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REVIEW 4 major objections 4 minor 76 references

Redshift Dependence of $H_0$ Dipole in Pantheon+ Supernovae

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims the locally measured Hubble constant carries a directional dipole of 1.16 ± 0.28 km/s/Mpc in the lowest-redshift Pantheon+ supernovae, a signal that disappears once supernovae below z ≈ 0.038 are excluded.

desk verdict A careful Pantheon+ dipole analysis with a novel z_min scan, but the significance hinges on mocks that may miss cosmic-variance bulk flows; worth refereeing with mandatory revisions. read the letter →

arxiv 2608.07209 v1 pith:SRTWKUYJ submitted 2026-08-07 astro-ph.CO astro-ph.GAgr-qc

M.H. Jalali-Kanafi , E. Ó Colgáin , S. Pourojaghi , M.M. Sheikh-Jabbari This is my paper · ORCID
classification astro-ph.COastro-ph.GAgr-qc PACS 98.80.Es
keywords HubbleconstantH0dipolePantheon+supernovaecosmologicalprincipleanisotropicexpansionbulkflowslow-redshiftuniversetypeIa
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

The paper asks whether the cosmic expansion rate measured close to us, the Hubble constant $H_0$, is the same in every direction of the sky, and argues that at the lowest redshifts it is not. Using the Pantheon+ catalogue of Type Ia supernovae, the authors divide the sky into 48 equal-area patches (a HEALPix grid), fit an expansion rate in each patch from the supernovae within a 75° circle around it, and then fit a dipole — a pattern that is higher along one sky axis and lower along the opposite one — to the resulting map. They find a dipole amplitude $A_{\rm dip} = 1.16 \pm 0.28$ km/s/Mpc when the sample starts at $z_{\rm min} = 0.015$, falling monotonically to $0.35 \pm 0.52$ km/s/Mpc, consistent with zero, at $z_{\rm min} = 0.045$. Because even an isotropic universe yields a nonzero positive amplitude by construction, significance is judged against 1,000 mock skies built from a ΛCDM model with the same sky positions, redshifts, and noise correlations; that comparison gives 2–3σ for $z_{\rm min} \lesssim 0.032$, and the dipole points within about 30° of both the Shapley supercluster and the CMB dipole. The authors conclude that the $H_0$ dipole is a low-redshift feature, plausibly tied to local bulk flows rather than to cosmology on large scales.

What carries the argument

The load-bearing device is the reconstructed sky map of $H_0$ combined with a dipole fit on top of it. The sky is discretized into 48 equal-area HEALPix pixels, and for each pixel a spherical cap of 75° radius collects the supernovae whose distance moduli are fit, through Monte Carlo sampling, to a flat ΛCDM model while the 77 Cepheid-calibrated host supernovae fix the absolute magnitude; this yields a value $H_0(\hat n)$ with an uncertainty for every pixel. The 48 values are then fit by generalized least squares to the dipole ansatz $H_0(\hat n) = H_0^{\rm mono} + \mathbf{D}\cdot\hat n$, where $\mathbf{D}$ is the dipole vector. The essential supporting object is the 48×48 covariance matrix $\Sigma$, not taken from the Pantheon+ catalogue directly but estimated from 1,000 MC-lcdm realizations: mock skies that keep the real supernovae's sky positions and redshifts, add Gaussian magnitude noise with the full Pantheon+ covariance matrix, and run the entire map-building pipeline. From this ensemble the authors get both $\Sigma$, which the dipole fit needs, and the null distribution of $A_{\rm dip}$ against which significance is read. Significance is quoted two ways — a p-value counting how many mocks exceed the observed amplitude, and a more conservative estimator $S$ that also propagates the quoted error — while a third estimator, $\Delta H_0^{\max}$, records the largest antipodal contrast between opposite patches, with $\beta$ quantifying whether that contrast is consistent with the fitted dipole alone.

