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The redshift dependence of the inferred $H_0$ in a local void solution to the Hubble tension

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A Gpc-scale local void can explain the Hubble tension, because the predicted decline of the inferred $H_0$ with redshift reproduces the uncorrelated measurements of Jia et al. for Gaussian and Exponential void profiles.

desk verdict A transparent and useful test of the local-void solution to the Hubble tension: the predicted H0(z) decline broadly matches Jia et al., but the approximate GR term in eq. (6) is a real caveat because it can shift the curves by about the size of the data error bars. read the letter →

arxiv 2412.12245 v2 pith:TQF4C4XR submitted 2024-12-16 astro-ph.CO

Sergij Mazurenko , Indranil Banik , Pavel Kroupa This is my paper · ORCID
classification astro-ph.CO PACS 98.80.-k98.80.Es
keywords HubbletensionlocalvoidKBCsupervoidH0redshiftdependencepeculiarvelocitiesgravitationalconstantMOND
topics The Hubble Tension
open problems The Hubble Tension
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

This paper argues that the Hubble tension—the roughly 8 percent mismatch between the locally measured expansion rate ($H_0 \approx 73$ km/s/Mpc) and the value inferred from the early universe ($H_0^{\rm Planck} = 67.4 \pm 0.5$ km/s/Mpc)—can be explained by our location inside the Gpc-scale KBC supervoid. Outflow from the underdense void adds extra redshift to nearby sources, so an observer who assumes a homogeneous universe overestimates $H_0$. The paper computes the resulting $H_0(z)$ curve, the expansion rate inferred from data in a narrow redshift bin around $z$, and shows it declines toward the Planck value at high redshift. Comparing with the uncorrelated redshift-binned measurements of Jia et al., the Gaussian and Exponential void profiles give broad agreement. If correct, the tension dissolves without modifying early-universe physics.

What carries the argument

The central object is the predicted $H_0(z)$ curve, constructed from Equation 6, which writes the total redshift of a source as the product of the cosmological factor $1/a(t)$, a special-relativistic Doppler factor $\sqrt{(c+v_{\rm int})/(c-v_{\rm int})}$ from the void outflow, and an approximate gravitational-redshift factor $\exp[(1/c^2)\int g_{\rm void}\,dr]$ for light climbing out of the potential hill. Emission times are found by intersecting particle trajectories with the observer's past lightcone via Equation 5. Three reconstruction methods convert this to $H_0(z)$; Method 3, which varies $H_0$ in a trial expansion history until $a = a_{\rm app}$ at the lookback time, most closely mirrors the JHW23 analysis. The near-identity of the Gaussian and Exponential curves arises because both place the void's deepest point at its centre, giving nearly the same combined Doppler-plus-GR boost.

What would settle it

Measure $H_0(z)$ in uncorrelated narrow bins across $z = 0.05$–$2$ with bin-level precision near $1$ km/s/Mpc: if the inferred value stays at the local $\approx 73$ km/s/Mpc beyond $z = 1$, or drops to the Planck value by $z = 0.3$, the predicted gradual decline for the Gaussian and Exponential profiles is ruled out. A direct probe of the void's potential, such as stacked lensing or the integrated Sachs-Wolfe imprint, that excludes the potential required for the gravitational-redshift term would similarly falsify the mechanism.

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Extended reading notes

Core claim

For the Gaussian and Exponential void density profiles of the HBK20 models, the predicted $H_0(z)$ at the void centre declines from the local $\approx 73$ km/s/Mpc to within $1\sigma$ of the Planck value by $z \approx 1.8$, tracking the declining trend observed by Jia et al. (2023) and the updated Jia et al. (2024) analysis. The gravitational-redshift contribution overtakes the Doppler term at $z \gtrsim 0.5$, meaning the void's potential hill, not only its outflow velocities, keeps the apparent $H_0$ elevated at intermediate redshifts. The Maxwell-Boltzmann profile converges too quickly and is already disfavoured by the observed bulk flow at $z < 0.1$, so the agreement is best when the void is deepest at its centre. The gradual rather than abrupt decline of $H_0(z)$ also speaks against pre-recombination solutions to the Hubble tension.

