MEASUREMENT OF COSMIC MICROWAVE BACKGROUND POLARIZATION POWER SPECTRA FROM TWO YEARS OF BICEP DATA

Deconvolving detector transfer functions is the first step in producing clean timestreams that are suitable for analysis. Relative gains are measured with elevation nods at 0.02 Hz, so the transfer functions must be characterized over a frequency range that spans at least 0.01–1 Hz in order to link the relative gains to the entire science band. The transfer functions were measured with a microwave source (Gunn oscillator or broadband noise source) that was placed near the telescope window and square-wave modulated at 0.01 Hz. The time-domain responses to the transitions were Fourier transformed, divided by the transform of the modulation waveform, and averaged for each detector in order to obtain the deconvolution kernel. The transfer functions have a measurement precision of 0.5% rms across the signal band and have sufficiently high signal-to-noise ratio (S/N) to be deconvolved directly from the bolometer timestreams. (Although the transfer functions can be described by a simple model, we do not rely on those fits.) The relative gain uncertainty that results from transfer function deconvolution is <0.3% over the range 0.01–1 Hz.

PSB relative gains are measured with elevation nods at the beginning and end of every set of constant-elevation azimuth scans. The nods inject an atmospheric signal into the bolometer timestreams, and the gains are obtained by fitting each timestream against the cosecant of the detector elevation. The resulting volts-per-air mass responsivity factors are normalized to the average of good detectors for each frequency band during the scan set. The common-mode rejection of CMB temperature anisotropies in gain-adjusted PSB pairs is measured to be better than 98.9% at degree angular scales.

We cross-correlate the CMB temperature fluctuations measured by WMAP and Bicep to obtain absolute gains, which relate CMB temperature to detector units. The WMAP maps are smoothed to Bicep's resolution by applying the ratio of the beam window functions, $B^{\rm BICEP}_\ell / B^{\rm WMAP}_\ell$. (We do not apply a correction for the pixelization of the WMAP maps since the effects are negligible.) The smoothed maps are then converted to simulated detector timestreams using the boresight pointing data. The timestreams are filtered and converted back into maps, thus creating a "Bicep-observed" version of the WMAP data. The Bicep map and processed WMAP maps, which have compatible beam and filter functions, are cross-correlated in multipole space to obtain the absolute gain

$$\begin{equation} g_b = { \sum _{\ell } P_\ell ^b {\big\langle a^{\rm{WMAP-1}}_{\ell m} {a^{*\: {\rm BICEP}}_{\ell m}} \big\rangle } \over {\sum _{\ell } P_\ell ^b \big\langle a^{\rm{WMAP-1}}_{\ell m} a^{*\: {\rm{WMAP-2}}}_{\ell m} \big\rangle } }, \end{equation} \tag{ 2 }$$

where P^b_ℓ is a top hat binning operator, and a_ℓm are the spherical harmonic expansion coefficients. To avoid noise bias, the a_ℓm coefficients in the denominator are taken from two different WMAP maps; for this analysis, we have used the Q- and V-band maps from the five-year data release (Hinshaw et al. 2009). The resulting gain calibration, g_b, is approximately flat over Bicep's ℓ range of 21–335, where the lower bound is set by the timestream filtering, and the upper bound is set by beam uncertainty. The absolute gain used for each of the Bicep frequency bands is a single number taken from the average of g_b over six uniform bins spanning a multipole window of 56 ⩽ ℓ ⩽ 265, and the absolute gain uncertainty is derived from the standard deviation of g_b. To assess the impact of errors in the beam window functions, we have calculated g_b using both the Q- and V-band WMAP data in the numerator of Equation (2). The average g_b values are consistent within errors, and we take the larger of the two standard deviation values, 2%, as a conservative estimate of the absolute gain uncertainty in temperature units.

The two components of detector pointing reconstruction are telescope boresight pointing and detector offsets relative to the boresight (focal plane coordinates). Raw boresight pointing timestreams are obtained from encoders located on the three mount axes, and corrections to the raw data are applied from a model describing axis tilts and encoder offsets. The pointing model is established with star observations from an optical camera located on the upper surface of the cryostat.

The focal plane coordinates of each detector are reconstructed from measurements of CMB temperature fluctuations. Temperature maps from each detector at each of the four boresight angles are cross-correlated with the full-season map, and the process is repeated with varying centroid adjustments until the maps are self-consistent. This method results in 003 rms centroid uncertainty.

To recover polarization information from detector timestreams, it is necessary to know the polarization orientation angle ψ and cross-polar leakage of each PSB. Note that is a property of the detector itself and is independent of the cross-polar beam, which is a property of the optical chain upstream of the PSB. For Bicep, dominates over the cross-polar beam. Both ψ and are measured with several different devices (e.g., rotating wire grid, dielectric sheet calibrator) that send polarized light into Bicep at many different angles with respect to the detectors. The phase of each PSB's sinusoidal response and the ratio of the minimum to maximum determine ψ and , respectively. The uncertainty in the measured orientation angles is ±07. The median for Bicep PSBs is about 0.04, with a measurement uncertainty of ±0.01.

