Perturbation Theory for BAO reconstructed fields: one-loop results in
  real-space matter density field
  We compute the power spectrum at one-loop order in standard perturbation
theory for the matter density field to which a standard Lagrangian Baryonic
acoustic oscillation (BAO) reconstruction technique is applied. The BAO
reconstruction method corrects the bulk motion associated with the
gravitational evolution using the inverse Zel'dovich approximation (ZA) for the
smoothed density field. We find that the overall amplitude of one-loop
contributions in the matter power spectrum substantially decrease after
reconstruction. The reconstructed power spectrum thereby approaches the initial
linear spectrum when the smoothed density field is close enough to linear,
i.e., the smoothing scale $R_s$ larger than around 10$h^{-1}$Mpc. On smaller
$R_s$,however, the deviation from the linear spectrum becomes significant on
large scales ($k\lt R_s^{-1}$) due to the nonlinearity in the smoothed density
field, and the reconstruction is inaccurate. Compared with N-body simulations,
we show that the reconstructed power spectrum at one loop order agrees with
simulations better than the unreconstructed power spectrum. We also calculate
the tree-level bispectrum in standard perturbation theory to investigate
non-Gaussianity in the reconstructed matter density field. We show that the
amplitude of the bispectrum significantly decreases for small $k$ after
reconstruction and that the tree-level bispectrum agrees well with N-body
results in the weakly nonlinear regime.