What would settle it

Recompute the $z_{\rm min} = 0.015$ dipole using a differently built null — shuffling supernova magnitudes among fixed sky positions instead of adding Gaussian noise, or using peculiar-velocity-corrected redshifts — and check whether the p-value stays below 0.001; if it does not, the signal is an artifact of the mock prescription or the redshift frame. A second check: the fitted direction near $(l, b) \approx (307°, 61°)$ predicts a coherent bulk flow of order a few hundred km/s over $z < 0.05$, so its absence in galaxy peculiar-velocity catalogues over that volume would contradict the local-structure interpretation.

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Extended reading notes

Core claim

On its own terms, the paper establishes that the local expansion rate reconstructed from Pantheon+ supernovae is anisotropic, and that the anisotropy is genuinely dipolar at low redshift. With maps built from overlapping 75° caps on a 48-pixel HEALPix grid and with redshifts in the CMB rest frame, the best-fit monopole-plus-dipole model reads $A_{\rm dip} = 1.16 \pm 0.28$ km/s/Mpc at $z_{\rm min} = 0.015$ and decreases monotonically as the lower redshift cut is raised, reaching $0.35 \pm 0.52$ km/s/Mpc at $z_{\rm min} = 0.045$. The paper does not read significance from the error bar alone, since a positive-definite amplitude is nonzero even under isotropy; instead the null distribution comes from 1,000 MC-lcdm realizations — mocks that keep the real sky positions and redshifts, add Gaussian magnitude noise with the full Pantheon+ covariance, and run the whole reconstruction pipeline. That comparison gives $p < 0.001$ (more than 3.1σ) at the lowest bin with the p-value estimator and 2.5σ with the more conservative $S$ estimator, and 2–3σ overall for $z_{\rm min} \lesssim 0.032$. For every threshold where the signal is significant, the direction clusters near the Shapley supercluster and the CMB dipole, and the amplitude becomes statistically indistinguishable from isotropy for $z_{\rm min} \geq 0.038$ — the redshift where Shapley itself begins. Residual antipodal contrast $\Delta H_0^{\max}$ remains nonzero at all thresholds, and because the $\beta$ estimator stays below 1, the authors read this as a dipole the data no longer constrains rather than as evidence for higher multipoles.

Load-bearing premise

The load-bearing premise is that the 1,000 mock skies used as the isotropic comparison — realizations that keep the real supernovae's sky positions and redshifts and add Gaussian magnitude noise with the Pantheon+ covariance matrix — faithfully represent what an isotropic universe with the same selection and correlations would look like; if those mocks misrepresent the noise, both the p-values and the dipole covariance matrix are biased in the same direction.

Editorial extensions

If this is right

  • At $z_{\rm min} = 0.015$ the expansion rate inferred from opposite patches of sky differs by up to about 3.8 km/s/Mpc, so the local $H_0$ is direction-dependent at a level current data can see.
  • Raising the lower redshift cut to $z_{\rm min} = 0.038$ erases the dipole, placing whatever produces it inside the very nearby universe, below redshift $\approx 0.038$.
  • Where the dipole is significant, its direction is stable across $z_{\rm min}$ cuts and points within about 30° of both the Shapley supercluster and the CMB dipole, tying the signal to known large-scale structure and bulk flows.
  • Because the anisotropy is dipole-dominated at low redshift ($\beta < 1$) while $\Delta H_0^{\max}$ stays nonzero where the dipole has faded, deeper data are needed to decide between an unconstrained dipole and higher multipoles.
  • If the local $H_0$ genuinely varies with direction, sub-percent $H_0$ values derived from low-redshift supernovae carry an angular selection effect that bears on the Hubble tension.

Reading between the lines

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

  • A direct test the authors mention but do not run: repeat the pipeline with peculiar-velocity-corrected redshifts ($z_{\rm HD}$) instead of CMB-frame redshifts; a surviving dipole would confirm it is a real velocity-field feature, while its disappearance would indict the redshift frame.
  • The quoted 2–3σ significance is conditional on the Gaussian-noise mock prescription; a null built by shuffling magnitudes among fixed sky positions, or one injecting simulated peculiar velocities, could shift the p-values either way — a comparison the authors defer to a companion paper.
  • Masking the Shapley region in the $z_{\rm min} = 0.015$ sample is a cheap, decisive experiment: a surviving dipole would implicate a broader bulk flow, while its vanishing would pin the anisotropy on the supercluster itself.
  • Future, larger low-redshift supernova samples should sharpen the dipole if it is physical; if its significance instead shrinks as more nearby supernovae are added, the present signal was a small-sample fluctuation.
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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 / 4 minor

