Peculiar Velocity Correlation Functions

Updated 7 January 2026

Peculiar velocity correlation functions are two-point statistics that measure the spatial coherence of cosmic velocity fields driven by density fluctuations.
They enable robust extraction of cosmological parameters such as the growth rate and dark energy properties by linking theoretical models with observations.
Observational techniques like line-of-sight estimators and optimized pair weighting minimize cosmic variance, enhancing precision in testing gravitational physics.

Peculiar velocity correlation functions characterize the spatial coherence and correlation structure of cosmic velocity fields sourced by large-scale density perturbations. They provide rigorous two-point statistics essential for extracting cosmological information from galaxy and supernova peculiar velocity (PV) surveys, directly probing the growth of structure and offering stringent tests for gravitational physics. The quantitative formalism underpins analyses in supernova cosmology, galaxy flows, kinetic Sunyaev–Zeldovich studies, and tests of ΛCDM and modified-gravity scenarios.

1. Fundamental Definitions and Mathematical Structure

The peculiar velocity field, $\mathbf{v}(\mathbf{x})$ , is taken as a 3D vector field in comoving coordinates relative to the cosmic rest frame. The two-point peculiar velocity correlation tensor is defined by

$\Psi_{ij}(r) = \langle v_i(\mathbf{x}) v_j(\mathbf{x} + \mathbf{r}) \rangle,$

where $r = |\mathbf{r}|$ and the averaging is over realizations of the cosmic density/velocity field. Statistical isotropy and homogeneity require the decomposition

$\Psi_{ij}(r) = [\Psi_\parallel(r) - \Psi_\perp(r)] \hat{r}_i \hat{r}_j + \Psi_\perp(r) \delta_{ij},$

with parallel and perpendicular correlation functions defined by

$\Psi_\parallel(r) = \langle [\mathbf{v} \cdot \hat{\mathbf{r}}][\mathbf{v}' \cdot \hat{\mathbf{r}}] \rangle, \quad \Psi_\perp(r) = \frac12 \left\langle [\mathbf{v} \cdot \hat{\bm{\theta}}][\mathbf{v}' \cdot \hat{\bm{\theta}}] + [\mathbf{v} \cdot \hat{\bm{\phi}}][\mathbf{v}' \cdot \hat{\bm{\phi}}] \right\rangle,$

where %%%%2%%%% is the separation unit vector and $\mathbf{v}',\,\mathbf{x}' = \mathbf{x} + \mathbf{r}$ .

For observable line-of-sight (LOS) velocities $u = \mathbf{v} \cdot \hat{n}$ , the ensemble-mean two-point correlation between galaxies A and B separated by $\mathbf{r}$ is

$\langle u_A u_B \rangle = \Psi_\parallel(r) (\hat{n}_A \cdot \hat{r}) (\hat{n}_B \cdot \hat{r}) + \Psi_\perp(r) \left[\hat{n}_A \cdot \hat{n}_B - (\hat{n}_A \cdot \hat{r})(\hat{n}_B \cdot \hat{r}) \right].$

These correlation functions are directly related to the velocity power spectrum $P_{vv}(k)$ via Hankel transforms: $\Psi_\parallel(r) = \int_0^\infty \frac{dk}{2\pi^2} P_{vv}(k) \left[j_0(kr) - 2 \frac{j_1(kr)}{kr}\right],$

$\Psi_\perp(r) = \int_0^\infty \frac{dk}{2\pi^2} P_{vv}(k) \frac{j_1(kr)}{kr},$

where $j_\ell$ are spherical Bessel functions (Blake et al., 2023, Wang et al., 2021, Wang et al., 2018).

2. Physical Origin and Theoretical Modeling

Peculiar velocity fields arise from gravitational response to density perturbations. In linear perturbation theory, velocity is sourced by the density field through

$\mathbf{v}(\mathbf{k}, z) = -i \frac{a D'(z)}{k^2} \mathbf{k} \delta(\mathbf{k},z=0),$

where $D(z)$ is the scale-independent growth factor and $D'$ its derivative with respect to conformal time (Davis et al., 2010). The velocity power spectrum,

$P_{vv}(k,z) = [a H f]^2 P_\delta(k, z) / k^2,$

is thus heavily weighted toward large-scale (low- $k$ ) modes, producing strong correlations on tens-to-hundreds of Mpc scales (coherence lengths $L_\delta \sim 100\,h^{-1}$ Mpc for the irrotational mode) (Zheng et al., 2013).

