Comparing approximate methods for mock catalogues and covariance
  matrices II: Power spectrum multipoles
  We study the accuracy of several approximate methods for gravitational
dynamics in terms of halo power spectrum multipoles and their estimated
covariance matrix. We propagate the differences in covariances into parameter
constrains related to growth rate of structure, Alcock-Paczynski distortions
and biasing. We consider seven methods in three broad categories: algorithms
that solve for halo density evolution deterministically using Lagrangian
trajectories (ICE-COLA, Pinocchio and PeakPatch), methods that rely on halo
assignment schemes onto dark-matter overdensities calibrated with a target
N-body run (Halogen, Patchy) and two standard assumptions about the full
density PDF (Gaussian and Lognormal). We benchmark their performance against a
set of three hundred N-body simulations, running similar sets of approximate
simulations with matched initial conditions, for each method. We find that most
methods reproduce the monopole to within $5\%$, while residuals for the
quadrupole are sometimes larger and scale dependent. The variance of the
multipoles is typically reproduced within $10\%$. Overall, we find that
covariances built from approximate simulations yield errors on model parameters
within $10\%$ of those from the N-body based covariance.