Summary. The paper reconstructs directional maps of the locally inferred Hubble constant from Pantheon+ Type Ia supernovae in the redshift interval z_min < z < 0.2, using HEALPix Nside=2 pixels with 75-degree spherical caps. A dipole is fitted to the 48-pixel H0 maps with generalized least squares, using a 48x48 covariance matrix estimated from 1000 Monte Carlo ΛCDM mocks. The authors report A_dip = 1.16 ± 0.28 km/s/Mpc at z_min = 0.015, a downward trend with increasing z_min, a dipole direction near the Shapley supercluster and CMB dipole, and claim a 2-3σ significance for z_min ≲ 0.032 that weakens at higher thresholds.

Significance. The paper addresses a timely and contested question: whether the local expansion rate is anisotropic. Its main strengths are the explicit treatment of the positive-definite nature of the dipole amplitude by comparing against an isotropic mock ensemble rather than A_dip = 0, and the careful propagation of the Pantheon+ covariance through the directional reconstruction into the patch covariance matrix. The tomographic z_min scan is a useful extension of earlier fixed-window analyses. If the mock-based significance is robust, the result would strengthen the case for a local bulk-flow-induced H0 dipole and would be relevant to the H0 tension debate. The analytical approximation for the patch covariance in Appendix A is a useful addition. However, the headline significance depends critically on the fidelity of the mock ensemble, which is not established.

major comments (4)
  1. [III C, Eqs. (16)-(18)] The MC-lcdm mocks add Gaussian magnitude noise with the Pantheon+ covariance to the same sky positions and redshifts, but they do not generate new realizations of the large-scale velocity field. At z < 0.2, the dominant ΛCDM contribution to a dipole in the locally inferred H0 comes from coherent peculiar velocities, whose correlations extend over tens to hundreds of Mpc. If, as in the public Pantheon+ release, the peculiar-velocity term in the covariance is effectively diagonal, these mocks will under-represent the cosmic-variance dipole width. Because the same mocks define both the null distribution in Fig. 5 and the GLS covariance matrix Σ in Eq. (18), the p-values and the quoted dipole uncertainties are biased in the same direction, and the 2-3σ significance is not yet supported. I recommend adding mocks that include a ΛCDM velocity-field realization (e.g., linear-theory realizations or N-body mocks) or an analytic covariance that includes the velocity power spectrum, and showing how the significance changes.
  2. [Abstract and IV A, Table I] The abstract states that A_dip decreases monotonically with z_min, but the values in Table I are not monotonic: A_dip increases from 0.87 at z_min = 0.022 to 0.91 at z_min = 0.025, and from 0.93 at z_min = 0.028 to 1.05 at z_min = 0.032. The overall trend is downward, but the non-monotonic features should be acknowledged and discussed, and the abstract should be rephrased accordingly.
  3. [Abstract, IV A, Table I, Fig. 6] The claim of a "2-3σ dipole pattern for z_min ≲ 0.032" overstates the results. The conservative S estimator in Eq. (22) exceeds 2σ only at z_min = 0.015 (2.5σ); at z_min = 0.018 it is 1.9σ, and at z_min = 0.022 it is 1.4σ. Even with the p-value estimator, z_min = 0.028 gives 1.7σ and z_min = 0.035 gives 1.2σ. The significance statement should be restricted to the lowest redshift bin or presented bin-by-bin without the aggregate "z_min ≲ 0.032" wording.
  4. [III C, Eq. (21)] The definition of the p-value is ambiguous: N_cross is described as "the number of mocks crossing the red line" in Fig. 5. It must be stated explicitly whether this is the number of mocks with A_dip ≥ A_obs, the number whose 1σ band crosses the observed line, or some other tail-count rule. A precise definition is needed for reproducibility, and the same sentence should clarify that the p-value and the S estimator are not independent because both derive from the same mock ensemble.
minor comments (4)
  1. [II, Table I] The text says the analysis uses "ten equally spaced values" of z_min in [0.015, 0.045], but the actual values in Table I are 0.015, 0.018, 0.022, 0.025, 0.028, 0.032, 0.035, 0.038, 0.042, 0.045, which are not equally spaced (differences alternate between 0.003 and 0.004). Please either use a regular grid or describe the chosen values as a set of ten specific thresholds.
  2. [III B, Eq. (22)] Equation (22) defines S as a signal-to-noise-like ratio, but it is not a standard Gaussian significance because σ_obs and σ_MC are not fully independent and the null distribution is non-Gaussian. The text labels S as conservative, but a sentence explicitly stating that S is a heuristic estimator rather than a formal significance would avoid misinterpretation.
  3. [IV D, Eq. (25)] The denominator in Eq. (25) is written as σ_{ΔH0/2}, which is ambiguous because the subscript is a ratio of two quantities. Please define it as the propagated uncertainty on ΔH0^{max}/2, for example σ_{Δ/2}, and state explicitly that the covariance between ΔH0^{max}/2 and A_dip is neglected, as the footnote already indicates.
  4. [Fig. 5] The caption of Fig. 5 states that the red line and band are "read from the fourth column in Table I," but it would be clearer to state that the band represents the 1σ uncertainty on A_dip and that the histograms are normalized probability densities.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the H0 dipole is a direct GLS fit to Pantheon+ maps; MC-lcdm mocks provide only null distribution and covariance, not the claimed signal.

