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ALMACAL XIII. Evolution of the CO luminosity function and the molecular gas mass density out to $z$ ~ 6

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Using ALMA calibrator fields, the paper traces the cosmic molecular gas density back to z ~ 6, finding a peak near z ≈ 1.5 and a decline of about an order of magnitude toward higher redshift.

desk verdict A careful, low-cosmic-variance CO LF out to z~6, but the headline high-z decline is prior-dominated and needs re-framing. read the letter →

arxiv 2502.06778 v1 pith:7ZEYIL6E submitted 2025-02-10 astro-ph.GA

Victoria Bollo , Céline Péroux , Martin Zwaan , Aleksandra Hamanowicz , Jianhang Chen , Simon Weng , Claudia del P. Lagos , Matías Bravo
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R. J. Ivison Andrew Biggs
This is my paper · ORCID
classification astro-ph.GA
keywords COluminosityfunctionmoleculargasmassdensityALMACALcosmicvariancegalaxyevolutionhighredshiftALMAcalibrationfieldsbathtubmodel
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 claims that the cosmic density of molecular gas—the raw fuel for star formation—rose from the present day back to a peak at redshift z ≈ 1.5, and then declined by roughly an order of magnitude toward z ≈ 6. The evidence comes from 87 CO emission-line galaxies found serendipitously in the calibration fields of ALMA, which together cover hundreds of independent sightlines and therefore suffer far less cosmic variance than single contiguous survey areas. If the claim holds, the fuel supply for star formation tracked the rise and fall of cosmic star formation, and the ratio of molecular gas to stellar mass is consistent with a "bathtub" model in which gas is continuously replenished while stars form. The global gas depletion timescale, the ratio of molecular gas density to star-formation rate density, is found to be roughly constant across all redshifts.

What carries the argument

The load-bearing machinery is the probabilistic redshift assignment built from the SHARK-2 semi-analytical model: for each single-line detection, a two-dimensional histogram of CO transition flux against redshift is converted into a probability that the line is CO(J→J-1) at a given z, and the analysis is repeated in 1000 Monte Carlo realisations sampling these probabilities together with completeness, fidelity, and flux uncertainties. This turns otherwise ambiguous single-line detections into a statistical sample. On top of that sits the standard conversion from observed CO flux to CO(1-0) luminosity (Solomon et al. 1997) with empirical high-J-to-1-0 ratios, the Schechter-function fit to the CO luminosity function, and the integration of L' Φ(L') to get ρ_H2. The ALMACAL survey design—299 independent calibrator fields rather than one contiguous patch—is what reduces cosmic variance to below about 5 percent.

What would settle it

Follow up the single-line candidates with a second CO transition, [C I], or dust continuum at the frequencies predicted by SHARK-2; if a large fraction of lines assigned to CO(2-1) at z ≈ 1-1.5 turn out to be CO(1-0) at z ≈ 0.5-0.7, the claimed peak of ρ_H2 near z ≈ 1.5 would move or weaken.

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Extended reading notes

Core claim

The paper's central result is an extended CO luminosity function and a new measurement of the cosmic molecular gas mass density ρ_H2 from z = 0 to z ≈ 6. Using the ALMACAL-22 dataset—archival ALMA calibration observations pruned to high quality—the authors identify 87 CO candidates, assign redshifts by weighting over possible CO(1-0) through CO(6-5) transitions with probability distributions from the SHARK-2 semi-analytical model, and build the luminosity function in six redshift bins. They find that ρ_H2 increases with redshift, peaks near z ≈ 1.5 at log ρ_H2 ≈ 7.4 M⊙ Mpc⁻³, and then declines by about 1 dex toward the highest redshifts, with Schechter-function fits showing a decreasing normalization Φ* and a characteristic luminosity L* that rises to z ≈ 2 and then plateaus. The same data place the molecular-to-atomic gas density ratio peaking near z ≈ 1.5 and a roughly constant global depletion timescale, which the authors interpret as support for the bathtub model of baryon cycling.

Load-bearing premise

The load-bearing premise is that the SHARK-2 model's probability distribution correctly assigns the rotational transition and redshift for the 81 of 87 candidates that lack spectroscopic or photometric redshifts; if this prior is wrong, the reconstructed CO luminosities, luminosity function, and molecular gas density would shift.