Load-bearing premise

Everything hangs on Equation 6, the redshift formula from HBK20 that multiplies the cosmological factor by a Doppler factor and an approximate gravitational-redshift integral; if that formula misestimates the non-cosmological redshift, all three $H_0(z)$ reconstruction methods are biased and the agreement with Jia et al. would be spurious.

Editorial extensions

If this is right

  • The observed decline of $H_0(z)$ toward the Planck value at high redshift is difficult to reconcile with early-time solutions that require a faster expansion throughout cosmic history.
  • A local-void resolution keeps the background $H_0$ at the Planck value, consistent with cosmic chronometer and stellar age constraints that disfavour a 10 percent younger universe.
  • BAO analyses that fix the standard ruler at 147.5 cMpc are effectively guaranteed to return the Planck value, so the void scenario must ultimately be tested with the raw BAO observables rather than $H_0(z)$ recovers.
  • The sensitivity of the curve to whether the void is deepest at its centre—rather than to its overall size—provides a new observational handle on the void's internal density profile.
  • Larger voids raise $H_0(z)$ only mildly, so the intermediate-redshift discrepancy with JHW23 cannot be removed by void size alone; other structures or systematics are needed.

Reading between the lines

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

  • The model's gravitational-redshift term is testable independently of the velocity field: an accurate reconstruction of the void's potential from weak lensing or the integrated Sachs-Wolfe effect would predict a redshift offset that can be compared with Equation 6.
  • An off-centre observer would see a line-of-sight-dependent $H_0(z)$ at low redshift, which the paper notes only qualitatively; quantifying this could connect the void scenario to reported anisotropies in supernova-based $H_0$ measurements.
  • The slight GR-induced floor in Method 1, which keeps $H_0(z)$ a few percent above Planck at all redshifts, offers a clean discriminant: comparing $z$-based and distance-based $H_0$ estimators at very high redshift would reveal whether such a floor is present.
  • If the void is real, it should leave a kinetic Sunyaev-Zel'dovich signal from the outflowing gas; upcoming CMB surveys could search for the corresponding velocity field at $z \sim 0.1$–$0.5$.
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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

2 major / 4 minor

Summary. This paper tests a late-time local-void resolution of the Hubble tension by computing how the inferred H0 would vary with redshift in the semi-analytic KBC void models of HBK20. The authors use three reconstruction methods, all based on the redshift formula of HBK20, and find that the resulting H0(z) curves decline from roughly the local SH0ES value to the Planck background value. For the Gaussian and Exponential void density profiles the decline is claimed to be in reasonable agreement with the redshift-binned H0 measurements of JHW23 and JHW24, while the Maxwell-Boltzmann profile converges too quickly. The authors also show that simply enlarging the Exponential void does not remove the intermediate-redshift discrepancy, and conclude that a local supervoid can solve the Hubble tension while keeping a Planck-compatible background H0.

Significance. The paper offers a clearly falsifiable prediction: if the KBC void is responsible for the Hubble tension, H0(z) must decline with redshift and approach the Planck value beyond z≈2, in the specific way set by the void profile. The high-redshift portion of this prediction is genuinely independent of the comparison data, since the HBK20 models were not fitted to the JHW23/JHW24 measurements, and the authors use a comparison set specifically designed to reduce bin-to-bin correlations. The paper is also transparent about its own limitations, including the central-observer assumption, the restricted redshift range z≥0.05, and the approximate nature of the GR term. However, the central comparison is made visually: the model curves have no uncertainty bands, no goodness-of-fit statistic is given, and the GR term in Eq. (6) is not error-controlled. These issues must be addressed before the 'broad agreement' can be regarded as established rather than suggestive.