full rationale

The central claim is a measured dipole in local H0 maps reconstructed from Pantheon+ data. The dipole amplitude A_dip is computed directly from the data via the GLS fit in Eqs. (9)-(15), and its statistical significance is assessed by comparing it to 1000 MC-lcdm realizations. The same mock ensemble is used both to estimate the 48x48 patch covariance matrix Sigma (Eq. 18) and to define the null distribution for the p-value (Eq. 21), but this is a standard parametric bootstrap: Sigma is an estimator of the noise covariance of the reconstructed H0 map, and the null is the sampling distribution under statistical isotropy. Neither equation defines A_dip in terms of the mocks; the observed dipole is not generated by the simulations. The paper explicitly handles the positive-definiteness of A_dip by using the mock distribution, which has a non-zero mean, rather than the naive A_dip = 0. Self-citations to earlier works by the authors (e.g., refs. [11], [13], [68], [76], [77], [63]) are contextual comparisons or methodological references, not load-bearing uniqueness claims or imported ansatze; removing them would not change the numerical dipole. Concerns that the mocks may under-represent cosmic variance from correlated peculiar velocities are validity questions about the null model, not circularity by construction.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

The paper is an observational data analysis; no new physics entities are introduced. The quantities the central claim depends on are either fitted to the SNe data (monopole, dipole components, M, Omega_m), chosen by hand (z_min grid, cap radius, HEALPix resolution), or assumed from the standard model and the Pantheon+ covariance. The list above records these inputs explicitly.

