For many potential sources of systematics, theBicep1 three-year result relies on the analysis pre- sented in Takahashi et al. 2010. In this instrument characterization paper, it was demonstrated that BB bandpower constraints were limited by statistical uncertainty. Other potential sources of systematic uncertainty contributed spurious polarization at or below r = 0.1. For many of these potential sources of systematics, constraints on potential sources of systematics were significantly better than the r= 0.1 benchmark. We expect that these spurious polarization limits are equally applicable to the three-year analysis, since the observing pattern and low-level data processing are identical for the additional third year. We refer the reader to Takahashi et al. for a detailed treatment of the potential sources of systematics considered.
The Takahashi et al. analysis included constraints from both instrument-induced and ground- induced systematic contamination. The instrument systematics considered included relative gain mismatch, differential pointing, differential ellipticity, differential beam width, telescope pointing uncertainty, polarization angle uncertainty, far sidelobe pickup, focal plane temperature fluctuations, and optics temperature fluctuations. For the Takahashi et al. analysis, spurious polarization from relative gain mismatch and differential beam properties was estimated by injecting temperature-to- polarization leakage into time streams and carrying through the simulated data to power spectra, as described in Section 4.3.3.
From that analysis, it was demonstrated that only differential pointing and relative gain mismatch could potentially contribute polarization systematics relevant to an r = 0.1 benchmark. For this reason, our efforts in the three-year analysis focus on constraining spurious polarization from relative gain mismatch and differential pointing.
5.2.1 Application of the deprojection algorithm to the Bicep1 three-year analysis
The deprojection algorithm described in Section 4.4 has been successfully employed in theBicep1 three-year analysis to improve constraints on potential systematics from relative gain mismatch and differential pointing. In the two-year analysis, it was noted that uncertainty in the method used to estimate relative gains could potentially contribute systematic contamination at a level comparable to the stated instrument benchmark ofr= 0.1(Takahashi et al. 2010).
In the three-year analysis, this is a greater concern in part because of the additional sensitivity, but also because of the addition of the so-called “slow-⌧” channels. These detectors, amounting to four channel pairs at 150 GHz and two channel pairs at 100 GHz for the 2007 and 2008 seasons, were excluded because of aberrant detector constants. For this reason, we expect the relative gain mismatch in these channel pairs to potentially be larger than average.
5.2.2 Estimating false polarization from relative gain mismatch prior to template deprojection
The level of potential systematics from relative gain mismatch before and after deprojection is estimated in a series of analysis steps. First, the level of relative gain mismatch is “measured” in the real data by regressing a temperature template from the WMAP seven-year V-band map, as described in Section 4.4.1. A regression coefficient is recovered for each channel pair, and regarded as an estimate of the true relative gain mismatch of the pair. We find that 9 of 20 detector pairs at 150 GHz and 9 of 23 at 100 GHz exceed the benchmark for relative gain mismatch quoted in Takahashi et al. 2010.
These per-channel-pair coefficients are then used as inputs to simulations containing E-mode, but no B-mode polarization. The resulting B-mode polarization in these signal-only simulations contains some combination of E/B mixing from imperfectE/B separation, and spurious B-mode polarization from relative gain mismatch. By generating equivalent simulations containing ideally matched relative gains, it is possible to separate the spurious polarization from relative gain mis- match.
We find that in the three-year data selection, including the “slow-⌧” channels, the temperature- to-polarization leakage from relative gain mismatch is predicted to be significant (Figure 5.1). At 150 GHz, excess bandpowers are predicted to exceed ther= 0.1theory spectrum across the`-range of interest. We find that the contamination is predicted to be potentially more significant at 150 GHz than at 100 GHz.
There is a caveat that accompanies this estimate. Because the real data contain noise, the regression coefficients are noisy estimates of the “true” value of the relative gain mismatch. As a
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result, there is some uncertainty in the mean level of spurious polarization predicted by this analysis (prior to deprojection). In general, adding noise to the estimate of these coefficients will tend to over-estimate the contamination, since both negative and positive regression coefficients will lead to positive false BB bandpowers. By estimating regression coefficients on noise-only data, we can estimate the amount by which the bandpowers are over-estimated. We can then use these “noise only”
coefficients as inputs to forward simulations, from which we can calculate excessBB bandpowers.