The Bicep beams were mapped by raster scanning the telescope over a bright source at various fixed boresight angles. The beams are well described by a Gaussian model, with fit residuals typically about 1% with respect to the beam amplitude. The beams at both frequencies are nearly circular, with ellipticities under 1.5%. We therefore approximate the beams as symmetric Gaussians with average full widths of 093 and 060 at 100 and 150 GHz, respectively. The distribution of beam widths varies by ±3% across the focal plane, and each width is measured to a precision of ±0.5%.

The data set is further reduced by omitting 9 hr observation phases with extremely poor weather quality. For each phase, the standard deviation of relative gains from elevation nods is calculated for each channel, and the median over the 100 and 150 GHz channels of those standard deviations yields two numbers per phase. An observation phase is cut if either of the median standard deviations is greater than 20% of the average relative gain. After this weather cut, the data set consists of 117 and 226 days for the 2006 and 2007 seasons.

Of the 49 optically active PSB pairs, several are excluded from each season due to anomalous behavior such as no response, excess noise, poorly behaved or time-dependent transfer functions, and exceptionally poor polarization efficiency. Six experimental pixels containing Faraday rotator modules (2006 season only) and the two 220 GHz pixels (2007 and 2008 seasons) are also excluded from CMB analysis. A total of 19 100 GHz and 14 150 GHz PSB pairs are used for 2006 analysis, and 22 100 GHz and 15 150 GHz PSB pairs are used for 2007.

The raw output of Bicep consists of voltage timestreams sampled at 50 Hz for 144 channels comprising 98 light bolometers, 12 dark bolometers, 16 thermistors, 10 resistors, and eight intentionally open channels. The low-level timestream cleaning begins with concatenating and trimming the raw data files for each 9 hr observation phase. Complete transfer functions are deconvolved for the 98 light bolometers, and all timestreams are low pass filtered at 5 Hz and downsampled to 10 Hz. Relative detector gains are derived from elevation nods, and horizon and celestial boresight coordinates are calculated using the pointing model. "Half-scans," or single sweeps in azimuth, are identified, and the turnarounds are excluded from further analysis. The remaining central portion of the half-scan, which has nearly constant velocity, makes up ∼75% of the total scan duration.

Each detector half-scan is subjected to three data quality checks. First, relative gains derived from elevation nods at the beginning and end of a set of scans are compared, and all of the half-scans in the scan set are excluded if the gains differ by more than 3%. Second, gain-adjusted PSB pairs are differenced over each half-scan, and the half-scan is rejected if the magnitude of the skew or kurtosis is abnormally high (PSB pair-difference timestreams are expected to be Gaussian white noise). Finally, half-scans that contain cosmic rays and other signal spikes are identified by points that are greater than 7 times the standard deviation of the smoothed timestream. On average, the combined half-scan flagging criteria exclude about 3% of all half-scans over all light detectors. Because the flagged percentage is small, the problematic half-scans are not gap-filled and are simply omitted from analysis.

After low-level cleaning, the bolometer timestreams are binned into temperature and polarization maps. We have developed two data analysis pipelines for Bicep that differ starting from the mapmaking stage. The code in each pipeline is completely independent of the other, and the only shared data products are the initial set of downsampled, cleaned detector timestreams, boresight pointing, and calibration data. Both pipelines have reproduced all the results reported in this paper, achieving similar statistical power and excellent agreement. The mapmaking algorithms used by the two pipelines are similar, although one (designated for this paper the "primary pipeline") bins in the Healpix (Górski et al. 2005) pixelization scheme, while the other (designated for this paper as the "alternate pipeline") produces maps in rectangular coordinates. In this section, we describe the primary pipeline's mapmaking procedure in detail.

Following Jones et al. (2007), the simplified timestream output d_ij of a single PSB can be expressed as

$$\begin{equation} d_{ij} = g_{ij} [T({\bf p}_j) + \gamma _{i} (Q({\bf p}_j)\cos 2\psi _{ij} + U({\bf p}_j)\sin 2\psi _{ij})], \end{equation} \tag{ 3 }$$

where g is the gain, T, Q, U are the beam-integrated Stokes parameters of the sky signal, γ ≡ (1 − )/(1 + ) is the polarization efficiency factor, and ψ is the PSB polarization orientation projected on the sky. The index i denotes the PSB channel number, j is the timestream sample number, and p_j is the map pixel observed at time j. The goal of mapmaking is to recover T, Q, U from the bolometer timestreams.

