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REVIEW 4 major objections 6 minor 47 references

Joint reconstruction of $H(z)$ and $f\sigma_8(z)$ with physics informed neural networks

T0 review · 4 major / 6 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Coupling the expansion history and structure growth through the GR growth equation during neural-network training is feasible and beneficial for model-independent late-universe reconstruction.

desk verdict Clean proof-of-concept that a dual-head PINN can couple H(z) and fσ8 through the growth ODE; feasibility holds, but fixed fiducials and H0 anchoring dominate the science claims. read the letter →

arxiv 2606.17614 v2 pith:E7UHERZO submitted 2026-06-16 astro-ph.CO gr-qc

Konstantinos F. Dialektopoulos This is my paper · ORCID
classification astro-ph.COgr-qc
keywords physics-informedneuralnetworksHubbleparameterreconstructiongrowthratefσ8linearequationH0tensionσ8Om(z)nulltestlate-timecosmology
topics The Hubble Tension
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 shows that a dual-head physics-informed neural network can reconstruct the Hubble expansion H(z) and the growth rate fσ8(z) jointly, rather than separately, by enforcing the linear growth equation of general relativity as a training penalty. The shared backbone feeds two output heads that are kept mutually consistent through the ODE residual evaluated by automatic differentiation at collocation points. With either of two local H0 anchors the reconstructed growth sits systematically below the flat ΛCDM prediction, matching the known σ8 tension, while the Om(z) diagnostic departs from a constant. The result matters because it turns a post-hoc consistency check into a built-in regularizer, letting the H0 prior propagate self-consistently into the growth sector and reducing ensemble uncertainty without assuming a dark-energy equation of state.

What carries the argument

The dual-head PINN with physics loss: a shared backbone maps redshift to two heads for H(z) and fσ8(z); the GR growth ODE residual, obtained by automatic differentiation at 1000 resampled collocation points, is added to the data losses with weight λ, forcing dynamical consistency during training rather than after the fact.

What would settle it

Re-train the same architecture after promoting Ωm,0 (and optionally σ8,0) to free parameters jointly optimized with the network weights; if the systematic deficit of fσ8 relative to ΛCDM disappears or reverses once those fiducials are marginalized, the claimed growth suppression is an artifact of the hard-coded values rather than a data-driven signal.

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Extended reading notes

Core claim

Coupling H(z) and fσ8(z) through the GR linear growth equation during PINN training is both feasible and beneficial: an ensemble of 100 dual-head networks recovers either local H0 prior exactly, produces indistinguishable fσ8 reconstructions that lie below the ΛCDM+GR curve at essentially all redshifts, and yields Om(z) profiles that are not flat, while the physics weight λ = 0.1 balances data fit against ODE residual without either term dominating.

Load-bearing premise

The growth equation is evaluated with fixed matter density and fluctuation amplitude (Ωm,0 = 0.3 and σ8,0 = 0.8) that are never learned from the data; the paper itself notes that varying the matter density over a plausible range moves the low-redshift growth amplitude by several times the reported ensemble uncertainty.

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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. The manuscript presents a proof-of-concept dual-head physics-informed neural network that jointly reconstructs H(z) and fσ8(z) by coupling the two heads through the GR linear growth ODE residual (Eq. 1.1 / 3.6), evaluated via automatic differentiation at 1000 collocation points. Training uses a 50-point H(z) compilation (CC+BAO) and a 63-point RSD fσ8 compilation, with an ensemble of 100 independently seeded networks. A λ scan motivates λ=0.1; primary results anchor H0 to SH0ES or H0dN, recovering the prior exactly and yielding nearly identical fσ8 reconstructions that sit below the ΛCDM+GR curve, with Om(z) and Omfσ8 null tests showing departures from flat ΛCDM. The paper argues that the physics coupling is feasible and beneficial as a regularizer and as a channel for H0–growth propagation.

