MEASUREMENTS OF SUB-DEGREE B-MODE POLARIZATION IN THE COSMIC MICROWAVE BACKGROUND FROM 100 SQUARE DEGREES OF SPTPOL DATA

The data from each detector in each observation are filtered prior to making maps for a number of reasons: to remove modes that are strongly affected by low-frequency noise, to prevent aliasing of high-frequency noise to lower frequencies when the data are binned into map pixels, and to eliminate potential contamination from detector coupling to the pulse-tube cooler used to cool the receiver. Data from each detector in each scan are fit to a combination of a first-order polynomial (mean and slope) and a set of low-order Fourier modes (sines and cosines). The best-fit polynomial and Fourier modes are removed, resulting in an effective high-pass filter. When maps are made from data filtered in this way, the effect of this time-domain filtering is a high-pass filter along the scan direction (equivalent to R.A. for observations from the South Pole) with a cutoff at angular scales of roughly 3° (or an effective multipole number in the scan direction of ${{\ell }}_{x}\sim 100$).³⁷ To avoid filtering artifacts around bright point sources, all sources above 50 mJy in unpolarized flux at 150 GHz are masked in the fitting process. An anti-aliasing low-pass filter is also applied to the data from each detector on each scan, resulting in a scan-direction low-pass filter in the maps with a cutoff of ${{\ell }}_{x}\sim 6600$. This filter cutoff is set by the size of the pixels used in mapmaking, which is 1 arcmin for this analysis. A frequency-domain filter with very narrow notches at the pulse-tube cooler frequency (and the second and third harmonics of this frequency) is applied to the data from each detector over the entire observation. A total of <0.01 Hz of bandwidth (<0.2% of the ${{\ell }}_{x}\lt 6600$ band) is removed with this filter, and we do not include its effect in the simulations of the filter transfer function described in Section 5.1.

The C14 analysis used slightly different filtering choices. The scan-direction high-pass filtering was accomplished in C14 using polynomial subtraction only (no Fourier modes); for this work, we found that the combination of polynomials and Fourier modes resulted in a lower level of spurious B-mode power created in the filtering step (see Section 5.5.1 for details). The anti-aliasing low-pass filter in C14 had a cutoff at ${{\ell }}_{x}\sim 10,000$, because the maps used in that analysis had 0.5 arcmin pixels.

Before the data from all detectors are combined into T, Q, and U maps, a factor is applied to the data from each detector to equalize the response to astronomical signal across each of the two detector arrays (95 and 150 GHz). The process of measuring and monitoring this relative calibration for each array is identical to the process described in C14 and used in earlier SPT analyses (see Schaffer et al. 2011); we describe the process briefly here.

The relative calibration process involves regular observations of the galactic H ii region RCW38 and of an internal chopped blackbody source. A 45-minute observation of the galactic H ii region RCW38 is conducted approximately once per day (shorter observations for pointing reconstruction are conducted more frequently), while 1-minute observations of the internal source are conducted at least once per hour. An effective temperature for the internal source is assigned individually for each detector (by comparing to the season average of response to RCW38). This value times the response of each detector to the internal source in the observation nearest a CMB field observation is used for relative calibration. We assume that RCW38 is unpolarized, and internal measurements suggest that it is <1% polarized. In Section 5.9.3 we discuss the potential for spurious B-mode polarization caused by small polarization in RCW38.

Drifts in the internal source temperature are accounted for by comparing the average source response across a detector module³⁸ in each observation to the module average over the entire season. Drifts in atmospheric opacity are addressed similarly using average RCW38 responses for each module.

Before combining the data from all detectors into T, Q, and U maps, we must know the polarization properties of the detectors. Each detector is designed to be sensitive to linearly polarized radiation at a particular angle and insensitive to radiation in the orthogonal polarization. The two numbers required to characterize each detector's polarization performance are the polarization angle θ (the angle of linearly polarized radiation at which the response of the detector is maximized) and the polarization efficiency η_p (a measure of the ratio of detector response to linearly polarized radiation at θ to radiation polarized at $\theta +\pi /2$).

These properties are measured for each detector in dedicated observations of a polarized calibration source during the Austral summer season. The calibration observations and the method by which θ and η_p are derived from the data are described in detail in C14. We briefly summarize the observations and data reduction here.

The polarization calibrator consists of a chopped thermal source located behind two wire grid polarizers (one at a fixed angle and one that can rotate), physically located 3 km from the telescope. One dual-polarization pixel is pointed at the calibrator, and the rotating polarizer is stepped through nearly 180° while the response of the two detectors is monitored. We fit the response as a function of rotating grid angle to a model including θ and η_p as free parameters. This procedure is repeated for all pixels, with multiple measurements per detector where possible. For detectors for which the measurements or model fits do not pass data quality cuts (∼25% of the detectors used in this work), we assign the median value measured from a subset of detectors, namely, those that are on the same detector module, have the same nominal angle, and did pass data quality cuts. We expect this to be a reliable substitution since these subsets of detectors were designed to have the same angles, and the successfully measured angles are consistent with the median angle.

The median statistical uncertainty on the detector polarization angles is 05 per detector, and the systematic uncertainty on these angles is estimated to be 1°. The mean measured polarization efficiency is 98%, and the median statistical error on the efficiency is 0.7% per detector. A correlated error in the estimation of all polarization angles will result in mixing between E and B modes, and this effect is addressed by the cleaning procedure described in Section 5.4.

We flag and ignore time-ordered data from individual detectors on a per-scan basis and a per-observation basis using cut criteria that are nearly identical to those used in C14. Briefly, we flag data from individual detectors on a per-scan basis based on detector noise and the presence of discontinuities in the data, including spikes (generally attributed to cosmic rays) and sharp changes in DC level (generally attributed to changes in the SQUID operating point). Data from a particular detector are not used for the entire scan if either of these types of discontinuities is detected, or if the rms of the data from that detector in that scan is greater than 3.5 times the median or less than 0.25 times the median. The median rms is calculated across all detectors on a module. These cuts remove roughly 5% of the data.

Data from individual detectors are flagged at the full-observation level based on response to the two types of calibration observations described in Section 4.2 and noise in the frequency band of interest. For a given CMB field observation, data from detectors with low signal-to-noise response to the closest observation of either RCW38 or the internal blackbody source are excluded from the map for that CMB field observation. The noise is calculated for each detector in each observation by taking the difference between left-going and right-going scans, Fourier transforming, and calculating the square root of the mean power between 0.3 and 2 Hz (corresponding to $400\lesssim {\ell }\lesssim 2000$ in the scan direction, roughly the signal band of interest to this work). Data from detectors with abnormally high or low noise in this band are cut from an entire observation. We also enforce that both detectors in a dual-polarization pixel pass all of these cuts; if one detector is flagged, then the pixel partner is flagged as well. We find empirically that this cut improves the low-frequency noise of the resulting maps, presumably by increasing the fidelity of the subtraction of unpolarized atmospheric signal.

After the cuts described in the previous section are applied, the filtered, relatively calibrated, time-ordered data from each detector are combined into T, Q, and U maps using the pointing information, polarization angle, and a weight for each detector. The procedure used in this analysis for polarized mapmaking is similar to earlier work, e.g., Couchot et al. (1999) and Jones et al. (2007). Briefly, for a single observation of the CMB field, weights are calculated for each detector based on detector polarization efficiency and noise rms—i.e., $w\propto {({\eta }_{p}/n)}^{2}$, where η_p is the polarization efficiency (one value per detector for all observations, based on the polarization calibration observations described in Section 4.3), and n is the noise rms, calculated as described in the previous section. The contributions to the weighted estimates of the T, Q, and U values for pixel α from detector i are then

$$\begin{eqnarray}{\hat{T}}_{i\alpha }^{W} & = & \displaystyle \sum _{t}{A}_{{ti}\alpha }\ {w}_{i}\ {d}_{{ti}}\\ {\hat{Q}}_{i\alpha }^{W} & = & \displaystyle \sum _{t}{A}_{{ti}\alpha }\ {w}_{i}\ {d}_{{ti}}\ \mathrm{cos}2{\theta }_{i}\\ {\hat{U}}_{i\alpha }^{W} & = & \displaystyle \sum _{t}{A}_{{ti}\alpha }\ {w}_{i}\ {d}_{{ti}}\ \mathrm{sin}2{\theta }_{i}\end{eqnarray} \tag{ 1 }$$

where t runs over time samples, d_ti is the value recorded by detector i in sample t, w_i is the weight for detector i (defined above), ${A}_{{ti}\alpha }$ is the pointing operator encoding when detector i was pointed at pixel α, and θ_i is the detector polarization angle, and the hat implies measured quantities.

