# Sigma(r) vs r plots in prep for the DSR, beyond naive combination.

B. Racine

In this posting, we update the $$\sigma(r)$$ vs. $$r$$ plots with a new combination of the hitmaps. We also made a few changes detailed below.
We show tables of $$\sigma(r)$$ constraints using different combinations of tubes.
WARNING, This is preliminary.
Note that in the first version of this posting (see here), we had not fully fixed the mistakenly used 20GHz channel for the shallow part of the Chile deep map.

## Introduction

In this recent posting, we proposed some $$\sigma(r)$$ vs. $$r$$ plots for the DSR.
In the current posting, we update these numbers and plot the different scalings and "$$A_L$$" delensing residual levels produced by Raphael: see Appendix A.
Changes since the last posting (as mentioned in this posting):

• We switched to rescaling from the BK inverse noise variance weightings (since they should be removed from the BK bpcm and noise levels), instead of the naive 1% scaling.
• We are also combining the Pole and Chile dataset at the hitcount level instead of adding in inverse quadrature as we used to in the post-CDT spreadsheet (see section 2).
• We are still using the mean site NET, and decided not to apply the relative efficiency corrections explained in this posting.
• Since the 20GHz channel is on the delensing LAT, it is only able to observe the patches that will be delensed, i.e. not the shallow chile one. Previously we had ignored that fact and were rescaling the 20GHz channel as the other channels. We are now accounting for the fact that there is only one 20GHz tube, at Pole. Note that we also take into account that if all the tubes are in Chile, we can still use the 20GHz from Pole on the deep delensed patch, as well as for the combined case, but of course never more than one tube in total.

## 1: $$\sigma(r)$$ vs r for different hitcount maps.

Here for now we focus on the 18 tubes configuration (slightly updated since the spreadsheet, now [2,2,6,6,6,6,4,4] tubes, each with [288, 288, 3524, 3524, 3524, 3524, 8438, 8438] detector per tube for [20, 30, 40, 85, 95, 145, 155, 220, 270] GHz, and 135 at 20GHz on a LAT).
We are progressively adding other configurations: 6 (1,2,2,1), 9 (1,3,3,2), 12 (1,4,4,3) in addition to 18 (2,6,6,4), where this notation shows the dichroic coupling.

In figure 1, we show the $$\sigma(r)$$ vs r plots for different cases, including or not a 50% yield penalty for Chile, and/or a foreground residual bias penalty.

The foreground penalty is the one that has been used in the spreadsheet and the current tables in the DSR. It is based on this posting. This is really a crude estimate, and there is room for improvement. We are basically using the foreground from BK at the foreground minimum at ell=80 as an estimate of the foregrounds amplitude and translating this into a r-equivalent. We then extrapolate it to different masks; for that we assume that the scaling is well described by what we get from map based analysis for the fitted $$A_d$$ for the 04b and 04c masks, model 7 having supposedly the most data-driven realistic foreground variation. We then use the more complex models to back up the fact that unmodeled foreground will leave a bias on r that will linearly depend on the foreground amplitude. We then consider the fg residual to be 0.5% of the total foreground, which is of course a “soft” choice here, and add it in quadrature to sigma(r).

## 2: $$\sigma(r)$$ Tables for combined observations.

In the post DSR spreadsheet, we were combining the constraints as a weighted average of independent results, i.e. summing the $$\sigma(r)$$ in inverse quadrature. Here instead, we are combining at the map level, by simply summing the hitmaps for the overlaping deep patch. These new combined observations then go throught Raphael's ILC to compute the residual signals and the corresponding "more optimal" scalings (see appendix A). For the Chile observations, we still add the shallow part in inverse quadrature (Since the patches don't overlap by definition, this is an ok approximation, even though they are measuring the same $$\ell$$ mode.)

## Appendix A: Scaling factors.

This part need to be documented more, but meanwhile, here are some notes from Raphael defining the new scaling factors, and how to rescale the BPCM: here (pdf scan of Raphael's notes).

The 4 scaling factors introduced in the notes above are plotted here as a function of the value of r for the different masks. They can be downloaded in this tarball.

• Note the fact that signal $$f_{\rm sky}$$ is larger for the shallow map than for the full chile map, which is maybe counter-intuitive at first. This is because we are here using definitions assuming a inverse noise variance weighting. This will wash out the signal in the full chile deep mask due to the shallow part.