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A Shallow Slope for the Stellar Mass--Angular Momentum Relation of Star-Forming Galaxies at $1.5 < z < 2.5$

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper finds that star-forming disk galaxies at z≈2 follow a Fall relation j_star ∝ M_star^{0.25}, about 3 sigma shallower than the local slope of 0.67.

desk verdict Careful measurement of a shallow Fall-relation slope at z~2, but the '3σ' headline relies on the wrong local baseline and unquantified M/L gradients; still worth refereeing. read the letter →

arxiv 2411.17312 v1 pith:L3XJ6Q45 submitted 2024-11-26 astro-ph.GA

Juan M. Espejo Salcedo , Karl Glazebrook , Deanne B. Fisher , Sarah M. Sweet , Danail Obreschkow , N. M. Förster Schreiber This is my paper · ORCID
classification astro-ph.GA
keywords angularmomentumFallrelationhigh-redshiftgalaxiesdiskgalaxykinematicsintegralfieldspectroscopyadaptiveopticscosmicnoon
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

Star-forming disk galaxies near the peak of cosmic star formation at z≈2 appear to spin up much more slowly as they grow in stellar mass than local spirals do. The paper measures the specific angular momentum (angular momentum per unit stellar mass) of 24 rotating disks among 41 galaxies at 1.5

What carries the argument

The load-bearing quantity is the specific angular momentum j_star = J/M, computed by azimuthally averaging the deprojected H-band surface brightness profile Σ(r_i) (assumed proportional to stellar mass with a constant mass-to-light ratio) and multiplying by a model rotation curve ṽ(r_i) from a joint fit to adaptive-optics and seeing-limited Hα kinematics: j_star = (2π Σ $r_i^{2}$ Σ(r_i) ṽ(r_i)) / (2π Σ r_i Σ(r_i)). This radial integration replaces the global Romanowsky–Fall approximation j̃_star ≈ k_n v_s r_eff, which depends on a single Sérsic fit and a single rotation velocity, and which the paper shows is biased for clumpy, compact, or poorly fitted galaxies. The same machinery yields the rotating-disk versus irregular classification and, together with halo masses from abundance matching, the angular momentum retention factors f_j.

What would settle it

Measure spatially resolved stellar mass maps for the same 24 disks, for example by fitting spectral energy distributions to deep rest-optical and near-infrared photometry, and recompute j_star by replacing the H-band light profile with the mass profile. If the slope moves from β≈0.25 toward β≈0.5–0.7, the shallow-slope claim is falsified; if the slope stays near 0.25, the claim survives.

Watch

Extended reading notes

Core claim

The central claim is that the stellar mass–specific angular momentum relation for rotating disk galaxies at 1.5<z<2.5 is much shallower than the local one: j_star ∝ $M_star^{{0.25±0.15}}$, compared with the commonly adopted β≈0.67 (2/3). The difference is significant at about 3 $\sigma$, and a fixed-slope fit with β=2/3 is statistically rejected for these data. The paper attributes the steeper slopes reported by earlier high-redshift studies to two systematic choices: classifying low-mass irregular systems as disks, which adds low-j_star points at low mass, and using the Romanowsky–Fall approximation j̃_star ≈ k_n v_s r_eff instead of integrating resolved radial mass and velocity profiles. Applying those same choices to this sample reproduces steeper slopes of β=0.48±0.21 and β=0.61±0.21 respectively. The paper also derives angular momentum retention factors f_j = j_star/j_h that decline with halo mass, with values above unity in low-mass halos, which it interprets as evidence of efficient angular momentum transport in gas-rich systems and loss of low-angular-momentum gas through feedback-driven outflows.

Load-bearing premise

The paper assumes that the near-infrared brightness of a galaxy, scaled uniformly, faithfully traces its stellar mass at every radius; if strong radial gradients in stellar population or dust exist, the measured low-mass angular momenta and the shallow slope could be artifacts.

