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Gamma Analytical Modeling Evolution (GAME) I: The physical implications of deriving the stellar mass functions from z=0 to z=8

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that a single gamma growth curve, with parameters already fixed by the cosmic star formation rate density, reproduces the observed galaxy stellar mass function from z=0 to z=8 without re-tuning to high-redshift data.

desk verdict A compact gamma-growth model that usefully maps the low/intermediate-mass galaxy stellar mass function from z=0 to z=8, but the high-mass failures are larger than the abstract admits and the 'prediction' rests on an untested universal-growth assumption. read the letter →

arxiv 2505.13301 v1 pith:7J5MAVBT submitted 2025-05-19 astro-ph.GA

Antonios Katsianis , Qingshan Wang , Xiaohu Yang , Xian Zhong Zheng , Pedro Cataldi , Nicola Napolitano , Weishan Zhu , Nicolas Tejos
show 8 more authors
Weiguang Cui Cheng Li Weipeng Lin Long-long Feng Junde Li Ying Tang Yuchang Li Hangxin Pu
This is my paper · ORCID
classification astro-ph.GA
keywords galaxystellarmassfunctiongammagrowthmodelcosmicstarformationratedensityevolutionhigh-redshiftgalaxiesJWSTobservationsassemblytimescale
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

The paper tries to establish that the evolution of the galaxy stellar mass function over 13.5 billion years is governed by the same few-parameter gamma growth pattern that already describes the cosmic star formation rate density. Starting from the present-day (z=0) mass function and the mass-history formula $M_{*}(T) = M_{*,0}\,\gamma(3.0, 0.5\,T)/\Gamma(3.0)$, with $\alpha = 3.0$ and $\beta = 0.5\,\mathrm{Gyr}^{-1}$ taken unchanged from cosmic star formation, the authors evolve each local mass bin backward in time and compare with observed mass functions from z=0 to z=8. They report agreement with low- and intermediate-mass galaxies across the full range, with typical scatter around 0.17 dex, while deviations appear only for the rarest, most massive galaxies at z>1.5. A sympathetic reader would care because, if correct, the result means the backbone of galaxy assembly, number counts in mass bins, needs no mass-dependent tuning or exotic feedback machinery, only accretion physics encoded in two parameters, plus one extra parameter to recover the high-mass end at early times.

What carries the argument

The load-bearing object is the gamma growth motif, in the specific form of the incomplete-gamma ratio $\gamma(\alpha,\beta T)/\Gamma(\alpha)$: the fraction of a galaxy's final (present-day) stellar mass that has been assembled after time $T$ since the first stars formed. The two parameters carry the physics: $\alpha = 3.0$ sets the early power-law rise $T^{\alpha-1} = T^{2}$, tied through $\alpha = 2Y/(3-Y)$ to the radial density profile of the accreting matter, while $\beta = 0.5\,\mathrm{Gyr}^{-1}$ sets the exponential decline, tied to the depletion timescale equated with the z=0 halo dynamical time $\tau_{\mathrm{dyn}} \approx 2\,\mathrm{Gyr}$. The scheme's working mechanism is a one-to-one mapping: every present-day mass bin is evolved backward along this universal curve, so galaxy numbers are conserved while bins shift to smaller masses; where the predicted and observed mass functions diverge, the divergence marks where the universal-growth assumption, rather than the gamma form itself, breaks down.

What would settle it

Follow the same galaxies across epochs, using merger trees from simulations or stellar-mass measurements of individual galaxies at two redshifts in deep JWST fields, and test whether fractional mass growth $M_{*}(T)/M_{*,0}$ is truly identical across mass bins; the paper's own comparison shows the universal curve overshoots the observed high-mass end by up to about 1 dex at z>1.5, so a decisive check is whether deeper high-redshift surveys of massive galaxies widen that gap beyond observational error.

