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Revisiting the temperature evolution law of the CMB with gaussian processes

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

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Pith's one-line read A model-free reconstruction of CMB temperature data finds hints of low-redshift deviations from the standard cooling law.

desk verdict A transparent, incremental GP re-analysis of CMB temperature data whose ~2 sigma low-z hints rest on unmodeled SZ systematics and a post-hoc kernel choice; worth a referee but not a discovery. read the letter →

arxiv 2505.24543 v1 pith:B5HVIFKQ submitted 2025-05-30 astro-ph.CO

Felipe Avila , Alexander Bonilla Rivera , Rafael C. Nunes , R.F.L. Holanda , Armando Bernui This is my paper
classification astro-ph.CO
keywords cosmicmicrowavebackgroundtemperature-redshiftrelationGaussianprocessesSunyaev-Zel'dovicheffectfine-structureconstantcosmologicalparameterinferencelow-redshiftdeviations
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

Using Gaussian Process regression on 49 compiled measurements of the CMB temperature, this paper reconstructs $T_{\rm CMB}(z)$ without assuming the standard linear law. It finds that the data are broadly consistent with $T_{\rm CMB}(z) = T_0(1+z)$ at the 2-$\sigma$ level, but the reconstruction hints at deviations at low redshifts $z<0.5$, where the tension with the COBE/FIRAS-based prediction reaches about $2\sigma$. The inferred present-day temperature is slightly high, $T_0 = 2.73987 \pm 0.00793$ K for the squared-exponential kernel, in mild tension with the direct COBE/FIRAS value. The paper also converts the temperature reconstruction into a possible variation of the fine-structure constant and finds no significant deviation, which it reads as support for standard physics in the electromagnetic sector.

What carries the argument

The central machinery is a Gaussian Process regressor, a non-parametric regression that assumes point-to-point Gaussian correlations governed by a covariance kernel, applied to the 49 compiled data points. The authors test squared-exponential, Matérn ($\nu=5/2,7/2,9/2$), and rational-quadratic kernels to reconstruct $T(z)$ and its derivatives. The derived quantity $\beta(z) = 1 - (1+z)T'(z)/T(z)$ converts the reconstructed curve into a direct measure of deviation from $T_0(1+z)$, and the tension function compares the reconstruction with the COBE/FIRAS prediction. The scalar-tensor relation (10) then maps the same reconstructed temperature curve into a constraint on $\Delta\alpha/\alpha$.

What would settle it

Running the same Gaussian Process pipeline on the 13 interstellar-medium-only high-redshift points, or on an SZ sample corrected for cluster asphericity and relativistic effects, would show whether the low-redshift deviation persists or vanishes.

Watch

Extended reading notes

Core claim

Applying Gaussian Process regression with several kernels to the 49-point compilation from [27], the authors reconstruct $T_{\rm CMB}(z)$, its first and second derivatives, and the deviation parameter $\beta(z)$ defined by $T_{\rm CMB}(z)=T_0(1+z)^{1-\beta}$. For the squared-exponential kernel they obtain $T_0 = 2.73987\pm0.00793$ K and $\beta_0 = 0.03301\pm0.01954$, i.e. about a $1.7\sigma$ departure from $\beta=0$; the tension function stays below $2\sigma$ everywhere and peaks near $z=0$. The reconstruction is consistent with the standard relation for $z\gtrsim0.5$, while the low-redshift behavior suggests either unaccounted systematics or new physics. Using relation (10) from scalar-tensor theories, they map the temperature reconstruction into $\Delta\alpha/\alpha$ and find no statistically significant variation, which they interpret as consistency with general relativity and the standard cosmological model.

Load-bearing premise

The reconstruction's low-redshift result depends on the cluster-based temperature measurements at $z<0.5$ being free of systematic biases; if those measurements carry errors from cluster asphericity, relativistic corrections, or calibration, the apparent deviation could disappear.