Editorial extensions

If this is right

  • The Fall-relation slope at z≈2 for disk galaxies is β=0.25±0.15, about 3 sigma below the local value, so specific angular momentum grows only weakly with stellar mass during cosmic noon.
  • Combining the R&F approximation with a sample that includes irregulars yields β≈0.77, matching the commonly adopted 2/3; previous high-redshift slopes therefore likely overestimate the true slope.
  • A fixed-slope fit with β=2/3 to the disks is statistically rejected (p≈4×10^-10), so analyses that assume 2/3 at z>1 miss real evolution.
  • Low-mass halos show angular momentum retention factors f_j>1 that decline with mass, implying efficient angular momentum transport and preferential loss of low-angular-momentum gas in low-mass galaxies.
  • The slope evolution seen here, shallow at z≈2 and steepening toward 0.67 at z=0, matches the trend seen in cosmological simulations, suggesting the local relation was largely assembled at z<1.

Reading between the lines

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

  • If the shallow slope is real, galaxy formation models must produce low-mass z≈2 disks with j_star comparable to their host halos; a direct test would compare molecular gas kinematics from millimetre interferometry in low-mass galaxies to see whether the gas also carries high angular momentum.
  • The method comparison implies that published high-redshift Fall-relation slopes based on the R&F approximation may need to be re-derived with radial integration; existing seeing-limited surveys may already contain the photometry needed for such a re-analysis.
  • The f_j>1 values at low mass may indicate that abundance-matching halo masses are too low for those systems rather than that stellar disks truly exceed halo angular momentum; independent halo mass estimates from weak lensing or satellite dynamics would separate these possibilities.
  • If radial mass-to-light gradients are present, the true slope could differ from 0.25; in particular, stellar mass profiles steeper than the H-band light would lower the low-mass j_star values and steepen the relation, a testable prediction for spatially resolved stellar mass maps.
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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 / 4 minor

Summary. The paper presents measurements of the specific stellar angular momentum j* for 41 star-forming galaxies at 1.5<z<2.5, based on HST near-IR surface brightness profiles and multi-resolution (AO plus natural seeing) IFS kinematics from KMOS, SINFONI, and OSIRIS. A sample of 24 rotating disks is used to fit the Fall relation j* ∝ M*^β, yielding β = 0.25 ± 0.15. The authors compare this with the local pure-disk slope β = 0.67, claim a roughly 3σ difference, and argue that previous steeper high-redshift slopes are driven by inclusion of irregular systems and by use of the R&F approximation j~ ≈ k_n v_s r_eff. They also derive angular-momentum retention factors f_j = j*/j_h using abundance matching and an assumed halo spin parameter, reporting high f_j at low halo masses. The analysis is careful in several respects: the multi-resolution kinematic modeling, Monte Carlo resampling, detailed PSF treatment, and direct radial integration of j* are strengths, and the authors explicitly flag several systematics, including the M/L-gradient issue and the construction-dependent nature of the f_j analysis.

Significance. If the measurement is robust, a shallow Fall-relation slope at z≈2 would be an important constraint on angular momentum acquisition and feedback at cosmic noon, and it would be broadly consistent with the IllustrisTNG-based prediction of Du et al. (2022). The paper's strengths are the unique AO+NS multi-resolution modeling, the radial-integration method rather than the global R&F approximation, the public release of the CONDOR modeling code, and the systematic comparison of methods and samples. However, the headline statistical claim is not currently supported: the paper's own §6.2 shows that against the selection-matched local baseline (β = 0.52 ± 0.04 for all rotating systems meeting their RD definition) the difference is only about 1.7σ, and the same section shows that doubling the j* uncertainties changes the slope to β = 0.53 ± 0.38. The unquantified M/L-gradient bias discussed in §6.1.2 could flatten the measured slope further. The measurement remains interesting, but the significance and error budget need to be revised before the central claim is established.