Nonlinear effects and galaxy biasing can modify the amplitude and scale-dependence of the correlation functions. 1-loop standard perturbation theory introduces $O(\delta_L^4)$ corrections, altering templates by $5$– $10\%$ at $r\lesssim 20$ – $30\,h^{-1}$ Mpc; biasing and density-weighting terms further contribute notably at these scales (Tonegawa et al., 2023, Okumura et al., 2013).

For galaxy surveys, only the density-weighted ("momentum") LOS component is observable: $p_\parallel(\mathbf{x}) = [1 + \delta(\mathbf{x})] v_\parallel(\mathbf{x}).$ Density weighting induces extra terms beyond pure velocity correlations, notably mixed bispectrum contributions (Tonegawa et al., 2023, Okumura et al., 2013).

3. Observational Estimation and Practical Methodologies

Peculiar velocity correlation functions are estimated using pairwise product statistics such as the Górski et al. LOS estimators

$\psi_1(r) = \frac{\sum_{pairs} u_1 u_2 \cos\theta_{12}}{\sum_{pairs} \cos^2\theta_{12}}, \quad \psi_2(r) = \frac{\sum_{pairs} u_1 u_2 \cos\theta_1 \cos\theta_2}{\sum_{pairs} \cos\theta_{12} \cos\theta_1 \cos\theta_2}$

with angular factors $\cos\theta_{12},\,\cos\theta_i$ given by the LOS and separation geometry (Wang et al., 2018, Dupuy et al., 2019). Survey-specific geometric normalization factors $A(r),B(r)$ relate these to $\Psi_\parallel,\Psi_\perp$ : $\psi_1(r) = A(r)\Psi_\parallel(r) + [1-A(r)]\Psi_\perp(r),\quad \psi_2(r) = B(r)\Psi_\parallel(r) + [1-B(r)]\Psi_\perp(r).$ Matrix inversion yields the physically defined parallel and perpendicular functions (Wang et al., 2018, Wang et al., 2021).

Volume-based weighting of galaxy pairs (e.g., $w_{mn} = (r_1 r_2)^p$ ) reduces local sample over-weighting, suppresses cosmic variance, and allows robust likelihood-based fits to the linear theory (Wang et al., 2021). Statistical uncertainties arise from combined measurement errors and cosmic variance, the latter dominating on large scales (Wang et al., 2018, Wang et al., 2021).

For supernova cosmology, a covariance-matrix formalism incorporates off-diagonal correlated velocity noise into magnitude-variance modeling: $C_{ij} = \sigma_i^2 \delta_{ij} + C_{ij}^{\rm vel},$ with $C_{ij}^{\rm vel}$ derived from the full configuration-space two-point function and model parameters (Davis et al., 2010).

4. Application Domains and Cosmological Inference

Peculiar velocity correlation functions are foundational to:

Direct growth-rate ( $f\sigma_8$ ) estimation: Both parallel and perpendicular correlation amplitudes scale as $(f\sigma_8)^2$ , making them sensitive probes of structure formation (Dupuy et al., 2019, Turner et al., 2 Dec 2025).
Testing gravity and dark energy: Distinctive signatures appear in modifications to the growth rate (scale dependence), captured in $f(R)$ and DGP gravity via scale- and redshift-dependent $f(k,a)$ in the theoretical templates for $\xi_{vv}$ , with forecasts establishing competitive parameter sensitivity (Lyall et al., 2022).
Supernova cosmology: Incorporation of velocity correlation matrices in SN Hubble diagram fits is required to control percent-level biases in the dark energy equation-of-state parameter $w$ (Davis et al., 2010).
Kinetic Sunyaev–Zeldovich/cross-correlation analysis: Inclusion of velocity correlation functions (and their cross-correlation with galaxy density) enables constraint on optical depth and velocity bias from kSZ-CMB data (Ma et al., 2017).
Standard-ruler definition: The scale $R_H$ at which $\Psi_\parallel(r)$ (or the bulk-in-sphere statistic) transitions from positive to negative marks a velocity-based homogeneity scale, providing an alternative cosmological ruler independent of galaxy bias (Giani et al., 6 Jan 2026).

