Bandpower estimation proceeds from calibrated maps in several steps: First, E and B raw power spectra are calculated from the two-dimensional fast Fourier transform (FFT) ofQandU(denotedQ˜ andU˜) and collapsed into 1-D angular power spectra as a function of multipole`. Next, the power
spectra are de-biased, for both noise and for E-to-B leakage from the pipeline’s imperfect E/B separation. Finally, bandpower window functions and suppression factor corrections are calculated for each of the spectra. We step through this procedure in detail.
4.2.1 Power spectra calculation
With the T, Q, and U maps in-hand, we proceed with angular power spectra generation. The maps are masked with a variance map, which is constructed from the time series variance. Next, two-dimensional FFTs of Q and U are calculated from co-added maps. In this step, a flat-sky approximation is made. We treat the RA/DEC-binned map as a set of rectilinear coordinates, taking the Fourier transform in thex, yplane. The choice to use a flat sky approximation is driven largely by convenience. Power spectrum estimation in the flat-sky limit is extremely simple, and does not require the complicated mathematics of spherical geometry. The flat sky approximation comes at minimal cost, since the resulting E/B mixing can be subtracted as a simple bias with additional variance that is small compared to the uncertainty due to noise.
TheE-mode andB-mode map transforms are calculated in the Fourier domain. Letting equal the angular position in theu, vplane (measured from North-going modes towards East-going modes, as per the International Astronomical Union convention), we calculate the Fourier transform of the E andB maps as:
E˜= ˜Qcos(2 ) + ˜Usin(2 ) (4.8)
B˜ = ˜Qsin(2 ) U˜cos(2 ). (4.9)
A heuristic description of the origin of these expressions can be found in Section 1.5.3.
After calculatingT ,˜ E˜ andB, we collapse the 2-D FFTs into 1-D auto- and cross-spectra. Auto-˜ spectra are generated by multiplying the 2-D FFT by its complex conjugate (calculated as XX†, where X may be T, E, or B) and then summing modes over annular bins centered at (0,0) in theu, v plane. This is effectively collapsing over m, so the resulting function depends only on the multipole moment,`. Cross-spectra are calculated the same way, summing modes in annular bins of XY†. HereX andY represent possible combinations of T, E, and B. Also, as per the usual CMB conventions, we multiply by a factor of `(`+ 1)/2⇡to arrive at the uncalibrated power spectra:
CbXY = `(`+ 1)
2⇡ CbXY. (4.10)
Theb subscript indicates that these are binned power spectra (to be distinguished from C`, which denotes the underlying theory spectrum).
This procedure for calculating the E andB power spectra results in imperfect E/B separation
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for two reasons. First, the finite sky coverage results in discretely sampled bins in the u, v plane.
Because of this pixelization, modes that would otherwise be pureE “bleed” intoB, and vice-versa3. Second, map distortion from the flat sky approximation produces some low-level mixing ofE andB.
As mentioned previously, this can be accounted for as a simple bias with tolerable additional sample variance. There are algorithmic solutions to improveE/B separation in pseudo-C` estimators; one such solution is described in Smith 2006 and may be included in future analyses ofBicep2data.
4.2.2 Noise de-bias
Auto-spectra naturally suffer from a noise bias. In order to present meaningful bandpower estimates, we must account for this bias. The correction of the bias is obtained from the simulation pipeline, which is described in detail in Section 4.3. We generate many realizations of simulated noise, which share the same statistical properties as the noise in the real data. We take the average over all realizations to be the noise bias, denoted as N`,b (calculated separately for T T, T E, etc.). This de-biasing requires a highly accurate noise model; over- or under-estimating the noise bias will lead directly to systematic uncertainty on bandpower estimates.
4.2.3 E-to-B de-biasing
As described in Section 4.2, the limited sky coverage and the flat-sky approximation invoked by the power spectrum estimator lead to imperfect E-to-B separation. This can be described as an E-to-B “bias” in the sense that signal-only simulations contain non-zeroCBBb power for zero input BB power. This is not, however, a systematic bias on theBBbandpowers, since it can be precisely accounted for with simulations. This is to be distinguished from sources of systematic bias onBB fromE-to-B mixing from instrument systematics, such as polarization angle uncertainty.
To account for imperfect E/B separation, we generate a suite of simulations containing cos- mological E-mode power, but no B-mode power (called “E-no-B” sims). Each simulation is an independent realization of the same underlying theory spectrum,C`. (The generation of these sims is described in more detail in Section 4.3). From the noiseless maps, we generate simulated TODs that are processed into maps and power spectra using the same analysis steps that are used for the real data. With the rawBB bandpowers in hand, the mean over all realizations is taken as theE- to-B bias. This is subsequently subtracted from the real data, thereby removing theE-to-B mixing from the imperfectE/Bseparation. In principle, this should be performed in the opposite direction as well: There is non-zeroB-to-E leakage from the power spectrum estimation technique as well.
However, because the cosmological BB power spectrum is known to be of much lower amplitude than theEEspectrum, we ignore this effect.
3One can imagine the extreme case where the Fourier plane has only four bins. In this case, any power inU˜ can only produceEpower.
This procedure does not account for sources ofE-to-Bleakage, resulting from instrument calibra- tion uncertainty. Similarly, this procedure does not force the observedEBspectrum to zero. While standard cosmological models predict zero E/B correlation, there are more “exotic” cosmological models, such as cosmic birefringence, which predict a non-zeroEBpower spectrum. Since the input simulations used in theE-to-B de-biasing procedure assume zeroB-mode power, any observedEB correlation in the real data will be preserved.
