$S_8$ from Tully-Fisher, fundamental plane, and supernova distances agree with Planck

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 exploit this by using 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 combine all three major classes of distance indicators within a common framework, enabling cross-checks of systematics across diverse galaxy populations. All three tracers yield consistent values of $S_8$ that are also in agreement with Planck. Our joint constraint is $S_8 = 0.819 \pm 0.030$, with the uncertainty dominated by the 2M++ galaxy field. These results demonstrate that peculiar velocity surveys provide a robust, consistent measurement of $S_8$, and support concordance with the cosmic microwave background.