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A unified Bayesian analysis of Tully–Fisher, fundamental-plane, and supernova distances finds all three tracers agree with the CMB on S8 = 0.819 ± 0.030.

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A unified Bayesian analysis of Tully-Fisher, fundamental plane, and supernova distances gives S8 = 0.819±0.030, consistent with the Planck CMB value of 0.832±0.013.

T0 review reviewed 2026-08-04 challenge →

load-bearing objection Useful unified peculiar-velocity analysis undermined by a direct abstract/body contradiction over the fundamental plane and the headline S8 value. the 3 major comments →

arxiv 2509.20235 v2 pith:J6YF2NG3 submitted 2025-09-24 astro-ph.CO

S₈ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane

Richard Stiskalek This is my paper
classification astro-ph.CO
keywords S8 tensionpeculiar velocitiesTully-Fisher relationfundamental planeType Ia supernovaehierarchical Bayesian modelf sigma8local velocity field reconstruction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Peculiar velocities — galaxies' deviations from the Hubble flow — encode the growth of cosmic structure and the amplitude parameter S8. The paper builds one hierarchical Bayesian model that simultaneously calibrates three independent distance indicators (the Tully–Fisher relation for spirals, the fundamental plane for ellipticals, and Type Ia supernovae) and rescales a whole-sky linear-theory reconstruction of the local velocity field. It finds that all five catalogues give consistent S8 values, and the joint constraint S8 = 0.819 ± 0.030 agrees with the cosmic microwave background rather than with the lower values weak-lensing surveys have reported. The paper also traces earlier, lower peculiar-velocity estimates to specific methodological choices, and flags that the fundamental-plane samples are sensitive to how inhomogeneous Malmquist bias is modeled.

Core claim

The central discovery is that low-redshift peculiar velocities are concordant with the cosmic microwave background. Treating distances as latent parameters, the analysis jointly calibrates each distance–indicator relation and the velocity scaling β* (the ratio of the growth rate to the galaxy bias of the density template) against a linear-theory reconstruction of the local velocity field. After accounting for each sample's own galaxy bias in the Malmquist-bias distance prior, the five samples agree on β* and hence on S8; the joint value is S8 = 0.819 ± 0.030. The fundamental-plane catalogues are the most precise but are unstable under the adopted inhomogeneous Malmquist-bias treatment, and a

What carries the argument

The load-bearing object is a whole-sky linear-theory reconstruction of the local universe, voxelized from a near-infrared galaxy catalogue with Gaussian smoothing on 4 h^-1 Mpc, used to predict each source's peculiar velocity from its inferred distance. The argument runs through the identity β* ≡ f/b, with f ≈ Ω_m^0.55 the linear growth rate and b the galaxy bias of the density template; peculiar velocities constrain β*, and multiplying by the externally measured galaxy fluctuation amplitude σg8 on 8 h^-1 Mpc gives f σ8^NL, which is converted to S8 using an emulator-based nonlinear-to-linear correction. Around this, the machinery includes a per-sample linear bias b1 in the distance prior (ph

Load-bearing premise

The claim hangs on the assumption that the true line-of-sight velocity of every tracer equals a single constant β* times the reconstructed linear-theory template plus a uniform external dipole; if the template's shape is wrong — due to catalogue incompleteness, the 4 h^-1 Mpc smoothing, or the constant-bias assumption — then β* and therefore S8 are biased by an amount not captured by the quoted error bars.

What would settle it

Run the identical inference on the same five catalogues against an independent reconstruction of the local velocity field (for example from a constrained simulation or a different galaxy catalogue). If the joint S8 moves by more than about 0.03, or if the fundamental-plane samples shift to roughly 0.75 while the Tully–Fisher and supernova samples stay near 0.8 under the quadratic Malmquist-bias term, then the claimed concordance is a property of the chosen template or model rather than of the data.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • A joint, self-consistent treatment of Tully–Fisher, fundamental-plane, and supernova distances gives S8 = 0.819 ± 0.030, consistent with the CMB; the quoted error is dominated by the uncertainty in the galaxy-field amplitude σg8, not by the velocity data.
  • Earlier peculiar-velocity measurements that reported lower S8 — in particular from fundamental-plane-only or Tully–Fisher-only analyses — are likely affected by methodological choices, such as fixing the galaxy bias to unity or modeling scatter in angular size rather than its logarithm.
  • The fundamental-plane samples give the tightest velocity-scaling constraints but are sensitive to the Malmquist-bias model; a quadratic extension preferred by the data shifts their S8 downward, so their systematics need further work.
  • If all samples are truly concordant, peculiar velocities do not support an early-versus-late S8 tension; they are consistent with the CMB and with recent weak-lensing values within current precision.