Watch

Extended reading notes

Core claim

The paper's central claim is that galaxy stellar mass grows, on average, along the same gamma growth curve as the cosmic star formation rate density: the growth rate $\frac{dM}{dT} = M_{z,0}\frac{\beta^{\alpha}}{\Gamma(\alpha)}T^{\alpha-1}e^{-\beta T}$ integrates to $M_{*}(T) = M_{*,0}\,\gamma(\alpha,\beta T)/\Gamma(\alpha)$, with $\alpha = 3.0$ and $\beta = 0.5\,\mathrm{Gyr}^{-1}$ left at the values already set by cosmic star formation rather than re-tuned. Applied bin-by-bin to the z=0 stellar mass function and evolved backward in time, this single curve reproduces the observed galaxy stellar mass function from z=0 to z=8 for galaxies up to the characteristic mass, with average scatter of roughly 0.17 dex, and the paper shows the same scheme agrees with recent JWST-based mass functions at z ≈ 4–8. Deviations appear only for very low-mass and exceedingly massive galaxies at z>1.5, which the paper traces to physical processes that shorten gas-consumption timescales at both extremes, chiefly mergers and low-angular-momentum collapse. A second version (GAME 1) lets $\beta$ vary with present-day stellar mass through a log-normal form centered at $\log_{10}(M_{*}/\mathrm{M}_{\odot}) \approx 10.85$, restoring agreement across the full mass range with one additional parameter calibrated at z=8.

Load-bearing premise

The load-bearing premise is that every galaxy in a given present-day mass bin grew along the same universal gamma curve, so a galaxy's mass today uniquely fixes its mass at any earlier time and galaxy numbers are conserved as bins are shifted backward, ignoring mass-dependent growth, mergers, and the fact that today's massive galaxies assembled from many smaller progenitors.

Editorial extensions

If this is right

  • The low- and intermediate-mass end of the mass function at any redshift z ≤ 8 becomes a prediction from local data alone: the present-day mass function plus two fixed parameters fixes the number density of these galaxies at all earlier epochs.
  • Because GAME 0 works with no mass-dependent tuning for more than 9 billion years, the paper's reading is that feedback mechanisms acting selectively at the low- and high-mass ends are not needed to explain the observed shape evolution of the mass function at z < 1.5.
  • The breakdown at the high-mass end for z > 1.5 identifies where the universal-growth picture fails, and the GAME 1 fix says the required adjustment is faster gas consumption (shorter depletion timescales) at both mass extremes at early times, not a late-time suppression of star formation.
  • The formalism's two parameters are tied to gravitational-collapse scales (the density profile of accreting matter and the z=0 halo dynamical time), so if the scheme holds, the mass function's evolution carries direct information about collapse physics rather than about tuned subgrid feedback; the paper reports GAME matches the observed z=0–8 mass function with average scatter comparable to or bett
  • The authors conclude that both their scheme and current simulations are consistent with JWST-derived mass functions at z ≈ 4–8, which they read as evidence that the early massive galaxies seen by JWST do not force a departure from standard growth physics once stellar masses are measured with near-infrared (MIRI) constraints included.

Reading between the lines

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

  • If the universal-curve mapping is right, it makes a strong statement about individual galaxies that the paper does not test: a galaxy's present-day stellar mass alone determines its entire assembly history. This is checkable with star-formation histories of local galaxies and is most likely to fail for the most massive systems, where merger-dominated assembly is expected, the same regime where GAM
  • The log-normal $\beta$ correction is implicitly a claim about the mass dependence of star-formation efficiency: efficiency peaks near $\log_{10}(M_{*}/\mathrm{M}_{\odot}) \approx 10.85$ and falls symmetrically toward dwarf and giant extremes. Direct measurements of molecular-gas depletion timescales as a function of stellar mass at z ≈ 2–4 could test that claim against Eq. (13).
  • The same gamma machinery could be pointed at other galaxy statistics, such as the star formation rate function or quenched fractions; the paper's observation that the star formation rate function evolves nearly in parallel across bins at low redshift suggests a gamma-based prediction for the SFR function at higher redshifts would be a natural and independent check.
  • Because everything is anchored to the z=0 mass function, the scheme inherits the systematics of local measurements; the paper averages several discrepant local determinations, so a better-constrained low-mass slope at z=0, for example from wider-area spectroscopy, would directly sharpen or challenge the high-redshift predictions.
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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 / 5 minor