Significance. If the methodological claim holds, this is a useful first demonstration of a multi-head PINN that enforces dynamical consistency between expansion and growth during training rather than post hoc. The architecture (shared backbone, positivity/bound output activations, AD residual), the λ sensitivity study (Table 2), dual local-H0 anchors, and standard Om/Omfσ8 diagnostics are concrete and reproducible in principle. The work is appropriately scoped as a proof of concept and is unusually candid about limitations (fixed fiducials, Planck rd inheritance, AP corrections, ensemble meaning). That honesty is a strength. The science interpretation of a σ8-like deficit and non-flat Om(z) is secondary to the method and, as written, not claimed as a high-significance detection.

major comments (4)
  1. §3.3 and §5: The growth residual hardcodes Ωm,0=0.3 and σ8,0=0.8. The paper itself states that Ωm,0∈[0.28,0.35] shifts fσ8(0) by ~0.05 (~7× the ensemble 1σ), so the fixed fiducials dominate the low-z growth normalization and the claimed systematic deficit relative to ΛCDM+GR (Figs. 4, 7). For the central science-facing statements about suppressed growth and Omfσ8, either promote Ωm,0 (and preferably σ8,0) to trainable parameters as the paper itself proposes, or reframe those statements strictly as conditional on the fiducials and report a sensitivity band. Leaving the dominant systematic unmarginalized while quoting ensemble-only errors is load-bearing for the growth conclusions.
  2. Table 2 and §4.2–4.3: Without an H0 prior the free reconstruction returns H0≈46–52 km s−1 Mpc−1 with large λ-dependent bias, so the method does not self-normalize the expansion history. The coupling reduces ensemble spread on H0 but does not recover a cosmologically sensible free H0. The primary results therefore depend on external local H0 anchors. The abstract and §5 should state more clearly that the free (unanchored) reconstruction is not viable for H0 inference, and that “beneficial coupling” is demonstrated mainly in the anchored setting and in reduced initialization scatter, not as a free joint inverse solution.
  3. §3.5, §4.5, §5: Uncertainty is quantified solely by an ensemble of independently seeded networks on the same fixed data; the paper acknowledges this captures initialization sensitivity rather than data-noise resampling and that no fully propagated posterior is available. Statements that “all 100 members lie systematically below ΛCDM+GR” are therefore not a statistical significance claim. Either implement parametric bootstrap / data resampling (or an equivalent) so that the band has a clear frequentist/Bayesian meaning, or systematically downgrade language in the abstract, §4.4, and §5 so that the deficit is presented only as a qualitative, initialization-robust trend conditional on the fixed fiducials and H0 prior.
  4. §3.1 and §5: Eighteen of the 50 H(z) points are BAO measurements converted with the Planck 2018 sound horizon rd=147.09 Mpc. The reconstruction is model-independent in the dark-energy equation of state but inherits early-universe ΛCDM calibration in the expansion sector. Combined with fixed Ωm,0 in the ODE, this weakens the “model-independent” framing used in the abstract and introduction. The restricted sense of model independence already noted in §5 should be stated up front (abstract/intro) and the BAO conversion should be stress-tested (e.g., rd variation or pure-CC runs) so that Om(z) features are not partly driven by the Planck rd prior.
minor comments (6)
  1. §3.4: Data losses are normalized mean absolute errors (L1) rather than χ². The robustness motivation is reasonable, but a short appendix or footnote comparing L1 vs χ² on the same ensemble would strengthen that the reconstructions and null tests are not loss-choice artifacts.
  2. §3.1: The BAO compilation deliberately omits DESI DR1/DR2 for comparability with Ref. [9]. Given the 2026 context and the paper’s own discussion of DESI-driven DE evolution, a brief quantitative note on how DESI would be expected to change the H(z)/Om(z) bands would help the reader gauge relevance.
  3. Fig. 6 / §4.6: The negative Om(z) at z≲0.2 is attributed to sparse low-z CC data. Consider adding the proposed monotonicity penalty (mentioned in §5) as a controlled test, or at least showing Om(z) with the H0 prior enforced more tightly, so readers can see the artifact’s sensitivity.
  4. Notation and presentation: Eq. (1.1) uses f while the network outputs fσ8; the conversion f=fσ8/σ8,0 should be stated once in the equation block for clarity. Table 1 lists hyperparameters cleanly; consider adding the total parameter count (~8×10^5) there as well.
  5. Reproducibility: No code or trained weights are mentioned. For a methods-focused PINN paper, a public repository (architecture, loss, seeds, data tables) would substantially increase impact and should be linked if available.
  6. Ensure the abstract and body describe the same datasets and free-H0 result. Any mismatch between a DESI/Pantheon+/bootstrap abstract and the CC+BAO+RSD body would need to be resolved before acceptance.

