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REVIEW 3 major objections 6 minor 34 references

Cosmography of CMB temperature measurements constrains violations of T(z)=T0(1+z) to the percent level, and a single extra parameter is enough.

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2026-07-30 19:48 UTC pith:ENYSRCB5

load-bearing objection Clean first cosmographic pass at T(z) that delivers a usable percent-level bound and a solid methodological warning; the quantitative claim is still hostage to untested SZ catalog systematics. the 3 major comments →

arxiv 2607.23535 v1 pith:ENYSRCB5 submitted 2026-07-26 astro-ph.CO gr-qchep-ph

Cosmic microwave background temperature cosmography

C. J. A. P. Martins This is my paper
classification astro-ph.CO gr-qchep-ph
keywords cosmic microwave backgroundtemperature-redshift relationcosmographySunyaev-Zeldovich effectphoton number non-conservationPadé approximantsLima model
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 standard rule that the cosmic microwave background cools strictly as T(z)=T0(1+z) is a pillar of the concordance model, yet many extensions (for instance photon non-conservation) break it. This paper applies model-independent cosmography—Taylor series, a rescaled redshift variable, and Padé approximants—to 63 Sunyaev-Zeldovich and high-resolution spectroscopic temperature measurements spanning 0 to z=6.34. Across every expansion choice the data require any deviation to stay at the percent level, and one non-standard coefficient is sufficient to describe the whole set. The same bound is recovered in the classic Lima power-law model. The work also shows that the usual cosmographic advice (prefer the y-variable or Padé forms) does not carry over to this observable, and that the directly fitted slope is compatible with, though slightly smaller than, the FIRAS value.

Core claim

Using 63 public CMB temperature measurements over 0≤z≤6.34, cosmographic expansions (Taylor in z or y, selected Padé approximants) and the Lima adiabatic model all constrain violations of T(z)=T0(1+z) to the percent level; a single extra parameter is enough, and standard scale-factor cosmography lore about the superiority of y or Padé does not apply to this observable.

What carries the argument

Cosmographic series for a multiplicative correction f(z) (or g(y) with y=z/(1+z)) and the matching Padé approximants P1,1 and P1,2, compared directly with the Lima form T(z)=T0(1+z)1−β; maximum-likelihood fits of the low-order coefficients against the same temperature data set.

Load-bearing premise

That the mixed Sunyaev-Zeldovich cluster bins and high-redshift spectroscopic points can be treated as unbiased, independent samples of one smooth T(z) function expandable to low order around z=0, even though most spectroscopic points lie far outside that neighbourhood.

What would settle it

A new suite of precise CMB temperature measurements at intermediate redshifts (0.5–2) that, when added to the present sample, drives any of the leading cosmographic coefficients or the Lima β more than a few percent away from zero at high significance.

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

If this is right

  • Percent-level bounds on T(z) violations translate into equally tight limits on photon-number non-conservation and related distance-duality breaches.
  • Only one beyond-standard-model parameter is currently needed to describe the temperature-redshift data.
  • Future high-z spectroscopic campaigns (e.g., ELT-ANDES) will tighten the higher-order coefficients that the present spectroscopic subset leaves loose.
  • The mild preference for a slope slightly below the FIRAS value can be checked against independent geometric T0–H0 degeneracies.

Where Pith is reading between the lines

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

  • Because T(z) and the distance-duality relation are tightly linked, the same cosmographic coefficients can be folded into joint analyses that currently rely only on luminosity and angular-diameter distances.
  • The persistent slight under-estimate of (dT/dz)0 relative to FIRAS, once FIRAS is removed, offers an independent cross-check on claims that a free T0 eases the Hubble tension.
  • The failure of Padé and y-variable advantages here suggests a practical diagnostic: observables whose standard-model series terminate after one or two terms should be expanded differently from scale-factor quantities.

