Pith. sign in

REVIEW 3 major objections 6 minor 32 references

The paper claims that a rational Padé approximant accurately reproduces the diffuse dispersion measure of fast radio bursts in flat ΛCDM and wCDM cosmologies, cutting computational cost by over an order of magnitude.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 18:59 UTC pith:LIEWNUVG

load-bearing objection Useful Padé approximation for DM_diff, but the 3.5% error claim is contradicted by the authors' own Table 2; needs fixing before publication. the 3 major comments →

arxiv 2512.02357 v3 pith:LIEWNUVG submitted 2025-12-02 astro-ph.HE astro-ph.CO

Pad\'e Approximants for cosmic Dispersion Measures

Marios Kalomenopoulos , Jiaming Zhuge This is my paper
classification astro-ph.HE astro-ph.CO
keywords Fast Radio Burstsdispersion measurePadé approximantsΛCDMwCDMdark energy equation of statecosmological parameter inferencenumerical integration
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 paper derives closed-form Padé approximants for the diffuse dispersion measure of fast radio bursts as a function of redshift, in both flat ΛCDM and wCDM universes. It claims that for redshifts 0.01–2, matter densities 0.2–1.0, and dark-energy equation-of-state parameters −3 to −0.5, the relative error stays below 3.5% compared with direct numerical integration. The approximant evaluates roughly 15–17 times faster in ΛCDM and about 2.5 times faster in wCDM, making it attractive for MCMC cosmological inference. Near the concordance cosmology the error is below 0.5%, so the approximation is essentially exact for current FRB data.

Core claim

The central discovery is a (3,3) Padé approximant for the integral F(a) = ∫₀ᵃ √Ωₘ / √(Ωₘ a'³ + (1−Ωₘ)a'⁶) da', expanded around a→0 (high redshift). This yields the closed-form expression DM_diff = DM_c_diff/√Ωₘ [Φ(x(0,Ωₘ)) − √(1+z) Φ(x(z,Ωₘ))] for flat ΛCDM, and a w-dependent analogue for wCDM. The approximant Φ is a ratio of cubic polynomials with explicitly tabulated coefficients. The paper's claim is that this single rational function reproduces the redshift integral well enough for FRB cosmology, replacing numerical quadrature with a few algebraic operations.

What carries the argument

Padé approximant: a ratio of polynomials fitted to the Taylor expansion of the dispersion-measure integral at high redshift (a→0), with coefficients given in Eqs. (3.6)–(3.12) for ΛCDM and Eqs. (4.5)–(4.17) for wCDM. The variable x=(1−Ωₘ)/Ωₘ·(1+z)⁻³ (or its wCDM generalisation) controls the transition between matter and dark-energy domination. The approximant does the work of the integral by producing a single rational function that is trivial to evaluate repeatedly.

Load-bearing premise

The paper's accuracy guarantee is an empirical observation on a sparse grid rather than a proven bound; its own Table 2 lists a case (z=0.01, Ωₘ=0.2, w=−0.5) with 4.93% error, which exceeds the claimed 3.5% ceiling.

What would settle it

Evaluate the approximant and the numerical integral on a dense grid covering the full stated range, especially near (z=0.01, Ωₘ=0.2, w=−0.5). If any point exceeds 3.5% relative error, the worst-case claim fails. A timing comparison in a realistic MCMC loop at matched accuracy tolerance would also settle the speed advantage.

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

If this is right

  • FRB cosmological pipelines can evaluate DM_diff millions of times inside MCMC chains without repeated numerical integration, making large-catalogue analyses practical.
  • Joint constraints on Ωₘ, w, and astrophysical parameters become computationally cheaper, potentially enabling higher-dimensional fits with current and upcoming FRB surveys.
  • Near the fiducial (Ωₘ, w) ≈ (0.31, −1), the error is below 0.5%, so the approximation introduces negligible bias in standard cosmological inference.
  • The formula can replace the exact hypergeometric-function solution in ΛCDM, which is about 3 times slower, with minimal loss of accuracy.

Where Pith is reading between the lines

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

  • Because the wCDM coefficients depend continuously on w, the approximant can be differentiated analytically, enabling fast Hessian-based forecasts and Fisher-matrix analyses without numerical derivatives.
  • The same Padé construction could apply to other line-of-sight cosmological integrals, such as angular diameter distance or volume elements, wherever the integrand has a similar a→0 expansion.
  • The stated 3.5% worst-case error bound is not a proven theorem; the paper's own Table 2 records a 4.93% error at (z=0.01, Ωₘ=0.2, w=−0.5), so the claimed ceiling should be re-verified with a denser grid before relying on it in the extreme corners.
  • A higher-order Padé approximant, or a piecewise form, could push the worst-case corner error below 1% if future FRB experiments demand better accuracy at low redshift and low Ωₘ.

