Cosmic curl -- Features and convergence of the vorticity power spectrum in $N$-body simulations

TL;DR

This work investigates the cosmic vorticity power spectrum in $N$-body simulations, showing that velocity statistics (divergence and vorticity) can be accurately captured with PM simulations, while vorticity converges slowly due to nonlinear mode coupling. The authors employ DTFE-based velocity interpolation and an extrapolation scheme across four simulations with different particle counts to estimate the fully converged $P_\omega(k)$ to about 5% accuracy over three decades in $k$. They analyze time evolution via $P_\omega(k,z) = C \cdot (D_+(z))^{\beta(k)}$, finding $\beta(k) \approx 6$ at small $k$, and quantify the ratio $P_\omega/P_\theta$, which remains small in the linear regime and grows to roughly 0.2 in the non-linear regime, with resolution-dependent artefacts at high $k$. The results provide a practical framework for predicting velocity-based observables in cosmology and informing the design of velocity surveys, while highlighting the challenges of extracting vorticity in small volumes.

Abstract

Observations of the cosmic velocity field could become an important cosmological probe in the near future. To take advantage of future velocity-flow surveys we must however have the theoretical predictions under control. In many respects, the velocity field is easier to simulate than the density field because it is less severely affected by small-scale clustering. Therefore, as we also show in this paper, a particle-mesh (PM) based simulation approach is usually sufficient, yielding results within a few percent of a corresponding P$^3$M simulation in which short-range forces are properly accounted for, but which also carry a much larger computational cost. However, in other respects the velocity field is much more challenging to deal with than the density field: Interpolating the velocity field onto a grid is significantly more complicated, and the vorticity field (the curl-part of the velocity field) is severely affected by both sample variance and discretisation effects. While the former can be dealt with using fixed amplitude initial conditions, the former makes it infeasible to run fully converged simulations in a cosmological volume. However, using the $N$-body code CONCEPT we show that one can robustly extrapolate the cosmic vorticity power spectrum from just 4 simulations with different number of particles. We expect our extrapolated vorticity power spectra to be correct within 5\% of the fully converged result across three orders of magnitude in $k$. Finally, we have also investigated the time dependence of the vorticity as well as the ratio of vorticity to divergence.

Figures (17)

• Figure 1: (a) The density field plotted in log-scale for $z = 0$ for the simulation with $N = 1024^3$ particles, $n_\text{grid} = 2048$, and $L_\text{box} = 256$ Mpc/h using the P$^3$M method. (b) The velocity field for the same simulation plotted in linear scale. Note that the variation in the velocity field are much more linear than the variation in the density field.
• Figure 2: (a) $\frac{P_\text{PM}}{P_{\text{P}^3\text{M}}}$ plotted for the power spectra of the velocity divergence and vorticity respectively. (b) $\frac{P_\text{PM}}{P_{\text{P}^3\text{M}}}$ plotted for the power spectra for the density field. It can be seen that the power spectra for the velocity divergence and vorticity for PM and P$^3$M respectively agree with each other within ten percent even on small scales. However, the power spectra for the density field for PM and P$^3$M respectively do not agree at all on small scales.
• Figure 3: (a) The divergence for the simulation with $N = 1024^3$ particles, $n_\text{grid} = 1024$, and $L_\text{box} = 128\, \text{Mpc}/h$ using the PM method. 'Gradient' and 'DTFE' is the divergence calculated with the velocity gradient method using the gradient and divergence as dtfe outputs respectively. 'Velocity' is the divergence calculated with the velocity field method, and 'CLASS' is the theoretical divergence power spectrum calculated with class. (b) Same simulation and methods, but with the vorticity plotted instead. Note that since the vorticity is non-linear, it cannot be calculated theoretically using class.
• Figure 4: Variance of the density, divergence and vorticity power spectra at redshift $z=0$. Left panels: $128\,\text{Mpc/h}$ box. Right panels: $512\,\text{Mpc/h}$ box. The vorticity power spectra are severely affected by sampling variance across a wide range of scales, especially in the $128\,\text{Mpc/h}$ box. Even in this case, however, the fixed amplitude simulation is consistent with the mean value.
• Figure 5: The variance for 5 simulations at $z = 5$ in the box with size 128 Mpc/h. It can be seen that the variance is much smaller here leading to the conclusion that the large variance in the 128 Mpc/h box at $z = 0$ is caused by strong non-linearity. As with the simulations at $z = 0$, the fixed amplitude simulation is consistent with the mean value.