (Fig. 6, right panels).The Lacey et al. (2016) GALFORM model assumes that atomic gas is distributed in a relatively compact disc (with the same scale-length as the stellar disc) in all galaxies.Cases where the emission is dominated instead by a much more diffuse disc, a ring, or even discrete clouds, such as is observed in some massive galaxies (e.g.Oosterloo et al. 1999;Serra et al. 2012), are not modelled.This leads to severe underestimates of  50 for some galaxies where a more realistic, radially extended H distribution would lead to a wider line.The deficit of galaxies at the highest H masses (visible in the H MF), which should be populated by galaxies with large total masses and correspondingly large H line widths, likely also plays a role.

SUMMARY AND OUTLOOK -THE H I WF AS A COSMOLOGICAL CONSTRAINT
The low line width end of the H WF is in principle sensitive to properties of the dark matter through its close relationship with the HMF.
Our H WF derived from the ΛCDM Sibelius simulations (Fig. 6) has a much shallower slope than expected from the HMF in the mocksurveyed volume, and is in reasonable qualitative agreement with that measured from ALFALFA observations at low line-widths.This reinforces previous work showing that the ALFALFA and HIPASS H WFs can be reconciled with a ΛCDM cosmology once relevant galaxy formation processes and observational biases are accounted for (Sawala et al. 2013;Brooks et al. 2017;Dutton et al. 2019).Specifically, this is a consequence of: (i) not every low-mass Sibelius-DARK halo hosting a GALFORM galaxy; (ii) not every galaxy having substantial amounts of atomic gas ( HI > 10 6 M ); and (iii) the nonlinear mapping between the maximum circular velocity and  50 .We have not, however, ruled out interpretations involving a low-mass truncation of the HMF, such as may arise from a WDM particle (e.g.Zavala et al. 2009).We plan to apply our mock survey methodology to a WDM version of the Sibelius-DARK volume in future work.
Our approach using a constrained N-body realisation of the entire ALFALFA survey volume has allowed us to investigate systematic effects inaccessible to previous studies.The Sibelius-DARK simulation makes the clear prediction that the spring and fall fields of the survey should have identical HMFs, and our modelling extends this to predicting identical H MFs and H WFs. Taken at face value, this is in tension with the observed differences between the spring and fall H MFs and H WFs measured from ALFALFA observations (Jones et al. 2018;Oman 2022).We identified two effects that can bias the H WF and plausibly explain the observed differences between the spring and fall ALFALFA H WFs: (i) The CLs in the spring and fall survey fields differ, but have previously been assumed to be identical in deriving the 1/ eff weights used in deriving the H WF. We tabulate updated measurements correcting for this in Appendix C.
(ii) The 1/ eff algorithm is intended to compensate for galaxy clustering (c.f. the classical 1/ max method assuming a uniform spatial distribution), but does so imperfectly.A foreground overdensity of H -rich galaxies in the spring ALFALFA field, and an underdensity in the foreground of the fall field, drive large systematic errors.
We note that the small remaining systematic offset between the spring and fall H WFs once the above two effects are accounted for is most likely also attributable to the limited ability of the 1/ eff to accurately compensate for differences between the galaxy clustering in distance in the two fields.In this interpretation, the true H WF is spatially invariant (within about 10 per cent) when sampled in volumes comparable to the volume of the ALFALFA survey; the observed differences between the spring and fall H WFs in ALFALFA are entirely spurious.Encouragingly, the influence of individual over/underdense regions on the calculation of 1/ eff weights can be mitigated by simply surveying a larger area on the sky; the ongoing WALLABY 8 survey (Koribalski et al. 2020) will cover an area about four times wider than that covered by ALFALFA.
The H WF has the potential to become a stringent test of cosmological models.Realising this potential will require a deeper understanding of the systematic biases influencing measurements, such as those due to spatial (or temporal) variability in survey sensitivity.Progress on theoretical issues is also needed.Combining semi-analytical models able to simulate directly the large volumes of current and future surveys with hydrodynamical simulations able to resolve in detail the internal structure of atomic gas in galaxies 8 Widefield ASKAP L-band Legacy All-sky Blind surveY seems a promising path forward.We look forward to the prospect of using the H WF as a constraint on the nature of dark matter.C1.Amplitudes and uncertainties of the measured .100 H WF for the spring and fall fields derived assuming the completeness limits given in Equations (A1-A6) and a cut in recessional velocity,  rec ≤ 15, 000 km s −1 , matching that used in Oman (2022).The combined measurement for the entire survey uses the 1/ eff weights calculated separately for the two fields.Note that although the logarithms of all amplitudes and uncertainties are given, the uncertainties are symmetric on a linear scale and should be interpreted as the 1 width of a Gaussian distribution (not a log-normal distribution).

