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Derivative hierarchy as the origin of kernel-dependent trends in Gaussian process reconstructions of the hubble parameter

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Afaq Maqsood

Cosmology and Nongalactic Astrophysics · Submitted 27 Aug 2026 · 44 references

Pith's verdict: A clear, honest demonstration that kernel smoothness moves GP-inferred H0 by roughly 3 km/s/Mpc; the main gap is the asserted but unshown constant-mean robustness check.

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arXiv ID2608.26774 (version 1)
Pith IDNFAGLFAF, Pith's permanent record of this paper and its review
CitationAfaq Maqsood. "Derivative hierarchy as the origin of kernel-dependent trends in Gaussian process reconstructions of the hubble parameter." arXiv:2608.26774 (2026). https://pith.science/paper/NFAGLFAF
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The paper

The paper claims

In Gaussian Process reconstructions, kernel smoothness alone shifts the inferred Hubble constant monotonically by about 3 km s$^{-1}$ Mpc$^{-1}$, with rougher kernels giving higher $H_0$.

The reading

The paper asks whether the covariance kernel—the smoothness assumption built into a Gaussian Process—is a harmless technical choice or a physically consequential prior when reconstructing the cosmic expansion history from 32 cosmic-chronometer measurements. The answer, on the paper's evidence, is the latter: across a well-defined differentiability hierarchy of kernels (Matérn 3/2, 5/2, 7/2, 9/2, and Squared Exponential), the reconstructed present-day Hubble constant falls monotonically from $70.1 \pm 6.6$ to $67.2 \pm 4.8$ km s$^{-1}$ Mpc$^{-1}$, while the reconstructed local slope $H'(0)$ rises. The paper argues this ordering is prior sensitivity, not model selection: the log marginal likelihood is nearly degenerate across kernels, and the $\chi^2$ hierarchy is descriptive of kernel flexibility rather than statistical preference. The trend survives jackknife resampling, an inflated uncertainty floor, and the addition of six DESI DR2 BAO measurements. If right, this matters because part of the spread between the SH0ES-like and Planck-like values of $H_0$ is attributable to the investigator's smoothness choice rather than to the data.

What carries the argument

The load-bearing object is the hierarchy of Matérn kernels with half-integer smoothness parameters $\nu = 3/2, 5/2, 7/2, 9/2$, completed by the Squared Exponential kernel as the infinitely differentiable endpoint. Its defining property, which the paper calls the derivative hierarchy, is that a Matérn kernel with parameter $\nu$ admits consistent reconstruction of derivatives only up to order $\lfloor \nu \rfloor$, while the Squared Exponential kernel is differentiable to all orders. The paper organizes the five kernels into a tree whose levels mark the highest reconstructible derivative order, so that comparisons along the tree are controlled changes in smoothness rather than arbitrary swaps. This machinery does two jobs: it explains why the $H_0$ ordering is monotonic rather than accidental, and it locates the mechanism in the propagation of differentiability constraints into the reconstructed slope $H'(0)$, whose anti-correlation with $H_0$ carries the interpretation. Because derivatives of a Gaussian Process are themselves Gaussian Processes, the same framework yields the derivative reconstructions that make the mechanism visible.

Extended reading notes

Core claim

Using only the 32 cosmic-chronometer measurements, the paper finds a systematic and monotonic ordering: as kernel smoothness increases from Matérn 3/2 through 5/2, 7/2, 9/2 to the Squared Exponential kernel, the reconstructed Hubble constant decreases monotonically, $H_0 = 70.1 \pm 6.6 \to 69.3 \pm 5.6 \to 68.8 \pm 5.2 \to 68.3 \pm 5.0 \to 67.2 \pm 4.8$ km s$^{-1}$ Mpc$^{-1}$, with uncertainties shrinking accordingly. The same smoothness hierarchy is visible in the first derivative: the slope $dH/dz$ at $z=0$ increases monotonically from $38.1 \pm 31.4$ to $47.2 \pm 16.2$ km s$^{-1}$ Mpc$^{-1}$, and the paper reads the anti-correlation between $H'(0)$ and $H_0$ as the mechanism linking differentiability to the lower $H_0$. The goodness-of-fit statistic $\chi^2$ follows the same ordering, $\chi^2_{3/2} < \chi^2_{5/2} < \chi^2_{7/2} < \chi^2_{9/2} < \chi^2_{\mathrm{SE}}$, both for the chronometers alone and after adding DESI DR2 BAO data, but the paper is explicit that this is descriptive: the spread in maximized log marginal likelihood ($\Delta\mathrm{LML} < 0.12$) and in leave-one-out scores is far too small for any kernel to be statistically preferred. The conclusion is that kernel smoothness acts as an effective prior that reshapes the reconstructed expansion history, not as a fingerprint of the true one.

