Modeling the HI-Halo Connection: Evolution, Scatter, and a Halo-based Prescription for 21-cm Mock Catalogs

Fabio Fontanot; Gabriella De Lucia; Lizhi Xie; Marta Spinelli; Michaela Hirschmann; Mohd Kamran

REVIEW 2 major objections 5 minor 296 references

Modeling the HI-Halo Connection: Evolution, Scatter, and a Halo-based Prescription for 21-cm Mock Catalogs

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The hydrogen content of a dark-matter halo is set as much by its assembly history as by its mass.

desk verdict A careful, honest fitting paper that gives 21-cm mock builders a practical prescription — the concentration-based scatter model has a real but acknowledged weak spot at high z, and the z50 alternative should be used there. read the letter →

arxiv 2607.20123 v1 pith:QXN4IA4P submitted 2026-07-22 astro-ph.CO astro-ph.GA

Mohd Kamran , Gabriella De Lucia , Marta Spinelli , Lizhi Xie , Fabio Fontanot , Michaela Hirschmann This is my paper · ORCID

classification astro-ph.COastro-ph.GA

keywords HI–halomassrelation21-cmintensitymappinghaloassemblybiasspinconcentrationsemi-analyticgalaxyformationneutralhydrogenmockcatalogs

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper shows that the neutral-hydrogen (HI) content of a dark-matter halo is not a single function of halo mass but depends systematically on how the halo assembled. At fixed mass, faster-spinning, later-forming, and less-concentrated halos are systematically HI-richer, and this assembly-driven variation accounts for a large part of the observed scatter. The authors distill the median relation and the scatter into a compact analytic prescription built from halo-catalog quantities (mass, spin, concentration), leaving only a residual intrinsic dispersion of about 0.3 dex. Because 21-cm intensity-mapping surveys require large dark-matter-only catalogs populated with HI, a physically grounded prescription with realistic scatter is a practical tool for building mock observations and interpreting current and upcoming surveys.

What carries the argument

The scatter plane: for each halo, the deviation of its HI content from the median relation at its mass is modeled as a linear function of standardized halo spin and concentration (Δ = C + A_λ x_λ + B_c x_c). This plane, combined with a Gaussian intrinsic scatter term, is what carries the argument that the scatter is predictable from catalog-level halo properties rather than being irreducible noise.

What would settle it

Fit NFW profiles to the z=0 snapshot particle data and compare the directly measured concentrations to the Vmax/Vvir-inverted values used here; if the two disagree systematically at the ~0.1–0.2 dex level for halos near 10^12 solar masses, the quoted intrinsic scatter and the concentration term are compromised.

Watch

Extended reading notes

Core claim

The central discovery is that the substantial scatter (about 0.5 dex) in the HI–halo mass relation is not random: at fixed halo mass, higher-spin, later-forming, and less-concentrated halos hold significantly more HI. Spin is the dominant secondary property along the gas-rich rising branch, while concentration (or equivalently formation time) matters most near and above the quenching scale, and a standardized linear plane in spin and concentration, supplemented by a Gaussian residual of about 0.3 dex, reproduces the full scatter of the model from z=0 to z=5. This assembly dependence is encoded in a five-parameter median fit plus a two-predictor scatter model, expressed entirely in quantities

Load-bearing premise

The concentration used to predict HI scatter is recovered by inverting the Vmax/Vvir relation assuming an NFW profile and restricting to the monotonic branch c_h ≥ 2.16; if that inversion is biased, particularly for unrelaxed halos and at high redshift where concentrations cluster near the degeneracy minimum, the scatter model's decomposition is unreliable.

