A MEASUREMENT OF THE COSMIC MICROWAVE BACKGROUND B-MODE POLARIZATION POWER SPECTRUM AT SUB-DEGREE SCALES WITH POLARBEAR

Individual detector maps are made from timestreams that are filtered to reduce the effect of atmospheric fluctuations. This filtering is accomplished with a masked polynomial: a first-order polynomial that is fit to the timestream everywhere outside of a 50' radius around the planet. The individual detector maps are combined to create a single-observation map with adequate coverage. The weighting used to combine individual detector maps is a noise weighting calculated from the rms of the map outside of the mask radius.

The azimuthally averaged Fourier components are then found for each single-observation map. We work in the flat-sky approximation, appropriate for our small patch sizes (each patch represents less than 0.03% of the sky). The two-dimensional Fourier transform of the temperature-calibrated map, M(ℓ, ϕ_ℓ), is averaged over ϕ_ℓ

$$\begin{equation} \tilde{M_{\ell }} = \mathbf {Re}\left \lbrace \frac{1}{2\pi }\int M(\ell, \phi _{\ell })d\phi _{\ell }\right \rbrace, \end{equation} \tag{ 1 }$$

and binned in ℓ, creating bins of width Δℓ ≈ 80. The beam can be approximated as azimuthally symmetric in intensity because of the rotation of the instrument beams as projected on the CMB patch due to changes in patch parallactic angle over the course of the day from our mid-latitude site (Fosalba et al. 2002).

The underlying beam model for each observation is found by dividing the azimuthally averaged map transforms by both the angular Fourier response function for a finite planetary disk, T_ℓ, and a correction for the map pixel size used, W_ℓ. The pixel window function for a pixel of size Δ is given by Wu et al. (2001) as

$$\begin{equation} W_{\ell } = e^{-(\ell \Delta)^{2.04}/18.1} [ 1-(2.72\times 10^{-2})({\ell }\Delta)^2], \end{equation} \tag{ 2 }$$

and T_ℓ of a planetary disk of known radius R and temperature T_p is

$$\begin{equation} T_{\ell } = 2\pi T_p R^2 J_1(\ell R)/(\ell R), \end{equation} \tag{ 3 }$$

where J₁ is the Bessel function of the first kind of order unity.

The multipole components and covariance matrix of the beam are estimated by the inverse-noise-weighted mean values and covariance matrix of the power spectra of many planet observations. The resulting B_ℓ and its uncertainty in each multipole bin from observations of Jupiter are shown in Figure 2. Results using Saturn observations are consistent.

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The effective beam $B_{\ell }^\mathrm{eff}$ used in our power spectrum analysis accounts for blurring of the co-added CMB maps that is introduced by inaccuracies in our pointing model. We model the effect of an rms pointing error in map space, σ_p, as

$$\begin{equation} B_{\ell }^{\mathrm{eff}} = B_{\ell } \times \ e^{-{\ell }({\ell }+1)\sigma _p^2/2}, \end{equation} \tag{ 4 }$$

where σ_p for each field is estimated by fitting each point source with the Jupiter beam profile convolved with a Gaussian smoothing kernel. We estimate a σ_p likelihood function for each individual point source in each map. We then combine the sources within each patch to find a joint likelihood for the rms pointing error on that patch. The results are shown in Table 2 and the patch-specific beam functions are displayed in Figure 2. Using these beams for each patch, and including the beam covariance, the individual patch $C_\ell ^{TT}$ power spectra, and the patch-combined power spectra (shown in Figure 4) are consistent with wmap-9 ΛCDM (Bennett et al. 2013), as described in Section 4.5.

We increased the uncertainty in $B_\ell ^{\rm eff}$ to account for discrepancies in the measurements of σ_p. The measurements of σ_p from the 10 point sources in RA4.5 exhibited statistically significant differences with one another, implying that the amount of blurring is not constant throughout the map, or that the model for the blurring is not capturing the entire effect; we do not understand the origin of this effect.

The beam uncertainty increases the uncertainty of our absolute gain calibration to $C_\ell ^{TT}$ (see Section 4.3) by a factor of 1.5, from 2.8% to 4.1%. The differences and uncertainties in the patch-specific beam functions primarily affect the ℓ-range at the high end of our reported band powers and beyond, having little effect on the constraint of A_BB (given in Section 8), where most of the significance comes from the low end of the reported ℓ-range. An analysis using simply the Jupiter beams, calibrated using $C_\ell ^{TT}$, results in an A_BB that differs by 0.3% from the reported result.

Polarbear observed Tau A several times per week throughout the observations reported here, except when Tau A was within 30° of the Sun, which occurred between May and July of each year. This resulted in 125 Tau A observations which were evaluated based on the same selection criteria as were implemented in the CMB analysis, where applicable (see Section 5.1). The observation pattern is a raster scan where the telescope tracks Tau A while executing a set of constant-velocity subscans with steps in elevation between them. Figure 3 shows a full-season co-added polarization map of Tau A.