major comments (2)
  1. [Section 2, Eq. (6)] The gravitational-redshift factor exp[(1/c^2)∫g_void dr] in Eq. (6) is evaluated as a static weak-field integral along the photon path. This neglects the time dependence of the void potential between emission and observation and the bending of the null geodesic. In the HBK20 models the void is still evolving at z<1, and Figure 3 shows that this GR term contributes about 1 km/s/Mpc near z≈0.5, comparable to the quoted JHW23/JHW24 errors. Because the same formula underlies all three reconstruction methods and all three profiles, an unquantified error in this term is load-bearing for the claimed agreement. Please provide a quantitative estimate of the neglected time-derivative and path-bending corrections, ideally by comparing with an exact light-cone calculation (e.g., an LTB or Szekeres model), or at least a bound showing that the corrections are small compared with the data uncertainties.
  2. [Section 3, Fig. 3] The central claim of 'reasonable agreement' with JHW23 and JHW24 is based exclusively on visual comparison. No uncertainty bands are shown for the model curves and no goodness-of-fit statistic is computed, even though the differences between curves and data are comparable to the 1–2 km/s/Mpc data uncertainties in the intermediate-redshift range. Please add a quantitative comparison, for instance a chi-square statistic evaluated against the binned JHW23/JHW24 points with their covariance, and propagate the HBK20 parameter uncertainties (or state clearly that the curves are deterministic). This would also make the conclusion in Section 4.1 about the limited effect of enlarging the void testable rather than qualitative.
minor comments (4)
  1. [Figure 3] The legend contains duplicated entries for 'Jia et al. 2023' and 'Jia et al. 2024', making it difficult to tell which marker style corresponds to which data set; please clarify the legend entries.
  2. [Equation (10)] The notation H_sim,1_0/H0 is ambiguous because H0 appears on both sides of the equation; please label the background value explicitly, e.g., H0,bg, to distinguish it from the inferred quantity.
  3. [Section 4.2] The central-observer assumption is acknowledged as a limitation, but the text does not quantify how the 100–150 Mpc offset inferred from the bulk flow would affect the H0(z) curves at the lowest redshifts shown (z≥0.05); a brief order-of-magnitude estimate would be helpful.
  4. [Section 4] The discussion of possible BAO circularity in JHW23 is qualitative; since the authors already note that SNe alone give a declining trend, a comparison restricted to non-BAO data would directly support the main claim.

Circularity Check

1 steps flagged · score 4.0 of 10

Low-redshift end of the predicted H0(z) curves inherits the HBK20 fit to local H0 and q0, but the intermediate/high-redshift decline is a genuinely independent prediction tested against JHW23 data.

  1. fitted input called prediction [Section 4, opening paragraph (DISCUSSION)]
    "The HBK20 models were constrained to fit the density profile of the KBC void (Keenan, Barger & Cowie 2013) and use outflows from it to also fit the locally measured H0 and q0 (Equation 9). ... Since the HBK20 models assume a local void with a background Planck cosmology, it is not surprising that the simulated H0(z) curves all show a transition from the high local cz′ down to the low Planck value at high redshift."

    The best-fitting void parameters in HBK20 were adjusted to reproduce the local H0 and q0 measurements at z < 0.15, and the predicted H0(z) curves are computed from the same non-cosmological redshift model (Equation 6) that produced that fit. Thus the low-redshift end of the predicted curves, and the fact that they start near the locally measured high H0, are not independent predictions but a restatement of the fit. The paper itself acknowledges that the high-redshift approach to the Planck value is unsurprising because the background cosmology is Planck. The genuinely predictive content is the shape and rate of the decline at intermediate and high redshift, which was not used to fit the model; this is why the circularity is only partial.

full rationale

The paper is primarily a forward test of the previously published HBK20 void models against the JHW23/JHW24 H0(z) compilations. The central comparison at z ≳ 0.2 is not fitted to those data: the non-cosmological contributions to the redshift are computed from the HBK20 trajectories via Equation 6, converted to H0(z) using three independent methods, and then compared with observations published after HBK20. This gives the central claim independent, falsifiable content, so the self-citation to HBK20 does not make the derivation circular. The low-redshift agreement is inherited from the HBK20 fit to local H0 and q0, and the high-redshift convergence to the Planck value is partly by construction given the assumed Planck background, but the shape of the transition is a real prediction. The approximate GR term in Equation 6 is a plausible systematic concern, but that is a correctness risk, not a circularity. Overall, the paper contains one significant self-citation chain with partial fitted-input content, while the main high-redshift result remains an independent prediction.

