# Delensing LAT relative Nhits v0p0

This posting documents initial stabs at answering the question: how should the Nhits map of the Delensing LAT be optimized in order to minimize <math>\sigma(r)</math>?

## Contents

## Background

So far (e.g. in low-ell BB Data Challenge 06), the relative Nhits maps of the Delensing LAT is taken to be identical to those of the Pole Deep SATs. Given that the lensing template is generated from CMB x phi(CMB, CMB) as opposed to direct imaging of map fluctuations in other components, one would expect the S/N of the lensing modes vs those of the other components to scale differently for the same input noise.

In this post, I use a very simple case to build intuition. Specifically, I answer this question: given a single-frequency SAT Nhits map, what should the Delensing LAT distribution of noise levels be such that S/N = 1 for the residual lensing modes.

In the regime where delensing is the limiting factor for sigma(r), this would give a close approximation of the actual distribution of noise levels (and hence Nhits) for the Delensing LAT. I list the next steps in order to make this more realistic and applicable to the survey design of the Delensing LAT.

## Method

- Generate 100 realizations of 1/ell noise given the DSR configuration of the SAT 95GHz Pole deep survey on flatsky (1 arcmin pixels) using this table ;
- Apply scale factor specified below table 1 of this posting ;
- Divide the noise map by the square-root of the SAT 95GHz relative hits map;
- Calculate the noise levels in uK-arcmin per pixel by measuring the standard deviation of each pixel across the 100 realizations of noise; this gives the white noise level.
- Scale the noise levels by ~1.17 in order to estimate the noise levels at ell~100 (factor is eyeballed from input 1/ell noise spectrum);
- Estimate (by interpolation of inputs to fig.68 of the DSR and extended to higher pol noise levels) the input Delensing LAT noise levels (at 95GHz, i.e. 2.6' FWHM) required in order for the residual lensing power to match the SAT noise power.

## Results

**Fig.1 DC05 95GHz Pole Deep SAT noise levels [uK-arcmin]**

- The color range is set to 0-5uK-arcmin. Where the color saturates, the lensing modes are imaged to S/N~1. The algorithm would not assign any weight for the LAT in this region.

**Fig.2 required LAT noise levels for residual lensing S/N=1 [uK-arcmin]**

- The color stretch is the same in this figure as the one above. There are some regions below 5uK-arcmin in the SAT noise map but don't have information in this LAT map. There will be the wide field LAT overlap for these regions (with Q/U noise levels of ~3uK-arcmin at 95/150 bands) so anything in LAT that doesn't require ~< 5uK-arcmin could be allocated to the wide-field LAT.

**Fig.3 SAT vs LAT noise level slices**

- This shows the noise level slices of the SAT vs the LAT. The Nhit map shape scales as 1/(nlev)^2.
- We see that the slope of change of the LAT noise is much faster than the SAT noise -- i.e. as the Nhits of the SAT decreases, the Nhits of the LAT should decrease more quickly to keep delensed residuals S/N=1.
- In other words, given the same total Nhits, we should concentrate more of the observing time closer to the center of the map for the LAT as compared to the SAT (in this simple setup).

## Next steps

- To account for foregrounds, and optimize the relative Nhit of the delensing LAT to minimize sigma(r), one can match the S/N of the delensed residuals to a map of SAT S/N of foregrounds from some component-separation methods.
- Use AL^res numbers using input noise-curves generated post ILC.
- Check the Nhits of the wide-field LAT in the Pole Deep patch and calculate the equivalent noise levels. This can inform us on how much of the Delensing LAT observing can be concentrated on the deepest of the Pole Deep patches.
- This analysis doesn't account for the deflection of modes. The LAT maps should cover slightly larger areas (~few arcmin) at the same depths in order to estimate the lensing B modes that get deflected from slightly outside the boundary definition of the 'deepest' part of the Pole Deep map.
- Map-based analysis.