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The parallelism between galaxy clusters and early-type galaxies: III. The Mass-Radius Relationship

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The mass-radius relation of galaxies and star clusters is the intersection of constant-density collapse tracks with a cosmic maximum-halo-mass boundary, the 'Cosmic Galaxy Shepherd'.

desk verdict A plausible but partially tuned explanation for the universal mass-radius locus; worth refereeing if the authors clearly separate fitting from prediction. read the letter →

arxiv 1908.08808 v2 pith:6O7D5ZB4 submitted 2019-08-22 astro-ph.GA

Cesare Chiosi , Mauro D'Onofrio , Emiliano Merlin , Lorenzo Piovan , Paola Marziani This is my paper · ORCID
classification astro-ph.GA
keywords mass-radiusrelationearly-typegalaxiesgalaxyclustersglobularhalogrowthfunctionCosmicShepherdvirialequilibriumpassiveevolution
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 asks why globular clusters, dwarf and early-type galaxies, and galaxy clusters and groups all crowd onto a narrow mass-radius relation instead of spreading across the plane. Its answer is that the relation is the combined product of two evolutions: the stellar component moves along collapse tracks of fixed initial density, while the dark-matter halo component follows a cosmological growth function that sets a maximum halo mass at each redshift. Translated onto the mass-radius plane, that maximum mass becomes a boundary the authors call the Cosmic Galaxy Shepherd, and the observed relation is the intersection of the collapse tracks with that boundary. Objects on the intersection are in virial equilibrium and are evolving passively, which explains both the changing slope of the relation and the existence of a zone of avoidance below it.

What carries the argument

The central machinery is the Cosmic Galaxy Shepherd: the locus obtained by intersecting the halo growth function $n(M_D,z)$ at a fixed number density $N_s$ with the analytic collapse relation of eq. (10), which ties a stellar system's half-mass radius to its halo mass, stellar-to-halo mass ratio, Sersic profile and formation redshift. A second ingredient is the manifold of constant-initial-density tracks, the M-MRRs, summarized by eq. (31) as lines whose slope and zero point depend on the formation redshift. The observed mass-radius relation is the intersection of these two families of curves, and can also be reproduced by the Press-Schechter cut-off mass as a function of redshift; along the intersection, systems satisfy the virial condition and evolve passively.

What would settle it

Take a mass-complete, volume-limited sample of quiescent early-type galaxies and compare its high-mass edge on the mass-radius plane with the Cosmic Galaxy Shepherd calculated from the survey's own halo number density. If the edge falls inside the zone of avoidance at the expected number density, or if the edge does not move with redshift as the adopted cut-off mass changes, the proposed mechanism is wrong. A simpler check is to recompute $N_s$ using only early-type galaxies; if the resulting boundary no longer tracks the observed relation, the coincidence rests on the one-galaxy-per-halo assumption.

Watch

Extended reading notes

Core claim

The paper claims that the observed mass-radius relation of stellar systems is not set by the virial theorem alone, nor by any single formation mechanism, but emerges where two ingredients cross. The first ingredient is a manifold of model mass-radius relations, each traced by objects of the same initial density (or formation redshift); the second is the redshift-dependent cut-off mass of the halo mass function, evaluated along a constant halo number density of $N_s = 10^{-2}$ halos per $(\mathrm{Mpc}/h)^3$. The cut-off line, named the Cosmic Galaxy Shepherd, runs close to the empirical mass-radius relation of the SDSS early-type galaxy sample and merges smoothly into the regions populated by globular clusters and galaxy clusters. Along this line objects are in virial equilibrium and passive evolution, and the slope of the line varies from about 0.5 to about 1 with increasing mass; galaxies with active star formation and strong winds, mainly dwarfs, lie above it.

Load-bearing premise

The construction assumes that every halo counted at the adopted number density $N_s=10^{-2}$ per $(\mathrm{Mpc}/h)^3$ hosts one and only one early-type galaxy of the kind in the comparison sample; because $N_s$ is taken from total SDSS galaxy counts rather than the early-type fraction alone, a lower true number density of early-type-hosting halos would move the Cosmic Galaxy Shepherd and could turn the agreement into a tuned coincidence.