Load-bearing premise

The argument stands on the full non-diagonal covariance matrix of the 32 cosmic-chronometer measurements being correct; if that matrix misrepresents the correlated systematic errors, the reconstructed $H_0$ values—and possibly the monotonic ordering itself—could shift.

Editorial extensions

If this is right

  • Quoted $H_0$ values from Gaussian Process reconstructions should be reported together with the kernel that produced them, since the kernel choice alone spans nearly 3 km s$^{-1}$ Mpc$^{-1}$—a shift comparable in size to the gap at the center of the Hubble-tension debate.
  • The apparent spread between SH0ES-like and Planck-like reconstructions is partly a smoothness effect: rougher kernels track local chronometer features and land at higher $H_0$, while smoother kernels average over those features and land at lower $H_0$.
  • Kernel selection should be guided by the derivative order a physical question requires: observables like the deceleration parameter or jerk, which depend on $H''$ or higher derivatives, cannot be consistently reconstructed with kernels below the matching differentiability level.
  • The near-degenerate log marginal likelihoods and leave-one-out scores mean the $\chi^2$ hierarchy is a descriptive measure of kernel flexibility, and it should not be quoted as evidence that one kernel is favored by the data.

Reading between the lines

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

  • An extension the paper leaves untested: adding the Matérn 1/2 kernel ($\nu = 1/2$, which admits no differentiable reconstruction) to the hierarchy should, by the paper's logic, yield an $H_0$ still higher than the Matérn 3/2 value and a shallower $H'(0)$; this is a cheap, direct test of the monotonicity claim.
  • If differentiability is truly the mechanism, the same monotonic ordering should appear in any low-redshift observable built from derivatives of $H(z)$—for instance the deceleration parameter—not only in $H_0$ itself.
  • The result suggests a calibration strategy for GP-based cosmology: fit the same kernel hierarchy to mock catalogs with known expansion histories and use the spread as an estimate of the systematic error budget that kernel choice contributes, rather than absorbing it into the physics.
  • Because the Squared Exponential kernel suppresses off-diagonal covariances most strongly, re-running the hierarchy on a much denser or sparser redshift sampling would test whether the ordering is tied to the specific 32-point footprint of the CC-32 compilation or is generic to the kernels themselves.
Original abstract

