small scale effects in the inversion procedure laid out in eq. (\[iteration\]) (see also eq. \[\[split\]\]). We “pretend” that we do not know the actual full distortion kernel (eq. \[\[Wfull\]\]), but instead assume only knowledge of the large scale distortion kernel (eq. \[\[Wlinear\]\]) when carrying out the inversion. Moreover, for method II, we have not been very careful in selecting the value of the constants $C_1$ and $C_2$: we simply set them to zero.

To estimate the effect of nonlinear distortions on our inversion procedure, we decrease $k^s_{\parallel}$ to $0.028
({\rm km/s})^{-1}$, and show the outputs of method I and II in Fig. \[pinvcomp\_ks\]. Here we are taking advantage of the fact that the factor of ${\rm exp} [- (k_\parallel / k^s_\parallel)^2]$ in eq. (\[Wfull\]) is commonly used to model nonlinear distortions in the case of galaxy surveys (see e.g. ). By raising the scale of nonlinear distortion by a factor of about $4$, we hope to gain an idea of how the as yet poorly understood nonlinear distortions on small scales might affect the inversion of the power spectrum on large scales. However, it remains to be checked using simulations how realistic this choice of scale, or this particular parametrization of nonlinear distortions, is. The agreement on large scales for method II is not as good as before, but is still within about $7 \%$, and is better than that of method I. It is possible to improve the agreement by playing with the input parameters $C_1$, $C_2$ or $k_{\star\star}$. We will not pursue that here.

We also show in Fig. \[pinvp\_ks\] what the inverted and input power spectra look like in a log-log plot. The subtle differences in the shapes of the power spectra are still possible, but harder, to discern in such a plot.

Lastly, we have assumed in all tests above that the input $\beta_f$ (eq. \[\[Wlinear\]\]) is known when performing the inversion. In practice, there is an uncertainty due to the lack of knowledge of the precise values of $\gamma$ and $f_\Omega$. In Fig. \[pinvcomp\