Measuring the homogeneity scale using the peculiar velocity field

We propose an innovative definition of the scale at which the Universe becomes homogeneous based on measurements of velocities rather than densities. When using the matter density field, one has to choose an arbitrary scale (e.g. within 1% of the average density) to define the transition to homogeneity. Furthermore, the resulting homogeneity scale is strongly degenerate with the galaxy bias. By contrast, peculiar velocities (PV) allow us to define an unambiguous scale of homogeneity, namely the distance at which the velocities between pairs of galaxies change from being on-average correlated to anti-correlated. Physically, this relates to when the motion of pairs of galaxies is influenced by the matter density between them, rather than beyond. The disadvantage is that peculiar velocities are more difficult to measure than positions, resulting in smaller samples with larger uncertainties. Nevertheless, we illustrate the potential of this approach using the peculiar velocity correlation functions obtained from the Sloan Digital Sky Survey PV catalog, finding an homogeneity scale of $R_H\approx 133\substack{+28 \ -52}, \rm{Mpc/h}$. Finally, we show that more precise measurements are within reach of upcoming peculiar velocity surveys, and highlight this homogeneity scale’s potential use as a standard ruler within the standard cosmological model.

💡 Research Summary

The paper introduces a novel, velocity‑based definition of the cosmic homogeneity scale, aiming to overcome two major shortcomings of traditional density‑based approaches. Conventional methods define the homogeneity scale (R_H) as the radius at which the number of galaxies inside a sphere scales with the volume to within a chosen tolerance (often 1 %). This definition is arbitrary, and the inferred (R_H) is strongly degenerate with the galaxy bias parameter (b), making it difficult to translate galaxy counts into the true matter homogeneity scale.

Peculiar velocities (PVs) provide a bias‑free tracer of the underlying matter distribution because, in linear theory, the velocity field is directly sourced by the gradient of the gravitational potential, which depends only on the matter density field. The authors therefore propose to define (R_H) as the separation at which the average parallel component of the velocity correlation function, (\Psi_{\parallel}(r)), changes sign from positive (correlated motions) to negative (anti‑correlated motions). Physically, this marks the distance beyond which the bulk flow of a pair of galaxies is no longer driven by mass fluctuations outside the sphere, i.e. the motion becomes insensitive to structures beyond that scale.

The theoretical framework starts from the linear‑perturbation expression for the velocity correlation tensor (\Psi_{ij}(r)), which can be decomposed into parallel (\Psi_{\parallel}(r)) and perpendicular (\Psi_{\perp}(r)) components. Observable quantities (\langle\Psi_1(r)\rangle) and (\langle\Psi_2(r)\rangle) are constructed from line‑of‑sight velocity measurements, incorporating survey geometry and optimal weighting to account for distance‑dependent errors. The authors also define a “total velocity” statistic (B_R) (the volume‑averaged sum of (\Psi_{\parallel}) and (2\Psi_{\perp})) and its derivative (S(R)=d(RB_R)/dR), showing that (S(R)) is proportional to (\Psi_{\parallel}) under the irrotational, scalar‑potential assumption. Consequently, the zero‑crossing of (\Psi_{\parallel}) is equivalent to a change in the slope of (B_R).

To test the method, the authors use the Sloan Digital Sky Survey (SDSS) peculiar‑velocity catalog, which contains ≈34 000 galaxy PVs over 7 000 deg² up to (z\approx0.1). The data are compressed into 25 radial bins (0–150 Mpc h⁻¹) for both (\Psi_{\parallel}(r)) and (\Psi_{\perp}(r)). A suite of 2 048 mock realizations, built from ΛCDM simulations, provides a covariance matrix for the measurements.

For model fitting, (\Psi_{\parallel}(r)) is described by a third‑order polynomial, while the product (R\Psi_{\perp}(R)) is modeled with a piecewise quadratic function that allows a turnover at (R_H). The parameter set ({C,\alpha,\beta,c_1,c_2,c_3,R_H}) is explored with an MCMC sampler (emcee) and analysed using ChainConsumer. Priors enforce physical limits (e.g., (\alpha,\beta\le1)).

Results from the mock mean recover the input ΛCDM transition scale of (R_H\simeq96) Mpc h⁻¹, confirming that the method works in an idealised setting. Applying the pipeline to the actual SDSS data yields (R_H=133^{+28}{-52}) Mpc h⁻¹. Independent fits to (\Psi{\parallel}) and (\Psi_{\perp}) give consistent but broader constraints, illustrating that the joint analysis is essential for a robust determination. The quoted uncertainty (~25 %) is dominated by the limited size of the PV sample and the relatively large measurement errors on individual velocities.

A sensitivity test where the covariance matrix is artificially reduced by a factor of five shows that tightening the statistical errors improves the constraints on the turnover parameters ((\alpha,\beta)) but only modestly refines (R_H). This reflects the fact that the zero‑crossing region contains few data points and is intrinsically less sensitive to noise reduction alone.

The authors argue that forthcoming PV surveys—such as DESI‑PV and the 4MOST Hemispheric Survey—will deliver orders of magnitude more high‑quality velocity measurements, potentially reducing the uncertainty on (R_H) to the 5–10 % level. With such precision, the homogeneity scale could serve as a new standard ruler, complementary to baryon acoustic oscillations and Type Ia supernovae. By measuring (R_H) at multiple redshifts, one could test the consistency of the expansion history and possibly alleviate current tensions in the Hubble constant and σ₈ measurements.

In summary, the paper presents a theoretically motivated, bias‑free definition of the cosmic homogeneity scale based on peculiar‑velocity correlations, demonstrates its feasibility with existing SDSS data, and outlines a clear path toward high‑precision measurements with upcoming surveys. The approach offers a promising new tool for probing large‑scale structure and testing the cosmological principle.