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Tau A is an extended source; IRAM observed that Tau A's polarization angle and polarized intensity are not uniform over its spatial extent. To compare the IRAM maps to Polarbear observations, a simulated timestream is generated for each bolometer by scanning the IRAM map with that bolometer's pointing. To create each sample of simulated timestream, the IRAM maps are convolved with that detector's beam, modeled for this as an elliptical Gaussian (note that the IRAM 30'' beam is negligibly small compared to the Polarbear beam). The simulated timestreams s_{i, ∥⊥}(t) are calculated from the IRAM I, Q, and U maps:

$$\begin{eqnarray} s_{i,\parallel /\perp } (t) & = & I_{{\rm sim},i,\parallel /\perp }(t) + Q_{{\rm sim},i,\parallel /\perp } (t) \cos 2\Theta _{i,\parallel /\perp (t)} \nonumber \\ & & + U_{{\rm sim},i,\parallel /\perp } (t) \sin 2\Theta _{i,\parallel /\perp }(t), \end{eqnarray} \tag{ 5 }$$

where Θ_{i, ∥⊥}(t) is the polarization angle projected on the sky in equatorial coordinates for the two orthogonal detectors (∥⊥) of pixel i. This is related to the detector polarization angle θ_{i, ∥⊥}, the HWP angle θ_HWP, and parallactic angle θ_PA by

$$\begin{equation} \Theta _{i,\parallel /\perp }(t) = \left(\frac{\pi }{2} - \theta _{i,\parallel /\perp } \right) + 2\theta _{\rm HWP}(t)+ \theta _{\rm PA}(t), \end{equation} \tag{ 6 }$$

and the simulated intensity and polarization signals X_sim(t) ∈ (I_sim(t), Q_sim(t), U_sim(t)), which are calculated from the IRAM maps $X(\boldsymbol{p})$ and the detector beam B_{i, ∥⊥}:

$$\begin{eqnarray} X_{{\rm sim}, i,\parallel /\perp } (t) & = & \int X(\boldsymbol{p}) B_{i,\parallel /\perp }(\boldsymbol{p} - \boldsymbol{p}_{i,\parallel /\perp }(t))\, {d\!\boldsymbol{p}}, \nonumber\\ &&\times X\in {I,Q,U}. \end{eqnarray} \tag{ 7 }$$

Polarization timestreams are created from Polarbear data by differencing the two orthogonally oriented detectors within one pixel (see Section 5.2 for the CMB polarization map-making process). Simulated difference timestreams of Tau A observations are calculated from individual detector timestreams s_{i, ∥⊥} using the fact that Θ_{i, ⊥} = Θ_{i, ∥} + π/2

$$\begin{eqnarray} d_{i} (t) &=& \frac{1}{2}(s_{i,\parallel }-s_{i,\perp }) \nonumber \\ & \approx & \frac{1}{2} (I_{{\rm sim},i,\parallel }(t) - I_{{\rm sim},i,\perp }(t)) \nonumber \\ &&+ \epsilon _i [Q_{{\rm sim},i}(t)\cos 2\Theta _i(t) + U_{{\rm sim},i}(t)\sin 2\Theta _i(t)] \nonumber \\ &&+ \frac{\Delta g_i}{2} (I_{{\rm sim},i,\parallel }(t) + I_{{\rm sim},i,\perp }(t)). \end{eqnarray} \tag{ 8 }$$

Here we have introduced the polarization efficiency _i and pixel-pair relative-gain error Δg_i.

The constant-velocity portion of each subscan is fit to a masked polynomial, as with observations of planets. In this case, the polynomial is fifth-order, and the mask is 10' in radius from the center of Tau A. The best-fit polynomial is then subtracted from the data. Detector differencing and polynomial baseline subtraction both act to remove atmospheric signals. The simulated observations of the IRAM maps undergo the same filtering, and then the data are fit to the simulations for detector pointing within 5' of the center of Tau A. The fit parameters for each pixel are the pixel polarization angle θ_i (implicitly in Θ_i), polarization efficiency _i, and the pixel-pair relative gain error Δg_i.

The polarization efficiency predicted due to the expected performance of the HWP over the finite Polarbear spectral band is between 0.97 and 0.98, slightly different from wafer to wafer due to the different measured spectral bands. The mean pixel polarization efficiency from the Tau A fit is 0.994  ±  0.036. Note that this measured polarization efficiency is strongly affected by any difference in polarization fraction between 90 and 148 GHz. In CMB map-making, the mean theoretical value of 0.976 is used as the polarization efficiency for every detector. The systematic uncertainty is constrained by the difference between the prediction and the mean value of the Tau A fit, which then encompasses other possible uncertainties than the HWP.

To monitor the time stability of the detector polarization angle and the accuracy of the reported HWP angle, the detector polarization angles fit to the full season of Tau A data, as described above, are used to make maps of Tau A for each observation. In this map-making process, pixel-sum and pixel-difference subscans are filtered with the same polynomial baseline subtraction as in the pixel polarization angle fit. Data from every pixel are co-added, and the polarization angle of Tau A for that observation, α_Tau A, is calculated from that observation's co-added Q and U maps:

$$\begin{equation} \alpha _{{\scriptsize {\rm Tau\, A}}} = \frac{1}{2}\arctan {\left(\frac{\sum {U_j}}{\sum {Q_j}}\right)}, \end{equation} \tag{ 9 }$$

where j is a map pixel within 10' of the center of Tau A. The variation of this polarization angle over the season is 12 (rms).