Editorial extensions

If this is right

  • If the low-redshift deviation is real, the standard $T_0(1+z)$ law would need revision at late times, meaning either the cluster-based temperature measurements carry unmodeled systematics or the photon temperature evolves non-standardly.
  • A positive $\beta_0\approx0.033$ corresponds to a slightly higher present-day CMB temperature than the FIRAS value, so analyses that calibrate CMB maps or subtract the dipole using the FIRAS temperature may be affected.
  • The null result for $\Delta\alpha/\alpha$ means that Einstein-equivalence-principle-violating scalar-field models of the type in [29] are not favored by these data, even though $\beta(z)$ shows hints of deviation.
  • The kernel dependence of the $z=0$ inference, especially the unstable Matérn $\nu=5/2$ result, implies that conclusions about $T_0$ should be tied to the chosen reconstruction kernel rather than treated as measurement-independent.

Reading between the lines

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

  • The low-redshift deviation is driven almost entirely by the Sunyaev-Zel'dovich cluster sample, so a Gaussian Process run on the 13 interstellar-medium points alone would show whether the signal disappears, pointing to SZ systematics.
  • Because Gaussian Process extrapolation to $z=0$ is sensitive to kernel choice and to the sparsity of data near $z=0$, the inferred $T_0$ tension is more a statement about the data's low-redshift behavior than an independent measurement of the present temperature.
  • Existing tight bounds on $\Delta\alpha/\alpha$ from quasar absorption spectra could be used as a prior to sharpen the reconstruction of $T_{\rm CMB}(z)$; the paper does not perform that joint analysis.
  • A direct measurement of the CMB temperature at $z\sim0.3$-$0.5$ by an independent technique, such as molecular excitation in a low-redshift absorber, would be a discriminating test of the reported deviation.
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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 / 5 minor

Summary. The paper uses Gaussian process (GP) regression to reconstruct the CMB temperature evolution T_CMB(z) from 49 measurements: 36 low-redshift Sunyaev-Zel'dovich (SZ) cluster points and 13 high-redshift interstellar medium (ISM) excitation points. The GP reconstruction is used to infer the present-day temperature T0, the logarithmic-deviation parameter beta(z), and, via Eq. (10), the implied variation of the fine-structure constant Delta alpha/alpha. The authors report a mild (~2 sigma) deviation from the standard law T(z) = T0(1+z) at z < 0.5, a corresponding tension between the GP-inferred T0 and the COBE/FIRAS value, and consistency of Delta alpha/alpha with zero. The analysis is transparent and uses public code (GaPP), but the central claims rest on untreated SZ systematics and a post-hoc kernel choice.

Significance. If the low-redshift deviations and the T0 tension were robust, they would indicate either unaccounted systematics in the SZ temperature measurements or new physics that modifies the CMB temperature evolution at late times. The paper's strengths are its transparent, reproducible GP methodology and its explicit comparison to the external COBE/FIRAS measurement. However, the significance is strongly limited by the fact that the low-redshift signal is entirely dominated by 36 SZ cluster points with known modeling and calibration systematics, and by the lack of a quantitative justification for the preferred kernel. A rigorous treatment of these issues would be needed before the claims can be considered established.

