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One set of equations links star formation history shape to feedback

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A unified empirical equation, fitted separately to three simulation suites, links galaxy star formation history shape to halo mass, baryon fraction, black hole mass, and feedback strength.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection A careful and useful survey of how feedback shapes SFHs across three CAMELS models, but the headline unified equations are in-sample fits to an emulator and should not be treated as validated predictions. the 4 major comments →

arxiv 2508.21152 v1 pith:MXCGDHLN submitted 2025-08-28 astro-ph.GA astro-ph.CO

How does feedback affect the star formation histories of galaxies?

Kartheik G. Iyer , Tjitske K. Starkenburg , Greg L. Bryan , Rachel S. Somerville , Juan Pablo Alfonzo , Daniel Angl\'es-Alc\'azar , Suchetha Cooray , Romeel Dav\'e
show 15 more authors
Austen Gabrielpillai Shy Genel Sultan Hassan Lars Hernquist Christian Kragh Jespersen Christopher C. Lovell Boon Kiat Oh Camilla Pacifici Lucia A. Perez Laura Sommovigo Joshua S. Speagle Sandro Tacchella Megan T. Tillman Francisco Villaescusa-Navarro John F. Wu
This is my paper
classification astro-ph.GA astro-ph.CO
keywords galaxy star formation historiesstellar feedbackAGN feedbackcosmological simulationsbaryon cyclingdouble power-law SFHCAMELSnormalizing flows
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 tries to show that average galaxy star formation histories can be summarized by a double power-law curve whose five shape parameters depend on a small set of physical quantities: halo mass, baryon fraction, black hole mass, and feedback strength. Using the three CAMELS simulation models, a single set of equations with model-specific coefficients reproduces the average SFHs across all three. In the equations, cosmology sets the early rise, halo mass sets the overall scale, and feedback plus baryon cycling set the late-time decline and width. This matters because it offers a path from observations of galaxy samples back to feedback physics.

Core claim

The paper's central discovery is a system of empirical equations, Eqn 18, describing the average SFH shape parameters in terms of Omega_m, sigma_8, halo mass, baryon fraction, relative black hole mass, and the CAMELS feedback parameters. The same functional form applies to IllustrisTNG, SIMBA, and ASTRID, with only coefficients changing (Table 2). The rising slope beta depends mainly on Omega_m; the falling slope alpha on halo mass with AGN and baryon terms; the peak/width tau on cosmology, halo mass, baryon fraction, and black hole mass; the normalization phi on halo mass times feedback corrections; and the start time eta on Omega_m. The paper further finds that stellar feedback is the domi

What carries the argument

The load-bearing machinery is a double power-law SFH parameterization plus a normalizing-flow emulator. The double power law, SFH(t) = phi * ([(t-eta)/tau]^alpha + [(t-eta)/tau]^(-beta))^(-1), compresses each average SFH into five interpretable numbers. The normalizing flow, trained on the CAMELS LH simulations, generates average SFHs anywhere in parameter space, enabling regression of those five numbers against halo mass, baryon fraction, black hole mass, and feedback scalings. Symbolic regression term frequencies and generalized additive model losses guide which physical variables enter each equation, with a bias toward galaxy state variables over direct feedback parameters.

Load-bearing premise

The entire analysis uses SFHs sampled from a machine-learning emulator rather than directly from the simulations, and the emulator is validated only qualitatively against a small single-parameter set; if it is biased, or if the double power-law fails for a substantial fraction of average SFHs, the derived equations and conclusions inherit that distortion.