We perform a model independent reconstruction of the cosmic expansion history using Gaussian Process regression, investigating how the smoothness properties of covariance kernels influence the inferred Hubble parameter. Using 32 cosmic chronometer measurements, we reconstruct $H(z)$ with the Mat\'ern $3/2$, $5/2$, $7/2$, $9/2$, and Squared Exponential kernels, which form a well-defined hierarchy of differentiability. Across this hierarchy, we observe a systematic and monotonic trend in which increasing kernel smoothness is associated with progressively lower reconstructed values of the present-day Hubble constant $H_0$, accompanied by reduced statistical uncertainties. Rather than indicating a statistical preference for a specific kernel, this behaviour reflects the sensitivity of non-parametric reconstructions to the assumed smoothness prior encoded in the covariance function. The same ordering is reflected in descriptive goodness-of-fit statistics and remains stable when incorporating recent DESI DR2 BAO measurements and performing jackknife resampling tests, demonstrating robustness against individual data points and dataset variations. Additional analysis of derivative reconstructions shows that differences in kernel differentiability propagate into the local slope of the expansion history, providing a consistent interpretation of the observed ordering in $H_0$. Our results highlight that kernel smoothness plays an important role in Gaussian Process reconstructions and should be carefully accounted for when interpreting non-parametric cosmological inferences.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,883 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.37 · compares with prior results 0.14 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We perform a model independent reconstruction of the cosmic expansion history using Gaussian Process regression, investigating how the smoothness properties of covariance kernels influence the inferred Hubble parameter. Using 32 cosmic chronometer measurements, we reconstruct $H(z)$ with the Mat\'ern $3/2$, $5/2$, $7/2$, $9/2$, and Squared Exponential kernels, which form a well-defined hierarchy of differentiability. Across this hierarchy, we observe a systematic and monotonic trend in which increasing kernel smoothness is associated with progressively lower reconstructed values of the present-day Hubble constant $H_0$, accompanied by reduced statistical uncertainties. Rather than indicating a statistical preference for a specific kernel, this behaviour reflects the sensitivity of non-parametric reconstructions to the assumed smoothness prior encoded in the covariance function. The same ordering is reflected in descriptive goodness-of-fit statistics and remains stable when incorporating recent DESI DR2 BAO measurements and performing jackknife resampling tests, demonstrating robustness against individual data points and dataset variations. Additional analysis of derivative reconstructions shows that differences in kernel differentiability propagate into the local slope of the expansion history, providing a consistent interpretation of the observed ordering in $H_0$. Our results highlight that kernel smoothness plays an important role in Gaussian Process reconstructions and should be carefully accounted for when interpreting non-parametric cosmological inferences.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We perform a model independent reconstruction of the cosmic expansion history using Gaussian Process regression, investigating how the smoothness properties of covariance kernels influence the inferred Hubble parameter. contribution 0.95 0.95 0.06 asserted 0.83
s1 Using 32 cosmic chronometer measurements, we reconstruct $H(z)$ with the Mat\'ern $3/2$, $5/2$, $7/2$, $9/2$, and Squared Exponential kernels, which form a well-defined hierarchy of differentiability. method 0.99 0.96 0.04 asserted 1.00
s2 Across this hierarchy, we observe a systematic and monotonic trend in which increasing kernel smoothness is associated with progressively lower reconstructed values of the present-day Hubble constant $H_0$, accompanied by reduced statistical uncertainties. result 1.00 0.96 0.59 asserted 1.00
s3 Rather than indicating a statistical preference for a specific kernel, this behaviour reflects the sensitivity of non-parametric reconstructions to the assumed smoothness prior encoded in the covariance function. interpretation 1.00 0.52 0.44 asserted 0.99
s4 The same ordering is reflected in descriptive goodness-of-fit statistics and remains stable when incorporating recent DESI DR2 BAO measurements and performing jackknife resampling tests, demonstrating robustness against individual data points and dataset variations. result 0.97 0.95 0.10 asserted 1.00
s5 Additional analysis of derivative reconstructions shows that differences in kernel differentiability propagate into the local slope of the expansion history, providing a consistent interpretation of the observed ordering in $H_0$. interpretation 0.73 0.94 0.09 asserted 0.98
s6 Our results highlight that kernel smoothness plays an important role in Gaussian Process reconstructions and should be carefully accounted for when interpreting non-parametric cosmological inferences. interpretation 0.95 0.62 0.79 asserted 0.94

What kind of paper the classifier reports this is: simulation study 0.52 · evidence from analysis of existing data 0.94.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. FIG. 1. Illustration of the kernel-dependent derivative hierarchy in Gaussian Process reconstruction. Panel (a) shows the
Figure 2. FIG. 2. Gaussian Process reconstruction of the Hubble parameter
Figure 3. FIG. 3. Hyperparameter space (
Figure 4. FIG. 4. Gaussian Process reconstruction of the first derivative of the Hubble parameter,
Figure 5. FIG. 5. Chi-square values for different Gaussian Process ker
Figure 6. FIG. 6. Leave-one-out cross-validation (LOO-CV) predictive

Figures reproduced from arXiv:2608.26774 by the authors.

References
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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 35 sentences without a confident label

Uses data or values from 11 papers
Compares with 2 papers
Cites as background 30 papers

Discussion

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