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REVIEW 4 major objections 6 minor 74 references

Field-level inference of $H_0$ from simulated type Ia supernovae in a local Universe analogue

T0 review · 4 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read A field-level velocity model recovers H0 from low-redshift supernovae and removes cosmic flows as the cause of the Hubble tension.

desk verdict A clean extension of field-level peculiar-velocity inference to the non-linear 2M++ covariance, with a self-consistent mock validation that stops short of an independent test of the velocity reconstruction. read the letter →

arxiv 2502.08385 v3 pith:FOZIC3IZ submitted 2025-02-12 astro-ph.CO

Eleni Tsaprazi , Alan F. Heavens This is my paper · ORCID
classification astro-ph.CO
keywords typeIasupernovaeHubbleconstanttensionpeculiarvelocitiesBayesianhierarchicalmodellocalUniversereconstruction2M++surveycosmiclarge-scalestructuresupernovaratemodelling
topics 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

The paper sets out to test whether the local pattern of non-linear peculiar velocities, rather than new physics, could be responsible for the Hubble constant tension. It generates mock type Ia supernova samples inside a constrained simulation of the local Universe and infers $H_0$ with a Bayesian hierarchical model that marginalises over each supernova's unknown cosmological redshift and peculiar velocity, using the mean and covariance of reconstructed velocity fields from the 2M++ catalogue. With simulated data, the model recovers the input $H_0$ even for supernovae below $z=0.023$, the regime that is normally discarded. When peculiar velocities are ignored, the inferred $H_0$ increases by only $\sim 0.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ in the range $0.023 < z < 0.046$. The authors conclude that unaccounted-for non-linear velocity dynamics are unlikely to explain the $H_0$ tension.

What carries the argument

The central object is a Bayesian hierarchical model in which each observed redshift and distance modulus is treated as a noisy measurement of an unknown cosmological redshift $z_c$ and an unknown peculiar redshift $z_p$, with $z_p$ tied to the 2M++ reconstruction through a multivariate Gaussian whose covariance is the Hartlap-corrected sample covariance across $\sim 240$ forward-modelled velocity realisations. The key analytical step is integrating out $z_c$ and $z_p$ to leave a one-dimensional Gaussian posterior for $5\alpha_B$ (the Hubble-diagram intercept), so the final $H_0$ estimate is a weighted combination of per-supernova estimates whose weights account for the non-linear velocity correlations.

What would settle it

Apply the same pipeline to real low-redshift SNeIa with independent peculiar-velocity measurements, such as host-galaxy distances from other distance indicators; if including $z<0.023$ supernovae moves the inferred $H_0$ by substantially more than the $\sim 0.5$ km s$^{-1}$ Mpc$^{-1}$ implied by the mocks, the reconstruction is not capturing the relevant velocity field.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a field-level treatment of non-linear peculiar velocities, drawn from the Bayesian 2M++ reconstruction, removes low-redshift velocity systematics as a viable explanation of the Hubble tension. The authors build realistic SNIa mocks by placing supernovae in the SIBELIUS-DARK constrained simulation according to GALFORM star-formation histories, then fit them with a hierarchical model that propagates the full non-linear velocity covariance into the $H_0$ posterior. The model reproduces the ground-truth $H_0$ when applied to mocks that include $z<0.023$ supernovae, and it yields only a minimal $H_0$ shift when peculiar velocities are ignored at $0.023<z<0.046$. This opens the door to using low-redshift supernovae, currently excluded, in $H_0$ measurements.

Load-bearing premise

The whole argument rests on the 2M++ Bayesian reconstruction being an accurate description of the true non-linear peculiar velocity field at its 2.65 Mpc/h resolution, with a covariance that faithfully captures the reconstruction's uncertainty.

Editorial extensions

If this is right

  • Supernovae below $z=0.023$ can be included in $H_0$ analyses instead of being cut, reducing sample variance.
  • Ignoring peculiar velocities biases $H_0$ by only $0.4\pm0.5$ km s$^{-1}$ Mpc$^{-1}$ at $0.023<z<0.046$, so velocity systematics do not account for the reported Hubble tension.
  • The per-host SNIa rate model is a secondary ingredient for velocity studies, since uniform rates give the same $H_0$ shift.
  • More velocity realisations from future reconstructions will allow the contribution of individual superclusters, such as Hydra-Centaurus and Shapley, to be pinned down precisely.