major comments (3)
  1. [Abstract and §5.2] The abstract claims that β = 0.25 ± 0.15 'deviates by approximately 3σ from the commonly adopted local value β = 0.67, indicating a statistically significant difference.' This significance is computed against the pure-disk slope of 2/3, but the sample is intentionally defined with a broad rotating-disk criterion. The paper itself acknowledges in §6.2 that for all rotating systems at z≈0 meeting this RD definition, the local slope is β = 0.52 ± 0.04 (Romanowsky & Fall 2012). Against that selection-matched baseline the difference is Δβ = 0.27 with a combined uncertainty of roughly 0.155, i.e., about 1.7σ, not 3σ. The headline significance is therefore unsupported as stated, independent of additional systematic biases. Please revise the abstract and §5.2 to compare against the appropriate local baseline and soften the significance claim accordingly.
  2. [§6.1.2] The mass-to-light-ratio gradient systematic is load-bearing for the central result. The fiducial j* measurement (Eq. 5, §4.2) assumes a radially constant stellar M/L and uses the H-band surface brightness as the mass profile. Section 6.1.2 states that stellar light profiles tend to be shallower than underlying mass profiles and that the resulting overestimation of M* at large radii and low masses 'could play a significant role in driving the high j★ for galaxies in the low-mass end,' which would flatten the fitted slope. This effect is not quantified or propagated into the quoted β = 0.25 ± 0.15. Please provide a quantitative estimate, for example by adopting a plausible radially varying M/L from spatially resolved SED fitting or by re-fitting after a conservative correction of the mass profiles, and include the resulting shift in β in the error budget or in the significance statement.
  3. [§6.2] The robustness test reported in §6.2 shows that doubling the j* uncertainties changes the best-fit slope from β = 0.25 ± 0.15 to β = 0.53 ± 0.38. This demonstrates that the measured slope is not robust to a plausible level of systematic error in the individual j* measurements. Because the abstract and conclusions quote only the statistical uncertainty, the precision of the central claim is overstated. The systematic contribution to the uncertainty in β should be propagated into the quoted value, or the significance of the shallow slope should be attenuated accordingly.
minor comments (4)
  1. [§5.4] The text says the fit to the clumpy systems shows a 'negative vertical offset from the less clumpy sample of Δα∼0.2,' but the quoted normalizations and Figure 12 (α = 3.07 ± 0.07 for the high-C sample vs α = 2.88 ± 0.10 for the low-C sample) show a positive offset. The sign should be corrected to be consistent with Figure 12 and with the conclusions bullet.
  2. [§7] The conclusions bullet on the method of measuring j* attributes β ≈ 0.26 ± 0.14 and β ≈ 0.64 ± 0.2 to the mock-galaxy experiment in §6.1.2, but that experiment reports β = 0.36 ± 0.06 for Eq. 5 and β = 0.57 ± 0.05 for the R&F approximation. The values quoted in the conclusions are the real-data fits from Table 4; please distinguish between the mock-derived and data-derived slopes.
  3. [Figure 11] The caption of Figure 11 gives the best-fit slope as β = 0.25 ± 0.14, while the abstract, §5.2, and Table 3 quote β = 0.25 ± 0.15. Please make the quoted uncertainty consistent throughout.
  4. [Appendix B] The summary figures in Appendix B list z = 1.29 for the OSIRIS subsample (e.g., COSMOS-110446, COSMOS-171407, COSMOS-130477, COSMOS-127977, UDS-124101, COSMOS-128904), whereas Table 1 lists their redshifts as 1.46–1.62. Please correct the captions to match the sample table.

Circularity Check

1 steps flagged · score 2.0 of 10

The central slope measurement is a direct fit, not a derivation; only the f_j interpretation is partially circular by construction, and the paper explicitly flags that construction.

  1. renaming known result [Section 6.3 and 6.3.1 (Eq. 9 and f_j = j_star/j_h)]
    "It is important to note that the various assumptions and approximations used to calculate M_h and j_h may introduce systematic biases and artificial trends. In particular, the explicit assumption that the angular momentum of the halo scales with halo mass as j_h ∝ M_h^{2/3} impacts the inferred f_j by construction. Therefore, caution is advised in interpreting these tentative outcomes, which are primarily qualitative in nature."

