{"id":"b40c48cf-469b-4e6a-93ec-88760da4658b","arxiv_id":"2412.12245","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A local-void model with a Planck-compatible background predicts a declining H0(z) that broadly matches the Jia et al. 2023 and 2024 measurements for Gaussian and Exponential void profiles.","lead":"This paper computes how the inferred expansion rate of the Universe (H0) changes with redshift if we live inside a giant, underdense 'void' that pushes nearby galaxies outward. It finds that the predicted decline in H0 with distance roughly matches recent measurements, supporting the idea that a local void, rather than new early-universe physics, may explain the Hubble tension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Load-bearing concern: the approximate GR term in Eq. 6 ignores time-dependent evolution of the void potential, with a plausible effect comparable to the JHW23 error bars.","rationale":"The reader's weakest-assumption analysis correctly identifies Eq. 6 as the most load-bearing component of the paper's central claim. The paper's headline result, that H0(z) approaches the Planck value in a way 'broadly agreeing' with JHW23/JHW24, is a quantitative comparison between model curves and data points. The model curves are completely determined by the choice of void profile and the redshift formula. Among the non-cosmological terms, the GR factor is the least securely grounded: it is an approximate line integral over a time-dependent, MOND-modified void potential, taken directly from HBK20 without independent re-derivation in this work. The paper itself stresses the importance of this term (solid vs dashed red curves in Figure 3), and the size of its effect is comparable to the statistical errors of the observational points being matched. If the approximation is wrong at the 0.5-1 km/s/Mpc level, the claimed agreement could become a disagreement at intermediate redshift, which is exactly where the paper already admits some tension with JHW23. The concrete test I propose would settle this by recomputing the predicted H0(z) using a fully relativistic treatment of the same density profiles. Given that the concern is specific, testable, and acknowledged in the reader's verdict, keeping the verdict at CONDITIONAL is appropriate; no change in verdict is needed based on this stress-test pass.","tokens_in":18973,"tokens_out":5498,"duration_ms":55111,"concrete_test":"Construct an LTB (Lemaître-Tolman-Bondi) model or a linearized-GR null-geodesic calculation using the same HBK20 void density profiles and Planck background, and recompute the exact gravitational redshift and hence H0(z) without the static approximation. Compare the exact H0(z) with the Eq. 6-based result over z = 0.05-2. If the difference exceeds ~0.5 km/s/Mpc in the intermediate-redshift range, the claimed agreement with JHW23/JHW24 must be re-evaluated; if it is below that, the approximation is adequate and the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The predicted H0(z) curves and hence the claimed agreement with JHW23/JHW24 hinge on Eq. 6, specifically on the factor exp[(1/c^2)∫ g_void dr]. This expression is a static-potential weak-field approximation: it integrates the instantaneous void radial gravity along the photon path, with no explicit correction for (i) the time dependence of the void potential between emission and observation, or (ii) the fact that the null path is bent and the integral should be evaluated on the actual trajectory. In the HBK20 model the void is still evolving at z<1 and the photon crossing time is several Gyr, so the potential at emission differs from that at reception. A fractional error of order (dot-Phi / H Phi) times the light-crossing time is naturally of the same order as the GR contribution itself. Figure 3 shows this term shifts H0 by about 1 km/s/Mpc near z≈0.5 and is the dominant non-Doppler contribution at high z; JHW23/JHW24 individual points have ~1-2 km/s/Mpc errors. Thus a realistic correction to this term moves the model curve by a substantial fraction of the data error bars and can decide whether the claimed 'broad agreement' holds or is actually a mild tension. Because the same formula is used for all three profiles and all three methods, it is the most load-bearing component of the comparison.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19217,"tokens_out":9147,"duration_ms":91499,"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":[{"comment":"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.","section":"Section 2, Eq. (6)"},{"comment":"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.","section":"Section 3, Fig. 3"}],"minor_comments":[{"comment":"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.","section":"Figure 3"},{"comment":"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.","section":"Equation (10)"},{"comment":"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.","section":"Section 4.2"},{"comment":"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.","section":"Section 4"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope and the comparison data are appropriate. The main risk is that the claimed agreement is currently stronger than the evidence because the GR term in Eq. (6) is not error-controlled and no statistical comparison is provided. I would not reject the paper, but the revisions needed are substantive rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read Mazurenko, Banik and Kroupa. The useful thing here is that they take the already-published HBK20 local void models and compute what an observer at the void centre would infer as H0 from data in a narrow redshift bin, using three different reconstruction methods. For the Gaussian and Exponential profiles, the predicted H0(z) declines from the SH0ES value at low z to Planck by z~1.8, and that decline tracks the decorrelated H0(z) compilations of Jia et al. (2023, 2024) at a qualitatively reasonable level. This is a genuine prediction from a previously published model, and it is the right kind of test for a local void solution to the Hubble tension.