Covariance matrix for peaks

[viajani]

Description of the issue

This is meant to discuss how to properly compute the covariance matrix:

• is it correct to take into account the sky coverage correction, i.e. f_{sky}=12.25/5000 to rescale the covariance matrix?

The current covariance matrix for peak counts looks like this:

with condition number ~ 10^4.

List of tasks to do within this issue:

1. Plot constraints with rescaled covariance
2. Plot constraints without rescaled covariance

[viajani]

1. Plot constraints with rescaled covariance

2. Plot constraints without rescaled covariance

[pettorin]

hi @viajani thanks, I have a couple of questions to better understand what you are doing:

• When you say 'The current covariance matrix for peak counts looks like this'; is this the non-rescaled one, and how did you get it?
• In the Euclid paper didn't you rescale for Euclid fsky?
• can you overplot Euclid constraints, so that we have a reference?

[viajani]

hi @pettorin

• the matrix shown above is the actually the correlation matrix (my mistake) corresponding to a covariance computed as

where x and mu are the peak counts for the massless cosmology of the MassiveNus simulations and the average is over the 10000 different realisation of such cosmology.
The rescaling factor is only a multiplicative pre-factor in front of the covariance as f_sky * Cov where for the size of our maps and CFIS sky coverage is 12.25/5000.

However, since we are using real data and these are not forecast I am not sure if using a rescaling is fair.

• yes for the Euclid paper we rescaled the covariance by multiplying it with a prefactor of (12.25/15000)

• I share here below the plot with the Euclid constraints however main differences to keep in mind are

1. The ones for Euclid are only forecasts with a fiducial cosmology from the simulations while for CFIS we are using real data and we build the likelihood with MassiveNus neglecting all systematics
2. The Euclid constraints are tomographic with four redshifts (z=[0.5, 1.0, 1.5, 2.0]) while for CFIS they are non-tomographic assuming n_gal=7 arcmin^2 at a single redshift z=0.5

I can compute the corresponding single redshift case (non-tomo) for Euclid forecast for a more fair comparison (but I guess that the huge bias shown here won't depend on that)

[viajani]

1. Try to extract several patches (not just one as I am doing now) and then sum/average the peak counts, then rescale the covariance for the area covered by those patches

[viajani]

Output for data array obtained by extracting 10 patches from the "big" data map, computing the peaks on each one of the 10 patches and then taking the average.
In this way the covariance is rescaled as: f_rescale=12.25/122.5 where 12.25 deg^2=size of a single map and 122.5 deg^2=coverage of the 10 patches

[viajani]

next: test the difference when increasing number of patches (try to take as many as I can extract)

[pettorin]

thanks @viajani: the plot above looks strange, Omegam is too high, it was actually already very high in the first plot (I hadn't noticed) when you had with 1 patch, but now with more patches it's getting even worse. Could you please remind me if the first plot above was already with peaks or was it from the PS? Did you check what you get with the PS before moving to peaks? It may just be that this is entirely dominate by (neglected) systematics or it might be something else which is wrong.

[pettorin]

Also, which are the units of As here?

[viajani]

Hi @pettorin , the units of As are 10^9*A_s while concerning the comparison with the PS I had opened the issue about this #3 and corresponding notebook https://github.com/CosmoStat/shear-pipe-peaks/blob/main/notebooks/compute_powerspectrum.ipynb but when I showed it during the last telecon I remember @martinkilbinger @aguinot explained me that it was even more complicated to get the contours with the Power Spectrum and that if we want to do second order statistics we should consider pseudo-Cls or COSEBIs, can you confirm on this @aguinot @martinkilbinger ?

[aguinot]

I don't know if it is more complicated to get contours using a power spectrum but I don't know why you first make a map and derive the PS from it when you can measure it directly from the catalogue. I feel like you would loose some informations.

Regarding your approach with multiple patch one problem you might have here might be due to masking. With one patch you have a problem at the border once, with 10 patches you have it 10 times.. Also the smaller the patch the more you will have effects at the borders.

Is there a way to test for the systematics (like the COSEBIs for the PS) on the mass maps? May be the smoothing scale is to small and you have only noise on your map..

[viajani]

Test to do discussed during 18/02 telecon:

1. Exclude extreme peaks: reduce number of bins to avoid bins with very few peaks.
2. See how the contours shift for larger scales, plot contours corresponding to different smoothing
3. Get constraints with full sky coverage rescaling for CFIS but with the 10 patches

N.B.

• MassiveNus not reliable after 10**3
• z=0.5 that we are using for the sims is not ideal given the actual redshift of CFIS. Further discussion on how to interpolate among redshifts needed