ImperfectE/Bseparation also results in increased statistical uncertainty on the finalBB band- powers. Since there is statistical uncertainty on EE due to cosmic variance, E-modes that leak into the BB spectrum will contribute statistical uncertainty to the BB estimates. For Bicep1, the additional statistical uncertainty from the imperfectE/B separation is a small fraction of the uncertainty due to noise. However, withBicep2’s additional sensitivity, improvedEandB estima- tion algorithms are motivated by the non-negligible statistical uncertainty inBB resulting from the Bicep1-styleE/Bestimator. These are currently being explored forBicep2.
4.2.4 Suppression factor and bandpower window functions
To calculate final, calibrated bandpowers, we must account for filtering and the bandpower window functions. Filtering, from both the data processing and the beam, is taken into account as a single suppression factor at each ` bin, calculated as the ratio of the mean of the simulated bandpowers to the input model bandpowers. The bandpower window function describes how power at different angular scales contributes to each ` bin b. We use a similar formal definition of the bandpower window function as in Knox 19994:
hCbi=⌃`W`,bC`. (4.11)
Here, hCbi is the expectation value for the observed bandpowers, W`,b is the bandpower window function, and C` is the input theory spectrum for some fiducial model. We briefly describe the procedure to calculate both the bandpower window function and the suppression factor.
The bandpower window function calculation begins with the mask window function. The mask window function accounts for correlation between`bins due to the limited sky coverage. The mask window function is calculated as in Pryke et al. 2008. The Fourier transform of the map mask is convolved with an annulus in the 2-D Fourier domain, where the annulus is defined by the chosen bin width used for the final power spectra calculation.
The mask window functionM`,b, which is computed strictly from the mask and the bin definitions, does not depend on the choice of filtering of the TODs. Because the filtering varies across a bin, not all modes contribute equally within the bin (as is assumed during the mask window function
4The thing we call the bandpower window function,W`,b, is actually different from the definition of the bandpower window functionW`,bdefined by Knox 1999. The two are related asW`,b=W`,b/`.
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0 0.01 0.02 0.03 0.04
TE
0 0.01 0.02 0.03
0.04 —150 GHz
- -100 GHz
TT
0 50 100 150 200 250 300 350
0 0.01 0.02 0.03 0.04
Multipole`
EE,BB,EB
Figure 4.1: Bandpower window functions for the 9 bins used in science analysis in the Bicep1 three-year analysis. The window function for each bin has been normalized to unity.
calculation). There is thus an additional step to compute the “true” bandpower window function from the mask window function. This calculation is performed using the following iterative procedure. To begin, the mask window function is computed to high resolution, typically `= 1. Next, a “naive”
suppression factorSb0is calculated for each individual binb as:
Sb0= hCsim`,bi
⌃`C`M`,b (4.12)
(the ‘0’ superscript will be incremented with subsequent iterations to our estimate of the suppression factor). The numerator is the binned signal-only simulated bandpowers Csim`,b averaged over all realizations for bin b. The denominator is the sum of the input model spectrum (denoted as C`) multiplied by the mask window functionM`,bfor binb, summed over all`. We call the denominator the “expected value.” (Equation 4.12 assumes that the mask window function has been normalized such that⌃`M`,b= 1).
The “naive” suppression factor, calculated at each binb, is then interpolated via a piecewise cubic Hermite polynomial interpolation. The mask window function is multiplied by the interpolated suppression factor (denoted S`0) and re-normalized. In this way, we account for the fact that the filtering is changing across each bin. We then iterate the procedure and recalculate the suppression
factor. For thej-th iteration, the suppression factor is calculated as:
Sbj+1= hC`,bsimi⌃`S`j
⌃`C`M`,bS`j. (4.13)
The denominator can be regarded as the iterated expectation value, that is, the modified expectation value that takes the changing suppression factor into account. The summation⌃`Sj`in the numerator is required to not “double count” the suppression of bandpowers. The procedure is iterated until the suppression factor is stationary with subsequent iterations. In practice, only one or two iterations are required; however, we will denote the final suppression factor asSbn.
Using the final iterated suppression factor, we can calculate the “true” bandpower window func- tion as:
W`,b= M`,b⌃`S`n
⌃`S`n . (4.14)
Again, we assume ⌃`M`,b = 1. Normalizing by ⌃`S`n ensures that the true bandpower window function is power preserving. The suppression factor for each bin is then calculated as:
Sb= hC`,bsimi
⌃`C`W`,b. (4.15)
At low`, the suppression factor is dominated by the polynomial filtering and ground subtraction described in Section 4.1.3. At high`, the suppression factor is dominated by the beam roll-off.
As a practical twist on this procedure, some care must be taken in choosing the appropriate bin width. The suppression factor interpolation procedure is made more precise by reducing the bin width (thereby increasing the number of interpolation points). However, this is unattractive for reporting final bandpowers because adjacent bins will be highly correlated. As a result, we use finely spaced bins to calculate the “true” bandpower window function, and afterwards merge the simulated bandpowers, real bandpowers, and bandpower window functions to more coarsely sampled, minimally correlated final bins.
4.2.5 Frequency combination
In the case of Bicep1, there are observations at both 100 and 150 GHz. In order to achieve the highest possible signal-to-noise measurement, bandpowers can be combined across frequencies. In the case of theT T, EE,andBB spectra, there are three unique spectra (100⇥100,150⇥150,and 100⇥150). For the cross-spectra, there is an additional unique spectrum calculated as150⇥100.
For each`bin, a covariance matrix is calculated from signal-plus-noise simulations across the three (or four) unique frequency combinations. Weights are then calculated as the column-wise sum of
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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.