A 3 × 3 matrix representing the T, Q, and U weights and the correlations between the three measurements is created for each map pixel using this same information. The contribution to the weight matrix for the estimate of T, Q, and U in pixel α from detector i is

$$\begin{eqnarray}&&{W}_{i\alpha }\;=\\ \left(\begin{array}{ccc}{w}_{i}{N}_{i\alpha } & {w}_{i}{N}_{i\alpha }\mathrm{cos}2{\theta }_{i} & {w}_{i}{N}_{i\alpha }\mathrm{sin}2{\theta }_{i}\\ {w}_{i}{N}_{i\alpha }\mathrm{cos}2{\theta }_{i} & {w}_{i}{N}_{i\alpha }{\mathrm{cos}}^{2}2{\theta }_{i} & {w}_{i}{N}_{i\alpha }\mathrm{cos}2{\theta }_{i}\mathrm{sin}2{\theta }_{i}\\ {w}_{i}{N}_{i\alpha }\mathrm{sin}2{\theta }_{i} & {w}_{i}{N}_{i\alpha }\mathrm{cos}2{\theta }_{i}\mathrm{sin}2{\theta }_{i} & {w}_{i}{N}_{i\alpha }{\mathrm{sin}}^{2}2{\theta }_{i}\end{array}\right)\\ &&\;\end{eqnarray} \tag{ 2 }$$

where ${N}_{i\alpha }$ is the number of samples in which detector i was pointed at pixel α. The contributions to the weighted T, Q, and U estimates and the weight matrix from each detector in a given observing band are summed, the weight matrix is inverted for each pixel, and the resulting 3-by-3 matrix is applied to the weighted T, Q, and U estimates to produce the unweighted map:

$$\begin{eqnarray}&&\{\hat{T},\hat{Q},\hat{U}\}{}_{\alpha }={W}_{\alpha }^{-1}\{{\hat{T}}_{\alpha }^{W},{\hat{Q}}_{\alpha }^{W},{\hat{U}}_{\alpha }^{W}\}.\end{eqnarray} \tag{ 3 }$$

We make maps using the oblique Lambert azimuthal equal-area projection with 1 arcmin pixels. This sky projection introduces small angle distortions, which we account for by rotating the Q and U Stokes components across the map to maintain a consistent angular coordinate system in this projection. The maps are in units of μK_CMB, the equivalent fluctuations of a 2.73 K blackbody that would produce the measured deviations in intensity.

The cross-spectrum analysis described in Section 5.3 assumes uniform noise properties across all maps used in the analysis and all pixels in the maps. Maps made from a one-half-hour observation (each) of the lead and trail halves of the SPTpol 100d field have sufficiently non-uniform coverage that the cross-spectrum analysis performed on these single maps would be significantly suboptimal. We choose to combine the single-observation maps into 41 subsets or "bundles" and do the cross-spectrum analysis on these bundles. As shown in Polenta et al. (2005), the excess variance on a cross-spectrum of data equally split N ways as compared to an auto-spectrum of the full data scales as $1/{N}^{3}$, so the excess variance with 41 bundles is negligible.

In a process similar to the detector-level data cuts described in Section 4.4, we calculate the distribution of a number of variables over the entire set of single-observation maps, and we only include in bundles the maps that lie within a certain distance of the median for all variables. As in C14, we cut maps based on the rms noise, the total map weight, the median map pixel weight, and the product of median weight times the square of the rms noise. In this analysis, we additionally cut maps based on the number of contributing bolometers and on the rms deviation of the time-ordered data averaged across all detectors from a model of that average time-ordered data (assuming that the primary signal is from the atmosphere). We include only those maps that pass cuts at both 95 and 150 GHz. A total of 4897 of the roughly 6000 individual observations of the full field are included, with approximately 120 observations included in each bundle.

The relative calibration process described in Section 4.2 results in single-observation or bundled T maps that are related to the true temperature on the sky by the filter and beam transfer function and a single overall number, the absolute temperature calibration. (The Q and U maps have an additional factor of the overall polarization efficiency, as discussed in Section 4.8.) The processes for estimating the beam and filter transfer function are described in Section 5.1; we obtain the absolute temperature calibration using the same method used in C14. We briefly describe the method here; for details, see C14.

The absolute calibration for the 150 GHz SPTpol data is derived by comparing full-season coadded temperature maps to temperature maps of the same field made previously with the SPT-SZ receiver, which were in turn calibrated through a comparison of the Story et al. (2013) SPT-SZ temperature power spectrum with the Planck Collaboration XVI (2014) temperature power spectrum. (The uncertainty on the SPT-SZ-to-Planck calibration is estimated to be 1.2% in temperature.) The 95 GHz SPTpol data are then calibrated by comparing to the 150 GHz SPTpol data.

We compare the 150 GHz SPTpol maps used in this work to the 150 GHz SPT-SZ maps of the same field by creating cross-power spectra, using the same method we use for cross-spectra of SPTpol bundle maps, described in Section 5.3. Specifically, we calculate the ratio of the cross-spectrum between a full-depth SPTpol map and a half-depth SPT-SZ map and the cross-spectrum of two half-depth SPTpol maps, corrected for the difference in beams between the two experiments. (The SPT-SZ maps are made with the same scan strategy and filtering as the SPTpol maps, so the filter transfer function divides out of the ratio.) We repeat this for many semi-independent sets of half-depth SPT-SZ maps and use the distribution of the cross-spectrum ratio to calculate the best-fit 150 GHz absolute calibration and the uncertainty on this calibration. A similar process is used to compare 95 GHz SPTpol maps to 150 GHz SPTpol maps and to obtain the 95 GHz absolute calibration. In the power spectrum pipeline, we multiply all SPTpol maps by these calibration factors before calculating cross-spectra.

We calculate the absolute temperature calibration separately for 2012 and 2013 data because the shape and overall width of the observing band changed slightly between the two seasons (see Section 3 for details). In all calculations, we treat the 150 GHz SPTpol full- and half-depth maps as noise-free compared to 95 GHz SPTpol and 150 GHz SPT-SZ full- and half-depth maps, because the roughly 3 times higher noise in the latter two data sets dominates the cross-spectrum uncertainty budget. The fractional statistical uncertainties on the calibration of SPTpol 150 GHz data to SPT-SZ 150 GHz data are 0.004 for both seasons (2012 and 2013). In quadrature with the SPT-SZ-to-Planck uncertainty, this results in a total calibration uncertainty of 1.3% at 150 GHz for both seasons, highly correlated between the two seasons. The statistical uncertainty on the comparison of SPTpol 95 GHz data to SPTpol 150 GHz data is <0.002 for both seasons, which contributes negligibly to the total uncertainty budget; thus, the total calibration uncertainty at 95 GHz is also 1.3% and nearly 100% correlated with the 150 GHz uncertainties. All of these uncertainties are much smaller than the fractional uncertainty with which we expect to measure the BB spectrum, and they will make a negligible contribution to the uncertainty on the power spectrum or any derived parameters.

Although the polarization angles and efficiencies are measured with a ground-based polarization source as described in Section 4.3, we allow for additional freedom in the normalization of the Q/U maps. This is to account for any mechanism that could alter the effective polarization calibration. One such mechanism, the impact of electrical crosstalk on our effective polarized beam, is discussed in Section 5.9.4. The polarization calibration is calculated by comparing the EE spectrum measured in this work to the EE spectrum from the best-fit ΛCDM model for the Planck+lensing+WP+highL constraint in Table 5 of Planck Collaboration XVI (2014). The calculation of the measured EE spectrum mirrors that of the BB spectrum as described in Section 5. The polarization calibration factor for spectral band $m\in \{95\;\mathrm{GHz},150\;\mathrm{GHz}\}$ is calculated as

$$\begin{eqnarray}&&{({P}_{\mathrm{cal}}^{m})}^{2}=\displaystyle \sum _{b}\left({C}_{b}^{{EE},m\times m,\mathrm{theory}}/{C}_{b}^{{EE},m\times m,\mathrm{data}}\right){w}_{b}\end{eqnarray} \tag{ 4 }$$

where b is the ℓ-bin index and w_b are simulation-based, inverse-variance weights that sum to 1. We perform this calculation after applying the absolute temperature calibration described in Section 4.7. We find ${P}_{\mathrm{cal}}^{95}=1.015\pm 0.024$ and ${P}_{\mathrm{cal}}^{150}=1.049\pm 0.014$, and we multiply the data polarization maps by these factors. In Section 5.9 we discuss the impact of the uncertainty of these calibration factors on the BB power spectrum measurement.

