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0 50 100 150 200 250 300 350
0 0.2 0.4 0.6 0.8 1
Multipole`
SuppressionFactor
T deprojection P3+AZ-fixed filter B`2150 GHz Total
Figure 4.2: Bandpower filter function forEEat 150 GHz, calculated from the suppression factor for theBicep1three-year analysis. Low-`attenuation is dominated by the P3 polynomial filtering and AZ-fixed filter, whereas the high-` modes are dominated by the beam roll-off (represented by the square of the beam window function,B2`). TheT deprojection represents the bandpower suppression resulting from the instrumental polarization regression analysis described in Section 4.4. TheT T andBB filter functions are very similar. The first bin, near`= 10, is not used for science analysis.
the inverse covariance matrix. The weights are then re-normalized such that the sum across all frequencies within a single`bin is unity. These weights are then used to calculate weighted averages for each` bin.
recent analyses from WMAP and Planck). A large number of realizations ofT,Q, andU are gener- ated, each with a unique distribution ofa`m’s (499 forBicep1). The synfastmaps are smoothed to the focal plane median beam width. Additionally, the first and second spatial derivatives are calculated using thesynfastpackage. ForBicep1, all of the maps are calculated at a resolution of
nside= 512. Future Bicep2analyses will likely take advantage of higher resolution maps.
Simulated TODs are generated by interpolating thesynfastmaps using the pointing data from the telescope. At each moment in time, each detector’s pointing center on the sky is calculated as a reckoning along a bearing angle✓and an angular distancerfrom the boresight, as defined in Figure 3.19. (The boresight pointing reconstruction is summarized in Section 3.7). Next, the algorithm identifies theHealpixpixel center nearest to the detector’s pointing center. The value of the map at the true pointing center is then approximated using a Taylor expansion from the nearestHealpix center using the first and second spatial derivative maps. This interpolation procedure is used for calculatingT, QandU all in the same manner. Simulations can span the entire observing period of the telescope, or just some subset of data.
The simulated TODs for A and B are assumed to be perfectly matched in gain (unless we explicitly inject mismatched gains), thereby obviating the el-nod relative gain correction step. The TODs, once generated, are carried forward in the analysis in the exact same way as the real data.
This includes using the same filtering and weighting procedure. Similarly, the same cuts are applied to the simulated data as the real data. Co-added maps and power spectra are calculated as in Sections 4.1.3 and 4.2, respectively.
Simulated maps containing E-mode power but no B-mode power (E-no-B sims) are used to calculate the E-to-B leakage from the power spectrum estimator (as in Section 4.2.3) and the suppression ofEE bandpowers (as in Section 4.2.4). Similarly,B-no-E sims are used to calculate the suppression ofBB bandpowers. This simulation set is also used in later stages of the analysis to calculate likelihood estimators forr(described in Section 4.6).
4.3.2 Simulated noise generation
The goal of the noise simulator is to produce TODs that, in both pair-sum and pair-difference, have the same statistical properties as the real data. The noise model is based on the simulator reported in Pryke et al. 2008, but has been somewhat modified for theBicep2andBicep1analyses.
The noise model is constructed in two different steps: The first step measures and then simulates the low-frequency polynomial modes, while the second step measures and subsequently simulates high-frequency Fourier modes. To begin, all scans within a scanset are concatenated and filtered with a simple first-order polynomial filter. From the polynomial coefficients (the mean and the slope), a joint covariance is calculated between all channels and modes, thereby preserving correlations between channels. The joint covariance then undergoes a Cholesky decomposition. This matrix
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is used to generate a set of randomized polynomial coefficients that share the same covariances as the data-derived polynomial modes. In the end, this is an overly complicated way of simply adding a randomized mean and slope to the scanset that is polynomial-filtered away before pairmap generation anyway. As a result, the noise realizations are insensitive to the re-injection of the low-order polynomial modes.
