Performance study of Lagrangian methods: Reconstruction of large-scale peculiar velocities and baryonic acoustic oscillations

No Action Method (NoAM) is a framework for reconstructing the past orbits of observed tracers of the large-scale mass density field. It seeks exact solutions of the equations of motion (EoM), satisfying initial homogeneity and the final observed particle (tracer) positions. The solutions are found iteratively reaching a specified tolerance defined as the RMS of the distance between reconstructed and observed positions. Starting from a guess for the initial conditions, NoAM advances particles using standard N-body techniques for solving the EoM. Alternatively, the EoM can be replaced by any approximation such as Zel'dovich and secondorder perturbation theory (2LPT). NoAM is suitable for billions of particles and can easily handle non-regular volumes, redshift space and other constraints. We implement NoAM to systematically compare Zel'dovich, 2LPT, and N-body dynamics over diverse configurations ranging from an idealized high-res periodic simulation box to realistic galaxy mocks. Our findings are: (i) non-linear reconstructions with Zel'dovich, 2LPT, and full dynamics perform better than linear theory only for idealized catalogues in real space. For realistic catalogues, linear theory is the optimal choice for reconstructing velocity fields smoothed on scales ≳5 h ^-1 Mpc; (ii) all non-linear back-in-time reconstructions tested here produce comparable enhancement of the baryonic oscillation signal in the correlation function.