Measuring the homogeneity scale using the peculiar velocity field

TL;DR

This paper introduces a PV-based measure of the cosmic homogeneity scale by defining $R_H$ as the radius where the line-of-sight velocity correlation $\Psi_{||}(R)$ crosses zero, leveraging the velocity correlation tensor $\Psi_{ij}(r)$ and its projections. The authors formalize the link between velocity statistics $\Psi_{||}$, $\Psi_\perp$, the bulk-velocity quantity $\mathcal{B}_R$, and a turnover observable $\mathcal{S}(R)$ to identify homogeneity, then test the method on SDSS PV data and mocks. They report a measured homogeneity scale $R_H \approx 133^{+28}_{-52}\,\mathrm{Mpc}/h$ (consistent with mocks and $\Lambda$CDM expectations) and argue that future PV surveys could tighten this to ~20% precision, enabling a PV-based standard ruler for cosmology. The work provides a complementary probe of large-scale structure and the cosmological principle, with potential extensions to bulk-flow observables and higher-redshift surveys.

Abstract

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.

Figures (6)

• Figure 1: Acting on small scales (purple dashed lines), density gradients on large scales source collective flows towards a common attractor (green ellipses), resulting in a positive correlation ($\Psi_\parallel > 0$). The lack of strong density perturbations on large scales, on the other hand, causes galaxies to move in opposite directions rather than together (red dotted lines), contributing a negative $\Psi_\parallel$. The homogeneity scale corresponds to the one at which $\Psi_\parallel=0$, i.e. the one at which galaxies motions with respect to each other are, on average, uncorrelated.
• Figure 2: The function $R\mathcal{B}_R$ and its derivative for a fiducial $\Lambda$CDM cosmology. We identify the homogeneity scale with the transition from correlated to anticorrelated averaged velocities along the vector separation of galaxy pairs $\Psi_\parallel=0$, corresponding to a change in slope in the radial evolution of $\mathcal{B}_R$ and the smoothed velocity variance $\sigma_v$.
• Figure 3: The functions $\Psi_\parallel$ and $\Psi_\perp$ for the SDSS data (pink), each mock realization (faint solid gray lines) together with their mean (green dots) and the fiducial cosmology used to build the mocks (orange). The solid pink and green lines correspond to the best fit results from our MCMC analysis (see Figs. \ref{['mock_validation']},\ref{['data_results']} and Sec. \ref{['SDSS']}) One can easily see that whilst the mean of the mocks recovers very well the input fiducial model, the distribution of the mocks around this mean is skewed and non-gaussian.
• Figure 4: The results of our MCMC explorations for the models defined in Eqs. \ref{['turnaround']},\ref{['model2']} fitted to the mock mean separately (orange and red) or jointly (blue).
• Figure 5: As in Fig. \ref{['mock_validation']}, but for the SDSS data. Compared to the results from the mocks, the parameters of the polynomial fitting of $\Psi_\parallel$ are much less constrained. This is non-suprising given the noticeable difference between the green and pink lines in the left panel of Fig. \ref{['scaling']}.