Sigma(r) vs r, using Fisher with new hitcount map rescaling.

  B. Racine

In this posting we look at Fisher forecasts for different configurations, including a proper signal and noise \(f_{\rm sky}\) rescaling in the covariance matrix. It is a modification from what Victor used for the google spreadsheet, but studying the same configurations.
Edited on April 23rd: added the new "Chile deep" and "Pole wide" masks.
Edited on April 29th: New plots for chile deep hybrid delensing.
Note that we are planning to rerun those numbers with a more optimal weighting (for now using inverse noise variance weighting).
Edited on April 30st: New plots using the new weighting computed by Raphael Flauger are now moved to a new posting.
WARNING, This is preliminary, some results might change.


Until now, we have been using an approximation to deal with the increase of \(f_{\rm sky}\) in Victor's Fisher code, as described in this posting for instance.
In the old version, the noise levels are based on the BK14 noise spectra, which correspond to roughly 1% of the sky. On top of the sensitivity rescaling, beam rescaling etc., we were applying a fixed factor of 3 (or 20), to take into account the fact that S4 would observe roughly 3% of the sky from Pole or 20% from Chile. After rescaling the noise by 3 (20) and the corresponding noise terms in the bandpower covariance matrix, the full bandpower covariance matrix was scaled down by a factor of 3 (20) to take into account the observations of more modes.
In the new version, we now use more realistic hitcount map, as we used in the map based re-analysis.
As it has been noted in different posting, for instance this one, the effective sky fraction corresponding to a given hitmap is not the same for signal and noise.

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

We then use these new noise levels and the rescaled bandpower covariance matrix within the Fisher framework to predict \(\sigma(r)\).

We use the configuration 5 described here, as well as the tube distribution shown here.
Note that for 4 years of observation, we model the delensing residual using \(A_L\) = 0.106 for Pole deep (04c), \(A_L\) = 0.161 for Pole wide, and \(A_L\) = 0.337 for Chile wide (04b), whereas for 7 years of observations, we use \(A_L\) = 0.081 for Pole deep, \(A_L\) = 0.123 for Pole wide, and \(A_L\) = 0.27 For Chile wide.
Note that for the Chile deep, we have an hybrid method, where we use the Pole deep delensing on the deepest patch, and the Chile wide delensing (from LAT) on the shallowest. We then combine the two.

Comments on the figure:

Figure 1: \(\sigma(r)\) as a function of r for the different hitcount maps.

Appendix A: Scaling factors.

Note here that we are using Inverse Noise Variance weighting. The weights derived here would be different for another weighting, see related posting from Raphael (see here).

In our simulations, we generate uniform noise maps using \(\texttt{synfast}\), which we then divide by the square root of these hitcount maps. We then also use these hitmaps as an observing mask when computing the power spectrum, since we use the inverse noise variance weighting. We showed on a 1D model how these weightings impact the number of degrees of freedom and the corresponding effective \(f_{\rm sky}\) for signal and noise. Raphael derived it at harmonic power spectrum level and showed how the bandpower covariance matrix needs to be rescaled in order to take the hitmap weighting into account: \begin{align} \Delta C_\ell^2 &\propto \left( \frac{C_\ell^2}{\rm f_{\rm sky,signal}}+ 2 \frac{C_\ell N_\ell}{\rm f_{\rm sky,cross}}+\frac{N_\ell^2}{\rm f_{\rm sky,noise}} \right), \end{align} where the effective \(f_{\rm sky}\) are defined below.
Note that in Victor's Fisher forecast, we start from the BK bandpower covariance matrix, including the non diagonal terms (see section on BPCM in this posting ). We will need to rescale the different signal x signal, noise x noise and signal x noise terms in the same way.

