The Hidden Role of Anisotropies in Shaping Structure Formation in Cosmological N-Body Simulations

TL;DR

This paper reveals that initial conditions constructed by displacing a cubic lattice with an isotropic random field introduce subtle, lattice-induced anisotropies that survive and are amplified by gravitational evolution, seeding persistent filamentary patterns even in the linear regime. By leveraging the Angular Distribution of Pairwise Distances (ADPD), the authors quantify directional anisotropies that escape standard angle-averaged statistics and demonstrate their growth through cosmic time in two CDM simulations. They show that these artifacts arise from the coupling between the non-isotropic pre-initial lattice and an isotropic displacement field, calling into question the isotropy assumption in ΛCDM simulations and highlighting the need for glass-like pre-ICs or alternative lattices. The work provides a practical diagnostic framework and advocates for systematic re-evaluation of IC construction to mitigate discretization-induced biases in the quasi-linear regime of structure formation.

Abstract

Initial conditions in cosmological $N$-body simulations are typically generated by displacing particles from a regular cubic lattice using a correlated field derived from the linear power spectrum, often via the Zel'dovich approximation. While this procedure reproduces the target two-point statistics (e.g., the power spectrum or correlation function), it introduces subtle anisotropies due to the underlying lattice structure. These anisotropies, invisible to angle-averaged diagnostics, become evident through directional measures such as the Angular Distribution of Pairwise Distances. Analyzing two Cold Dark Matter simulations with { varying resolutions and box sizes}, we show that these anisotropies are not erased but are amplified by gravitational evolution. They seed filamentary structures that persist into the linear regime, remaining visible even at redshift $z = 0$. Our findings demonstrate that such features are numerical artifacts -- emerging from the anisotropic coupling between the displacement field and the lattice -- not genuine predictions of an isotropic cosmological model. These results underscore the importance of critically reassessing how initial conditions are constructed, particularly when probing the large-scale, quasi-linear regime of structure formation.

Figures (18)

• Figure 1: Power spectrum (left) and corresponding two-point correlation function in linear-log (middle) and log-log (right) plots for a typical early-time Cold Dark Matter model (see Eq. \ref{['pk']}). The vertical red line marks the zero-crossing scale $r_t \approx 120 \mathrm{Mpc}$/$h$. Amplitudes are shown in arbitrary units.
• Figure 2: Point distribution (upper panel) and heatmap of the Angular Distribution of Pairwise Distances (ADPD, lower panel) for a system composed of multiple bars. Each bar is characterized by an orientation angle $\theta_i$ relative to the $x$-axis and a finite length $\ell_i$. Length scales are expressed in units of the average distance between nearest neighbors $\Lambda$.
• Figure 3: Angular Distribution of Pairwise Distances (ADPD) for four configurations: a Poisson distribution (top left), a perfect cubic lattice (top right), a shuffled lattice with displacement amplitude $\delta = 0.6$ (bottom left), and with $\delta = 1.0$ (bottom right). For improved visual interpretation, the color scale in each panel has been individually adjusted to enhance contrast and highlight relevant structures in the ADPD. Length scales are expressed in units of the average distance between nearest neighbors $\Lambda$.
• Figure 4: Angular variance of the Angular Distribution of Pairwise Distances for Poisson distributions in a box of side $L=400$ Mpc/$h$ with the different number of particles (see labels). A line $\propto 1/r$ (see Eq.\ref{['ang_var_adpd_Poisson']}) is also shown as a reference.
• Figure 5: Angular variance of the Angular Distribution of Pairwise Distances for four different configurations: a single bar, a perfect cubic lattice, a shuffled lattice with displacement amplitude $\delta = 0.2$, and with $\delta = 0.6$. All distributions are generated in a box of side $L=400$ Mpc/$h$ and have the same number of points $N= 10^5$. As reference the angular variance of the ADPD for a Poisson with same number of particles is also reported.