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 applies cosmographic methods — Taylor expansions in z and in y=z/(1+z), Padé approximants, and the canonical Lima (1+z)^{1−β} phenomenology — to 63 measurements of the CMB temperature over 0≤z≤6.34 (FIRAS at z=0, 45 SZ-binned cluster measurements, 17 high-z spectroscopic points). The author finds no significant deviation from T(z)=T0(1+z), with violations constrained to percent level, and concludes that a single non-standard parameter suffices. Two further results are reported: (i) the directly fitted slope (dT/dz)0 is compatible with but systematically below the FIRAS value, a skew that persists when FIRAS is removed; and (ii) standard scale-factor cosmography lore (advantages of the y variable and of Padé approximants) does not transfer to this observable, because the standard law has only one non-zero expansion coefficient. A FIRAS-removal analysis and separate SZ/spectroscopic fits (Table 4) quantify the leverage of each subset.

Significance. This is the first systematic cosmographic treatment of the temperature-redshift relation, a fundamental consistency test of the standard model, and it is executed cleanly: multiple parameterizations are cross-checked against each other and against the full Lima power law on the same data, the leverage of each data subset is explicitly quantified (Table 4), and the T0–H0 degeneracy connection (Ivanov et al. 2020) gives the persistent slope skew a concrete cosmological motivation. The negative methodological result — that y-variable and Padé lore, derived from scale-factor expansions, fails here because the standard T(z) law has a single non-zero coefficient — is argued from the structure of the observable rather than the data and is a genuinely useful correction to the cosmography literature. The analysis is simple, transparent, and directly falsifiable with forthcoming ELT-ANDES data.

major comments (3)
  1. [§1 (data compilation) and §4, Tables 1–4] The headline percent-level bound rests on treating all 63 points as independent Gaussian measurements, but 30 of them are binned products of just two catalogs (18 Planck bins from 815 clusters, 12 SPT bins from 158 clusters). Bins within a catalog share tSZ spectral modeling, bandpass calibration, and the X-ray/tSZ temperature cross-calibration, so their errors are correlated at some level, and a coherent multiplicative catalog offset would propagate directly into (dT/dz)0 and β rather than averaging down — Table 4 shows the SZ subset dominates the joint constraint. This matters concretely because the paper's own Tables 1, 3 and 4 show a persistent, unexplained downward skew of the fitted slope relative to FIRAS across every parameterization, which is exactly the signature a small catalog-level offset would produce. The manuscript should either (a) state what bin-to-bin covariance inform
  2. [§3, Table 3, P2,1 row] For P2,1 the third derivative is reported as 'Unconstrained' after imposing the prior (d³T/dz³)0 ∈ [−10⁻³, +10⁻³], chosen to regulate the 0/0 behaviour of Eq. (15) in the standard-model limit. The width of this prior is arbitrary, and since the posterior is prior-dominated the P2,1 row conveys no information about the data. The paper is candid about this, but the reader is left without a sense of how the other P2,1 entries (slope and second derivative) depend on the prior width. A short statement quantifying this sensitivity — or relegating P2,1 to a remark — would make Table 3 more honest; as presented, the P2,1 row sits in the table on equal footing with data-driven results.
  3. [§2, Table 2, cubic f(z) row] The cubic truncation of the f(z) series gives β = −0.001 ± 0.004, an uncertainty ~2.5× smaller than the full power-law fit (±0.010) and smaller than the linear and quadratic truncations. This is counterintuitive for a series with known convergence problems at z>1 and is presumably an artifact of the cubic truncation of (1+z)^{−β} misrepresenting the function over the fitted range, so that the apparent precision is spurious. The text notes 'well-known convergence issues' in passing, but this specific number — the tightest β constraint in the paper — risks being quoted out of context. A sentence explaining why the cubic-truncation error bar should not be interpreted as a genuine improvement is needed.
minor comments (6)
  1. [Data Availability] The data are stated to be public, but the likelihood code and the assembled T(z) table are not provided. Given the simplicity of the analysis, releasing the compiled dataset and a minimal likelihood script would make the results trivially reproducible and is standard practice for MNRAS.
  2. [§2, Table 1] The y-series coefficients in Table 1 have much larger uncertainties than the z-series ones (e.g. (d³g/dy³)0 = 1.0 ± 2.6 vs (d³f/dz³)0 = −0.003 ± 0.019). The text says the two are 'superficially comparable'; it would help to state explicitly that the uncertainties map onto one another under the variable change at the relevant order, so the comparison is not misread as the z-series being more constraining.
  3. [§4, Table 4] The Spec-only P1,1 fit gives T0 = 2.50 (+0.48/−0.90), notably asymmetric and low relative to FIRAS, while the Spec-only T2 fit gives 2.92 ± 0.57. A brief comment on this difference between the two functional forms for the same subset would be useful, since it bears on the stability of the Spec-only inferences.
  4. [Various] Typographical: 'auggested' (§3, first paragraph); 'separatrly' (Table 4 caption); 'a a single additional parameter' (§3); 'the combination of the the two' (Fig. 4 caption); 'SZ data, Forthcoming' (§5, comma splice/capitalization).
  5. [§1] The 15 'earlier measurements' from Battistelli et al. (2002) and Luzzi et al. (2009) overlap in instrument and analysis with each other and possibly with clusters in the Planck sample; a footnote clarifying whether any clusters enter twice would remove ambiguity about double-counting.
  6. [Figures 1–4] The figures are clear, but axis ranges differ between top and bottom panels (noted only for Fig. 1); a consistent note in each caption, or matched ranges where feasible, would ease comparison.