Where Pith is reading between the lines

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

  • A direct test: repeat the joint inference with an independent reconstruction of the local velocity field; if the five samples remain concordant and S8 shifts by less than roughly 0.03, the result is a property of the local velocity field rather than of the chosen template.
  • The fundamental-plane instability hints that the phenomenological distance prior r^2 [1 + b1 δ(r)] is too crude for early-type samples; a survey-specific selection model based on actual apparent-magnitude limits could either confirm the concordance or expose a real FP-specific bias.
  • If the galaxy-field amplitude σg8 were remeasured with better data, the joint S8 uncertainty would drop below 0.02, making peculiar velocities competitive with weak lensing as a late-time probe.
  • The inferred external velocity dipole differs in direction between the two fundamental-plane samples; this may signal residual large-scale flow not captured by the reconstruction, and its origin could be tested by splitting the samples by sky region.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper presents a unified Bayesian hierarchical forward model for inferring the cosmological parameter combination fσ8 (and S8) from peculiar velocities measured with three distance indicators: the Tully–Fisher relation (two CosmicFlows-4 samples), the fundamental plane (SDSS and 6dF), and Type Ia supernovae (Pantheon+). The velocity field is modelled with the Carrick et al. (2015) 2M++ reconstruction, with a free velocity scaling β*, an external dipole, and a phenomenological inhomogeneous Malmquist-bias prior with a free tracer bias b1. The main result stated in the full-text abstract and Section 5 is that all three tracers yield mutually consistent S8 values in agreement with Planck, with a joint constraint S8 = 0.819 ± 0.030; the paper further argues that previous lower S8 estimates from peculiar velocities are affected by methodological problems. The paper also reports tracer-by-tracer constraints and compares with external S8 probes.

Significance. If the central result holds, it is an important contribution: it provides the first analysis that treats TFR, FP, and SNe within one framework, cross-checks systematics across galaxy populations, and suggests that peculiar velocities do not add to the S8 tension. The model is well specified, the checks on Ωm sensitivity are useful, and the use of a Pantheon+ covariance with peculiar-velocity terms removed is a careful detail. The paper also gives concrete methodological criticisms of earlier work (Said et al.; Boubel et al.). However, the current manuscript contains a direct contradiction between the abstract preceding the full text and the full-text abstract/body regarding the FP stability and the joint S8 value. This must be resolved before the significance can be accepted; the underlying analysis appears potentially sound, but the manuscript is not internally consistent as written.

major comments (3)
  1. [Abstract (preceding text) vs §5 (Table 1, Fig. 2)] The abstract preceding the full text states a different central result from the full-text abstract and body. It reports S8 = 0.798 ± 0.035 from TFR+SNe, calls the FP constraints 'unstable under the inhomogeneous Malmquist bias treatment', and mentions a quadratic extension preferred by FP data. The full-text abstract, Section 5, Fig. 2, and Table 1 report S8 = 0.819 ± 0.030 from all five samples and state that all three tracers agree with Planck, with no quadratic FP extension anywhere in the body. This is not a typo: the two abstracts disagree on which samples enter the joint constraint and on whether FP is robust. The manuscript must be revised so that the abstract, body, and tables present one consistent result, and any FP quadratic-extension test must be either reported and interpreted or explicitly removed.
  2. [§5, Table 1; §7] The headline uncertainty S8 = 0.819 ± 0.030 omits the sample-variance estimate from Hollinger & Hudson (2024), which the paper itself cites in §7 and then dismisses as 'likely overly conservative'. Since the abstract and Table 1 quote only the smaller error, a reader cannot tell whether the agreement with Planck is robust to this variance. Please either include the Hollinger & Hudson variance in the headline error budget or provide a quantitative justification (e.g., mock-based) for excluding it.
  3. [§5, Fig. 2, Table 1] The assertion that all three tracers are 'in excellent agreement' with Planck and with each other is supported only by visual inspection of Fig. 2. The 6dF FP value, S8 = 0.782 ± 0.033, is about 1.4σ from Planck, and the five samples span a range of 0.78–0.86. A quantitative consistency statistic (e.g., chi-square between sample posteriors or a joint posterior predictive check) is needed to substantiate the central claim, especially given the abstract conflict about FP stability.
minor comments (5)
  1. [Table 1] The 'Joint' row appears to show only two values, with '-0.476±0.005' in the b1 column. Since each sample has its own b1, the Joint row should list β* and S8 with a blank or explanatory note for b1.
  2. [References] Stiskalek et al. 2025b and 2025c are listed with the same arXiv identifier (arXiv:2509.09665). One of these entries is likely erroneous; please correct the duplicate.
  3. [Figure 4 caption] 'The contours are1and2σ' is missing spacing; also specify whether these are 1σ and 2σ contours of the two-dimensional posterior.
  4. [Eq. (24)] The notation σ_L^8 and σ_NL^8 is easy to misread. Consider defining 'σ8 computed from the linear/non-linear power spectrum' explicitly before Eq. (24).
  5. [§4, selection term] The sentence 'This term modifies the model probability density ... by a factor [p(S=1|η_obs)]^{-n}' is terse. A one-line derivation or reference to Kelly et al. (2008) would improve readability.