Circularity Check

1 steps flagged · score 2.0 of 10

No load-bearing circular derivation; only minor by-construction H0 recovery when the prior is appended as a data point, which the paper itself treats as implementation validation rather than a prediction.

  1. fitted input called prediction [Abstract; §3.4; §4.3; Table 2; §5]
    "With either prior the Hubble constant is recovered exactly at the prior value, with an ensemble spread below 0.001 km s−1 Mpc−1 — the prior effectively pins the normalization. ... An H0 prior, when used, is incorporated as an additional data point at z=0 appended to the H(z) compilation with the prior central value and uncertainty."

    The SH0ES/H0dN central value is inserted into the H data loss as a z=0 point; minimizing L_data,H then forces H(z=0) to that value (spread <0.001). Reporting exact recovery is therefore by construction of the loss, not an independent prediction. The paper mostly frames this as implementation validation, so the circularity is minor and not load-bearing for the coupling/feasibility claim.

full rationale

The paper is a proof-of-concept methodology demonstration: a dual-head PINN is trained on external CC+BAO and RSD compilations with an external GR linear-growth residual (Eq. 1.1 / L_physics) as a regularizer. The reconstructions and null tests are outputs of that optimization, not algebraic rearrangements of the inputs. Fixed fiducials Ωm,0=0.3 and σ8,0=0.8, flat geometry, and Planck rd conversion of BAO points are stated assumptions/limitations (correctness and model-independence scope issues), not circular steps that make a claimed prediction equal its input by definition. Self-citations (chiefly Ref. [9] as the independent ANN comparison point, plus related prior ANN reconstructions) are used for benchmarking and context, not as uniqueness theorems that force the present result. The only mild by-construction element is that, once an H0 prior is appended as a z=0 data point, H(z=0) is recovered exactly at the prior value—an expected consequence of the data loss, which the paper presents as a check that the implementation responds correctly, not as an independent cosmological prediction. That does not elevate the score above 2. Central claims (feasibility of ODE coupling, robustness of fσ8 to the two local H0 anchors, systematic deficit relative to ΛCDM+GR under those anchors) remain non-tautological given the data and the external growth equation.

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

The central claim rests on GR growth dynamics with fixed matter and σ8 normalizations, flat geometry, Planck-calibrated BAO distances, and a hand-chosen physics weight λ. No new physical entities are postulated; the dual-head PINN is a computational construct. Free parameters and domain assumptions dominate the ledger; the data-driven heads do not remove the fiducial cosmology baked into the ODE and BAO conversion.

free parameters (5)
  • physics coupling weight λ = 0.1 (selected)
    Scalar trade-off between data fit and ODE residual; scanned over {0, 0.01, 0.1, 1.0} and fixed at 0.1 for primary runs (§4.2).
  • Ωm,0 in growth ODE = 0.3
    Hardcoded in Eq. (1.1) residual; not learned. Dominant systematic per §5.
  • σ8,0 converting fσ8 to f = 0.8
    Fixed conversion f = fσ8/σ8,0 in physics loss (§3.3).
  • H_fid softplus scale = 70 km s^{-1} Mpc^{-1}
    Output-layer scale for H head positivity and convergence (§3.2).
  • network architecture widths/depth = 3×512 backbone
    Backbone 3×512, heads 512→256→1; ~8e5 parameters chosen by design (Table 1).
assumptions (5)
  • domain assumption Linear growth equation of GR with smooth dark energy and flat geometry (Eq. 1.1) holds for the reconstructed pair.
    Physics loss is exactly the residual of this ODE; modified gravity is not accommodated (§3.3, §5).
  • domain assumption BAO H(z) points may be converted using Planck 2018 sound horizon rd = 147.09 Mpc.
    18 of 50 H(z) points inherit early-universe calibration (§3.1); limits model independence.
  • domain assumption No Alcock–Paczyński corrections needed for the 63-point fσ8 compilation.
    Explicit choice matching Ref. [9]; coherent ~5–10% bias possible when H0~73 (§5).
  • ad hoc to paper Normalized L1 (mean absolute) data losses are appropriate substitutes for χ².
    Chosen for robustness to low-σ BAO points; sensitivity not verified (§3.4).
  • standard math ELU activations and automatic differentiation yield a well-defined ODE residual at collocation points.
    Standard PINN practice; justified vs ReLU in §2–3.2.

