# Bias on r from additive systematics

*Steve Palladino, Justin Willmert, Colin Bischoff -- 2017-08-31*

In this post, we compare estimates of *r* between data challenges 02.00 and 03.00. These two sets of simulations contain identical noise, CMB signal (lensed-ΛCDM with r=0.003 for even realizations only), and Gaussian foregrounds. The only difference between the two sets is the inclusion of additive systematics for experiment definition 03. For a description of the additive systematics, see **2017 July 10: Additive systematics for data challenge 03** and the table in the **Experiment Definition 03, 03b, 03c** section of the **Experiment Definitions** page. As shown in that table, the type of additive systematic (white or 1/ℓ spectrum, correlated or uncorrelated across frequencies) is different for each block of 125 realizations. The first 125 realizations have no systematic, so they should be identical between 02.00 and 03.00. For the other seven blocks of realizations, the amplitude of the systematics were all chosen to correspond to a bias on *r* of 1e-4.

The simulated maps were processed through to BB bandpowers by Justin as described in **2017 Aug 18: BK-style power spectra for 1000 realizations of v03.00,.03 CMB-S4 simulation maps**. Next, the bandpowers were plugged into a quadratic estimator for *r* that is constructed to have zero response to dust and synchrotron foregrounds. Steve calculated this estimator using a bandpower covariance matrix constructed from the 02.00 signal and noise sims, so it doesn't know anything about the additive systematics.

The figure below shows the difference in the *r* estimate between 02.00 and 03.00 for the various delensing levels. For realizations 0–124, the difference is identically zero as expected. For realizations 125–999, we see a small bias on *r* that is consistent with the 1e-4 target. The results are much noisier at higher values of A_L because of random correlations between the systematics maps and CMB realizations. Red points are for odd realizations with *r*=0 and blue points are for even realizations with *r*=0.003, but there is little difference.

The table below shows the average of each set of data points, split according to type of systematic, level of delensing, and tensor signal. Results seem consistent across each row of the table, so the magnitude of the bias doesn't depend on the lensing and tensor signal levels. Steve's estimator was used to select the benchmark systematics levels, so it is not surprising that we recover biases of the expected magnitude, but this is still a useful closed-loop test. If we look at the A_L=0.03 column, which has the least noise, then we do see some variations. Correlated noise with a white spectrum seems to produce larger biases (rlz 375–499 and 750–874) while uncorrelated noise with a 1/ℓ spectrum produces smaller biases (rlz 250–374 and 625–749).

rlz | systematic | A_L=0.03 | A_L=0.1 | A_L=0.3 | A_L=1.0 |
---|---|---|---|---|---|

000–124 | none | 0, 0 | 0, 0 | 0, 0 | 0, 0 |

125–249 | uncorrelated, white spectrum | 0.99e-4, 1.07e-4 | 0.85e-4, 1.00e-4 | 0.66e-4, 0.91e-4 | 0.41e-4, 0.87e-4 |

250–374 | uncorrelated, 1/ℓ spectrum | 0.68e-4, 0.73e-4 | 0.75e-4, 0.75e-4 | 0.87e-4, 0.73e-4 | 1.01e-4, 0.66e-4 |

375–499 | correlated, white spectrum | 1.53e-4, 1.47e-4 | 1.35e-4, 1.35e-4 | 1.11e-4, 1.20e-4 | 0.87e-4, 1.10e-4 |

500–624 | correlated, 1/ℓ spectrum | 1.01e-4, 0.86e-4 | 1.09e-4, 0.85e-4 | 1.20e-4, 0.85e-4 | 1.35e-4, 0.86e-4 |

625–749 | uncorrelated, white + 1/ℓ spectrum | 0.74e-4, 0.90e-4 | 0.71e-4, 0.87e-4 | 0.68e-4, 0.83e-4 | 0.64e-4, 0.79e-4 |

750–874 | correlated, white + 1/ℓ spectrum | 1.39e-4, 1.26e-4 | 1.37e-4, 1.21e-4 | 1.31e-4, 1.18e-4 | 1.25e-4, 1.21e-4 |

875–999 | uncorrelated + correlated with white + 1/ℓ spectrum | 1.05e-4, 1.05e-4 | 1.00e-4, 1.07e-4 | 0.91e-4, 1.14e-4 | 0.78e-4, 1.29e-4 |