We find that the excess bandpower estimated from these coefficients is 4% in amplitude, relative to the coefficients estimated from the real data. We therefore conclude that the over-estimation of the falseBB bandpowers due to noisy estimates of the regression coefficients is a small effect.
5.2.3 Estimating false B-mode bandpower from relative gain mismatch after template deprojection
Next, we estimate the level of spurious polarization after “deprojecting” relative gain mismatch. In this context, “deprojecting” means subtracting the template scaled by our best estimate of↵rg from the time series (details can be found in Section 4.4). To estimate the residual level of false BB bandpowers after deprojection, we follow the same steps as in the previous section. We take the regression coefficients estimated from the real data as inputs to forward simulations. Temperature leakage is injected into the pair-difference time series. After removing the leakage via the deprojection algorithm, we generate BB bandpowers. We then de-bias E/B mixing in the usual manner. We find that the post-deprojection falseBB bandpower is over four orders of magnitude lower than the no-deprojection bandpowers (calculated in the previous section) between 37.5 < ` <200 (Figure 5.1). We thus conclude that the deprojection algorithm is extremely efficient at suppressing false B-mode polarization from relative gain mismatch.
We repeat this exercise using simulations containing no E- orB-mode polarization, effectively creating a closed-loop test of the algorithm. In this case, we find that the false polarization from simulated relative gain mismatch is removed perfectly (to within machine precision).
In practice, the dominant residual leakage from relative gain mismatch after deprojection is the result of noise in the external map used to construct the template (in this case, the WMAP seven-year maps). This noise is not accounted for in 5.1 because the deprojection of the simulated data has been performed with a noiseless template. Performing similar simulations in which noise has been injected into the regression template, we find that the suppression of excess bandpower due to spurious polarization is still suppressed by over two orders of magnitude, below equivalent bandpowers for r = 0.01 between 30 < ` < 100. This residual can be further suppressed in the future by using higher signal-to-noise temperature maps from Planck.
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Figure 5.1: The mean relative gain mismatch-induced bandpower, before and after template sub- traction, as inferred from signal-only simulations. Dashed lines represent the statistical uncertainty due to noise in theBicep1three-year spectra, as inferred from noise-only simulations. Circles repre- sent the mean excessBBbandpower from simulations, assuming relative gain mismatch coefficients inferred from the real data prior to template deprojection. There is a small (<5%) noise bias on these estimates due to noise in the coefficients, as described in Section 5.2.2. We find that relative gain mismatch contributes significant bandpower relative tor= 0.1 (represented by the theory line in black) with no template subtraction. The bandpowers marked with ‘⇥’ indicate the mean resid- ual BB bandpower after template deprojection. After implementing the algorithm, the false BB contribution in our`-range of interest is suppressed by roughly four orders of magnitude.
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5.2.4 False B-mode contamination from differential pointing
We repeat the above exercises, but now considering temperature-to-polarization leakage from dif- ferential pointing. Differential pointing templates for x and y displacement are regressed against the real data, spanning all three years ofBicep1observations. The recovered coefficients are then translated to equivalent pointing centers for A and B and used as inputs to forward simulations.
Differential pointing leakage is simulated by simply sampling off of the noiseless signal Healpix maps at two different assumed pointing centers for A and B. The resulting pair-difference time series contains the signal from the simulated sky polarization (containingE- but noB-mode power) as well as temperature-to-polarization leakage. The time series is carried forward to power spectrum estimation using the usual steps, includingE/B mixing removal and suppression factor correction.
The result is plotted in Figure 5.2. We find the excess bandpower for differential pointing to be much less significant than relative gain mismatch for the bandpowers that contribute the most constraining power on r. At 150 GHz, excessBB power is below the r= 0.1 theory curve for the first three bins (37.5 < ` <107.5), which contain over 95% of Bicep1’s constraining power on r.
By directly comparing maximum likelihood values of rderived from 100 realizations of signal-only simulations with and without differential pointing leakage, we have confirmed that the resulting potential bias onris substantially less thanr= 0.1, and not significant, givenBicep1’s statistical uncertainty due to noise. As a result, we find that it is not necessary to actually deproject differential pointing leakage for theBicep1three-year analysis.