Editorial extensions

If this is right

The prescription allows dark-matter-only N-body catalogs to be populated with HI, including a halo-property-dependent scatter, enabling more realistic 21-cm mock catalogs for intensity-mapping surveys.
Accounting for the assembly dependence of HI content changes the predicted clustering and shot noise of the 21-cm field, because HI is preferentially found in later-forming, higher-spin halos.
At high redshift (z≳3), the median HI–halo mass relation becomes a nearly single power law, and the spin-plus-formation-time scatter model remains predictive, whereas concentration loses discriminating power.
The scatter peaks near the quenching scale (~10^12 solar masses), where the HI distribution is bimodal; a symmetric Gaussian residual under-represents the gas-poor tail, indicating limits of the simple scatter prescription.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the assembly-driven scatter is real, HI-selected samples should exhibit assembly bias in clustering: HI-rich halos of a given mass are likely more clustered if they assembled later, a prediction that could be tested by cross-correlating 21-cm intensity maps with galaxy surveys.
The same scatter model could be propagated into the predicted 21-cm auto-power spectrum, especially the shot-noise term, which the paper does not explicitly compute but whose amplitude depends directly on the halo-to-halo HI variance.
Because concentration becomes unreliable as a predictor at z≳3 due to the Vmax/Vvir inversion degeneracy, a practical hybrid prescription would use concentration at low redshift and formation time at high redshift, a choice the paper's own comparison supports.
The residual intrinsic scatter of about 0.3 dex may partly arise from baryonic feedback processes not captured by halo properties; if so, it represents a fundamental floor for any halo-based HI prescription, motivating future work that incorporates baryon-sensitive features.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Kamran et al. does exactly what it says: it characterizes the HI-halo mass relation in GAEA2023 and packages it as a prescription for populating dark-matter halos. The median relation and fitting form are inherited from Spinelli et al./Baugh et al., but the systematic scatter analysis — spin, concentration, formation time, redshift evolution, residual scatter — is the new part, and it is done carefully. The z=0 comparison with ALFALFA/HIPASS, the conditional HIMF against Jones et al., and the comparison with other models are all transparent and honest.

The paper is also candid about its own limitations. It openly states that the z=0 HIMF is a calibration target, so agreeing with it is partly circular. It flags that the scatter model is a fit to the same simulation it is compared against. It reports that high-redshift fit parameters for the median relation are unconstrained. I appreciate that — the text does not oversell.

The most substantive concern is the concentration inversion. Eq. (4) recovers c_h from Vmax/Vvir assuming NFW and relaxed halos. At z~3, where most halos sit at c_h~3-5 close to the minimum of that curve, small errors in the input map to large errors in c_h. That will attenuate B_c and inflate sigma_int, so the fading concentration signal at high z is at least partly an artifact of the proxy. The paper acknowledges this, but it does not do the obvious z=0 check: direct NFW fits are possible for the z=0 snapshot and would validate or calibrate Eq. (4) without waiting for higher-z particle data. That feels like a missed opportunity.

That said, the concern does not sink the paper. The spin term is robust at all redshifts, and the formation-time model in Appendix C.2 (Table C.3) captures assembly information out to z=5 with nearly constant sigma_int ~0.3 dex. So the broader 'assembly matters' claim survives, and the user of the prescription has a better tool for high z. I would trust the z=0 concentration model and use the z50 version beyond z~2.

This is a useful, solid paper for the 21-cm mock community. It is not a breakthrough and it does not resolve a long-standing question, but it is a genuine step up from the usual single-scatter-number prescriptions. The math and the fitting procedures are straightforward and reproducible; the tabulated parameters make implementation trivial.

My recommendation: send it to peer review. It deserves a serious referee. I would suggest the referee push for the z=0 direct NFW validation (or an explicit statement of why it cannot be done) and for a clearer caveat that the sigma_int values at high z are partly proxy-limited. The paper will be better for it.

Referee Report

2 major / 5 minor

Summary. The paper characterizes the HI-halo connection using the GAEA2023 semi-analytic model applied to the Millennium-I and Millennium-II simulations, from z=0 to z~5. It presents the HI mass function, the median M_HI(M_h) relation, and an analysis of the scatter about that relation. The central claims are that the scatter at fixed halo mass is not random but correlates with halo assembly, that higher-spin and less-concentrated (later-forming) halos are HI-richer, and that a compact prescription based on halo mass, spin, and concentration, with a Gaussian residual of sigma_int ~0.3 dex, reproduces the full scatter and can be used to populate dark-matter-only catalogs for 21-cm mocks.