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Pith. "Pith review of Joint reconstruction of $H(z)$ and $f\sigma_8(z)$ with physics informed neural networks." pith.science (2026). https://pith.science/paper/E7UHERZO

@misc{pith2026260617614,
  author       = {Pith},
  title        = {Pith review of: Joint reconstruction of $H(z)$ and $f\sigma_8(z)$ with physics informed neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E7UHERZO}},
  note         = {Machine review of arXiv:2606.17614}
}
abstract

We present a model-independent joint reconstruction of the Hubble parameter $H(z)$ and the growth rate $f\sigma_8(z)$ using a dual-head physics-informed neural network trained on four complementary late-universe datasets: Cosmic Chronometers, Redshift-Space Distortions, the DESI DR2 BAO mean vector and full covariance, and the Pantheon$+$SH0ES supernova compilation. The two output heads share a backbone and are coupled through the linear growth equation of general relativity, penalizing the ODE residual at 1000 collocation points per training step via automatic differentiation. Uncertainty is quantified by an ensemble of 100 networks, each trained on an independent parametric-bootstrap resample of the data and its own draw of the fiducial cosmological parameters from Planck 2018 priors, so that the ensemble spread captures data-noise, initialization, and fiducial-cosmology systematics simultaneously. The physics coupling weight $\lambda$ is selected via an L-curve analysis over six values; the curve is nearly flat in total data $\chi^2$, indicating that the joint dataset is intrinsically consistent with the growth ODE. Without any $H_0$ prior, the free reconstruction yields $H_0 = 69.0 \pm 4.7$\,km\,s$^{-1}$\,Mpc$^{-1}$, consistent with the Planck 2018 CMB value and with the DESI DR2 inverse distance-ladder determination, and approximately $0.9\sigma$ below the SH0ES local measurement. The reconstructed $H(z)$ lies systematically below the flat $\Lambda$CDM prediction at $z \sim 0.7$-$0.8$, consistent with the dark energy evolution suggested by DESI DR2. As conditional analyses, we also anchor $H_0$ to the SH0ES value $73.04 \pm 1.04$\,km\,s$^{-1}$\,Mpc$^{-1}$ and the Local Distance Network consensus $73.50 \pm 0.81$\,km\,s$^{-1}$\,Mpc$^{-1}$; both anchored reconstructions yield a suppressed $f\sigma_8$, illustrating the propagation of the $H_0$--$\sigma_8$ link through the ODE coupling.

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Works this paper leans on

47 extracted references

  1. [1]

    Riess et al

    Adam G. Riess et al. Cosmic Distances Calibrated to 1% Precision with Gaia EDR3 Parallaxes and Hubble Space Telescope Photometry of 75 Milky Way Cepheids Confirm Tension with ΛCDM.Astrophys. J. Lett., 908:L6, 2021. doi: 10.3847/2041-8213/abdbaf

    doi:10.3847/2041-8213/abdbaf 2021

  2. [2]

    The Local Distance Network: A community consensus report on the measurement of the Hubble constant at∼1% precision.Astron

    Stefano Casertano et al. The Local Distance Network: A community consensus report on the measurement of the Hubble constant at∼1% precision.Astron. Astrophys., 708:A166, 2026. doi: 10.1051/0004-6361/202557993

    doi:10.1051/0004-6361/202557993 2026

  3. [3]

    Aghanim et al

    N. Aghanim et al. Planck 2018 results. VI. Cosmological parameters.Astron. Astrophys., 641:A6, 2020. doi: 10.1051/0004-6361/201833910

    doi:10.1051/0004-6361/201833910 2018

  4. [4]

    KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints.Astron

    Catherine Heymans et al. KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints.Astron. Astrophys., 646:A140, 2021. doi: 10.1051/0004-6361/202039063

    doi:10.1051/0004-6361/202039063 2021

  5. [5]

    T. M. C. Abbott et al. Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing.Phys. Rev. D, 105:023520, 2022. doi: 10.1103/PhysRevD.105.023520

    doi:10.1103/physrevd.105.023520 2022

  6. [6]

