Cosmic web, multistream flows, and tessellations

Understanding the structure of the matter distribution in the Universe due to the action of the gravitational instability—the creation of the “cosmic web”—is complicated by lack of direct analytic access to the complex nonlinear domain of structure formation. Here, we suggest and apply a novel tessellation method designed for cold dark matter (CDM) N-body cosmological simulations. The method is based on the fact that the initial CDM state can be described by a 3-dimensional manifold (in a 6-dimensional phase space) that remains continuous under evolution. Our technique uses the full phase space information and has no free parameters; it can be used to compute multistream and density fields, the main focus of this paper. Using a large-box $\Lambda \mathrm{CDM}$ simulation we carry out a variety of illustrative initial analyses with the technique. These include studying the correlation between multistreaming and density, the identification of structures such as Zel’dovich pancakes and voids, and statistical measurements of quantities such as the volume fraction as a function of the number of streams—where we find a remarkable scaling relation. Cosmological implications are briefly discussed.

Figure 1

Two-dimensional illustration of the difference in representing density fields by using particles (left column) and tessellations (right column). The top panels exhibit Lagrangian space where the density is uniform while the bottom panels display the system at a nonlinear stage of the evolution. The nodes of the triangulation in the right panel are at the positions of the particles in the left panel. The density field can be derived from the particle/node coordinates.

Figure 2

Decomposition of a cube into five tetrahedra. The central tetrahedron shown by a heavy line is regular and its volume equals a third of the volume of the cube. Each of the remaining four tetrahedra is half of the volume of the central one, or one-sixth of the volume of the cube.

Figure 3

Particle distribution, i.e., positions of tetrahedron vertices in a small subcube of size $8h^{-1}\mathrm{Mpc}$ extracted from the full simulation.

Figure 4

Distribution of the tetrahedron centroids in the same region as in Fig. 3. The size of the dots is reduced in order to approximately compensate for their larger number.

Figure 5

Same box as in Figs. 3, 4. The top and bottom panels show the density and multistream fields, respectively. The fields in the left column are sampled on an $8^3$ grid, in the middle column on a ${32}^3$ grid, and in the right column on a ${128}^3$ grid. The three density contours are: $3\overline{\rho }$ (red), $10\overline{\rho }$ (green), and $100\overline{\rho }$ (blue), and the three multistream contours are: 3 streams (red), 7 streams (green), and 15 streams (blue). One can clearly see how filaments and pancakes emerge with increasing resolution from the left to the right.

Figure 6

Number of sampling mesh nodes as a function of density in the regions with number of streams shown next to each curve.

Figure 7

Particles lying in regions with different numbers of streams compared to the density. The box is the same as in the previous figures. All four panels show the density contour $\rho /\overline{\rho }=10$ (gray). The panels show particles (black dots) lying in $\ge 9$-, 5- or 7-, 3-, and one-stream flows.

Figure 8

Bulk of the CDF of the volumes of the tetrahedra in the range from 1% to 99%. The black and gray lines show the CDF, $F(V)$, and $1-F(V)$ respectively.

Figure 9

CDFs $F(T)$, $F(B)$, and $F(L)$ are shown as the black dotted, dashed and solid lines, respectively. The corresponding gray lines represent $1-F$ for each size parameter.

Figure 10

CDF, $F(V)$, of 3% of the least voluminous tetrahedra. The gray straight line is the power law $F\propto V$.

Figure 11

CDFs, $F(T)$, $F(B)$, and $F(L)$ of the smallest 3% of the corresponding values are shown by the dotted, dashed, and solid black lines, respectively. The gray solid straight lines show $F(T)\propto T$, $F(B)\propto B^{3.8}$ and $F(L)\propto L^{6.2}$ respectively.

Figure 12

Cumulative distribution function, $F(V)$, of 3% of the most voluminous tetrahedra. The gray straight lines are the exponential fits, $F\propto \exp (-V/V_c)$, with $V_c=0.41h^{-3}{\mathrm{Mpc}}^3$ and $V_c=2.25h^{-3}{\mathrm{Mpc}}^3$.

Figure 13

CDFs, $F(T)$, $F(B)$, and $F(L)$ of the largest 3% of the corresponding values are shown by dotted, dashed, and solid black lines, respectively. The gray solid straight lines show the exponential fits with $T_c\approx 0.08h^{-1}\mathrm{Mpc}$, $B_c\approx 0.215h^{-1}\mathrm{Mpc}$, $L_{c,1}\approx 0.77h^{-1}\mathrm{Mpc}$, and $L_{c,2}\approx 1.15h^{-1}\mathrm{Mpc}$.

Figure 14

Fraction of volume as a function of the number of streams. The data points with statistical error bars are shown in gray. The black solid and two dashed lines show the fit $f_v(n_s)=0.93n_s^{-2.82\pm 0.05}$.

Figure 15

Number of sampling mesh nodes as a function of density in regions with a fixed number of streams (values shown next to each curve).

Figure 16

Fraction of mass as a function of the number of streams (dots). The solid line is $F_m=1.19n_s^{-1.46}$.