Let's first denote our hitmaps as \(h_i\), where i is the pixel number. One can show that \begin{align} {\rm f_{\rm sky,noise}} &= \frac{\Omega_{pix}}{4\pi} \frac{(\sum_i h_i)^2}{\sum_i h_i^{2}}, \end{align} \begin{align} {\rm f_{\rm sky,signal}} &= \frac{\Omega_{pix}}{4\pi} \frac{(\sum_i h_i^2)^2}{\sum_i h_i^{4}}, \end{align} \begin{align} {\rm f_{\rm sky,cross}} &= \frac{\Omega_{pix}}{4\pi} \frac{(\sum_i h_i^2)(\sum_i h_i)}{\sum_i h_i^{3}}, \end{align} (note that computationally, if we have an array h with NaNs for unobserved pixels, we can define a function W(i) = nansum(h^i)/size(h), and we get f_nn=W(1)^2/W(2), f_ss=W(2)^2/W(4), and f_ns=W(1)W(2)/(2*W(3)))

Once we compute these factors, we can rescale the bandpower covariance matrix. Remember that the covariance matrix has 3 types of terms (see equation 1), as described in this posting: the \(\texttt{sig}\), \(\texttt{noi}\), and cross terms: \(\texttt{sn1}\), \(\texttt{sn2}\), \(\texttt{sn3}\) and \(\texttt{sn4}\).
To account for the noise boost due to the dilution of the effort, we rescale the noise terms by the ratio of the \(f_{\rm sky,noise}\): \begin{align} \texttt{noi}^{\rm New\;mask} &= \left(\frac{\rm f_{\rm sky,noise}^{\rm New\;mask}}{f_{\rm sky,noise}^{\rm Old\;mask}}\right)^2 \texttt{noi}^{\rm Old\;mask}, \end{align} \begin{align} \texttt{sn}^{\rm New\;mask} &= \frac{\rm f_{\rm sky,noise}^{\rm New\;mask}}{f_{\rm sky,noise}^{\rm Old\;mask}} \texttt{sn}^{\rm Old\;mask}. \end{align} We then need to take into account the fact that we observe more modes, and rescale the different terms accordingly: \begin{align} \texttt{noi}^{\rm New\;mask} &= \frac{\rm f_{\rm sky,noise}^{\rm Old\;mask}}{f_{\rm sky,noise}^{\rm New\;mask}} \texttt{noi}^{\rm Old\;mask}, \end{align} \begin{align} \texttt{sn}^{\rm New\;mask} &= \frac{\rm f_{\rm sky,cross}^{\rm Old\;mask}}{f_{\rm sky,cross}^{\rm New\;mask}} \texttt{sn}^{\rm Old\;mask}. \end{align} \begin{align} \texttt{signal}^{\rm New\;mask} &= \frac{\rm f_{\rm sky,signal}^{\rm Old\;mask}}{f_{\rm sky,signal}^{\rm New\;mask}} \texttt{signal}^{\rm Old\;mask}. \end{align}

The different factors, as well as the ratios, are reported in the table 1 and 2 below for different hitmaps.

Table 1: \(f_{\rm sky}\) factors for the different hitcount maps, for the signal, noise and cross terms. We also show the "flat" case, where we consider the ratio of non 0 pixels.
Mask\type binary noise signal cross
04 0.054 0.036 0.029 0.033
04b 0.349 0.185 0.114 0.153
04c 0.048 0.027 0.018 0.023
04d 0.024 0.013 0.010 0.012
Pole wide 0.120 0.065 0.043 0.055
Chile deep 0.516 0.182 0.059 0.116
Chile deep (deepest) 0.050 0.034 0.024 0.030
Chile deep (shallow) 0.466 0.204 0.100 0.156
Table 2: Ratios of the \(f_{\rm sky}\) factors to the BK ones, as used to rescale in the code.
Mask\type binary noise signal cross
04/04 1.000 1.000 1.000 1.000
04b/04 6.451 5.144 3.984 4.660
04c/04 0.896 0.741 0.643 0.701
04d/04 0.446 0.368 0.338 0.356
Pole wide/04 2.210 1.815 1.496 1.682
Chile deep/04 9.541 5.063 2.053 3.556
Chile deep (deepest)/04 0.927 0.944 0.834 0.903
Chile deep (shallow)/04 8.613 5.680 3.494 4.777

Comments on figure 2:

Figure 2: hitcount maps and the corresponding scale factors, as shown above. Note the different normalization in the plots here.