The polarization model is based on a hierarchical understanding of our polarization calibration, consisting of a global reference polarization angle, the wafer-averaged polarization angles relative to that global angle, and the individual pixel angles relative to the wafer-averaged angle. We consider uncertainty in each of these. The systematic uncertainty in the global reference angle and wafer-averaged angle are shown in Table 3. These are dominated by uncertainties in the pixel-pair relative gain and in the non-axisymmetric beam model and the substructure of Tau A at 148 GHz. Non-idealities in the HWP over the finite Polarbear spectral bandwidth are also an important source of uncertainty, both in rotation angle of linear polarization and in the mixing of circular polarization into linear polarization. Using the upper limit on Tau A's circular polarization fraction of 0.2% (Wiesemeyer et al. 2011), the systematic error from the circular polarization of Tau A is estimated to be 009. The individual pixel polarization angle uncertainty in each wafer is estimated to be 10 from the spread of the pixel polarization angle distribution from the Tau A measurement. The other systematic effects we evaluated, listed in Table 3, are all negligible. The impact of all these uncertainties on the $C_\ell ^{BB}$ and $C_\ell ^{EB}$ power spectra are addressed in Section 7.

Table 3. Systematic Uncertainties in Global Reference and Wafer-averaged Polarization Angle, as Measured Using Tau A

Angle Uncertainty	Global Reference	Wafer-averaged
Absolute pointing uncertainties	012	⋅⋅⋅
Beam uncertainties	021	023
Relative gain uncertainties	022	042
Non-ideality of HWP	021	064
Circular polarization of Tau A	009	≪01
HWP angle uncertainties	015	013
Pixel pointing uncertainties	≪01	018
Bolometer time constant	≪01	≪01
Filtering effect	≪01	≪01
Polarized dust	≪01	≪01
Total	043	083

Notes. Uncertainty in the global reference angle as measured using $C_\ell ^{EB}$ is addressed in Section 4.5.

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Polarbear observations of Cen A follow the same raster scan as the observations of Tau A. Single-observation maps of Cen A are produced and the polarization angle of Cen A is measured from those maps as was done for Tau A using Equation (9). To calculate the polarization angle of Cen A, all map pixels within 12' of its center are used. The QUaD experiment measured the polarization of Cen A at 100 and 150 GHz (Zemcov et al. 2010). Polarbear measured a Cen A polarization angle of 1479 ± 06(stat.) ± 10(sys.) with the Tau A-derived polarization angle. Using the $C_\ell ^{EB}$-derived polarization angle described in Section 4.5 results in a measured Cen A polarization angle of 1490 ± 06(stat.) ± 09(sys.). The Polarbear and QUaD measurements of Cen A agree within their measurement uncertainties.

Bolometer TOD are low-pass filtered with an anti-aliasing filter matched to the digitization sample rate of 190.7 Hz. The TOD-based data selection described in Section 5.1 is then applied. The telescope azimuth and elevation encoders are queried at 95.4 Hz, synchronous but offset with every other sample of the bolometer timestream. The telescope pointing is reconstructed using the procedure described in Section 4.1 and linearly interpolated to the bolometer sample times. Because this analysis focuses on ℓ < 2100, corresponding to time-domain frequencies less than 4.2 Hz, we downsample the bolometer TOD and reconstructed pointing by a factor of six to 31.8 Hz. The bolometer optical response times, which are between 1–3 ms, are small enough that deconvolving these transfer functions is not necessary (Arnold et al. 2012).

Individual bolometer timestreams are calibrated as described in Section 4.3, then pixel-pair bolometers are summed and differenced to derive temperature and polarization timestreams from each pixel. The polarization timestreams contain a linear combination of Stokes Q and U signal depending on the polarization orientation of the detector projected onto the sky.

Timestreams are noise weighted with a filter F which projects out three types of low signal-to-noise-ratio modes: high frequencies, low-order polynomials per subscan, and scan synchronous signals.

Because map making, pixelized at a Nyquist frequency of ℓ = 5400, is a decimation operation from the timestreams, pixelized at a Nyquist frequency of ℓ = 7950, high frequencies can alias into our science band. We apply a convolutional low-pass filter to the timestreams to eliminate aliasing. The frequency profile of the filter is

$$\begin{equation} F_{\rm lpf}(f) = e^{-\log (\sqrt{2}) \left(\frac{f}{f_{\rm lpf}}\right)^6}, \end{equation} \tag{ 13 }$$

where the 3 dB frequency of the filter is f_lpf = 6.3 Hz, corresponding to an angular multipole of ℓ = 3150.