major comments (4)
  1. [Section II] The 36 SZ cluster measurements are treated as independent Gaussian errors with no systematic covariance or nuisance parameters. These data are the only constraints at z < 0.5, and they are known to be subject to cluster asphericity, relativistic corrections, pressure-profile assumptions, and Planck/SPT calibration uncertainties. A common multiplicative offset of only ~0.3% would shift T0 by about 0.008 K, which is the quoted uncertainty and the size of the claimed ~2 sigma tension. To support the central claim, the authors should either include a systematic covariance matrix (e.g., a common calibration nuisance parameter) or perform a sensitivity analysis that demonstrates the low-z deviations and T0 tension survive when a global offset is added or when each SZ subsample is removed in turn.
  2. [Section III, Table I] The choice of the Squared Exponential (SE) kernel over the Matern 5/2 kernel is not justified by any model-comparison test. Table I shows that the inferred T0 differs by about 0.016 K between these two kernels, and the Matern 5/2 result (T0 = 2.75562 ± 0.01166) would make the tension with COBE/FIRAS stronger (~2.6 sigma) than the SE result (~1.8 sigma). The statement that the Matern 5/2 kernel is 'less reliable' is not supported by quantitative evidence. The authors should use leave-one-out cross-validation, the marginal likelihood, or a Bayes factor to select among kernels, rather than rejecting the kernel that yields a stronger deviation.
  3. [Section III, Figure 3 and Table I] The maximum tension in Figure 3 occurs at z = 0, which is an extrapolation of the GP below the lowest data point at z = 0.037. The GP prediction at z = 0 depends on the prior mean function (which is not stated) and on the optimized kernel hyperparameters (which are not reported). The authors should report the optimized length scale and amplitude for each kernel and test robustness to the choice of prior mean (e.g., zero mean versus a mean given by T0(1+z) with T0 = 2.72548 K). This is load-bearing because the T0 tension claim relies on the extrapolated value.
  4. [Section II, dataset selection] The exclusions of the five upper limits, the HFLS3 point, and the z = 0 direct measurement are described but not fully justified. In particular, including the COBE/FIRAS z = 0 measurement as a data point in the GP would anchor the low-redshift reconstruction and likely remove the reported tension. The authors should test the sensitivity of their results to these exclusions, or at least provide a quantitative argument for why the z = 0 point cannot be included (e.g., double counting or a different error model).
minor comments (5)
  1. [Section III, Figures 1 and 2] The figure captions contain the phrase 'expect value', which should be 'expected value'. In addition, the right panel of Figure 1 would benefit from a label indicating that the dashed line is the standard-law derivative T0.
  2. [Section III, Eq. (8)] The tension function in Eq. (8) is introduced without a descriptive name; calling it the 'tension function' in the text would improve readability.
  3. [Section IV, Eq. (10)] Equation (10) is derived from a specific scalar-tensor theory by Hees et al.; the paper should state more explicitly that this relation is not model-independent and that the reconstructed Delta alpha/alpha should be interpreted within that theoretical framework, not as a general probe of varying alpha.
  4. [Section III] The paper reports ~2 sigma deviations in beta(z) for z < 0.5 without discussing the look-elsewhere effect. Since multiple redshifts and several kernels are examined, the statistical significance of a deviation at a particular redshift should be corrected for the number of independent tests or presented with this caveat.
  5. [Section II and Table I] Table I would be more informative if it also reported the optimized GP hyperparameters (length scale l and amplitude sigma_f) for each kernel, since the kernel behavior is central to the reconstruction and extrapolation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reconstruction is fit to external temperature data, the T0 comparison is a holdout extrapolation against COBE/FIRAS, and the alpha relation is an external mapping.

full rationale

The paper's derivation chain is: (i) take 49 published T_CMB(z) measurements from SZ clusters and ISM excitation; (ii) fit a Gaussian Process to those data; (iii) read off T(z), derivatives, beta(z), and the extrapolated T(z=0); (iv) compare the fitted curve and extrapolated T0 to the external COBE/FIRAS value and to T = T0(1+z); (v) map the fitted T(z) through the external Hees et al. relation, Eq.(10), to Delta alpha/alpha. No step identifies an output with an input by definition, and no fitted parameter is renamed as a prediction. Beta(z) is defined in Eq.(4) as 1 - (1+z)T'/T, so a nonzero beta is a restatement of the fitted slope; however, the paper presents it as a reconstruction rather than as an independent prediction, and the deviation claim is anchored to the external FIRAS value and the standard law. The z=0 temperature is not fitted to T0: the FIRAS point is explicitly excluded from the GP sample, so the mild tension is a genuine holdout extrapolation versus a direct measurement. Eq.(8) is a test statistic, not a derivation. Eq.(10) is an external theoretical relation applied to the fitted curve, not an input fitted from the data. Self-citations such as [30], [60], and [61] are contextual or definitional and are not load-bearing for the central reconstruction. The principal vulnerability is unmodeled systematics in the 36 SZ cluster points, which is a data-quality and correctness concern, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the reliability of the input data and on the suitability of the GP reconstruction; no new physical entity is introduced. The GP hyperparameters are the only fitted numbers, and the paper relies on an external theory relation for the alpha interpretation.

