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arxiv: 2510.11555 · v2 · submitted 2025-10-13 · 🌌 astro-ph.CO

O_k null test with multi-task Gaussian processes: cosmic curvature and data compatibility

Yungui Gong , Qing Gao , Xuchen Lu , Zhu Yi This is my paper

Pith reviewed 2026-05-18 07:35 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords cosmic curvaturenull testDESI BAOAlcock-Paczynski parameterGaussian processesdata compatibilitydistance ratios
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The pith

A ratio-only null test for cosmic curvature works without absolute distances and finds DESI BAO and supernova data compatible.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a null test for cosmic curvature that uses only relative distance information from DESI baryon acoustic oscillation measurements, avoiding the absolute distance scales that most traditional versions demand. It combines the Alcock-Paczynski parameter F_AP with ratios such as D_V prime over D_V or D_M prime over D_M and reconstructs these quantities with multi-task Gaussian processes. This same framework also checks whether BAO and Type Ia supernova datasets are mutually consistent. A reader would care because a confirmed zero curvature removes a degeneracy that otherwise mixes with dark energy parameters in many analyses, while any detected curvature would either indicate new physics or point to inconsistencies among current observations. The results show the datasets agree but note a roughly two-sigma hint of nonzero curvature below redshift 0.5 where data remain sparse.

Core claim

The O_k null test is reformulated to use only ratio quantities available in DESI BAO data, specifically by combining the Alcock-Paczynski parameter F_AP with distance-measure ratios such as D_V'/D_V or D_M'/D_M. Multi-task Gaussian processes are developed to reconstruct the required functions and carry out the null test on the observed data. When the method is applied to DESI BAO measurements alone or jointly with SNe Ia, the two datasets prove compatible, although an approximately 2 sigma deviation from zero curvature appears at redshifts z less than or equal to 0.5, a result limited by the scarcity of observations in that range.

What carries the argument

The O_k null test built from the Alcock-Paczynski parameter F_AP together with distance ratios such as D_V'/D_V or D_M'/D_M, performed by multi-task Gaussian process regression that jointly reconstructs the necessary functions from ratio data.

If this is right

  • If the null test is satisfied, cosmic curvature is consistent with zero and thereby reduces degeneracies between curvature and other cosmological parameters such as the dark energy equation of state.
  • The ratio-based construction supplies a model-independent check that DESI BAO and SNe Ia datasets are compatible with each other.
  • The mild 2 sigma indication of nonzero curvature at low redshift can be tested or refuted once more observations become available in that redshift range.

Where Pith is reading between the lines

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

  • The same ratio-only approach could be applied to other surveys that measure the Alcock-Paczynski parameter, extending curvature tests to datasets that lack absolute distance calibrations.
  • The multi-task Gaussian process technique may be adapted to other cosmological null tests that involve several correlated functions reconstructed from overlapping observations.
  • A persistent low-redshift curvature signal would motivate dedicated follow-up observations or checks for systematics in the current low-z data releases.

Load-bearing premise

The multi-task Gaussian process reconstruction recovers the true underlying distance and expansion functions without adding biases that could create or conceal curvature signals, especially at low redshifts where observations are sparse.

What would settle it

Additional low-redshift BAO ratio measurements or independent distance-ratio data below z = 0.5 that either drive the O_k statistic significantly away from zero or bring it into statistical agreement with zero.

Figures

Figures reproduced from arXiv: 2510.11555 by Qing Gao, Xuchen Lu, Yungui Gong, Zhu Yi.

Figure 1
Figure 1. Figure 1: The details of the GP reconstruction by considering the correlations are presented in Appendix. From [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The reconstructed [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows Ωk ̸= 0 at z <∼ 0.5. Unlike U3 and PP, the D5 sample also indicates a deviation from Ωk = 0 at z >∼ 0.5. This result suggests that the apparent deviation may be driven by incomplete BAO coverage at low redshift. Additional observations, especially more low-z BAO measurements, are therefore needed to robustly test spatial flatness. 1 2 z 1.0 0.5 0.0 0.5 1.0 k ( z ) DR2+D5 DR2+U3 DR2+PP FIG. 3. The rec… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The reconstructed [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The reconstructed [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
read the original abstract

The $O_k$ null test can not only assess whether the cosmic curvature is zero, therefore if true reducing degeneracies between cosmic curvature and other cosmological parameters, but also provide a model-independent check of compatibility between different data sets. However, traditional implementations often require absolute distance data from Type Ia supernovae (SNe Ia) or baryon acoustic oscillation (BAO) measurements, limiting their applicability because such absolute distance data usually are not accessible. The BAO Alcock Paczynski (AP) parameter $F_{AP}$ is a measurement of a distance ratio, making the Dark Energy Spectroscopic Instrument (DESI) AP measurements particularly well suited for the $O_k$ null test because no absolute distance measurements are required. We propose a novel null test of cosmic curvature tailored to DESI BAO data that combines $F_{AP}$ with ratios such as $D_V'/D_V$ or $D_M'/D_M$. Crucially, this construction eliminates the need for absolute distance measurements. We further develop multi-task Gaussian processes to perform the null test. This approach can also be applied to a joint DESI BAO and SNe Ia dataset, and we find that DESI BAO and SNe Ia data are compatible. Although there is $\sim 2\sigma$ evidence of nonzero curvature at low redshift $z\lesssim 0.5$, this result is not conclusive largely due to the lack of observational data in the corresponding redshift range.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a model-independent null test for the cosmic curvature parameter O_k that combines the BAO Alcock-Paczynski parameter F_AP with distance ratios such as D_V'/D_V or D_M'/D_M, thereby avoiding any requirement for absolute distance measurements. Multi-task Gaussian processes are introduced to reconstruct the relevant functions from DESI BAO data, with an extension to a joint analysis with SNe Ia. The results indicate compatibility between the DESI BAO and SNe Ia datasets and report an approximately 2σ hint of nonzero curvature at z ≲ 0.5, while explicitly noting that this hint is not conclusive owing to sparse data coverage in the low-redshift regime.