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

3 major / 6 minor

Summary. The manuscript derives rational-function (Padé) approximants for the diffuse dispersion measure integral of fast radio bursts, for flat ΛCDM and flat wCDM cosmologies. The coefficients are obtained by Taylor-expanding the integrand at high redshift (a→0) and constructing (3,3) Padé approximants. The authors claim that for 0.01≤z≤2, 0.2≤Ωm≤1.0, and −3.0≤w≤−0.5 the relative error with respect to numerical quadrature is always below 3.5%, and that the closed forms are about 17 times (ΛCDM) and 2.5 times (wCDM) faster than numerical integration. The abstract additionally promises a simulated-data MCMC analysis demonstrating unbiased parameter recovery, and Appendix A compares the ΛCDM approximant with a hypergeometric closed form.

Significance. If the accuracy and speed claims held as stated, these formulas would offer a practical, low-cost substitute for numerical integration in FRB likelihood pipelines, particularly for MCMC analyses. A genuine strength is that the Padé coefficients are fixed by a Taylor expansion of the same integral, not fitted to mock data, and the validation is against an independent numerical evaluation; this is not circular. The algebraic derivation is standard and mostly transparent. However, the headline 'always below 3.5%' is internally falsified by the paper's own Table 2, which reports 4.93% inside the claimed parameter range. The abstract also advertises an MCMC analysis that does not appear in the submitted text. The approximation itself may be salvageable with a corrected error bound or a restricted parameter range, but the manuscript as it stands does not support its central advertised claims.

major comments (3)
  1. [§5, Table 2; Abstract; §6] The central accuracy claim is contradicted by the authors' own validation grid. Table 2 lists ΔE=4.93% for (w,Ωm,z)=(−0.5,0.2,0.01), all within the stated ranges 0.01≤z≤2, 0.2≤Ωm≤1.0, −3.0≤w≤−0.5. Table 1 also reports 3.51% at (Ωm,z)=(0.2,0.01), which is not 'smaller than 3.5%'. The abstract and Section 6 restate the 3.5% ceiling. The error bound must be relaxed (e.g., <5%), the parameter ranges restricted (e.g., Ωm≥0.3 or z≥0.02), or the claim reworded to 'on the tested grid' — the current wording is false.
  2. [Eq. (1.3) vs §3, Eqs. (3.6)–(3.12)] The sign convention in Eq. (1.3) is wrong. Eq. (1.3) gives Φ(x)=−2.0+2.856x+1.095x²+0.0913x³ over (1.0+1.3280x+0.4486x²+0.0277x³), but the coefficients quoted in §3 are b1=−2.85592665, b2=−1.0945641, b3=−0.0913347 — all negative. A reader implementing from Eq. (1.3) will obtain incorrect DM values. Correct the signs or replace Eq. (1.3) with the accurate rounded coefficients.
  3. [Abstract vs §5–§6] The abstract states: 'we perform a cosmological analysis of simulated FRB data and show that our approximation gives robust and unbiased results, even when applied in regions of parameter space where its relative error becomes larger than 3%.' No MCMC or parameter-recovery analysis appears anywhere in the manuscript (Sections 1–6 and Appendix A). This advertised result is missing from the text. It must either be added with full details (simulation setup, likelihood, priors, coverage checks) or removed from the abstract and any summary claims.
minor comments (6)
  1. [§6 vs Abstract and Table 3] Speedup numbers are inconsistent: the abstract says 'more than 15 (2) times faster', Table 3 gives ~17 and ~2.5, while §6 says 'more than 10 times faster for ΛCDM and more than 2 times faster for wCDM'. Harmonize these values and specify the evaluation conditions (hardware, quadrature tolerance, array sizes).
  2. [Eq. (3.5)] The typesetting '1p a(x) Φ(x)' is garbled; it should read (1/√a)Φ(x) or similar. Please fix this notation so the Padé form is unambiguous.
  3. [Table 3] The header 'ΛCDM (Num) wCDM (Num)' is confusing. Clarify that the entries are speedup factors Δt=t_Num/t_App and state the numerical integration tolerance/algorithm used for the comparison.
  4. [Figure 2 caption] The caption mentions a red dashed line (ΔE=1%) and a white dotted line (ΛCDM values); ensure these features are clearly visible in the printed figure and use distinguishable line styles/colors.
  5. [Software availability] The code is not provided; 'will become available together with our upcoming work' is not a firm availability statement. For reproducibility, provide a versioned repository or Zenodo DOI at submission.
  6. [Appendix A] When citing the hypergeometric closed form [31], please include the exact expression from that reference or verify that Eq. (A.1) is correctly transcribed, including the evaluation limits and the argument of ₂F₁.