    The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics.Phys

    Eleonora Di Valentino et al. The CosmoVerse White Paper: Addressing observational tensions in cosmology with systematics and fundamental physics.Phys. Dark Univ., 49: 101965, 2025. doi: 10.1016/j.dark.2025.101965

    doi:10.1016/j.dark.2025.101965 2025

  7. [7]

    Challenges for ΛCDM: An update.New Astron

    Leandros Perivolaropoulos and Foteini Skara. Challenges for ΛCDM: An update.New Astron. Rev., 95:101659, 2022. doi: 10.1016/j.newar.2022.101659

    doi:10.1016/j.newar.2022.101659 2022

  8. [8]

    Reconstruction of dark energy and expansion dynamics using Gaussian processes.JCAP, 06:036, 2012

    Marina Seikel, Chris Clarkson, and Mathew Smith. Reconstruction of dark energy and expansion dynamics using Gaussian processes.JCAP, 06:036, 2012. doi: 10.1088/1475-7516/2012/06/036

    doi:10.1088/1475-7516/2012/06/036 2012

Show all 47 references
  1. [9]

    Neural network reconstruction of late-time cosmology and null tests

    Konstantinos Dialektopoulos, Jackson Levi Said, Jurgen Mifsud, Joseph Sultana, and Kristian Zarb Adami. Neural network reconstruction of late-time cosmology and null tests. JCAP, 02(02):023, 2022. doi: 10.1088/1475-7516/2022/02/023. – 18 –

    2022 doi

  2. [10]

    Eric V. Linder. Cosmic growth history and expansion history.Phys. Rev. D, 72:043529,

  3. [11]

    doi: 10.1103/PhysRevD.72.043529

    doi

  4. [12]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.J

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.J. Comput. Phys., 378:686–707, 2019. doi: 10.1016/j.jcp.2018.10.045

    2019 doi

  5. [13]

    I. E. Lagaris, A. Likas, and D. I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations.IEEE Trans. Neural Netw., 9(5):987–1000, 1998. doi: 10.1109/72.712178

    1998 doi

  6. [14]

    Chantada, Susana J

    Augusto T. Chantada, Susana J. Landau, Pavlos Protopapas, Claudia G. Sc´ occola, and Cecilia Garraffo. Cosmology-informed neural networks to solve the background dynamics of the Universe.Phys. Rev. D, 107(6):063523, 2023. doi: 10.1103/PhysRevD.107.063523

    2023 doi

  7. [15]

    Chantada, Susana J

    Augusto T. Chantada, Susana J. Landau, Pavlos Protopapas, Claudia G. Sc´ occola, and Cecilia Garraffo. Faster Bayesian inference with neural network bundles and new results for f(R) models.Phys. Rev. D, 109(12):123514, 2024. doi: 10.1103/PhysRevD.109.123514

    2024 doi

  8. [16]

    Cosmo-PINN: A Physics-Informed Neural Network for Cosmological Reconstruction

    Andronikos Paliathanasis. Cosmo-PINN: A Physics-Informed Neural Network for Cosmological Reconstruction. 5 2026

    2026

  9. [17]

    Aluri, and David F

    Anshul Verma, Shashwat Sourav, Pavan K. Aluri, and David F. Mota. Cosmology-informed Neural Networks to infer dark energy equation-of-state. 8 2025

    2025

  10. [18]

    Inferring Cosmological Parameters with Evidential Physics-Informed Neural Networks.Universe, 11(12):403, 2025

    Hai Siong Tan. Inferring Cosmological Parameters with Evidential Physics-Informed Neural Networks.Universe, 11(12):403, 2025. doi: 10.3390/universe11120403

    2025 doi

  11. [19]

    Reconstructing the Type Ia Supernova Absolute Magnitude with Two-Probe Physics-Informed Neural Networks

    Denitsa Staicova. Reconstructing the Type Ia Supernova Absolute Magnitude with Two-Probe Physics-Informed Neural Networks. 3 2026. doi: 10.1016/j.dark.2026.102342

    2026 doi

  12. [20]

    M. P. Bento, H. B. Cˆ amara, and J. F. Seabra. Unraveling particle dark matter with Physics-Informed Neural Networks.Phys. Lett. B, 868:139690, 2025. doi: 10.1016/j.physletb.2025.139690

    2025 doi

  13. [21]

    SPINN: Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics Informed Neural Networks.Astrophys