Editorial extensions

If this is right

  • The slope and curvature of the observed mass-radius relation, from roughly 0.5 for low-mass systems to about 1 for the most massive ones, are inherited from the shape of the halo growth function rather than from the virial theorem itself.
  • The zone of avoidance below the relation is a cosmic-time statement: at each redshift, halos massive enough to populate that part of the plane are statistically too rare to be seen.
  • Compact massive galaxies at high redshift should lie near the Cosmic Galaxy Shepherd for their epoch, above the present-day relation, rather than contradicting it; the paper indicates this will be examined in a companion study.
  • Dwarf galaxies are expected to sit above the relation while star formation and galactic winds inflate them, and to settle onto it once they quench and reach mechanical and thermal equilibrium.
  • The same construction spans stellar masses from globular clusters to galaxy clusters, about eleven orders of magnitude, with a single physical origin.

Reading between the lines

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

  • If the Shepherd is recalibrated using only the number density of early-type galaxy hosts rather than all SDSS galaxies, its position will shift; this turns the paper's claimed coincidence into a quantitative, testable prediction rather than a posterior fit.
  • The same boundary logic should apply to any population with a known collapse relation and a formation cutoff, such as globular cluster systems within a single galaxy or satellites within a cluster; checking whether their mass-radius envelopes show analogous shepherds would extend the idea beyond homogeneous galaxy samples.
  • Surveys with different volumes or depths should see the high-mass edge of the mass-radius relation move: deeper volume coverage reaches rarer, more massive halos at fixed redshift and should push the envelope to larger masses and radii.
  • A direct numerical convolution of the full halo mass function with the manifold of collapse tracks, rather than the linear intersection used in Section 5, would give a sharp test of the predicted slope and zero point across the whole mass range.
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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

4 major / 6 minor

Summary. This paper (the third in a series) aims to explain the observed Mass-Radius Relation (MRR) spanned by globular clusters, early-type galaxies, and galaxy clusters/groups. The authors assemble literature data (Burstein et al. 1997; Bernardi et al. 2010; WINGS) and compare them with hydrodynamical models (Illustris, plus older monolithic and early-hierarchical simulations). They propose that the observed MRR is the envelope of two complementary effects: (i) lines of constant initial collapse density (M-MRRs, described by eqs. 10 and 31) that map halo mass and redshift to stellar mass and radius, and (ii) a redshift-dependent upper mass limit for halos, which they call the Cosmic Galaxy Shepherd (CGS), derived from the halo growth function via the condition n(MD,z)=Ns (eqs. 12-14). The intersections of the M-MRR manifold with the CGS boundary are claimed to reproduce the empirical ETG MRR (eq. 2), implying that objects along the MRR are in virial equilibrium and passive evolution. The paper is explicitly exploratory: several assumptions are labeled 'strong,' 'crude,' or 'provisional,' and the authors caution that the quantitative location of the CGS depends on uncertain ingredients.

Significance. If the proposed picture is correct, it provides a conceptually appealing and unifying explanation for why objects spanning ~11 orders of magnitude in stellar mass lie on a single, relatively narrow MRR: the relation emerges from the convolution of collapse physics (initial density) with cosmological halo growth (maximum halo mass at each epoch). The paper's qualitative argument that the MRR is a boundary rather than a fundamental scaling relation is interesting and worth testing. The authors deserve credit for using a wide range of observational data, comparing independent simulation schemes, and providing explicit analytical fits (e.g., eqs. 12-14, 31-33) that make the proposed mechanism falsifiable. However, the quantitative claims are not yet parameter-free: the CGS location depends on a number density Ns estimated from the total SDSS galaxy count rather than the ETG-hosting halo density, on a provisional z-independent stellar-to-halo mass relation, and on a free normalization in the Press-Schechter cut-off mass. These admitted uncertainties directly affect the central figures (Figs.

