Pith. sign in

REVIEW 4 major objections 5 minor 38 references

Computing Nonlinear Power Spectra Across Dynamical Dark Energy Model Space with Neural ODEs

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that a neural ODE trained only on synthetic LambdaCDM power spectra can compute nonlinear power spectra for any dynamical dark energy model parameterised by $w(z)$, to within 4% accuracy up to $k=5\,h\,\mathrm{Mpc}^{-1}$.

desk verdict A plausible neural-ODE emulator for nonlinear P(k) in w(z)CDM, but the 'any w(z)' headline outruns the validation; still deserves refereeing. read the letter →

arxiv 2506.09128 v2 pith:7WEXMF6Z submitted 2025-06-10 astro-ph.CO

Peter L. Taylor This is my paper
classification astro-ph.CO
keywords neuralODEnonlinearmatterpowerspectrumdynamicaldarkenergyw(z)CDMcosmologicalemulationstructuregrowthgeneralisation
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 proposes that the redshift evolution of any summary statistic of the matter density field satisfies a first-order ordinary differential equation whose forcing depends only on the current summary, the Hubble rate, the mean matter density, and redshift. A neural network learns this forcing from LambdaCDM training spectra, and the resulting neural ODE is then numerically integrated to predict nonlinear power spectra for arbitrary smooth dark energy histories. The central accuracy claim is that this generalises to $w(z)$CDM within 4% up to $k=5\,h\,\mathrm{Mpc}^{-1}$, without running new simulations for each model. This would let survey analyses compute nonlinear growth predictions across the full dynamical dark energy model space, and the method is argued to extend beyond the power spectrum to other growth-sensitive statistics.

What carries the argument

The load-bearing object is the closure ansatz for summary statistics: $\mathrm{d}s/\mathrm{d}z = f(s(z), H(z), \ln\bar{\rho}_m(z), z)$, which turns power-spectrum prediction into an initial-value problem. Once the forcing function $f$ is approximated by a neural network trained on LambdaCDM data, the same network is integrated from high redshift, where $w(z)$CDM and LambdaCDM are indistinguishable, down to $z=0$. A second load-bearing piece is the matching argument that any point in $w(z)$CDM parameter space has a corresponding LambdaCDM point with approximately equal power spectrum, Hubble rate, and mean matter density, so the network never leaves the region it was trained on.

What would settle it

Run an N-body simulation of a smooth $w(z)$ model far from the training prior, such as $w \approx -1.3$ at $z=1.5$, and compare the $z=0$ nonlinear power spectrum with the neural ODE prediction; if the discrepancy exceeds 4% on any scale up to $k=5\,h\,\mathrm{Mpc}^{-1}$, the generalisation claim fails. A faster test is to feed the network a $w(z)$ that oscillates on a timescale shorter than the training redshift sampling, which the paper itself predicts should break the model.

Watch

Extended reading notes

Core claim

The paper's central claim is that the instantaneous change in the nonlinear matter power spectrum, $\mathrm{d}P/\mathrm{d}z$, is a function only of the current power spectrum $P(z)$, the Hubble rate $H(z)$, the mean matter density $\bar{\rho}_m(z)$, and redshift. Training a neural network to approximate this forcing function on synthetic LambdaCDM data yields a neural ODE that can be integrated from $z=1.5$ to $z=0$ to predict power spectra for any dynamical dark energy model. The model generalises because, to first order, any $w(z)$CDM state can be mapped onto a LambdaCDM state with matching $P$, $H$, and $\bar{\rho}_m$ by rescaling $\sigma_8$, $\Omega_m$, and $H_0$. The paper validates the claim on $w_0$-$w_a$ models and on a custom equation of state resembling current BAO reconstructions, finding errors within 4% up to $k=5\,h\,\mathrm{Mpc}^{-1}$.

Load-bearing premise

The instantaneous evolution of the nonlinear power spectrum is fully determined by the current power spectrum, the Hubble rate, and the mean matter density, even though the power spectrum is not a sufficient statistic for the full nonlinear density field.

