Modeling iterative reconstruction and displacement field in the large
  scale structure
  The next generation of galaxy surveys like the Dark Energy Spectroscopic
Instrument (DESI) and Euclid will provide datasets orders of magnitude larger
than anything available to date. Our ability to model nonlinear effects in late
time matter perturbations will be a key to unlock the full potential of these
datasets, and the area of initial condition reconstruction is attracting
growing attention. Iterative reconstruction developed in Ref. [1] is a
technique designed to reconstruct the displacement field from the observed
galaxy distribution. The nonlinear displacement field and initial linear
density field are highly correlated. Therefore, reconstructing the nonlinear
displacement field enables us to extract the primordial cosmological
information better than from the late time density field at the level of the
two-point statistics. This paper will test to what extent the iterative
reconstruction can recover the true displacement field and construct a
perturbation theory model for the postreconstructed field. We model the
iterative reconstruction process with Lagrangian perturbation theory~(LPT) up
to third order for dark matter in real space and compare it with $N$-body
simulations. We find that the simulated iterative reconstruction does not
converge to the nonlinear displacement field, and the discrepancy mainly
appears in the shift term, i.e., the term correlated directly with the linear
density field. On the contrary, our 3LPT model predicts that the iterative
reconstruction should converge to the nonlinear displacement field. We discuss
the sources of discrepancy, including numerical noise/artifacts on small
scales, and present an ad hoc phenomenological model that improves the
agreement.
