Next-to-leading resummation of cosmological perturbations via the Lagrangian picture: 2-loop correction in real and redshift spaces

We present an improved prediction of the nonlinear perturbation theory (PT) via the Lagrangian picture, which was originally proposed by Matsubara (2008). Based on the relations between the power spectrum in standard PT and that in Lagrangian PT, we derive analytic expressions for the power spectrum in Lagrangian PT up to 2-loop order in both real and redshift spaces. Comparing the improved prediction of Lagrangian PT with $N$-body simulations in real space, we find that the 2-loop corrections can extend the valid range of wave numbers where we can predict the power spectrum within 1% accuracy by a factor of 1.0 ($z=0.5$), 1.3 (1), 1.6 (2) and 1.8 (3) vied with 1-loop Lagrangian PT results. On the other hand, in all redshift ranges, the higher-order corrections are shown to be less significant on the two-point correlation functions around the baryon acoustic peak, because the 1-loop Lagrangian PT is already accurate enough to explain the nonlinearity on those scales in $N$-body simulations.