Circularity Check

0 steps flagged

No circularity: S8 is derived from the fitted velocity scaling beta* and the external 2M++ clustering amplitude sigma_g8; neither quantity is defined in terms of the target S8.

full rationale

The derivation chain is self-contained and non-circular. The paper infers the velocity scaling beta* from the peculiar-velocity likelihood (Eqs. 14-16) using the Carrick et al. (2015) reconstruction, then converts beta* to S8 via Eq. (24): S8 = sigma_L8 (beta* sigma_g8 / Omega_m^0.55) sqrt(Omega_m/0.3). S8 never appears in the likelihood, priors, or distance-indicator calibrations, so the result is not an input recycled as an output. The external inputs (sigma_g8, the Planck cosmology used for the conversion) are not fitted to the S8 target; the paper explicitly tests Omega_m = 0.25 and 0.35 and finds <1 sigma changes. The self-citations (Stiskalek et al. 2025a,b,c) are methodological references for the hierarchical forward model, but Section 4 reproduces the full likelihood equations, and the nonlinear-to-linear sigma8 mapping is computed with the external syren emulator (Bartlett et al. 2024; Sui et al. 2024), not merely asserted from the self-citation. I also flag a serious internal inconsistency that is not circularity: the abstract supplied at the top states 'The fundamental plane constraints are instead unstable under the inhomogeneous Malmquist bias treatment; the quadratic extension preferred by the fundamental plane data drives their S8 values lower,' whereas the full-text abstract and Section 5/Fig. 2/Table 1 report 'All three tracers yield consistent values of S8 that are also in agreement with Planck' with a joint S8 = 0.819 ± 0.030 and make no mention of a quadratic FP extension. This contradiction concerns which result is the paper's central claim, but it does not make the derivation circular, as neither version feeds S8 back into the inference.

Axiom & Free-Parameter Ledger

6 free parameters · 7 axioms · 0 invented entities

The analysis introduces no new physical entities. All free parameters are standard calibration and nuisance parameters of a peculiar-velocity forward model. The heaviest external inputs are the C15 reconstruction, its sigma_g8 measurement, and the syren-new emulator, all treated as trusted references.

free parameters (6)
  • beta* (velocity scaling) = 0.452±0.010 to 0.500±0.029 per sample
    Scales the C15 velocity field to the observed data; central to the S8 measurement.
  • b1 (tracer bias in Malmquist prior) = 1.088±0.013 to 1.539±0.005 per sample
    Linear bias of each distance-indicator sample against the 2M++ density field; fitted jointly with beta*.
  • Vext (external dipole) = ~140±7 to ~200±20 km/s, direction varies
    Constant external velocity flow; inferred to account for structure beyond the reconstruction volume.
  • sigma_v (residual velocity scatter) = not quoted in paper
    Gaussian scatter of residual redshifts; fitted with scale-invariant prior.
  • p, q, R (distance prior shape) = not quoted in paper
    Parameters of the phenomenological completeness prior of Eq. (11).
  • Distance-indicator calibrations (a,b,c, sigma_int) and hyperparameters = not quoted in paper
    TFR zero-point/slope/curvature/scatter; FP coefficients/scatter; SN absolute magnitude; hyperpriors.
axioms (7)
  • domain assumption The galaxy density field traces matter with constant linear bias delta_g = b delta (Eq. 2)
    Used to replace matter field in Eq. (1) with the 2M++ galaxy density field.
  • standard math Linear LambdaCDM continuity equation v ~ grad^-2 delta (Eq. 1)
    The reconstruction's velocity field follows linear theory; nonlinear contributions are absorbed into the fitted scatter.
  • domain assumption f ~ Omega_m^0.55 (LambdaCDM growth approximation)
    Needed to convert f sigma_8 to S8 in Eq. (24); explicitly adopted with a stated insensitivity to Omega_m.
  • domain assumption C15 reconstruction and sigma_g8 = 0.99±0.04 are correct
    The whole template and the dominant error bar come from Carrick et al. (2015) and Westover (2007).
  • domain assumption Residual peculiar velocities are uncorrelated and Gaussian (Eq. 14)
    The likelihood for z_obs assumes diagonal covariance with variance sigma_v^2; Blake & Turner (2024) argue this may underestimate beta* errors.
  • ad hoc to paper Phenomenological distance prior of Eq. (11) and quadratic smoothing of Eq. (13)
    Selection is not modelled directly; the paper itself calls this approach 'phenomenological, approximating the effect of selection'.
  • domain assumption Nonlinear-to-linear sigma_8 mapping via syren-new emulator
    The emulator is used to convert fitted sigma_NL8 to the linear sigma_L8; no in-paper validation of the emulator's accuracy.