To remove excess low-frequency noise, for each detector we subtract a polynomial per CES subscan. Linear polynomials are used for difference (polarization) timestreams, and cubic polynomials for sum (temperature) timestreams. Point sources (described further in Sections 4.2.2 and 6) are masked with a 10' diameter mask during fitting to keep the point source power localized. The exception is Mars, which traverses the RA12 patch multiple times during the season. Because of Mars's brightness, TOD are masked within a 30' diameter disk centered on it during the polynomial filtering, and this masked data is also excluded from the maps.

Our scan strategy is designed to concentrate scan-synchronous signals, such as a far sidelobe scanning the ground, into a small number of modes which can be easily filtered. During a CES, scan-synchronous signals will repeat in azimuth for the duration of the scan. These signals are projected out by averaging the timestreams in 008 azimuth bins for each bolometer to build a scan-synchronous signal template; the template is then subtracted from the timestreams. Because the CMB patch moves by about 3° in this time, the effect of this filter on sky signal has only a small impact on the multipole range reported here, 500 < ℓ < 2100, as shown in Figure 7.

The pointing matrix and noise weights used in Equation (12) are split into components acting on the sum and difference timestreams of the two orthogonally polarized detectors in one focal plane pixel. The pointing matrix for the sum component maps only Stokes intensity I from the sky into the detector timestream $d_t^{\rm sum}$,

$$\begin{equation} d^{\rm sum}_t = \mathbf { A }_{tp} I_p + n^{\rm sum}_t. \end{equation} \tag{ 14 }$$

The pointing matrix for the difference timestream is assumed to have no temperature response and only Stokes Q and U response,

$$\begin{equation} d^{\rm diff}_t = \mathbf { A }_{tp} \cos (2 \psi) Q_p + \mathbf { A }_{tp} \sin (2 \psi) U_p + n^{\rm diff}_t. \end{equation} \tag{ 15 }$$

The assumption of no temperature response is described in detail in Section 7. Detector pointing is projected onto a flat map using the cylindrical equal area projection centered at the nominal patch center (Snyder 1926). The map pixels have a width of 2'.

The noise weights N⁻¹ used for co-adding the pixel maps are estimated under an idealized model that the noise is white. The time-domain power spectral density are averaged from 1–3 Hz to estimate the detector noise variance. This frequency band corresponds approximately to ℓ of 500–1500.

We co-add the single-CES, all-detector maps into a set of single-day I, Q, and U maps. The patches RA4.5, RA12, and RA23 have 148, 139, and 189 daily maps respectively. The map making procedure produces inverse noise covariance estimates for each pixel describing the covariance of Stokes I, Q, and U. Polarization maps are apodized using the minimum eigenvalue of the 2 × 2 Q and U block of the pixel noise inverse covariance matrix. This eigenvalue is an upper bound for the variance in either Q or U. Temperature maps are apodized with an inverse noise variance apodization at the edge of the map, and with a flat function where the CMB fluctuations are measured at a large signal-to-noise ratio.

Pixels with an apodization window value below 1% of its peak are set to zero, as are pixels within 3' of point sources. To reduce E/B leakage, the apodization edges are modified using the C² taper described in Grain et al. (2009), which is applied inward with a 30' width.

Q and U maps are transformed to E and B maps. The B maps are estimated from the pure B-mode transform (Smith 2006). Before Fourier transforming, the maps are padded to a width of 384 pixels or 128 to eliminate spurious periodic effects. Co-added maps from the entire season are not used in the analysis; however we show them here in Figure 6 for Q and U on RA23, which results in a polarization white noise level of 6 μK-arcmin (8 μK-arcmin with the beam and filter transfer function divided out. See Sections 4.2 and 5.3).

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$M_{\ell \ell ^\prime }$ is computed analytically, by co-adding the temperature and polarization apodization windows from the daily maps for the entire season. The resulting window map is used to calculate $\mathbf {M}_{\ell \ell ^{\prime }}$ (Louis et al. 2013).

We estimate the transfer function F_ℓ of the time-domain filters from a suite of Monte Carlo simulations. The input to the Monte Carlo simulations is a set of 1'-resolution Gaussian realizations of a 10° × 10° patch of the CMB from the best-fit wmap-9 power spectra, C_ℓ (Bennett et al. 2013). We use the pointing data from observations to produce TOD from the simulated maps, and apply the pseudo-power spectrum estimation procedure. We then estimate the filter transfer function from

$$\begin{equation} F_\ell ^n = F_\ell ^{n-1} + \frac{\tilde{C}_\ell - \sum _{\ell ^{\prime }} \mathbf {M}_{\ell \ell ^{\prime }} F_{\ell ^{\prime }}^{n-1} C_{\ell ^{\prime }} B_{\ell ^{\prime }}^2}{C_\ell B_\ell ^2}, \end{equation} \tag{ 19 }$$

with $F_\ell ^0 = 1$, and convergence achieved within the 10 iterations used to calculate $F_\ell = F_\ell ^{10}$.