Summary. The paper presents GAME0 and GAME1, analytical models for the evolution of the galaxy stellar mass function (GSMF) from z=0 to z=8. GAME0 assumes that every present-day stellar mass bin grows according to the same incomplete-gamma function previously used to fit the cosmic star formation rate density (CSFRD), Eq. (4), with α=3.0, β=0.5 Gyr^-1, and no further tuning to high-redshift mass functions. The predicted GSMFs are compared with a large compilation of observed GSMFs, with cosmological simulations (EAGLE, IllustrisTNG, Simba, ELUCID+L-Galaxies), and with empirical models such as UniverseMachine and Leja et al. (2020). The authors report good agreement for low- and intermediate-mass galaxies, with an average RMSD of 0.17 dex up to M*, and propose GAME1, which introduces a mild mass-dependent adjustment of β at z>2, to improve the high- and low-mass ends. They conclude that the evolution of the GSMF is set by simple and physically motivated growth with few parameters.

Significance. If the central claim holds, the result is valuable: a two-parameter, no-re-tune description of the low- and intermediate-mass GSMF over 13.5 Gyr, with parameters tied to accretion and gas-consumption physics. The paper's strengths include the breadth of the observational comparison, the quantitative RMSD and AIC tables, the inclusion of recent JWST data, and the explicit acknowledgement of modeling assumptions in Section 3.1. However, because Eq. (4) is a rigid log-shift of the z=0 GSMF with exactly conserved number densities, the empirical agreement at low and intermediate masses may largely reflect the shape of the adopted z=0 GSMF rather than a validated universal growth law for individual galaxies. The total-mass part of the prediction is also inherited from the earlier CSFRD fit, so the genuinely new content is the shape evolution of the mass function, which is exactly where the model shows its largest failures.

