The anisotropy of the power spectrum in periodic cosmological simulations

The classical gravitational force on a torus is anisotropic and always lower than Newton's $1/r^2$ law. We demonstrate the effects of periodicity in dark matter only $N$-body simulations of spherical collapse and standard $\Lambda$CDM initial conditions. Periodic boundary conditions cause an overall negative and anisotropic bias in cosmological simulations of cosmic structure formation. The lower amplitude of power spectra of small periodic simulations are a consequence of the missing large scale modes and the equally important smaller periodic forces. The effect is most significant when the largest mildly non-linear scales are comparable to the linear size of the simulation box, as often is the case for high-resolution hydrodynamical simulations. Spherical collapse morphs into a shape similar to an octahedron. The anisotropic growth distorts the large-scale $\Lambda$CDM dark matter structures. We introduce the direction-dependent power spectrum invariant under the octahedral group of the simulation volume and show that the results break spherical symmetry.