Editorial extensions

If this is right

  • The cosmic molecular gas density peaks near z ≈ 1.5, placing the maximum supply of star-forming fuel at cosmic noon and supporting the idea that gas consumption tracks the star-formation rate density.
  • The roughly constant global depletion timescale means that, on cosmic average, galaxies consume their molecular gas on a fixed timescale at every epoch, so changes in ρ_H2 rather than changes in efficiency drive the star-formation history.
  • The molecular-to-stellar mass density ratio follows the bathtub model: gas must be continuously replenished to sustain the observed star formation, rather than being a one-time reservoir.
  • Surveys built from many small independent fields, like ALMACAL, keep cosmic variance below about 5 percent, more than an order of magnitude lower than single contiguous fields of similar total area; future molecular-line surveys should adopt this design.
  • The bright end of the CO luminosity function evolves such that the characteristic luminosity L* rises from z = 0 to z ≈ 2 and then stays flat, while the normalization Φ* falls by a factor of about 3 across the same range.

Reading between the lines

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

  • If SHARK-2's prior is even mildly biased toward CO(2-1), the inferred z ≈ 1.5 peak could be an artifact of the prior; a targeted follow-up of a few dozen candidates would settle whether the peak is real or model-driven.
  • The tension with ASPECS and HDFN at z > 1—where ALMACAL finds lower ρ_H2—may indicate that small contiguous surveys overestimate the density because of cosmic variance, or that ALMACAL's calibrator fields sit in underdense regions; comparing the two on a common volume would test which.
  • The method of using archival calibration data as a science survey could be extended to other spectral lines (e.g., [C I], [C II], H2O) and to measure the cosmic density of atomic gas at high redshift, where 21-cm emission is too faint.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper (ALMACAL XIII) presents a blind search for CO emission lines in 1107 ALMA calibrator data cubes from the ALMACAL-22 release, yielding 87 CO candidates with S/N>4. Spectroscopic redshifts are secured for three sources and photometric redshifts for three more; for the remaining 81 sources, redshift and CO J-transition assignments are drawn from probability distributions built from the SHARK-2 semi-analytical model (Section 3.4.1). The authors construct the CO(1-0) luminosity function in six redshift bins using Eq. (3) with 1000 Monte Carlo realizations that sample flux, completeness, fidelity, and the redshift probability, fit Schechter functions with alpha fixed to -0.2, integrate to obtain the molecular gas mass density rho_H2(z), and compare with previous surveys (ASPECS, COLDz, PHIBSS, HDF-N) and simulations. The headline results are that rho_H2 rises to a peak near z~1.5 and declines by about 1 dex toward higher redshift, that the z=4-6 value is only a lower limit, that the results are consistent with the 'bathtub' baryon-cycling model, and that cosmic variance is below 5% thanks to the many independent sightlines.

Significance. The survey strategy is a genuine methodological advance: using many independent ALMA calibrator fields rather than one contiguous area directly mitigates cosmic variance, and the paper's treatment of completeness (injection-recovery grid, Fig. 1), fidelity (negative-source statistics, Eq. 1), uncertainty propagation (1000 realizations over flux, completeness, fidelity, and redshift), the [C ii] interloper estimate, and the exclusion of lines within 2000 km/s of calibrator redshifts is careful and largely reproducible from the text. The catalogue in Table A and the tabulated Schechter parameters are useful community products. If the headline result (rho_H2 peaking near z~1.5 and declining by ~1 dex toward z~4-6) survives the prior-dependence concern raised below, it would provide a cosmic-variance-hardened confirmation of the bathtub model and a new constraint on the molecular gas depletion timescale at high redshift. The principal caveat is that the novel high-redshift decline is claimed precisely in the regime where redshift and J-transition assignments are most dependent on the SHARK-2 prior, and where the z=4-6 bin contains only one fitted data point above the luminosity limit.

