Maximum likelihood search results for Data Challenge 04b and 04c, model 00, 01, 02, 03, 07, 08 and 09

Introduction

This posting summarizes results from analysis of CMB-S4 Data Challenge 04b and 04c using a BICEP/Keck-style parametrized foreground model. It is an update on this posting, that focused on models 0 and 7.
The method is analogous to this previous posting, which reports DC4 results for model 00 to 09. We are using proper bandpass conventions, as explained in this posting, section 3, and a new log-likelihood cut as explained in this posting, section 2.

In the current posting, we analyse simulations with nominal Chile (DC04b) and Pole (DC04c) masks, as described here, which we compare to the previous circular idealized f_sky 3% mask from the CDT report (DC04). More information about DC4 can be found in the Experiment Definition page.
We are now using models 0, 1, 2, 3, 7, 8 and 9, that are described with a bit more details in this posting, where the analysis on the 3% circular mask is reported.

• Model 0 has Gaussian foregrounds, with no non-Uniform variation of the intensity on the sky, following the same law as the ones we use the our multicomponent model.
• Model 1, 2 and 3 are PySM versions, with more realistic variations over the sky, with different complexities (1: varying \(\beta_d\), 2:+ another dust component, curved synchrotron, 2% polarization fraction AME, 3: Hensly/Draine model)
• Model 7: Amplitude modulated Gaussian foregrounds, modified version of model 00, where the brightness of dust varies over the sky, based on real data.
• Model 8: Multi-layer phenomenological 3D model, naturally producing decorrelation and flattening at low frequency.
• Model 9: Vansyngel model, where a multi-layer magnetic field is simulated to produce non-Gaussian polarized dust maps.

In section 1, we show the main results concerning the "r" results.
In section 2, we show the ML parameters' distributions, including foreground parameters, as histograms or as "triangle plots".
In section 3, we report tables of r constraints for different sky models, masks, lensing residuals, and with and without decorrelation in the ML search.

Note about the Model:
In this analysis, for each realization, we find the set of model parameters that maximizes the likelihood multiplied by priors on the dust and sync spectral index parameters (\(\beta_d\) and \(\beta_s\)). These priors are based on Planck data, so they are quite weak in comparison with CMB-S4 sensitivity. However, in principle foreground models may violate them potentially leading to biases (e.g. DC4 model 03 where the preferred value of \(\beta_d\) is outside the prior range - see this posting, Figure 2).

The model includes the following parameters:

• \(r\): tensor-to-scalar ratio
• \(A_d\): \(BB\) power spectrum amplitude of dust, in \(\mu K_{CMB}^2\) units at \(\nu\)=353 GHz and \(\ell\)=80
• \(\beta_d\): dust emissivity spectral index; Gaussian prior on \(\beta_d\) is centered at 1.6 and has width 0.11
• \(A_s\): \(BB\) power spectrum amplitude of synchrotron, in \(\mu K_{CMB}^2\) units at \(\nu\)=23 GHz and \(\ell\)=80
• \(\beta_s\): synchrotron emissivity spectral index; Gaussian prior on \(\beta_s\) is centered at -3.1 and has width 0.3
• \(\alpha_d\): power law index for dust \(\mathcal{D}_\ell\) scaling in \(\ell\); limited to range [-2,2]
• \(\alpha_s\): power law index for synchrotron \(\mathcal{D}_\ell\) scaling in \(\ell\); limited to range [-2,2]
• \(\epsilon\): frequency-independent spatial correlation of dust and synchrotron; limited to range [-1,1]
• \(\Delta_d\): dust correlation between 217 and 353 GHz; not included when “Decorrelation model” is set to “none”

For the decorrelation model, we assume that the cross-spectrum of dust between frequencies \(\nu_1\) and \(\nu_2\) is reduced by factor \(\exp\{log(\Delta_d) \times [\log^2(\nu_1 / \nu_2) / \log^2(217 / 353)] \times f(\ell)\}\). For the \(\ell\) dependence we fix the scaling to take a linear form (pivot scale is \(\ell\)=80).

1. Summary Plots

Figure 1 shows how the constraints on r evolve for the different observation masks, for different sky models. Figure 2 shows the evolution of \(\sigma(r)\) as a function of residual lensing \(A_L\) for the 3 masks.

Comments on Figure 1:

• For model 00 (and 02 and 07), swtiching between the circular mask (04) and the Chile mask (04b) or the Pole mask (04c), the mean value doesn't change much, but the error bar change in a \(A_L\)-dependent way. Figure 2 is a better way of seeing it.
• For model 01 (and 03), swtiching between the circular mask (04) and the Chile mask (04b), we see that the bias get worsened. The Pole mask (04c) doesn't get such a strong hit.
• For model 08 (and 09), swtiching between the circular mask (04) and the Chile mask (04b), the effect is even more striking.

Comments on Figure 2:

• Switching between all the models, there is no drastic change in the slope of the \(A_L\)-dependence.
• The slope is steeper for the Pole mask, meaning that the de-lensing is a great source of improvement. Overall, it seems like a Chile type observation mask would perform better if we don't delens. But once the necessary delensing is applied, a Pole mask performs better. We circled plausible delensing levels for Chile (\(A_L\)=0.3, in green) and for Pole (\(A_L\)=0.1, in red), to be checked with more realistic delensing.
• Models 8 and 9 have an overall higher \(\sigma(r)\), and somehow for model 9, the Chile mask takes an even bigger hit on sensitivity. We will see in figure3 that this might be due to a stronger decorrelation.

2: Parameter distributions

In figure 3, we report the full distribution of the ML parameters, in the form of histograms or of "Triangle plots".
Note the overall level of dust amplitude for 04.07 peaking at 14.8±0.8 \(\mu K^2\), 04b.07 peaking at 72.9±2.5 \(\mu K^2\), and 04c.07 peaking at 22.6±1.8 \(\mu K^2\), compared to 4.3±0.3 \(\mu K^2\) for 04.00. Despite this high level of dust amplitude, \(\sigma(r)\) doesn't significantly change, as we have seen in figure 2.
Swtiching between 04.09 04b.09, and 04c.09, we see such a brightness variations, but this time, we have a non-negligible decorrelation. This might explain why this model take a larger hit on sensitivity when using wider masks. Somehow, model 8, which also has a strong decorrelation and some realistic bightness variations, doesn't take such a big hit, only an overall increase of \(\sigma(r)\).

3: DC4, DC4b and DC4c results table

00: Gaussian foregrounds

The mean values and standard deviations of \(r\) for simulations with simple Gaussian foregrounds are summarized in Table 00, Table 00b and Table 00c, respectively for the circular 3% mask, the nominal Chile mask and the nominal Pole mask.

Figure 1 shows these results in a plot.

01: PySM a1d1s1f1

For more details about this model, see this previous posting

Figure 1 shows these results in a plot.

02: PySM a2d4f1s3

For more details about this model, see this previous posting

Figure 1 shows these results in a plot.

03: PySM a2d7f1s3

For more details about this model, see this previous posting

Figure 1 shows these results in a plot.

07: Amplitude modulated Gaussian foregrounds

The mean values and standard deviations of \(r\) for simulations with amplitude modulated Gaussian foregrounds are summarized in Table 07, Table 07b and Table 07c, respectively for the circular 3% mask, the nominal Chile mask and the nominal Pole mask.

Figure 1 shows these results in a plot.

08: MKD model (3D multi-layer)

For more details about this model, see this previous posting

Figure 1 shows these results in a plot.

09: Vansyngel model

For more details about this model, see this previous posting

Figure 1 shows these results in a plot.