Circularity Check

0 steps flagged

No circularity: Padé coefficients are fixed by a Taylor expansion of the same integral, with validation against an independent numerical evaluation.

full rationale

The central derivation in Sections 3 and 4 starts from the DM_diff integral (Eq. 2.3 / 4.1), defines F(a) or \tilde F(a), expands as a→0, and computes a (3,3) Padé approximant with coefficients (3.6)–(3.12) and (4.5)–(4.17). These coefficients come from the expansion of the integrand itself, not from fitting to DM data or to the numerical DM values used for validation. The accuracy test (Eq. 5.1) compares the approximant against a separate numerical evaluation of the same integral; this is an independent check of an approximation, not a circular reuse of the fitted quantity. The only self-citations are methodological references to [15] and [17] for the Padé technique and a forward reference [30] for code release; neither supplies the central result nor forbids alternatives. No uniqueness theorem is imported. The known hypergeometric solution [31] is acknowledged and used only as a benchmark, not renamed as new. The abstract's stated 'always smaller than 3.5%' appears contradicted by the paper's own Table 2 (ΔE=4.93% at (w,Ωm)=(-0.5,0.2), z=0.01), but that is a correctness/claim-validation issue, not circularity. Overall: the derivation is self-contained and the approximation is benchmarked against an independent numerical integral; circularity score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The approximation itself introduces no free parameters: all Padé coefficients are fixed by Taylor expansion of the same integral (Secs. 3–4). The load-bearing axioms are the standard flat cosmological model, the constancy of χ and f_diff, and the empirical validity of the Padé approximant over the stated grid. No new physical entities are introduced.

axioms (4)
  • domain assumption Flat ΛCDM/wCDM FLRW metric with Ωm+ΩΛ=1 (or Ωm+ΩDE=1) and dark energy equation of state p=wρc²
    Used throughout Eqs. (2.1)–(2.3) and (4.1); standard cosmological model from refs [18–20], not derived in this paper.
  • domain assumption Homogeneous distribution and ionization of baryons, with χ(z)=χ_{7/8}≈7/8 constant for z<3
    Invoked in Sec. 2 to take χ out of the integral; supported by refs [9,11,12,14]. If χ were not constant, the DM integral and approximation would need modification.
  • domain assumption f_diff is constant and equal to ~0.84, independent of redshift
    Sec. 2, from [13]; taken out of the integral. This is an astrophysical input, not derived.
  • ad hoc to paper The (3,3) Padé approximant from the a→0 Taylor expansion remains accurate down to z=0.01 across the stated parameter ranges
    This is the load-bearing validity assumption; tested on a grid but contradicted at (w,Ωm)=(-0.5,0.2), z=0.01 (Table 2).

pith-pipeline@v1.3.0-alltime-deepseek · 7971 in / 14802 out tokens · 135139 ms · 2026-08-03T18:59:33.078912+00:00 · methodology

0 comments
read the original abstract

Fast Radio Bursts (FRBs) have become an indispensable tool for studying the ``missing baryons'', the Universe's ionisation properties, as well as the cosmological parameters. This is achieved by analysing the diffuse dispersion measure (${\rm DM}_{\rm diff}$) of FRBs as a function of redshift. However, the rapidly increasing data size requests more and more computational resources. In this work, we first develop an accelerated method for any cosmic dispersion measure by deriving an analytical approximation formula for flat, $\Lambda$CDM and $w$CDM universes. Focusing on FRBs, we show that our approximation works well for the ranges $0.01 \leq z \leq 2$, $0.2 \leq \Omega_m \leq 1.0$ and $-3.0 \leq w \leq -0.5$, with relative error to a numerically evaluated ${\rm DM}_{\rm diff}$ always smaller than $3.5 \%$ (in the worst case scenario). This error remains below observationally relevant ${\rm DM}$ scatter and is especially small near the concordance $\Lambda$CDM cosmology. Additionally, we perform a cosmological analysis of simulated FRB data and show that our approximation gives robust and unbiased results, even when applied in regions of parameter space where its relative error becomes larger than $3\%$. Finally, the approximation is more than $15$ ($2$) times faster than the numerical solution of $\Lambda$CDM ($w$CDM), and can reach a timing improvement of a factor of $25$ when used in an MCMC cosmological inference.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 23 linked inside Pith