major comments (4)
  1. [Section 3.1, Eq. (4)] The mapping M(T)=M0*g(T) with a mass-independent g(T) is a rigid log-shift of the z=0 GSMF: the shape and low-mass slope are unchanged and the number of galaxies in every bin is conserved. The paper's own Section 3.1 (second bullet) concedes that massive z=0 galaxies have lower-mass progenitors, i.e., the number-conserving assumption is violated; Table B3 shows the expected consequence, with GAME0 total RMSD reaching 1.20 dex at z=3 and 1.74 dex at z=7. The authors do not quantify progenitor bias for intermediate-mass bins, so the low/intermediate-mass success cannot be claimed as direct evidence for universal gamma growth. A test using median M(T|M0) growth tracks in EAGLE, IllustrisTNG, or Simba, or a continuity-model comparison with mass-dependent growth, is needed.
  2. [Section 2, Fig. 1] The input z=0 GSMF is a hand-drawn 'average' of several measurements, and GAME0 inherits its low-mass slope and normalization exactly. Because the high-z low-mass slope is the same as the z=0 low-mass slope under the rigid-shift mapping, the claimed agreement at low and intermediate masses is sensitive to this choice; adopting the steeper Wang et al. (2024b) or Xu et al. (2022) low-mass slopes could change the high-z predictions by more than the quoted RMSD. The gray uncertainty region in Figs. 2-4 is not propagated into the GAME0 curves, and no sensitivity test is presented. This is load-bearing for the central claim and should be addressed.
  3. [Abstract and Section 3.2 vs. Appendix B] The statement that deviations appear 'solely' at z>1.5 and 'specifically for very small and exceedingly massive objects' is not supported by the paper's own tables. Table B3 shows that the total RMSD including the high-mass end is already 0.60 dex at z=1.5 and 0.61 dex at z=2.0, while the up-to-M* RMSD is about 0.18 dex at both redshifts, so massive galaxies dominate the deviation well within the claimed regime. At z=4, the total RMSD is 0.79 dex versus 0.177 dex up to M*, again a high-mass effect. No table or figure isolates a systematic failure at the very low-mass end; the 'up to M*' statistics include low masses and show RMSD values of 0.2-0.3 dex at z=5-8. The claim should be qualified to the high-mass end, or a separate low-mass RMSD should be presented.
  4. [Sections 3-4] The parameters α and β are inherited from fits to the CSFRD in earlier work (Katsianis et al. 2021b, 2023). Since the CSFRD is the time derivative of the total stellar mass density, and GAME0's total stellar mass density is g(T) times the z=0 value by construction, the total-mass-growth part of the 'prediction' is not independent: it is forced once α and β reproduce the CSFRD. The genuinely predictive content is the shape evolution of the GSMF, i.e., mass-dependent growth and number conservation, which is precisely where GAME0 fails at high masses. The '0 parameters re-tuned' claim should be reframed to acknowledge this inheritance, so that readers can correctly identify the novel and the constrained parts of the model.
minor comments (5)
  1. [Section 5, Eqs. (12)-(13)] The expression for τ⋆ in Eq. (12) is written ambiguously with what appears to be a missing multiplication sign; the simplified form τ⋆ = 0.5 + 1.5 exp(-0.5 (log10 M - 10.85)^2) stated in the text is much clearer and should be used in the displayed equation.
  2. [Section 3.1, Eq. (5)] The role of f0 is not made concrete: the example sets the z=0 mass equal to the final mass, yet with βT = 6.775 the factor γ(3, βT)/Γ(3) is about 0.965, not 1. The authors should specify f0 explicitly and use it consistently in the worked example.
  3. [Figures 8-10 and general text] There are numerous typographical errors: 'sold green line' should be 'solid green line'; Fig. 9 z=0.9 panel legend says 'GAME1, z=2.0'; 'Marchensini' should be 'Marchesini'; 'EALGE' should be 'EAGLE'; 'tunned' should be 'tuned'; and 'consenus' should be 'consensus'. A careful proofreading pass is needed.
  4. [Appendix B tables] The RMSD values are reported with inconsistent precision (e.g., 0.2075 in Table B2 versus 0.21 in the text and 0.17 in Table B1 summaries). Uniform significant figures should be used throughout the tables and text.
  5. [Fig. 7 caption] The meaning of the 'x' symbols and open circles in the top panels is not defined in the caption; the caption should state explicitly which mass bins are classified as not well captured by GAME0.

Circularity Check

2 steps flagged · score 4.0 of 10

GAME0's GSMF 'prediction' is partly a rigid shift of the z=0 GSMF along the authors' prior CSFRD fit, and GAME1 calibrates β on the z=8 GSMF it later uses for validation; the shape comparison retains independent content, so partial circularity.

  1. fitted input called prediction [Section 3.1, Eq. (4); see also Abstract and Eq. (2)]
    "Beginning with a mass bin of the stellar mass function at z = 0, M∗,z0, we aim to employ the basic Γ growth model, identical to the CSFRD (in form and parameters), and estimate the mass value of this mass bin at higher redshifts/earlier eras. Thus, in our formalism we adopt that all the factors that contribute to galaxy growth aggregate to follow a Γ growth pattern of the form (Katsianis et al. 2021b, 2023): M∗(T) = M∗,final × γ(α, β × T)/Γ(α) = M∗,z0/f0 × γ(α, β × T)/Γ(α)"

    Because the factor g(T)=γ(α,βT)/Γ(α) is mass-independent, Eq. (4) maps every z=0 bin by M(T)=M0 g(T); number conservation makes the predicted high-z GSMF exactly the input z=0 GSMF shifted rigidly in log M by −log10 g(T). The same g(T) is the integral of Eq. (2), the CSFRD fit from Katsianis et al. (2021b, 2023), so the evolution of the total stellar mass density (the zeroth moment of the GSMF) is inherited from that fit by construction. The 'prediction' of the GSMF level and knee shift is therefore a re-expression of the CSFRD fit plus the assumed no-crossing mapping, not an independent test of the gamma growth law.

