Lagrangian displacement field estimators in cosmology

The nonlinear Lagrangian displacement field and initial linear density field are highly correlated. Therefore, reconstructing the nonlinear displacement field could better extract the primordial cosmological information from the late time density field. Continuing from Ref. [1], we investigate to what extent the iterative displacement reconstruction in Ref. [2] can recover the true displacement field with a particular emphasis on improving the numerical discreteness effect and improving the perturbation theory model for the postreconstructed field. We propose two new displacement field estimators to overcome the numerical discreteness effect in measuring the displacement field; the normalized momentum estimator (NME) and the rescaled resumed estimator (RRE). Furthermore, by adequately accounting for the UV-sensitive term in a reconstruction procedure with an effective field theory approach, we manage to decrease the discrepancy between the theoretical model and the simulation result almost five times, to the level of a few \% at $k\sim 0.2h/{\rm Mpc}$ for the redshift $z=0.6$. We conclude that the previous displacement field reconstruction cannot fully reconstruct the shape of the nonlinear displacement field at $k\sim 0.2h/{\rm Mpc}$ at $z=0.6$, while it is still an efficient BAO reconstruction method.