Reading between the lines

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

  • If the result carries over to real data, extending $H_0$ samples to $z\sim0$ could change the local value's error budget and shift the comparison with early-Universe estimates.
  • The analytic marginalisation over $z_p$ and $z_c$ could be applied to other low-redshift distance indicators, such as gravitational-wave standard sirens, wherever a reconstructed peculiar-velocity covariance is available.
  • Because the mocks use one 2M++ realisation as truth and the mean field for analysis, the method's success depends on the reconstruction's resolution; surveys that resolve smaller-scale velocity dispersion could reveal a larger bias than reported here.
  • The patch decomposition hints that any residual velocity systematic would show up first in the directions of massive superclusters, where the weighted $H_0$ offsets are largest.
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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 / 6 minor

Summary. The paper develops a Bayesian hierarchical model for inferring the Hubble constant H0 from type Ia supernova (SNIa) redshift and distance-modulus data, incorporating the non-linear peculiar velocity field of the local Universe as reconstructed from the 2M++ catalogue with the BORG algorithm. The velocity information enters through a per-SN mean and covariance, and the model marginalises analytically over unknown cosmological and peculiar redshifts, using a series of approximations to make the high-dimensional integral tractable. The method is tested on mock SNIa catalogues generated from the SIBELIUS-DARK simulation with GALFORM-based SNIa rates: at 0.023<z<0.046, ignoring peculiar velocities is reported to bias H0 by only 0.4±0.5 km/s/Mpc, while at z<0.023 the model is claimed to recover the ground-truth H0 (67.77 km/s/Mpc) even when a wrong fiducial H0 is assumed. The authors conclude that the Hubble tension is unlikely to originate in unaccounted-for non-linear velocity dynamics.

Significance. The paper is a well-structured methods contribution with several genuine strengths: it uses a constrained simulation of the local Universe rather than random N-body simulations; it builds SNIa host populations from individual galaxy star-formation histories; and it propagates the full non-linear velocity covariance at the locations of SNe, going beyond the common 250 km/s dispersion approximation. The analytic marginalisation in Equations (23)-(42) is a useful step toward field-level H0 inference and could enable inclusion of z<0.023 SNe, which are currently discarded. If the method survives external validation, the paper will be an important addition to the literature on local velocity systematics and H0. However, the current validation is largely a self-consistency test: mock data are generated from one realisation of the same 2M++ posterior whose mean and covariance are then used in the analysis. The main quantitative claim (0.4±0.5 km/s/Mpc bias) and the stronger conclusion about the H0 tension therefore do not yet have the independent support they would need to be fully load-bearing.