  1. [1]

    Lorimer, M

    D.R. Lorimer, M. Bailes, M.A. McLaughlin, D.J. Narkevic and F. Crawford,A Bright Millisecond Radio Burst of Extragalactic Origin,Science318(2007) 777 [0709.4301]

    Pith/arXiv arXiv 2007

  2. [2]

    Cordes and S

    J.M. Cordes and S. Chatterjee,Fast Radio Bursts: An Extragalactic Enigma, ARA&A57 (2019) 417 [1906.05878]

    Pith/arXiv arXiv 2019

  3. [3]

    Petroff, J.W.T

    E. Petroff, J.W.T. Hessels and D.R. Lorimer,Fast radio bursts, A&A Rev.27(2019) 4 [1904.07947]

    Pith/arXiv arXiv 2019

  4. [4]

    Petroff, J.W.T

    E. Petroff, J.W.T. Hessels and D.R. Lorimer,Fast radio bursts at the dawn of the 2020s, A&A Rev.30(2022) 2 [2107.10113]

    Pith/arXiv arXiv 2022

  5. [5]

    Wu and F.-Y

    Q. Wu and F.-Y. Wang,Statistical Properties and Cosmological Applications of Fast Radio Bursts,Chinese Physics Letters41(2024) 119801 [2409.13247]

    Pith/arXiv arXiv 2024

  6. [6]

    Glowacki and K.-G

    M. Glowacki and K.-G. Lee,Cosmology with Fast Radio Bursts,arXiv e-prints(2024) arXiv:2410.24072 [2410.24072]

    Pith/arXiv arXiv 2024

  7. [7]

    Zhang,Multiwavelength and Multimessenger Counterparts of Fast Radio Bursts,Annual Review of Nuclear and Particle Science74(2024) 89 [2410.02216]

    B. Zhang,Multiwavelength and Multimessenger Counterparts of Fast Radio Bursts,Annual Review of Nuclear and Particle Science74(2024) 89 [2410.02216]

    Pith/arXiv arXiv 2024

  8. [8]

    Thornton, B

    D. Thornton, B. Stappers, M. Bailes, B. Barsdell, S. Bates, N.D.R. Bhat et al.,A Population of Fast Radio Bursts at Cosmological Distances,Science341(2013) 53 [1307.1628]

    Pith/arXiv arXiv 2013

  9. [9]

    Deng and B

    W. Deng and B. Zhang,Cosmological Implications of Fast Radio Burst/Gamma-Ray Burst Associations, ApJ783(2014) L35 [1401.0059]

    Pith/arXiv arXiv 2014

  10. [10]

    Prochaska and Y

    J.X. Prochaska and Y. Zheng,Probing Galactic haloes with fast radio bursts, MNRAS485 (2019) 648 [1901.11051]

    Pith/arXiv arXiv 2019

  11. [11]

    Ioka,The Cosmic Dispersion Measure from Gamma-Ray Burst Afterglows: Probing the Reionization History and the Burst Environment, ApJ598(2003) L79 [astro-ph/0309200]

    K. Ioka,The Cosmic Dispersion Measure from Gamma-Ray Burst Afterglows: Probing the Reionization History and the Burst Environment, ApJ598(2003) L79 [astro-ph/0309200]

    Pith/arXiv arXiv 2003

  12. [12]

    Inoue,Probing the cosmic reionization history and local environment of gamma-ray bursts through radio dispersion, MNRAS348(2004) 999 [astro-ph/0309364]

    S. Inoue,Probing the cosmic reionization history and local environment of gamma-ray bursts through radio dispersion, MNRAS348(2004) 999 [astro-ph/0309364]

    Pith/arXiv arXiv 2004

  13. [13]

    Z. Li, H. Gao, J.J. Wei, Y.P. Yang, B. Zhang and Z.H. Zhu,Cosmology-insensitive estimate of IGM baryon mass fraction from five localized fast radio bursts, MNRAS496(2020) L28 [2004.08393]

    Pith/arXiv arXiv 2020

  14. [14]

    Fukugita, C.J

    M. Fukugita, C.J. Hogan and P.J.E. Peebles,The Cosmic Baryon Budget, ApJ503(1998) 518 [astro-ph/9712020]

    Pith/arXiv arXiv 1998

  15. [15]

    Adachi and M

    M. Adachi and M. Kasai,An Analytical Approximation of the Luminosity Distance in Flat Cosmologies with a Cosmological Constant,Progress of Theoretical Physics127(2012) 145 [1111.6396]