    The f_j trend is not independently measured; it is computed as j_star/j_h, with j_h set by the assumed scaling j_h ∝ M_h^{2/3} and M_h derived from M_star via abundance matching. Given the tight M_star-M_h link, the decline of f_j with mass is an algebraic rescaling of the same measured j_star-M_star relation whose shallow slope it is invoked to explain. The text then states that low-mass haloes 'could possess a higher capacity to retain angular momentum... This observation potentially contributes to the shallower slope of the Fall relation,' attributing physical cause to a quantity that is, by construction, the residual of the observed slope after removing the assumed halo scaling.

full rationale

The central claim, beta = 0.25 +/- 0.15, is a direct power-law fit (Eq. 7) to j_star measurements obtained by the radially integrated method (Eq. 5) for 24 disk galaxies. That measurement is self-contained: it depends on HST photometry, kinematic modelling, and Monte Carlo uncertainties, not on the f_j analysis or on any self-cited theorem. The self-citation of ES22 introduces the CONDOR multi-resolution modelling method but does not smuggle in the beta result; it is a methodological citation and the slope is fitted from the paper's own data. The abstract's 'approximately 3 sigma' comparison is weakened inside the paper itself: Section 6.2 states that for all rotating systems meeting the paper's RD definition the local slope is beta = 0.52 +/- 0.04, so the tension 'is not as large as initially suggested,' and doubling j_star uncertainties changes the fit to beta = 0.53 +/- 0.38. These are correctness and robustness concerns, not circularity. Similarly, the unquantified M/L-gradient caveat in Section 6.1.2 could flatten the measured slope but does not make the measurement circular. The one genuinely circular-by-construction element is the f_j interpretation, where the assumed j_h scaling reshapes the observed j_star-M_star slope into an apparent retention-factor trend and that trend is then offered as an explanation of the shallow slope. Because this step is explicitly caveated and is not load-bearing for the primary measurement, the overall circularity score is low.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particles or forces. Its physical interpretation rests on standard assumptions: constant M/L for mass profiles, H-alpha tracing stellar kinematics, flat rotation curves, abundance matching, and a fixed spin parameter. These are domain assumptions rather than invented entities. The fitted slope and normalization are the target measurements; the per-galaxy rotation curve parameters are fitted to data. The assumed spin parameter lambda=0.035 is the only externally fixed number that directly shapes the f_j discussion.

free parameters (5)
  • beta (Fall relation slope) = 0.25 ± 0.15
    Power-law index fitted to the 24 disk galaxies with hyper-fit MCMC. Central result of the paper.
  • alpha (Fall relation normalization) = 3.00 ± 0.06
    Intercept at log10(M*/Msun)=10.5, fitted alongside beta.
  • lambda (halo spin parameter) = 0.035
    Assumed mean spin parameter from Bullock et al. 2001, used in Eq. 9 to compute j_h and f_j. Not measured here.
  • q0 (intrinsic disk axis ratio) = 0.2
    Assumed for thick disks to derive inclinations from observed axis ratios; affects deprojection and velocity model.
  • Per-galaxy v_flat and r_flat = Table 2
    Boissier rotation curve parameters fitted for each galaxy; j_star depends on them. Represent 24 disk fits.
assumptions (7)
  • domain assumption Constant mass-to-light ratio with no radial gradient
    Used when converting H-band surface brightness to mass profile in Section 4.2 and Eq. 5. Author-flagged in Section 6.1.2 as a potential bias.
  • domain assumption H-alpha gas kinematics trace stellar kinematics
    Used to measure rotation curves; asymmetric drift estimated as about 0.1 dex from Cortese et al. 2016. Section 4.3.
  • domain assumption Rotation curves follow the Boissier flat model
    Assumed v(r)=v_flat(1-exp(-r/r_flat)) for all disks; rising or dropping curves not accounted for beyond uncertainties. Section 4.3.
  • domain assumption Inclination derived from photometric axis ratio
    Uses Holmberg relation with q0=0.2; inclination is fixed, not fitted. Section 4.2.
  • domain assumption Abundance matching (Moster et al. 2013) maps stellar to halo mass
    Used to estimate M_h and calculate j_h and f_j in Section 6.3.
  • domain assumption Isothermal spherical halo with j_h proportional to M_h^(2/3)
    Underlying model for halo angular momentum in Eq. 9. Authors note this affects f_j by construction.
  • domain assumption Cylindrical symmetry in the radial integration of j_star
    Eq. 5 assumes axisymmetric mass and velocity fields; only applied to classified disks. Section 4.4.

