Empirical signatures of velocity and density cascades in the Local Universe probed by CosmicFlows4 dataset

Abstract

Aims: We aim to characterise the multiscale statistical properties of the reconstructed velocity and density fields of the nearby universe, identify possible scaling regimes, quantify intermittency, and assess indications for the transition toward large-scale homogeneity within the range probed by current data. Methods: We analyse the CosmicFlows4 three-dimensional velocity and density-contrast cubes using absolute structure functions of arbitrary order, $q$. The analysis is performed within a volume extending to $z \lesssim 0.08$ ($\simeq 350~\mathrm{Mpc}$ $h^{-1}$). Structure function scaling exponents $ζ(q)$ are estimated from configuration-space statistics. Intermittency is characterised using the Universal Multifractal formalism, and probability density functions of increments are examined. Results: Two regimes are detected. Small separations are dominated by reconstruction smoothing and show nearly linear $ζ(q)$ behaviour. At larger separations, a scaling regime appears with $ζ_ρ(1)\simeq0.3$ ($D_ρ\approx3.7$) and $ζ_v(1)\simeq0.4$. The correlation function follows $ξ(r)\sim r^{-1.4}$ over $[45,250]~\mathrm{Mpc}\,h^{-1}$, implying $D_2\simeq1.6$. Non-linear $ζ(q)$ and Lévy-stable increment PDFs indicate intermittency and strong non-Gaussianity. Velocity increments show a systematic negative skewness suggestive of a cascade-like asymmetry associated to amplification of negative, compressive gradients.

Figures (12)

• Figure 1: Top: Morphology of the density contrast field $\delta$ in the $SGZ=0$ plane (red: overdensities; blue: underdensities). Bottom: Profiles of $\delta$ (red) and of the velocity magnitude $||\mathbf{v}||=\sqrt{v_x^2+v_y^2+v_z^2}$ (blue, km s$^{-1}$) along the red dashed line shown in the top panel. The line is chosen arbitrarily and serves only to illustrate typical variations of the density and velocity fields.
• Figure 2: First-order structure function ($q=1$) of the artificial MDPL2 density field. Red dots show $S_1^\rho(r)=\langle|\delta(\mathbf{x}+\boldsymbol{r})-\delta(\mathbf{x})|\rangle$ computed from $3.2\times10^{7}$ pairs, and blue dots from $6.4\times10^{7}$ pairs. Two regimes appear: (i) a limited small-scale range with slope $\sim0.25$--$0.30$, and (ii) a broader regime starting at $\sim40$--$50\,\mathrm{Mpc}\,h^{-1}$ with slope $\approx0$. Red lines mark these regimes; blue lines show linear fits to the $6.4\times10^{7}$-pair sample (the first has slope $0.26$). The close overlap of the two estimates indicates convergence for $\gtrsim3.2\times10^{7}$ sampled pairs.
• Figure 3: Structure functions $S_q^v(r)$ of the velocity field for the MDPL2 mock cube, shown for several orders $q$ ($0.5\le q\le2$), using $64\times10^{6}$ randomly sampled point pairs. Two regimes are visible: (i) a narrow small-scale range below $\sim40$--$50\,h^{-1}\,\mathrm{Mpc}$ that does not support genuine scaling, and (ii) a second regime spanning nearly one decade in scale. The latter shows multiaffine behaviour with $H=\zeta_v^{\mathrm{MDPL2}}(1)\approx0.11$ and $\zeta_v^{\mathrm{MDPL2}}(2)\approx0.16$; the condition $\zeta_v^{\mathrm{MDPL2}}(2)<2\,\zeta_v^{\mathrm{MDPL2}}(1)$ indicates statistically significant intermittency. Solid lines show linear fits with their slopes indicated above each line.
• Figure 4: Scaling exponents $\zeta_v^{\mathrm{MDPL2}}(q)$ derived from the second (large-scale) regime of the velocity field. The concave dependence of $\zeta(q)$ on $q$ demonstrates clear departures from monofractal scaling and confirms the presence of statistical intermittency. The dashed curve shows the best-fit UM model, obtained with parameters $\alpha \approx 2$ (log-normal multiplicative cascade), $C_1 \approx 0.03$, and $H \approx 0.11$, indicating weak but non-zero intermittency.
• Figure 5: Empirical curve of $q\zeta'(0) - \zeta(q)$ as a function of $q$ (dots) for the MDPL2 velocity field, shown on a log–log plot, from which the UM parameters $\alpha$ and $C_1$ are directly inferred: $\alpha$ corresponds to the slope of the linear regime, while $C_1$ is estimated from the intercept. We obtain $\alpha \approx 2$ and $C_1 \approx 0.03$.