  We test third-order standard perturbation theory (SPT) as an approximation to
non-linear cosmological structure formation. A novel approach is used to
numerically calculate the three-dimensional dark matter density field using SPT
from the initial conditions of two high-resolution cosmological simulations.
The calculated density field is compared to the non-linear dark matter field of
the simulations both point-by-point and statistically. For smoothing scales
above 8 Mpc/h it shows a good agreement up to redshift 0. We present a simple
fitting formula to relate the linear and non-linear density contrast that
accurately recovers the non-linear time evolution for 0 <= z <= 10 at the per
cent level. To address the problem of biasing between the matter field and the
haloes identified in the simulation, we employ the Eulerian local bias model
(ELB), including non-linear bias up to the third order. The bias parameters are
obtained by fitting a scatter plot of halo and matter density (both from the
simulation and from SPT). Using these bias parameters, we can reconstruct the
halo density field. We find that this reconstruction is not able to capture all
the details of the halo distribution. We investigate how well the large scale
bias can be described by a constant and if it corresponds to the linear bias
parameter b_1 of the local bias model. We also discuss how well the halo-halo
power spectrum and the halo-mass cross spectrum from the reconstructed halo
density field agree with the corresponding statistics from the simulation. The
results show that while SPT is an excellent approximation for the matter field
for suitably large smoothing scales even at redshift 0, the ELB model can only
account for some of the properties of the halo density field.