To distinguish between leakage and transfer function effects, the filter transfer functions for E and B are computed from separate TT+EE and TT+BB simulations. The TE, TB, and EB spectra filter transfer functions are estimated as the geometric mean of the respective auto spectra. TT, EE, and BB transfer functions are shown in Figure 7.

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Polynomial filtering and scan-synchronous signal subtraction create leakage from $C_\ell ^{EE}$ to $C_\ell ^{BB}$. The leakage transfer function is estimated from the Monte Carlo simulations and leakage is subtracted in power spectrum estimation. Equation (19), TT+EE simulations, with the EE theory for C_l and BB pseudospectra and mode mixing matrix are used to estimate $F^{E \rightarrow B}_\ell$. Before subtraction, the leakage is largest in the lowest bin centered at ℓ = 700 where it is 9% of the $C_\ell ^{BB}$ band power. The power is subtracted in pseudospectrum space with an amplitude of

$$\begin{equation} \tilde{C}^{E \rightarrow B}_\ell = \frac{F^{E \rightarrow B}_{\ell }}{F^{E \rightarrow E}_{\ell }} \tilde{C}^{E}_\ell. \end{equation} \tag{ 20 }$$

The uncertainty associated with this subtraction is calculated via Monte Carlo simulations that include TT, EE, and BB power. The residual bias and its uncertainty, including sample variance, is calculated as $\ell (\ell +1)C_\ell ^{BB}/(2\pi) = (6.1\pm 39.9)\times 10^{-4}$ μK² at ℓ = 700. This uncertainty is included in the presented limits on A_BB.

F_ℓ and $\mathbf {M}_{\ell \ell ^{\prime }}$ are calculated with a resolution of Δℓ = 40. We solve for the unbiased estimates of the sky spectra on four coarser bins of width Δℓ = 400 within 500 < ℓ < 2100 using binning and interpolation operators P_bℓ and Q_ℓb. The unbiased, binned estimator for the Δℓ = 400 binned power spectra is

$$\begin{eqnarray} &&\hat{C}_b = \sum _{b^{\prime }\ell } \mathbf {K}^{-1}_{bb^{\prime }} \mathbf {P}_{b^{\prime }\ell } \tilde{C}_{\ell }, \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} &&\mathbf {K}_{bb^{\prime }} = \sum _{\ell \ell ^{\prime }} \mathbf {P}_{b\ell } \mathbf {M}_{\ell \ell ^{\prime }} F_{\ell ^{\prime }} B_{\ell ^{\prime }}^2 \mathbf {Q}_{\ell ^{\prime }b^{\prime }}. \end{eqnarray} \tag{ 22 }$$

The functional dependence of the binned band powers $\hat{C}_b$ on the true high-resolution power spectra is given by the band power window functions w_bl, where

$$\begin{eqnarray} &&\hat{C}_b = \sum _{l} \mathbf {w}_{b\ell } C_\ell, \end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray} &&\mathbf {w}_{bl} = \sum _{b^{\prime }\ell ^{\prime }} \mathbf {K}^{-1}_{bb^{\prime }} \mathbf {P}_{b^{\prime }\ell ^{\prime }} \mathbf {K}_{\ell ^{\prime }\ell }. \end{eqnarray} \tag{ 24 }$$

The resulting band power window functions for temperature and polarization are shown in Figure 8.

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All pixel polarization angles are referenced to the instrument's global reference angle. How this angle maps to the sky is the sum of different contributions, described in Equation (6). The miscalibration of this angle has been studied analytically by Keating et al. (2013). Simulations with an incorrect instrument polarization angle were consistent with these analytic results. We simulated 100 realizations of miscalibration of the instrument polarization angle, and found that they produced bias in $C_\ell ^{EB}$ and $C_\ell ^{BB}$ expected from the analytical calculation. For the Polarbear $C_\ell ^{BB}$ results, the global reference angle was measured using $C_\ell ^{EB}$ with an uncertainty of 020 as discussed in Section 4.5. All of the relative polarization angle uncertainties were simulated 100 times, and in each simulation the instrument angle was measured from $C_\ell ^{EB}$, with realistic variance in the angle added to the noiseless simulations. In this way, we calibrated the instrument angle for the systematic uncertainty simulations in the same way it is calibrated in the analysis.

The relative polarization angles of each pixel are measured using Tau A, as described in Section 4.4. We simulate the noise in this measurement with two random components: one component which is common across the detectors in each wafer (the uncertainty of wafer-averaged polarization angles) and a component that is a pixel-by-pixel random uncertainty within each wafer (the individual pixel polarization angle uncertainty). The amplitude of each is based on the measurement uncertainty. We also include day-to-day variations in the instrument polarization angle at the largest level allowed by measurements of Tau A. Note that we do not expect or see evidence for day-to-day variations, however this treatment accounts for any possible rotation (jitter) of the stepped and fixed HWP. The combined uncertainty in $C_\ell ^{EB}$ and $C_\ell ^{BB}$ due to polarization angle calibration uncertainty, after self-calibration using $C_\ell ^{EB}$, is shown in Figures 9 and 10. We found that the global reference angle uncertainty has a somewhat larger contribution to this uncertainty than the relative pixel polarization angle uncertainty. The gray-shaded region in Figures 9 and 10 shows the 1σ bounds on the cumulative bias limit for the polarization angles from 100 realizations after the self-calibration procedure.

