Maximum likelihood search results for Data Challenge 04

  V. Buza

Note that there was an issue in the noise debiasing in the spectra used for this posting: a fixed posting is available here


This posting summarizes results from analysis of CMB-S4 Data Challenge 04 using a BICEP/Keck-style parametrized foreground model. It is analogous to this previous posting by C. Bischoff, V. Buza, J. Willmert and B. Racine -- in places where it makes sense, I adapt some of the same text and templates. In this analysis, for each realization, I 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. In addition to that I also run a case with no priors on the spectral indices, and in which I allow \(A_L\) to vary with a tight prior, though I do not quote results from these runs.

The model includes the following parameters:

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

There are two additional changes from the previous analysis: 1. I fix the lensing spectrum to the input one. In the previous posting it uses a lensing spectrum that differs from the input. 2. I zero out the theory spectrum below ell=30, since that's what was done in the sims. The previous posting does not take this into account.

00: Gaussian foregrounds

The mean values and standard deviations of \(r\) for simulations with simple Gaussian foregrounds are summarized in Table 00. With a 10% lensing residual, we don't quite achieve \(\sigma(r) = 5 \times 10^{-3}\) for sims with \(r = 0\).

Turning on dust decorrelation in the model doesn't cause any bias in \(r\) and the recovered \(\rho_d\) values are centered around 1 (i.e. analysis recovers zero decorrelation). Adding this parameter does increase \(\sigma(r)\) somewhat.

Table 00:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with simple Gaussian foregrounds.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none -0.633±2.954 -0.186±1.158 -0.070±0.538 -0.0267±0.307
linear -0.738±2.801 -0.227±1.135 -0.075±0.609 -0.0375±0.417
Input \(r\) = 0.003
none 2.285±2.884 2.779±1.296 2.924±0.739 2.990±0.509
linear 2.255±2.991 2.787±1.473 2.949±0.933 2.997±0.682

01: PySM a1d1s1f1

As has been previously noted, dust power is much higher in this model (\(A_d \sim 12.5 \mu K^2\)) than for the Gaussian foreground sims (\(A_d = 3.75 \mu K^2\)). The PySM d1 dust model does feature a spatially varying spectral index, but we don't find any detectable decorrelation in this analysis. The PySM s1 synchrotron model yields \(A_s \sim 0.5 \mu K^2\) and there is \(\sim 6\)% correlation between dust and sync.

Table 01:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with PySM a1d1f1s1 foregrounds.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none 1.175±2.559 1.545±0.993 1.578±0.540 1.601±0.380
linear -2.296±2.604 -0.821±1.082 -0.234±0.660 -0.201±0.489
Input \(r\) = 0.003
none 4.219±2.744 4.599±1.223 4.745±0.774 4.951±0.615
linear 0.624±2.837 2.003±1.285 2.632±0.832 2.738±0.659

02: PySM a2d4f1s3

The d4 version of PySM dust adds a second dust component (with different blackbody temperature and emissivity power law) based on Meisner & Finkbeiner (2014). Not sure what type of \(\beta_d\) spatial variations are included in this model, but Colin thinks it is more or less the same as for d1. The s3 synchrotron model adds curvature to the synchrotron spectral index: \(\beta_s \rightarrow \beta_s + C \ln (\nu / \nu_C)\). The a2 AME model uses a 2% polarization fraction for AME, which seems very high, but there is no attempt to model AME in this analysis.

Results for this model show that \(A_d\) is even larger (\(\sim 32.5 \mu K^2\)) than for the d1 dust model. The mean value of \(\beta_d\) decreases from 1.59 (for PySM d1 model) to 1.55, which is probably a sign of the two component dust. The mean value of \(\beta_s\) decreases from -3.05 (for PySM s1 model) to -3.13, which is probably due to synchrotron spectral curvature (and perhaps polarized AME?). Dust–sync correlation is higher, at \(\sim 10\)%, which could be from polarized AME.

Table 02:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with PySM a2d4f1s3 foregrounds.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none -0.230±2.548 0.210±0.954 0.301±0.478 0.316±0.299
linear -0.542±2.716 -0.00567±1.156 0.154±0.659 0.177±0.445
Input \(r\) = 0.003
none 2.603±2.538 3.142±1.078 3.281±0.645 3.335±0.480
linear 2.348±2.660 2.981±1.235 3.162±0.795 3.196±0.613

03: PySM a2d7f1s3

The next PySM version uses the Hensley/Draine dust model, which has additional complexity in the dust SED (perhaps described in arXiv:1709.07897?). The level of dust power is similar to sky model 01 (PySM d1 model), but we find that the emissivity power law is even flatter than the last case, with \(\beta_d \sim 1.44\).

The recovered means seem quite wacky, and \(A_L\) dependent.

Table 03:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with PySM a2d7f1s3 foregrounds.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none 1.074±2.574 1.505±0.998 1.594±0.546 1.657±0.401
linear -4.116±2.668 -1.767±1.149 -0.871±0.699 -0.702±0.508
Input \(r\) = 0.003
none 4.089±2.712 4.538±1.187 4.740±0.754 4.975±0.610
linear -1.205±2.938 1.004±1.393 1.913±0.906 2.120±0.704

04: Ghosh dust model

The Ghosh dust model (described here) is based on GASS HI data with a model for the Galactic magnetic field. For these sims, it is combined with the PySM a2, f1, and s3 components (same as the two previous models).

