Reconstruction of primordial density fields

The Monge–Ampère–Kantorovich (MAK) reconstruction is tested against cosmological N ‐body simulations. Using only the present mass distribution sampled with particles, and the assumption of homogeneity of the primordial distribution, MAK recovers for each particle the non‐linear displacement field between its present position and its Lagrangian position on a primordial uniform grid. To test the method, we examine a standard Lambda cold dark matter (ΛCDM) N ‐body simulation with Gaussian initial conditions and six models with non‐Gaussian initial conditions: a χ 2 model, a model with primordial voids and four weakly non‐Gaussian models. Our extensive analyses of the Gaussian simulation show that the level of accuracy of the reconstruction of the non‐linear displacement field achieved by MAK is unprecedented, at scales as small as ∼3  h −1 Mpc . In particular, it captures in a non‐trivial way the non‐linear contribution from gravitational instability, well beyond the Zel'dovich approximation. This is also confirmed by our analyses of the non‐Gaussian samples. Applying the spherical collapse model to the probability distribution function of the divergence of the displacement field, we also show that from a well‐reconstructed displacement field, such as that given by MAK, it is possible to accurately disentangle dynamical contributions induced by gravitational clustering from possible initial non‐Gaussianities, allowing one to efficiently test the non‐Gaussian nature of the primordial fluctuations. In addition, we test successfully a simple application of MAK using the Zel'dovich approximation to recover in real space the present‐day peculiar velocity field on scales of 8  h −1 Mpc . Although non‐trivial observational issues yet remain to be addressed, our numerical investigations suggest that MAK reconstruction represents a very promising tool to be applied to three‐dimensional Galaxy catalogues.