Editorial extensions

If this is right

  • Nonlinear power spectra for any smooth $w(z)$ model can be produced in seconds per cosmology without running new N-body simulations.
  • The reported accuracy (3% in LambdaCDM, 4% in $w_0$-$w_a$CDM) is below current baryonic feedback modelling uncertainties, so the method is positioned as survey-ready modulo feedback corrections.
  • Because the forcing function does not depend on cosmology, the same trained network could be extended to non-Gaussian summary statistics, provided suitable training data exist.
  • The requirement of broad LambdaCDM priors implies that simulation suites should be run over wider parameter ranges than current data prefer, so that the matching argument remains valid.

Reading between the lines

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

  • An implication left implicit is that residuals of the neural ODE could be used as a diagnostic of how non-sufficient the power spectrum is as a state variable: if the closure assumption fails, the error should trace the missing higher-order information.
  • A testable extension would be to map the volume of $w(z)$ space whose triples $(P, H, \bar{\rho}_m)$ lie inside the LambdaCDM training hull; models outside that hull are predicted to fail, which would sharpen the paper's stated failure mode.
  • One could append additional statistics, such as the bispectrum or redshift-space multipoles, to the state vector; the architecture would remain unchanged as long as training data for those statistics are available.
  • Starting the integration from a higher redshift where linear theory is reliable, rather than from emulator outputs at $z=1.5$, would remove a potential source of systematic bias; the paper notes this as future work.
Share X Bluesky LinkedIn Reddit HN

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 proposes a neural ODE approach for computing nonlinear matter power spectra in dynamical dark energy models. The author trains a neural network on synthetic ΛCDM power spectra from the BACCO emulator to model the redshift evolution dP/dz = f(P, H, ln ρ̄m, z), then integrates the learned ODE from z=1.5 to z=0. The central claim is that the model, trained only on ΛCDM data, generalizes to any smooth w(z)CDM model with 4% accuracy up to k=5 h/Mpc. Validation is presented for held-out ΛCDM cosmologies, for w0waCDM cosmologies drawn from a BACCO prior, and for one custom w_c(z) equation of state, with the last checked against linear theory only on large scales.

Significance. If the central generalization claim holds, the method would offer a practical way to generate nonlinear power spectra for arbitrary dark energy histories without running new N-body simulations, and the neural-ODE framework naturally extends to other summary statistics. The paper is clearly written, the ΛCDM and w0waCDM tests use held-out cosmologies and report residuals that are small and consistent with the claimed accuracy, and the computational cost is modest. The author also correctly identifies the closure approximation in Eq. (5) and the matching argument in Sec. II.D as the key assumptions, which is a strength in framing. However, the manuscript's headline claim is stronger than the empirical evidence presented, for the reasons detailed below.