The analysis of this model yields smaller still values of \(\beta_d \sim 1.3-1.4\). Dust-sync correlation is still present, but smaller (2–3%), which is probably due to the fact that the Ghosh dust sims don't know anything about the PySM synchrotron or AME components. The fact that they are correlated at all probably happens because both models are based on data at larger scales.

Dust decorrelation is small in absolute terms, but detected at high significance. Using a model without dust decorrelation leads to a large positive bias on \(r\) in the range \(4-5 \times 10^{-3}\). Dust decorrelation with linear \(\ell\) scaling produces the smallest biases, but still quite large compared to other sky models.

Table 04:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with Ghosh dust model plus PySM a2f1s3 foregrounds.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none 2.177±2.644 3.196±1.189 4.105±0.784 5.251±0.603
linear -1.080±2.723 -0.038±1.280 -0.622±0.813 -0.898±0.617
Input \(r\) = 0.003
none 5.596±3.038 6.657±1.432 7.837±0.994 9.371±0.903
linear 2.088±3.036 2.770±1.504 2.552±1.032 2.261±0.840

05: Gaussian decorrelated dust model

We only have 100 realizations (50 each for \(r\) = 0, 0.003) to analyze for this Gaussian decorrelated model, so the statistics aren't great. However, this model has extremely large dust decorrelation (15% between 217 and 353 GHz at \(\ell\) = 80) and it exactly follows the assumed functional form of decorrelation with linear \(\ell\) scaling, so we can still draw some useful conclusions.

When we choose decorrelation with linear \(\ell\) scaling to match the sims, then we find no bias on \(r\) and recover \(\rho_d\) = 0.85.

An important point to note from this model is that, even for the unbiased case where the decorrelation is correctly modeled in both \(\nu\) and \(\ell\), we find \(\sigma(r) \sim 1.4\), much larger than the target sensitivity of CMB-S4. This shows that, for extreme levels of foreground decorrelation, we lose the ability to clean foregrounds from the maps because the foreground modes are significantly independent between the various CMB-S4 frequencies. Regardless of whether you are doing map-based cleaning or fitting the power spectra as we do here, the only way to improve sensitivity would be use more observing bands that are more closely spaced. It also makes the point that our Fisher forecasts should assume some non-zero level of decorrelation. Adding decorrelation as a free parameter to a forecast that assumes \(\rho_d = 1\) only captures part of the statistical penalty.

Table 05:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 50 realizations with Gaussian decorrelated dust foreground.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
linear -1.836±3.290 -0.962±1.794 -0.568±1.379 -0.383±1.253
Input \(r\) = 0
linear 1.613±2.804 2.561±1.735 2.885±1.409 2.997±1.278

06: Flauger MHD foregrounds

Out understanding is that this model uses MHD simulations to consistently model polarized dust and synchrotron in the Galactic magnetic field. This makes it quite interesting that this analysis finds negative dust-sync correlation with \(\epsilon \sim -0.36\). The dust power is similar to the Gaussian sims, and \(\beta_d\) matches the Planck value of 1.59. This analysis finds a synchrotron SED power law that is much flatter than usual, \(\beta_s \sim -2.6\), which is inconsistent with the prior at about \(1.5 \sigma\).

This model does not show any significant dust decorrelation. In general, the results for this model look nearly as good as the simple Gaussian foregrounds (sky model 00).

Table 06:
Mean \(r \times 10^3\) and \(\sigma(r) \times 10^3\) from sets of 500 realizations with Flauger MHD foreground model.
Decorrelation model \(A_L\) = 1 \(A_L\) = 0.3 \(A_L\) = 0.1 \(A_L\) = 0.03
Input \(r\) = 0
none 0.225±2.476 0.779±1.004 -0.984±0.569 1.127±0.399
linear -3.229±2.579 -1.518±1.109 -0.693±0.674 -0.506±0.490
Input \(r\) = 0.003
none 3.291±2.902 3.851±1.288 4.141±0.795 4.425±0.613
linear -0.195±2.990 1.361±1.401 2.178±0.926 2.408±0.732

Appendix A: Results for CDT report

As can be seen in Table 00, we seem to have biases even in the case of the Gaussian foreground simulations. This is being investigated.

Just as for the CDT report, we remove this "algorithmic bias" to focus on the bias produced by the different dust simulations. We also chose to report results using the linear \(\ell\) dependence for the decorrelation model. See caption of Table 07.

Table 07:
Bias on \(r\), obtained by subtracting the "algorithmic bias" to the mean of \(r\) for each of the 6 complex foreground models, for the case \(A_L\) = 0.1, assuming linear decorrelation for the Gaussian model. The "algorithmic bias" is estimated from the Gaussian simulations (-0.075±0.609) (see Table 00). For this Gaussian foreground case, we report a bias based on the absolute value of the sample variance on the mean for 1000 sims, which acknowledges statistical limitations exist even for closed-loop tests calibrated by MC sims.
\(r\) bias \(\times 10^4\) \(\sigma(r) \times 10^4\)
Input \(r\) = 0
02.00 0.2 6.1
02.01 -1.6 6.6
02.02 2.3 6.6
02.03 -8.0 7.0
02.04 -5.5 8.1
02.05 -4.9 13.8
02.06 -5.2 6.7
Input \(r\) = 0.003
02.00 0.3 9.3
02.01 -3.2 8.3
02.02 3.3 6.5
02.03 -10 9.1
02.04 -4.0 10.3
02.05 -0.6 14.1
02.06 1.1 9.9