The mapmaking procedure for Bicep begins with the formation of gain-adjusted sum and difference timestreams for each PSB pair:

$$\begin{equation} d^\pm _{ij} = {1\over 2} (d_{2i,j} / g_{2i,j} \pm d_{2i+1,j} / g_{2i+1,j}). \end{equation} \tag{ 4 }$$

To reduce atmospheric 1/f noise, a third-order polynomial is subtracted from the sum and difference timestreams for each half-scan in azimuth. Azimuth-fixed and scan-synchronous contamination are removed by subtracting a template signal, which is formed by binning the polynomial-filtered detector timestreams in azimuth over each set of fixed-elevation scans. There are slight differences in the scan-synchronous signal between left- and right-going half-scans, so separate templates are calculated for each case. The scan-synchronous contamination removed in this step is very small; Q and U maps typically change by 100–400 nK at the largest scales after its removal.

The temperature T at each map pixel p is recovered from the filtered sum timestreams with

$$\begin{equation} \left(\sum ^{\rm{No. of }pairs}_i \sum _{j \in {\bf p}} w^+_{ij} d^+_{ij} \right) {\Bigg /} \left(\sum ^{\rm{No. of }pairs}_i \sum _{j \in {\bf p}} w^+_{ij} \right) \simeq T({\bf p}), \end{equation} \tag{ 5 }$$

where w⁺ is the weight assigned to each pair sum, and we have assumed that the effects of polarization leakage are negligible. In other words, the temperature is obtained simply by binning the filtered detector timestreams into map pixels. Stokes Q and U are calculated from linear combinations of the difference timestreams using the matrix equation

$$\begin{eqnarray} \nonumber && \sum ^{\rm{No. of }pairs}_i \sum _{j \in {\bf p}} w^-_{ij} \left(\begin{array}{@{}c@{}} d^-_{ij} \alpha _{ij} \\[3pt] d^-_{ij} \beta _{ij} \\ \end{array}\right) \qquad\qquad \qquad\qquad \qquad\quad \\ &&\qquad = {1\over 2} \sum ^{\rm{No. of }pairs}_i \sum _{j \in {\bf p}} w^-_{ij} \left(\begin{array}{@{}cc@{}} \alpha ^2_{ij} & \alpha _{ij}\beta _{ij} \\ \alpha _{ij}\beta _{ij} & \beta ^2_{ij}\\ \end{array}\right) \left(\begin{array}{@{}c@{}} Q({\bf p}) \\ U({\bf p}) \\ \end{array}\right). \end{eqnarray} \tag{ 6 }$$

Here, w⁻ is the weight assigned to each pair difference, and α, β are PSB pair orientation angle factors defined as

$$\begin{eqnarray} &&\alpha _{ij} = \gamma _{2i}\cos 2\psi _{2i,j} - \gamma _{2i+1}\cos 2\psi _{2i+1,j}, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} &&\beta _{ij} = \gamma _{2i}\sin 2\psi _{2i,j} - \gamma _{2i+1}\sin 2\psi _{2i+1,j}. \end{eqnarray} \tag{ 8 }$$

The 2 × 2 matrix in Equation (6) is singular for a single pair of ψ_2i,j and ψ_2i+1,j, and the equation can be solved only by accumulating more than one timestream sample in a given map pixel p. As p is observed with many detector polarization angles ψ, the off-diagonal α_ijβ_ij terms average to zero, and the matrix becomes invertible. Although only two different polarization angles are required to invert the matrix, some instrumental systematics average down as the number of observation angles increases. By examining the determinant of the matrix, pixels (typically at the edge of the map) with insufficient polarization angle coverage or low integration time are identified and masked from analysis.

We choose the pair sum and difference weights w^± to be proportional to the inverse variance of the filtered timestreams. The weights are evaluated from power spectral densities averaged over each set of azimuth scans (every 50 minutes), a period during which the noise properties are approximately stationary. For each channel pair, the sum/difference weight for a scan set is calculated from the inverse of the average value of the auto-correlation between 0.5 and 1 Hz.

The first step in power spectrum estimation is calculating and subtracting the noise bias ${\hat{N}}_\ell$. We estimate ${\hat{N}}_\ell$ with Monte Carlo simulations of instrument noise: starting from a noise model, we generate simulated timestreams that are filtered, co-added into maps, and processed with Spice. The resulting simulated noise spectra are averaged over many realizations to form ${\hat{N}}_\ell$.

The Bicep noise model is derived from gain-adjusted PSB pair sum and difference timestreams (Equation (4)) under the assumption that the timestream S/N is negligible, i.e., the signal is the noise. The S/N is ⩽10% and ⩽0.1% for the sum and difference timestreams, respectively. The sum and difference timestreams are polynomial-filtered and Fourier transformed over each half-scan, and all auto-correlations and cross-correlations between channel pairs are computed to form the complex frequency-domain noise covariance matrix ${{ \tilde{\bm N}}}(f)$. To form the noise model, we average ${{ \tilde{\bm N}}}(f)$ over the 100 half-scans within each scan set and then average into 12 logarithmically spaced bins spanning 0.05–5 Hz.

