Peculiar Velocity Measurements in Cosmology

Updated 9 December 2025

Peculiar velocity measurements are methods that determine galaxy velocities by comparing redshift-independent distance indicators with observed redshifts.
They bypass galaxy bias and directly map the local cosmic velocity field, offering critical insights into the growth of large-scale structure.
Advanced statistical modeling and DESI DR1 calibrations yield precise constraints on the growth rate and tests of gravity in the low-redshift Universe.

Peculiar velocity measurements provide direct constraints on the local cosmic velocity field and, by extension, the underlying distribution and growth of large-scale structure in the Universe. Unlike redshift-space distortion measurements, which infer velocity statistics indirectly from galaxy clustering anisotropy, peculiar velocity surveys—using redshift-independent distance indicators such as the Fundamental Plane (FP) of early-type galaxies or the Tully–Fisher (TF) relation for spirals—provide direct, line-of-sight velocities that are insensitive to galaxy bias. Their statistical power, systematics control, and complementarity with density surveys make them a key cosmological probe, particularly at low redshifts.

1. Foundations and Physical Principles

Peculiar velocities arise as the component of an object's velocity relative to the uniform Hubble flow, generated by the gravitational potential fluctuations of large-scale structure. The observed redshift of a galaxy, $z_{\rm obs}$ , is a combination of the cosmological redshift due to expansion and the Doppler shift from the peculiar velocity $v_p$ , such that

$1 + z_{\rm obs} = (1 + z_{\rm cos}) (1 + z_p).$

The line-of-sight peculiar velocity can then be written non-relativistically as

$v_p = c\,\frac{z_{\rm obs} - z_{\rm cos}}{1+z_{\rm cos}},$

where $z_{\rm cos}$ is the cosmological redshift corresponding to the true comoving distance of the system inferred from a redshift-independent distance indicator (Davis et al., 2014). On large scales, the velocity divergence $\theta = \nabla \cdot \mathbf{v}$ is directly connected to the growth rate of structure as

$\theta(\mathbf{x},a) = -aH(a)f(a)\delta(\mathbf{x},a),$

where $f(a) = d\ln D/d\ln a$ is the linear growth rate, with $D$ the linear growth function (Howlett et al., 2016).

2. Observational Strategies and Distance Indicators

Contemporary peculiar velocity surveys exploit the Fundamental Plane and Tully–Fisher relations to obtain redshift-independent galaxy distances out to $z \approx 0.1$ . The Fundamental Plane is parameterized as

$\log R_e = a\,\log \sigma_0 + b\,\log I_e + c,$

where $R_e$ is the effective radius, $\sigma_0$ is the velocity dispersion, and $I_e$ is the mean surface brightness (Ross et al., 2 Dec 2025, Howlett et al., 2022).

For each galaxy, the deviation from the calibrated FP or TF relation yields a log-distance ratio $\eta = \log_{10}[d_{\rm FP}/d(z_{\rm cmb})]$ , where $d(z_{\rm cmb})$ is the comoving distance for the observed CMB-frame redshift (Ross et al., 2 Dec 2025). This log-distance ratio is mapped to a peculiar velocity via

$\hat{v} = c\ln 10\,\bigg[\frac{(1+z_{\rm cmb})c}{H(z_{\rm cmb})d(z_{\rm cmb})} - 1 \bigg]^{-1} \eta,$

a generalization of the naive $v_p \approx cz - H_0 D$ expression that is accurate at all observed redshifts and avoids systematic biases exceeding 100 km/s at $z \sim 0.04$ (Davis et al., 2014, Ross et al., 2 Dec 2025).

The Dark Energy Spectroscopic Instrument (DESI) DR1 provides the largest homogeneous FP-based sample to date—98,292 unique early-type galaxies with mean random distance errors of 26%, doubling the previous z < 0.1 FP-based sample size (Ross et al., 2 Dec 2025).