What would settle it

Recompute average SFHs directly from the CAMELS LH simulation particle data, fit double power laws, and compare the resulting alpha, beta, tau, phi, eta to Eqn 18 on a grid of parameter values; systematic deviations beyond sampling noise would disprove universality. Specifically examine ASTRID galaxies with late-time star-formation tails: if those SFHs are not well described by a double power law, the falling-slope equation is biased.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Observed average SFHs could constrain cosmology and stellar feedback strength using Eqn 18 without rerunning simulations.
  • The same equation form should apply to other simulation codes, turning cross-model calibration into a coefficient-fitting exercise.
  • Because stellar feedback changes black hole growth, SFH-based constraints on stellar and AGN feedback will remain partially degenerate.
  • Reparameterizing winds by mass loading and energy per unit SFR makes the three models' SFH responses qualitatively consistent, clarifying interpretations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Eqn 18 is fitted on emulated SFHs; applying it to other simulation suites like SWIFT-EAGLE or CAMELS-SAM would test whether the shared functional form reflects physical regularity or a property of these three models.
  • AGN feedback is almost unconstrained from SFH shape alone, so combining these equations with baryon fraction or black hole mass observations should sharpen late-time feedback constraints.
  • The double power-law's poor fit to ASTRID's sustained late-time tails suggests a modified form with an added plateau could alter the predicted falling slopes and the conclusion that ASTRID galaxies quench fastest.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies how variations in stellar feedback, AGN feedback, and cosmology affect the average star formation histories (SFHs) of galaxies at z~0 in the CAMELS IllustrisTNG, SIMBA, and ASTRID simulations. The authors train normalizing-flow emulators on the CAMELS Latin Hypercube (LH) sets, use the emulators to sample average SFHs, and fit these with a double power-law (Eq. 13) to obtain shape parameters {α, β, τ, φ, η}. They then use random forests, generalized additive models, and symbolic regression to construct a common set of equations (Eq. 18) with model-specific coefficients (Table 2) that relate SFH shape parameters to halo mass, baryon fraction, black hole-to-halo mass ratio, and CAMELS feedback/cosmology parameters. The paper also presents interaction analyses, a reparameterization of supernova feedback in terms of mass/energy loading, an SBI-based proof-of-concept inference of CAMELS parameters from SFHs, and several supplementary observational diagnostics.

Significance. If the central claim holds—that one functional form with model-dependent coefficients describes SFH shape across three different hydrodynamical codes—this would be a useful empirical framework for connecting SFH observations to feedback physics and for interpreting CAMELS parameter-space studies. The paper's strengths include its use of the public CAMELS suite, explicit treatment of galaxy selection and resolution limits, the combination of multiple ML methods, and the extensive diagnostic figures. However, the equations in Eq. 18 are fitted to emulator-sampled SFHs, the emulator validation is only qualitative, and the coefficients carry no reported uncertainties or out-of-sample tests. These gaps are load-bearing for the 'single set of equations' claim, so the result is currently a promising but not fully supported empirical summary.

major comments (4)
  1. [§3.1, Appendix A (Fig. 19)] All SFHs used to build Eqn. 18 are generated by sampling the trained normalizing flows ('Unless otherwise mentioned, all the SFHs in the following sections are generated by sampling the trained normalizing flows', §3.1). The only comparison to actual simulation outputs is the qualitative 1P validation in Fig. 19, which states 'qualitative agreement' but reports no quantitative coverage, calibration, or held-out error statistic. Since Eqn. 16 minimizes loss on emulator-sampled SFHs, any emulator bias in under-sampled regions of the 6D parameter space propagates directly into the DPL parameters and hence into every coefficient in Table 2. I request a quantitative validation of the emulator against held-out LH boxes and/or the 1P runs, with per-parameter residuals and coverage diagnostics, and a demonstration that the derived Eqn. 18 coefficients are stable when the emulator is retrained or
  2. [§6.1.1, Eq. (18), Table 2] The central equations are presented without any measure of predictive accuracy. The coefficients in Table 2 have no uncertainties, and the loss defined in Eq. (16) is minimized on the same emulator-generated data used to select the terms. No residual plots, R² values, or held-out predictions are shown for the DPL parameters. The paper even uses the GAM loss as a 'proxy of the Bayes risk' (§3.3.2), but does not compare Eq. 18 against that benchmark. Without out-of-sample validation, the claim that 'a single set of equations ... can describe the SFHs across all three CAMELS models' is supported only by in-sample agreement. Please provide bootstrap/subsample coefficient uncertainties and a held-out evaluation (e.g., fitting coefficients on a training subset and evaluating on a withheld LH subset, or predicting one model's coefficients from another).
  3. [§3.2, §4.2, §6.1.1] The paper acknowledges that the double power-law 'does not describe a subset of ASTRID SFHs with sustained late-time tails' (§3.2). These ASTRID SFHs are nevertheless included in the fits that determine the ASTRID coefficients in Table 2, where α and τ are the least constrained parameters. Because Eqn. 18 is claimed to hold across all three models, the fraction of average SFHs that are poorly represented by the DPL form must be quantified, and the sensitivity of the derived coefficients to excluding or reparameterizing these cases should be shown. If the DPL failure is non-negligible in ASTRID, the corresponding rows of Table 2 may encode an artifact of the fitting form rather than the feedback response.
  4. [§6.1.1, Eq. (16), Eq. (18)] The claim of a 'single set of equations' is weakened by the fact that the coefficients are free to vary per model in Eq. (16). If the functional form is the same but every coefficient differs, the statement reduces to 'each model can be fit by a member of a parametric family'. The paper needs to demonstrate what is shared beyond the functional form—for example, that the same terms remain important across models, that coefficients can be predicted from model properties, or that the equations generalize to held-out models/parameters. As written, the evidence for universality is largely the symbolic-regression term frequencies in Fig. 12, which are qualitative and in-sample.
minor comments (6)
  1. [§3.1, Appendix A] The validation text in Appendix A repeats 'qualitative agreement' twice. Please state explicitly which quantitative metrics were computed (e.g., coverage, calibration, chi-square) and whether any failed.
  2. [§3.2, Eq. (13)] The text defines α as the falling slope and β as the rising slope, but Figure 1 and Eq. (13) can be misread because the two power-law terms are symmetric. Consider adding a sentence explicitly defining the relation between α, β and the t<τ versus t>τ behavior.
  3. [§4.3, Eq. (17)] The χ² metric in Eq. (17) has unusual units (SFR² over SFR² integrated over time). Please clarify whether the integrand is intended to be a dimensionless ratio or whether the normalization is meant to produce a time-averaged statistic.
  4. [§3.3.1] The random forest feature importances in Fig. 13 are shown without error bars or sensitivity checks. Given that they are used as a 'sanity check' for Eq. 18, a bootstrap estimate would strengthen the comparison.
  5. [§6.1.1, Eq. (18)] The equation for η is written separately with no coefficients in Table 2. For completeness, state explicitly that η = 12.5Ωm − 3 is used for all three models, and whether a coefficient uncertainty was estimated.
  6. [General] There are several typographical and formatting issues, including 'early rimes' in §6.1.1, inconsistent use of 'Mhalo' (log Mhalo vs Mhalo) in Eq. (18) and Table 2, and missing figure cross-references in the text. A careful proofread is recommended.