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Cite this review

Pith. "Pith review of A Shallow Slope for the Stellar Mass--Angular Momentum Relation of Star-Forming Galaxies at $1.5 < z < 2.5$." pith.science (2026). https://pith.science/paper/L3XJ6Q45

@misc{pith2026241117312,
  author       = {Pith},
  title        = {Pith review of: A Shallow Slope for the Stellar Mass--Angular Momentum Relation of Star-Forming Galaxies at $1.5 < z < 2.5$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3XJ6Q45}},
  note         = {Machine review of arXiv:2411.17312}
}
abstract

We present measurements of the specific angular momentum $j_\star$ of 41 star-forming galaxies at $1.5<z<2.5$. These measurements are based on radial profiles inferred from near-IR \textit{HST} photometry, along with multi-resolution emission-line kinematic modelling using integral field spectroscopy (IFS) data from KMOS, SINFONI, and OSIRIS. We identified 24 disks (disk fraction of $58.6\pm 7.7\%$) and used them to parametrize the $j_\star$ \textit{vs} stellar mass $M_\star$ relation (Fall relation) as $j_\star\propto M_\star^{\beta}$. We measure a power-law slope $\beta=0.25\pm0.15$, which deviates by approximately $3\sigma$ from the commonly adopted local value $\beta = 0.67$, indicating a statistically significant difference. We find that two key systematic effects could drive the steep slopes in previous high-redshift studies: first, including irregular (non-disk) systems due to limitations in spatial resolution and second, using the commonly used approximation $\tilde{j}_\star\approx k_n v_s r_\mathrm{eff}$, which depends on global unresolved quantities. In our sample, both effects lead to steeper slopes of $\beta=0.48\pm0.21$ and $\beta=0.61\pm0.21$, respectively. To understand the shallow slope, we discuss observational effects and systematic uncertainties and analyze the retention of $j_\star$ relative to the angular momentum of the halo $j_h$ (angular momentum retention factor $f_j =j_\star/j_h$). For the $M_\star$ range covered by the sample $9.5 <\log_{10} (M_\star/M_\odot) < 11.5$ (halo mass $11.5 < \log_{10} (M_h/M_\odot) < 14$), we find large $f_j$ values ($>1$ in some cases) in low-mass haloes that decrease with increasing mass, suggesting a significant role of efficient angular momentum transport in these gas-rich systems, aided by the removal of low-$j_\star$ gas via feedback-driven outflows in low-mass galaxies.

Figures

Figures reproduced from arXiv: 2411.17312 by the authors.