    Ashutosh Kumar Mishra and Emma Tolley. SPINN: Advancing Cosmological Simulations of Fuzzy Dark Matter with Physics Informed Neural Networks.Astrophys. J., 988(1):114, 2025. doi: 10.3847/1538-4357/ade43e

    2025 doi

  14. [22]

    Solving the cosmological Vlasov–Poisson equations with physics-informed Kolmogorov–Arnold networks.Mon

    Nicolas Cerardi, Emma Tolley, and Ashutosh Mishra. Solving the cosmological Vlasov–Poisson equations with physics-informed Kolmogorov–Arnold networks.Mon. Not. Roy. Astron. Soc., 545(4):staf2241, 2026. doi: 10.1093/mnras/staf2241

    2026 doi

  15. [23]

    Physics-informed neural networks in the recreation of hydrodynamic simulations from dark matter.Mon

    Zhenyu Dai, Ben Moews, Ricardo Vilalta, and Romeel Dave. Physics-informed neural networks in the recreation of hydrodynamic simulations from dark matter.Mon. Not. Roy. Astron. Soc., 527(2):3381–3394, 2023. doi: 10.1093/mnras/stad3394

    2023 doi

  16. [24]

    Dialektopoulos, Purba Mukherjee, Jackson Levi Said, and Jurgen Mifsud

    Konstantinos F. Dialektopoulos, Purba Mukherjee, Jackson Levi Said, and Jurgen Mifsud. Neural network reconstruction of cosmology using the Pantheon compilation.Eur. Phys. J. C, 83(10):956, 2023. doi: 10.1140/epjc/s10052-023-12124-3

    2023 doi

  17. [25]

    Dialektopoulos, Purba Mukherjee, Jackson Levi Said, and Jurgen Mifsud

    Konstantinos F. Dialektopoulos, Purba Mukherjee, Jackson Levi Said, and Jurgen Mifsud. Neural network reconstruction of scalar-tensor cosmology.Phys. Dark Univ., 43:101383,

  18. [26]

    doi: 10.1016/j.dark.2023.101383

    2023 doi

  19. [27]

    Neural network reconstruction of – 19 – H’(z) and its application in teleparallel gravity.JCAP, 12:029, 2022

    Purba Mukherjee, Jackson Levi Said, and Jurgen Mifsud. Neural network reconstruction of – 19 – H’(z) and its application in teleparallel gravity.JCAP, 12:029, 2022. doi: 10.1088/1475-7516/2022/12/029

    2022 doi

  20. [28]

    Dialektopoulos, Jackson Levi Said, and Jurgen Mifsud

    Purba Mukherjee, Konstantinos F. Dialektopoulos, Jackson Levi Said, and Jurgen Mifsud. A possible late-time transition of M B inferred via neural networks.JCAP, 09:060, 2024. doi: 10.1088/1475-7516/2024/09/060

    2024 doi

  21. [29]

    Constraining cosmological parameters based on relative galaxy ages.Astrophys

    Raul Jimenez and Abraham Loeb. Constraining cosmological parameters based on relative galaxy ages.Astrophys. J., 573:37–42, 2002. doi: 10.1086/340549

    2002 doi

  22. [30]

    Improved constraints on the expansion rate of the Universe up to z∼1.1 from the spectroscopic evolution of cosmic chronometers.JCAP, 08:006, 2012

    Michele Moresco et al. Improved constraints on the expansion rate of the Universe up to z∼1.1 from the spectroscopic evolution of cosmic chronometers.JCAP, 08:006, 2012. doi: 10.1088/1475-7516/2012/08/006

    2012 doi

  23. [31]

    Raising the bar: new constraints on the Hubble parameter with cosmic chronometers atz∼2.Mon

    Michele Moresco. Raising the bar: new constraints on the Hubble parameter with cosmic chronometers atz∼2.Mon. Not. Roy. Astron. Soc., 450:L16–L20, 2015. doi: 10.1093/mnrasl/slv037

    2015 doi

  24. [32]

    A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration.JCAP, 05:014, 2016

    Michele Moresco et al. A 6% measurement of the Hubble parameter atz∼0.45: direct evidence of the epoch of cosmic re-acceleration.JCAP, 05:014, 2016. doi: 10.1088/1475-7516/2016/05/014

    2016 doi

  25. [33]