Significance. If the prescription is robust, it is a useful and practical contribution: it is compact, expressed entirely in terms of quantities available in N-body halo catalogs, and it directly targets intensity-mapping mock construction. The paper is careful in several respects: it discloses that the z=0 HIMF is partly a calibration target, it tests the sensitivity of the median fit to the MSI/MSII stitching scale with bootstrap uncertainties, and it provides an alternative formation-time-based scatter model in Appendix C.2. The redshift-dependent median fit and the comparison with existing prescriptions and observations are valuable. However, the main scatter model rests on a halo-concentration proxy that is never validated against direct NFW fits, and the 'reproduces the full scatter' claim is, by the authors' own statement, partly by construction. These issues are load-bearing for the quantitative claims but appear fixable.

major comments (2)

[Sect. 5.2, Eq. (4); Table 3] The central scatter model, Eqs. (7)-(9), is specified in terms of c_h recovered by inverting the Vmax/Vvir-c relation under the assumptions of relaxed halos and an NFW profile, restricted to the monotonic branch c_h>=2.16. This inversion is most fragile at high redshift, where typical c~3-5 lie near the shallow minimum of Eq. (4). Small Vmax/Vvir errors from mergers, triaxiality, substructure, or resolution can then produce large c_h errors, which would attenuate B_c and inflate sigma_int. The paper explicitly states in footnote 2 that particle data are available for the z=0 snapshot but does not use them to validate Eq. (4). Because the high-z fading of the concentration signal and the redshift growth of sigma_int are among the main quantitative results, the authors should provide a direct NFW-fit comparison at z=0 and, if possible, at one or two higher redshifts, and propagate the unce
[Sect. 6.2, Eqs. (7)-(9); Fig. 10] The statement that the prescription 'reproduces the full scatter' is circular as presented: the scatter model is fitted to the GAEA2023 simulation and then compared with the same simulation. The authors acknowledge this in the text ('By construction the two components reproduce the overall HI scatter'), but the central claim in the abstract and Sect. 7 is stronger. Since the paper's purpose is to provide a predictive prescription for mocks, the authors should either reframe the claim as a self-consistency check or, preferably, add an out-of-sample test: e.g., fit on MSII and test on MSI (or a split sample in mass/redshift), or compare the resulting 21-cm clustering/shot noise against observations or an independent model. At minimum, the mass-dependent performance should be quantified with formal variance-reduction statistics, rather than the r2_lambda and r2_c values quoted in the text,

minor comments (5)

[Table 3] The table lists A_lambda, B_c, standardization constants, and sigma_int without uncertainties. Given the bootstrap treatment used for the median fit, the scatter-fit parameters should carry similar error estimates; this is particularly important for sigma_int and B_c because of the concentration-proxy issue.
[Sect. 6.2] The symbols r2_lambda and r2_c are introduced in the text but never formally defined. Please define them as variance reductions relative to the scatter about the median relation, and state over which halo-mass range and redshift they are evaluated.
[Table 2] For z>=3.1, M_break is railed against its prior and a_2 is unconstrained. This is noted and reasonable, but the table should mark these entries more prominently as unphysical, or move them to a separate table, to avoid misuse by readers constructing mocks at high redshift.
[Appendix C.2 / Fig. C.2] The caption of Fig. C.2 says 'Scatter fit (λ+ z50)' while the legend refers to 'Median fit'; please harmonize the labels. Also clarify how many halos have undefined z50 and how the sample restriction affects the comparison with Table 3.
[Section 3.1] The sentence about the convolution with 0.25 dex is clear, but consider adding a sentence in the figure captions of Figs. 1-3 and A.1 stating that the observational-resolution convolution is applied only in that comparison, to avoid readers interpreting the model scatter in terms of this convolution.