To construct simulated correlated noise timestreams, we take the Cholesky decomposition ${{\tilde{ \bm N}}}(f) = {\bf L}(f){\bf L}^{\dagger} (f)$ of the noise covariance matrix and multiply a vector of pseudo-random complex numbers ρ(f) by L(f):

$$\begin{equation} {{\tilde{\bm v}}}(f) = {\bf L}(f){\bm \rho }(f). \end{equation} \tag{ 10 }$$

The complex numbers in ρ(f) have Gaussian-distributed real and imaginary components and are normalized such that the magnitude has a standard deviation of one. The resulting product ${{\tilde{\bm v}}}(f)$ has the same statistical properties as the data and is inverse Fourier transformed to obtain a set of simulated noise timestreams. Scan-synchronous templates are calculated and subtracted from each set of azimuth scans, and the filtered noise timestreams are then co-added into maps.

The noise bias, ${\hat{N}}_\ell$, is estimated by averaging the power spectra from an ensemble of noise-only maps. Figure 4 shows ${\hat{N}}_\ell$ (red dashed lines), calculated from the average over 500 realizations, in comparison to raw power spectra. Bicep measures TT and TE with high S/N, and the noise contribution is negligible up to ℓ ∼ 330. The noise is mostly uncorrelated between temperature and polarization, so ${\hat{N}}_\ell$ for TB is distributed around zero, and the same is true for EB. In contrast, noise comprises the bulk of the EE and all of the BB amplitude at ℓ>150, which illustrates the need for careful noise modeling and subtraction. We have studied the impact of noise misestimation by intentionally scaling the noise spectra (Takahashi et al. 2010), and we find that the largest possible error in ${\hat{N}}_\ell$ is ±3%.

Zoom In Zoom Out Reset image size

The finite resolution of the telescope and pixelization of the maps result in suppression of the observed power spectra at small angular scales. This suppression of power is described by ${\mycal B}^2_\ell = B^2_\ell H^2_\ell$, the product of the beam and pixel window functions, and is illustrated in Figure 5. In this analysis, we approximate B_ℓ as the Legendre transform of the average beam within a given frequency band, which is assumed to be a circular Gaussian. The full widths at 100 and 150 GHz are 093 and 060, respectively. The pixel window function H_ℓ is supplied by Healpix and corresponds to nside=256. In the case of Bicep, ${\mycal B}_\ell$ varies slowly with respect to the Spice kernel, $\kappa ^X_{\ell \ell ^{\prime }}$, and can therefore be pulled out of the convolution. The observed power spectra, after noise subtraction, are divided by ${\mycal B}_\ell ^2$ to correct for the effects of beam suppression and pixelization:

$$\begin{equation} \big(\hat{C}_\ell ^{X} - \hat{N}_\ell ^X\big) \big/ {\mycal B}_\ell ^2 \simeq \sum _{\ell ^{\prime }} \kappa ^X_{\ell \ell ^{\prime }} F^X_{\ell ^{\prime }} C_{\ell ^{\prime }}^{X}. \end{equation} \tag{ 11 }$$

Zoom In Zoom Out Reset image size

The bolometer timestreams are cleaned by subtracting a third-order polynomial and scan-synchronous template; this cleaning procedure also has the effect of removing large-scale CMB signal. The amount of signal loss is described by F_ℓ, the ℓ-space transfer function imposed by timestream filtering. In addition, although the Spice estimator is unbiased in the mean, the timestream processing removes spatial modes from the observed T, Q, and U maps, generically introducing couplings between the observed E and B spectra. These couplings are small, but are of interest for the leakage of the relatively large E-mode signal into B-mode polarization. We characterize both F_ℓ and E-to-B leakage with signal-only simulations.

The signal simulation procedure begins by generating model power spectra using the CAMB (Lewis et al. 2000) software package. We use ΛCDM parameters derived from WMAP five-year data (Hinshaw et al. 2009) and a tensor-to-scalar ratio of zero. From the model spectra, we use the synfast utility in Healpix to create an ensemble of simulated CMB skies pixelized at 011 resolution (nside=512) and convolved with the Bicep beams. Actual pointing data are used to generate smoothly interpolated PSB timestreams from the simulated T, Q, U maps and their derivatives. A PSB timestream sample d(p) that falls into a pixel centered at p is expressed as a convolution of the beam ${\mycal P}({\bf r}-{\bf r}_b)$, which is centered at r_b, with a second-order Taylor expansion of the sky signal m around p (E. F. Hivon & N. Ponthieu 2010, in preparation):

$$\begin{eqnarray} \nonumber d({\bf p}) = && \int d{\bf r} \: {\mycal P}({\bf r}-{\bf r}_b) {\Big [} m({\bf p}) + {\vec{\nabla }}m({\bf p})({\bf r}-{\bf p}) \\ && + {1\over 2} ({\bf r}-{\bf p})^T D^2m({\bf p}) ({\bf r}-{\bf p}) {\Big ]}. \end{eqnarray} \tag{ 12 }$$

