MUSCLE-UPS: Improved Approximations of the Matter Field with the Extended Press-Schechter Formalism and Lagrangian Perturbation Theory

Lagrangian algorithms to simulate the evolution of cold dark matter (CDM) are invaluable tools to generate large suites of mock halo catalogues. However, their precision may not be up to the challenge of upcoming surveys. In this paper, we first show that the main limitation of current schemes is their inability to model the evolution of overdensities in the initial density field, a limit that can be circumvented by detecting halo particles in the initial conditions. We thus propose a new scheme, dubbed \muscleups, which reproduces the results from Lagrangian perturbation theory on large scales, while improving the modelling of the overdensities on small scales. \muscleups\ uses the extended Press and Schechter (EPS) formalism, adapted to a Lagrangian picture. For regions exceeding a collapse threshold in the density smoothed at a radius $R$, we consider all particles within a radius $R$ collapsed. Exploiting a multi-scale smoothing of the initial density, we build a halo catalogue on the fly by optimizing the selection of halo candidates. This allows us to generate a density field with a halo mass function that matches one measured in $N$-body simulations. We further explicitly gather particles in each halo together in a profile, providing a numerical, Lagrangian-based implementation of the halo model. Compared to previous semi-analytical Lagrangian methods, we find that \muscleups\ improves the recovery of the statistics of the density field at the level of the probability density function (PDF), the power spectrum, and the cross correlation with the $N$-body result.