3. Statistical Analysis and Covariance Modeling

The extraction of cosmological information from peculiar velocity surveys requires careful statistical modeling. Line-of-sight velocities $v_p$ at positions $\mathbf{r}_m$ are treated as a realization of a Gaussian random field, with their auto- and cross-covariances determined by the underlying matter and velocity power spectra. For a full catalogue $S_m = \hat{\mathbf{r}}_m \cdot \mathbf{v}(\mathbf{r}_m)$ , the likelihood is

$\mathcal{L}(S|P_v) = \frac{1}{(2\pi)^{N/2} |R|^{1/2}} \exp\left( -\frac{1}{2} S^T R^{-1} S \right),$

where $R$ is the sum of the cosmic covariance and measurement noise (Macaulay et al., 2011). The cosmic covariance is

$R_{mn}^{(v)} = \int \frac{4\pi k^2 dk}{(2\pi)^3} P_v(k) f_{mn}(k),$

with $P_v(k) = (H_0 f / k)^2 P_{mm}(k)$ the velocity power spectrum and $f_{mn}(k)$ the angular kernel. Uncorrelated noise (measurement error and small-scale random motions) enters the diagonal.

Advanced analyses (e.g., DESI DR1) extend this to joint modeling of the velocity field $\eta$ and galaxy density $\delta_g$ fields using block covariances, with detailed analytic expressions for each covariance component. Maximum-likelihood fits and a suite of simulated mock catalogues are used for unbiased parameter estimation of the growth rate $f\sigma_8$ and validation of method uncertainty (Lai et al., 2 Dec 2025).

4. Cosmological Applications and Constraints

Peculiar velocity measurements are a direct probe of the growth rate of structure, $f\sigma_8(z)$ . In joint field analyses of the DESI DR1 FP and TF catalogues with the galaxy density field, consensus $f\sigma_8$ determinations at $z_{\rm eff} = 0.07$ achieve $\sim12\%$ fractional precision (Lai et al., 2 Dec 2025, Turner et al., 2 Dec 2025), fully consistent with $\Lambda$ CDM predictions ( $f\sigma_8 = 0.449 \pm 0.008$ from Planck).

These measurements are cosmic-variance limited on the largest accessible scales, directly test the gravitational instability paradigm, and are critical for discriminating between General Relativity and modified gravity (via the gravitational growth index $\gamma$ ). Combining low-redshift $f\sigma_8$ from peculiar velocities with higher redshift redshift-space distortion measurements yields constraints on $\gamma$ at the level $\gamma = 0.58 \pm 0.11$ , consistent with GR's $\gamma \approx 0.55$ (Lai et al., 2 Dec 2025, Turner et al., 2 Dec 2025). The independence from galaxy bias and the minimal modeling of the mass–velocity connection are distinctive advantages over clustering-only approaches (Koda et al., 2013, Howlett et al., 2016).

Beyond constraining dynamical parameters, peculiar velocity catalogues are used for direct power spectrum estimation—including on scales inaccessible to galaxy clustering—and for measurements of the local bulk flow. The local bulk flow amplitude and direction measured in SDSS and DESI PV catalogues are found to be consistent with $\Lambda$ CDM once survey geometry, measurement error, and sample variance are correctly propagated (Howlett et al., 2022, Ross et al., 2 Dec 2025).

5. Systematic Errors and Calibration Techniques

Systematics control is central in peculiar velocity cosmology. In the FP-based approach, systematic effects include:

