Lagrangian Perturbation Theory : Exact One-Loop Power Spectrum in General Dark Energy Models

Recently, we find that the correction for the EdS assumption on the one-loop matter power spectrum for general dark energy models using the standard perturbation theory is not negligible \cite{LPB}. Thus, we investigate the same problem by obtaining the exact displacement vector and kernels up to the third order for the general dark energy models in the Lagrangian perturbation theory (LPT). Using these exact solutions, we investigate the present one-loop matter power spectrum in the $\Lambda$CDM model with $\Omega_{m0} = 0.25 (0.3)$ to obtain about 0.2 (0.18) \% error correction compared to that obtained from the EdS assumption for $k = 0.1 \rm{h\, Mpc}^{-1}$ mode. If we consider the total matter power spectrum, the correction is only 0.05 (0.03) \% for the same mode. It means that EdS assumption is a good approximation for $\Lambda$CDM model in LPT theory. However, one can use this method for general models where EdS assumption is improper.