Circularity Check

0 steps flagged

No circularity: Eqn 18 is an explicitly empirical fit, not a first-principles derivation; the real weaknesses are missing out-of-sample validation and qualitative emulator checks, not a definitional loop.

full rationale

The claimed 'derivation' of Eqn 18 is openly empirical: Section 3.3.3 leaves coefficients free and minimizes L = sum_model (Theta_SFH - sum_i c_i,model f(Theta_sfh,i))^2 (Eqn 16) on Theta_SFH parameters obtained by MCMC-fitting the same normalizing-flow-sampled SFHs, and Section 6.1.1 calls the result 'empirical equations.' Fitting a response surface to its own training data is in-sample regression, not circularity; the paper does not present Eqn 18 as a held-out prediction or as a consequence of a self-citation or uniqueness theorem. The normalizing flow itself is anchored to the LH simulations and has an admittedly qualitative 1P validation in Appendix A (Fig 19), and the acknowledged DPL failure for some ASTRID SFHs with late-time tails (Section 3.2) is a stated modeling limitation. The feedback-trend conclusions in Section 4 and the mass/energy-loading reparametrization in Section 6.2 are separate empirical findings. The genuine gap is the absence of a quantitative held-out test of Eqn 18 against direct simulation SFHs; that is a validation/correctness weakness, not a circular step. No load-bearing self-citation or ansatz-smuggling was found.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The central equations rest on freshly fitted coefficients (Table 2), the emulator, the double power-law functional form, and the sufficiency of present-day state variables. No new physical entities are introduced. The free parameters are the equation coefficients; the axioms include the averaging procedure, the functional-form choice, emulator reliability, and the state-variable reduction.

free parameters (1)
  • SFH equation coefficients (Table 2) = Multiple values, e.g., tau_TNG: c1=4.571, c2=0.891, c3=-0.173, c4=0.930, c5=3.236
    Coefficients c1-c5 for each SFH shape parameter (tau, alpha, beta, phi, eta) are fitted per model by minimizing the joint loss across CAMELS models (Eqn 16). These are the principal fitted quantities in the paper.
axioms (5)
  • domain assumption The average SFH of 100 randomly selected galaxies in a box is a stable proxy for the mean SFH in that box and mass range.
    Introduced in Section 2.4; the entire analysis uses this averaged quantity rather than individual galaxies.
  • ad hoc to paper The double power-law form (Eqn 13) adequately captures the shape of average SFHs.
    Adopted in Section 3.2; the paper notes it does not describe a small fraction of ASTRID SFHs with sustained late-time tails.
  • domain assumption Simulation-based inference with normalizing flows trained on the LH datasets provides unbiased SFH samples across the parameter space.
    Used throughout Section 4 and in deriving Eqn 18; validation against 1P runs (Appendix A) is only qualitative.
  • ad hoc to paper The z=0 values of baryon fraction and black hole-to-halo mass ratio are sufficient state variables to characterize the integrated feedback history relevant to SFH shape.
    Used in Section 5 and in Eqn 18; these state variables are themselves outputs of the same simulations, so using them as inputs assumes they summarize the relevant physics without needing full time evolution.
  • domain assumption The CAMELS parameter variations (ASN1, ASN2, AAGN1, AAGN2, Omega_m, sigma_8) span the physically interesting feedback and cosmology space.
    Implicit in using the CAMELS LH sets; if the parameter priors miss relevant physics, the derived equations will not generalize.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of How does feedback affect the star formation histories of galaxies?." pith.science (2026). https://pith.science/paper/MXCGDHLN