major comments (4)
  1. [Sec. III.C and Table I] The printed w0wa prior is 'w_a ∈ [−0.3,−0.3]', which is degenerate in w_a and cannot support the statement that the model generalises to w0waCDM. If this is a typo for [−0.3,0.3], the text should say so explicitly; if it is literal, the validation does not vary w_a at all and the 4% claim applies only to w0 = −1.15 to −0.85 at fixed w_a = −0.3. Please correct the prior and, if possible, extend the validation to a non-degenerate w_a range.
  2. [Sec. III.D and Fig. 4] The custom w_c(z) test is validated only against linear theory below k=0.07 h/Mpc. Agreement on linear scales cannot test the nonlinear small-scale prediction, which is exactly what the headline 4%-up-to-k=5 claim concerns. Without a nonlinear ground truth for a non-CPL w(z), the 'any w(z)' generalization is an extrapolation of the untested Eq. (5) closure ansatz. I recommend either running (or obtaining) an N-body simulation for the w_c(z) model and comparing on nonlinear scales, or clearly restricting the claim to CPL models and presenting the w_c(z) case as a proof-of-concept only.
  3. [Footnote 10] Footnote 10 states that for w0waCDM the initial power spectrum at z=1.5 is taken from BACCO's w0wa output rather than from a ΛCDM spectrum at higher redshift. The paper's advertised pipeline is to start from ΛCDM at high z, where w(z)CDM and ΛCDM are indistinguishable, and integrate forward. This end-to-end pipeline is therefore not exercised in the w0wa test. The distinction matters because the matching argument in Sec. II.D depends on the initial state being a ΛCDM state; using the true w0wa initial condition bypasses any approximation in the initial-state mapping. Please clarify whether the reported residuals include the initial-condition approximation, and if not, provide a test that does include it.
  4. [Eq. (5) and Sec. II.D] The central generalization argument relies on two load-bearing premises: (i) the closure in Eq. (5) that (P, H, ρ̄m, z) is sufficient for the instantaneous evolution of P(k), and (ii) the Sec. II.D matching claim that every w(z)CDM state corresponds to a ΛCDM point in the training prior. The paper notes in Sec. III.B that the power spectrum is not a sufficient statistic, so the closure is an approximation. The paper also notes in Sec. II.D that rapid oscillations or non-differentiable w(z) will cause failure. These caveats are appropriate, but the abstract and Sec. III.C-D present the 4% 'any w(z)' result without these qualifications. I ask the author to state the precise conditions under which the 4% claim is asserted and to adjust the abstract and conclusion accordingly.
minor comments (5)
  1. [Sec. II.A] In the sentence after Eq. (4), 'Ω_DE is the dark matter density' should read 'dark energy density'.
  2. [Sec. I] The sentence 'In is difficult, albeit not impossible, to extend ReACT...' contains a typo: 'In is' should be 'It is'.
  3. [Table I] The w_a range '[-0.3,-0.3]' is almost certainly a typo; please confirm the intended range and also specify whether w0waCDM uses the same σ8, Ωm, Ωb, ns, h ranges as ΛCDM.
  4. [Sec. III.D] The validation paragraph reads 'To validate the model, the linear theory prediction down to scales of 0.07hMpc−1. As expected, the model is accurate to within 4% over the entire range.' The first clause lacks a verb; please rephrase, e.g., 'To validate the model, I compare to the linear theory prediction down to scales of 0.07 h/Mpc.'
  5. [Sec. III.A] The text says 'Derivatives are computed using a two-sided finite difference with Δz=0.001 at 20 equally-spaced redshifts over this range.' It would be helpful to state how the training data are generated from BACCO at those redshifts (e.g., whether the emulator is evaluated at each redshift and how the finite difference is applied to the emulated spectra).

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the neural ODE is trained on external LambdaCDM emulator output and validated on held-out w0wa and custom w(z) predictions; the generalization claim rests on a heuristic argument, not on recycling fitted inputs.

full rationale

The paper's derivation chain is: (i) assume the Eq. (5) closure dP/dz = f(P, H, ln rho_m, z); (ii) train f on 1.2e5 LambdaCDM BACCO spectra with finite-difference derivatives; (iii) integrate the trained ODE from z=1.5 to z=0 for held-out LambdaCDM, w0waCDM, and a custom w_c(z); (iv) compare predictions against BACCO (for LambdaCDM and w0wa) or linear theory (for w_c). No fitted parameter is later renamed as a prediction: the network weights are fixed on LambdaCDM data, and the w0wa validation uses BACCO's true initial power spectrum at z=1.5 (footnote 10) as the IVP input, not a quantity inferred from the network. The generalizability argument in Sec. II.D is heuristic, but asserting an unproven interpolation argument is a correctness gap, not circularity. There is no uniqueness theorem or load-bearing self-citation; the only self-citation, ref. [31], appears in a passing future-work sentence and carries no argumentative weight. The Eq. (5) ansatz is explicitly flagged as an approximation in Sec. III.B ('the power spectrum is not a sufficient statistic'), so the closure assumption is acknowledged rather than smuggled in. No circular step can be exhibited from the paper's equations or citations.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a learned closure relation, a generalization heuristic, and the accuracy of the BACCO emulator used as ground truth. No new physical entities are introduced.

