Maximum likelihood search results for Data Challenge 02.00 with an ideal lensing template

 (C. Umiltà)
Appendix added on 2018-11-13

Introduction

This posting summarizes results from analysis of CMB-S4 Data Challenge 02.00 using a BICEP/Keck-style parametrized foreground model and an ideal lensing template LT. In principle this should lead to recovering, realization by realization, the same r as in an unlensed case.

Summary

The simulations used for this maximum likelihood (ML) searches are presented here. These are a subset of a hundred 02.00 simulations, each with its respective lensing template. Half of these sims have \(r\)=0 and half have \(r\)=0.003. The lensing template is obtained as the difference of lensed and unlensed sims. We should compare these results with unlensed sims, since the ideal template should allow, realization by realization, perfect removal of the lensing contribution. We do not have sims without lensing, but \(A_{lens}\)=0.03 sims have very little lensing contribution. I run the searches with fixed \(A_{lens}\) and no decorrelation; other paramters are set as usual.

Fig. 1 shows, realization by realization, the \(r\) results of ML search. Orange and blue points compare ML searches with and without a lensing template. Without a lensing template, I obtain the blue points, which have a biased mean but large \(\sigma(r)\). The points in orange are those with the lensing template added: ideally these points should be on the black line. Instead, their mean is biased at a similar level as sims without the lensing template, but their \(\sigma(r)\) is much reduced (at the same level of \(A_{lens}\)=0.03 sims).

Figure 1:
Recovered maximum-likelihood r values for 100 sims. The x-axis values come from sims with \(A_{lens}\)=0.03. The y-axis values are derived from sims with \(A_{lens}\)=1 contribution. The unlensed sims have all been analyzed with standard configuration, with fixed \(A_{lens}\). The lensed sims (y-axis values) are analyzed using the same standard configuration with (orange points) and without (blue points) an ideal lensing template, constructed realization by realization.

Version 1, 2 and 3 show the same points plotted with different scales on the x-y axes. Version 3 also shows the mean and standard deviation of points (solid line and shaded region).

In Fig.2 I plot an histograms of the distributions. The smaller \(\sigma(r)\) and the bias are evident from this figure. The blue and the orange distributions come from simulations with the same \(A_{lens}\), only with and without the idela LT in the analysis.

Figure 2:
Histograms of recovered maximum-likelihood \(r\) for \(A_{lens}\)=0.03 (green), and \(A_{lens}\)=1 with (orange) and without (blue) an ideal lensing template.

Conclusion

The addition of a lensing template decreases \(\sigma(r)\) to the expected level, but there is a residual bias in the recovered value of \(r\).

Appendix

Table 1:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 50 realizations with simple Gaussian foregrounds.
Decorrelation model = none \(A_L\) = 1 \(A_L\) = 1, with LT \(A_L\) = 0.03
Input \(r\) = 0-0.891±2.777-1.485±0.279-0.069±0.326
Input \(r\) = 0.0031.866±2.2041.276±0.4402.942±0.476

I quote here the mean and std for the simulations with and without LT. The results with and without LT are biased by a different quantity, and the difference in bias is \(~6 \cdot 10^{-4}\) for both values of input \(r\). Considering the error on the mean \(\bar{\sigma}\) (i.e. the std divided by \(\sqrt{N_{sims}}=\sqrt{50}\)), we see that the average of the two distributions differ by 1.5\(\bar{\sigma}\) and 1.9\(\bar{\sigma}\) for \(r\) = 0 and 0.003 respectively. (When considering the full 500 sims for AL=1 without LT, the differences jump to 6\(\bar{\sigma}\) and 6.7\(\bar{\sigma}\) respectively). Interestingly, the bias when adding the LT is larger, even though \(\sigma(r)\) is strongly reduced by using a LT.