major comments (4)
  1. [Section 4.2, eqs. (12)-(14) and Fig. 8] The placement of the Cosmic Galaxy Shepherd uses Ns = 10^-2 halos per (Mpc/h)^3, derived from the total SDSS galaxy count (~10^6 galaxies over ~10^8 Mpc^3). The text itself calls the assumption that each halo hosts exactly one early-type galaxy 'a strong assumption.' Because the SDSS sample includes satellites and a large spiral/dwarf population, the number density of halo central galaxies that are early types is plausibly an order of magnitude or more lower. Since the condition n(MD,z)=Ns is steeply decreasing in MD, reducing Ns by a factor of 10-30 shifts the CGS to lower halo masses by roughly 0.5-1 dex, moving the predicted locus away from the Bernardi et al. (2010) points. Thus, the apparent coincidence between the CGS and the observed MRR in Fig. 8 is a tuning statement unless a separate, observationally motivated estimate of the ETG-hosting halo number density is provided.
  2. [Section 5, eqs. (32)-(33) and Fig. 9] Equation (32) defines the cut-off mass as MCO_T = MN × (1+z)^(-6/(n+3)), with MN described only as 'a suitable normalization mass scale.' In the Press-Schechter formalism the normalization is set by the power-spectrum amplitude (e.g., sigma_8) and the collapse threshold, so MN is not a free parameter; here it is left undetermined. Additionally, the text of Section 5 states that the curves are computed for gamma = 10 (MT = gamma * MCO_T), while the caption of Fig. 9 states that the dashed black lines correspond to 'total mass equal to 50 × MCO(z).' This discrepancy means that the maximum-mass boundary can be slid in the mass-radius plane by changing either MN or gamma. With two effective free normalizations in this step, the 'analytical demonstration' in Fig. 9 is not a parameter-free prediction; it is an exercise in matching the data with adjustable constants.
  3. [Appendix A, eq. (A.5), and Section 4.1] The stellar-to-halo mass ratio is a load-bearing ingredient because it converts the CGS halo masses into stellar masses via eq. (10). The adopted relation, log m = 0.062 log MD + 0.429, is fitted to Illustris galaxies at z = 0 and has no redshift dependence. The authors acknowledge this is provisional and note that other relations (e.g., Fan et al. 2010, Shankar et al. 2006, Girelli et al. 2020) disagree strongly at low halo masses and at high redshifts. Because the CGS is built from intersections over a wide redshift range, the z=0-only m(MD) may introduce a systematic bias in the predicted Ms and hence in the shape of the CGS in the MR-plane. The paper should quantify how the CGS changes when the redshift-dependent relations (A.1) or the Girelli et al. (2020) relations are used instead.
  4. [Section 5, eqs. (23)-(27)] The dissipation-less collapse derivation reproduces the slope of the observed MRR (eq. 23), but the zero-point constant k_gamma in eqs. (25)-(27) is explicitly left undetermined ('cannot be determined unless the constant k is specified by fixing the initial conditions'). Consequently, the agreement in slope between eq. (26) and the Bernardi et al. (2010) relation does not by itself constrain the MRR zero point; the latter is absorbed into k_gamma. This limits the strength of the claim that eq. (2) represents the locus of ideal dissipation-less collapse, because the zero-point remains a free parameter in the derivation.
minor comments (6)
  1. [Abstract] The abstract contains several typographical errors, including 'comsic' for 'cosmic' and 'evolution' in the final sentence where 'evolutions' or 'evolution processes' would be clearer. These should be corrected.
  2. [Introduction, Section 3 heading] The paper uses 'Section 3' twice: once for the description of simulations and once for 'The MRR of collapsing proto-galaxies.' The section numbering in the text should be made consistent.
  3. [Section 4.2, eq. (13)] Equations (13) and (14) are cubic polynomials in log Ms with many digits; it would be helpful to state the valid mass range for each fit and to give the fitting uncertainties, as is customary for empirical scaling relations.
  4. [Figure 9 caption] The caption states that the dashed black lines correspond to '50 × MCO(z),' while the body text says the calculation uses gamma = 10. This inconsistency should be resolved to avoid confusion about which parameter value was actually used.
  5. [Section 6, item (vi)] The conclusions section skips from item (iv) to item (vi); the numbering should be corrected or the missing item should be restored.
  6. [Appendix B.1] The description of the SCDM cosmology in Chiosi & Carraro (2002) could be confusing to readers because the rest of the paper adopts the Illustris Λ-CDM parameters; the distinction between the cosmological parameters used in the older models and those used in the present analysis should be clarified.