  2. fitted input called prediction [Section 5, Eqs. (12)-(13), and Fig. 10]
    "Besides for the z = 0 GSMF, this time, for the case of GAME 1 we pick as our constrain the stellar mass function at z = 8.0 to constrain the high redshift correction (bottom left panel of Fig. 7). ... At z = 7, z = 6, z = 5 we are able to reproduce the observed results."

    The GAME1 β⋆(M⋆,0) log-normal form (Eq. 13) is adjusted so that the model reproduces the z=8 GSMF, and the paper then presents GAME1 against z=8 JWST measurements as evidence of success. Because the same observable used for calibration is used for validation, the z=8 agreement is enforced by the fit rather than independently predicted. The paper concedes this: 'GAME1 results for the high mass end should be considered also as tuned rather than predictive at z = 4-8.'

full rationale

The central GAME0 test is not wholly circular: α=3, β=0.5 are taken from fits to the cosmic star formation rate density (an external, if author-overlapping, constraint), and the predicted GSMF shape is compared to high-redshift observations over 13.5 Gyr. The paper is also transparent about the main failure (high-mass end, z>1.5, RMSD up to 1.74 dex) and about GAME1 being tuned. However, two reductions are present. First, Eq. (4) imposes one mass-independent growth factor on every z=0 bin, so the predicted high-z GSMF is a rigid log-shift of the input z=0 GSMF; the shift is the integral of the CSFRD fit (Eq. 2), meaning the global normalization and knee evolution are inherited from that fit by construction. What remains genuinely new is the assumption that the shape is unchanged, which is an ansatz validated only by aggregate curve comparison, not by direct individual-galaxy growth histories. Second, GAME1's mass-dependent β is calibrated on the z=8 GSMF and then used to demonstrate agreement at z=8, a circular validation that the authors themselves label 'tuned rather than predictive.' These issues are partial, not total: the low/intermediate-mass shape comparison is not numerically forced by the CSFRD fit alone, so a score of 4 is appropriate.

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

The central claim rests on the choice of a universal gamma growth curve (Eq. 4) whose parameters were fit to a related observable (the CSFRD) in prior work, plus the assumption of number conservation. The purported first-principles derivation of α and β in Section 4 is post-hoc: the paper starts from the fitted values and selects collapse scalings that reproduce them. No genuinely new physical entity is introduced.