Here, m(p) = g[T(p) + γ(Q(p)cos 2ψ + U(p)sin 2ψ)], and ${\vec{\nabla }}$ and D² denote the first and second derivatives in spherical coordinates. Assuming Gaussian beams for each frequency band, Equation (12) reduces to

$$\begin{eqnarray} \nonumber d({\bf p}) = && m({\bf p}) + {\vec{\nabla }}m({\bf p}) ({\bf r}_b - {\bf p}) \\ && + {1\over 2} {\rm Tr} {\Bigg [} D^2 m({\bf p}) \left(\begin{array}{@{}cc@{}} (\Delta \phi)^2 & \Delta \phi \Delta \theta \\ \Delta \phi \Delta \theta & (\Delta \theta)^2 \\ \end{array}\right) {\Bigg ]}, \end{eqnarray} \tag{ 13 }$$

where Δϕ and Δθ are the longitude and latitude offsets between the pointing vector r_b and the pixel center p. We apply Equation (13) to simulate signal-only detector timestreams according to Bicep's scan strategy. Measured PSB pair centroids, detector orientation angles, and cross-polar leakage values are included in the simulations. For the purpose of characterizing the effects of timestream filtering, all differential beam systematic effects are turned off so that there is no mixing between temperature and polarization. The simulated timestreams are filtered and weighted in exactly the same way as the real data and then co-added into maps.

Once we have "Bicep-observed" signal-only maps in hand, we compute the power spectra with Spice and average the results over many realizations. Because the input CAMB spectra have r = 0, any non-zero BB power in the simulation outputs is interpreted as contamination from E-mode polarization that is induced by timestream filtering. We therefore apply a correction,

$$\begin{equation} \big(\hat{C}_\ell ^{BB} - \hat{N}_\ell ^{BB} - \hat{C}_\ell ^{BB}\vert _{EE {\rm only}}\big) / {\mycal B}_\ell ^2 \simeq \sum _{\ell ^{\prime }} \kappa ^{BB}_{\ell \ell ^{\prime }} F^{BB}_{\ell ^{\prime }} C_{\ell ^{\prime }}^{BB}, \end{equation} \tag{ 14 }$$

to the BB power spectrum of the data, where $\hat{C}_\ell ^{BB}\vert _{EE {\rm only}}$ is the ensemble average BB spectrum from the r = 0 signal simulation outputs. The amplitude of this correction is roughly $\ell (\ell +1)\hat{C}_\ell ^{BB}\vert _{EE {\rm only}}/(2\pi) \simeq 3\times 10^{-3} \mu {\rm K}^2$ at ℓ ∼ 100, comparable to inflationary BB power for r = 0.05, and the sample variance is about 3 × 10⁻³μK². The correction factors for the other spectra are negligibly small, so only the BB spectrum is adjusted with this procedure.

To quantify the suppression of large-scale power from timestream filtering, we begin by assuming that F^X_ℓ, like ${\mycal B}^2_\ell$, varies slowly in comparison to the Spice kernel and can be pulled out of the convolution. We choose to calculate the binned filter suppression factor, F^X_b, defined as

$$\begin{equation} \sum _{\ell } P_\ell ^b \frac{\ell (\ell +1)}{2\pi } \frac{\big(\hat{C}_\ell ^{X} - \hat{N}_\ell ^X\big)}{{\mycal B}_\ell ^2} \,{\simeq}\, F_b^X \sum _\ell P_\ell ^b \frac{\ell (\ell +1)}{2\pi } \sum _{\ell ^{\prime }} \kappa _{\ell \ell ^{\prime }} C_{\ell ^{\prime }}^X, \end{equation} \tag{ 15 }$$

where the binning operator P^b_ℓ is a top hat. (In the case of the BB spectrum, the left hand side of the equation also contains $\hat{C}_\ell ^{BB}\vert _{EE {\rm only}}$, as in Equation (14).) We obtain the filter suppression factors by comparing the power spectra of the Bicep-observed signal-only maps to those of the input synfast maps; F^X_b is the ratio of the spectra, after multiplying by ℓ(ℓ + 1)/(2π) and binning.

Figure 5 shows F^X_b averaged over 500 signal simulations using Bicep's timestream filtering. In each simulation, signal-only timestreams are generated from the full two years of pointing data. At ℓ ∼ 100, the value of F^X_b is about 0.7 for all spectra and rises slowly as ℓ increases. Identical filter functions are used for the EE and BB spectra, and the filter functions of the cross-spectra are calculated as the geometric mean of those determined from the auto-spectra. The validity of this approach for obtaining the BB and cross-spectra filter functions has been confirmed with simulations over a limited multipole range. Dividing by F^X_b is the final step in obtaining the Bicep band power estimates:

$$\begin{eqnarray} \hat{\mathscr{C}}^X_b &\equiv & \frac{1}{F^X_b} \sum _\ell P^b_\ell \frac{\ell (\ell +1)}{2\pi } \frac{\big(\hat{C}_\ell ^{X} - \hat{N}_\ell ^X\big)}{{\mycal B}_\ell ^2}, X \ne B\!B \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \hat{\mathscr{C}}^{BB}_b &\equiv & \frac{1}{F^{BB}_b} \sum _\ell P^b_\ell \frac{\ell (\ell +1)}{2\pi } \frac{\big(\hat{C}_\ell ^{BB} - \hat{N}_\ell ^{BB} - \hat{C}_\ell ^{BB}\vert _{EE {\rm only}}\big)}{{\mycal B}_\ell ^2}.\nonumber\\ \end{eqnarray} \tag{ 17 }$$