The high-frequency modes are generated in much the same way, but are decomposed into Fourier modes, rather than polynomial modes. The data are again concatenated across half-scans within a scanset, and then filtered to remove the polynomial modes captured by the first part of the noise simulation. Next, the data are Fourier transformed, and then binned into logarithmically-spaced frequency bins. In this way, the noise simulator preserves correlations between channels, but treats each Fourier mode as independent. The final steps proceed exactly as in the case of the low-frequency modes: The Cholesky decomposition is calculated and used to generate random Fourier coefficients that have the same joint covariance as the real data. By performing an inverse discrete Fourier transform, the randomized coefficients are converted back into the time domain, creating TODs that are carried forward for later analysis.
An example of a randomly selectedBicep1phase is plotted in Figure 4.3. The simulated noise power spectra is plotted against the real data. The noise model relies on the assumption that in any given half-scan, the detector time series is noise-dominated. This is a safe assumption: atmospheric fluctuations are typically of order0.1 1K, whereas temperature anisotropies in the CMB are4 5 orders of magnitude lower in amplitude.
4.3.3 Spurious polarization generation
In addition to generating noise and signal simulations, the pipeline can also generate simulated spurious polarization from specific classes of instrument systematics. This aspect of the pipeline was expanded for theBicep2analysis to include a wider variety of potential sources of false polarization.
In the simplest case, the pipeline code can be used to generate spurious polarization that results from mismatched detector gains. Relative gain mismatch induced leakage is injected into the TODs by simply artificially adjusting the relative gain corrections forAandB such that the temperature signal does not exactly vanish in the pair difference. The simulator can either set a randomized relative gain correction for each scanset (with some specified distribution), or inject a uniform mismatch in the simulated data over the entire timeframe of the simulation.
Similarly, differential pointing can be simulated by displacing the assumed pointing centers for Aand B from the common centroid. When differenced, the unpolarized signal measured from two nearby points forAandBwill not difference away perfectly, resulting in temperature-to-polarization leakage. This can be simulated as a fixed displacement betweenAandB, or a randomized displace- ment that varies with time.
10 1 100 10 6
10 5 10 4
V/p Hz
Pair sum, real Pair sum, sim
10 1 100
10 6 10 5 10 4
Frequency [Hz]
V/p Hz
Pair difference, real Pair difference, sim
Figure 4.3: Example PSD of Bicep1 real data (black) and simulated noise (dashed red) for a randomly selected detector pair and phase. For clarity, the plotted PSD is the average across all per-half-scan spectra within a single phase. This averaging reduces sample variance, thereby making a visual comparison of the power spectra more clear.
Spurious polarization from instrument mis-calibration, such as polarization orientation uncer- tainty, can also be simulated by varying the assumed instrument properties. When carried through to angular power spectra, potential levels of spurious polarization can be assessed.
In the simulation pipeline reported in Takahashi et al. 2010, differential beam width and differ- ential ellipticity were simulated by performing various beam convolutions on a flat-sky projection of the simulated curved-skyHealpixmap. Signal-only TODs were then calculated by interpolating these flat-sky maps at locations corresponding to the detector pointing centers, as in Section 4.3.1.
In the three-year analysis, as well as in theBicep2analysis, the interpolation algorithm was changed to sample theHealpixmap directly, thereby skipping the intermediate flat-sky projected map. This modification required a change to the simulation procedure used to calculate differential beam width and differential ellipticity.
The Bicep2 simulation pipeline has been expanded to approximate leakage from a number of
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potential sources of instrumental polarization on the curved sky, including differential ellipticity and differential beam width. These are in active development, so a finalized description of the method is not yet possible. We will, however, give a general description of the procedure. In all cases, spurious polarization is calculated by taking a linear combination of external template maps, taken as curved-sky Healpix temperature maps, and injecting some fraction of that linear combination into the TODs. Simulations of differential beam width, for instance, use a combination ofHealpix maps smoothed to different effective beam widths. The signal inA and B is calculated by taking different linear combinations of the pre-smoothed maps to simulate the signal appropriate for the beam widths for Aand B, respectively. Ellipticity is simulated in much the same way, using maps that are both smoothed to some nominal beam width and also symmetrically displaced about the common centroid. Multiple circular Gaussians with displaced centroids are used to approximate an elliptical beam. This additional capability of the Bicep2 pipeline will be required to accurately assess potential sources of systematics resulting from beam mismatch.