free parameters (6)
  • Dipole vector components D_x, D_y, D_z = A_dip = 1.16 ± 0.28 km/s/Mpc at z_min=0.015
    Central claim is the amplitude |D|, obtained by the GLS fit in Eq. (15).
  • Monopole H0_mono = About 71-75 km/s/Mpc depending on z_min
    Fitted jointly with the dipole in Eq. (9); needed to define the H0 map but not the target of the paper.
  • Absolute magnitude M of SNe Ia = Not quoted
    Calibrated globally with the 77 Cepheid-host SNe in Eqs. (5)-(8); its uncertainty propagates into every patch H0.
  • Matter density Omega_m = Not quoted
    Fitted per patch, but enters the distance modulus only weakly at z<0.2; still affects the reported H0 errors.
  • Lower redshift thresholds z_min = Ten values from 0.015 to 0.045
    Hand-chosen scanning variable; the significance claim is tied to the lowest thresholds.
  • Spherical cap radius and HEALPix resolution = 75 degrees, Nside=2 (48 pixels)
    Hand-chosen; controls how many SNe share each patch and how correlated the patches are.
assumptions (6)
  • domain assumption Spatially flat LambdaCDM is used to convert redshifts to distance moduli (Eqs. 2-4).
    The paper claims model independence at z<0.2, but the distance relation is still LambdaCDM; other models could slightly shift H0 values.
  • domain assumption CMB-frame redshifts without peculiar-velocity corrections are the appropriate input for the H0 maps.
    Section II states z_CMB is used, not z_HD; the paper argues conventional corrections would not change the result but does not test this.
  • domain assumption The Pantheon+ covariance matrix correctly describes all statistical and systematic uncertainties, including cross-SNe correlations.
    Every patch fit and every mock realization truncates this matrix; an underestimated covariance would inflate significances.
  • domain assumption The monopole plus dipole ansatz (Eq. 9) adequately captures the angular structure of H0 maps.
    The beta test in Section IV D finds beta<1, but the test has large errors and cannot exclude higher multipoles.
  • domain assumption The MC-lcdm mocks, with fixed sky positions and redshifts and Gaussian magnitude noise drawn using the Pantheon+ covariance, represent the isotropic null hypothesis.
    Section III C uses these mocks both for the null distribution and for the covariance matrix Sigma; any mismatch biases the p-values.
  • domain assumption Uniform priors on H0, Omega_m, M yield unbiased posterior medians.
    Section III A specifies uniform priors; the posterior median is assigned to each pixel.

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Cite this review

Pith. "Pith review of Redshift Dependence of $H_0$ Dipole in Pantheon+ Supernovae." pith.science (2026). https://pith.science/paper/SRTWKUYJ

@misc{pith2026260807209,
  author       = {Pith},
  title        = {Pith review of: Redshift Dependence of $H_0$ Dipole in Pantheon+ Supernovae},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SRTWKUYJ}},
  note         = {Machine review of arXiv:2608.07209}
}
abstract

We examine the dipole structure in the local cosmic expansion rate $H_0$ using the Pantheon+ compilation of Type Ia supernovae. We reconstruct directional maps of $H_0$ across the sky within the low-redshift regime, $z_{\rm min} < z < 0.2$, evaluated in the CMB rest frame. To perform a tomographic analysis, we choose 10 different values for $z_{\rm min}\in [0.015 , 0.045]$ range. We find a dipole amplitude $A_{\rm dip}=1.16\pm0.28\ {\rm km/s/Mpc}$ for the lowest bin, monotonically decreasing to $A_{\rm dip} =0.35\pm0.52\ {\rm km/s/Mpc}$, as we increase $z_{\rm min}$. Since $A_{\rm dip}$ is positive-definite, by construction $A_{\rm dip} > 0$ even in an isotropic universe, thus to assess the statistical significance, we compare to $A_{\rm dip}$ generated from 1000 Monte Carlo simulations based on the $\Lambda$CDM model and the same redshift ranges. Our results reveal a $2-3\sigma$ dipole pattern (depending on how the significance level is computed) for $z_{\rm min}\lesssim 0.032$. For redshift thresholds where the signal remains statistically significant ($>2\sigma$), the inferred dipole direction points close to Shapley supercluster and CMB dipole directions. As $z_{\rm min}$ increases, the dipole amplitude diminishes in line with expectations of a higher redshift isotropic Universe, but the difference in maximal antipodal $H_0$ determinations, $\Delta\text{H}_0^{\rm max}$, retain a signal of an anisotropy that disappears in the dipole ansatz. The decreasing statistical significance of all estimators with increasing $z_{\rm min}$ suggests that any $H_0$ dipole is a low-redshift feature.

Figures

Figures reproduced from arXiv: 2608.07209 by the authors.

Figure 1
Figure 1. FIG. 1: Angular distribution of the Pantheon+ SNe in galactic coordinates for different redshift selections. As [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Schematic illustration of the computational pipeline we adopted to probe anisotropies in the local cosmic [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Directional maps of the locally inferred [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Dipole amplitude best-fit values as a function [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Distribution of the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Statistical significances of the dipole [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The effect of the covariance matrix on the dipole amplitude [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Dipole direction as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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