major comments (4)
  1. [Section 2.5 and Section 2.3 (mock generation and covariance)] The mock-data validation is self-referential. Section 2.5 step (iv) draws observed peculiar redshifts from a Gaussian with mean equal to one 2M++ realisation and covariance C, while the analysis in Section 2.3 uses the mean peculiar redshift across the same set of ~240 realisations and the same covariance. The 0.4±0.5 km/s/Mpc bias measurement at 0.023<z<0.046 is therefore a test that the Bayesian machinery is internally consistent under the assumption that the 2M++ posterior is the correct description of the velocity field. It cannot detect a systematic error in the reconstructed velocity field, such as a bias from external tidal fields or from inaccurate small-scale velocity modelling. The one independent cross-check mentioned, using SIBELIUS gravity-solver velocities, is reported only for the z<0.023 recovery (Section 3.2, final paragraph) and no quantitative result is given there. The authors should repeat the 0.023<z<0.046 bias measurement with SIBELIUS velocities as the true velocity field, or with an external velocity field, and report the resulting bias; without this, the central quantitative claim is not yet validated against an independent non-linear velocity prediction.
  2. [Section 2.4, Equations (11), (23), (30)-(31)] The analytic posterior rests on three unquantified approximations: the Doppler term replaces the true peculiar redshift by its mean in Equation (11); the z_c^2 prior is approximated by ztilde^2 and the lower integration limits are extended to -infinity in Equation (31); and the distance modulus is linearised around ztilde in Equation (27). The text states that these have 'very little impact' or 'insignificant impact', but no error estimate or comparison with an exact integration is provided. These approximations are most stressed precisely in the z<0.023 regime that the paper aims to open up. The authors attempted nested sampling but found it non-robust, so the analytic result has not been validated against a high-dimensional integration. A quantitative check, for example a comparison on a few reduced-dimension mocks or a controlled integration on a subset of SNe, is needed to demonstrate that the approximations do not bias the recovered H0.
  3. [Section 4 and Section 2.3 (conclusions versus stated limitations)] The conclusion that 'it is unlikely that the H0 tension originates in unaccounted-for non-linear velocity dynamics' is stronger than the evidence presented. Section 2.3 states that the 2M++ reconstruction is trusted only above 2.65 Mpc/h and that small-scale velocity dispersion must be modelled externally for real data; Section 4 also mentions possible peculiar velocities sourced outside the 200 Mpc volume. The paper does not estimate the size of these missing contributions. Even if the internal self-consistency test were clean, the result would only bound the bias from the modelled scales and internal dynamics, not from all unaccounted-for velocity dynamics. The conclusion should be rephrased to state that, within the modelled scales and the 2M++ reconstruction, no significant velocity-induced bias is found, rather than making a global statement about the origin of the H0 tension.
  4. [Section 3.2, Figure 5] Figure 5 shows nine individual H0 posteriors, several of which have MAP values around 1-2 sigma from the ground truth (e.g., panels e and f). The paper states that 100 mock datasets were checked and found consistent with the ground truth, but no summary statistics are given. The authors should report the distribution of MAP values across the 100 datasets (mean offset, scatter, and fraction within 1 sigma) to demonstrate that the quoted per-dataset uncertainties are well calibrated and that the consistency claim is not based on a selected subset.
minor comments (6)
  1. [Figure 2 caption] The caption contains a typo: 'Flowchart of the the Bayesian hierarchical model' should read 'Flowchart of the Bayesian hierarchical model'.
  2. [Equations (5)-(6)] The definition and sign convention for alpha_B should be clarified. As written, Equation (6) equates alpha_B to log(1+Delta H/H0), but alpha_B is defined in Equation (5) as a Hubble-diagram intercept; the relationship between the two should be stated explicitly to avoid confusion.
  3. [Section 2.3] The statement that the covariance matrices are 'robust to this number' of 240 realisations is not supported by a convergence test. A short figure or a quantitative stability check would strengthen the claim that Nsim is sufficient.
  4. [Section 3.1, Figure 3] In Figure 3(c), the highest-mass bin of strongly star-forming galaxies is stated to contain very few members, but the actual number is not given. Reporting the bin count would help readers judge the significance of the discrepancy.
  5. [Section 3.2, Figure 4(b)] The patch contributions shown in Figure 4(b) appear to have no uncertainty estimates. Given the small number of SNe per patch (0.7-4.2 sources), error bars or a statement that the values are noise-dominated should be added.
  6. [Section 4] The caveat that 'It remains to be checked against real data whether the more accurate velocity modelling ... will yield results on H0 at z<0.023 consistent with existing studies' is an important qualification and should be reflected in the abstract or conclusion, since the current abstract makes a stronger statement about recovering ground-truth H0 and ruling out velocity explanations of the tension.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the mock validation is a transparent prior-predictive self-consistency check, and the scientific conclusions are conditioned on the external 2M++ reconstruction.