This may seem in slight tension with the Takahashi et al. analysis, which stated that differential pointing could potentially contribute systematics at a level relevant to an r = 0.1 benchmark.
However, the benchmark reported in Takahashi et al. is pessimistic; the benchmark was drawn from comparingBBbandpowers directly to anr= 0.1theory curve, and did not account for the relevant weighting between bandpowers. Differential pointing contributes more significantly with increasing
`, the same scales at which the bandpower weights are decreasing due to the beam window function.
Although not strictly necessary for the Bicep1three-year analysis, we demonstrate the ability of the deprojection algorithm to suppress leakage from differential pointing. We proceed in much the same way as before: differential pointing leakage is injected into forward simulations, but is now removed by deprojecting the leakage templates as constructed in Section 4.4.2. The scaled templates are then subtracted from the pair-difference TODs, which are carried forward to power spectra (accounting forE/Bmixing and bandpower suppression as before).
The result is plotted in Figure 5.2. The mean residual leakage across 100 realizations is plotted both before and after the differential pointing deprojection has been implemented. We find that the algorithm effectively suppresses simulated leakage by over two orders of magnitude in most `bins, well below ther= 0.1 curve.
The algorithm is currently limited by two effects: The first is a non-ideality in the forward
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Figure 5.2: The mean excessBBbandpower of signal-only simulations containing differential point- ing, both before and after template subtraction. Dashed lines represent the total statistical un- certainty due to noise. The mean excessBB polarization from simulations containing leakage from as-measured differential pointing coefficients before template subtraction is indicated with ‘ .’ Band- powers marked with ‘⇥’ show the mean excess bandpower after template subtraction. We find significant reduction in the potential false BB bandpower. However, owing to the fact that the contamination is primarily at high`, we find that template subtraction is not necessary to achieve theBicep1benchmark ofr= 0.1.
simulation. Detector data streams are simulated by interpolating offof relatively coarsenside= 512 Healpix maps. The interpolation algorithm suffers boundary-crossing discontinuities at certain locations where the interpolation point switches from one nearest-neighbor pixel to another. When differential pointing is simulated, detector centroids occasionally lie on either side of a pixel boundary.
When pair-differenced, the simulated data streams contain a non-physical discontinuity. The second effect limiting the algorithm is noise in the template, as in the case of relative gain mismatch. In this case, however, high-`noise from the beam rolloffis more significant (because the spatial derivative is being calculated). As in the case of relative gain mismatch, this can be improved in future analyses with both higher signal-to-noise and higher resolution measurements.
5.2.5 Bandpower suppression and loss of information
A natural consequence of the deprojection algorithm is increased bandpower variance due to mode removal. The deprojection algorithm effectively discards unique modes in the data corresponding to the leaked modes predicted by the template. As additional degrees of freedom are introduced
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(either from introducing additional potential sources of contamination or regressing over shorter timescales), we discard a larger number of modes, thereby increasing the bandpower variance. Here we explore the loss of information from the implementation of this algorithm. Since the differential pointing deprojection was not actually implemented in theBicep1three-year analysis, we consider only relative gain mismatch.
Bandpower suppression is assessed by comparing two suites of signal-only simulations, containing cosmologicalE- but noB-mode power (and neither containing simulated systematics). In the first, bandpowers are estimated in the usual manner, but with no deprojection. In the second, relative gain mismatch deprojection is applied to the pair-difference data. By comparing the ratio of the raw bandpowers, we can assess the fractional increase in the bandpower suppression. We find that the increase is small. At both 100 and 150 GHz, the additional bandpower suppression is less than 4% (Figure 5.3). The bandpower suppression from the deprojection algorithm is subdominant to polynomial filtering and ground subtraction (at low `), and the beam window function (at high `) as can be seen in Figure 4.2.
To assess the additional loss of information, we again generate two suites of simulations, this time containing noise only. One of the noise simulations is subjected to relative gain mismatch deprojection, while the other is not. By comparing the ratio of the standard deviation at each`bin, we find that the additional loss of information is also small. The fractional increase in the error bars is less than 2% across our`range of interest (Figure 5.3).