Circularity Check

0 steps flagged

No circularity: constraints are direct likelihood fits of free expansion coefficients (or Lima β) to external T(z) data against an independent standard-law null.

full rationale

The paper's load-bearing chain is: compile 63 external CMB temperature measurements (FIRAS, Planck/SPT/earlier SZ bins, high-z spectroscopy) → maximize likelihood for Taylor/y-series coefficients, Padé coefficients, or the single Lima parameter β → report that higher-order coefficients and β are consistent with zero at the percent level, and that y-rescaling and Padé bring no meaningful gain for this observable. The standard law T(z)=T0(1+z) (vanishing higher derivatives, β=0) is an external theoretical benchmark, not defined by the fit. Self-citations (Rocha & Martins 2022; Martins et al. 2022; Martins 2025) only supply prior methodological applications of cosmography to other observables; they do not justify uniqueness, force the truncation, or enter the likelihood. Comparisons of truncated series to the full power-law Lima model are consistency checks on the same data, not predictions constructed from fitted inputs. No step reduces by construction to its own input.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The claim rests on standard FLRW redshift, the blackbody preservation argument that yields T∝(1+z) when photon number is conserved, public astrophysical T measurements treated as unbiased, and the usual truncation of a Taylor/Padé series around z=0. No new physical entity is postulated; free parameters are the series coefficients (or β) and T0.

free parameters (4)
  • T0 (present CMB temperature) = 2.7255±0.0006 K (FIRAS); ~2.74±0.02 K without FIRAS (joint)
    Free in all fits; pinned to FIRAS when included, or left free when FIRAS is dropped.
  • cosmographic derivatives (df/dz)_0, (d²f/dz²)_0, … or (dT/dz)_0, (d²T/dz²)_0, … = e.g. (dT/dz)_0 ≈ 2.65–2.69 K (with FIRAS); higher derivatives consistent with 0
    Coefficients of the Taylor/Padé expansions fitted to the 63-point sample; central objects of the constraints.
  • Lima β = β = 0.008±0.010 (full); 95% 0.008±0.020
    Single-parameter adiabatic extension T=T0(1+z)^{1-β} fitted for comparison with the series.
  • Padé P2,1 third-derivative prior = prior range only; parameter left unconstrained
    Ad-hoc narrow prior (d³T/dz³)_0 ∈ [−10^{-3}, +10^{-3}] imposed to avoid 0/0 numerical pathology in the standard-model limit.
axioms (4)
  • domain assumption Under adiabatic expansion and an initially blackbody spectrum with conserved photon number, T(z)=T0(1+z) exactly.
    Stated in §1 as the standard-law baseline whose violations are being bounded.
  • ad hoc to paper A low-order Taylor or Padé expansion of f(z) or T(z) about z=0 (or y=0) adequately captures allowed departures over 0≤z≤6.34.
    Core cosmographic premise; paper itself notes convergence issues for the z-series and limited gain from y/Padé (§2–3).
  • domain assumption Published SZ cluster-bin and high-resolution spectroscopic temperatures are unbiased estimators of the CMB T at those redshifts for the purpose of a joint likelihood.
    Data compilation in §1; no additional systematic floor is marginalized.
  • standard math Standard maximum-likelihood / Gaussian error propagation on the reported measurement uncertainties is sufficient.
    Implicit throughout the constraint tables; no hierarchical or systematic-marginalized likelihood is introduced.