free parameters (3)
  • Neural network weights (10 networks x 5 layers x 512 neurons) = Unknown; optimized with Adam, not reported
    All predictive power comes from these weights fitted to 120,000 BACCO LambdaCDM spectra; no analytic derivation is provided.
  • Network hyperparameters (depth, width, learning rate, batch size, early stopping) = 5 x 512, lr=1e-3, batch 2048, patience 20
    Chosen by hand; the claimed accuracy depends on this configuration.
  • Input standardization statistics for H(z) and ln rho_m(z) = Mean and standard deviation over the training set
    Used to normalize inputs before training; fitted to the LambdaCDM training distribution.
assumptions (6)
  • domain assumption The evolution of a summary statistic s(z) is described by a first-order ODE ds/dz = f(s, H, rho_m, z) (Eq. 5).
    Load-bearing ansatz; the author admits in Sec. III.B that the power spectrum is not a sufficient statistic, so this closure is approximate.
  • domain assumption For any w(z)CDM model, there exists a point in the LambdaCDM training prior with approximately matching P, H and rho_m at each redshift (Sec. II.D).
    Used to justify generalization from LambdaCDM-only training; argued by analogy with linear theory, not proven or bounded.
  • domain assumption The BACCO emulator provides accurate ground-truth nonlinear power spectra over the prior.
    Both training and validation use BACCO outputs; any emulator error is inherited by the neural ODE claims.
  • domain assumption Initial conditions at z=1.5 can be obtained from an external source (here BACCO) for the target dark energy model.
    For arbitrary w(z), the paper does not specify how to generate these initial conditions beyond 'provided by BACCO'; BACCO is restricted to z<1.5 and to its w0wa parameterization.
  • domain assumption w(z) is smooth and sampled finely enough relative to the 20 redshift nodes.
    The paper states the model fails for rapid oscillations or non-differentiable w(z), so the 'any w(z)' claim is restricted to smooth histories.
  • standard math Standard neural network training and numerical ODE integration are reliable for this problem.
    Relies on universal approximation and standard optimizers; no formal guarantees are provided.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Computing Nonlinear Power Spectra Across Dynamical Dark Energy Model Space with Neural ODEs." pith.science (2026). https://pith.science/paper/7WEXMF6Z

@misc{pith2026250609128,
  author       = {Pith},
  title        = {Pith review of: Computing Nonlinear Power Spectra Across Dynamical Dark Energy Model Space with Neural ODEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7WEXMF6Z}},
  note         = {Machine review of arXiv:2506.09128}
}
abstract

I show how to compute the nonlinear power spectrum across the entire $w(z)$ dynamical dark energy model space. Using synthetic $\Lambda$CDM data, I train a neural ordinary differential equation (ODE) to infer the evolution of the nonlinear matter power spectrum as a function of the background expansion and mean matter density across $\sim$$9 {\rm \ Gyr}$ of cosmic evolution. After training, the model generalises to {\it any} dynamical dark energy model parameterised by $w(z)$. With little optimisation, the neural ODE is accurate to within $4\%$ up to k = $5 \ h {\rm Mpc}^{-1}$. Unlike simulation rescaling methods, neural ODEs naturally extend to summary statistics beyond the power spectrum that are sensitive to the growth history.

Figures

Figures reproduced from arXiv: 2506.09128 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. FIG. 4. The nonlinear power spectrum found using the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. FIG. 3. A custom dark energy equation of state not de [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

38 extracted references · 8 canonical work pages

  1. [1]

    A. G. Adameet al.(DESI), JCAP02, 021 (2025), arXiv:2404.03002 [astro-ph.CO]

    arXiv 2025

  2. [2]

    Abdul Karimet al.(DESI), (2025), arXiv:2503.14738 [astro-ph.CO]

    M. Abdul Karimet al.(DESI), (2025), arXiv:2503.14738 [astro-ph.CO]

    arXiv 2025

  3. [3]

    Scolnicet al., Astrophys

    D. Scolnicet al., Astrophys. J.938, 113 (2022), arXiv:2112.03863 [astro-ph.CO]

    arXiv 2022

  4. [4]

    Rubinet al., (2023), arXiv:2311.12098 [astro- ph.CO]

    D. Rubinet al., (2023), arXiv:2311.12098 [astro- ph.CO]

    arXiv 2023

  5. [5]