free parameters (6)
  • alpha (α) = 3.0
    Power-law growth exponent in Eq. (4); inherited from the authors' prior fit to the cosmic star formation rate density (Katsianis et al. 2021b), not derived in this paper.
  • beta (β) = 0.5 Gyr^-1
    Inverse depletion timescale in Eq. (4); same prior CSFRD fit. Paper argues it equals 1/tau_dyn(z=0), but the adopted value comes from fitting.
  • T_Lag = 0.17 Gyr
    Delay between Big Bang and first star formation, adopted from ΛCDM literature; sets the zero point of T in Eqs. (4)-(5).
  • f0 = ~1
    Fraction of final stellar mass already in place at z=0; chosen from literature claims of 95% mass assembly, normalizes Eq. (4).
  • z=0 reference GSMF = Not tabulated
    The averaged observed GSMF (black dashed line in Fig. 1) used as the model's boundary condition. The exact functional form or table is not provided.
  • GAME1 log-normal beta parameters = center log10 M* = 10.85; width and amplitude implicit
    Eqs. (12)-(13) introduce a mass-dependent tau_star for z>2, fitted so that the z=8 GSMF is reproduced. The paper says only the central mass is a new parameter, but the width and amplitude of the log-normal are also set by the fitting.
assumptions (6)
  • standard math The incomplete gamma function integral is the standard one with Γ(α) normalization
    GAME0 uses the identity M(T) = M_z0 γ(α, βT)/Γ(α); this is a mathematical definition.
  • domain assumption Every galaxy follows the same gamma-shaped growth curve as the cosmic average, with mass-independent α and β
    Section 3.1 states 'The approach followed in GAME 0 maintains the same α, β and T for all mass bins regardless of mass. This is a simplification.' This is the key modeling assumption.
  • domain assumption Number density of galaxies is conserved under backward evolution; mergers only rescale masses
    The mapping from z=0 bins to high-z bins assumes one-to-one correspondence; the paper acknowledges this fails for massive galaxies in Section 3.1.
  • domain assumption The CSFRD is well described by the gamma form with α=3.0 and β=0.5 Gyr^-1
    Inherited from Katsianis et al. 2021b; used without re-fitting to define the growth curve.
  • domain assumption The gas depletion timescale equals the z=0 halo dynamical time (2 Gyr) at all redshifts
    Section 4.2, Eq. (8): ⟨τ*⟩ = τ_dyn(z=0) = 2 Gyr, with arguments that redshift dependence is weak. This sets beta.
  • domain assumption Power-law density profiles and accretion-rate scalings from star-forming clouds (Murray and Chang 2015; Wang et al. 2018) apply to galaxy-scale growth
    Section 4.1 uses M-dot ∼ M^{1/2} or Eq. (7) to argue α = 2Y/(3-Y) is physically motivated; the transfer from cloud-scale to galaxy-scale is assumed.

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Pith. "Pith review of Gamma Analytical Modeling Evolution (GAME) I: The physical implications of deriving the stellar mass functions from z=0 to z=8." pith.science (2026). https://pith.science/paper/7J5MAVBT

@misc{pith2026250513301,
  author       = {Pith},
  title        = {Pith review of: Gamma Analytical Modeling Evolution (GAME) I: The physical implications of deriving the stellar mass functions from z=0 to z=8},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J5MAVBT}},
  note         = {Machine review of arXiv:2505.13301}
}
abstract

The $\Gamma$ growth model is an effective parameterization employed across various scientific disciplines and scales to depict growth. It has been demonstrated that the cosmic star formation rate density (CSFRD) can also be described broadly by this pattern, i.e. $\frac{dM(T)}{dT} = M_{z,0}\, \times \frac{\beta^{\alpha}}{\Gamma(\alpha)} \, T^{\alpha-1} e^{-\beta \, T }$ M$_{\odot}$ Gyr$^{-1}$, where $M_{z,0}$ is the stellar mass at $z$ = 0, $\alpha = 3.0$, $\beta = 0.5 $ Gyr$^{-1}$ and $T$ describes time. We use the identical $\Gamma$ growth pattern given by the CSFRD to extend the present day (z = 0) stellar mass bins $M_{\ast}(T)$ of the Galaxy Stellar Mass Function (GSMF) and investigate if we are able to reproduce observations for the high redshift GSMFs. Surprisingly, our scheme describes successfully the evolution of the GSMF over 13.5 Gyrs, especially for objects with intermediate and low masses. We observe some deviations that manifest {\it solely} at very high redshifts ($z > 1.5$, i.e. more than 9.5 Gyr ago) and {\it specifically} for very small and exceedingly massive objects. We discuss the possible solutions (e.g. impacts of mergers) for these offsets. Our formalism suggests that the evolution of the GSMF is set by simple (few parameters) and physically motivated arguments. The parameters $\beta$ and $\alpha$ are theoretically consistent within a multi-scale context and are determined from the dynamical time scale ($\beta$) and the radial distribution of the accreting matter ($\alpha$). We demonstrate that both our formalism and state-of-the-art simulations are consistent with recent GSMFs derived from JWST data at high redshifts.

Figures

Figures reproduced from arXiv: 2505.13301 by the authors.