pith-pipeline@v1.2.0-grok45-kimik3 · 13568 in / 3000 out tokens · 56984 ms · 2026-07-30T19:48:31.371588+00:00 · methodology

0 comments
read the original abstract

The temperature-redshift relation, $T(z)=T_0(1+z)$, is a cornerstone of the standard cosmological model, but it is violated in many physically motivated extensions thereof, e.g. when photon number is not conserved. We apply the methodology of cosmography, a model-independent approach to cosmology, to this observable, and use astrophysical Sunyaev-Zeldovich and high-resolution spectroscopic measurements spanning the redshift range $0\le z\le6.34$, to constrain violations of the standard relation. We explore the impact of different cosmographic approaches and expansion variable choices and also compare the cosmographic constraints with those obtained, with the same data, in the canonical adiabatic extension model of Lima. While these choices have some impact, overall we find that such violations are constrained to percent level, and that a single non-standard parameter suffices to describe them. We also show that the directly fitted slope of the temperature-redshift relation is compatible with (but slightly smaller than) the one inferred from FIRAS, and that standard cosmography lore, based on scale factor expansions, does not apply to this observable, e.g. Pad\'e approximants have no meaningful advantage over plain Taylor series.

Figures

Figures reproduced from arXiv: 2607.23535 by C. J. A. P. Martins.

Figure 1
Figure 1. Figure 1: Constraints on the parameters of the cosmographic series of Eq. (5) and Eq. (6), in the top and bottom panels respectively, as a function of the truncation order. In two-dimensional panels, one, two and three sigma confidence levels are shown. Note the different axis ranges in the top and bottom panels [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Constraints on the Lima phenomenological temperature-redshift relation Eq. (2) and its Taylor series approximation, truncated at various orders. Left and right panels correspond to the 𝑓 (𝑧) and 𝑔(𝑦) series respectively. and is evidently unphysical, given that observationally 𝑇0 ∼ (𝑑𝑇/𝑑𝑧)0. Indeed, all approximants of the form 𝑃0,𝑚 will have anal￾ogous terms in the denominator, and are thus unfit for purpo… view at source ↗
Figure 3
Figure 3. Figure 3: Constraints on the derivatives (𝑑 𝑖𝑇/𝑑𝑧𝑖 )0 of Eq. (12) for various choices of Taylor series (top panels) and Padé approximants (bottom panels). In the two-dimensional panels, one, two and three sigma confidence levels are shown. Other parameters, when they exist, have been marginalized. The dotted lines correspond to the FIRAS value. Note that the third derivative is not plotted for 𝑃2,1, cf. the main tex… view at source ↗
Figure 4
Figure 4. Figure 4: Constraints on 𝑇0 and its derivatives for 𝑇2 (𝑧) (top panels) and 𝑃1, 1(𝑧) (bottom panels), without including the FIRAS value, and considering separately the Sunyaev-Zeldovich and spectroscopic data, as well as the combination of the the two. In the two-dimensional panels, one, two and three sigma confidence levels are shown. Other parameters have been marginalized, and the dotted lines correspond to the F… view at source ↗

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Reference graph

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