We construct an apodization mask, ${\mathcal{W}}$, to downweight the high-noise regions at the boundary of the SPTpol coverage area and to mitigate mode coupling. The apodization mask is constructed as follows. Each map bundle has an associated weight array that is approximately inversely proportional to the square of the white-noise level at each pixel of the temperature map. We smooth each weight array with a σ = 10' Gaussian kernel, threshold the smoothed array on 0.05 of its median value, and take the intersection of all thresholded arrays across all bundles. We then pad the boundary of the intersection map with ${5}^{\prime }$ of zeros and smooth with a $\lambda \ =\ {60}^{\prime }$ cosine kernel. We perform this process at 95 and 150 GHz, and the final apodization mask ${\mathcal{W}}$ is the product of the individual 95 and 150 GHz masks.

We perform two map-space cleaning operations on the bundle $\{T,Q,U\}$ maps. These operations are performed on both the data maps and the simulation maps described in Section 5.1.

First, to reduce our sensitivity to bright, emissive sources, we interpolate over regions of the map near all sources with unpolarized fluxes ${S}_{150}\gt 50$ mJy. For each source, we replace the values of the map within $r\lt {6}^{\prime }$ of the source with the median value of the map in an annulus defined by ${6}^{\prime }\lt r\lt {10}^{\prime }$.

Second, we filter the maps to reduce our sensitivity to scan-synchronous signals. We see evidence for a polarized scan-synchronous signal with ∼5 μK rms at 95 GHz and one with ∼0.6 μK rms at 150 GHz. We clean these signals by removing from each map a template that is only a function of R.A. Owing to SPT's polar location and constant-elevation scan strategy, R.A. is essentially equivalent to the scan coordinate, and any signal that depends only on the scan coordinate will depend only on R.A. We construct the scan-synchronous template for each map by averaging the map in bins of R.A. and then smoothing this binned function with a 1°-in-R.A. Hann function. The resulting one-dimensional template is expanded to a two-dimensional template and then subtracted from the original, two-dimensional map.

A number of steps in the power spectrum analysis rely on mock maps, and here we describe the process by which the mock maps are generated.

First, we generate noise-free sky maps that form the input to time-ordered-data simulations. The input sky maps are Gaussian realizations of $\{T,Q,U\}$ generated in the HEALPix pixelized sphere format (Górski et al. 2005) with 043 $({n}_{\mathrm{side}}=8192)$ resolution. The sky maps are generated using $\{{TT},{TE},{EE},{BB}\}$ CMB power spectra computed using the CAMB Boltzmann code (Lewis et al. 2000). The ΛCDM cosmological parameters input to CAMB are taken from the Planck+lensing+WP+highL best-fit model in Table 5 of Planck Collaboration XVI (2014). There is no tensor power (r = 0) and no foreground power. There are two sets of input sky maps.

• 1.  
  TEB—These sky maps use all of the non-zero ΛCDM CMB power spectra: $\{{TT},{TE},{EE},{BB}\}$. The BB spectrum is due to gravitational lensing of E-mode polarization.
• 2.  
  TE—These sky maps are identical to those described above, including identical random seeds, but do not include B-mode polarization power: ${C}_{{\ell }}^{{BB}}=0$.

We generate 100 realizations of each set of input sky maps and convolve them with an azimuthally symmetric beam function. The beam function is measured using observations of planets and PKS 2355–534, the brightest source in the 100d field at these observing frequencies. We use four distinct beam functions, one for each combination of spectral band and year of observations, to generate input sky maps for each spectral band and year of observation.

Next, we use the SPTpol pointing information to generate time-ordered data representing mock observations of the input sky maps. Before reducing the time-ordered data to maps, we simulate the effect of "crosstalk," the mixing of time-ordered data between different detectors. We have used observations of the galactic H ii region RCW38 to measure percent-level crosstalk between some detectors. The crosstalk is believed to originate in the readout electronics. We define a crosstalk matrix V_ab to denote the coupling between detectors a and b, and we use V_ab to mix the simulated time-ordered data of different detectors. The main effect of crosstalk for the BB analysis presented here is $T\to B$ leakage, which we discuss further in Section 5.5.1. After introducing the effect of crosstalk, the simulated time-ordered data are reduced to bundle maps in a manner that matches the reduction of the real data as described in Section 4.

Finally, we add realistic noise to the simulated bundle maps using differences of the data maps. We generate a single "realization" of noise for a specific simulated map bundle by combining a random half of its ∼120 constituent data maps into one coadded map, the remaining half into another coadded map, and differencing the two. The resulting difference has no true sky signal and should be statistically representative of the noise in this particular map bundle, modulo one small difference. When making the normal map bundles, we combine the two halves with their respective weights, but when making the noise bundles, we combine with equal weights. We estimate that this results in the power in the noise bundles being ∼0.1% larger than the true noise power.

We generate 200 realizations of noise for each bundle in this manner. The constituent data maps are differenced differently in each noise realization. In total there are 200 realizations of noisy map bundles. Each of the 100 input sky realizations is used twice, while each of the 200 noise realizations is used once.

Here we describe the process by which ${E}_{{\boldsymbol{\ell }}}$ and ${B}_{{\boldsymbol{\ell }}}$, the harmonic-space representations of the E-mode and B-mode polarizations, are constructed from the real-space $\{Q,U\}$ maps. First, we address small angle distortions introduced by our use of the Lambert azimuthal equal-area projection. We account for these distortions by rotating the Q and U Stokes components across the map to maintain a consistent angular coordinate system.

To construct the harmonic-space representation of the E-mode polarization, we use the standard convention (Zaldarriaga 2001)

$$\begin{eqnarray}&&{E}_{{\boldsymbol{\ell }}}={Q}_{{\boldsymbol{\ell }}}\mathrm{cos}2{\phi }_{{\boldsymbol{\ell }}}+{U}_{{\boldsymbol{\ell }}}\mathrm{sin}2{\phi }_{{\boldsymbol{\ell }}}\end{eqnarray} \tag{ 5 }$$

where ${\phi }_{{\boldsymbol{\ell }}}$ is the azimuthal angle of ${\boldsymbol{\ell }}$, ${P}_{{\boldsymbol{\ell }}}\equiv {\mathcal{F}}\{P{\mathcal{W}}\}$ is the Fourier transform of each apodized map $P\in \{Q,U\}$, and ${\mathcal{W}}$ is the apodization window.

To construct the harmonic-space representation of the B-mode polarization, we use the χ_B method described in Smith & Zaldarriaga (2007). In this method, an intermediate χ_B map is constructed as the sum of convolutions of the unapodized Q and U maps. The kernels for these convolutions are compact in real space; each pixel in the χ_B map is a curl-like linear combination of all Q/U pixels within the surrounding square of 5 × 5 pixels. Finally, we construct ${B}_{{\boldsymbol{\ell }}}$, the harmonic-space representation of the B-mode polarization,

$$\begin{eqnarray}&&{B}_{{\boldsymbol{\ell }}}={(({\ell }-1){\ell }({\ell }+1)({\ell }+2))}^{-\frac{1}{2}}{\mathcal{F}}\{{\chi }_{B}{\mathcal{W}}\}\end{eqnarray} \tag{ 6 }$$

where ${\mathcal{F}}\{{\chi }_{B}{\mathcal{W}}\}$ is the Fourier transform of the apodized χ_B map, and the ∼ℓ⁻² factor accounts for the second-order spatial derivative present in the curl-like linear combination of the Q and U maps.