The uncertainties in the power spectra consist of two components, one that is proportional to the signal itself (sample variance), and another that depends on the instrumental noise. We estimate the errors by examining the variance of power spectra from simulated signal-plus-noise maps, which exactly encode time-dependent correlated noise, scan strategy, and sky coverage. We add the simulated noise-only and signal-only maps, described in Sections 6.1 and 6.3; and power spectra are calculated for each realization using the same ${\hat{N}}_\ell$, ${\mycal B}_\ell ^2$, and F^X_b as applied to the real data. If the simulations include a reasonable model of the signal and faithfully reproduce all the noise properties of the experiment, then the data and simulations should be indistinguishable. Figure 4 shows raw power spectra, uncorrected for noise and filter bias, from 500 signal-plus-noise simulations at 150 GHz. The raw spectra of the actual data, shown by the black points, lie within the scatter of the simulations. We calculate the band power covariance matrix from the ensemble of simulations, after applying noise, filter, and beam corrections. The band power errors are obtained from square root of the diagonal terms of the matrix.

In order to compare our band power estimates to a theoretical model, we need a method to calculate expected band power values from theoretical power spectra. The relationship between the model and the expected band powers is described by band power window functions, ω^b_ℓ, defined as

$$\begin{equation} {\mathscr{C}}_b = \sum _\ell \frac{(\ell +\frac{1}{2})}{\ell \big(\ell +1\big)} \omega _\ell ^b \mathscr{C}_\ell, \end{equation} \tag{ 18 }$$

where $\mathscr{C}_\ell \equiv \ell (\ell +1) C_\ell / (2\pi)$. The window functions are given by

$$\begin{equation} \omega _\ell ^b = \frac{2}{2\ell +1} \sum _{\ell ^{\prime }} P^b_{\ell ^{\prime }} \ell ^{\prime } (\ell ^{\prime }+1) \kappa _{\ell ^{\prime }\ell } \end{equation} \tag{ 19 }$$

and are normalized such that

$$\begin{equation} \sum _\ell \frac{(\ell +\frac{1}{2})}{\ell (\ell +1)} \omega _\ell ^b = 1. \end{equation} \tag{ 20 }$$

The window functions depend on the Spice kernel, which depends on the apodization function applied to the correlation functions. For this analysis, we apodize the correlation functions with a cosine window that spans 50°. We choose to use a uniform binning of width Δℓ = 35, spanning a multipole range of 21 ⩽ ℓ ⩽ 335. This choice of bin width provides minimally correlated band powers while preserving the spectral resolution determined by the width of the Spice kernel. Figure 6 illustrates Bicep's band power window functions for the nine ℓ bins.

Zoom In Zoom Out Reset image size

Because the BB power spectrum is debiased with the procedure described in Equation (14), we set the EE-to-BB window functions to zero. This is a valid approximation as long as the EE spectrum of the signal under consideration is statistically consistent with the measured Bicep band powers (which, as shown in Section 12, are well described by a concordance ΛCDM cosmology).

We check the self-consistency of the power spectra by performing jackknives, statistical tests in which the data are split in two halves and differenced. The split is performed at the mapmaking stage, and the resulting differenced map should have power spectra that are consistent with the expected residual signal level (nearly zero) after subtracting noise bias. The interaction of timestream filtering with the details of the split causes imperfect signal cancellation when forming jackknife maps, but in practice, this residual signal is small.

The data are tested with six jackknives that are sensitive to different aspects of the instrument's performance. In the scan direction jackknife, the data are split into left- and right-going azimuth half-scans. Failures generally point to a problem in the detector transfer function deconvolution, or thermal instabilities created at the scan endpoints. The elevation coverage jackknife is formed from the two CMB observations in each 48 hr cycle; each observation covers the same azimuth range but starts from a different elevation. This jackknife is sensitive to ground-fixed or scan-synchronous contamination. Bicep observes at four fixed boresight orientation angles that can be split into two pairs, {−45°, 0°} and {135°, 180°}, to form a boresight angle pair jackknife. This test is perhaps the most powerful of the jackknives performed and is sensitive to many factors, including thermal stability, atmospheric opacity, relative gain mismatches, differential beam pointing, and ground pickup. In the temporal jackknife, the 8 day observing cycles—48 hr at each of the four boresight angles—are interleaved to form the two halves, and this jackknife tests sensitivity to weather changes. The season split jackknife simply divides the data into the two observing seasons, and failures reflect any changes made to the instrument between the two years. In particular, the focal plane thermal architecture was improved for the 2007 season, and the temperature control scheme was changed. The focal plane QU jackknife splits the detectors into two groups according to their polarization orientation within the focal plane (approximately alternating hextants) and is a method of probing instrumental polarization effects.

