PureB by Messenger Method
Michael Ray, Colin Bischoff, 2019-09-xx
This posting describes an alternate pure-B estimator applied to the 95 GHz DC 04.00 maps. It uses the "messenger method" (Elsner & Wandelt 2012) to Wiener filter the maps. We treat the E-mode part of the signal covariance as if it was noise covariance (Bunn & Wandelt 2017), so filter equation becomes:
mapwf = (SB/(SB+SE+N)) * map,
where SE,B are the E-mode and B-mode parts of the signal covariance.
For these simulations with unfiltered skies, the signal covariance is diagonal in the spherical harmonic basis. We treat the noise covariance as diagonal in the pixel basis (not strictly true because of the white + 1/ℓ noise spectrum in the data challenge maps). The messenger method iteratively solves fo the Wiener-filtered map by introducing a messenger field, t, with covariance matrix, T, that is proportional to the identity matrix, so it can be written easily in any basis. For each step of the algorithm, we transform back and forth between pixel space and spherical harmonic space, with the messenger field carrying information between the two bases.
As a first demonstration, we applied this technique to the 95 GHz DC 04.00 maps (Gaussian foregrounds, circular 3% fsky mask). The signal covariance matrices (in spherical harmonic space) are set to the standard lensed-LCDM E and B-mode spectra (purely diagonal). We did not include any foreground contribution to the signal covariance. The noise covariance is diagonal in pixel space, with values equal to XX μK2 times the inverse of the mask. For unobserved pixels, the noise covariance is set to a very large but finite value.
The messenger method includes parameter, λ, which scales the covariance of the messenger field. Convergence proceeds more quickly for large values of λ, but the final solution is obtained for λ = 1. This means that we want to implement a "cooling schedule" for efficient filtering. In our implementation, λ starts at a value of 1300, is decreased to 100 on the second iteration, and then is reduced by factor η = 0.825 on each subsequent iteration. When λ reaches 1, we iterate an additional 5 times. We have experimented fairly extensively with other cooling schedules and found that this method gives good convergence in a reasonable amount of time. (We are able to filter a single set of Q and U maps in 34 minutes on a single processor.)
Figures 1 and 2 show a map before and after filtering. The input maps is dominated by LCDM E modes. The output map contains B modes from foregrounds and lensing. The output map color scale has been zoomed in so that we can see that the pure-B filtering operation has extended the map into the unobserved region.
The algorithm takes in a set of two maps, corresponding to a Q and U measurement, and outputs a B only weiner filtered set of Q and U maps. One can visually see that although the input map has a defined edge, the algorithm extends the output map into the unobserved region. This is because the code is coming up with a full sky weiner filtered map and thus any power detected at low ell will result in map fluctuations at large angular scales. It is also visually apparent that the input Q map contains mostly E modes (fluctuations that appear straight up and down or straight across), while the output Q map looks like it's all B modes (diagonal fluctuations). This is exactly what we would hope to see from a pure-B estimator. Note that there is a scale difference between the two plots. This is because the input map contains E signal, so the Q map will have a larger amplitude in each pixel for the input map than the output map which contains only B modes.
There is still some noise bias in the output map, however we can remove that through using monte carlo methods. There is also a suppression factor involved in the output data which is the reason for the scale difference in plots. This factor is corrected for through the use of band power window functions. Both the noise debiasing and suppression factor correction are made at the power spectrum level only. We are not capable of applying noise and suppression factor corrections to maps.
Shown below is a plot of 100 spectra which are cleaned signal plus noise maps at 95 GHz taken from the 04.00 CMBS4 experiment configuration. The bins used in this analysis began at ell of 20, and went up through ell of 370 with a bin size of 35. So, bin number zero is ell of 20 - 55. Bin 9 (last bin) is ell of 335 - 370. Also included is the mean of these 100 spectra and the expectation value which was calculated using band power window functions and theory BB spectra.
As one can see, the expectation value for our estimator runs directly through the middle of our simulations and is very close to our mean. It is worth noting that there is definitely something out of the ordinary happening in our last bin. In this bin, we routinely get results that do not agree with the rest of the data. Because of this, we generally ignore results from the last bin given that we are not too worried with what is happening at that ell range anyway. Shown below are plots of BB to BB and EE to BB band power window functions for both the messenger method estimator and the S2hat estimator. Band power window functions for the messenger estimator were calculated with an input signal C_ell = 1 on the sky. This means that the actual input power for deriving band power window functions is 1 * B_ell squared, where B_ell squared contains the effect of the beam. We also ran 6 simulations at each ell with this input power and then took the mean of the output across these six simulations to get the final band power window functions. Summing the window functions across ell values gives us the total suppression factor.
Again, it is visually apparent that there is something odd going on in the last bin for the messenger method estimator.
Below are plots the variance in each bin across the 100 realizations which were filtered. The exact same simulations were used as input to the two methods compared (signal plus noise simulations at 95 GHz using the 04.00 experiment configuration).
As shown above, the messenger method beats the S2hat estimator in terms of variance per bin. We believe the main reason for this is that the S2hat estimator uses a simple weighting of 1/N to filter the maps. Since the signal plus noise simulations have a sizeable amount of BB lensing signal contained within them, the fact that we are doing a weiner filter (and thus taking into account the signal covariance) means that we would expect to get a large improvement over an estimator which only knows about noise covariance. For delensed simulations, we would expect to see less of an improvement from the messenger method over the S2hat filtering.