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

The central claim inherits the full HBK20 model: a MOND/nuHDM background, three fitted density profiles, spherical symmetry, and a central observer. No new entities are introduced. The free parameters are the HBK20 best-fit void parameters, adopted without re-fitting. The most fragile assumptions are the approximate GR redshift formula and the neglect of off-centre observer effects and void anisotropy.

free parameters (1)
  • HBK20 void profile parameters (density contrast, scale radius, sharpness for Maxwell-Boltzmann, Gaussian, Exponential… = Best-fitting values from HBK20, not restated in this paper
    The predicted H0(z) curves are computed directly from these previously fitted parameters; the paper adopts them without re-fitting, so the agreement with low-z data is partly inherited from the fit.
assumptions (4)
  • domain assumption The HBK20 void models, evolved under the MOND force law in a nuHDM background, correctly describe the local density and velocity field.
    The entire prediction chain relies on this non-standard physics; the paper states the KBC void is a 6 sigma problem in LambdaCDM and that MOND provides the needed enhanced structure growth (Section 1.1).
  • domain assumption Equation 6 correctly gives the total redshift as the product of cosmological, Doppler, and gravitational redshift contributions, with the GR term evaluated via the approximate integral from HBK20.
    All three methods reconstruct H0 from z using this equation; no independent derivation or numerical check of the GR approximation is given in this paper (Section 2, equation 6).
  • domain assumption The observer is at the void centre.
    Stated in Section 2; the paper restricts figures to z >= 0.05 to limit the impact of this assumption, but it still affects the shape of the H0(z) curve.
  • domain assumption The void is spherically symmetric and its systemic velocity cancels when averaging over sky directions.
    Stated in Sections 2 and 4.2; anisotropies and the whole-void motion are neglected, which could bias the inferred H0(z) if sky coverage is incomplete.

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Pith. "Pith review of The redshift dependence of the inferred $H_0$ in a local void solution to the Hubble tension." pith.science (2026). https://pith.science/paper/TQF4C4XR

@misc{pith2026241212245,
  author       = {Pith},
  title        = {Pith review of: The redshift dependence of the inferred $H_0$ in a local void solution to the Hubble tension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TQF4C4XR}},
  note         = {Machine review of arXiv:2412.12245}
}
abstract

Galaxy number counts suggest that we are located within the Gpc-scale KBC void. The Hubble tension might arise due to gravitationally driven outflow from this void, as explored in detail by Haslbauer et al. We explore how the impact of the void on redshift decays at large distances. We define $H_0(z)$ as the present expansion rate $H_0$ that would be inferred from observations in a narrow redshift range centred on $z$. We find $H_0(z)$ in three different ways, all of which give similar results. We then compare these results with the observations of Jia et al., who were careful to minimise the impact of correlations between $H_0$ measurements from data in different redshift bins. We find reasonable agreement with their results for the Gaussian and Exponential void underdensity profiles, although the agreement is less good in the Maxwell-Boltzmann case. The latter profile causes severe disagreement with the observed bulk flow curve at $z < 0.1$ (Mazurenko et al.), so the tension with higher redshift data further highlights that the deepest part of the KBC void is probably near its centre. The observations show a decline of $H_0(z)$ towards the background $Planck$ value in qualitative agreement with the considered models, even if we use a larger void. The good overall agreement with the recent results of Jia et al. suggests that the local supervoid evident from the galaxy luminosity density out to a Gpc might also solve the Hubble tension while retaining a low background $H_0$ consistent with $Planck$ data, assuming enhanced structure formation on $>100$ Mpc scales.

Figures

Figures reproduced from arXiv: 2412.12245 by the authors.

Figure 1
Figure 1. Illustration of how we find the intersection of our past lightcone (black; Equation 5) with the trajectories of five different particles at different initial distances from the void centre (coloured lines). Notice that particles close to the void centre have a more appreciable increase in their comoving radius due to the larger gravitational impact of the void. Results are available only for 𝑎 > 0.1 (𝑧 < 9) as that … view at source ↗
Figure 2
Figure 2. The red lines show the predicted 𝐻0 (𝑧) in the local void models of HBK20, as found using three different methods (Section 2) and indicated in the legend. For clarity, results are shown here only for their best-fitting void parameters assuming a Gaussian underdensity profile. The grey points with error bars are observational results from table 4 of JHW23, with the horizontal uncertainty being the width of each redsh… view at source ↗
Figure 3
Figure 3. Similar to [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Similar to [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    astro-ph.CO 2026-07 conditional novelty 5.0 of 10

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  3. Hubble tension: a short review of theoretical explanations

    astro-ph.CO 2026-07 accept novelty 4.0 of 10

    A comprehensive review finds no theoretical Hubble-tension solution yet passes all consistency tests; new early-dark-energy chains reach high H0 only when the SH0ES calibration is added.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.