Power spectra of jackknife maps are computed with the method described in Section 6, using simulated jackknife noise and signal-plus-noise maps to subtract noise bias and assign error bars. The filter function F^X_b is applied so that the magnitude of any non-zero jackknife band powers can be compared to the amplitude of the non-jackknife spectra. For each jackknife spectrum, we calculate χ² over nine bins spanning 21 ⩽ ℓ ⩽ 335. To account for the expected level of residual signal, the χ² values are evaluated with respect to the average jackknife spectra from an ensemble of signal-only simulations. The criteria for jackknife success or failure are based on the probability to exceed (PTE) the χ² value, which is calculated from the distribution of χ² in 500 signal-plus-noise simulations. Jackknife victory is declared when (1) none of the PTEs are abnormally high or low, given the number of χ² tests performed, and (2) the PTEs are consistent with a uniform distribution between zero and one.

The jackknife spectra for the six different data splits appear similar in that the band powers are all distributed around zero. We therefore show only the spectra from the boresight pair jackknife (Figure 7, gray points), which is arguably the most stringent of the six tests. Table 1 lists the PTE values from all of the jackknife χ² tests for Bicep's polarization-only spectra (EE, BB, EB). Out of all the PTEs, the lowest value is 0.014, in the 150 GHz focal plane QU jackknife. This low value is, however, consistent with expectations from uniformly distributed PTEs over the 60 polarization-only jackknives. Figure 8 shows that the PTEs are consistent with a uniform distribution between zero and one.

Zoom In Zoom Out Reset image size

The polarization jackknife tests are the most powerful probe of the accuracy of the noise model. In addition to the χ² tests, we have also expressed the jackknife spectra as band power deviations with respect to the mean of signal-only simulations. The sum of the band power deviations in each jackknife spectrum provides an additional and more precise gauge of the correct estimation of instrument noise. We have verified that the band power deviation sums in the data are consistent with signal-plus-noise simulations and are not systematically biased high or low, thus confirming that the noise levels are correctly estimated. Furthermore, we have probed the sensitivity to systematic variations in the modeled noise amplitude over a range that is comparable to the S/N of the sum and difference timestreams, and find no significant changes in the jackknife spectra. The relative insensitivity of the jackknifes to the amplitude of the noise model validates the procedure described in Section 6.1.

In contrast to the polarization data, the temperature data display significant jackknife failures. There is an excess of small PTE values in the TE and TB jackknives, and most of the TT jackknife PTEs are smaller than 0.002, which is the resolution from 500 simulations. The TT, TE, and TB jackknife PTEs are therefore not listed in Table 1. We attribute these failures to the fact that Bicep's temperature maps have high S/N (see TT plot in Figure 4), and the jackknives are therefore extremely sensitive to small gain calibration errors or imperfections in modeling and subtracting unpolarized atmospheric emission. (As described in Section 6.1, the ⩽10% S/N in the pair-sum timestreams is a known imperfection of the Bicep noise model for temperature data.) Although the TT, TE, and TB jackknife failures are statistically significant, Figure 7 illustrates that the amplitudes of the jackknife spectra are small compared to both the amplitude and errors of the signal spectra. The magnitudes of the TT and TE/TB jackknife band powers are typically 1–10 μK² and 0.1–1 μK², respectively. In all cases, the error bars of the non-jackknife spectra are at least a factor of 10 larger, except in the highest ℓ bin.

We have performed the same jackknives with both analysis pipelines, and the results are in excellent agreement. The alternate pipeline confirms that the polarization data pass the χ² tests, and although the TT, TE, and TB spectra do not pass, the amplitudes of the jackknife spectra are small.

We are mostly concerned with systematic errors that mix the bright temperature signal into polarization and thus induce a false B-mode signal. We define a benchmark for each systematic error such that the false B-mode amplitude is no greater than the peak of the inflationary BB spectrum. For Bicep's target of r = 0.1, this requirement corresponds to ℓ(ℓ + 1)C^BB_ℓ/(2π) = 7 × 10⁻³μK² at ℓ ∼ 100.

The primary source of potential temperature leakage into polarization is differential gains within PSB pairs. Gain mismatch effects can also arise from other systematics, such as errors in the bolometer transfer functions, which act as frequency-dependent gains. We have characterized the common-mode rejection of PSBs by examining the cross-correlation of pair-sum and pair-difference maps. There is no statistically significant evidence for gain mismatch in the data, and we set an upper limit of Δ(g₁/g₂)/(g₁/g₂) < 1.1% on differential gains.