Circularity Check

1 steps flagged · score 2.0 of 10

One admitted calibration step; the central HI–halo and scatter results are independent model outputs.

fitted input called prediction [Section 3, first paragraph (HIMF at z=0); also Abstract and Section 7, item 1]
"In GAEA2023, the partitioning of the cold gas into its atomic and molecular components is tuned so that the model reproduces the local (z=0) HIMF against the ALFALFA measurement of Haynes et al. (2011). The resulting agreement at the high-mass end, above the blind-survey completeness limit (log(M_HI/M⊙)≳9), is therefore partly by construction, whereas the low-mass end is less directly constrained by the calibration and therefore offers a more independent test of the model."

The abstract says 'At z=0, the model reproduces the observed HIMF' and Section 7 item 1 says 'The GAEA2023 local HIMF reproduces the ALFALFA and HIPASS measurements above their completeness limits.' But the high-mass end of that same HIMF is the explicit target of the cold-gas partitioning calibration, so the agreement is imposed by the fit, not derived. The paper immediately discloses this ('partly by construction') and redirects to uncalibrated parts (low-mass end, host-halo decomposition), so the step is minor and does not undermine the central independent claims about the HI-halo relation and scatter.

full rationale

The paper is substantially self-contained against external benchmarks. The central results—the median M_HI(Mh) relation, its redshift evolution, and the scatter's secondary dependences on spin, concentration, and formation time—are not calibrated to the M_HI(Mh) relation or its scatter; they are emergent outputs of the GAEA2023 semi-analytic model, which was calibrated to other data (stellar mass function, local HIMF, AGN luminosity function). The paper explicitly states that 'the detailed dependence of HI content on halo mass and the associated scatter were not directly calibrated' (Sect. 6.1), and it validates the halo-mass-dependent parts against independent group-catalog measurements. The only true circular step is the z=0 high-mass HIMF 'reproduction,' which is a calibration target and is honestly disclosed as such; the abstract's unqualified phrasing overstates it slightly, but the body correctly labels it 'partly by construction.' The scatter prescription is a fit to the simulation and is presented as such ('By construction the two components reproduce the overall HI scatter of the simulation'), so it is a goodness-of-fit rather than a circular prediction. The concentration inversion via Eq. (4) is an approximation/assumption, not a circular step. Self-citations (Spinelli et al. 2020; De Lucia et al. 2024) supply the fitting form and model description, but they are not load-bearing in place of independent evidence. Overall, the derivation chain is not circular in its core claims; the minor admitted calibration step warrants a low score of 2.

Assumptions & free parameters 11 free parameters · 4 assumptions · 0 invented entities

The central prescription rests on a large number of free parameters fitted to the GAEA2023 output, for both the median relation (5 parameters per redshift plus two fixed values) and the scatter model (A_lambda, B_c, sigma_int, plus standardization constants). These are not derived from first principles. The axioms are the standard assumptions of semi-analytic modeling and N-body halo catalogs; none are new entities introduced by this paper.

free parameters (11)