Figure 1 .
Figure 1.Top panel: Distribution of .100 ALFALFA sources (red points) within the footprint on the sky of the spring field.Black points show sources contained in our ALFALFA-like mock survey catalogue constructed from the Sibelius-DARK + GALFORM simulation.Middle panel: Same as top panel, but for the fall field.Bottom panel: Cone diagram for the survey catalogues over all declinations as a function of distance from the Milky Way and right ascension on the sky.

Figure 3 .
Figure3.Two examples illustrating our model for generating a mock 21-cm velocity spectrum for Sibelius-DARK + GALFORM galaxies.Left panels: examples of the circular velocity curve shown within 3  disc for a 'low' line width (top) and an 'intermediate' line width galaxy (bottom), including the (1 −  mol ) atomic fraction profile beneath.The arrow indicates the scale-length of the cold gas disc,  disc .Contributions from the dark matter halo (red line), stellar bulge (purple dotted line), stellar disc (green dash-dot line), hot gas (orange dashed line) and cold gas (blue solid line) are shown.The sum in quadrature of these components gives the model velocity curve (thick brown line).Right panels: the 21-cm velocity spectrum resulting from the circular velocity curves and H gas surface density profiles for both the case before (black open histogram) and after (red filled histogram) including the gas velocity dispersion,   .The left-hand y-axis shows the distribution by H mass, while the right-hand y-axis shows the distribution by 21-cm flux,  21 .The randomly chosen inclination to the line of sight , for each galaxy is given in the top-left corners of the panels.'Low' line width galaxy properties: M vir = 4.4 × 10 10 M , stellar disc mass, M ★,disc = 1.3 × 10 8 M , stellar bulge mass, M ★,bulge = 1.2 × 10 7 M , hot gas mass, M hot = 1.1 × 10 10 M and cold gas mass, M cold = 6.9 × 10 8 M .'Intermediate' line width galaxy properties: M vir = 1.4 × 10 12 M , M ★,disc = 3.3 × 10 10 M , M ★,bulge = 2.8 × 10 10 M , M hot = 1.3 × 10 11 M , M cold = 3.7 × 10 10 M .

Figure 4 .
Figure 4. Left panel: the observed counts per unit volume as a function of H mass for the .100 (dashes) and GALFORM (crosses) catalogues.The spring (green) and fall (purple) fields are shown separately.Shaded bands and error bars show Poisson uncertainties.Right panel: similar to the left panel but showing the observed counts per unit volume as a function of 21-cm line width,  50 .

