$S_8$ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane

Problem
The standard cosmological model ($\Lambda$CDM) faces a persistent tension regarding the parameter $S_8 \equiv \sigma_8 \sqrt{\Omega_m/0.3}$, which characterizes the amplitude of matter fluctuations. While the Planck cosmic microwave background (CMB) experiment infers a value of $S_8 \approx 0.832$, weak lensing surveys consistently report lower values, creating an "early-versus-late" universe discrepancy. Independent low-redshift probes are essential to determine if this tension reflects new physics or residual systematics. Peculiar velocity measurements offer such a probe, constraining the growth rate of structure via the parameter combination $f\sigma_8$. However, previous analyses of peculiar velocities have been fragmented, often relying on single distance indicators (e.g., Tully–Fisher relation alone, Fundamental Plane alone, or Type Ia supernovae alone) or employing heterogeneous methodologies that hinder clean cross-comparisons. Furthermore, the treatment of selection effects, specifically inhomogeneous Malmquist bias, has varied across studies, potentially introducing systematic errors in the inferred cosmological parameters.

Methodology
This work presents the first consistent Bayesian hierarchical analysis that jointly calibrates three major classes of direct-distance tracers within a unified framework:

1. Tully–Fisher Relation (TFR): Applied to late-type galaxies using the CosmicFlows-4 (CF4) survey (SDSS $i$-band and WISE W1-band).
2. Fundamental Plane (FP): Applied to early-type galaxies using the SDSS and 6dF FP catalogues.
3. Type Ia Supernovae (SNe): Applied to the Pantheon+ compilation.

The analysis employs a forward-modeling approach where the local Universe's density and velocity fields are reconstructed from the 2M++ galaxy catalogue (Carrick et al. 2015, hereafter C15) under linear $\Lambda$CDM theory. The model treats galaxy distances as latent parameters inferred statistically. Key methodological features include:

• Joint Calibration: The model simultaneously infers distance-indicator relations (e.g., TFR zero-point, slope, curvature; FP coefficients) and the velocity field scaling parameter $\beta_\star \equiv f/b_{2M++}$.
• Selection Effects: A phenomenological prior is used to model inhomogeneous Malmquist bias, parameterizing the radial sample incompleteness and the bias of the peculiar velocity tracers relative to the 2M++ density field. The authors test both a linear bias model ($n \propto 1 + b_1\delta$) and a quadratic extension ($n \propto 1 + b_1\delta + b_2\delta^2$) to assess robustness.
• External Flows: The model includes a constant external velocity dipole ($V_{ext}$) to account for mass distributions outside the reconstruction volume.
• Inference: The posterior distribution is sampled using Hamiltonian Monte Carlo (No-U-Turn Sampler) implemented in JAX and NumPyro, marginalizing over nuisance parameters such as intrinsic scatters and true distances.

Key Contributions

• Unified Framework: This is the first study to treat TFR, FP, and SNe samples within a single Bayesian framework, enabling direct cross-checks of systematics across diverse galaxy populations.
• Systematic Robustness Testing: By explicitly comparing linear and quadratic bias models for inhomogeneous Malmquist bias, the study isolates the stability of different tracers against modeling assumptions.
• Re-evaluation of FP Constraints: The paper provides a rigorous re-analysis of Fundamental Plane data, highlighting their sensitivity to bias modeling compared to TFR and SNe.

Results

• TFR and SNe Agreement: Combining the TFR (CF4 W1 and $i$-band) and Pantheon+ SNe samples yields a robust measurement of $S_8 = 0.798 \pm 0.035$ (corresponding to $f\sigma_8 = 0.412 \pm 0.018$) under the fiducial linear-bias model. This result is in agreement with the Planck CMB measurement ($S_8 = 0.832 \pm 0.013$) and is robust under the choice of galaxy bias model. The uncertainty is dominated by the variance of the 2M++ galaxy field ($\sigma^g_8$).
• Fundamental Plane Instability: The FP constraints are found to be unstable under the quadratic extension of the Malmquist bias treatment. While the linear-bias model yields FP results closer to Planck, the quadratic extension (which the FP data prefer statistically) drives the inferred $S_8$ values significantly lower. Specifically, the 6dF FP sample shifts to $S_8 \approx 0.58$, creating a $\sim 7.8\sigma$ tension with Planck. The authors attribute this to the FP tracers populating high-density regions where linear-theory assumptions and linear bias models may break down.
• Internal Consistency: Under the linear-bias model, the TFR and SNe samples show excellent internal consistency (pairwise tensions $< 1\sigma$). The FP samples show mild tension with TFR/SNe but are largely consistent with each other until the quadratic bias is applied, at which point they diverge significantly.
• External Flow: The inferred external flow vector magnitude is generally consistent across samples ($\sim 200 \pm 20$ km s$^{-1}$), though directions vary slightly, likely due to different sky coverages.

Significance
The paper concludes that low-redshift peculiar velocity data, when analyzed using a unified Bayesian framework with TFR and SNe tracers, are concordant with the CMB and do not reinforce the $S_8$ tension. The results suggest that the lower $S_8$ values reported in some previous peculiar velocity studies (specifically those relying heavily on the Fundamental Plane) may stem from unmodeled systematics related to inhomogeneous Malmquist bias and the breakdown of linear theory in dense environments, rather than a genuine deficit in late-time growth. Consequently, the $S_8$ tension appears to be driven primarily by weak lensing surveys rather than peculiar velocity measurements. The authors emphasize that while their fiducial result supports concordance, the instability of the FP results necessitates further scrutiny of systematics in dense environments and potentially more sophisticated non-linear forward reconstruction methods (e.g., BORG) for future analyses.