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arxiv: 2509.15453 · v2 · submitted 2025-09-18 · 🌌 astro-ph.CO · astro-ph.IM

CosmoGen: A genetic algorithm framework for the exploration of dark energy dynamics

D. Castel\~ao , I. Tereno This is my paper

Pith reviewed 2026-05-18 15:16 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.IM
keywords dark energycosmological tensionsgenetic algorithmssymbolic regressionH0 tensionS8 tensionBayesian analysisevolutionary algorithms
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The pith

A genetic algorithm framework can mechanically generate dark energy models that alleviate the H0 and S8 cosmological tensions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces CosmoGen, a computational system that applies evolutionary algorithms to create mathematical expressions for dark energy behavior. It guides the evolution by scoring candidates according to how well they reproduce observed expansion history and the growth of cosmic structures. The aim is to produce alternatives to the standard model that reduce specific mismatches in measurements of the Hubble constant and the amplitude of matter fluctuations. This mechanical, data-driven search supplements manual model building by exploring functional forms that may not occur to theorists first.

Core claim

CosmoGen implements evolutionary algorithms for symbolic regression where fitness is determined by how well the model reproduces structure formation and background cosmological quantities. Applied to dark energy fluid models aimed at mitigating the S8 and H0 tensions, the framework produces models with high fitness values. Bayesian analysis of one such illustrative model confirms that it alleviates the tensions, although the Bayes factor still shows a weaker preference compared to the standard LambdaCDM model.

What carries the argument

CosmoGen, a genetic algorithm framework for symbolic regression of dark energy dynamics, with fitness evaluation based on computations of structure formation and background cosmological quantities.

If this is right

  • High-fitness models can be passed to standard Bayesian pipelines for detailed parameter constraints and model comparison.
  • The generated dark energy expressions demonstrate the capacity to ease discrepancies in Hubble constant and structure growth measurements.
  • The same evolutionary procedure can be reused for other cosmological sectors once appropriate fitness functions are defined.

Where Pith is reading between the lines

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

  • Adding explicit physical constraints such as stability or causality to the fitness function could improve the Bayesian ranking of evolved models.
  • Running the framework on larger populations or with additional tension metrics might surface models that outperform LambdaCDM more decisively.
  • The approach could be tested on early-universe data to check whether models evolved for late-time tensions remain viable across cosmic history.

Load-bearing premise

The evolutionary process guided by structure formation and background quantities will automatically produce models that are mathematically consistent and avoid introducing new conceptual problems.

What would settle it

A full Bayesian parameter estimation and evidence comparison on multiple generated models using current cosmological datasets that finds no consistent reduction in the H0 or S8 tensions relative to LambdaCDM would falsify the central claim.

Figures

Figures reproduced from arXiv: 2509.15453 by D. Castel\~ao, I. Tereno.

Figure 1
Figure 1. Figure 1: Flowchart showing the sequence of steps of a CosmoGen run. The machine learning component is enclosed in the solid red border and the cosmological component in the dashed blue border. The sequence of steps moves through the two components in a loop. GP and MP stand for genetic programming and MontePython, respectively. See text for a full description. – A compiled version of CLASS, which is used for cosmol… view at source ↗
Figure 2
Figure 2. Figure 2: Representation of the function A/ [Da − ln(Da)] of complexity 10, as a tree generated from the GP algorithm. also define the maximum complexity of the initial population, the maximum complexity increase from generation to genera￾tion (small values favouring simples functional forms), and the tournament size (the number of individuals that compete for a single spot). The chosen parameter values are given in… view at source ↗
Figure 5
Figure 5. Figure 5: Top panel: Matter power spectrum for the CG dark energy fluid for different values of D. Bottom panel: Deviation from ΛCDM [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Top panel: CMB temperature angular power spectrum for the CG dark energy fluid for different values of D. Bottom panel: Deviation from ΛCDM. As and ns ; baryon density (Ωb); CDM density (ΩCDM) and the Hubble constant (h). A derived parameter ΩDE is calculated for the CG model, as is ΩΛ for the ΛCDM model. The nuisance parameters used are the APlanck for CMB and the amplitude of the intrinsic alignment mode… view at source ↗
Figure 8
Figure 8. Figure 8: WL KV-450 analyses. Marginalized two-dimensional 1- and 2- σ contours of the posterior and one-dimensional marginalized posterior for the CG model (green) and ΛCDM (gold). 6. Conclusions We have presented a new type of cosmological tool that in￾tegrates traditional methods of numerical cosmology with ML techniques. In the current work, we have shown an example of how this tool can provide new insights into… view at source ↗
Figure 9
Figure 9. Figure 9: CG model. Marginalized two-dimensional 1- and 2-σ contours of the posterior and one-dimensional marginalized posterior for the pa￾rameters relevant for the Hubble and S 8 tensions, for CMB Planck 2018 (gold) and WL KV-450 (green) [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: ΛCDM model. Marginalized two-dimensional 1- and 2-σ con￾tours of the posterior and one-dimensional marginalized posterior for the parameters relevant for the Hubble and S 8 tensions, for CMB Planck 2018 (gold) and WL KV-450 (green). The results presented were obtained using a standard lap￾top with 16 threads. This computational limitation restricted our ability to explore the full range of mathematical fu… view at source ↗
read the original abstract