    T. M. C. Abbottet al.(DES), Astrophys. J. Lett.973, L14 (2024), arXiv:2401.02929 [astro-ph.CO]

    arXiv 2024

  6. [6]

    T. M. C. Abbottet al.(DES), (2025), arXiv:2503.06712 [astro-ph.CO]

    arXiv 2025

  7. [7]

    E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D15, 1753 (2006), arXiv:hep-th/0603057

    arXiv 2006

  8. [8]

    Chevallier and D

    M. Chevallier and D. Polarski, Int. J. Mod. Phys. D10, 213 (2001), arXiv:gr-qc/0009008

    arXiv 2001

Show all 38 references
  1. [9]

    E. V. Linder, Phys. Rev. Lett.90, 091301 (2003), arXiv:astro-ph/0208512

    2003 arXiv

  2. [10]

    Holsclawet al., Physical Review D—Particles, Fields, Gravitation, and Cosmology84, 083501 (2011)

    T. Holsclawet al., Physical Review D—Particles, Fields, Gravitation, and Cosmology84, 083501 (2011)

    2011

  3. [11]

    Lodhaet al.(DESI), (2025), arXiv:2503.14743 [astro-ph.CO]

    K. Lodhaet al.(DESI), (2025), arXiv:2503.14743 [astro-ph.CO]

    2025 arXiv

  4. [12]

    Zhao and X.-m

    G.-B. Zhao and X.-m. Zhang, Phys. Rev. D81, 043518 (2010), arXiv:0908.1568 [astro-ph.CO]

    2010 arXiv

  5. [13]

    Zhaoet al., Nature Astron.1, 627 (2017), arXiv:1701.08165 [astro-ph.CO]

    G.-B. Zhaoet al., Nature Astron.1, 627 (2017), arXiv:1701.08165 [astro-ph.CO]

    2017 arXiv

  6. [14]

    Lodhaet al.(DESI), Phys

    K. Lodhaet al.(DESI), Phys. Rev. D111, 023532 (2025), arXiv:2405.13588 [astro-ph.CO]

    2025 arXiv

  7. [15]

    Mellieret al.(Euclid), (2024), arXiv:2405.13491 [astro-ph.CO]

    Y. Mellieret al.(Euclid), (2024), arXiv:2405.13491 [astro-ph.CO]

    2024

  8. [16]

    Blanchardet al.(Euclid), Astron

    A. Blanchardet al.(Euclid), Astron. Astrophys.642, A191 (2020), arXiv:1910.09273 [astro-ph.CO]

    2020 arXiv

  9. [17]

    Eifleret al., Mon

    T. Eifleret al., Mon. Not. Roy. Astron. Soc.507, 1514 (2021), arXiv:2004.04702 [astro-ph.CO]

    2021 arXiv

  10. [18]

    J. Xu, T. Eifler, E. Huff, P. R. S., H.-J. Huang, S. Ev- erett, and E. Krause, Mon. Not. Roy. Astron. Soc.519, 2535 (2022), arXiv:2201.00739 [astro-ph.CO]

    2022 arXiv

  11. [19]

    Abateet al.(LSST Dark Energy Science), (2012), arXiv:1211.0310 [astro-ph.CO]

    A. Abateet al.(LSST Dark Energy Science), (2012), arXiv:1211.0310 [astro-ph.CO]

    2012 arXiv

  12. [20]

    Villaescusa-Navarroet al., Astrophys

    F. Villaescusa-Navarroet al., Astrophys. J. Suppl.250, 2 (2020), arXiv:1909.05273 [astro-ph.CO]

    2020 arXiv

  13. [21]

    Knabenhanset al.(Euclid), Mon

    M. Knabenhanset al.(Euclid), Mon. Not. Roy. Astron. Soc.505, 2840 (2021), arXiv:2010.11288 [astro-ph.CO]

    2021 arXiv

  14. [22]

    Knabenhanset al.(Euclid), Mon

    M. Knabenhanset al.(Euclid), Mon. Not. Roy. Astron. Soc.484, 5509 (2019), arXiv:1809.04695 [astro-ph.CO]

    2019 arXiv

  15. [23]