    Setting the Stage for Cosmic Chronometers in the Sloan Digital Sky Survey.Astrophys

    Michele Moresco et al. Setting the Stage for Cosmic Chronometers in the Sloan Digital Sky Survey.Astrophys. J., 898:82, 2020. doi: 10.3847/1538-4357/ab9eb0

    2020 doi

  26. [34]

    Toward a Better Understanding of Cosmic Chronometers: A New Measurement ofH(z) atz∼0.7.Astrophys

    Nicola Borghi, Michele Moresco, and Andrea Cimatti. Toward a Better Understanding of Cosmic Chronometers: A New Measurement ofH(z) atz∼0.7.Astrophys. J. Lett., 928:L4,

  27. [35]

    doi: 10.3847/2041-8213/ac3fb2

    doi

  28. [36]

    Clustering of luminous red galaxies

    Enrique Gazta˜ naga, Anna Cabr´ e, and Lam Hui. Clustering of luminous red galaxies. IV. Baryon acoustic peak in the line-of-sight direction and a direct measurement ofH(z).Mon. Not. Roy. Astron. Soc., 399:1663–1680, 2009. doi: 10.1111/j.1365-2966.2009.15405.x

    2009 doi

  29. [37]

    Blake et al

    C. Blake et al. The WiggleZ Dark Energy Survey: joint measurements of the expansion and growth history atz <1.Mon. Not. Roy. Astron. Soc., 425:405–414, 2012. doi: 10.1111/j.1365-2966.2012.21473.x

    2012 doi

  30. [38]

    The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample.Mon

    Shadab Alam et al. The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample.Mon. Not. Roy. Astron. Soc., 470:2617–2652, 2017. doi: 10.1093/mnras/stx721

    2017 doi

  31. [39]

    The Completed SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations with LyαForests.Astrophys

    H´ elion du Mas des Bourboux et al. The Completed SDSS-IV Extended Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations with LyαForests.Astrophys. J., 901: 153, 2020. doi: 10.3847/1538-4357/abb085

    2020 doi

  32. [40]

    Quasar-LyαForest Cross-Correlation from BOSS DR11: Baryon Acoustic Oscillations.JCAP, 05:027, 2014

    Andreu Font-Ribera et al. Quasar-LyαForest Cross-Correlation from BOSS DR11: Baryon Acoustic Oscillations.JCAP, 05:027, 2014. doi: 10.1088/1475-7516/2014/05/027

    2014 doi

  33. [41]

    A. G. Adame et al. DESI 2024 VI: cosmological constraints from the measurements of baryon acoustic oscillations.JCAP, 02:021, 2025. doi: 10.1088/1475-7516/2025/02/021

    2024 doi

  34. [42]

    Abdul Karim et al

    M. Abdul Karim et al. DESI DR2 results. II. Measurements of baryon acoustic oscillations and cosmological constraints.Phys. Rev. D, 112(8):083515, 2025. doi: 10.1103/tr6y-kpc6

    2025 doi

  35. [43]

    The Pantheon+ Analysis: Cosmological Constraints.Astrophys

    Dillon Brout et al. The Pantheon+ Analysis: Cosmological Constraints.Astrophys. J., 938 (2):110, 2022. doi: 10.3847/1538-4357/ac8e04. – 20 –

    2022 doi

  36. [44]

    Evolution of thef σ 8 tension with the Planck15/ΛCDM determination and implications for modified gravity theories.Phys

    Lavrentios Kazantzidis and Leandros Perivolaropoulos. Evolution of thef σ 8 tension with the Planck15/ΛCDM determination and implications for modified gravity theories.Phys. Rev. D, 97:103503, 2018. doi: 10.1103/PhysRevD.97.103503

    2018 doi

  37. [45]

    Kingma and Jimmy Ba

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. 2014

    2014

  38. [46]

    PyTorch: An imperative style, high-performance deep learning library

    Adam Paszke et al. PyTorch: An imperative style, high-performance deep learning library. In H. Wallach et al., editors,Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019

    2019

  39. [47]

    Starobinsky

    Varun Sahni, Arman Shafieloo, and Alexei A. Starobinsky. Two new diagnostics of dark energy.Phys. Rev. D, 78:103502, 2008. doi: 10.1103/PhysRevD.78.103502. – 21 –

    2008 doi

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