The standard Lambda cold dark matter ($\Lambda$CDM) paradigm of the physical Universe suffers from well-known conceptual problems and is challenged by observational data. Alternative models exist in the literature, both phenomenological and physically motivated, but many of them suffer from similar or new problems. We propose a method to mechanically generate alternative models in a data-informed procedure tuned to mitigate specific problems. We implemented a computational framework, dubbed CosmoGen, based on evolutionary algorithms for symbolic regression. The evolutionary process is guided by the computation of structure formation and background cosmological quantities. As a proof-of-concept, we applied the procedure to the specific case of dark energy fluid models and asked the framework to generate models capable of alleviating the cosmological tensions $S_8$ and $H_0$. The system generated models with high fitness values, and through a Bayesian analysis of an illustrative model, we show that the model indeed alleviates the tensions, even though the Bayes factor indicates a weaker preference for $\Lambda$CDM.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces CosmoGen, a genetic algorithm framework for symbolic regression to generate alternative dark energy fluid models. Guided by computations of background cosmology and structure formation, the method targets alleviation of the S8 and H0 tensions. As a proof-of-concept, it reports generation of high-fitness models and presents a Bayesian analysis of one illustrative model showing tension reduction, although the Bayes factor indicates a weaker preference relative to ΛCDM.

Significance. If the generated models satisfy basic physical consistency requirements, the framework provides a systematic, data-driven alternative to manual construction of dark energy models, potentially useful for exploring dynamics that address cosmological tensions. The approach is novel in its use of evolutionary algorithms tuned to observables, but its impact depends on demonstrating that high-fitness outputs remain viable beyond the fitness criteria.

major comments (2)
  1. [Fitness function (methods section describing evolutionary process and fitness criteria)] Fitness function (methods section describing evolutionary process and fitness criteria): The fitness is defined via matching to background and structure-formation observables for S8/H0 alleviation, but lacks explicit penalties or constraints for perturbation-level instabilities such as c_s² < 0, ghost modes, or gradient instabilities. This is load-bearing for the central claim because symbolic regression can yield expressions that satisfy coarse fitness yet produce unphysical linear perturbation equations, undermining the viability of the generated models.
  2. [Bayesian analysis of illustrative model (results section)] Bayesian analysis of illustrative model (results section): The claim of tension alleviation is presented for one selected model, but the manuscript does not detail the post-generation selection criteria, any filtering for mathematical consistency, or error handling in the evolutionary outputs. This weakens the robustness of the proof-of-concept demonstration.
minor comments (1)
  1. [Abstract] The abstract states that the Bayes factor indicates a weaker preference for ΛCDM; including the numerical value of the Bayes factor would improve clarity and allow readers to assess the strength of the comparison directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. We have carefully considered the major comments and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns and improve the clarity and robustness of the presented framework.

read point-by-point responses

  1. Referee: Fitness function (methods section describing evolutionary process and fitness criteria): The fitness is defined via matching to background and structure-formation observables for S8/H0 alleviation, but lacks explicit penalties or constraints for perturbation-level instabilities such as c_s² < 0, ghost modes, or gradient instabilities. This is load-bearing for the central claim because symbolic regression can yield expressions that satisfy coarse fitness yet produce unphysical linear perturbation equations, undermining the viability of the generated models.

    Authors: We agree that explicit safeguards against perturbation-level instabilities are important for ensuring the physical viability of models generated by symbolic regression. The current fitness function prioritizes agreement with background cosmology and structure-formation observables relevant to the S8 and H0 tensions, but does not yet incorporate direct penalties for conditions such as c_s² < 0 or the presence of ghost or gradient instabilities. In the revised manuscript we will augment the fitness function in the methods section with additional penalty terms that penalize these instabilities, thereby guiding the evolutionary process toward physically consistent solutions. We will also report the fraction of generated models that survive these checks. revision: yes

  2. Referee: Bayesian analysis of illustrative model (results section): The claim of tension alleviation is presented for one selected model, but the manuscript does not detail the post-generation selection criteria, any filtering for mathematical consistency, or error handling in the evolutionary outputs. This weakens the robustness of the proof-of-concept demonstration.