The effect of incorrect pointing reconstruction can be evaluated in this simulation pipeline by scanning the noiseless map into timestreams using one pointing model—the scanning pointing model—but then reconstructing the map using a second pointing model—the mapping pointing model. Measuring the spurious $C_\ell ^{EB}$ and $C_\ell ^{BB}$ created by this procedure is a measure of how different the pointing models are, referenced to power spectrum space.

The covariance matrix of the five parameters in the pointing model describes the constraints that the pointing data has put on the model parameters. For the pointing model to be precise enough, inaccuracies in these parameters within the space of this covariance matrix must be acceptable. One hundred realizations of the pointing model within the parameter covariance matrix were generated and used as the mapping pointing model in the simulation. The mean of the spurious signal created in these simulations was found to be negligible compared to the systematic uncertainty in the pointing model described in the next paragraph.

As described in Section 4.1, systematic differences in pointing reconstruction were noticed depending on the sources used to create the pointing model. To ensure that these differences were unimportant in this measurement of $C_\ell ^{BB}$, simulations were done with each of these systematically different pointing models used as the mapping pointing model. The largest spurious $C_\ell ^{BB}$ and $C_\ell ^{EB}$ found in these simulations is shown in Figures 9 and 10. This level of systematic error in pointing model would have been responsible for more beam-smearing than was observed in the Polarbear maps (as described in Section 4.2), but the spurious $C_\ell ^{EB}$ and $C_\ell ^{BB}$ are still small compared to the statistical uncertainties in the measurement. While sufficiently accurate for the measurement of $C_\ell ^{BB}$ reported here, in the future, we plan to establish a more precise and consistent pointing model for Polarbear through more detailed pointing calibration observations.

The differential pointing between two detectors in a pixel is measured from observations of planets. It is estimated independently for each HWP position. The mean differential pointing magnitude is 5''. This is one of the most important instrumental systematic effects because it creates spurious polarization proportional to the derivative of the CMB intensity.

Data averaged over different angles between the sky polarization and the differential pointing vector act to average out the effect of differential pointing. This averaging out is provided by sky rotation as the CMB patch rises and sets, and HWP rotation from one angle to another (Miller et al. 2009). The simulations show that a majority of this leakage-mitigation provided by the Polarbear observation strategy has occurred after several days of observation. Figures 9 and 10 show the spurious $C_\ell ^{BB}$ and $C_\ell ^{EB}$ signals created by the differential pointing in an entire season.

Miscalibrations of relative bolometer gains in a pixel pair will "leak" temperature signal into Q or U. A systematic miscalibration between two bolometers does not necessarily lead to a significant systematic bias in polarization maps.

The relative gain model we use has a term motivated by the polarization of the thermal calibration source which depends on the angle of the HWP, and a term motivated by variations in detector properties which has no HWP dependence. Uncertainty in either of these terms can lead to leakage of temperature into polarization. We evaluate our uncertainty in the term that changes with HWP position by comparing two different gain models with an independent determination of this term, as described in more detail below. We evaluate uncertainty in the term that does not change when the HWP rotates in two ways: via a comparison of different gain models with separate measurements of this HWP-independent term, and also via differential gain map-making described in Section 7.2.1.

The simulation pipeline described above is used to compare relative gain models. In each case, a simulated map with no B-modes is "observed," producing timestreams using the gain model under question, and then reconstructed using the standard analysis gain model. The level of resulting $C_\ell ^{BB}$ quantifies the difference in these gain models in power spectrum space.

As explained in Section 4.4, each pixel's polarization angle relative to the instrument frame is measured using Tau A. This fit also returns a single value for the average relative gain miscalibration over the course of the year, which we use to independently determine our non-HWP dependent gain model term. The difference between this measurement and the measurement by planets that is used to calibrate the pixel-pair relative gain was analyzed with the simulation pipeline. The resulting $C_\ell ^{BB}$ bias is shown in Figure 9 and enumerated in Table 5.

Elevation nods can also be used to establish the relative gain between detectors. We use this technique to determine our HWP-dependent relative gain model term and compare to the planet-derived term with the same simulation process. We find the difference to be small, as shown in Figures 9 and 10 and Table 5.

Our normal procedure to correct for gain drift over the duration of a scan is to interpolate our gains between measurements of the thermal calibration source taken at the beginning and end of hour-long observation periods. In order to understand the impact of potential errors in this interpolation, we constructed a set of gains based only on the measurements taken at the beginning of every hour and thus use no interpolation. We find the impact, evaluated through a simulation comparing these two models, to be negligible as shown in Table 5. All four probes of the relative gain model described here show that the uncertainty in the Polarbear relative gain model is small compared to the statistical uncertainty in $C_\ell ^{BB}$. We perform a further systematic check on all sources of differential gain via a cross-correlation of temperature maps with B-mode maps, described in Section 7.2.2, and find consistency with zero leakage to within this test's statistical power.