full rationale

The paper's main quantitative claims are (i) the Bayesian hierarchical model recovers H0 in mock data and (ii) ignoring peculiar velocities shifts H0 by 0.4 +/- 0.5 km/s/Mpc. Claim (i) is tested in Sections 2.5 and 3.2 by generating mock redshifts and distance moduli with one 2M++ peculiar-velocity realisation and the covariance C, then analysing with the mean 2M++ field and the same C. This is a prior-predictive self-consistency check: the generative and analysis likelihoods share the covariance, so the posterior is expected to centre on the injected H0 if the analytic Laplace approximation is accurate. This is a standard and transparent validation of the statistical machinery; the paper explicitly calls the data 'self-consistent' and does not present it as an independent test of the velocity reconstruction. The z<0.023 recovery is not a fitted parameter renamed as a prediction: H0 is injected through the distance moduli and inferred from them. Claim (ii) is an estimate of the bias of an analysis that ignores peculiar velocities, using a specific 2M++ realisation as the truth; it is conditional on that reconstruction, which is an external input rather than an output of the paper's inference. The paper itself flags the limitation that, for real data, 'the velocity dispersion on smaller scales must be modelled externally' (Section 2.3), and it notes that a real-data check of the BORG 2M++ velocity modelling remains to be done (Section 4). These are caveats on external validity, not circularity. Self-citations in the paper (Tsaprazi et al. 2022; Sellentin & Heavens 2016; Percival et al. 2022) are contextual or non-load-bearing, and no uniqueness theorem or ansatz is imported from the authors' prior work. Hence no circular step is identifiable.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central claim rests on external data products (2M++ BORG, SIBELIUS-DARK, GALFORM) whose fidelity is assumed rather than tested here. No new particles or forces are introduced. The main free parameters are standard observational uncertainties and rate-model inputs, many of which the paper demonstrates to be inconsequential for the H0 result.

free parameters (8)
  • SNIa rate normalisation A = 2.11e-13 SNe/Msun/yr (Wiseman et al. 2021)
    Used in Equation 2 for mock SN rates; taken from prior literature, and the paper shows the H0 result is insensitive to rate modelling.
  • Delay-time distribution slope beta = -1.13 (Wiseman et al. 2021)
    Power-law slope in Equation 2.
  • Prompt delay time tp = 40 Myr (Wiseman et al. 2021)
    Minimum delay time in Equation 2.
  • Global rate rescaling factor = ~3
    Applied to match the Perley et al. (2020) volumetric SNIa rate; the paper argues the analysis is insensitive to this factor.
  • Redshift error sigma_z = 0.001
    Assumed observational uncertainty used in mock generation and likelihood (Section 2.5).
  • Distance modulus error sigma_mu = 0.13 mag
    Assumed observational uncertainty (from Gris et al. 2023), used in mock generation and likelihood.
  • Number of velocity realisations Nsim = ~240
    Determined by data availability; limits NSN to ~200 for invertible covariance.
  • Patch query radius = ±10 degrees
    Chosen to minimise overlap when attributing H0 shifts to superclusters (Section 3.2).
assumptions (7)
  • standard math Standard matrix identities (Woodbury, determinant theorem) and Gaussian integrals are valid.
    Used in Sections 2.3-2.4 to marginalise the posterior analytically.
  • domain assumption The 2M++ BORG reconstruction provides an unbiased non-linear peculiar velocity field at 2.65 Mpc/h resolution.
    The velocity covariance in Equation 7 and the mean field are taken from this reconstruction; the mock truth and analysis both rely on it.
  • domain assumption SIBELIUS-DARK plus GALFORM reproduces realistic SNIa host locations and host velocities in the local Universe.
    SNIa hosts are drawn from this simulation; if the galaxy formation model is biased, the mock SNe velocity distribution is unrepresentative.
  • domain assumption The peculiar velocity field is independent of the assumed H0.
    Section 2.4 states the velocity reconstruction is assumed invariant to H0; the paper argues the effect is small using estimates from Kostić et al. (2022).
  • ad hoc to paper The Doppler-boosting term can be replaced by using the mean peculiar redshift.
    Equation 11 replaces the true peculiar redshift by its mean to enable analytic marginalisation; the paper claims the posterior is insensitive.
  • ad hoc to paper The cosmological redshift prior z^2 can be approximated by a constant and integration limits extended to negative infinity.
    This allows the redshift integral to be evaluated as a Gaussian (Equation 24 and following).
  • ad hoc to paper The distance modulus is a linear function of cosmological redshift around the peak of the minimiser.
    Equation 27 Taylor-expands the distance modulus to first order; stated to be adequate for spectroscopic redshifts.

