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, where we remove the priors on the \(\beta\)'s and \(A_L\) parameters.
The method is analogous to this posting, which reports DC4 results for model 00 to 09, where we introduced a new cut on the bad simulations based on map std outliers, which didn't have a significant impact.

In the current posting, we analyze 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 using models 0, 1, 2, 3, 7, 8 and 9, that are described with a bit more details here.

• 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: Hensley/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, and their dependence on \(A_L\)
In section 2, we show the ML parameters' distributions, including foreground parameters, in the form of histograms.
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 the former analyses, for each realization, we found 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).
In the current analysis, we remove all the remaining parameter priors step-by-step:

• 1. Gaussian prior on \(\beta\)'s, see below, fixed \(A_L\)
• 2. flat priors on \(\beta\)'s: \(\beta_s\in[-4.5,-1.5]\), \(\beta_d\in[0.8,2.4]\), fixed \(A_L\)
• 3. flat priors on \(\beta\)'s, gaussian priors on \(A_L\), centered on the fiducial value, with \(\sigma\) 5% of that mean value.
• 4. flat priors on \(\beta\)'s, flat priors on \(A_L\): \(A_L\in[0,2]\).

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”
• \(A_L\): Amplitude of the residual lensing \(BB\) power spectrum, simple way to study how delensing would help the constraints on \(r\).

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 all models except model 4, releasing the prior on \(A_L\) from a fixed \(A_L\) to a free \(A_L\), all the constraints on \(r\) shift up. For model 4 (which only exists for DC04 mask), it is reversed.
• For all models, releasing the priors on \(\beta\) as negligible effect, with model 4 and 9 seeing the strongest effect.
• For model 00, for the DC04 mask, we see how the bias on r gets reduced for all \(A_L\) when we release the priors from this case, to this case. For the other masks, the effect is not as striking. For the Pole mask, switching between this case and this case, it seems to reduce the bias in the case where \(r=0\) but not if \(r=0.003\). For the Chile mask, the shift is larger.
• Releasing the priors has an interesting effect on model 4, where the bias dependence on \(A_L\) disappears. There is still an over all bias for the no decorrelation case. For model 8 and 9, where there is also a dependence of the bias on \(A_L\), the effect is still there after opening the prior.
• For model 08 and 09 (and with less amplitude for 01 and 03), switching 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.

In figure 2, we show the evolution of \(\sigma(r)\) as a function of \(A_L\). We also do a linear fit that we report in the legend, and show as a faded dashed line, which can be used to estimate the \(\sigma(r)\) at other \(A_L\) under that assumption.

Comments on Figure 2:

• Switching between all the models, there is no drastic change in the slope of the \(A_L\)-dependence.
• Opening the priors on \(A_L\) shifts the \(\sigma(r)\) up and the slope raises.
• Opening the decorrelation shifts the \(\sigma(r)\) up but the slopes are not changed significantly.
• We circled plausible delensing levels for Chile (\(A_L\)=0.3, in green) and for Pole (\(A_L\)=0.1, in red), which are close to the forecasted delensing capabilities.
• 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.

Comments on figure 3 (More comments in the case of DC04 can be seen in this posting):

• 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.
• Switching 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 brightness variations, doesn't take such a big hit, only an overall increase of \(\sigma(r)\).
• For these models 8 and 9 that have a reasonable amplitude variation on the sky and enough complexity, we see that the bias on r is proportional to the amplitude of the foreground. This is further discussed in this posting.

3: DC4, DC4b and DC4c results table

In the following tables, we report the \(r\) results for the case where all the parameters have generous flat priors.
For the other cases, see the tables in the following links:
Gaussian priors on \(\beta\)'s, fixed \(A_L\)
free \(\beta\)'s, fixed \(A_L\)
free \(\beta\)'s, Gaussian 5% prior on \(A_L\)
free \(\beta\)'s, free \(A_L\) (as below)

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.