@misc{pith2026250821152,
  author       = {Pith},
  title        = {Pith review of: How does feedback affect the star formation histories of galaxies?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MXCGDHLN}},
  note         = {Machine review of arXiv:2508.21152}
}
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abstract

Star formation in galaxies is regulated by the interplay of a range of processes that shape the multiphase gas in the interstellar and circumgalactic media. Using the CAMELS suite of cosmological simulations, we study the effects of varying feedback and cosmology on the average star formation histories (SFHs) of galaxies at $z\sim0$ across the IllustrisTNG, SIMBA and ASTRID galaxy formation models. We find that galaxy SFHs in all three models are sensitive to changes in stellar feedback, which affects the efficiency of baryon cycling and the rates at which central black holes grow, while effects of varying AGN feedback depend on model-dependent implementations of black hole seeding, accretion and feedback. We also find strong interaction terms that couple stellar and AGN feedback, usually by regulating the amount of gas available for the central black hole to accrete. Using a double power-law to describe the average SFHs, we derive a general set of equations relating the shape of the SFHs to physical quantities like baryon fraction and black hole mass across all three models. We find that a single set of equations (albeit with different coefficients) can describe the SFHs across all three CAMELS models, with cosmology dominating the SFH at early times, followed by halo accretion, and feedback and baryon cycling at late times. Galaxy SFHs provide a novel, complementary probe to constrain cosmology and feedback, and can connect the observational constraints from current and upcoming galaxy surveys with the physical mechanisms responsible for regulating galaxy growth and quenching.

Figures

Figures reproduced from arXiv: 2508.21152 by Austen Gabrielpillai, Boon Kiat Oh, Camilla Pacifici, Christian Kragh Jespersen, Christopher C. Lovell, Daniel Angl\'es-Alc\'azar, Francisco Villaescusa-Navarro, Greg L. Bryan, John F. Wu, Joshua S. Speagle, Juan Pablo Alfonzo, Kartheik G. Iyer, Lars Hernquist, Laura Sommovigo, Lucia A. Perez, Megan T. Tillman, Rachel S. Somerville, Romeel Dav\'e, Sandro Tacchella, Shy Genel, Suchetha Cooray, Sultan Hassan, Tjitske K. Starkenburg.