    DeRose, R

    J. DeRose, R. H. Wechsler, J. L. Tinker, M. R. Becker, Y.-Y. Mao, T. McClintock, S. McLaughlin, E. Rozo, and Z. Zhai, Astrophys. J.875, 69 (2019), arXiv:1804.05865 [astro-ph.CO]

    2019 arXiv

  16. [24]

    Cataneo, L

    M. Cataneo, L. Lombriser, C. Heymans, A. Mead, A. Barreira, S. Bose, and B. Li, Mon. Not. Roy. Astron. Soc.488, 2121 (2019), arXiv:1812.05594 [astro-ph.CO]

    2019 arXiv

  17. [25]

    R. E. Angulo, M. Zennaro, S. Contreras, G. Aric` o, M. Pellejero-Iba˜ nez, and J. St¨ ucker, Mon. Not. Roy. Astron. Soc.507, 5869 (2021), arXiv:2004.06245 [astro- ph.CO]

    2021 arXiv

  18. [26]

    Contreras, R

    S. Contreras, R. E. Angulo, M. Zennaro, G. Aric` o, and M. Pellejero-Iba˜ nez, Mon. Not. Roy. Astron. Soc.499, 4905 (2020), arXiv:2001.03176 [astro-ph.CO]

    2020 arXiv

  19. [27]

    Fluri, T

    J. Fluri, T. Kacprzak, A. Lucchi, A. Schneider, A. Re- fregier, and T. Hofmann, Phys. Rev. D105, 083518 (2022), arXiv:2201.07771 [astro-ph.CO]

    2022 arXiv

  20. [28]

    R. T. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, Advances in neural information processing systems31(2018)

    2018

  21. [29]

    Lanzieri, F

    D. Lanzieri, F. Lanusse, and J.-L. Starck, in39th Inter- national Conference on Machine Learning Conference (2022) arXiv:2207.05509 [astro-ph.CO]

    2022 arXiv

  22. [30]

    Mousset, E

    L. Mousset, E. Allys, M. A. Price, J. Aumont, J.-M. Delouis, L. Montier, and J. D. McEwen, Astronomy & Astrophysics691, A269 (2024)

    2024

  23. [31]

    P. L. Taylor, T. D. Kitching, J. Alsing, B. D. Wandelt, S. M. Feeney, and J. D. McEwen, Phys. Rev. D100, 023519 (2019), arXiv:1904.05364 [astro-ph.CO]

    2019 arXiv

  24. [32]

    Jeffreyet al.(DES), Mon

    N. Jeffreyet al.(DES), Mon. Not. Roy. Astron. Soc. 536, 1303 (2024), arXiv:2403.02314 [astro-ph.CO]

    2024 arXiv

  25. [33]

    The DeepMind JAX Ecosystem,

    DeepMind, I. Babuschkin,et al., “The DeepMind JAX Ecosystem,” (2020)

    2020

  26. [34]

    D. P. Kingma, arXiv preprint arXiv:1412.6980 (2014)

    2014 arXiv

  27. [35]

    Kidger,On Neural Differential Equations, Ph.D

    P. Kidger,On Neural Differential Equations, Ph.D. the- sis, University of Oxford (2021)

    2021

  28. [36]

    Tsitouras, Computers & Mathematics with Applica- tions62, 770 (2011)

    C. Tsitouras, Computers & Mathematics with Applica- tions62, 770 (2011)

    2011

  29. [37]

    Schneider, R

    A. Schneider, R. Teyssier, J. Stadel, N. E. Chisari, A. M. C. Le Brun, A. Amara, and A. Refregier, JCAP 03, 020 (2019), arXiv:1810.08629 [astro-ph.CO]

    2019 arXiv

  30. [38]

    Huang, T

    H.-J. Huang, T. Eifler, R. Mandelbaum, and S. Do- delson, Mon. Not. Roy. Astron. Soc.488, 1652 (2019), arXiv:1809.01146 [astro-ph.CO]

    2019 arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.