S 8 from peculiar velocities: agreement with Planck for Tully–Fisher and supernovae, tension for the fundamental plane

Peculiar velocity measurements constrain the parameter combination fσ8, the product of the linear growth rate f and the fluctuation amplitude σ8. Under the approximation that f is a monotonic function of Ωm, this can be related to S₈ ≡ σ ₈ √Ω 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 S8 = 0.798 ± 0.035 (fσ8 = 0.412 ± 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 S8 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 S8 tension, though the fundamental plane results call for further scrutiny of their systematics.