a1 (Eq. 5, Table 2) = 1.3e-3 (z=0) to 3.4e-3 (z=4.9)
Amplitude of the rising cooling branch of the median M_HI(Mh); fitted to the GAEA2023 stitched MSI+MSII median at each redshift.
a2 (Eq. 5, Table 2) = 5.5e-4 (z=0) to ~0 (z=4.9)
Amplitude of the high-mass satellite-dominated term; fitted to the median relation.
alpha (Eq. 5, Table 2) = 0.63 (z=0) to 0.23 (z=4.9)
Sharpness of the exponential truncation in the cooling branch; fitted.
beta (Eq. 5, Table 2) = 1.17 (z=0) to 0.41 (z=4.9)
Slope of the rising branch (M_HI ∝ M_h^(1+beta)); fitted.
M_break (Eq. 5, Table 2) = 10^11.03 Msun (z=0); railed at 10^10.5 for z>=3
e-folding mass of the truncation; fitted, but unconstrained at z>=3 where the high-mass downturn is absent.
M_min (Eq. 5) = 10^8 Msun
Low-mass cut-off mass; held fixed because it is below the resolved halo mass range and not constrained by the fit.
gamma (Eq. 5) = 0.5
Exponent of the low-mass cut-off; fixed following Spinelli et al. (2020).
A_lambda, B_c (Eq. 7, Table 3) = A_lambda 0.259 (z=0) to 0.222 (z=5); B_c -0.027 (z=0) to -0.036 (z=5)
Coefficients of standardized spin and concentration in the scatter plane; fitted by volume-weighted least squares to the GAEA output.
sigma_int (Eq. 9, Table 3) = 0.311 dex (z=0) to 0.361 dex (z=5)
Residual scatter about the fitted plane; interpreted as the 'intrinsic' dispersion, but actually the residual of the linear model.
Standardization constants (tilde_lambda, s_lambda, tilde_c, s_c, Table 3) = e.g. lambda_tilde 0.037, s_lambda 0.023 (z=0)
Medians and spreads used to standardize spin and concentration in Eq. 8; computed from the halo sample.
Central and satellite fit parameters (Appendix C, Tables C.1, C.2) = Various
Separate fits to central (Eq. 5) and satellite (Eq. C.1) relations, each with several free parameters fitted to GAEA output.

assumptions (4)

domain assumption Halos are relaxed NFW systems; concentration is recovered from the Vmax/Vvir-c inversion restricted to c_h >= 2.16 (Eq. 4).
Used to obtain c_h for the scatter prescription; explicitly noted as approximate for unrelaxed systems and near the non-monotonic branch at high redshift.
domain assumption GAEA2023 is a faithful model of galaxy formation, especially its treatments of satellite stripping, AGN feedback, and atomic/molecular gas partitioning.
All predictions of the HI-halo relation and scatter depend on the SAM's physics; the paper acknowledges model-dependence but does not test against hydrodynamical simulations for this specific claim.
domain assumption The Millennium I and II simulations with WMAP1 cosmology (Omega_m=0.25, sigma_8=0.9) adequately represent the dark-matter halo population.
The halo catalogs and merger trees are taken from these simulations; the paper notes the single-cosmology limitation for cosmological applications.
domain assumption Halo spin and concentration from the halo catalog are error-free predictors in the scatter fit.
The paper treats lambda and c as fixed and solves Eq. 7 by volume-weighted least squares in Delta, ignoring measurement error in the predictors.

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Cite this review

Pith. "Pith review of Modeling the HI-Halo Connection: Evolution, Scatter, and a Halo-based Prescription for 21-cm Mock Catalogs." pith.science (2026). https://pith.science/paper/QXN4IA4P

@misc{pith2026260720123,
  author       = {Pith},
  title        = {Pith review of: Modeling the HI-Halo Connection: Evolution, Scatter, and a Halo-based Prescription for 21-cm Mock Catalogs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXN4IA4P}},
  note         = {Machine review of arXiv:2607.20123}
}

