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arxiv: 2606.12822 · v1 · pith:5SHY7ALGnew · submitted 2026-06-11 · 🌌 astro-ph.CO

Geometric obstruction to resolving the Hubble tension: orthogonality of scale and shape in distance measurements

Zhihuan Zhou , Zhuang Miao , Sheng Bi , Chaoqian Ai , Hongchao Zhang This is my paper

Pith reviewed 2026-06-27 06:21 UTC · model grok-4.3

classification 🌌 astro-ph.CO
keywords Hubble tensionBAOsupernovaesound horizondark energy equation of statecosmological parameters
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0 comments X

The pith

The gap in matter density inferred from BAO versus supernovae stays fixed under any uniform sound-horizon rescaling and cannot be removed by smooth late-time dark-energy adjustments.

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

The paper establishes that in standard cosmology the difference between the matter density preferred by baryon acoustic oscillation data and that preferred by supernova data remains exactly the same no matter how the sound horizon is uniformly rescaled. Adjustments to the dark-energy equation of state at late times also leave this gap intact because the two datasets respond in nearly opposite directions to those adjustments. As a result, no combination of early-time rescaling and late-time smooth deformation can bring the two probes into agreement while still matching the local distance-ladder measurement of the Hubble constant.

Core claim

Within ΛCDM the BAO–SN matter-density gap ΔΩ_m = 0.037 is exactly invariant under the sound-horizon rescaling α ≡ r_s^mod / r_s^ΛCDM, and late-time w(z) deformations cannot eliminate this gap either: reconciling the two datasets requires opposite deformations—phantom (w < −1) for BAO, quintessence (w > −1) for SN at z < 0.5—an anti-alignment quantified by cosθ = −0.97 in w(z) space.

What carries the argument

The invariance of the BAO–SN ΔΩ_m gap under sound-horizon rescaling α, together with the near-orthogonal response of the two datasets to late-time w(z) deformations (cosθ = −0.97).

If this is right

  • The optimal rescaling α* = 0.992 produces a joint fit with H0 = 70.3 ± 0.3 km s^{-1} Mpc^{-1}, still 3.2σ below the local ladder value.
  • The deformation space already spans 93 percent of the Ω_m response direction, yet the two datasets continue to disagree through independent channels.
  • The remaining Hubble-constant deficit is anchored in the absolute distance scale, which late-time w(z) can reshape but cannot uniformly rescale.

Where Pith is reading between the lines

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

  • The claimed orthogonality between scale (sound-horizon) and shape (redshift-dependent distances) implies that any resolution must introduce either scale-dependent early-universe physics or modifications that affect the absolute calibration outside the current deformation space.
  • If future data tighten the BAO and supernova constraints without closing the gap, the tension would point to a fundamental mismatch in how the two probes measure distance rather than to missing parameters.

Load-bearing premise

That all allowed changes consist of a uniform sound-horizon rescaling plus a smooth late-time w(z) function whose effect stays inside the 93 percent of the matter-density direction already covered by that space.

What would settle it

A direct measurement showing that the BAO and supernova matter-density inferences move toward each other under a uniform sound-horizon rescaling, or that a single smooth w(z) function brings both inferences into agreement without requiring opposite signs.

Figures

Figures reproduced from arXiv: 2606.12822 by Chaoqian Ai, Hongchao Zhang, Sheng Bi, Zhihuan Zhou, Zhuang Miao.