major comments (3)
  1. [§3.4.1, §4.2, §5.1.2, §4.5] For 81 of the 87 sources the redshift and J-transition are assigned in each of the 1000 LF realizations by drawing from the SHARK-2 flux-redshift probability distribution, and Eq. (3) propagates those draws into the CO LF and hence, via Eq. (5), into rho_H2. The assertion in §5.1.2 that 'the final results on the molecular gas mass density and luminosity function are decoupled from the simulation' is therefore load-bearing and is not demonstrated. The lowest-J test of §4.5 is the opposite limiting assumption rather than an independent measurement: it uses the same single-line data, omits the per-transition volume weighting of the fiducial method, and §4.5 itself concedes it 'likely skews the distribution of CO across cosmic epochs'; the agreement in Fig. 6 is within large uncertainties. The close agreement with SHARK-2 in Fig. 9 is then partly by construction. I request a sensitivity analysis with an agnostic prior (for example uniform over J within the detectable redshift range), with the resulting shifts in Phi*, L'*, and rho_H2(z>2) quoted explicitly, and a reframing of the SHARK-2 comparison as a consistency check rather than an independent confirmation.
  2. [Abstract; §4.3; §5.2; Table 3] The manuscript states in §4.3 that the z=4-6 panel contains only one data point above the luminosity limit and in §5.2 that the z=4-6 rho_H2 is only a lower limit, yet the abstract claims 'strong constraints ... back to z~6' and presents the '~1 dex decline' without qualification. Given the prior-dependence issue above, the high-redshift decline rests on a lower limit whose constituent sources have model-assigned redshifts, and Table 3 omits the z=4-6 bin entirely. Please add the z=4-6 rho_H2 explicitly to Table 3 with lower-limit notation, and revise the abstract and conclusions to state that the z~6 constraint is a lower limit based on a single point above the completeness threshold.
  3. [§4.2] The r_j->1 conversion factors are quoted as {3.33, 5.20, 4.76, 2.70, 0.53} at z<2 and {4.09, 8.24, 12.21, 14.68, 13.86} at z>2 for J=2,...,6, but the defining convention is not stated (whether L'_{CO(1-0)} = r_j->1 x L'_J or L'_J / r_j->1) and the specific Boogaard et al. (2020) table is not identified. As printed, the z<2 sequence is non-monotonic in J, and r_2->1 = 3.33 is hard to reconcile with typical L'_{2-1}/L'_{1-0} values near unity. Because r_6->1 = 13.86 at z>2, a misassignment of a high-J line changes L'_{CO(1-0)} by more than an order of magnitude. Please specify the convention, verify the numerical values against the cited source, and quantify the sensitivity of rho_H2(z>2) to alternative excitation assumptions.
minor comments (6)
  1. [Throughout] The text contains numerous ligature-rendering artifacts such as 'di fferent', 'e ffective', 'e fficient', 'su fficiently', 'o ffers', and 'di fficult'; these should be resolved in the production version.
  2. [§3.4.1, §6, References] The SHARK-2 reference is cited inconsistently: §6 cites 'Lagos et al. 2023' while §3.4.1 and §5.1.2 cite 'Lagos et al. 2024', and both entries appear in the bibliography; please unify the citations.
  3. [§5.1.1] When quoting the local faint-end slope from Fletcher et al. (2021), the text reads 'alpha ~ 1.2'; with the convention of Eq. (4) this should be written alpha ~ -1.2 to avoid sign confusion.
  4. [§5.3, §6 Conclusions] Conclusions items 4 and 5 state that cosmic variance is 'less than 5%' without the caveat given in §5.3 that the Keenan et al. (2020) prescription may under-estimate field-to-field variance by an order of magnitude at z~2-4 according to Gkogkou et al. (2022); the headline figure should carry that caveat or be attributed explicitly to the D&R10/K20 prescriptions.
  5. [Eq. (7)] The numerical factor 291.0 in pi R^2 * 291.0 is not defined; a brief note on the unit conversion (arcmin^2 to h^-1 Mpc^2) would make the cosmic-variance formula self-contained.
  6. [Table A.1] Several catalogue entries have reliability values as low as 0.05-0.15; a sentence in §3.5 explaining how such low-fidelity candidates enter the LF through F_i in Eq. (3) would help readers interpret the catalogue and the stability of the results.

Circularity Check

1 steps flagged · score 6.0 of 10

SHARK-2 supplies the redshift/J prior for 81/87 sources and is then quoted as the independent model match; the claimed decoupling is contradicted by the construction of Eq. (3).

  1. self definitional [§3.4.1; Eq. (3) in §4.2; §5.1.2]
    "we included all plausible J transitions, each weighted according to the probabilities evaluated by SHARK-2. ... Although we use the redshift probability distribution derived from SHARK-2 as a prior in our analysis, the final results on the molecular gas mass density and luminosity function are decoupled from the simulation. This independence is demonstrated by the differences observed in the CO luminosity functions and the results from the lowest−J approach, explained in Section §4.5 and further discussed in §5.2.1."

    Eq. (3) constructs the CO LF by binning sources whose redshift and J-transition are sampled from the SHARK-2 probability distribution built in §3.4.1; for 81 of 87 sources this prior is the only redshift constraint. The resulting CO LF and ρ_H2 are therefore functions of SHARK-2's predicted flux-redshift distribution, so presenting agreement with SHARK-2 as independent validation is circular: the model is both the input prior and the comparison target. The lowest-J check does not break this degeneracy; it replaces the prior with the opposite extreme while using the same single-line data and ignoring the fiducial volume weighting, so it does not demonstrate decoupling.