Zero-point calibration: Inter-survey offsets are eliminated by zero-point matching to external velocity field reconstructions (e.g., CosmicFlows-III), achieving stability at the 0.004 dex level (Howlett et al., 2022).
Group/cluster environment correction: Correlations between FP residuals and group richness are removed by refitting the FP parameters in richness bins, empirically flattening biases (Ross et al., 2 Dec 2025, Howlett et al., 2022).
Photometric and spectroscopic systematics: DESI/SDSS comparisons of band calibrations, cross-checks of internal error inflation, and external validation against independent surveys enforce photometric and velocity-dispersion robustness (Ross et al., 2 Dec 2025, Howlett et al., 2022).
Redshift-independent mapping: Non-relativistic approximations for velocity conversion are avoided; the accurate mapping $v_p = c(z_{\rm obs} - z_{\rm cos})/(1+z_{\rm cos})$ is universally applied to eliminate significant redshift-dependent bias (Davis et al., 2014).
Small-scale nonlinearities: Velocity power spectrum fits marginalize over nuisance parameters (e.g., velocity dispersion, additional nonlinear terms), preventing biases from scale-dependent velocity bias or shot-noise misestimation (Lai et al., 2 Dec 2025, Howlett et al., 2016).

Systematics from FP parameter uncertainties (e.g., $\alpha_{ap}, Q$ ) in DESI DR1 contribute less than 0.005 dex to the log-distance uncertainty across the bulk of the sample (Ross et al., 2 Dec 2025). For all next-generation analyses, extensive suites of mocks propagate the effect of survey selection, measurement noise, and environmental dependencies on the derived velocities (Howlett et al., 2022).

6. Extensions: Transverse Velocities, Multi-probe Synergy, and Prospects

Peculiar velocities can be measured both radially and, with precise astrometry, in the plane of the sky. Transverse peculiar velocities, $\mathbf{v}_{\perp}$ , are accessible via extragalactic proper motion measurements from VLBI or Gaia and offer geometric, distance-ladder-independent probes of the matter power spectrum. The two-point transverse velocity correlation function $\xi_{v,\perp}$ provides sensitivity to structure on $10-50\,{\rm Mpc}$ scales, and, in synergy with traditional radial velocity measurements, enables three-dimensional mapping of cosmic flows (Darling et al., 2018, Truebenbach et al., 2018).

Future surveys incorporating both weak lensing and peculiar velocity measurements will exploit complementarity: while lensing is sensitive to the lensing potential ( $\Phi - \Psi$ ) and matter fluctuations, peculiar velocities directly test the Newtonian potential ( $\Psi$ ), allowing scale- and time-dependent modifications to GR to be robustly probed (Song et al., 2010).

Moreover, the inclusion of peculiar velocity likelihoods in joint BAO+velocity+CMB Fisher matrices enhances constraints on the dark energy equation of state ( $\sigma(w) \approx 0.06$ , $\sigma(w_a) \approx 0.20$ ), beyond what is achievable by BAO or velocities alone (Song, 2010). For neutrino mass, low- $z$ peculiar velocity data substantially improve constraints over galaxy redshifts alone in absence of external CMB priors (Whitford et al., 2021).

The DESI PV program, expanding to $>180,000$ peculiar velocity measurements, will drive percent-level constraints on the growth rate at $z<0.1$ , provide the definitive low-redshift test of cosmic acceleration, and decisively test structure growth across gravity scenarios (Ross et al., 2 Dec 2025, Lai et al., 2 Dec 2025, Turner et al., 2 Dec 2025).

Parameter	Value/Range	Notes
Redshift Range	0.0033 < z < 0.1	Early-type galaxies (FP), BGS footprint
Number of FP distances (DESI DR1)	98,292	Doubles all prior z < 0.1 FP data
Distance Uncertainty (mean, σ_η)	0.12 dex (26%)	Comparable to previous largest samples
Consensus $f\sigma_8(z_{\rm eff}=0.07)$	$0.450 \pm 0.055$	Consistent with Planck+ $\Lambda$ CDM
Systematics (zero-point, selection, etc.)	$<$ 0.005 dex	Controlled by environmental/photometric corrections
Astrophysical Footprint	$\sim9000$ deg $^2$	Within DESI LSS area

Peculiar velocity measurements, rigorously calibrated and statistically controlled, deliver unbiased probes of the amplitude and growth of density fluctuations, directly trace cosmic flows, and offer robust tests of gravity and dark energy in the low-redshift Universe. The latest DESI and SDSS analyses have set new standards in sample size, systematics control, and cosmological impact.