How it works
A galaxy's observed redshift contains two contributions: the Hubble flow (from cosmic expansion) and a peculiar velocity driven by the local gravity from nearby overdensities. If we can measure the galaxy's distance independently — via a standardisable distance indicator like a Type Ia supernova, the Tully–Fisher relation, or the Fundamental Plane — we recover its peculiar velocity directly.
The amplitude of peculiar velocities at z ≈ 0 is set by the growth rate of structure, f(z) ≈ Ωm0.55(z), times the clustering amplitude, σ8. We measure the combination fσ8(z ≈ 0) by predicting the velocity field from a galaxy density-field reconstruction (see Mapping LSS) and minimising the residuals against observed velocities. The same machinery reproduces the Local Group's motion relative to the CMB rest frame as a by-product.
The technique cleanly complements high-redshift cosmology: cosmic flows are a local probe of the growth rate, sensitive to modified gravity in a regime distinct from the CMB or weak lensing.
Distance indicators
Type Ia supernovae
Standardised SNe Ia deliver ~5–7% distances per object — the cleanest peculiar-velocity tracer available, but historically limited by sample size. In Boruah, Hudson & Lavaux 2020 we combined a much larger SNe Ia peculiar-velocity sample (drawn from Pantheon, Foundation, and other surveys) with the 2M++ density-field reconstruction and obtained fσ8 = 0.400 ± 0.017. Future analyses with the 4HS peculiar velocity survey, the Zwicky Transient Facility (ZTF), and the LSST SNe stream will shrink the error by an order of magnitude.
Tully–Fisher relation
The Tully–Fisher (TF) relation links the luminosity (or baryonic mass) of a spiral galaxy to its rotational velocity width. TF distances are noisier per object than SNe Ia (~20% scatter) but the catalogues are far larger. In Carrick et al. 2015 we used the SFI++ and A1 TF catalogues, again combined with the 2M++ density-field reconstruction, to obtain fσ8(z ≈ 0) = 0.401 ± 0.024. The 2MTF, ALFALFA, and forthcoming WALLABY (ASKAP) HI surveys provide a complementary radio Tully–Fisher stream.
Fundamental Plane
For early-type galaxies, the Fundamental Plane (FP) relates the effective radius, surface brightness, and central velocity dispersion in a tight three-parameter relation that delivers ~20% distances per object. The FP gave the first generation of cluster peculiar-velocity measurements (early-to-mid 1990s) and continues to drive low-redshift cosmology — the 6dF Galaxy Survey, SDSS, and the forthcoming 4HS will be mined for FP velocities. These older datasets are what extended the cosmic-flow baseline to clusters out to ~150 h−1 Mpc before SNe Ia samples were large enough to do so on their own.
Bias calibration and the latest fσ8
The fσ8 values quoted above (~0.40 ± 0.02–0.04) assume the density-field reconstruction is unbiased. In practice it isn't: finite survey volume, flux-limited selection, and Galactic-plane obscuration leave residual systematics that bias the recovered fσ8 high. In Hollinger & Hudson 2024 we characterised these effects on 2M++-like mocks and found a 1.04 ± 0.01 high bias in the 100–180 h−1 Mpc range. Correcting for it and applying the method to recent peculiar-velocity samples gives fσ8lin = 0.362 ± 0.023 — below the uncorrected Carrick (0.401 ± 0.024) and Boruah (0.400 ± 0.017) values, and in tension with Planck CMB extrapolations. The corresponding S8 ≈ 0.69 sits ~2σ below the UNIONS-3500 cosmic-shear values, reinforcing the broader late-time S8 tension.
The underlying methodology is set out in Hollinger & Hudson 2021; the dependence on tracer choice, smoothing scale, and noise model is described on the Mapping LSS page.
Looking ahead
The 4HS peculiar velocity survey will deliver O(105) FP velocities and a substantially expanded SNe Ia sample. Combined with the next generation of density-field reconstructions (DESI BGS, 4HS) and joint analyses with cosmic shear from UNIONS and Euclid, the goal is a percent-level fσ8(z) measurement and a clean test of the growth-rate predictions of ΛCDM versus modified gravity.