Figure 1
Figure 1. Figure 1: Pairwise tension decomposition as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Best-fit trajectories in (Ωm, H0) as α varies over [0.970, 1.000]. BAO traces a vertical path (Ωm = 0.297 con￾stant), SN is a fixed point, and Planck moves diagonally due to Ωmh 2 ≈ const. The BAO–SN Ωm gap is exactly α￾invariant. 0.3 km s−1 Mpc−1 and Ωm = 0.290±0.004—still 3.2σ be￾low SH0ES. The Planck penalty for α = 0.992 is minimal: χ 2 Planck rises by only 1.0 relative to α = 1, because the full param… view at source ↗
Figure 3
Figure 3. Figure 3: Two-dimensional ∆χ 2 contours in the (Ωm, H0) plane at α ∗ = 0.992. BAO (blue), Planck (green, Fisher ellipse), and SN (red) show fully separated 1σ regions. The joint minimum (black star) lies outside all individual preferred regions. Right panel: zoom on the Planck ellipse. V. WHY w(z) CANNOT CLOSE THE GAP A. w(z) recovery curves To visualize the anti-alignment, we invert the FPA re￾sponse relation [Eq. … view at source ↗
Figure 4
Figure 4. Figure 4: Dark energy equation of state w(z) required by each dataset to reach the joint best-fit from its own optimum, at α ∗ = 0.992. BAO (blue) needs phantom dark energy (w < −1) at low redshift, while SN (red) needs quintessence (w > −1). The curves lie on opposite sides of ΛCDM (w = −1, dashed), confirming the gradient anti-alignment. to w(z), and the coefficients needed to shift it scale inversely with this sm… view at source ↗
Figure 5
Figure 5. Figure 5: Ωm profile χ 2 for BAO (blue), Pantheon+ (red), and DES-SN5YR (orange), with H0 profiled out. Both SN datasets prefer Ωm ≈ 0.330, separated from BAO’s Ωm = 0.297 by ∆Ωm ≈ 0.035. The obstruction is SN-dataset inde￾pendent. Planck spectrum is already well-fit (χ 2/dof = 0.90), leav￾ing no room for damping modifications. VII. ROBUSTNESS A. SN dataset independence Replacing Pantheon+ with DES-SN5YR (1820 SNe) … view at source ↗
read the original abstract

We identify a geometric obstruction to resolving the Hubble tension by combining early-time sound-horizon reduction with late-time smooth dark energy. Within $\Lambda$CDM, the BAO--SN matter-density gap $\Delta\Omega_m = 0.037$ is exactly invariant under the sound-horizon rescaling $\alpha \equiv r_s^{\rm mod}/r_s^{\Lambda{\rm CDM}}$, and late-time $w(z)$ deformations cannot eliminate this gap either: reconciling the two datasets requires \emph{opposite} deformations -- phantom ($w < -1$) for BAO, quintessence ($w > -1$) for SN at $z < 0.5$ -- an anti-alignment quantified by $\cos\theta = -0.97$ in $w(z)$ space. A full MCMC analysis of DESI DR2 BAO, Planck plik\_lite, and Pantheon+ bears this out: the optimal $\alpha^* = 0.992$ ($0.8\%$ $r_s$ reduction) brings the joint fit to $H_0 = 70.3 \pm 0.3\;\mathrm{km\,s^{-1}\,Mpc^{-1}}$, still $3.2\sigma$ below SH0ES, with the inter-dataset tension reduced but not removed. The obstruction reflects not a shortage of model freedom but an irreducible disagreement between probes. The deformation space $\{\alpha, \beta_{\rm damp}, w(z)\}$ already spans $93\%$ of the $\Omega_m$ response direction; nonetheless BAO and SN constrain $\Omega_m$ through independent channels and disagree, while the residual $H_0$ deficit, anchored by the local distance ladder, resides in the absolute distance scale that $w(z)$ reshapes but cannot rescale.

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 paper claims a geometric obstruction to resolving the Hubble tension: within ΛCDM the BAO–SN matter-density gap ΔΩ_m = 0.037 is exactly invariant under sound-horizon rescaling α ≡ r_s^mod / r_s^ΛCDM, while smooth late-time w(z) deformations cannot close the gap because they require opposite (phantom for BAO, quintessence for SN at z<0.5) modifications, quantified by cosθ = −0.97. An MCMC fit to DESI DR2 BAO + Planck plik_lite + Pantheon+ yields α* = 0.992, H0 = 70.3 ± 0.3 km s^{-1} Mpc^{-1} (still 3.2σ below SH0ES), with the {α, β_damp, w(z)} space spanning 93 % of the Ω_m response direction yet leaving an irreducible inter-probe disagreement.

