# Performances of perturbation theory at NNLO

To conclude this section, we present the performances of the perturbation theory at two-loop order. In Fig. 2.17, we present the performances of RegPT up to two-loop

**Fig. 2.17 ****Comparison of perturbation theory results with Ж-body results for the power spectrum at (a) г = 1 and (b) г = 0.35. The dotted curve is the linear prediction; the dashed curves are the standard perturbation theory and RegPT NLO predictions; and the solid curves are the NNLO predictions. The shaded areas show the departure between these predictions at one-loop order (light gray) and two-loop order (darker gray).**

**Fig. 2.18 **A tentative chart of the accuracy domain of the perturbation theory results with contour plots of the 1% and 2% accuracy regions for the linear, NLO, and NNLO predictions for the power spectrum. The calculations have been made for the *WMAP-5* cosmological parameters.

order compared with standard perturbation theory and with numerical results. The solid curves show the two-loop-order predictions for the standard perturbation theory and RegPT schemes. They agree with one another up to wave modes that are significantly larger than the validity regime of the linear theory. They also extend significantly the regime of validity of the one-loop results. This is particular striking at redshift 1 and above.

These findings are summarized in Fig. 2.18, which tentatively charts the performances of the linear, one-loop and two-loop predictions. The dotted curves correspond to the 1% and 2% accuracy lines of the linear theory. The contour lines are obtained from a comparison between the linear and the standard two-loop results. The dashed curves show the same results for the standard one-loop predictions, and the solid curves show an estimate of those curves from a consistency comparison between different two-loop schemes.