abstract

The redshifted 21-cm line of neutral hydrogen (HI) is a powerful tracer of large-scale structure, and post-reionization HI intensity mapping is emerging as a competitive cosmological probe whose interpretation requires a description of how HI populates galaxies and dark matter halos. We characterize the HI--halo mass relation, its redshift evolution, and its intrinsic scatter, identifying its secondary dependences. We use the updated GAlaxy Evolution and Assembly (GAEA) semi-analytic model, applied to the Millennium-I and Millennium-II simulations, to predict the HI mass function (HIMF) and the HI--halo mass relation from the present day to redshift $z\simeq5$. At $z=0$, the model reproduces the observed HIMF and its decomposition by host-halo mass. The median HI--halo mass relation rises with halo mass, peaks near $10^{11.7}\,M_\odot$, declines as central galaxies are quenched by feedback from active galactic nuclei, and rises again where satellites dominate, approaching a single power law at high redshift. We show that the substantial scatter, of about 0.5 dex, is not random but is governed by halo assembly: at fixed mass, higher-spin, later-forming, and less-concentrated halos are systematically HI-richer, with spin together with either concentration or formation time accounting for part of this scatter and leaving an intrinsic dispersion of about 0.3 dex. We encode the median relation, these secondary trends, and the intrinsic scatter in a compact, physically motivated prescription expressed entirely in terms of quantities available in dark-matter halo catalogs. This prescription reproduces the full scatter and enables the construction of large-volume 21-cm mock catalogs for interpreting ongoing intensity-mapping measurements with SKA precursor facilities, such as MeerKAT, and for preparing for forthcoming surveys with the SKA.

Figures

Figures reproduced from arXiv: 2607.20123 by the authors.

**Figure 1.** The galaxy Hi mass function (HIMF) at z = 0 from the MSI (solid curves) and MSII (dashed curves) simulations. Black curves show the HIMF of all galaxies (centrals and satellites combined); magenta and green curves show the central- and satellite-galaxy contributions, respectively. Observational data points from HIPASS (Zwaan et al. 2005) and ALFALFA (Martin et al. 2010; Haynes et al. 2011; Jones et al. 2018) are sh… view at source ↗

**Figure 2.** Evolution of the galaxy HIMF from z = 0 to z = 1. Upper panel: Total HIMF for all galaxies from MSI (solid) and MSII (dashed) simulations. Lower panel: Satellite-only HIMF from the same simulations. Colored lines correspond to different redshifts as indicated, with matching symbols showing observational constraints: at z = 0, ALFALFA blind survey shown by salmon annuli (Jones et al. 2018) and HIPASS blind survey s… view at source ↗

**Figure 3.** Hi conditional mass function at z = 0 for different ranges of host halo mass. Solid curves show the predictions of the GAEA2023 model for all halos and for the three halo-mass bins indicated in the legend. Navy squares show the total ALFALFA HIMF of Martin et al. (2010) (i.e. the all-halos case), while the cyan, magenta, and gold symbols show the ALFALFA group-catalog measurements of Jones et al. (2020) in restrict… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Median Hi–halo mass relation, MHI(Mh), at z = 0 for all galaxies within dark matter halos. The black solid curve is the GAEA2023 median; the vertical dotted line (lower panel only) marks the MSII–MSI stitching scale, log(Mh/M⊙) ≃ 11.9. The black shaded band, and the er…

**Figure 5.** Figure 5: Redshift evolution of the median Hi–halo mass relation in GAEA2023, for all galaxies, centrals, and satellites. The satellite relation is taken from MSI alone, whose larger volume best samples the satellite-dominated, high-mass halos. Solid curves show the median MHI(…

**Figure 6.** Figure 6: Distribution of the halo spin parameter, λh, in the Mh–MHI plane for GAEA2023 at z = 0 (upper) and z = 3.1 (lower). The binned quantity is the total Hi mass of each parent halo (centrals plus satellites), and the hexagonal bins are colored by the median λh of the host…

**Figure 8.** Figure 8: Same as [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: Total Hi–halo mass relation, MHI(Mh) (central plus satellite Hi per parent halo), for GAEA2023 at z = 0 (black) and z = 3.1 (dark cyan). For each redshift, the solid line shows the GAEA median and the dashed line the best-fitting Eq. 5 ( [PITH_FULL_IMAGE:figures/full_…

**Figure 10.** Figure 10: Total Hi–halo mass relation and its scatter at z = 0 (upper) and z = 3.1 (lower). The solid line is the median fit. The gray band is the 16–84 percentile scatter measured in the simulation; the colored band is the scatter produced by our fit (Eq. 9) – the median plus …

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