Crosstalk from one bolometer to another due to coupling in the multiplexed readout will lead to polarization and temperature leakage from a point outside the main beam, creating a localized, polarized near side lobe. Crosstalk is strongest in bolometer channels that share a SQUID in the frequency-domain multiplexed readout, and are closest together in bias frequency, as described in Dobbs et al. (2012). Polarbear bolometers of this type show nominal crosstalk of about 1%. Simulations were done with 2% crosstalk between all nearest-neighbor bolometers; we ran 10 simulations of 5 typical days of Polarbear observations. Because of the frequency schedule in the 8× multiplexed readout, at least half of the pixels will see crosstalk from another pixel that entirely leaks temperature to polarization; in practice the fraction is greater than half because the existence of non-functioning bolometers also creates unbalanced crosstalk. All of this means that these simulations should overestimate the effect in $C_\ell ^{BB}$ of an average 1% electrical crosstalk. The simulation showed temperature to polarization leakage that produces a maximum spurious polarized signal of $\ell \left(\ell + 1 \right) C_\ell ^{BB} / (2\pi)$<1.7 × 10⁻⁴ μK². This simulation was done without simulating stepping of the HWP; observing at several HWP angles mitigates this already small effect.

In Section 4.2, we described measurements of the individual beam shapes of each detector, which can be used to measure differences in beam shape between the two bolometers measuring orthogonal polarization in one pixel. These differences can create spurious polarization signals by "leaking" temperature into polarization (Miller et al. 2009).

The differential beam shape leakage in Polarbear can be modeled as additional terms in the pointing matrix corresponding to leakage from the zero, first, and second derivatives of the temperature map to polarization. We fit our elliptical Gaussian beam parameters to a model of derivatives of the beam, and then run simulations to measure the systematic bias in the measured power spectra. Differential beam size couples to the zeroth and second derivatives, while differential pointing couples to the first derivative, and differential ellipticity to the second derivative. The measured Polarbear beam and pointing over a representative period of five days were used to sample a simulated ΛCDM realization of the CMB and it's derivative maps. The leakages into $C_\ell ^{EB}$ and $C_\ell ^{BB}$, found to be negligible, are shown in Figures 9 and 10.

Differential gain maps are constructed by extending the model for polarized timestreams to include a temperature component due to a possible mismatched calibration of the two detectors in a pixel-pair,

$$\begin{equation} d_t = G_t + Q_t \cos (2\phi _t) + U_t \sin (2\phi _t). \end{equation} \tag{ 33 }$$

We construct maps individually for each wafer's Q and U pixels. This is motivated by the idea that the dominant systematic relative gain error could result from differential spectral band pass, which might arise from common fabrication errors within similarly oriented pixels on one wafer. Correlation of G maps with temperature maps can be used to estimate differential gain leakage. No power is detected in any of the TG cross-spectra; the measured leakage value for each pixel type on each wafer is consistent with zero. Simulations that include wafer common mode differential gain leakage at the level constrained by the TG cross spectra show negligible systematic bias.

We investigated the Polarbear maps for unexpected correlations between temperature and polarization. Specifically, we looked at correlations between T and B maps in two-dimensional Fourier space. Any temperature leakage into Q or U is expected to average out to first order in $C_\ell ^{{TB}}$. However, if the Fourier modes are weighted by sine or cosine of the Fourier plane azimuthal angle before performing the azimuthal averaging, the temperature to polarization leakage signal is preserved. This weighting scheme has the added benefit of averaging out any E to B leakage due to polarization angle rotation error, making it easier to isolate the effects of gain leakage.

We form estimators for temperature to polarization leakage based on TB using this weighting and compare simulations without gain leakage to our real data. The signal found in these correlations is consistent with zero-leakage with a minimum PTE of 27% across the three patches. We ran 100 additional simulations at each of two different levels of uniform leakage from temperature to polarization. One hundred percent of these simulations with a 0.5% leakage show a larger leakage than seen in the real data, and 86% of simulations with a 0.3% leakage show larger values than observed with the real data. These simulations demonstrate the real leakage is less than 0.5% and likely less than 0.3%, corresponding to at most 75% (and likely less than 35%) of the expected BB power at ℓ = 700. Note that although these numbers are potentially large, this test is consistent with zero gain leakage and, of the several tests presented, has the least constraining power on gain leakage.

A scan-synchronous template is computed for every 15 minute CES and subtracted from the individual azimuth scans (Section 5.2.2); however, some residual contamination that changes in time could remain. We make maps for each wafer in ground coordinates and then use these maps to add contamination to the standard CMB signal-only simulations. Taking the difference between these ground-contaminated simulations and the standard signal-only simulations, applying the scan-synchronous signal filter to both, we constrain the expected residual power in ground signal not removed by the filter to be $\ell (\ell +1)C_\ell ^{BB}/(2\pi) = 1.7\times 10^{-4}$ μK² in the band powers reported here.

