July 14th, 2019 - CIB Lensing: Smoothing and Interpolation

Before moving on to creating a new halo catalog (which depicts the lensing by displacing the positions of the halos using my own script), Pavel and Alex suggested that I try smoothing the CIB and produce lensed null and lensed maps using these smoothed maps.

There are still many more tests and sanity checks that could be (and should be) done, but here I present what I have so far.

Smoothing vs. Lensed Null Histograms

Yesterday, I realized that the histograms of the smoothed maps (of order several arc-seconds) greatly resemble the null lensed histogram.

Although we can only see that the 12" smoothed histogram and null lensed histogram are a bit different on the left plot, it is more convincing that the 1" smoothed histogram and null lensed histogram match on the right plot.

Smoothing vs. Lensed Null Cartview

As Pavel pointed out that I should check my hypothesis using cartview, I took a look at 0.5 degree by 0.5 degree patches for the I) Original CIB, II) Lensed Null CIB, III) FWHM 12" smoothed, IV) FWHM 5s" smoothed, V) FWHM 1" smoothed maps (left to right, top to bottom). If you take a look at each of these maps by making new tabs and switching back and forth, you will find that all but the original CIB will completely match.

[math]\displaystyle{ b_{\ell} }[/math] of smoothed maps

However, top-hat filtering these smoothed maps and plotting the reduced bispectra gives completely different results. (Left: reduced bispectra, Right: reduced bispectra fractional differences)

Except for the 1" smoothed map, the fractional differences are noticeable at high ell.

[math]\displaystyle{ b_{\ell} }[/math] Interpolation (1 shell)

To simplify the investigation of the interpolation effect, I chose to lens the whole CIB with one null kappa map. Below is a comparison of the interpolation effect between the I) original CIB and II) 5" smoothed maps.

The interpolation effect does not seem to change whether it is the original CIB (with the oddly shaped histogram) or the 5" smoothed CIB that is lensed by the null kappa map.

What we can note here however, is that the interpolation effect seems much smaller when the whole CIB is lensed by one kappa shell (less than 0.1%) opposed to when each of the 18 CIB shells (of equal comoving distances from 400 to 7600 Mpcs) are lensed by null kappa maps (more than 3% on average). It could be that a rogue shell is causing this large discrepancy, or simply be that the interpolation effect adds up cumulatively. Below is the plot of the interpolation effect on the shell-split then lensed CIB.

[math]\displaystyle{ b_{\ell} }[/math] Comparison: Smoothing vs. Interpolation

Now, I will compare the effect of smoothing (1") and the interpolation on the [math]\displaystyle{ b_{\ell} }[/math].

It can be said that smoothing with a small enough beamwidth (1" for example) distorts [math]\displaystyle{ b_{\ell} }[/math] comparable to or less than the interpolation effect. Whether this holds for the shell-split then lensed CIB must be investigated separately.

Conclusion

Smoothed CIB maps' histograms are quite similar to the lensed null CIB. While they look the same (or almost exactly the same) on the map scale, their [math]\displaystyle{ b_{\ell} }[/math]'s show noticeable differences; if 'lenspix' can be used properly, it seems plausible to use the 1" smoothed CIB as the standard for the unlensed CIB (further checks are needed though).

One puzzling thing is that the odd peak around [math]\displaystyle{ I_{\nu} \approx 0 }[/math] for the original CIB and other differences between the original CIB and the smoothed (1") or lensed null histograms do not seem to have an effect on the bispectra.

Another puzzling thing is that the smoothed histograms and maps look nearly the same when the beamwidths are small enough (on the order of arc-seconds).

We also need to figure out a way to make 'lenspix' do the lensing properly for NSIDE 4096 maps; I will cover this in another post.