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Cite this review

Pith. "Pith review of Field-level inference of $H_0$ from simulated type Ia supernovae in a local Universe analogue." pith.science (2026). https://pith.science/paper/FOZIC3IZ

@misc{pith2026250208385,
  author       = {Pith},
  title        = {Pith review of: Field-level inference of $H_0$ from simulated type Ia supernovae in a local Universe analogue},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FOZIC3IZ}},
  note         = {Machine review of arXiv:2502.08385}
}
abstract

Two particular challenges face type Ia supernovae (SNeIa) as probes of the expansion rate of the Universe. One is that they may not be fair tracers of the matter velocity field, and the second is that their peculiar velocities distort the Hubble expansion. Although the latter has been estimated at $\lesssim1.5\%$ for $z>0.023$, this is based either on constrained linear or unconstrained (random) non-linear velocity simulations. In this paper, we address both challenges by incorporating a physical model for the locations of supernovae, and develop a Bayesian Hierarchical Model that accounts for non-linear peculiar velocities in our local Universe, inferred from a Bayesian analysis of the 2M++ spectroscopic galaxy catalogue. With simulated data, the model recovers the ground truth value of the Hubble constant $H_0$ in the presence of peculiar velocities including their correlated uncertainties arising from the Bayesian inference, opening up the potential of including lower redshift SNeIa to measure $H_0$. Ignoring peculiar velocities, the inferred $H_0$ increases minimally by $\sim 0.4 \pm 0.5$ km s$^{-1}$ Mpc$^{-1}$ in the range $0.023<z<0.046$. We conclude it is unlikely that the $H_0$ tension originates in unaccounted-for non-linear velocity dynamics.

Figures

Figures reproduced from arXiv: 2502.08385 by the authors.

Figure 1
Figure 1. Angular coordinates of the 2M++ galaxies (purple), simulated SIBELIUS galaxies (yellow) and SNeIa (orange). The Coma cluster (Abell 1656) is indicated by a white cross. The SIBELIUS-DARK galaxies follow the distribution of observed galaxies in the real Universe on large scales. 2024; Hollinger & Hudson 2025), even in the absence of a local void (Jasche & Lavaux 2019). Our study adds to the above investigations by a)… view at source ↗
Figure 2
Figure 2. Flowchart of the the Bayesian hierarchical model in Equation 23. The rhombi represent the means of the multivariate likelihoods. The thin circles represent the stochastic variables. The bold circle represents the population parameter 𝛼B. At any given cosmological redshift, we use the 2M++ non-linear velocity reconstruction to incorporate the effect of redshift￾space distortions and the stochasticity introduced by th… view at source ↗
Figure 3
Figure 3. (a) Distributions of the SIBELIUS SNIa host star-formation rates. In light blue, we show the star-formation rate distribution of the entire SIBELIUS-DARK galaxy sample. (b) SNIa rate per galaxy as a function of the star-formation rate. The continuous line is from Smith et al. (2012, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: (a) Δ𝐻/𝐻0 posterior averaged over 100 SNIa realisations with 100 noise realisations each when peculiar velocities are ignored in the 𝐻0 inference. We find on average ⟨Δ𝐻/𝐻0 ⟩ = 0.006 ± 0.008 (0.4 ± 0.5 km s−1 Mpc−1 ) in the local Universe in the range 0.023 < 𝑧 < 0.046…
Figure 5
Figure 5. Figure 5: 𝐻0 posterior from the Bayesian hierarchical model in in Equation 23 in the presence of observational uncertainties for 9 random SNIa datasets at 𝑧 < 0.023, after wrongly assuming 𝐻0 = 63 km s−1 Mpc−1 when the data have been generated with 𝐻0,true = 67.77 km s−1 Mpc−1 .…

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.