We employ a cross-spectrum pseudo-C_ℓ approach to estimate the BB power spectrum, as described below. The first step is to define the two-dimensional cross-spectrum between two CMB fields $(\alpha ,\beta )\in \{T,E,B\}$, each coming from a spectral band $({\nu }_{m},{\nu }_{n})\in \{95,150\;\mathrm{GHz}\}$:

$$\begin{eqnarray}&&{\widehat{C}}_{{\boldsymbol{\ell }}}^{{\alpha }^{m}{\beta }^{n}}\equiv \displaystyle \frac{1}{{n}_{\mathrm{pairs}}}\displaystyle \sum _{(i,\ j),i\ne j}\mathrm{Re}\{{({\alpha }_{{\boldsymbol{\ell }}}^{m,i})}^{*}{\beta }_{{\boldsymbol{\ell }}}^{n,j}\}\end{eqnarray} \tag{ 7 }$$

where (i, j) are indices of distinct map bundles. These quantities are defined on a two-dimensional Fourier grid with resolution $\delta {\ell }=25$. The next step is to calculate one-dimensional bandpowers. These are the weighted sums of the two-dimensional spectra within a set of ℓ-bins $\{b\}$,

$$\begin{eqnarray}&&{\widehat{C}}_{b}^{{\alpha }^{m}{\beta }^{n}}\equiv \displaystyle \sum _{{\boldsymbol{\ell }}\in b}{\widehat{C}}_{{\boldsymbol{\ell }}}^{{\alpha }^{m}{\beta }^{n}}{H}_{{\boldsymbol{\ell }}}^{{mn}}.\end{eqnarray} \tag{ 8 }$$

The bandpowers are defined on five ℓ-bins, each $\delta {\ell }=400$ wide, running from ${{\ell }}_{\mathrm{min}}=300$ to ${{\ell }}_{\mathrm{max}}=2300$. The two-dimensional Fourier weight ${H}_{{\boldsymbol{\ell }}}^{{mn}}$ is defined as

$$\begin{eqnarray}&&{H}_{{\boldsymbol{\ell }}}^{{mn}}\equiv {W}_{{\boldsymbol{\ell }}}^{{mn}}{Z}_{{\boldsymbol{\ell }}}^{{mn}}\end{eqnarray} \tag{ 9 }$$

where ${W}_{{\boldsymbol{\ell }}}^{{mn}}$ is a Wiener filter optimized for detecting a ΛCDM lensed BB power spectrum and ${Z}_{{\boldsymbol{\ell }}}^{{mn}}$ is a binary mask. The Wiener filter is defined as

$$\begin{eqnarray}&&{W}_{{\boldsymbol{\ell }}}^{{mn}}\equiv \displaystyle \frac{{\langle {\widehat{C}}_{{\boldsymbol{\ell }}}^{{B}^{m}{B}^{n},\mathrm{TEB}}-{\widehat{C}}_{{\boldsymbol{\ell }}}^{{B}^{m}{B}^{n},\mathrm{TE}}\rangle }_{\mathrm{sims}}}{\mathrm{Var}{\left\{{\widehat{C}}_{{\boldsymbol{\ell }}}^{{B}^{m}{B}^{n},\mathrm{TE}}\right\}}_{\mathrm{sims}}}\end{eqnarray} \tag{ 10 }$$

where both the mean and variance are taken across the set of 200 noisy simulations. The numerator is the mean two-dimensional power spectrum of the signal of interest, the lensed BB power spectrum, and the denominator is the two-dimensional variance in the absence of this signal. We note that this filter is optimized to reject the hypothesis of no lensed BB power, while the filter optimized to constrain the amplitude of the lensed BB power would use $\mathrm{TEB}$ simulations in the denominator. Given that the data presented here are noise-variance-limited, the two filters are quite similar. Both the numerator and denominator are smoothed by a Gaussian kernel with ${\mathrm{FWHM}}_{{\ell }}=150$ prior to taking their ratio.

The binary mask ${Z}_{{\boldsymbol{\ell }}}^{{mn}}$ is used to mask out modes satisfying $| {{\ell }}_{x}| \lt 175$ or $| {{\ell }}_{y}| \lt 150$ for any spectrum involving 95 GHz data. This mask is necessary to pass some of the jackknife tests described in Section 5.8.2.

We note that while ${H}_{{\boldsymbol{\ell }}}^{{mn}}$ has been defined to optimize sensitivity to the lensed BB spectrum, we simply use the same weight for the other CMB field combinations (EB, EE, etc.), as the BB spectrum is the focus of interest for this work. In Figure 1 we show the real-space representation of the B modes analyzed in this work, namely, ${B}^{m}=\mathrm{Re}\{{{\mathcal{F}}}^{-1}\{\sqrt{{H}_{{\boldsymbol{\ell }}}^{{mm}}}{B}_{{\boldsymbol{\ell }}}^{m}\}\}$. Each map B^m has been multiplied by the apodization mask ${\mathcal{W}}$ for the purpose of visualization.

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Given that the CMB temperature and E-mode polarization signals are expected to contain significantly more power than the B-mode polarization signal, the potential for leakage from T or E into B is a central issue for any BB analysis. The leakage from a number of instrumental effects—gain mismatch, differential pointing, differential beam ellipticity, incorrect polarization angles, crosstalk, etc.—appears (to first order) as a spatially local leakage of the T or E skies into B. In other words, the leakage appears as a convolution of T or E into B. Here we describe a process, "convolutional cleaning," by which we reduce the impact of convolutional $(T,E)\to B$ leakage. The basic idea is to use the CMB T and E modes themselves to estimate and project out convolutional leakage for O(100) convolution kernels per spectral band. After applying this cleaning, we are largely insensitive to any potential parity-violating cosmological TB or EB signal.

We note that the cleaning process described here is similar to a number of other techniques used to clean convolutional T and E leakage. For example, a global polarization angle offset causes convolutional $E\to B$ leakage, and a number of recent works (BICEP2 Collaboration 2014; POLARBEAR Collaboration 2014a; Naess et al. 2014; Keck Array & BICEP2 Collaborations et al. 2015) use the EB spectrum, and to a lesser degree the TB spectrum, to fit for (and in some cases correct for) this offset. This technique fits the one-dimensional EB spectrum to a single parameter, the polarization angle offset. Another technique, "deprojection" (BICEP2 Collaboration 2014; Keck Array & BICEP2 Collaborations et al. 2015), cleans any potential convolutional $T\to B$ leakage caused by differential gain, differential pointing, or differential beam ellipticity. This technique takes advantage of the well-measured temperature sky and is performed on a per-detector-pair basis in the time-ordered data. Finally, a previous SPTpol analysis (C14) measured and corrected for convolutional $T\to (Q,U)$ leakage using TQ and TU spectra and corrected the TE and EE spectra for the additional effects of crosstalk. The convolutional cleaning process described below allows for a more general and more aggressive form of cleaning than that used in C14. This change was motivated by the more stringent cleaning requirements of a B-mode analysis.

We consider leakage of T or E into B that is described by a real-space convolution kernel, or equivalently, a multiplication in two-dimensional Fourier space by a complex coupling function ${f}_{{\boldsymbol{\ell }}}$. For example, the leakage from T to B for a particular coupling function ${f}_{{\boldsymbol{\ell }}}$ looks like

$$\begin{eqnarray}&&{B}_{{\boldsymbol{\ell }}}=A\;{f}_{{\boldsymbol{\ell }}}{T}_{{\boldsymbol{\ell }}}.\end{eqnarray} \tag{ 11 }$$

The strength of the leakage in the data is set by the parameter A, and we estimate A in each ℓ-bin b for each coupling function. In other words, we make the approximation that the amplitude of a particular form of leakage is constant over one ℓ-bin. There is still freedom for ℓ-dependence of the leakage because the amplitude is allowed to vary between ℓ-bins.

We use coupling functions of the form

$$\begin{eqnarray}&&{f}_{{\boldsymbol{\ell }}}^{k\theta }=\mathrm{cos}(k{\phi }_{{\boldsymbol{\ell }}}+\theta )\;{i}^{k}\end{eqnarray} \tag{ 12 }$$

where k is a small integer $k\in \{0,1,\ldots ,{k}_{\mathrm{max}}\}$, ${\phi }_{{\boldsymbol{\ell }}}$ is the azimuthal angle in two-dimensional Fourier space, $\theta \in \{0,\pi /2\}$, and the i^k factor ensures that f corresponds to a purely real (rather than imaginary) real-space kernel. We use this form because the coupling functions for well-known types of leakage can be expressed as linear combinations of couplings with the form described above. For example, differential gain has a TB coupling ${f}_{{\boldsymbol{\ell }}}\propto \mathrm{cos}(2{\phi }_{{\boldsymbol{\ell }}})$, differential pointing has a TB coupling ${f}_{{\boldsymbol{\ell }}}\propto i{\ell }\left(\mathrm{cos}({\phi }_{{\boldsymbol{\ell }}})+\mathrm{cos}(3{\phi }_{{\boldsymbol{\ell }}})\right)$, and a global polarization angle offset has an EB coupling ${f}_{{\boldsymbol{\ell }}}\propto 1$. We emphasize that we do not assume a particular physical mechanism (such as differential pointing or differential ellipticity) for this leakage. Rather, we assume that the leakage is varying slowly in ℓ and has an azimuthal dependence described by $\mathrm{cos}(k{\phi }_{{\boldsymbol{\ell }}}+\theta )$ for small integer k. We use ${k}_{\mathrm{max}}=4$, which allows for the cleaning of any of the types of leakage described above. We find that the main result of this work, the constraint on the amplitude of lensing B-mode power, is insensitive to the particular choice of ${k}_{\mathrm{max}}$; the lensing amplitude shifts by ∼0.1σ when trying ${k}_{\mathrm{max}}\in \{2,4,6\}$. Conversely, we find that convolutional cleaning is an important step in this analysis. In the absence of convolutional cleaning, the nominal estimate of the lensing amplitude shifts significantly upward, by ∼3σ. This shift is driven mainly by changes in the first two bins of the 150 GHz × 150 GHz BB spectrum, which contain most of the sensitivity to lensing B modes.