Figure 1
Figure 1. Figure 1: Motivation: The star formation histories (SFHs) of galaxies act as tracers of the overall gas regulation in the galaxy, which depend sensitively on the adopted prescription and strength of stellar and AGN feedback. By analyzing how the shape of the average SFH at a given mass (described using normalization (ϕ), peak/width (τ ), and rising and falling slopes (α, β)) depends on the strength and nature of the… view at source ↗
Figure 2
Figure 2. Figure 2: Median SFHs in different stellar mass bins from the three CAMELS models we use in this work, compiled using the CV datasets. The plot shows the SFHs of galaxies starting at 109.25M⊙ in increments of 0.2 dex up to 1010.85M⊙, with darker colors indicating higher mass bins. Feedback type TNG SIMBA ASTRID ASN1 Energy per unit SFR Mass loading factor Energy per unit SFR (Galactic Winds; range: (0.25,4)) (Galact… view at source ↗
Figure 3
Figure 3. Figure 3: Changes in the average SFHs of galaxies with variations in the double power-law parameters τ (peak/width), α (falling slope), β (rising slope), and ϕ (overall normalization), as described in Eqn. 13. plain them needs to account for both the effects of in￾dividual parameter variations as well as interactions. In this section, we briefly describe the three methods we use to examine the relation between the a… view at source ↗
Figure 4
Figure 4. Figure 4: Variation in the average SFHs of 100 randomly selected galaxies with stellar masses 109.5 <M∗ < 1011.5M⊙ caused by varying the cosmology and feedback strengths across the three CAMELS models (ASTRID, TNG, SIMBA), predicted using the trained normalizing flows. Solid lines and shaded regions show the median SFH and corresponding 1σ uncertainties from the bootstrap sampling used to train the normalizing flow.… view at source ↗
Figure 5
Figure 5. Figure 5: Similar to [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Similar to [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Similar to [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Interactions between different feedback and cosmology channels (thicker lines) can sometimes affect the SFHs beyond the linear additive effects of variations in the individual parameters (thin dashed lines). The three examples illustrate the effects of the interactions between the galactic wind parameters (ASN1 and ASN2) in TNG (top), between cosmology and feedback (Ωm and ASN1) in SIMBA (middle) and betwe… view at source ↗
Figure 9
Figure 9. Figure 9: Baryon content as a function of halo mass across the three CAMELS models [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Similar to [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Similar to [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: The frequency with which different terms occur in the symbolic regression solutions to the SFH parameters across the three CAMELS models. While different from traditional metrics of feature importance (e.g. using saliency maps or random forests), the occurrence of certain recurring terms (e.g. Mhalo being an important feature for ϕ across all three models) in the set of equations generated by symbolic reg… view at source ↗
Figure 13
Figure 13. Figure 13: (Top) Summary of the various factors impacting average galaxy SFHs across the three CAMELS models. (Bottom) Feature importance of the cosmology, feedback scaling factors and galaxy properties as defined in Eqn. (18) on the star formation rate as a function of lookback time. Cosmology dominates the average SFR of galaxies at early times, giving way to halo mass as they assemble. At late times, the SFR is a… view at source ↗
Figure 14
Figure 14. Figure 14: Changes in the SFHs for all three CAMELS models while varying ASN1 and ASN2 such that the energy per unit SFR is varied while keeping the mass loading factor fixed, at low (top), intermediate (middle) and high (bottom row) halo masses (using the normalizing flow emulator developed in Section 3.1). Note that (except for the high mass halos where BH feedback dominates), increasing the energy per units SFR (… view at source ↗
Figure 15
Figure 15. Figure 15: Same as [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Variations in the baryon fraction (top panels) and black hole mass (bottom panels) for all three simulations (going from left to right: CAMELS/TNG, CAMELS/SIMBA, and CAMELS/ASTRID) as a function of halo mass. Within each set of six panels, the top column shows the result of varying the mass loading while keeping the energy per SFR constant while the bottom column is the opposite. time star formation. This… view at source ↗
Figure 17
Figure 17. Figure 17: Varying the energy per unit SFR (ASN1) and wind speed (ASN2) in CAMELS/TNG have distinct effects on the baryon fraction and stellar-to-gas mass ratio (SGMR). This allows us to disentangle the effects of factors that change the gas content of halos from factors that change the efficiency of forming stars from that gas. The black lines show lines of constant baryon-fraction to stellar-to-gas mass ratio. Whi… view at source ↗
Figure 18
Figure 18. Figure 18: Inferring the CAMELS box parameters (both cosmology and feedback strengths) from a sample of 100 SFHs using SBI across the three models. Top: Corner plots showing the inferred posteriors for a single sample of SFHs corresponding to a realization of cosmology and feedback strength. Bottom: Average results across 1000 realizations across the full CAMELS parameter space. The cosmology is well constrained, as… view at source ↗
Figure 19
Figure 19. Figure 19: Validation plots comparing median (solid lines) and 16-84th percentiles (shaded regions) of the SFHs sampled from the trained normalizing flows (labeled NN) compared against SFHs from the CAMELS 1P dataset that vary a single parameter in the CAMELS box while keeping the others fixed at the fiducial value. The different panels show the 1P dataset for all six CAMELS box parameters. While the 1P datasets sho… view at source ↗
Figure 20
Figure 20. Figure 20: The distribution of χ 2 SFH values across the CAMELS parameter space when comparing the SFHs varying 2 parameters to the linear combination of the corresponding 1 parameter variations for the three CAMELS models: TNG (blue), SIMBA (red) and ASTRID (green). Values in each cell show the χ 2 SFH value, which also correspond the intensity of the colormaps [PITH_FULL_IMAGE:figures/full_fig_p042_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Similar to [PITH_FULL_IMAGE:figures/full_fig_p042_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The impact of varying cosmology and feedback strengths on the gas phase metallicity of galaxies as a function of halo mass at z ∼ 0 in the three models. Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167, doi: 10.3847/1538-4357/ac7c74 Ayromlou, M., Nelson, D., & Pillepich, A. 2023, MNRAS, 524, 5391, doi: 10.1093/mnras/stad2046 Bagley, M. B., Pirzkal, N., Finkelstein, S. L.,… view at source ↗
Figure 23
Figure 23. Figure 23: Similar to [PITH_FULL_IMAGE:figures/full_fig_p044_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Similar to [PITH_FULL_IMAGE:figures/full_fig_p044_24.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 5, 2026.