Figure 1
Figure 1. Left: The stellar mass functions from Baldry et al. (2012), Driver et al. (2022), Bernardi et al. (2018) and Wang et al. (2024b). The gray area represents the uncertainty of the observations at z = 0 and thus the uncertainty of our formalism. According to the updated observations the results of Baldry et al. (2012) were unable to capture successfully the low mass end. Right: The stellar mass function at z = 0 in the… view at source ↗
Figure 2
Figure 2. Comparison of the GAME0 model (black dashed line) and observations up to T = 7.8 Gyr, z = 0.615 (6 Gyr ago). The comparison with respect observations complements the simplicity of our formalism. We consider that the number of free parameters for the model is 0 since the α and β values are well motivated both by theory (subsection 4) and observations (CSFRD). GAME0 is the start / backbone for our family of models. We… view at source ↗
Figure 3
Figure 3. Comparison of GAME0 (black dashed line) and observations up to T = 1.54 Gyr (z = 4). The comparison with respect observations up to T = 4.3 Gyr (9.5 Gyr ago) particularly highlights the success of the simple scheme we adopt. We note that GAME0 reproduce GSMFs in agreement with observations for 70% of the age of the Universe. However at redshift 2-4 it is evident that the high mass end is not well reproduced. Cosmolo… view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Comparison of GAME0 (dashed black line)) and observations up to T = Gyr (z =8). We note that GAME0 produces GSMFs that are in good agreement with respect observations, especially for low mass and intermediate mass galaxies. IllustisTNG (red dot-dashed line), EAGLE (ora…
Figure 5
Figure 5. Figure 5: Comparison of the GAME0 model with α = 3.0 (black dashed line) and predictions of equation 3 when α = 1.0 (magenta dot-dashed line), α = 2.0 (green dotted line) and α = 4.0 (red dashed line). 4.1 Early growth, power law and the α parameter Power law growths (t n ) have…
Figure 6
Figure 6. Figure 6: Comparison of the GAME0 model with β = 0.5 (black dashed line) and predictions of equation 3 when β = 0.25 (magenta dashed line), β = 0.75 (green dot-dashed line) and β = 1.0 (red dotted line). τdyn(disk) is the dynamical timescale of the disk and Mcrit is the threshol…
Figure 7
Figure 7. Figure 7: Top panels: The evolution of GAME0 from redshift 0 to 8. The formalism besides its simplicity is able to reproduce 88% of the data points present in this work. The x symbols represent the mass bins that are not well captured by GAME0 while the open circles represent th…
Figure 8
Figure 8. Figure 8: Comparison of GAME0 (black dashed line) and GAME1 (sold green line) at z > 3. The small tuning of the β parameter improves the comparison at the low and high mass ends. tribution of that variable is that it follows a log-normal distribution. Mutch et al. (2013) suggest…
Figure 9
Figure 9. Figure 9: Comparison of GAME0 (black dashed line) and GAME1 (sold green line) at z < 2.0. The small tuning of the β parameter improves the comparison at the low and high mass ends while the predictions of both models are the same at z < 0.9. reasonable approaches could be applie…
Figure 10
Figure 10. Figure 10: Comparison of GAME1 (sold green line) with respect recent JWST observations Wang et al. (2024a, open diamonds) and Navarro-Carrera et al. (2024, open circles) and cosmological simulations. which can be written as τ⋆ = 0.5 + 1.5 × e (−0.5×(log10(M⋆,0)−10.850)2) while β…
Figure 11
Figure 11. Figure 11: Comparison of GAME1 (sold green line) with respect recent observations and IllustisTNG (red dot-dashed line), EAGLE (orange solid line) Elucid+L-galaxies (blue dashed line) and Simba (green dotted line). scope of our current work which focus on reproducing the GSMF in…
Figure 1
Figure 1. Figure 1: Comparison of the GSMFs produced by the models GAME0 (black dashed line), GAME1 (green solid line), Leja et al. (2020) (blue dotted line) and UniverseMachine (yellow dot-dashed line). We note that UniverseMachine was constrained to reproduce the GSMF at z = 0-4 while t…
Figure 2
Figure 2. Figure 2: Comparison of the GSMFs produced by the models GAME0 (black dashed line), GAME1 (green solid line), Leja et al. (2020) (blue dotted line) and UniverseMachine (yellow dot-dashed line). We note that UniverseMachine was constrained to reproduce the GSMF at z = 0-4 while t…