The linear estimate of the amplitude A of a particular form of leakage ${f}_{{\boldsymbol{\ell }}}^{k\theta }$ in ℓ-bin b of the data ${B}_{{\boldsymbol{\ell }}}^{m}$ for spectral band m is

$$\begin{eqnarray}&&{\hat{A}}_{b}^{k\theta {Xm}}=\displaystyle \frac{{\displaystyle \sum }_{{\ell }\in b}\mathrm{Re}\{{({B}_{{\boldsymbol{\ell }}}^{m})}^{*}{X}_{{\boldsymbol{\ell }}}{f}_{{\boldsymbol{\ell }}}^{k\theta }\}{H}_{{\boldsymbol{\ell }}}^{{mm}}}{{\displaystyle \sum }_{{\ell }\in b}{| {X}_{{\boldsymbol{\ell }}}{f}_{{\boldsymbol{\ell }}}^{k\theta }| }^{2}{H}_{{\boldsymbol{\ell }}}^{{mm}}}\end{eqnarray} \tag{ 13 }$$

where $X\in \{T,E\}$ and the weight ${H}_{{\boldsymbol{\ell }}}^{{mm}}$ is defined in Section 5.3. The numerator is linear in the data ${B}_{{\boldsymbol{\ell }}}^{m}$, and the denominator is used to normalize the estimate. Because the 150 GHz data are deeper than the 95 GHz data, we use the 150 GHz ${T}_{{\boldsymbol{\ell }}}$ and ${E}_{{\boldsymbol{\ell }}}$ modes to estimate the leakage in the 95 and 150 GHz ${B}_{{\boldsymbol{\ell }}}$ modes. We use a cross-spectrum approach to estimate both $\mathrm{Re}\{{({B}_{{\boldsymbol{\ell }}}^{m})}^{*}{X}_{{\boldsymbol{\ell }}}{f}_{{\boldsymbol{\ell }}}^{k\theta }\}$ and ${| {X}_{{\boldsymbol{\ell }}}{f}_{{\boldsymbol{\ell }}}^{k\theta }| }^{2}$. For our baseline value of ${k}_{\mathrm{max}}=4$, there are a total of 18 leakage modes per ℓ-bin, or 90 modes for all five ℓ-bins, per spectral band.

Ideally, we would use these estimates of the leakage amplitudes to remove from the data ${B}_{{\boldsymbol{\ell }}}^{m}$ the inferred leaked B mode,

$$\begin{eqnarray}&&{B}_{{\boldsymbol{\ell }}}^{\mathrm{leak},m}=\displaystyle \sum _{k\theta {Xb}}{\hat{A}}_{b}^{k\theta {Xm}}{X}_{{\boldsymbol{\ell }}}{f}_{{\boldsymbol{\ell }}}^{k\theta }{q}_{b{\boldsymbol{\ell }}}\end{eqnarray} \tag{ 14 }$$

where the two-dimensional binning operator ${q}_{b{\boldsymbol{\ell }}}$ is one for ${\boldsymbol{\ell }}\in b$ and zero otherwise. In practice, however, as a result of the non-orthogonality of the different modes being removed, the cleaning process does not converge after one iteration. We address this issue using a damped, iterative scheme. In each iteration we estimate the amplitude of each form of leakage; subtract a damped version of the inferred leaked mode, $0.2{B}_{{\boldsymbol{\ell }}}^{\mathrm{leak},m}$, from ${B}_{{\boldsymbol{\ell }}}^{m}$; and repeat for all forms of leakage. We iterate over these steps 20 times, at which point the process has converged. We use the accumulated estimate for each leakage amplitude to construct the final inferred leakage mode, ${B}_{{\boldsymbol{\ell }}}^{\mathrm{leak},m}$. Finally, we subtract ${B}_{{\boldsymbol{\ell }}}^{\mathrm{leak},m}$ from ${B}_{{\boldsymbol{\ell }}}^{m,i}$ for each bundle i. To be clear, the same ${B}_{{\boldsymbol{\ell }}}^{\mathrm{leak},m}$ is subtracted from all bundles. In other words, all two-dimensional ${B}_{{\boldsymbol{\ell }}}$ that appear in this work have been cleaned in this way, with one exception: the Fourier weights ${H}_{{\boldsymbol{\ell }}}$ are calculated using ${B}_{{\boldsymbol{\ell }}}$ that have not been cleaned.

We find that the convolutional cleaning process strongly suppresses a form of convolutional leakage that we expect to be present in the SPTpol data: crosstalk-induced leakage. We have used our simulated maps to determine that convolutional cleaning reduces the additive bias to the BB spectrum introduced by crosstalk by factors of ∼20–100. The cost of this improvement, however, is the introduction of a small, additive noise bias, as discussed below in Section 5.5.1 and in the appendix.

Here we describe the process by which the raw BB bandpowers described in Section 5.3 are processed into unbiased bandpowers. The raw bandpowers are subject to additive and multiplicative biases, and we correct for these biases using simulated bandpowers.

5.5.1. Additive Bias

Our strategy for dealing with additive bias—any BB power that is present in the absence of a true B-mode polarization signal—is straightforward. We measure the mean BB power present in noisy, simulated bandpowers generated using $\mathrm{TE}$ input skies (i.e., no true BB power), and we subtract this bias from the data bandpowers and the simulated $\mathrm{TEB}$ bandpowers. This subtraction is performed prior to correcting for any multiplicative bias. The additive biases can to some degree be organized as follows.

• 1.  
  $E\to B$ from geometry and filtering—imperfect separation of E and B on a small area of sky and the high-pass filtering of the time-ordered data result in $E\to B$ leakage. The resulting bias in BB is approximately +0.6σ in the lowest ℓ-bin of the 150 × 150 spectrum, and smaller than 0.1σ in other ℓ-bins of this spectrum or any ℓ-bin of the other spectra.
• 2.  
  $(T,E)\to B$ from crosstalk—the electrical crosstalk between detectors results in $(T,E)\to B$ leakage, predominantly $T\to B$. Although the raw crosstalk-induced additive bias is as high as 1.0σ in the lowest ℓ-bin of 150 × 150, convolutional cleaning reduces this to $\lt 0.1\sigma$.
• 3.  
  Negative noise bias from convolutional cleaning—we find that the convolutional cleaning step introduces a negative bias. By analyzing two sets of simulations that have convolutional cleaning turned on and off, we can isolate this effect. We find that its magnitude and shape are in broad agreement with a simplified analytic calculation given in the appendix. The amplitude of the bias is approximately −0.4σ in the lowest two ℓ-bins of the 95 × 95 and 150 × 150 spectra, smaller elsewhere, and approximately zero at 95 × 150. The convolutional cleaning process has strongly attenuated biases arising from convolutional leakage (crosstalk, beam systematics, etc.) in exchange for a small noise bias that we can characterize effectively perfectly.

The additive biases are shown in Figure 2. We subtract these biases from the measured spectra, and we address systematic uncertainties associated with the additive biases in Section 5.9.

