  We obtain approximations for the CDM particle trajectories starting from
Lagrangian Perturbation Theory. These estimates for the CDM trajectories result
in approximations for the density in real and redshift space, as well as for
the momentum density that are better than what standard Eulerian and Lagrangian
perturbation theory give. For the real space density, we find that our proposed
approximation gives a good cross-correlation (>95%) with the non-linear density
down to scales almost twice smaller than the non-linear scale, and six times
smaller than the corresponding scale obtained using linear theory. This allows
for a speed-up of an order of magnitude or more in the scanning of the
cosmological parameter space with N-body simulations for the scales relevant
for the baryon acoustic oscillations. Possible future applications of our
method include baryon acoustic peak reconstruction, building mock galaxy
catalogs, momentum field reconstruction.
