Maximum likelihood search results: dependence on smallest angular scales.

2019-03-03 B. Racine

This short posting reports the dependence of the maximum likehood searches on the highest multipoles.
This could be important for the matrix pipeline analysis where we might have to use \(N_{\rm side}=128\), and \(\ell_{\rm max} \simeq 256\).

Introduction

The posting is based on past analyses of the DC4 maps with the maximum likelihood pipeline. Here we focus on the Gaussian foreground model (sky model 00), in the 4 sky regions studied here.
More information about the sky models, the parametric model as well as the dependence on priors can be found in older postings.

In the current posting, we study how the multicomponent analysis depends on the bandpowers used.
This is motivated by the matrix pipeline analysis, which is computationally intensive and might require to limit our analysis to maps with \(N_{\rm side}=128\).
Since it is recommended to restrict the analysis to multipoles below \(2\times N_{\rm side}\), as explained in the healpix anafast function's manual, we need to restrict to \(\ell_{\rm max} \simeq 256\), though pushing between 256 and 384 could be done with care.
In this short posting, I plotted the bandpower window functions for the 04 a, b and c regions.

In section 1, we show the main results concerning the scale dependence of the \(\sigma(r)\).
In section 2, we show the usual results concerning the "r" results, and their dependence on \(A_L\), for different bandpower selections.
In section 3, we show the ML parameters' distributions, including foreground parameters, in the form of histograms.

1. Dependence of \(\sigma(r)\) on the bandpower selection.

In figure 1, we show the evolution of \(\sigma(r)\) as a function of the maximum bandpower used in the multicomponent analysis. Out of the 9 bandpowers used in the usual likelihood, whose bandpower window functions are shown here, we restrict the range to the 4, 5, 6, 7, or 8 first bandpowers, and compare it to the full range.

Comments on Figure 1:

When we look at the plot normalized at the full range value, we see that in most cases, we have an increase of \(\sigma(r)\) up to 40 percents if we restrict to the first 4 bandpowers.
It seems like restricting to the 6 first multipole would result in a \(\simeq 10\) percent hit on sensitivity, whereas as can be seen on bandpower window functions, this could probably be done with \(N_{\rm side}=128 \) maps.
The hit is slightly higher for r=0.003 than for r=0.0, especially for the smaller masks (04c and 04d).

**Figure 1**: Evolution of \(\sigma (r)\) as a function of the highest bandpower used in the ML search for the 4 masks: In blue the circular 3% mask, in green the nominal Chile mask , red, the nominal Pole mask and magenta, the BK14 mask. The error bars are an estimate of the error on the std, estimated as \(\sigma/\sqrt{2 N_{sims}}\).

2. Usual summary Plots

We show the usual summary plots and how the vary with bandpower selection.
Note that in the standard analysis, the first bandpower is not used, so here [2,10] corresponds to the 9 usual bandpowers.
Figure 2 shows how the constraints on r evolve for the different observation masks, for sky model 00.
Figure 3 shows the evolution of \(\sigma(r)\) as a function of residual lensing \(A_L\) for the 3 masks.

**Figure 2**: Summary of the results for different values of \(A_L\). In red, the analysis of the simulations with r=0.003, in green, the r=0 case. On the left, the case without decorrelation in the parametrization, on the right, with linear-\(\ell\) decorrelation. The outer error bars show the standard deviation \(\sigma\) of the \(N_{sims}\) simulations' ML results (\(N_{sims}\)=150), and the inner error bars show \(\sigma/\sqrt{N_{sims}}\) ; (04 has 1000 realizations and 04b/c/d have only 300).

In figure 3, we show the evolution of \(\sigma(r)\) as a function of \(A_L\).

Comments on Figure 3:

While restricting the multipole range increases \(\sigma(r)\) in all cases, here we see that it changes the slope of the \(A_L\) dependence. In figure 1, we saw that the relative increase is similar for all cases.

**Figure 3**: Evolution of \(\sigma(r)\) as a function of \(A_L\), for the 4 masks. In blue the circular 3% mask, in green the nominal Chile mask , red, the nominal Pole mask and magenta, the BK14 mask. The error bars are an estimate of the error on the std, estimated as \(\sigma/\sqrt{2 N_{sims}}\).

3: Parameter distributions

In figure 4, we report the full distribution of the ML parameters.

Comments on Figure 4:

Using the fixed limits, you can see that restricting the multipole range has a stronger effect on the scale dependence parameters \(\alpha\)'s as well as the dust-synchrotron correlation parameter than on the amplitude parameters, which change very little, as well as on the frequency Dependence parameters \(\beta\)'s.
Looking at the reported mean and std of the lensing amplitude, we can see that it takes a big hit when restricting to lower multipoles.

**Figure 4**: Figure representing the ML parameters histograms for model 00.

In the histograms, we report the mean of the distribution as a red line, and the fiducial input value as a **black line** (if defined: see this table). We also report the mean±standard deviation of the distribution.

Conclusion

In this analysis, we varied the multipole range used in the multicomponent analysis by cutting the highest bandpowers.
When restricting to the scales exploitable from maps at \(N_{\rm side}=128\), i.e. only keeping the bandpowers [2,7], we have an increase in \(\sigma(r)\) of roughly 10%.
Note that the current posting only considers the simple Gaussian model (00) and the penalty might be stronger for some more complex models.
We need to keep this in mind if we want to go ahead and use that resolution for the matrix pipeline.