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5.5.2. Multiplicative Bias

After removing the additive bias, we correct for the multiplicative bias. The bandpowers ${\widehat{C}}_{b}$ are a biased estimate of the true binned sky power, ${C}_{{b}^{\prime }}$, due to effects such as time-ordered data filtering, beam smoothing, finite sky coverage, and mode-mode mixing from the source and apodization mask. The biased and unbiased estimates are related by

$$\begin{eqnarray}&&{\widehat{C}}_{b}^{{B}^{m}{B}^{n}}\equiv {K}_{{{bb}}^{\prime }}^{{B}^{m}{B}^{n}}{C}_{{b}^{\prime }}^{{B}^{m}{B}^{n}}\end{eqnarray} \tag{ 15 }$$

where the K matrix accounts for the effects of the instrumental beam and time-ordered data filtering and the application of the apodization mask (${\mathcal{W}}$). K can be expanded as

$$\begin{eqnarray}&&{K}_{{{bb}}^{\prime }}={P}_{b{\ell }}\left({M}_{{\ell }{{\ell }}^{\prime }}[{\mathcal{W}}]\;{F}_{{{\ell }}^{\prime }}\right){Q}_{{{\ell }}^{\prime }{b}^{\prime }}.\end{eqnarray} \tag{ 16 }$$

where ${F}_{{\ell }}$ is the effective transfer function from the beam and the filtering of the time-ordered data, ${P}_{b{\ell }}$ is the binning operator, and ${Q}_{{{\ell }}^{\prime }{b}^{\prime }}$ its reciprocal (Hivon et al. 2002). The mode-coupling kernel ${M}_{{\ell }{{\ell }}^{\prime }}[{\mathcal{W}}]$ accounts for the mixing of power between bins due to the apodization mask. The mode-coupling kernel is calculated analytically, as described in the Appendix of C14. The calculation corrects only for $B\to B$ coupling and effectively assumes that there is no $E\to B$ coupling. The latter is, of course, not true, but we correct for $E\to B$ leakage by subtracting off an additive bias, as described in Section 5.5.1. As in C14, the effective transfer function ${F}_{{\ell }}$ is calculated by comparing the mean simulated spectra to the theory spectrum input to the simulations. The calculation is iterative and converges after two iterations.

We approximate the covariance between BB bandpowers as completely diagonal, and we take the variance of bandpowers from the set of 200 noisy simulations as our estimate of the diagonal variances. There are two sets of covariances: those that use $\mathrm{TE}$ input skies and thus naturally account for the noise variance and the variance from T and E leakage, and those that use $\mathrm{TEB}$ input skies, which additionally account for variance from true BB power. The $\mathrm{TE}$ variances are useful for rejecting the ${C}^{{BB}}=0$ hypothesis. The $\mathrm{TEB}$ variances are only slightly larger than the $\mathrm{TE}$ variances, at most 20% larger, implying that noise power dominates over signal power, as expected.

We have used the noisy simulated bandpowers to assess the validity of the diagonal covariance approximation. We attempted to measure the covariance between off-diagonal spectral combinations, e.g., (95 GHz × 95 GHz) and (95 GHz × 150 GHz), or between neighboring ℓ-bins. In each case, the distribution of covariance estimates is consistent with zero. This is not surprising given that the covariance is dominated by noise power and the noise is expected to be uncorrelated between spectral bands and between ℓ-bins. We have also explicitly tested the importance of the off-diagonal covariance terms at (95 × 95) × (95 × 150) and (150 × 150) × (95 × 150). We estimated these terms using an analytic approximation, altered the covariance matrix accordingly, and found that the main result of this work, the constraint on the amplitude of lensing B-mode power, changed by ∼0.1σ. For simplicity, we do not include these off-diagonal terms.

We note that our simulation input skies contain Gaussian realizations of lensed BB power and therefore do not account for the ∼20% inter-ℓ-bin correlation present for non-Gaussian, truly lensed B modes (Benoit-Lévy et al. 2012). Given that the non-Gaussian structure leads to a 20% bin-bin correlation on a source of power (lensed BB power) that itself contributes only 10%–20% of the bandpower variance, we ignore the non-Gaussian contribution in our $\mathrm{TEB}$ variances.

Finally, we note that, because our simulations were free of foreground power, we have slightly underestimated the BB variance. Power from randomly distributed, polarized point sources should be negligible; even the upper limits of the Poisson powers considered in Section 6 are <1% of the data noise power. Variance from polarized galactic dust should also be small compared to the noise power. For our nominal dust model described in Section 6.1, the dust power is approximately 1% of the noise power in the lowest ℓ-bin of the 150 GHz × 150 GHz spectrum, and smaller elsewhere.

We compute bandpower window functions to allow the measured bandpowers to be compared to theoretical power spectra. The window functions, ${w}_{{\ell }}^{b}/{\ell }$, are defined through the relation

$$\begin{eqnarray}&&{C}_{b}=({w}_{{\ell }}^{b}/{\ell }){C}_{{\ell }}.\end{eqnarray} \tag{ 17 }$$

Following the formalism described previously in Section 5.5.2, this can be rewritten as

$$\begin{eqnarray}&&{C}_{b}={({K}^{-1})}_{{{bb}}^{\prime }}{P}_{b\prime {\ell }\prime }{M}_{{\ell }\prime {\ell }}{F}_{{\ell }}{C}_{{\ell }},\end{eqnarray} \tag{ 18 }$$

which implies that

$$\begin{eqnarray}&&{w}_{{\ell }}^{b}/{\ell }={({K}^{-1})}_{{{bb}}^{\prime }}{P}_{b\prime {\ell }\prime }{M}_{{\ell }\prime {\ell }}{F}_{{\ell }}.\end{eqnarray} \tag{ 19 }$$

In this section we test the internal consistency of the data by considering two types of null tests. First, we consider the TB and EB spectra. Second, we consider a suite of "jackknife" tests that are sensitive to potential instrumental systematics.

5.8.1. TB and EB Spectra

5.8.2. Jackknives

We perform a suite of "jackknife" tests to further validate the consistency of the data. In each jackknife test the bundle maps are sorted according to a metric designed to trace a potential systematic effect, and "jackknife maps" are constructed by differencing high-metric bundle maps with low-metric bundle maps. The resulting jackknife maps should not contain sky signal and should have a power spectrum, the jackknife spectrum, consistent with noise. We construct the jackknife spectrum by taking the mean cross-spectrum among the jackknife bundle maps. We note that we do not apply the convolutional cleaning step to the bundle maps going into the jackknife maps. The convolutional cleaning step would have no effect, as it would remove the same modes identically from each of the maps prior to taking their difference.

We perform four jackknife tests.

• 1.  
  Left–Right: Jackknife maps are made by differencing data from left-going and right-going scans. This tests for any power or systematic effect that is present more strongly in one scan direction, such as spurious power caused by the step in telescope elevation at the end of each left–right scan pair.
• 2.  
  Ground: Jackknife maps are made by dividing the maps into two subsets thought to have different susceptibility to ground contamination. We use the same azimuthal range metric used in SPT-SZ power spectrum analyses such as Shirokoff et al. (2011). This tests for spurious power from nearby buildings or other sources of contamination that are fixed to the ground.
• 3.  
  1st half–2nd half: Jackknife maps are made by differencing data from the first and second halves of the set of chronologically ordered map bundles. This tests for any systematic associated with the changes made to the SPTpol receiver between 2012 and 2013, systematics associated with small changes to the scan strategy made in late 2012, or any other slowly varying systematic.
• 4.  
  Visual Inspection: Jackknife maps are made by differencing map bundles that show visually identified anomalies (e.g., faint stripes or spots) and those that do not. This tests for systematic power associated with these features.

We use a slightly different notation for the jackknife bandpowers than is used for the main bandpowers. Each jackknife bandpower is C^fsj_b, where $f\in \{{BB},{EB},{TB}\}$ denotes a CMB field combination, $s\in \{$ 95 × 95, 95 × 150, 150 × 150 $\}$ denotes a spectral combination, and $j\in \{\mathrm{LR},\ \mathrm{GROUND},\mathrm{TIME},\ \mathrm{VIS}\}$ denotes a jackknife.

We test the consistency of the jackknife spectra with the null hypothesis as follows. We estimate the diagonal covariance of the jackknife bandpowers, ${\sigma }^{2}\left({C}_{b}^{{fsj}}\right)$, using the variance among the cross-spectra divided by the number of unique cross-spectra. Next, we define jackknife "χ bandpowers,"

$$\begin{eqnarray}&&{\chi }_{b}^{{fsj}}\equiv \displaystyle \frac{{C}_{b}^{{fsj}}}{\sigma \left({C}_{b}^{{fsj}}\right)}\end{eqnarray} \tag{ 20 }$$

and calculate four test statistics to determine the compatibility of this set of spectra with the null hypothesis. These test statistics are:

• 1.  
  ${\mathrm{max}}_{{fsj}}\left(| {\sum }_{b}{\chi }_{b}^{{fsj}}| \right)$—this tests for spectra that are preferentially positive or negative across the full multipole range;
• 2.  
  ${\mathrm{max}}_{{fsj}}\left({\sum }_{b}{({\chi }_{b}^{{fsj}})}^{2}\right)$—this tests for spectra that preferentially have outlying bandpowers;
• 3.  
  ${\mathrm{max}}_{{bfsj}}\left({({\chi }_{b}^{{fsj}})}^{2}\right)$—this tests for any particularly strong outlying bandpower;
• 4.  
  ${\displaystyle \sum }_{{bfsj}}{({\chi }_{b}^{{fsj}})}^{2}$—this tests for a general tendency to have outlying bandpowers.