Significance. If the algebraic invariance and anti-alignment hold, the result demonstrates that standard early-time rescaling plus late-time smooth DE extensions are geometrically insufficient to reconcile BAO and SN constraints on Ω_m, implying that further progress requires either scale-dependent modifications or explicit acknowledgment of probe-level tension rather than parameter-space enlargement. The direct MCMC validation on current datasets adds concrete support.

major comments (2)
  1. [abstract and §4] The statement that the deformation space {α, β_damp, w(z)} spans 93 % of the Ω_m response direction is presented without an explicit derivation or formula; please supply the linear-response calculation or matrix projection that yields this fraction (abstract and §4).
  2. [§2] §2, Eq. (definition of ΔΩ_m): the exact invariance under α is asserted as a geometric consequence of distance definitions; confirm that this holds after marginalization over the full set of nuisance parameters used in the BAO and SN likelihoods, not only in the idealized ΛCDM limit.
minor comments (2)
  1. Clarify the precise redshift range and functional form assumed for the w(z) parameterization when computing the response vectors and cosθ (currently stated only qualitatively).
  2. Table or figure showing the individual BAO and SN Ω_m posteriors before and after the α rescaling would help readers visualize the claimed invariance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful review and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [abstract and §4] The statement that the deformation space {α, β_damp, w(z)} spans 93 % of the Ω_m response direction is presented without an explicit derivation or formula; please supply the linear-response calculation or matrix projection that yields this fraction (abstract and §4).

    Authors: We agree that an explicit derivation was omitted. The 93% figure is obtained from a linear-response calculation: we construct the matrix whose columns are the partial derivatives of the model predictions (BAO and SN distances) with respect to the deformation parameters in {α, β_damp, w(z)}, then project the column space onto the Ω_m response direction and compute the fraction of the squared norm explained by that projection. We will add this matrix projection and the resulting formula to the revised §4 (and note it in the abstract). revision: yes

  2. Referee: [§2] §2, Eq. (definition of ΔΩ_m): the exact invariance under α is asserted as a geometric consequence of distance definitions; confirm that this holds after marginalization over the full set of nuisance parameters used in the BAO and SN likelihoods, not only in the idealized ΛCDM limit.

    Authors: The invariance of ΔΩ_m under α follows directly from the definitions of the comoving distances and the sound-horizon rescaling, which enter both probes identically; this geometric property is independent of the likelihood details. We confirm that it remains exact after marginalization over the full nuisance sets in the DESI DR2 BAO and Pantheon+ SN likelihoods (including damping, bias, and calibration parameters), as these nuisances are orthogonal to the uniform α rescaling. We will add a brief clarifying sentence in §2. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation establishes invariance of ΔΩ_m under α rescaling as a geometric property arising directly from the definitions of BAO (r_s/D_V) and SN (luminosity distance) observables in terms of the expansion history; this orthogonality of scale and shape follows from the distance-redshift relations without parameter fitting or self-citation. The cosθ = −0.97 anti-alignment is obtained by projecting the linear response vectors of smooth w(z) deformations onto the Ω_m direction, an explicit computation independent of the target gap value. The MCMC (α* = 0.992) serves only as numerical confirmation. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the result is self-contained against the distance definitions and external data.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that distance measures for BAO and SN can be deformed only through a uniform sound-horizon factor α and a smooth late-time w(z); no new particles or forces are introduced, but the deformation space itself is treated as exhaustive for the purpose of the obstruction argument.

free parameters (2)
  • α
    Sound-horizon rescaling factor fitted in the MCMC; optimal value 0.992 reported.
  • w(z) parameters
    Coefficients or nodes of the dark-energy equation-of-state function varied to minimize tension.
axioms (1)
  • domain assumption BAO and SN constrain Ω_m through independent channels whose response directions in w(z) space are anti-aligned.
    Invoked to conclude that the datasets disagree even after maximal deformation.

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discussion (0)

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Reference graph

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