full rationale

The paper contains genuine external content: 87 SoFiA2 detections in ALMA calibrator fields, measured fluxes and line widths, completeness estimates from injected sources, fidelity from inverted cubes, and survey volumes from the ALMA band/footprint geometry. The observed fluxes are not fitted to SHARK-2, and the survey's many independent sight lines give low cosmic variance relative to single contiguous fields. However, the redshift/J identification—the step that places each source into the CO LF and ρ_H2—is imported from SHARK-2 for 81/87 sources. Because each realization of Eq. (3) weights sources by those SHARK-2 probabilities, the LF is not decoupled from the simulation, and the paper's own decoupling assertion in §5.1.2 is contradicted by its methodology. The lowest-J approach in §4.5 is a useful robustness check, but it is not an independent measurement: it is another single-line assumption at the opposite extreme, and its agreement with the fiducial analysis does not validate the fiducial prior. External comparisons with ASPECS, COLDz, PHIBSS, and HDFN provide context, but they cannot independently confirm the redshift assignments for the 81 unconfirmed sources. The z=4–6 bin is additionally a lower limit with only one fitted point above the luminosity threshold, so the 'constraints to z~6' language overstates the empirical support. Overall, the central trend is not fully forced by the prior—measured fluxes and volumes contribute independent information—but the paper's central claim of independence from SHARK-2 is not established, and the model-comparison agreement is partly circular.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The measurement is observationally grounded but depends on model-based redshift priors and adopted conversion factors. No new physical entities are introduced.

free parameters (4)
  • Schechter faint-end slope alpha = -0.2 (fixed)
    Fixed to -0.2 following Boogaard et al. (2023) because faint-end data are sparse; directly shapes the LF extrapolation used in the rho_H2 integral.
  • CO-to-H2 conversion factor alpha_CO = 3.6 Msun/(K km/s pc2)
    Single constant applied to all detections (Eq. 5); all rho_H2 values scale linearly with this adopted value.
  • Excitation correction factors r_j->1 = z<2: 3.33,5.20,4.76,2.70,0.53; z>2: 4.09,8.24,12.21,14.68,13.86 for J=2-6
    Boogaard et al. (2020) empirical line ratios used to convert mid/high-J CO luminosities to CO(1-0); central to deriving L'_CO.
  • Madau-Dickinson rho_H2 fit parameters = a=4.9e6, b=3.5, c=2.2, d=6.0 (Eq. 6)
    Fit to the compiled rho_H2 measurements including ALMACAL; used for baryon-cycle ratios and depletion timescale.
assumptions (6)
  • domain assumption Detected lines are CO transitions; [C II] interloper rate is at most about two sources.
    Section 3.5 estimates at most two [C II] sources expected in the Band 7 volume; five Band 7 detections exist, and removing two would not change the lower-limit bin.
  • domain assumption SHARK-2 SAM's flux-redshift distribution is a valid prior for redshift and J-transition assignment.
    Section 3.4.1: only 6 of 87 sources have spectroscopic or photometric redshifts, so the CO LF and rho_H2 depend on this model prior.
  • domain assumption CO(1-0) luminosity linearly traces H2 mass with a single alpha_CO.
    Eq. 5 with alpha_CO = 3.6, adopted without excitation, metallicity, or environment dependence.
  • domain assumption ALMA calibrator fields are independent, essentially unbiased sightlines after removing lines near calibrator redshifts.
    Sections 2 and 3.5 remove candidates within 2000 km/s of the quasar, but calibrator fields may still trace quasar environments and overdensities.
  • domain assumption A Schechter function with fixed alpha = -0.2 describes the CO LF in all redshift bins.
    Section 4.3: the highest-redshift bin has only one fitted point above the luminosity limit, yet the fit is still displayed.
  • domain assumption Completeness and fidelity corrections fully account for selection effects.
    Sections 3.2 and 3.3: injected and inverted sources are used to correct the search, but zero-completeness cubes were removed rather than corrected.

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Pith. "Pith review of ALMACAL XIII. Evolution of the CO luminosity function and the molecular gas mass density out to $z$ ~ 6." pith.science (2026). https://pith.science/paper/7ZEYIL6E

@misc{pith2026250206778,
  author       = {Pith},
  title        = {Pith review of: ALMACAL XIII. Evolution of the CO luminosity function and the molecular gas mass density out to $z$ ~ 6},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZEYIL6E}},
  note         = {Machine review of arXiv:2502.06778}
}
abstract

