Cosmology in one dimension: Vlasov dynamics

Numerical simulations of self-gravitating systems are generally based on $N$-body codes, which solve the equations of motion of a large number of interacting particles. This approach suffers from poor statistical sampling in regions of low density. In contrast, Vlasov codes, by meshing the entire phase space, can reach higher accuracy irrespective of the density. Here, we perform one-dimensional Vlasov simulations of a long-standing cosmological problem, namely, the fractal properties of an expanding Einstein–de Sitter universe in Newtonian gravity. The $N$-body results are confirmed for high-density regions and extended to regions of low matter density, where the $N$-body approach usually fails.

Figure 1

Phase-space distribution functions (left) and matter densities (right) at time $\theta =325$. The top panels show the entire domain, and the center and bottom panels show consecutive zooms. The contour levels of the distribution functions are distributed logarithmically in the interval: ${10}^{-8}\le F\le 1$. The actual maximum value of $F$ is around $F_{\max }=185$. All quantities are expressed in dimensionless units, as explained in the text following Eq. (13).

Figure 2

Matter densities at time $\theta =325$ on a semilogarithmic scale. All quantities are expressed in dimensionless units, as explained in the text following Eq. (13).

Figure 3

Power spectrum of the matter density $|{\rho }_k{|}^2$ for different times from $\theta =0$ to $\theta =300$. Later times correspond to larger values of $|{\rho }_k{|}^2$. A moving average over 41 points is taken in order to smooth the fluctuations. All quantities are expressed in dimensionless units, as explained in the text following Eq. (13).

Figure 4

High-density power spectrum at $\theta =300$. For each curve, density values below the corresponding ${\rho }_{\min }$ have been removed before computing the spectrum. All quantities are expressed in dimensionless units, as explained in the text following Eq. (13).

Figure 5

Low-density power spectrum at $\theta =300$. For each curve, density values above the corresponding ${\rho }_{\max }$ have been removed before computing the spectrum. All quantities are expressed in dimensionless units, as explained in the text following Eq. (13).

Figure 6

Computation of the fractal dimension $D_q$ from Eq. (25) for three values of $q$, $q=-3,-1$, and 3, and several cutoffs, ${\rho }_{th}={10}^{-4}$, ${10}^{-2}$, and 1. The thick lines indicate the ranges over which the slope was computed (these ranges are the same for different values of $q$ at a given cutoff).

Figure 7

Fractal dimension $D_q$ for various values of the exponent $q$ and different cutoffs of the matter density. Open circles correspond to the $N$-body results obtained with $N\approx 262000$ particles. The black dashed curve corresponds to the full Vlasov results (no cutoff). Other lines correspond to Vlasov results with cutoff thresholds at ${\rho }_{\min }={10}^{-4}$, ${10}^{-2}$, 1, and 4.

Figure 8

Evolution of the scaled thermal velocity ${\eta }_{th}$ as a function of the scaled time ${\theta }_{\mathrm{old}}$ normalized to ${\omega }_J^{-1}$ (standard scaling with $\beta =1$) for an $N$-body simulation with the initial condition similar to that used for the Vlasov case. The measured slope is 0.361, close to the theoretical value of 1/3 (straight line).