Circularity Check

2 steps flagged · score 6.0 of 10

The CGS boundary is partly calibrated: Ns is chosen so the curve sits at the ETG edge, and the Sec. 5 intersection proof contains an unspecified normalization and an inconsistent gamma; the slope/shape from the HGF is still independent.

  1. fitted input called prediction [Section 4.2, Eqs. (12)-(13), Fig. 8]
    "It turns out that with the Ns corresponding to 10−2 halos per (Mpc/h)3 (that roughly corresponds to the volume surveyed by the SDSS), the curve is just at the edge of the observed distribution of ETGs on the MR-plane. Higher values of Ns would shift it to larger halos (baryonic galaxies), the opposite for lower values of Ns. ... Lower or higher values of the halo number density would predict loci in the MR-plane too far from the observational MRR."

    Ns is the parameter that turns the Lukic halo growth function into the Cosmic Galaxy Shepherd on the MR-plane. The paper selects the value that places the curve 'just at the edge' of the Bernardi et al. ETG data and states that any other value moves it 'too far'; the CGS is then used to 'predict' the same observed MRR. The zero point of the prediction is therefore re-imported from the data, while only the slope/curvature is independently determined. The nominal SDSS count behind Ns = 10^-2 also counts all galaxies, not ETG central halos, an assumption the paper itself calls 'strong'.

  2. other [Section 5(vii)(b), Eqs. (32)-(33), and Fig. 9 caption]
    "MCO_T = MN × (1+z)^−6/(n+3) (32) ... MN is a suitable normalization mass scale. ... limited to the case with MT = γ MCO_T for γ = 10 ... we get Rs = 16.9 × 10^12 × γ^−0.79 × (1+z)^3.96 × Ms (33). [Fig. 9 caption:] MRRs expected for galaxies with total mass equal to 50 × MCO(z)."

    The 'analytical demonstration' that constant-density tracks intersect the cutoff-mass line on the observed MRR depends on two adjustable choices: the normalization MN, whose value is never specified, and the multiplicative factor γ, which is 10 in the text and 50 in the Fig. 9 caption. Since γ enters with exponent -0.79, changing it slides the boundary along the mass-radius plane, and with MN also free the intersection can be moved onto eq. (2). The statement 'This indeed is what we see in Fig. 9' is thus a consistency check made by the authors' choices, not a parameter-free first-principles prediction.

full rationale

The paper's qualitative claim, that the MR-plane distribution results from combining the collapse tracks of stellar systems with the redshift-dependent cutoff of the halo mass function, is a genuinely independent idea: the rising cutoff mass with time (Lukic et al. 2007; Press-Schechter) and the Fan et al. (2010) collapse relation are external inputs, and the curved slope change from ~0.5 to ~1 is not directly fitted to the Bernardi et al. (2010) fit. However, the quantitative CGS curve has a free normalization Ns that the authors effectively choose so that the curve is 'just at the edge' of the ETG data, and their Sec. 5 intersection proof leaves MN unspecified while using gamma=10 in the text and gamma=50 in the figure caption. Both degrees of freedom slide the predicted boundary along the MR-plane. The agreement in Figs. 8-9 is therefore partially a tuning statement, which is circular in the sense that a parameter selected after comparison with the target data is later presented as confirming the prediction. The heavy use of the authors' own earlier hydrodynamical models to build Eq. (31) is a self-citation, but those models are also compared with Illustris and with independent data, so I do not treat that as an independent circular step. Overall: the slope and shape derivation retains independent content, but the zero-point and the intersection demonstration are calibrated; score 6, not higher, because the central mechanism is not equivalent to the observed relation by definition.

