Dark matter statistics for large galaxy catalogues: power spectra and covariance matrices

Large-scale surveys of galaxies require accurate theoretical predictions of the dark matter clustering for thousands of mock galaxy catalogues. We demonstrate that this goal can be achieved with the new Parallel Particle-Mesh (PM) N-body code GLAM at a very low computational cost. We run ∼22 000 simulations with ∼2 billion particles that provide ∼1 per cent accuracy of the dark matter power spectra P(k) for wavenumbers up to k ∼ 1 h Mpc^-1. Using this large data set we study the power spectrum covariance matrix. In contrast to many previous analytical and numerical results, we find that the covariance matrix normalized to the power spectrum C(k, k')/P(k)P(k') has a complex structure of non-diagonal components: an upturn at small k, followed by a minimum at k ≈ 0.1-0.2 h Mpc^-1, and a maximum at k ≈ 0.5-0.6 h Mpc^-1. The normalized covariance matrix strongly evolves with redshift: C(k, k') ∝ δ^α(t)P(k)P(k'), where δ is the linear growth factor and α ≈ 1-1.25, which indicates that the covariance matrix depends on cosmological parameters. We also show that waves longer than 1 h^{-1}{ Gpc} have very little impact on the power spectrum and covariance matrix. This significantly reduces the computational costs and complexity of theoretical predictions: relatively small volume {∼ } (1 h^{-1}{ Gpc})^3 simulations capture the necessary properties of dark matter clustering statistics. As our results also indicate, achieving ∼1 per cent errors in the covariance matrix for k < 0.50 h Mpc^-1 requires a resolution better than ɛ ∼ 0.5 h^{-1}{ Mpc}.