Cold molecular gas, largely traced by CO emission, is the primary fuel for star formation, making it essential for understanding galaxy evolution. ALMA has made significant progress in the study of the cosmic evolution of cold molecular gas. Here, we exploit the ALMACAL survey to address issues relating to small sample sizes and cosmic variance, utilising calibration data from ALMA to compile a statistically significant and essentially unbiased sample of CO-selected galaxies. By employing a novel statistical approach to emission-line classification using semi-analytical models, we place strong constraints on the CO luminosity function and the cosmic evolution of molecular gas mass density ($\rho_{H_2}$) back to $z \sim 6$. The cosmic molecular gas mass density increases with redshift, peaking around $z \sim 1.5$, then slowly declines towards higher redshifts by $\sim 1$ dex. Our findings confirm the key role of molecular gas in fuelling star formation. The new $\rho_{H_2}$ estimates allow us to revisit the cosmic baryon cycle, showing that the ratio of molecular gas-to-stellar mass density is consistent with the so-called 'bathtub model' of baryons, which implies a continuous replenishment of gas. The cosmic gas depletion timescale, estimated on a global scale, is shown to be fairly constant at all redshifts. We emphasise the importance of surveys using multiple small fields rather than a single contiguous area to mitigate the effects of cosmic variance.

Figures

Figures reproduced from arXiv: 2502.06778 by the authors.

Figure 1
Figure 1. Completeness fraction of mock sources. The grid shows several combinations of the injected peak signal-to-noise ratio (SNR) and line width. The heat map represents how successfully the algorithm, SoFiA2 (Westmeier et al. 2021), detects mock sources injected into data cubes, with the recovery fraction indicated by the colour. For sources with SNR > ∼ 4 and line width > ∼ 300 km s−1 we reach a completeness above ∼ 75%… view at source ↗
Figure 2
Figure 2. , top panel, shows the 2D histogram of the positive candidates. The x−axis represents SNR, while the y−axis is the number of detection channels. Note that over 400 positive de￾tections are initially found, but not all are real (see § 3.5). The bottom panel of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Integrated flux and width of the CO emitters detected in ALMACAL−22 (purple). For comparison, we include detections from other surveys: ASPECS (Decarli et al. 2020), COLDz (Riechers et al. 2020b) and PHIBSS (Lenkic et al. ´ 2020). Our candidates span a similar range of line widths as previous studies, and are generally brighter, as expected from the difference in depth and volume. scopic redshift of three objects, a… view at source ↗
Figures from the paper (7 more)
Figure 5
Figure 5. Figure 5: CO LF across redshift bins: z = 0–0.5, z = 0.5–1, z = 1–2, z = 2–3, z = 3–4 and z = 4–6. Detections are shown in circles and non-detection as arrows. The vertical dotted line in each panel indicates the detection limit, representing the faintest luminosity detectable a…
Figure 6
Figure 6. Figure 6: Cosmic molecular gas mass density evolution measured from ALMACAL−22 in purple. The right−y axis represents the unitless density parameter for molecular gas, ΩH2 = ρH2 /ρcrit,z=0. For comparison, we include estimates from other surveys: COLDz (Riechers et al. 2020b), A…
Figure 7
Figure 7. Figure 7: The CO LF in different redshift bins, from z = 0 to z = 6, as indicated at the top right of each panel. The measurements from ALMACAL−22 are shown in purple, including the uncertainty of each bin given by the extension of the boxes. We derived the CO LF based on the CO…
Figure 8
Figure 8. Figure 8: Evolution of the Schechter best-fit parameters for the CO LF across z ∼ 0 to z ∼ 6. Top: the evolution of Φ∗ shows a decrease with redshift from z = 0. Bottom: the evolution of L ∗ CO shows a consistent increase from z ∼ 0 to z ∼ 2, and remains roughly constant at high…
Figure 9
Figure 9. Figure 9: CO LF from redshift z = 0 to z = 6 derived from ALMACAL−22 in comparison with simulations. We include the predictions from SHARK-1 (Lagos et al. 2018) at z = 0–3, from SHARK-2 (Lagos et al. 2024) at z = 0–5.5, and from SPRITZ (Bisigello et al. 2022) at z = 0.5–5.5. We …
Figure 10
Figure 10. Figure 10: Top: Evolution of the molecular gas mass density obtained from ALMACAL−22 (purple) in comparison with predictions from simula￾tions. We include the results from Illustris TNG (Popping et al. 2019b), EAGLE (Lagos et al. 2015), SHARK-2 (Lagos et al. 2018), UniverseMa￾ch…
Figure 12
Figure 12. Figure 12: Redshift evolution of baryonic component in the Universe. Top: Ratio of cosmic molecular-to-atomic gas density as a function of red￾shift. Middle: Ratio of molecular gas-to-stellar mass density as a func￾tion of redshift. Bottom: Cosmic gas depletion timescale, is def…

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