Assumptions & free parameters 6 free parameters · 7 assumptions · 1 invented entities

The central claim rests on several chosen or fitted ingredients: Ns, the stellar-to-halo mass relation m(MD), f_sigma, and the zero-point constant k_gamma. The CGS is a derived construct rather than a new physical entity. The cosmological background, halo growth function, and collapse relations are taken from prior literature and simulations.

free parameters (6)
  • Halo number density Ns = 10^-2 halos per (Mpc/h)^3
    Chosen so the CGS curve falls at the observed upper edge; justified by a crude SDSS galaxy count versus volume estimate, but higher or lower values shift the predicted locus.
  • Stellar-to-halo mass ratio m(MD) = log m = 0.062 log MD + 0.429
    Fitted to Illustris stellar and halo masses at z=0 in Appendix A (eq. A.5); has no redshift dependence and is extrapolated over 10^4 to 10^15 solar masses.
  • Velocity dispersion ratio f_sigma = 1
    Assumed in the Fan et al. (2010) relation (eq. 10); the paper acknowledges a different value would shift the baryonic locus.
  • Zero-point constant k_gamma in eq. (26) = unspecified
    The paper states k cannot be determined without fixing the initial conditions, leaving the derived MRR with a free zero point.
  • Gamma factor for maximum halo mass in eq. (33) = 10
    Introduced as M_T = gamma times the Press-Schechter cutoff mass; the paper computes the boundary for gamma=10.
  • Primordial power spectrum slope n = -1.8
    Standard CDM value used in eqs. (20)-(23) and (32), taken from the literature.
assumptions (7)
  • domain assumption Lambda-CDM cosmology with WMAP-7 parameters is the correct background for structure formation
    Adopted throughout, from the Illustris values in Section 1 and the halo growth function of Lukić et al. (2007).
  • domain assumption The halo growth function n(MD,z) from Lukić et al. (2007) accurately describes the number density and cutoff of dark matter halos
    Used as external input in Section 4.2; the paper interpolates the published curves and does not re-derive them.
  • domain assumption The Fan et al. (2010) relation (eq. 10) gives the stellar half-mass radius from halo mass and collapse redshift
    The paper notes it is strictly valid for monolithic infall of baryons into a dark halo, yet uses it as the general M-MRR.
  • ad hoc to paper Each dark matter halo hosts one and only one galaxy, and that galaxy is an early-type galaxy
    Explicitly called a strong assumption in Section 4.2; the morphological fraction of ETGs is ignored.
  • domain assumption The dissipationless spherical collapse model RT proportional to M_T^(5+n)/6 applies to galaxy formation
    Used in Section 5(vii) through Gott & Rees (1975) and Blumenthal et al. (1984).
  • ad hoc to paper The stellar-to-halo mass relation fitted to Illustris at z=0 (eq. A.5) is valid for all redshifts and masses used in the CGS calculation
    Appendix A admits the relation is provisional and independent of redshift, while eq. (10) requires m(MD,z).
  • standard math The Press-Schechter cutoff mass M_CO(M) as a function of redshift (eq. 32) represents the maximum galaxy mass at each epoch
    Used in Section 5(vii) to derive the boundary lines; follows the classical Press-Schechter formalism.
invented entities (1)
  • Cosmic Galaxy Shepherd (CGS)
    purpose: Boundary curve on the mass-radius plane traced by halos at a fixed number density Ns, interpreted as the redshift-dependent maximum mass cutoff
    A named conceptual locus, not a physical object. It predicts no observable beyond the mass-radius boundary itself, and its location depends on the chosen Ns.

