Statistical Properties of Paired Fixed Fields

The initial conditions of cosmological simulations are commonly drawn from a Gaussian ensemble. The limited number of modes inside simulations gives rise to sample variance: statistical fluctuations that limit the accuracy of the simulation predictions. Fixed fields offer an alternative initialization strategy; they have the same power spectrum as Gaussian fields but no intrinsic amplitude scatter. Paired fixed fields consist of two fixed fields with opposite phases that cancel phase correlations. We study the statistical properties of those fields for 19 different quantities at different redshifts through a large set of 600 N-body and 530 state-of-the-art magnetohydrodynamic simulations. We find that paired fixed simulations do not introduce a bias on any of the examined quantities. We quantify the statistical improvement brought by these simulations on different power spectra - matter, halos, cold dark matter, gas, stars, galaxies, and magnetic fields - finding that they can reduce their variance by factors as large as 10⁶. We quantify the improvement achieved by fixing and by pairing, showing that sample variance can be highly suppressed by pairing after fixing. Paired fixed simulations do not change the scatter in quantities such as the probability distribution function or the halo, void, or stellar mass functions. We argue that procedures aiming at reducing the sample variance of those quantities are unlikely to work. Our results show that paired fixed simulations do not affect either mean relations or scatter of galaxy properties and suggest that the information embedded in one-point statistics is highly complementary to that in clustering.