# Sigma(r) vs r plots in prep for the DSR.

B. Racine

In this posting, we show how the sigma(r) vs r plot changes if we add a foreground penalty, or if we don't assume Chile's yield to be equal to the one of Pole. We also use a more optimal weighting in the definition of our $$f_{\rm sky}$$ rescaling used to compute the bandpower covariance matrix.
All the pots here are for 18 tubes, 7 years of observation.
WARNING, This is preliminary, some numbers will slightly change once the weights converge more, and some number are already updated. The posting will be updated in the next couple of days.

## Introduction

In a recent posting, we introduced an update to the Fisher forecasts that includes more realistic survey strategies, which are taken into account via rescalings of the different terms of the bandpower covariance matrix: $$f_{\rm sky, noise}$$, $$f_{\rm sky, signal}$$ and $$f_{\rm sky, cross}$$. In this posting we used a definition of these factors that considered an inverse noise variance weighting in the analysis. This would be optimal if we were completely noise dominated, but since we are studying the variation of $$\sigma(r)$$ as a function of r, and since we have some lensing (and foreground) residuals, that solution is not optimal.
This was explain in more details in this short posting. Note that the effect is expected to be non negligible for $$r\gtrsim0.003$$ for Chile, but already for r=0 for Pole, where the noise level is below the lensing residuals. This means that more delensing is needed for this survey strategies to be optimal.

One can instead use a more optimal weight, including the signal sample variance. In this posting, we derived for a simple 1D model what the $$f_{\rm sky}$$ factors are for an arbitrary weighting. Raphael has derived these scalings more carefully in harmonic space, to rescale the bandpower covariance matrix (see Appendix A)

In this short posting, we are using the scalings produced by Raphael to rescale the BK bandpower covariance matrix, using Victor's performance based Fisher framework.

## 1: $$\sigma(r)$$ vs r.

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. In figure 2 and figure 3, we show bands of $$\sigma(r)$$ values including or not some foreground residuals or the yield 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).

• We still show the old inverse noise variance weighting, compared to the new noise variance weighting
• In this click, we show the curves corresponding to the Chile shallow and deepest part of the Chile deep map.
• Note that in the current version forecasted here, we don't take into account the penalty on survey weight coming from observing at low elevation, with higher NET.

## 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).

Table 1: Pole deep.
$$f_{\rm sky}$$ factors for the signal, noise and cross terms, as well as the corresponding optimized $$A_L$$ and the $$f_{\rm eff}$$ (see Raphael's notes)
r 0.000 0.003 0.010 0.030
$$A_L$$ 0.076 0.079 0.082 0.085
$$f_{\rm sky,noise}$$ 0.038 0.039 0.039 0.037
$$f_{\rm sky,cross}$$ 0.036 0.039 0.043 0.048
$$f_{\rm sky,signal}$$ 0.026 0.029 0.031 0.033
$$f_{\rm eff}$$ 0.036 0.040 0.046 0.055
Table 2: Pole wide.
$$f_{\rm sky}$$ factors for the signal, noise and cross terms, as well as the corresponding optimized $$A_L$$ and the $$f_{\rm eff}$$ (see Raphael's notes)
r 0.000 0.003 0.010 0.030
$$A_L$$ 0.114 0.118 0.122 0.128
$$f_{\rm sky,noise}$$ 0.092 0.096 0.100 0.103
$$f_{\rm sky,cross}$$ 0.082 0.089 0.099 0.112
$$f_{\rm sky,signal}$$ 0.060 0.064 0.069 0.076
$$f_{\rm eff}$$ 0.084 0.091 0.101 0.118
Table 3: Chile deepest patch.
$$f_{\rm sky}$$ factors for the signal, noise and cross terms, as well as the corresponding optimized $$A_L$$ and the $$f_{\rm eff}$$ (see Raphael's notes)
r 0.000 0.003 0.010 0.030
$$A_L$$ 0.080 0.083 0.086 0.091
$$f_{\rm sky,noise}$$ 0.041 0.043 0.046 0.046
$$f_{\rm sky,cross}$$ 0.037 0.040 0.045 0.050
$$f_{\rm sky,signal}$$ 0.029 0.032 0.035 0.038
$$f_{\rm eff}$$ 0.038 0.041 0.045 0.052
Table 4: Chile shallow patch.
$$f_{\rm sky}$$ factors for the signal, noise and cross terms, as well as the corresponding optimized $$A_L$$ and the $$f_{\rm eff}$$ (see Raphael's notes)
r 0.000 0.003 0.010 0.030
$$A_L$$ 0.270 0.270 0.270 0.270
$$f_{\rm sky,noise}$$ 0.279 0.287 0.301 0.324
$$f_{\rm sky,cross}$$ 0.237 0.247 0.264 0.296
$$f_{\rm sky,signal}$$ 0.161 0.168 0.180 0.201
$$f_{\rm eff}$$ 0.253 0.260 0.274 0.304

• 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.