We confirmed that the additional absorptive shielding installed above the primary mirror in the middle of the season did in fact eliminate the far side lobe it was designed to mitigate, reducing the amplitude of the scan synchronous signal by more than two orders of magnitude. We apply scan-synchronous signal filtering even after the visor installation to minimize the effect of any residual ground pickup.

The null tests are performed for several interesting splits of the data, chosen to be sensitive to various sources of systematic contamination or miscalibration. We also required that the data divisions for different null tests be reasonably independent. The data splits are as follows.

• 1.  
  "First half versus second half": probes time variation on month-long timescales. This test is sensitive to systematic changes in the calibration, beams, telescope, and detectors, and effects due to the mid-season addition of absorptive shielding above the primary mirror (see Sections 2.1 and 7.2.3).
• 2.  
  "Rising versus setting": checks for systematic bias due to poor sky rotation. This is also sensitive to residual ground signal via the far sidelobe, which for RA23 sees a nearby hill in only the setting scans (Section 5.2.2).
• 3.  
  "High elevation versus low elevation": tests for contamination caused by noise or glitches due to the faster telescope motion required at higher elevation.
• 4.  
  "High gain versus low gain": probes for problems due to linearity or saturation power of the detectors, and checks for miscalibration.
• 5.  
  "Good versus bad weather": checks for residual problems after the PWV cut (Section 5.1) is made.
• 6.  
  "Pixel type": each detector wafer is fabricated with pixels at two different polarization orientation angles. We split the data into the two individual types of pixels to check for systematic contamination or miscalibration by different cross-linking, bandwidth, or microfabrication differences.
• 7.  
  "Left-side versus right side": checks for optical distortion on one side of the focal plane versus another, or for different map coverage.
• 8.  
  "Left- versus right-going subscans": probes for residual atmosphere (which is asymmetric in telescope direction due to wind), and for contamination due to vibration, which may be asymmetric in velocity.
• 9.  
  "Moon distance": checks for residual contamination after setting the moon proximity threshold for an observation to be considered for analysis.

We also established a "Sun distance" null test, but it was highly correlated with the "high gain versus low gain" test for RA4.5, and also correlated with the "first half versus second half" test for RA12 and RA23, so we did not include it. This left nine null tests to analyze as described below.

For each null power spectrum bin b, we calculate the statistic $\chi _{\rm null}(b)\equiv \hat{C}_b^{\rm null}/\sigma _b$, where σ_b is a Monte Carlo–based estimation of the corresponding standard deviation, and its square $\chi _{\rm null}^2(b)$. χ_null(b) is sensitive to systematic biases in the null spectra, while $\chi _{\rm null}^2(b)$ is more sensitive to outlier bins.

To probe for systematic contamination that is focused in a particular power spectrum or null test data split, we calculate the sum of $\chi _{\rm null}^2(b)$ over 500 < b < 2100 for EB and BB separately, ("$\chi ^2_{\rm null}$ by spectrum"), and the sum of both these spectra for a specific test ("$\chi ^2_{\rm null}$ by test"). Figure 11 shows the PTE distribution of the $\chi ^2_{\rm null}$ by (a) bin, (b) spectrum, and (c) test for the three patches. We require that each of these sets of PTEs each be consistent with a uniform distribution, as evaluated using a K-S test, requiring a p-value (probability of seeing deviation from uniformity greater than that which is observed given the hypothesis of uniformity) greater than 5%. These distributions are consistent with a uniform distribution from zero to one.

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We create test statistics based on these quantities to search for different manifestations of systematic contamination. The five test statistics are (1) the average value of χ_null; the extreme value of $\chi ^2_{\rm null}$ by (2) bin, (3) spectrum, and (4) test; and (5) the total $\chi ^2_{\rm null}$ by summing up the nine null tests. In each case, the result from the data is compared to the result from simulation, and PTEs are calculated. Finally, we combine each of the test statistics, and calculate the PTE of that final test statistic, requiring it to be greater than 5%. Table 7 shows summary of the PTE values of each test statistic for each patch. Comparing the most significant outlier from the five test statistics with that from simulations, we get PTEs of 32.8%, 55.6%, and 18.0% for RA4.5, RA12, and RA23 respectively. We achieve the requirements described above, finding no evidence for systematic contamination or miscalibration in the Polarbear data set and analysis.

Table 7. PTEs Resulting from the Null Test Framework

Patch	Average of	Extreme of	Extreme of	Extreme of	Total
	χnull(b)	$\chi ^2_{\rm null}(b)$	$\chi ^2_{\rm null}$ by EB/BB	$\chi ^2_{\rm null}$ by Test	$\chi ^2_{\rm null}$
RA4.5	11.6%	16.6%	20.6%	21.8%	14.0%
RA12	92.4%	84.2%	60.8%	23.8%	52.6%
RA23	75.2%	61.6%	6.0%	7.0%	18.6%

Notes. No significantly low or high PTE values are found, consistent with a lack of systematic contamination or miscalibration in the Polarbear data set and analysis. Note that the PTE values in each patch are not independent from each other.

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