Significance. If the multi-task GP reconstruction proves robust, the method supplies a practical route to curvature tests that sidesteps absolute-distance calibrations and can serve as a cross-check on dataset consistency. The construction that eliminates absolute distances is a clear technical strength, as is the paper’s explicit caveat on the inconclusiveness of the low-z signal. The approach could help reduce parameter degeneracies in future analyses once low-z data improve.

major comments (2)
  1. [§4] §4 (multi-task GP reconstruction): the claim that the ~2σ low-z O_k deviation is driven solely by data sparsity would be strengthened by an explicit mock-data validation in which catalogs are generated with O_k = 0, realistic DESI-like sampling (especially sparse at z ≲ 0.5), and the same multi-task kernel; without such a test it remains possible that kernel mismatch or hyperparameter marginalization shifts the reconstructed derivatives enough to produce a spurious 2σ signal precisely where leverage is weakest.
  2. [Results section] Results on the O_k null-test statistic (around Eq. (15)–(18)): the propagation of GP hyperparameter uncertainties into the final O_k error budget is not shown in sufficient detail; a table or figure quantifying the contribution of each hyperparameter to the reported 2σ deviation would clarify whether the significance is robust or dominated by prior volume.
minor comments (2)
  1. [§2] The notation for the primed quantities (D_V', D_M') is introduced without an explicit derivative definition; adding one line of mathematics would remove ambiguity for readers implementing the test.
  2. [Figures] Figure 3 (or equivalent) showing the reconstructed functions would be clearer if the individual BAO and SNe data points with error bars were over-plotted rather than only the GP mean and bands.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestions that will help strengthen our analysis. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (multi-task GP reconstruction): the claim that the ~2σ low-z O_k deviation is driven solely by data sparsity would be strengthened by an explicit mock-data validation in which catalogs are generated with O_k = 0, realistic DESI-like sampling (especially sparse at z ≲ 0.5), and the same multi-task kernel; without such a test it remains possible that kernel mismatch or hyperparameter marginalization shifts the reconstructed derivatives enough to produce a spurious 2σ signal precisely where leverage is weakest.

    Authors: We agree that an explicit mock-data validation would provide additional support for our interpretation that the low-redshift deviation arises from data sparsity. We will generate mock catalogs assuming O_k = 0 with realistic DESI-like redshift sampling and noise properties (with particular attention to the sparse coverage at z ≲ 0.5), apply the identical multi-task GP kernel and marginalization procedure, and include the results in a new subsection of the revised §4. This will allow direct comparison with the real-data reconstruction to assess whether the observed ~2σ feature is consistent with statistical fluctuations under the null hypothesis. revision: yes

  2. Referee: [Results section] Results on the O_k null-test statistic (around Eq. (15)–(18)): the propagation of GP hyperparameter uncertainties into the final O_k error budget is not shown in sufficient detail; a table or figure quantifying the contribution of each hyperparameter to the reported 2σ deviation would clarify whether the significance is robust or dominated by prior volume.

    Authors: We acknowledge that a more granular breakdown of hyperparameter uncertainty propagation would improve transparency. We will add a new figure and associated discussion in the Results section that decomposes the variance contributions from the individual GP hyperparameters (length scales, signal variances, and noise terms) to the final O_k uncertainties at different redshifts. This will clarify the relative impact of hyperparameter marginalization versus data constraints on the reported ~2σ deviation. revision: yes

Circularity Check

0 steps flagged

Null test constructed from observable ratios; GP reconstruction is a standard fitting tool with no reduction to input by construction

full rationale

The paper defines the O_k null test directly from the AP parameter F_AP combined with derivative ratios D_V'/D_V or D_M'/D_M, which are constructed to eliminate absolute distance requirements. Multi-task Gaussian processes are then applied as a nonparametric reconstruction method to these observables. No equation shows the null-test output being equivalent to a fitted hyperparameter or prior result by definition. The reported ~2σ low-z deviation is explicitly caveated by data sparsity rather than claimed as a forced prediction. Self-citations, if present, are not load-bearing for the central compatibility or curvature-null claims. The derivation chain remains self-contained against external data benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The paper relies on standard cosmological assumptions about distance measures and the validity of the null test framework, with GP hyperparameters as free parameters fitted during analysis.

free parameters (1)
  • Gaussian process hyperparameters
    Multi-task GPs typically have kernel parameters fitted to data.
axioms (2)
  • domain assumption The O_k null test can assess cosmic curvature and data compatibility in a model-independent way.
    Stated in abstract as the basis for the test.
  • domain assumption BAO AP parameter F_AP and distance ratios do not require absolute distance measurements.
    Key to the novel construction.

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