Cylindrical cosmological simulations with StePS

Abstract

The global topology of the Universe can affect the long-range gravitational forces through the boundary conditions. To study non-trivial topologies in detail, simulations that natively adopt such geometries are required. Cosmological $N$-body simulations typically evolve matter in a periodic cubic box. While numerically convenient, this imposes a non-trivial 3-torus topology that affects long-range gravitational forces, potentially biasing large-scale statistics. We introduce a compactified simulation framework that is only periodic along a single axis, while having infinite topology with isotropic boundary conditions towards the perpendicular directions, i.e. an $\mathrm{S}^1\times\mathbb{R}^2$ ("slab") topology. This new simulation geometry is ideal for simulating systems with cylindrical symmetries like filaments or certain anisotropic cosmological models. We compactify comoving space via inverse stereographic projection along the radial direction of a periodic cylinder, and evolve particles with Newtonian dynamics. A smoothly varying spatial and mass resolution with radius suppresses edge artefacts at the free outer boundary. Our implementation in the StePS (STEreographically Projected cosmological Simulations) framework uses a direct $\mathcal{O}(N^2)$ force calculation that maps efficiently to GPUs, and Octree $\mathcal{O}(N \log N)$ force calculation that can be used on large CPU-clusters. The cylindrical domain's topology enables fully self-consistent simulations in the $\mathrm{S}^1\times\mathbb{R}^2$ manifold and mitigates periodic-image artefacts for targets whose symmetries are mismatched to a cubic box. The main trade-off is radially varying resolution with distinct systematics and analysis requirements. We demonstrate the accuracy of the new simulation method in a standard $Λ$CDM cosmological simulation.

Figures (6)

• Figure 1: Comparison of gravitational force laws between the periodic cubic ($\mathrm{T}^3$) and the periodic cylindrical ($\mathrm{S}^1\times\mathbb{R}^2$) geometry. Each plot demonstrates the deviation from the Newton's law of universal gravitation due to the periodic topology. In both cases, a test particle was placed to the centre of the simulation volume, and the gravitational force field was calculated both in the corresponding and in $\mathbb{R}^3$ topology. The heat maps are showing the $|\boldsymbol{F}|/|\boldsymbol{F}_{\mathbb{R}^3}|$ ratio of the force field magnitudes. The vector plots are demonstrating the $\boldsymbol{F}-\boldsymbol{F}_{\mathbb{R}^3}$ difference of the gravitational force fields around the particle. The length of the arrows is proportional to the magnitude of the difference between the forces.
• Figure 2: Mass resolution as a function of radial coordinate $\varrho$ in an example cylindrical glass. The parameters of this glass are the following: $D_S=70\,\mathrm{Mpc}$, $R_{sim}=500.0\,\mathrm{Mpc}$, $\omega_c=1.63973\,\mathrm{RAD}$, $L_z = 500\,\mathrm{Mpc}$, and the total particle number is $N_{part} = 1.2\cdot10^6$. The glass has constant resolution for $\varrho \, < \, \varrho_c = D_S\tan(\omega_c/2) = 75.0\,\mathrm{Mpc}$ to avoid unwanted mixing of particles with widely different masses in the simulations.
• Figure 3: Thin slices of the dark matter density field from the first $\mathrm{S}^1\times\mathbb{R}^2$StePS simulation at $z=11$, $z=3$, and $z=0$ redshifts in comoving coordinates. Each slice has a thickness of $15.0\,\mathrm{Mpc}$, and each column shows a different slice orientation. The late-time non-linear cosmic web is visually similar to that found in $\mathrm{T}^3$ and $\mathbb{R}^3$ simulations: voids, walls, filaments, and dark matter haloes are clearly visible. Due to the zoom-in nature of the StePS$\mathrm{S}^1\times\mathbb{R}^2$ simulation method, the spatial resolution is highest along the central axis, while only the largest structures are resolved near the $R_{\textrm{sim}}$ simulation radius.
• Figure 4: The dark matter power spectra of four selected snapshots (solid lines) with the corresponding theoretical linear power spectra (dashed lines) and the non-linear 2021MNRAS.502.1401M halofit power spectra (dotted dashed lines). Our $\mathrm{S}^1\times\mathbb{R}^2$StePS simulation is capable of following both linear and non-linear structure formation.
• Figure 5: Spatial distribution of dark matter haloes in the new $\mathrm{S}^1\times\mathbb{R}^2$StePS simulation at different redshifts, in comoving coordinates. Due to the zoom-in geometry, and the fact that low-mass haloes form first, haloes first appear close to the central axis, where the resolution is highest, and the smallest haloes can be resolved. As structure formation proceeds towards higher masses, haloes can later be found at larger radii, where the resolution is lower, and only more massive haloes can be resolved. Toward larger $\varrho=\sqrt{x^2+y^2}$ only increasingly massive haloes are resolved, so the apparent decline in number density at large $\varrho$ is a selection effect.