Jan 23, 2019 - Comparison of mine and Jason's filters

It is somewhat surprising we are seeing such big differences between skewness/kurtosis we are seeing with Jason. The primary suspect is the different set of filters we are using. To check what the filters are actually doing, we look at [math]\displaystyle{ F_\ell = C_\ell^\mathrm{filtered}/C_\ell^\mathrm{total} }[/math], fraction of power kept in the filtered map (in my case after applying smoothed top-hat filter, for Jason difference of two maps smoothed by different Gaussian filters).

In my case we get the expected picture

In Jason's case what we see is somewhat strange

This does not look like top hat filters at all, for the last bins we even get most of the power from the very smallest scales. This could be reason why we see such high skewness and kurtosis in Jason's analysis - it is picking up info from the smallest scales, which might not be well modeled. Also, our original idea of having filters localized in ell does not seem to be that well realized here.

The situation is clearer from taking a look at power spectra of Jason's map filtered with the Gaussian filters, again relative to the total power spectrum, [math]\displaystyle{ R_\ell = C_\ell^\mathrm{filtered}/C_\ell^\mathrm{total} }[/math] where now we are not looking at difference maps. We checked that [math]\displaystyle{ R_\ell = \exp\left(- \frac{\ell(\ell+1)\theta_{FWHM}^2}{8 \log 2}\right) }[/math].

Theoretically, for Jason's filters we should have [math]\displaystyle{ F^i_\ell = \left(\sqrt{R^{i+1}_\ell} - \sqrt{R^i_\ell}\right)^2, }[/math] because the Gaussian smoothing does [math]\displaystyle{ a_{\ell m} \rightarrow \sqrt{R_\ell} a_{\ell m} . }[/math] We checked this is a very good approximation; Jason's filters are thus acting as expected (on the power spectrum level).