View source for Additive systematics for data challenge 03

''Colin Bischoff, Steve Palladino, Victor Buza, John Kovac -- 2017-07-10''

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This post follows up on [http://bicep.rc.fas.harvard.edu/CMB-S4/analysis_logbook/20170525_s4syst/ Victor's 2017-06-09 post] and provides a recipe for including the additive systematics in [[Experiment_Definitions|Experiment Definition 03]].

We consider two classes of additive systematics: 
* Uncorrelated systematic, which biases the auto-spectra but not cross-spectra between maps at different frequencies.
* Correlated systematic with common amplitude across all frequencies, i.e. looks like a CMB signal

For both classes of systematics, we write the power spectrum of the additive systematic as white plus 1/ell components. There are two free parameters: amplitude of the white component, '''A'''; and amplitude of 1/ell component, '''B'''. We use a quadratic estimator for ''r'' that is constructed to be insensitive to foregrounds (developed by Steve Palladino) and calculate the values of '''A''' and '''B''' that yield a bias on ''r'' equal to 1e-4. Note that the bias on ''r'' is linear in these parameters, so it is easy to adjust the systematic level to achieve a different spec. Setting both '''A''' and '''B''' to the quoted values would bias ''r'' by 2e-4. The quadratic estimator used here was cross-checked against Victor's maximum likelihood results and was found to yield very similar results for bias and &sigma;(''r''), but other analysis methods might yield different results.

One caveat: The additive systematic is defined below for nine frequency bands, including the 20 GHz band that was added in experiment config 02. However, the quadratic estimator used for the bias calculation was derived for experiment config 01 and does not use 20 GHz. I doubt this makes much difference on the results.

== Uncorrelated systematic ==

For the uncorrelated systematic, we add power independently to each frequency map that is scaled by the noise power in the map, using the [http://bicep.rc.fas.harvard.edu/CMB-S4/analysis_logbook/20170515_chkS4/ noise levels from Experiment Definition 02]. For the 1/ell component, we also use the BB ell knee and slope values from the table in that post. This systematic is &ldquo;noise-like&rdquo;, so the power in the map does not roll off at high ell due to beam smoothing. The expression for the auto-spectrum of the systematic is:

<strong style="font-size: large;">C<sub>&#x2113;</sub> = N [A + B (&#x2113; / &#x2113;<sub>knee</sub>)<sup>&gamma;</sup>]</strong>

The parameters, <strong>N, &#x2113;<sub>knee</sub>, &gamma;</strong>, used at each frequency are copied here for easy reference.

{| class="wikitable"
! frequency !! map noise [&mu;K-arcmin] !! N [&mu;K<sup>2</sup>] !! &#x2113;<sub>knee</sub> !! slope, &gamma;
|-
| 20 GHz || 14.69 || 1.83e-5 || 50 || -2
|-
| 30 GHz || 9.36 || 7.41e-6 || 50 || -2
|-
| 40 GHz || 8.88 || 6.67e-6 || 50 || -2
|-
| 85 GHz || 1.77 || 2.65e-7 || 50 || -2
|- 
| 95 GHz || 1.40 || 1.66e-7 || 50 || -2
|-
| 145 GHz || 2.19 || 4.06e-7 || 60 || -3
|-
| 155 GHz || 2.19 || 4.06e-7 || 60 || -3
|-
| 220 GHz || 5.61 || 2.66e-6 || 60 || -3
|- 
| 270 GHz || 7.65 || 4.95e-6 || 60 || -3
|}

Since we are scaling the systematic power spectrum from the noise power definition, the units of parameters A and B are "fraction of noise power" for the white and 1/ell components, respectively. We find bias corresponding to ''r'' = 1e-4 for a systematic with white power spectrum amplitude <strong>A = 0.0328</strong> (3.3% of the white noise) or 1/ell power spectrum amplitude <strong>B = 0.0689</strong> (6.9% of the 1/ell noise).

== Correlated systematic ==

For the correlated systematic, we add a common signal at the same amplitude to all frequency maps. Since this signal is common, it affects both auto-spectra and cross-spectra. As before, this systematic has a white component with amplitude <strong>A</strong> and a 1/ell component with amplitude <strong>B</strong>. In place of the per-frequency noise power levels, we define the overall level of this systematic relative to <strong>N<sub>comb</sub> = 1.26e-7 &mu;K<sup>2</sup></strong>, which is the inverse quadrature sum of the per-frequency noise power levels. We used a common <strong>&#x2113;<sub>knee</sub> = 50</strong> and <strong>&gamma; = -2</strong>, as shown in the table below. Since this systematic is &ldquo;CMB signal-like&rdquo;, we calculated the bias assuming that it has been smoothed in each map according to the beamsize of that frequency.

<strong style="font-size: large;">C<sub>&#x2113;,ij</sub> = N<sub>comb</sub> [A + B (&#x2113; / &#x2113;<sub>knee</sub>)<sup>&gamma;</sup>] B<sub>&#x2113;,i</sub> B<sub>&#x2113;,j</sub></strong>

{| class="wikitable"
|-
! frequency !! N<sub>comb</sub> [&mu;K<sup>2</sup>] !! &#x2113;<sub>knee</sub> !! slope, &gamma; !! beam FWHM [arcmin]
|-
| 20 GHz 
| rowspan="9"| 1.26e-7 
| rowspan="9" | 50 
| rowspan="9" | -2 
| 76.6
|-
| 30 GHz || 76.6
|-
| 40 GHz || 57.5
|-
| 85 GHz || 27.0
|-
| 95 GHz || 24.2
|-
| 145 GHz || 15.9
|-
| 155 GHz || 14.8
|-
| 220 GHz || 10.7
|-
| 270 GHz || 8.5
|}

The amplitudes that correspond to bias of ''r'' = 1e-4 are <strong>A = 0.0584</strong> (5.8%) for a correlated systematic with white spectrum or <strong>B = 0.1049</strong> (10.5%) for a correlated systematic with 1/ell spectrum. It may be surprising that these amplitudes appear higher than the limits we found for the correlated case, but remember that it is scaled by <strong>N<sub>comb</sub></strong>, which is a very small noise power level.