5. Breakdown of Linear Theory and Nonlinear Corrections

Linear theory provides an accurate description for $r \gtrsim 20$ –$30$ $h^{-1}$ Mpc, but breaks down on smaller scales due to:

Nonlinear density fluctuations, leading to breakdowns in both Fourier and configuration space ( $>10\%$ errors at $k\gtrsim0.05$ – $0.1\,h/$ Mpc and $r\lesssim30$ – $50\,h^{-1}$ Mpc) (Okumura et al., 2013, Zheng et al., 2013).
Density weighting, which induces bispectrum-like contributions that must be modeled to avoid $\sim 10\%$ systematic bias in $f\sigma_8$ for next-generation survey volumes when fitting down to $r_{\rm min} \sim 20\,h^{-1}$ Mpc (Tonegawa et al., 2023).

One-loop Eulerian PT and hybrid simulation-calibrated templates accurately reproduce the N-body simulation results to these scales (Okumura et al., 2013, Turner et al., 2 Dec 2025).

6. Covariance, Cosmic Variance, and Limitations

Covariance estimation for pairwise statistics incorporates:

Survey selection functions, observational noise, and geometry (Blake et al., 2023).
Large-scale bulk flow covariance, which produces high correlation between separation bins.
Analytic models for covariance matrices that accurately reproduce the variance and bin–bin correlation structure seen in thousands of survey mocks (to $\lesssim$ 5% in $f\sigma_8$ errors), enabling precision cosmological fits without requiring excessive numbers of computationally expensive mock catalogs (Blake et al., 2023, Wang et al., 2018).

Cosmic variance is the fundamental limiting factor in velocity-based cosmology, dominating statistical error budgets for present and near-future PV surveys, especially at large separations where the number of independent modes is necessarily small and the distribution of correlation estimators is strongly non-Gaussian (Wang et al., 2018). Strategies to combat this include deeper/higher-density surveys, optimized pair weighting, and multi-statistic joint analyses.

7. Extensions: Angular, Transverse, and Higher-Order Correlations

Generalizations of the two-point formalism include:

Velocity direction correlation functions, $C_v(r)$ , which measure alignment of normalized velocity vectors—providing bias- and normalization-independent tests of gravity law (e.g., rejecting MONDian $r^{-1}$ force from lack of excess coherence at large scales) (Mould et al., 2015).
Two-point transverse velocity statistics, probing cosmological information through projected (proper-motion–based) velocity correlations sensitive to the linear matter power spectrum and evading classical radial distance-degeneracy (Darling et al., 2018).
Higher-point streaming models, such as the $n$ -point streaming formalism, offering an exact mapping between real- and redshift-space statistics via the joint PDF of relative LOS velocities. For the 3-point case (and bispectrum applications), this enables robust theoretical predictions for redshift-space data analyses (Kuruvilla et al., 2020).

These extensions open new statistical channels for upcoming wide/deep peculiar velocity and proper-motion surveys, providing complementary constraints to galaxy clustering and deepening the capacity for model-independent cosmic tests across gravity and dark energy models.

In sum, peculiar velocity correlation functions—rigorously constructed and precisely modeled—are indispensable for analysis and interpretation of cosmic flows, supernova Hubble diagrams, redshift-space distortions, and kinetic SZ cross-correlations. Their robust connection to fundamental cosmological parameters, relative insensitivity to galaxy bias on large scales, and sensitivity to both gravitational physics and structure growth make them a central tool in contemporary and future precision cosmology (Davis et al., 2010, Tonegawa et al., 2023, Lyall et al., 2022, Turner et al., 2 Dec 2025, Blake et al., 2023, Wang et al., 2018, Wang et al., 2021, Zheng et al., 2013, Giani et al., 6 Jan 2026).