Figure 1
Figure 1. Star-formation rate (SFR) vs stellar mass 𝑀★ of the full sample. Dots represent the disks and triangles represent those identified as either irregulars or mergers. Blue markers correspond to galaxies at 𝑧 > 2 and orange markers at 𝑧 ∼ 1.5, while the grey dots correspond to the KMOS3D sample, which is one of the parent seeing-limited samples. The broken laws are extracted from Whitaker et al. (2014), and the scatter … view at source ↗
Figure 2
Figure 2. Clump detection method for Q2343-BX610. a) Original 𝐽110 HST image with the kinematic centre indicated with the yellow cross. b) Convolved image with the Gaussian kernel. c) “Detection image” where the unsharp features have been removed (Equation 1). d) Original image with the identified clumps in green circles/ellipses. 2.2.2 OSIRIS observations (adaptive optics) We selected the seven galaxies above based on the pr… view at source ↗
Figure 3
Figure 3. with their corresponding H- and J-band data. There is some overlap between our analysis and other studies that measure the clump properties of some galaxies in the sample. In Genzel et al. (2011), they used the resolved H𝛼 line detections from the SINS observations (AO) to detect large clumps in five galax￾ies that overlap with this sample (Q1623-BX599, Q2346-BX482, Deep3a-15504, ZC407302, and ZC406690). They based … view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Size dependence of the number of clumps 𝑁clumps (bottom) and galaxy “clumpiness” C (top) as a function of galaxy size 𝑟eff. The dots and triangles distinguish the disks from the Irregulars/mergers as discussed in §5.1. The Spearman correlation coefficients are shown on…
Figure 6
Figure 6. Figure 6: Comparison between the concentration expected from simulated exponential disks (red line) and from the sample of real galaxies (blue dots). The green stars correspond to galaxies ZC400528 and ZC400569, where a bulge component can be easily identified from a visual insp…
Figure 7
Figure 7. Figure 7: Summary of galaxy Q2343-BX389 which is identified as a rotating disk (RD): a) H𝛼 intensity fields at high- (top) and low-resolution (bottom) where the white circles represent the PSF FWHM, b) velocity fields with the main kinematic axes indicated by the dashed green li…
Figure 8
Figure 8. Figure 8: Position-velocity (P-V) diagrams for the full sample, where the red line is the best-fit model from our kinematic modelling and the shaded region represents the uncertainties in the fit. The blue (AO) and orange (NS) dots are extracted from a slit along the kinematic a…
Figure 9
Figure 9. Figure 9: Difference between 𝑗★ from the the radial measurement (Equation 5) and pixel-by-pixel measurements (Equation 6). Grey triangles represent irregular galaxies, and blue circles represent the disks. The grey dashed line represents the one-to-one correspondence. No signifi…
Figure 10
Figure 10. Figure 10: Specific angular momentum 𝑗★ vs stellar mass 𝑀★ “Fall relation” where the measurements of 𝑗★ come from the radially integrated method using Equation 5. The large blue dots correspond to the disk galaxies, which yield a hyper-fit solution with slope 𝛽 = 0.25 ± 0.15 and…
Figure 11
Figure 11. Figure 11: Comparisons between different power law slopes in the Fall relation for different analyses. The blue line on both panels indicates the best fit (𝛽 = 0.25 ± 0.14) using the disks with the integrated measurement of 𝑗★ (blue dots). Left: Red line indicates the fit using …
Figure 12
Figure 12. Figure 12: Left: Specific angular momentum 𝑗★ vs stellar mass 𝑀★ “Fall relation” for the disk galaxies coloured depending on whether they are below (blue) or above (green) the median “clumpiness” C = 11.72% as defined in §3.1. The solid red line indicates the 𝑗★ ∝ 𝑀 𝛽 ★ relation…
Figure 14
Figure 14. Figure 14: Comparison between the 𝑗★ measurements from the integrated method and from the R&F approximation ˜𝑗★ ≈ 𝑘𝑛𝑣𝑠𝑟eff as a function of the effective radius where the blue dots correspond to the galaxies where the light profile is well described by a single Sérsic profile an…
Figure 15
Figure 15. Figure 15: Fits to the 𝑗★ vs 𝑀★ plane (Fall relation) using mock disk galaxies. The blue line corresponds to the fit from the integrated method in Equation 5 using the velocity and mass profiles, which results in a slope 𝛽 = 0.36±0.05. The red line indicates the fit using ˜𝑗★ ≈ …
Figure 16
Figure 16. Figure 16: Angular momentum retention factor 𝑓 𝑗 as a function of halo mass 𝑀ℎ (left) and stellar mass 𝑀★ (right) for the 24 rotating disks in the sample. Blue dots correspond to the estimations based on the Moster et al. (2013) abundance matching relations. The red line corresp…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.