We compare the data values of these test statistics to those obtained using a set of 10,000 zero-mean, unit-width Gaussian realizations of each ${\chi }_{b}^{{fsj}}$. We calculate the probability to exceed (PTE) the value of each data test statistic given the values in the set of simulated test statistics. Finally, we define a global test statistic, ${P}_{\mathrm{joint}}$, the probability to simultaneously exceed all of the test statistics, and calculate the probability to exceed $1-{P}_{\mathrm{joint}}$, again using simulations.

As the focus of this work is the BB spectrum, our nominal set of jackknife CMB field combinations is simply $\{{BB}\}$. We find that the PTEs for the four test statistics are 0.29, 0.60, 0.15, and 0.09, and the global PTE is 0.22. We take this as evidence that the BB spectra do not have significant contamination under these jackknife tests.

We have also repeated these jackknife tests using an expanded set of CMB field combinations, $\{{BB},{EB},{TB}\}$. Owing to the high signal-to-noise imaging of T and E modes, these tests are in principle more difficult to pass. In this case, the PTEs for the four test statistics are 0.41, 0.59, 0.39, and 0.02, and the global PTE is 0.14. We take this as further evidence that the B-mode data used in this work do not have significant contamination under these jackknife tests.

Here we discuss several potential sources of systematic uncertainty in the BB power spectrum measurement. In all cases we demonstrate that the systematic uncertainty is much smaller than the statistical uncertainties and can be safely ignored.

5.9.1. Uncertainty in Additive BB Bias

5.9.2. Uncertainty in Multiplicative Bias in BB

5.9.3.  $T\to Q/U$ Leakage

5.9.4. Effect of Crosstalk on Beams

5.9.5. Time-dependent Crosstalk

5.9.6. Small-scale Beam Features

5.9.7. Large-scale Beam Features

5.9.8. Systematics Summary

The final BB bandpowers are provided in Table 1 and shown in Figure 3. The bottom panel of Figure 3 shows the inverse-variance-weighted combination of the three sets of bandpowers. The spectrally combined bandpowers are for visualization purposes only; the likelihood employed in Section 6 uses the original set of three bandpowers. The bandpowers, covariance matrix, and bandpower window functions are available at the SPT Web site.³⁹

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Table 1.  BB Bandpowers, ${{\ell }}_{\mathrm{center}}{C}_{{\ell }}$ (${10}^{-3}\mu$K²)

		95 × 95		95 × 150		150 × 150		Combined
${{\ell }}_{\mathrm{center}}$	ℓ Range	${\ell }{C}_{{\ell }}$	$\sigma ({\ell }{C}_{{\ell }})$	${\ell }{C}_{{\ell }}$	$\sigma ({\ell }{C}_{{\ell }})$	${\ell }{C}_{{\ell }}$	$\sigma ({\ell }{C}_{{\ell }})$	${\ell }{C}_{{\ell }}$	$\sigma ({\ell }{C}_{{\ell }})$
500	300–700	3.4	1.4	0.88	0.55	0.57	0.33	0.76	0.28
900	700–1100	2.9	1.1	1.18	0.50	0.51	0.32	0.82	0.26
1300	1100–1500	0.4	1.3	0.27	0.50	1.07	0.37	0.77	0.29
1700	1500–1900	0.2	1.5	−0.23	0.55	0.16	0.38	0.04	0.30
2100	1900–2300	−1.7	1.6	−0.59	0.61	−0.29	0.47	−0.47	0.36

Note. BB bandpowers and uncertainties measured in this work. The last two columns give results for the inverse-variance-weighted combination of the three sets of bandpowers.

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The bandpowers from this work shown in Figure 3 show a clear preference for non-zero (positive) power. Using a multivariate Gaussian likelihood and the bandpower covariance matrix defined in Section 5.6, we find that the χ² of the data to the null hypothesis of zero BB power is 41.5, with an associated PTE for 15 degrees of freedom (dof) of $3\times {10}^{-4}$. The data are not well fit by zero sky power. A far better fit is the predicted lensing BB spectrum from (for instance) the Planck+Lens best-fit model from Table II of Planck Collaboration XVI (2014). With zero free parameters, the χ² of our data to this model is 18.1 for 15 dof, for a PTE of 0.25, and a ${\rm{\Delta }}{\chi }^{2}$ relative to zero BB power of 23.4.

To translate this preference into a detection significance, we introduce a single parameter, ${A}_{\mathrm{lens}}$, which we use to artificially scale the predicted lensing BB power from our fiducial cosmological model. We explore this parameter space (and all parameter spaces discussed subsequently in this work) using the Markov chain Monte Carlo method provided by the CosmoMC⁴⁰ software package (Lewis & Bridle 2002). The χ² for the best-fit model is 17.6 ($\mathrm{PTE}=0.23$). The data prefer the addition of this single parameter by a ${\rm{\Delta }}{\chi }^{2}$ value of 23.9, translating to a 4.9σ detection of lensing, under the assumption of no other sky components. Note that all of these χ² values are calculated with noise variance only (i.e., no sample variance from sky signals), because we are only asking at what level the data in this patch of sky prefer a component that looks like lensed B modes, and are not attempting to relate the amplitude of this component to any global cosmological parameter.

We do expect other sky components to contribute to the measured BB power, so we also fit for the amplitude of the lensing B modes in the presence of other signals. We know that there will be contributions from polarized emission from extragalactic sources and from dust emission within our own Galaxy, and we add parameters to the fit describing each of these.

The area of sky used in this work is contained within the BICEP2 area and should thus have a similar level of galactic polarized dust emission. Motivated by the recent Planck results (Planck Collaboration et al. 2014a), we model the contribution from Galactic dust as

$$\begin{eqnarray}&&{D}_{{\ell }}^{\mathrm{dust}}({\nu }_{1}\times {\nu }_{2})\equiv \displaystyle \frac{{\ell }({\ell }+1)}{2\pi }\ {C}_{{\ell }}^{\mathrm{dust}}({\nu }_{1}\times {\nu }_{2})\\ &&=\;{A}_{\mathrm{dust}}\ {D}_{80}^{\mathrm{dust}}({\nu }_{0}\times {\nu }_{0})\;\displaystyle \frac{{S}_{\mathrm{dust}}({\nu }_{1}){S}_{\mathrm{dust}}({\nu }_{2})}{{S}_{\mathrm{dust}}^{2}({\nu }_{0})}{\left(\displaystyle \frac{{\ell }}{80}\right)}^{-0.42}\end{eqnarray} \tag{ 21 }$$

where ${C}_{{\ell }}^{\mathrm{dust}}({\nu }_{1}\times {\nu }_{2})$ is the contribution of dust to the 95 GHz × 95 GHz, 95 GHz × 150 GHz, or 150 GHz × 150 GHz spectrum; ${A}_{\mathrm{dust}}$ is an overall scaling of the amplitude of the dust power spectrum at all frequencies; ${\nu }_{0}=150\;\mathrm{GHz}$; ${D}_{80}^{\mathrm{dust}}({\nu }_{0}\times {\nu }_{0})$ is the best-fit value of the dust BB spectrum in the BICEP2 observing region at 150 GHz and ℓ = 80, according to Planck Collaboration et al. (2014); and S_dust is the Planck Collaboration et al. (2014) assumption for the spectral behavior of the dust (in units of CMB temperature). We integrate the dust spectrum over the measured SPTpol bandpasses. The only free parameter in our fit to this model is ${A}_{\mathrm{dust}}$. For reference, the nominal value of ${A}_{\mathrm{dust}}$ = 1 corresponds to powers of ${D}_{80}^{\mathrm{dust}}=$ (0.00169, 0.00447, 0.0118) $\mu {{\rm{K}}}^{2}$ in the SPTpol spectra at (95 × 95, 95 × 150, and 150 × 150).

We model the contribution from extragalactic sources as a ${C}_{{\ell }}=\mathrm{constant}$ term, as would be expected if the emission was dominated by the Poisson noise in the number of sources in our observing region, rather than the angular clustering of those sources. We allow the amplitude of the Poisson term in our three frequency combinations to vary independently, giving us three free parameters, ${A}_{\mathrm{PS},95\times 95}$, ${A}_{\mathrm{PS},95\times 150}$, and ${A}_{\mathrm{PS},150\times 150}$.