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Pith. "Pith review of The parallelism between galaxy clusters and early-type galaxies: III. The Mass-Radius Relationship." pith.science (2026). https://pith.science/paper/6O7D5ZB4

@misc{pith2026190808808,
  author       = {Pith},
  title        = {Pith review of: The parallelism between galaxy clusters and early-type galaxies: III. The Mass-Radius Relationship},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6O7D5ZB4}},
  note         = {Machine review of arXiv:1908.08808}
}
read the original abstract

Context. This is the third study of a series dedicated to the observed parallelism of properties between Galaxy Clusters and Groups(GCGs) and early-type galaxies (ETGs). Aims. Here we investigate the physical origin of the Mass-Radius Relation (MRR). Methods. Having collected literature data on masses and radii for objects going from Globular Clusters (GCs) to ETGs and GCGs, we set up the MR-plane and compare the observed distribution with the MRR predicted by theoretical models both for the monolithic and hierarchical scenarios. Results. We argue that the distributions of stellar systems in the MR-plane is due to complementary mechanisms: (i) on one hand, as shown in paper II, the relation of the virial equilibrium does intersect with a relation that provides the total luminosity as a function of the star formation history; (ii) on the other hand, the locus predicted for the collapse of systems should be convolved with the statistical expectation for the maximum mass of the halos at each comsic epoch. This second aspect provides a natural boundary limit explaining either the curved distribution observed in the MR-plane and the existence of a zone of avoidance. Conclusions. The distribution of stellar systems in the MR-plane is the result of two combined evolution, that of the stellar component and that of the halo component.

Figures

Figures reproduced from arXiv: 1908.08808 by the authors.

Figure 1
Figure 1. The log(Rs) versus log(Ms) relation for all the samples under consideration. Rs is the half-mass radius, and Ms the total stellar mass. Although Rs is not strictly identical to the effective radius Re, they are very close to each other. Throughout this paper we will always use Rs , which is easier to calculate for hydrodynamical models of galaxies, and assume Rs ≃ Re. Burstein et al. (1997) sample: the filled red ci… view at source ↗
Figure 2
Figure 2. Left Panel: The Ms − MD relations at different redshifts (z=4, blue; z=2, green; z=1, yellow; z=0, red). Masses are in solar units. The solid lines are the best fits, the coefficients of which are given in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. The stellar half-mass radius Rs vs the mass Ms of the galaxy models of the Illustris database at different values of the redshift, i.e. z=4 (blue), z=2 (green), z=1 (yellow) and z=0 (red). The best fit of the data at redshift z=0 using the relationship Rs = ηMǫ s (where masses and radii are in M⊙ and kpc, respec￾tively) yields the values listed in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 5
Figure 5. Figure 5: The composite Rs vs Ms relation and comparison between theory and observations. The light powder-blue dots are the data of Burstein et al. (1997) for ETGs, DGs, GCs and GCGs: the dashed line of the same colour is the best fit of the sole ETGs however extended to the do…
Figure 6
Figure 6. Figure 6: Comparison of the Fan et al. (2010) lines and the theoretical models by Merlin et al. (2012) (green filled squares) with the observa￾tional data of Burstein et al. (1997) from GCs (small red squares), to DGs (small blue triangles), and GCGs (light blue filled circles) …
Figure 7
Figure 7. Figure 7: The growth function of halos n(MD,z) reproduced from Lukic et al. (2007). ´ Given a certain number density of halos Ns , on the n(MD,z)− z plane of [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: The Cosmic Galaxy Shepherd (CGS) and the corresponding lo￾cus of DM parent halos(the black solid thick and thin lines, respectively) for the number density of Ns = 10−2 halos per (Mpc/h)3 . In addition to this we show the case with Ns = 10−8 halos per (Mpc/h)3 (the mag…
Figure 9
Figure 9. Figure 9: The Mass-Radius Relationship: comparison between data and theory. Radii Rs and stellar masses Ms are in kpc and M⊙, respectively. The pale-blue filled circles are all the data considered in this study, the pale green filled circles the models of Illustris. The dark-red…

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    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.