  We study the mapping from Lagrangian to Eulerian space in the context of the
Effective Field Theory (EFT) of Large Scale Structure. We compute Lagrangian
displacements with Lagrangian Perturbation Theory (LPT) and perform the full
non-perturbative transformation from displacement to density. When expanded up
to a given order, this transformation reproduces the standard Eulerian
Perturbation Theory (SPT) at the same order. However, the full transformation
from displacement to density also includes higher order terms. These terms
explicitly resum long wavelength motions, thus making the resulting density
field better correlated with the true non-linear density field. As a result,
the regime of validity of this approach is expected to extend that of the
Eulerian EFT, and match that of the IR-resummed Eulerian EFT. This approach
thus effectively enables a test of the IR-resummed EFT at the field level. We
estimate the size of stochastic, non-perturbative contributions to the matter
density power spectrum. We find that in our highest order calculation, at
redshift z=0 the power spectrum of the density field is reproduced with an
accuracy of 1 % (10 %) up to k=0.25 h/Mpc (k=0.46 h/Mpc). We believe that the
dominant source of the remaining error is the stochastic contribution.
Unfortunately, on these scales the stochastic term does not yet scale as $k^4$
as it does in the very low-k regime. Thus, modeling this contribution might be
challenging.