    Pith/arXiv arXiv 2012

  16. [16]

    Bender and A.S

    M.C. Bender and A.S. Orszag,Advanced mathematical methods for scientists and engineers I: asymptotic methods and perturbation theory., Springer-Verlag (1999)

    1999

  17. [17]

    Wei, X.-P

    H. Wei, X.-P. Yan and Y.-N. Zhou,Cosmological applications of Pad´ e approximant, J. Cosmology Astropart. Phys.2014(2014) 045 [1312.1117]

    Pith/arXiv arXiv 2014

  18. [18]

    Hobson, G.P

    M.P. Hobson, G.P. Efstathiou and A.N. Lasenby,General Relativity, Cambridge University Press (2006), 10.2277/0521829518

    doi:10.2277/0521829518 2006

  19. [19]

    Linder,Exploring the Expansion History of the Universe, Phys

    E.V. Linder,Exploring the Expansion History of the Universe, Phys. Rev. Lett.90(2003) 091301 [astro-ph/0208512]

    Pith/arXiv arXiv 2003

  20. [20]

    Frieman, M.S

    J.A. Frieman, M.S. Turner and D. Huterer,Dark energy and the accelerating universe., ARA&A46(2008) 385 [0803.0982]. – 10 –

    Pith/arXiv arXiv 2008

  21. [21]

    Escamilla, W

    L.A. Escamilla, W. Giar` e, E.D. Valentino, R.C. Nunes and S. Vagnozzi,The state of the dark energy equation of state circa 2023, J. Cosmology Astropart. Phys.2024(2024) 091 [2307.14802]

    Pith/arXiv arXiv 2023

  22. [22]

    Aghanim, Y

    N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al.,Planck 2018 results. VI. Cosmological parameters, A&A641(2020) A6

    2018

  23. [23]

    Macquart, J.X

    J.-P. Macquart, J.X. Prochaska, M. McQuinn, K.W. Bannister, S. Bhandari, C.K. Day et al., A census of baryons in the Universe from localized fast radio bursts, Nature581(2020) 391 [2005.13161]

    Pith/arXiv arXiv 2020

  24. [24]

    Robitaille, E.J

    Astropy Collaboration, T.P. Robitaille, E.J. Tollerud, P. Greenfield, M. Droettboom, E. Bray et al.,Astropy: A community Python package for astronomy, A&A558(2013) A33 [1307.6212]

    Pith/arXiv arXiv 2013

  25. [25]

    Price-Whelan, B.M

    Astropy Collaboration, A.M. Price-Whelan, B.M. Sip˝ ocz, H.M. G¨ unther, P.L. Lim, S.M. Crawford et al.,The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package, AJ156(2018) 123 [1801.02634]

    Pith/arXiv arXiv 2018

  26. [26]

    Harris, K.J

    C.R. Harris, K.J. Millman, S.J. van der Walt, R. Gommers, P. Virtanen, D. Cournapeau et al., Array programming with NumPy,Nature585(2020) 357

    2020

  27. [27]

    Virtanen, R

    P. Virtanen, R. Gommers, T.E. Oliphant, M. Haberland, T. Reddy, D. Cournapeau et al., SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,Nature Methods17 (2020) 261

    2020

  28. [28]

    Hunter,Matplotlib: A 2d graphics environment,Computing in Science & Engineering9 (2007) 90

    J.D. Hunter,Matplotlib: A 2d graphics environment,Computing in Science & Engineering9 (2007) 90

    2007

  29. [29]

    Meurer, C.P

    A. Meurer, C.P. Smith, M. Paprocki, O. ˇCert ´ ık, S.B. Kirpichev, M. Rocklin et al.,Sympy: symbolic computing in python,PeerJ Computer Science3(2017) e103

    2017

  30. [30]

    Zhuge, M

    J. Zhuge, M. Kalomenopoulos, C.J. Haster and B. Zhang,Cosmology with FRBs and GWs associations without redshift information,The Astrophysical Journal (To be submitted)(2025)

    2025

  31. [31]

    Jahns-Schindler and L.G

    J.N. Jahns-Schindler and L.G. Spitler,Breaking the Baryon Density-Hubble Constant Degeneracy in Fast Radio Burst Applications with Associated Gravitational Waves,arXiv e-prints(2025) arXiv:2508.14434 [2508.14434]

    Pith/arXiv arXiv 2025

  32. [32]

    Abramowitz and I.A

    M. Abramowitz and I.A. Stegun,Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover (1965). 11

    1965