free parameters (1)
  • GP hyperparameters (length scale l and amplitude sigma_f) = optimized per kernel via marginal likelihood (values not reported)
    The Gaussian Process reconstruction of T(z) and its derivatives depends on these hyperparameters. They are fitted to the data and are not physically motivated. The paper does not report the optimized values, which hampers exact reproduction.
assumptions (4)
  • domain assumption The 49 CMB temperature measurements from Riechers et al. (2022), especially the SZ data, are unbiased and independent.
    The low-z deviation claim is driven by the SZ data; if these measurements suffer from unmodeled systematics (cluster physics, calibration), the deviation could be an artifact. This is implicit throughout the GP analysis in Section II.
  • domain assumption The chosen GP kernel and optimized hyperparameters yield a faithful reconstruction of T(z) and its derivatives, including at z=0 where there is no data.
    The GP extrapolation to z=0 and the derivative reconstruction assume the kernel correctly models the correlation structure. The paper tests several kernels but does not rigorously justify discarding one.
  • domain assumption The parameterization T = T0(1+z)^(1-beta) is a valid description of deviations, so equation (4) for beta is applicable.
    This is the standard phenomenological parameterization in the literature, but it assumes a single power-law exponent at each redshift. Section II.A introduces it.
  • domain assumption Equation (10) from Hees et al. (2014) correctly relates CMB temperature variations to fine-structure constant variations in scalar-tensor theory.
    The reconstruction of Delta alpha/alpha depends entirely on this external relation. If the relation is incorrect, the alpha result is invalid. Section IV uses it without modification.

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Cite this review

Pith. "Pith review of Revisiting the temperature evolution law of the CMB with gaussian processes." pith.science (2026). https://pith.science/paper/B5HVIFKQ

@misc{pith2026250524543,
  author       = {Pith},
  title        = {Pith review of: Revisiting the temperature evolution law of the CMB with gaussian processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B5HVIFKQ}},
  note         = {Machine review of arXiv:2505.24543}
}
abstract

In this work, we perform a statistical inference of the classical background law governing the evolution of the temperature of the cosmic microwave background radiation (CMB), given by $T_{\rm CMB}(z) = T_0(1 + z)$. To this end, we employ Gaussian Process (GP) regression techniques to reconstruct the temperature evolution based on two observational datasets: (i) CMB-Sunyaev-Zel'dovich (SZ) cluster measurements and (ii) CMB-interstellar medium (ISM) interaction data. Our analysis reveals interesting results that may suggest potential deviations from the standard temperature-redshift relation, particularly at low redshifts ($z < 0.5$), where discrepancies up to $\sim$2$\sigma$ are observed. Additionally, we identify a mild but noteworthy tension, also at the $\sim$2$\sigma$ level, between our GP inferred value of the present-day CMB temperature, $T_{\rm CMB}(z=0)$, and the precise direct measurement from the COBE/FIRAS experiment. We also explore possible phenomenological implications of our findings, including interpretations associated with possible variations in fundamental constants, such as the fine-structure constant $\alpha$, which could provide a physical explanation for the observed deviations at low redshift.

Figures

Figures reproduced from arXiv: 2505.24543 by the authors.

Figure 1
Figure 1. FIG. 1. Left panel: GP reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Left panel: GP reconstruction of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Evolution of the degree of tension between the sta [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. GP reconstruction of the fine-structure constant us [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Pith tools

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