Based on the arguments in BICEP2 Collaboration (2014) and BICEP2/Keck & Planck Collaborations et al. (2015), we expect the contribution from Galactic synchrotron emission to be far below our ability to detect it, and we ignore this contribution in our analysis. We also ignore the contribution from the polarized emission by clustered extragalactic sources. We expect this signal to be far less important relative to the Poisson signal in polarization measurements than it is in temperature measurements, because the clustered signal is dominated by dusty, star-forming galaxies (DSFGs), which we expect to have a much smaller polarization fraction than the synchrotron-emitting AGNs expected to contribute the bulk of the Poisson signal (e.g., Seiffert et al. 2007; Battye et al. 2011).

Finally, we add a component to our fit that has the shape of the expected IGW B-mode signal, and we use the free parameter r to control the amplitude of this component. We do not expect to be able to distinguish this component from Galactic dust in the ℓ range and sensitivity level in this work; we include the IGW component primarily to facilitate direct comparison between our lensing results and those from BICEP2 (see Section 6.2 below).

Our nominal fit includes priors on all five nuisance parameters (${A}_{\mathrm{PS},95\times 95}$, ${A}_{\mathrm{PS},95\times 150}$, ${A}_{\mathrm{PS},150\times 150}$, ${A}_{\mathrm{dust}}$, and r). The nominal prior on ${A}_{\mathrm{dust}}$ is a Gaussian centered on 1.0, with $\sigma =0.3$. The nominal priors on Poisson power are uniform between zero and four times the value calculated for each parameter by integrating source models (De Zotti et al. 2010 for AGNs; Negrello et al. 2007 for DSFGs) up to the unpolarized flux cut of 50 mJy and assuming polarization fractions of 5% for AGNs and 2% for DSFGs (the upper bounds from Seiffert et al. 2007 and Battye et al. 2011, respectively). For reference, the upper edges of Poisson ${C}_{{\ell }}$ priors are $1.9\times {10}^{-7}\;\mu {{\rm{K}}}^{2}$, $9.1\times {10}^{-8}\;\mu {{\rm{K}}}^{2}$, and $4.4\;\times {10}^{-8}\;\mu {{\rm{K}}}^{2}$ at 95 × 95, 95 × 150, and 150 × 150, respectively. The prior on r is uniform between 0 and 0.4. We also place a prior on ${A}_{\mathrm{lens}}$ (uniform between 0 and 3.0), but the posterior on ${A}_{\mathrm{lens}}$ in all fits is dominated by the data, not the prior.

The best-fit point in this nominal six-parameter space (including priors) has a χ² value of 17.9. If we fix ${A}_{\mathrm{lens}}=0$ and re-fit, the best-fit point has a χ² value of 36.8. Thus, the data show a preference of ${\rm{\Delta }}{\chi }^{2}=18.9$, or 4.3σ, for lensing B modes when we marginalize over foreground parameters.

To report a value of ${A}_{\mathrm{lens}}$ that can be used to assess the validity of the assumed cosmological model and to compare to other values in the literature, we repeat the six-parameter fit including BB sample variance in the uncertainty budget. The best-fit value and $1\sigma$ uncertainty from this fit are

$$\begin{eqnarray}&&{A}_{\mathrm{lens}}=1.08\pm 0.26\end{eqnarray} \tag{ 22 }$$

consistent with the value of 1.0 that we would expect if the cosmological model we assumed were correct. The constraint on ${A}_{\mathrm{lens}}$ is not strongly dependent on the foreground priors: if we increase the upper limit of the Poisson priors to 100 times the nominal power, the best-fit value of ${A}_{\mathrm{lens}}$ shifts by less than 1σ, and the error bar increases by less than 10%. Similar behavior is seen when the ${A}_{\mathrm{dust}}$ prior is changed to a uniform prior between 0 and 3.

Figure 4 demonstrates that the measurement of BB power in this work is at least visually consistent with previous measurements in the same ℓ range. The most straightforward way to compare the results quantitatively is to use the reported constraints on ${A}_{\mathrm{lens}}$. This comparison is complicated somewhat by the fact that these values are in general reported with respect to BB spectra predicted using different values of cosmological parameters. The differences between the predicted amplitudes are typically much smaller than the 1σ uncertainty on ${A}_{\mathrm{lens}}$ from any of the measurements, however, and we will ignore them in the following comparison.

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First, we compare to other B-mode power spectrum measurements. The best-fit value of ${A}_{\mathrm{lens}}$ from POLARBEAR measurements of the B-mode power spectrum was 1.12 ± 0.61 (statistical only, POLARBEAR Collaboration 2014a), consistent with the value we find here. The ACTPol collaboration does not report a value of ${A}_{\mathrm{lens}}$ from their B-mode power spectrum (Naess et al. 2014). BICEP2 Collaboration (2014) reports a 5.5σ detection of lensed B modes and a best-fit value of ${A}_{\mathrm{lens}}\simeq 1.75$, roughly 2σ above 1.0, with no marginalization over foregrounds. We have used the publicly available BICEP2 likelihood module⁴¹ to repeat this analysis with a marginalization over foreground and IGW tensor power that mirrors the SPTpol analysis presented here. We obtain a constraint from the BICEP2 data of ${A}_{\mathrm{lens}}=1.45\pm 0.38$, roughly 1.2σ above 1.0. More recently, BICEP2/Keck & Planck Collaborations et al. (2015) reported a strong (7σ) detection of lensing B modes at ${\ell }\sim 200$: ${A}_{\mathrm{lens}}=1.13\pm 0.18$. This constraint was obtained after marginalizing over contributions from IGW tensor power and polarized galactic dust emission and represents the most precise direct measurement of lensing B modes to date. To summarize, these BB power spectrum measurements are consistent with each other and with the ΛCDM prediction. These measurements suggest that the BB spectra in these particular fields and at these observing frequencies are dominated by lensing B modes, at least at ${\ell }\gtrsim 200$. The SPTpol BB spectrum presented here is particularly useful in that regard, in the sense that ${A}_{\mathrm{lens}}\ \geqslant 2$—a rough proxy for a scenario in which other sources of B modes dominate—is rejected at $\gt 3\sigma$ independent of whether or not we marginalize over foregrounds.

Next, we compare to measurements of the lensing B-mode power spectrum that rely on cross-correlation with tracers of the CMB lensing potential ϕ. In these analyses, a lensing B-mode template is constructed by lensing measured E modes by an estimate of ϕ derived from CIB maps or CMB lensing. The B-mode template is then correlated with the measured B modes to estimate the lensing BB power spectrum. The linear relationship between the CIB and ϕ is, in turn, based on the ${C}_{{\ell }}^{\phi -\mathrm{CIB}}$ spectrum measured by Planck, and it would therefore be surprising if these CIB cross-correlation analyses gave results that were not consistent with the ϕϕ spectrum measured by Planck (which itself is consistent with the ΛCDM prediction). The best-fit value of ${A}_{\mathrm{lens}}$ from the SPTpol CIB cross-correlation analysis in H13 was 1.092 ± 0.141. The POLARBEAR collaboration does not report a value of ${A}_{\mathrm{lens}}$ from their CIB cross-correlation analysis. The ACTPol collaboration measured ${A}_{\mathrm{lens}}=1.30\pm 0.40$ in their cross-correlation analysis with Planck CIB data (van Engelen et al. 2014). Planck Collaboration et al. (2015) also detect lensed B modes in this fashion and report best-fit values of ${A}_{\mathrm{lens}}=0.93\pm 0.10$ when using the CIB as a ϕ tracer or ${A}_{\mathrm{lens}}=0.93\pm 0.08$ when using the quadratic-estimator-derived ϕ. These measurements are consistent with the ΛCDM prediction and with our measured value.

Finally, we note that our constraint on ${A}_{\mathrm{lens}}$ is also consistent with determinations of this parameter from quadratic-estimator reconstructions of the lensing potential using the four-point function of CMB temperature and polarization data. The strongest such constraint (Planck Collaboration et al. 2015) uses full-mission Planck data to yield ${A}_{\mathrm{lens}}=0.983\pm 0.025$. Additionally, three recent works use polarization-only quadratic estimators to constrain ${A}_{\mathrm{lens}}$. POLARBEAR Collaboration (2014c) finds ${A}_{\mathrm{lens}}=1.06\pm 0.47$ (statistical only), Story et al. (2014) use SPTpol data to measure ${A}_{\mathrm{lens}}=0.92\pm 0.25$ (statistical only) on the same $100\ {\mathrm{deg}}^{2}$ field used in this work, and Planck Collaboration et al. (2015) find ${A}_{\mathrm{lens}}=1.252\pm 0.350$. Again, these measurements of ${A}_{\mathrm{lens}}$ are consistent with the ΛCDM prediction and with our measured value.