Maximum likelihood search results for Data Challenge 04b and 04c, model 00 and 07

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 analogous to this previous posting, which reports DC4 results for model 00 to 06, where we introduced a method to flag flawed simulations and used fixed bandpass conventions.

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 can be found in the Experiment Definition page.
For now we only have 300 simulations using Gaussian foregrounds (model 00) for the new masks. Since we are here studying the effect of variations of the sky coverage, it is important to keep in mind that these simulations are uniform over the sky.
A new set of 300 simulations has been produced for model 07, that incorporates a data driven variation of the dust amplitude over the sky (see here).

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, both for sky model 00 and 07. Figure 2 shows the evolution of \(\sigma(r)\) as a function of residual lensing \(A_L\) for the 3 masks.
While the 07 model has variations of dust amplitude, and an overall higher dust level (see Section 2), the constraints on r are not deteriorated significantly. A real sky with some level of decorrelation would probably take a bigger hit on sensitivity. Future work will help testing this hypothesis.
For reasonable delensing factors, \(A_L=0.3\) for the Chile mask, \(A_L=0.1\) for the Pole mask, it seems like a Pole-like observation leads to tighter constraint, despite the caveat raised above.

Comments on Figure 1:

• Swtiching between Sky model 00 and Sky model 07, we see that there is no significant change in our constraints, especially for Pole mask (04c).
• Switching between 04, 04b and 04c, we see that the change in \(\sigma(r)\) depends on the level of delensing. This is better shown in figure 2.

Comments on Figure 2:

• As in figure 1, swtiching between Sky model 00 and Sky model 07, we see that there is no significant change in our constraints, especially for Pole mask (04c).
• We also see that going from the 3% circular mask (04) to the nominal Chile mask (04c) there is a non negligible improvement in the \(\sigma(r)\) for the case with no delensing (\(A_L\)=1.0). But as we delens more, \(\sigma(r)\) becomes worse than the circular mask case. For the (\(A_L\)=0.1) case, used in the CDT report, using the Chile mask deteriorates the constraint.
• Going from the 3% circular mask (04) to the nominal Pole mask (04c), there is no improvement in the (\(A_L\)=1.0) case, but for the targeted (\(A_L\)=0.1) and below, the deterioration is weaker than the (04b) case.
• 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. In figure 2, we circled plausible delensing levels for Chile (\(A_L\)=0.3, in green) and for Pole (\(A_L\)=0.1, in red).

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. A more realistic model including some level of decorrelation might allow us to test the effect of mask variation more fairly.

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.

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.