Matter power spectrum and the challenge of percent accuracy

Abstract

Future galaxy surveys require one percent precision in the theoretical knowledge of the power spectrum over a large range including very nonlinear scales. While this level of accuracy is easily obtained in the linear regime with perturbation theory, it represents a serious challenge for small scales where numerical simulations are required. In this paper we quantify the precision of present-day N -body methods, identifying main potential error sources from the set-up of initial conditions to the measurement of the final power spectrum. We directly compare three widely used N -body codes, Ramses , Pkdgrav3 , and Gadget3 which represent three main discretisation techniques: the particle-mesh method, the tree method, and a hybrid combination of the two. For standard run parameters, the codes agree to within one percent at k ⩽1 h Mpc −1 and to within three percent at k ⩽10 h Mpc −1 . We also consider the bispectrum and show that the reduced bispectra agree at the sub-percent level for k ⩽ 2 h Mpc −1 . In a second step, we quantify potential errors due to initial conditions, box size, and resolution using an extended suite of simulations performed with our fastest code Pkdgrav3 . We demonstrate that the simulation box size should not be smaller than L =0.5 h −1 Gpc to avoid systematic finite-volume effects (while much larger boxes are required to beat down the statistical sample variance). Furthermore, a maximum particle mass of M p =10 9 h −1 M ⊙ is required to conservatively obtain one percent precision of the matter power spectrum. As a consequence, numerical simulations covering large survey volumes of upcoming missions such as DES , LSST , and Euclid will need more than a trillion particles to reproduce clustering properties at the targeted accuracy.