June 29th, 2020 - CIB Lensing: Lenspix - Smoothing and Negative Pixels

Since we saw that 'lenspix' reduces the variance of the Websky CIB map, we agreed I should try smoothing the unlensed map with beam comparable to the size of a pixel. However, using an FWHM a bit smaller than the width of an average pixel would not preserve the power spectrum at high [math]\displaystyle{  \ell  }[/math]'s, and the variance would become much smaller than the 'lenspix'-lensed map.

So, I subsequently made the beam smaller and smaller, until the smoothing had virtually no effect on the 2, 3, 4-point statistics and the variance of the smoothed map was comparable to that of the 'lenspix'-lensed map. (FWHM = [math]\displaystyle{  10^{-7}   }[/math] rad)

Map Characteristics
	Unlensed	Smoothed (FHWM = 1e-7)	Lensed (Lenspix)
Max values (MJy/sr)	6.408424 (= 400 mJy)	5.959061	6.408424
Min values (MJy/sr)	0.000000	-1.321831	-0.322438
# of pixels smaller than 0	0	1167158	1477969
Mean	0.3179518	0.3179518	0.3180850
Variance	0.0331789	0.0258813	0.0264728
Skewness	1.720313	1.150339	1.776150
Kurtosis	23.366768	11.763371	35.614250

There are discrepancies in the minimum, maximum, skewness and kurtosis values of the smoothed map with the 'lenspix'-lensed map - I believe part of this is because 'lenspix' must smooth the map with an even smaller beam, and also because I smoothed the unlensed map which already had the 400mJy fluxcut imposed (rather than smoothing, and then removing the bright pixels).

Histograms
  

However, the unlensed-smoothed histogram quite resembles the 'lenspix'-lensed histogram (except for a slight shift to the right for the lensed histogram). 

N-point statistics (2, 3, 4-point)

Below is a sanity check to show that the smoothing did not change the n-point statistics. 

Summary
It is now clear that the negative pixels in the 'lenspix'-lensed map won't affect our statistical analysis - whether 'lenspix' lenses point sources accurately may be a different matter (hopefully we don't need to worry about this).