Figure 5 .
Figure 5. Upper left: Number of galaxies across the  50 −  HI plane in the Sibelius-DARK + GALFORM catalogue, including all galaxies (detected and undetected) in the simulation at  = 0.The two peaks in the distribution correspond to the contributions from central (peak near (  HI ,  50 ) ∼ (10 8 M , 50 km s −1 )) and satellite galaxies (peak near (  HI ,  50 ) ∼ (2 × 10 6 M , 70 km s −1 )).The colour scale represents the number of galaxies per bin in  50 −  HI .Upper right: Number of galaxies across the  50 −  HI plane in the mock GALFORM catalogue (sources passing the ALFALFA selection criteria).The colour scale is the same as in the upper left panel.Lower left: Number of galaxies across the  50 −  HI plane in the ALFALFA catalogue.The colour-scale is the same as in the upper left and upper right panels.Lower right: Relative abundance of sources between the ALFALFA and mock GALFORM surveys across the  HI −  50 plane.The two distributions are normalised by their respective integrals across the entire  50 −  HI plane, subtracted in each bin, then compared to the normalised abundance of ALFALFA sources in the same bin.Red regions indicate an excess of ALFALFA sources; blue indicates an excess in GALFORM.

Figure 6 .
Figure 6.Upper left panel: the H mass function (H MF) calculated by adopting the same completeness limit in the spring and fall fields.The main panel shows the H MF measured from the .100 (dashes with shaded 1 uncertainties) and GALFORM catalogues (crosses with 1 uncertainties shown with error bars), separately for the two ALFALFA fields, spring (green) and fall (purple).Additionally, the 'true' GALFORM H MF (solid lines with 1 uncertainty shown with shaded band) is shown for the spring and fall fields.The lower sub-panel shows the ratio of the spring and fall GALFORM (black crosses) and ALFALFA (red crosses) H MFs with 1 uncertainties.Upper right panel: The H width function (H WF) calculated by adopting the same completeness limit in the spring and fall fields.Lines and symbols are as in upper left panel.Lower left and right panels: similar to upper panels, but showing the H MF and H WF calculated by adopting separate completeness limits in the spring and fall fields.

Figure 7 .
Figure 7. Upper panel:The spring (green) and fall (purple) .100 AL-FALFA H WF as derived using the 1/V eff weights assigned independently (described in Sec.3.3; dashes with shaded 1 uncertainties) and from the full survey catalogue (points with error bars showing 1 uncertainties).The completeness limit adopted has been raised by 0.10 dex above that empirically derived for the fall field.Sources detected at all distances from the Milky Way are included.The lower sub panel shows the spring-to-fall ratio for the independent (red points) and full survey (black points) 1/V eff weights.Middle panel: The spring (green) and fall (purple) H WF derived using the 1/V eff (dashes) and 1/V max (open circles) weights assigned independently but including only those sources detected at distances  mw > 30 Mpc.The lower sub panel again shows the spring-to-fall ratio for the independent 1/V eff (red points) and independent 1/V max (open black circles) 1/V eff weights.Lower panel: The median H WF ratio for spring-to-fall regions (derived using the 1/V eff weights) as a function of the minimum distance from the Milky Way at which sources are retained, d mw .

Figure 8 .Figure 9 .
Figure 8. Logarithmic ratio of the abundance per unit volume of sources in the  50 −  HI plane between the spring and fall survey fields.Left panel: Sibelius-DARK + GALFORM catalogue data including all galaxies (detected and undetected).The approximately uniform overdensity in the fall relative to the spring region is due to the different HMFs in the two regions as detailed in Sec.2.2.Middle panel: mock GALFORM catalogue data (detected sources only) weighted by 1/ eff derived independently in each survey field.Right panel: ALFALFA catalogue data weighted by 1/ eff derived independently in each survey field.In the centre and right panels, only sources above a completeness limit 0.10 dex above that empirically derived for the fall field are included.

Table 1 .
Characteristics of the .100ALFALFA and mock Sibelius-DARK + GALFORM survey catalogues.Additionally shown is the information of the GALFORM galaxy population before any selection criteria are applied.The assumed distance limit is 200 Mpc.The area of the spring and fall fields are 1.240 sr and 0.752 sr, respectively.

Table B1 .
The sky coordinate position definition and number counts of all dark matter haloes contained within bounds of each Sibelius-DARK octant.The median octant contains 2.42 million dark matter haloes.A distance cut keeping only sources within d mw ≤ 200 Mpc is applied.