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Works this paper leans on

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    Comparison of the GSMFs produced by the models GAME 0 (black dashed line), GAME 1 (green solid line), Leja et al. (2020) (blue dotted line) and UniverseMachine (yellow dot-dashed line). We note that Uni verseMachine was constrained to reproduce the GSMF at z = 0-4 while the empirical model of Leja et al. (2020) was constrained at z = 0.2, 1.6 and 3.0. The...

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    1 and 2 we present the GSMFs generated by Uni- verseMachine, represented by the yellow dot-dashed line, a t z = 0.23-4 and 5-8, respectively

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    Last, similarly to other efforts (like EAGLE) we only tune our results for the GS MF at z = 0 in order to evaluate if our model is actually predictiv e at higher redshifts

    going beyond a pure empirical approach . Last, similarly to other efforts (like EAGLE) we only tune our results for the GS MF at z = 0 in order to evaluate if our model is actually predictiv e at higher redshifts. We then explore, what further elements we need to add (GAME 1 employs 1 more parameter) to improve further the comparison with observations. Le...

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    Besides cosmological hydr ody- namic simulations, the evolution of the GSMF has also been fo l- lowed by empirical models

    We demonstrated that each model has different strengths and shortcomings but overall most models considered produc ed a sensible evolution for the GSMF. Besides cosmological hydr ody- namic simulations, the evolution of the GSMF has also been fo l- lowed by empirical models. For example, two baseline models are described in Behroozi et al. (2019) and Leja...

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    2020, (C, z = 0.2,1.6,3) Observations GAME0, z = 0.23, (C, z=0) GAME1, z = 0.23, (C, z=0,8) UniverseMachine, (C, z=0 to

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    Leja et al. 2020, (C, z = 0.2,1.6,3) 107 108 109 1010 1011 1012 1013 M∗ [M⊙] 10−7 10−6 10−5 10−4 10−3 10−2 10−1 dn/dlog10(M∗) [ Mpc−3] z = 0.4 Observations GAME0, z = 0.4, (C, z=0) GAME1, z = 0.4, (C, z=0,8) UniverseMachine, (C, z=0 to

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    Leja et al. 2020, (C, z = 0.2,1.6,3) 107 109 1011 1013 M∗ [M⊙] 10−7 10−6 10−5 10−4 10−3 10−2 10−1 dn/dlog10(M∗) [ Mpc−3] z = 0.615 Observations GAME0, z = 0.615, (C, z=0) GAME1, z = 0.615, (C, z=0,8) UniverseMachine, (C, z=0 to

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    Comparison of the GSMFs produced by the models GAME 0 (black dashed line), GAME 1 (green solid line), Leja et al. (2020) (blue dotted line) and UniverseMachine (yellow dot-dashed line). We note that UniverseMachine was constrained to reproduce the GSMF at z = 0-4 while the emp...

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    The continuity model from Leja et al. (2020) also performs well, especially at selected redshifts where it wa s con- strained. Interestingly, the Simba simulation, which exhi bits rel- atively high RMSD values at lower redshifts, shows competit ive performance at z = 4 and z =...

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    360 130.262 Leja et al. 2020 (11) 45.662 60.837 38.905 52.056 32.704 32.496 48.316 Table B8. Continuation of table B7 at z = 4.0 − 8.0. Model z = 4.0 z = 5.0 z = 6.0 z = 7.0 z = 8.0 GAME0 (3) 80.873 103.557 74.679 99.517 74.311 UniverseMachine (44) 109.418 143.781 98.347 105.5...

    2020

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