reviewed 2026-08-04 · how reviews work

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Cite this review

Pith. "Pith review of $S_8$ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane." pith.science (2026). https://pith.science/paper/J6YF2NG3

@misc{pith2026250920235,
  author       = {Pith},
  title        = {Pith review of: $S_8$ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J6YF2NG3}},
  note         = {Machine review of arXiv:2509.20235}
}
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read the original abstract

Peculiar velocity measurements constrain the parameter combination $f\sigma_8$, the product of the linear growth rate $f$ and the fluctuation amplitude $\sigma_8$. Under the approximation that $f$ is a monotonic function of $\Omega_{\rm m}$, this can be related to $S_8 \equiv \sigma_8 \sqrt{\Omega_{\rm m}/0.3}$, enabling direct comparison with weak lensing and cosmic microwave background results. We use three classes of direct-distance tracers -- the Tully--Fisher relation, the fundamental plane, and Type Ia supernovae -- to infer peculiar velocities. A unified hierarchical forward model jointly calibrates each distance indicator and a linear theory reconstruction of the local Universe. This is the first consistent Bayesian analysis to treat all three major classes of distance indicators within a common framework, enabling cross-checks of systematics across diverse galaxy populations. Combining the Tully--Fisher and Type Ia supernova samples, we obtain $S_8 = 0.798 \pm 0.035$ ($f\sigma_8 = 0.412 \pm 0.018$), in agreement with Planck and robust under the choice of galaxy bias model, with the uncertainty dominated by the variance of the 2M++ galaxy field. The fundamental plane constraints are instead unstable under the inhomogeneous Malmquist bias treatment; the quadratic extension preferred by the fundamental plane data drives their $S_8$ values lower. These findings indicate that low-redshift peculiar velocity data are concordant with the cosmic microwave background and do not reinforce the early-versus-late $S_8$ tension, though the fundamental plane results call for further scrutiny of their systematics.

Figures

Figures reproduced from arXiv: 2509.20235 by Richard Stiskalek.

Figure 1
Figure 1. Figure 1: Directed acyclic graph of our TFR model. Analogous model applies to the FP and SNe. The implications of truncation in ηobs are discussed in our earlier work Stiskalek et al. (2025a), which we apply con￾sistently here, though the effect on the results is negligible. In brief, following Kelly et al. (2008), we introduce a term p(S = 1 | ηobs) that describes the fraction of retained sam￾ples after selection i… view at source ↗
Figure 2
Figure 2. Figure 2: Posterior predictive distributions of S8 computed from the inferred β ⋆ following Eq. (24). All samples (TFR, FP, and Type Ia SNe from Pantheon+) are in mutual agreement and consistent with the Planck measurement (0.832 ± 0.013; Planck Collaboration et al. 2020a). with the TFR, we treat the galaxies as independent. We ex￾plicitly sample both log σ0,true and log Ie,true of each galaxy, and numerically margi… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of fσL 8 (z). Both our and literature measurements (Beutler et al. 2012; Boruah et al. 2020; Said et al. 2020; Stahl et al. 2021; Boubel et al. 2024) are plotted at the effective redshifts of the samples, defined as their mean redshifts. Error bars denote 1σ uncertainties, and the Planck prediction is shown as a 1σ shaded band (Planck Collaboration et al. 2020a). As our measurements are restrict… view at source ↗
Figure 4
Figure 4. Figure 4: Posterior distribution of the external flow vector Vext, shown in terms of its magnitude Vext and direction (ℓext, bext) in Galactic coordinates. The inferred magnitudes are broadly consis￾tent across samples, whereas the directions show mild discrepan￾cies: the two TFR samples agree with each other but differ from the two FP samples, which themselves are not mutually consis￾tent. The contours are 1 and 2σ… view at source ↗
Figure 5
Figure 5. Figure 5: The S8 parameter inferred by calibrating the Carrick et al. (2015) linear field against the peculiar velocity samples. We compare to literature results using peculiar velocities (Huterer et al. 2017; Nusser 2017; Boruah et al. 2020; Said et al. 2020), weak lensing , clustering (DESI Collaboration et al. 2024; Porre￾don et al. 2022), cluster abundance (Ghirardini et al. 2024; Boc￾quet et al. 2019; Planck Co… view at source ↗
Figure 6
Figure 6. Figure 6: Inferred β ⋆ as a function of the linear galaxy bias parameter b1 of the Carrick et al. (2015) field for the CF4 W1 and SDSS FP samples. In the fiducial analysis both b1 and β ⋆ are free parameters, while here b1 is fixed to illustrate its impact on the inferred β ⋆ [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.