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.
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