\n\nWhat is new: the three methods are cleanly defined and agree with each other; the paper shows that the gravitational redshift term is what keeps H0(z) inflated at z>0.5, not the Doppler outflow; and it shows the Maxwell-Boltzmann profile, already in trouble from the z<0.1 bulk flow, also converges to Planck too fast to match the intermediate-z data. The comparison data are appropriate because Jia et al. tried to remove bin-to-bin correlations. The paper is honest that the low-z end of the curves is not a fresh prediction—the HBK20 parameters were fit to the local H0 and q0 measurements that define the tension.\n\nThe soft spots, in order of importance. First, the GR contribution in eq. (6) is a static-potential integral over the void's acceleration, with no explicit treatment of the time-dependent potential along the photon path. The void is still evolving at z<1 and the light crossing time is several Gyr, so the fractional error in this term can be order one. In Figure 3 the GR shift is about 1 km/s/Mpc, comparable to the Jia et al. error bars, so a less approximate treatment could move the model curves by a substantial fraction of the data uncertainty. The paper flags the approximation only by a pointer to HBK20, not with a quantitative error estimate. Second, the model curves have no uncertainty bands; the HBK20 parameter uncertainties are not propagated, and the agreement is judged visually. A binned likelihood or chi-square would make the comparison much stronger. Third, the circularity concern is real but mild: the low-z part of the curve inherits the HBK20 fit to local measurements, so the independent content is mostly in the high-z decline.\n\nNone of this sinks the paper. The central qualitative claim—that a local void with a Planck background cosmology predicts a declining H0(z) in rough agreement with the decorrelated data—survives. The GR caveat deserves a closer look, but given the current error bars it is a refinement, not a fatal flaw.\n\nWho this is for: cosmologists working on the Hubble tension who want a concrete, testable late-time alternative to early-time solutions. It deserves a serious referee; the methods are transparent and the prediction is falsifiable with a more careful GR calculation. If I were the editor, I would send it out.","headline":"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.","tokens_in":19773,"tokens_out":4209,"would_cite":true,"duration_ms":38352,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.-k","98.80.Es"],"model":"deepseek-v4-flash","headline":"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.","keywords":["Hubble tension","local void","KBC supervoid","H0 redshift dependence","peculiar velocities","gravitational redshift","Hubble constant","MOND"],"falsifier":"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.","tokens_in":18747,"feed_emoji":"🌌","tokens_out":8058,"duration_ms":64816,"temperature":0.7,"pith_summary":"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.","feed_headline":"Local void model predicts H0 falling to Planck value at high z","feed_subtitle":"Inferred expansion rate declines with redshift, matching Jia et al. and supporting a local-void fix to the Hubble tension.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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 $z \\sim 0.1$–$0.5$."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the three void density profiles, their best-fitting parameters, and the redshift formula (Equation 6) used to compute predicted redshifts.","marker":"HBK20"},{"why":"Provides the uncorrelated redshift-binned $H_0(z)$ observations that the model predictions are compared against.","marker":"JHW23"},{"why":"Updated version of the JHW23 analysis with a more rapidly declining $H_0(z)$, also compared in Figures 2–4.","marker":"JHW24"},{"why":"Provides the bulk flow constraints that rule out the Maxwell-Boltzmann profile and locate the observer within 100–150 Mpc of the void centre.","marker":"Mazurenko et al. 2024"},{"why":"The bulk flow measurements that the profiles must reproduce at $z<0.1$ and that justify the central-observer assumption.","marker":"Watkins et al. 2023"},{"why":"Sets the background expansion rate $H_0^{\\rm Planck}=67.4\\pm0.5$ km/s/Mpc used in the HBK20 models and as the high-redshift asymptote.","marker":"Planck Collaboration VI 2020"},{"why":"Establishes the observed KBC supervoid density profile used to constrain the models.","marker":"Keenan, Barger & Cowie 2013"},{"why":"Age-based $H_0$ constraint used as independent evidence that the background expansion rate is close to the Planck value.","marker":"Cimatti & Moresco 2023"}],"fun_headline_variants":["H0 falls toward Planck as redshift grows in void model","Void model's H0(z) decline aligns with Jia et al.","Local void predicts H0 dropping to Planck value by z≈2","Gravitational void redshift explains inferred H0 decline"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["H0 falls toward Planck as redshift grows in void model","Void model's H0(z) decline aligns with Jia et al.","Local void predicts H0 dropping to Planck value by z≈2","Gravitational void redshift explains inferred H0 decline"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001321,"raw_usage":{"total_tokens":5439,"prompt_tokens":1063,"completion_tokens":4376,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":679,"completion_tokens_details":{"reasoning_tokens":4305}},"tokens_in":679,"tokens_out":4376,"duration_ms":30465,"temperature":1.0,"reasoning_tokens":4305,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:15:26.316759+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}