    Authors: We acknowledge that additional transparency regarding model selection would strengthen the proof-of-concept section. In the revised manuscript we will expand the results section to describe the post-generation selection criteria applied to the illustrative model. This will include the specific filters used to enforce mathematical consistency (e.g., removal of expressions containing singularities or undefined operations), the criteria for choosing the model among the high-fitness population, and the procedures employed to handle invalid or erroneous outputs from the genetic algorithm. A short discussion of the robustness of the selected model with respect to these filters will also be added. revision: yes

Circularity Check

0 steps flagged

No significant circularity: computational search framework with external fitness criteria

full rationale

The paper presents CosmoGen as an evolutionary symbolic regression framework that mechanically generates dark energy fluid models by optimizing against externally defined fitness functions computed from background cosmology and structure formation observables. The goal is to produce high-fitness models targeting S8 and H0 tension alleviation, followed by separate Bayesian analysis on an illustrative example. No derivation chain reduces by construction to its inputs; fitness is defined from independent cosmological data rather than self-referential parameters, and the method is explicitly a data-informed search procedure rather than a first-principles derivation or renamed empirical fit. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing steps in the provided description, rendering the approach self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim depends on standard cosmological background and perturbation calculations as fitness drivers, plus the assumption that symbolic regression outputs remain physically interpretable.

axioms (1)
  • domain assumption Standard LambdaCDM background evolution and linear structure formation calculations provide sufficient guidance for model fitness.
    Invoked when the evolutionary process is guided by computation of structure formation and background cosmological quantities.

pith-pipeline@v0.9.0 · 5704 in / 1209 out tokens · 35248 ms · 2026-05-18T15:16:10.374395+00:00 · methodology

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Works this paper leans on

38 extracted references · 38 canonical work pages · 3 internal anchors

  1. [1]

    Abroshan, M., Mishra, S., & Khalili, M. M. 2023, in Proceedings of the AAAI Conference on Artificial Intelligence

    work page 2023

  2. [2]

    2006, arXiv e-prints, astro

    Albrecht, A., Bernstein, G., Cahn, R., et al. 2006, arXiv e-prints, astro

    work page 2006

  3. [3]

    & Nesseris, S

    Arjona, R. & Nesseris, S. 2020, Phys. Rev. D, 101, 123525

    work page 2020

  4. [4]

    Audren, B., Lesgourgues, J., Benabed, K., & Prunet, S. 2013, J. Cosmology As- tropart. Phys., 2013, 001

    work page 2013

  5. [5]

    J., Desmond, H., & Ferreira, P

    Bartlett, D. J., Desmond, H., & Ferreira, P. G. 2024, IEEE Transactions on Evo- lutionary Computation, 28, 950

    work page 2024

  6. [6]

    Blas, D., Lesgourgues, J., & Tram, T. 2011, J. Cosmology Astropart. Phys., 2011, 034

    work page 2011

  7. [7]

    G., Michalski, R

    Carbonell, J. G., Michalski, R. S., & Mitchell, T. M. 1983, in Machine Learning, ed. R. S. Michalski, J. G. Carbonell, & T. M. Mitchell (San Francisco (CA): Morgan Kaufmann), 3–23

    work page 1983

  8. [8]

    2019, Reviews of Modern Physics, 91, 045002

    Carleo, G., Cirac, I., Cranmer, K., et al. 2019, Reviews of Modern Physics, 91, 045002

    work page 2019

  9. [9]

    DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

    Delgado, A. M., Wadekar, D., Hadzhiyska, B., et al. 2022, MNRAS, 515, 2733 DESI Collaboration, Abdul-Karim, M., Aguilar, J., et al. 2025, arXiv e-prints, arXiv:2503.14738 Di Valentino, E., Mena, O., Pan, S., et al. 2021, Classical and Quantum Gravity, 38, 153001 Euclid Collaboration, Mellier, Y ., Abdurro’uf, et al. 2025, A&A, 697, A1

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  10. [10]

    Fluke, C. J. & Jacobs, C. 2020, WIREs Data Mining and Knowledge Discovery, 10, e1349

    work page 2020

  11. [11]

    Goldberg, D. E. 1989, Genetic Algorithms in Search, Optimization and Machine Learning (Addison-Wesley)

    work page 1989

  12. [12]

    & Haupt, S

    Haupt, R. & Haupt, S. E. 2004, Practical genetic algorithms, 2nd edn. (Wyley)

    work page 2004

  13. [13]

    2021, Astronomy & Astrophysics, 646, A140

    Heymans, C., Tröster, T., Asgari, M., et al. 2021, Astronomy & Astrophysics, 646, A140

    work page 2021

  14. [14]

    L., et al

    Hildebrandt, H., Köhlinger, F., van den Busch, J. L., et al. 2020, A&A, 633, A69

    work page 2020

  15. [15]

    2017, MNRAS, 465, 1454

    Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454

    work page 2017

  16. [16]

    Koksbang, S. M. 2023, Phys. Rev. D, 107, 103522

    work page 2023

  17. [17]

    Koza, J. R. 1992, Genetic Programming: On the Programming of Computers by Means of Natural Selection (Cambridge, MA: MIT Press)

    work page 1992

  18. [18]

    2023, Machine Learn- ing: Science and Technology, 4, 045002

    Lemos, P., Jeffrey, N., Cranmer, M., Ho, S., & Battaglia, P. 2023, Machine Learn- ing: Science and Technology, 4, 045002

    work page 2023

  19. [19]

    The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview

    Lesgourgues, J. 2011, arXiv e-prints, arXiv:1104.2932

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  20. [20]

    Li, Y ., Ni, Y ., Croft, R. A. C., et al. 2021, Proceedings of the National Academy of Science, 118, e2022038118

    work page 2021

  21. [21]

    Linder, E. V . 2003, Phys. Rev. Lett., 90, 091301 LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, arXiv e-prints, arXiv:0912.0201

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  22. [22]

    2019, BAAS, 51, 14

    Ntampaka, M., Avestruz, C., Boada, S., et al. 2019, BAAS, 51, 14

    work page 2019

  23. [23]

    1999, ApJ, 517, 565 Planck Collaboration, Ade, P

    Perlmutter, S., Aldering, G., Goldhaber, G., et al. 1999, ApJ, 517, 565 Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A16 Planck Collaboration, Aghanim, N., Akrami, Y ., et al. 2020, A&A, 641, A6

    work page 1999

  24. [24]

    G., Filippenko, A

    Riess, A. G., Filippenko, A. V ., Challis, P., et al. 1998, AJ, 116, 1009

    work page 1998

  25. [25]

    G., Macri, L

    Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016, ApJ, 826, 56

    work page 2016

  26. [26]

    J., Kelly, D

    Riess, A. G., Yuan, W., Macri, L. M., et al. 2022, ApJ, 934, L7 Roman ROTAC and CCSDC. 2025, arXiv e-prints, arXiv:2505.10574

    work page arXiv 2022

  27. [27]

    & Barto, A

    Sutton, R. & Barto, A. 2018, Reinforcement Learning, second edition: An Intro- duction, Adaptive Computation and Machine Learning series (MIT Press)

    work page 2018

  28. [28]

    2008, Contemporary Physics, 49, 71–104

    Trotta, R. 2008, Contemporary Physics, 49, 71–104

    work page 2008

  29. [29]

    Turing, A. M. 1950, Mind, LIX, 433

    work page 1950

  30. [30]

    & Tegmark, M

    Udrescu, S.-M. & Tegmark, M. 2020, Science Advances, 6, eaay2631 Article number, page 9 A&A proofs:manuscript no. main

    work page 2020

  31. [31]

    2017, in Advances in Neural Infor- mation Processing Systems (NeurIPS), 5998–6008

    Vaswani, A., Shazeer, N., Parmar, N., et al. 2017, in Advances in Neural Infor- mation Processing Systems (NeurIPS), 5998–6008

    work page 2017

  32. [32]

    2024, ARA&A, 62, 287

    Verde, L., Schöneberg, N., & Gil-Marín, H. 2024, ARA&A, 62, 287

    work page 2024

  33. [33]

    C., et al

    Wadekar, D., Thiele, L., Hill, J. C., et al. 2023, MNRAS, 522, 2628

    work page 2023

  34. [34]

    2023, The European Physical Journal C, 83 [arXiv:2304.05807]

    Wang, D., Koussour, M., Malik, A., Myrzakulov, N., & Mustafa, G. 2023, The European Physical Journal C, 83 [arXiv:2304.05807]

    work page arXiv 2023

  35. [35]

    Webb, S. A. & Goode, S. R. 2025, in IAU Symposium, V ol. 19, IAU Symposium, ed. J. McIver, A. Mahabal, & C. Fluke, 40–49

    work page 2025

  36. [36]

    1989, Reviews of Modern Physics, 61, 1

    Weinberg, S. 1989, Reviews of Modern Physics, 61, 1

    work page 1989

  37. [37]

    H., Stölzner, B., Asgari, M., et al

    Wright, A. H., Stölzner, B., Asgari, M., et al. 2025, KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey

    work page 2025

  38. [38]

    Zlatev, I., Wang, L., & Steinhardt, P. J. 1999, Phys. Rev. Lett., 82, 896 Article number, page 10 D. Castelão and I. Tereno: CosmoGen: a cosmological model generator Appendix A: Robustness tests We show the top ranked models from two other runs of CosmoGenin Tables A.1 and A.2. There are some models in common between the results of the various runs. We no...

    work page 1999