A second source of potential temperature and polarization mixing is beam mismatch within PSB pairs. We describe beam mismatch with three quantities: differential beam size, pointing, and ellipticity. Of these three, the dominant effect in Bicep is differential pointing, which is stable over time and has been measured with a median amplitude of $({\bf r}_1-{\bf r}_2)/\bar{\sigma } = 1.3 \, \pm \, 0.4\%$, where $\bar{\sigma }$ is the average Gaussian beam size within a pair of PSBs. Measured upper limits on differential size and ellipticity are negligible.

Most systematic errors interact with the scan strategy in complex ways, and the exact effects on the power spectra can be computed only through signal simulations. We follow the formalism presented in E. F. Hivon & N. Ponthieu (2010, in preparation) and Shimon et al. (2008) to calculate the expected level of false BB in Bicep. Starting from synfast maps with r = 0, Equation (13) is used to generate simulated signal timestreams that include the effects of gain mismatch and differential pointing. The timestreams are filtered and co-added into maps, and the amplitude of the BB power spectrum at ℓ ∼ 100 is compared with the 7 × 10⁻³μK² benchmark.

In the differential gain simulations, we randomly assign 1.1% rms gain mismatch to PSB pairs, and we find that the expected false BB amplitude is 7.4 × 10⁻³μK² and 2.9 × 10⁻²μK² at ℓ ∼ 100 for 100 and 150 GHz, respectively. Although this amplitude exceeds the r = 0.1 benchmark, Figure 9 illustrates that it is small compared to the statistical error in this analysis of the two-year data. In a future analysis of the entire Bicep data set, we anticipate placing tighter constraints on PSB gain mismatch as the noise levels integrate down. With further work, we are confident that uncertainty in gain mismatch can be substantially reduced from the current 1.1%.

Differential pointing has been precisely characterized for each of Bicep's PSB pairs, so we run signal simulations using the measured centroid offset vectors, rather than randomized distributions. The false BB from differential pointing has an ℓ ∼ 100 amplitude of 2.7 × 10⁻³μK² and 4.2 × 10⁻³μK² at 100 and 150 GHz, respectively. These amplitudes are slightly smaller than the r = 0.1 benchmark and are well below the noise level of the initial two-year data set. In a future analysis, it may be possible to use the measured centroid offsets to correct for systematic effects.

We emphasize that, in addition to the differential gain and pointing discussed here, most uncertainties in instrument characterization create false positive BB signal. The fact that Bicep's BB spectra are consistent with zero and pass jackknives demonstrates that we have achieved sufficient control over systematic errors in this analysis. Furthermore, until a positive B-mode detection is made, the presence of systematic effects that produce spurious polarization could only make the reported BB upper limits higher (more conservative) than they would be otherwise.

The scaling of the power spectra is determined by the absolute gain factors that convert detector units to temperature, and the 2% uncertainty in this gain (Section 3.2) translates into a 4% uncertainty in the power spectrum amplitude. The polarized spectra have additional amplitude uncertainty that arises from errors in the cross-polar leakage. A systematic error of Δ = 0.01 corresponds to 3.9% amplitude uncertainty for polarization-only spectra (EE, BB, EB) and 2.0% for temperature–polarization cross-spectra (TE, TB). Therefore, the combined amplitude uncertainty G^X ≡ ΔC^X_ℓ/C^X_ℓ from gain and errors is 4% for TT, 4.5% for TE/TB, and 5.6% for EE/BB/EB. These uncertainties are shown in Figure 9, assuming ΛCDM spectra with r = 0.

For Gaussian beams, measurement errors in the beam widths introduce a fractional uncertainty,

$$\begin{equation} S_\ell \equiv \Delta C^X_\ell \big/ C^X_\ell = e^{\sigma ^2 \ell (\ell +1)(\delta ^2+2\delta)} - 1 - {\bar{S}}, \end{equation} \tag{ 21 }$$

in the power spectrum amplitude as a function of ℓ. Here, $\sigma = {\rm FWHM} / \sqrt{8\ln (2)}$, and δ is the fractional beam width error Δσ/σ. Because this band power uncertainty is degenerate with absolute gain error, we subtract the mean, ${\bar{S}}$, calculated over 56 ⩽ ℓ ⩽ 265, the angular scales over which we perform absolute calibration. For this analysis, we use average beam widths of 093 and 060 at 100 and 150 GHz, respectively. Although the distribution of beam widths in the focal plane varies by ±3%, the measurement precision is ±0.5%; we therefore expect the effective δ to lie somewhere in between. We calculate the maximum beam width error allowed by our calibration cross-spectra (Equation (2)), which are very close to flat, and we constrain δ < 1.2% and δ < 2.8% at 100 and 150 GHz, respectively. Figure 9 shows the expected power spectrum errors from these maximum allowed beam uncertainties, which are most likely conservative. The systematic error is smaller than the statistical error in all cases except the TT spectra at high ℓ, where the levels of statistical and beam systematic uncertainty are comparable.

Polarized dust emission in the Bicep CMB field is estimated from "FDS Model 8" (Finkbeiner et al. 1999). We assume a 5% polarized fraction, guided by a study of WMAP data that shows that high-latitude dust has a mean fractional polarization of 3.6% (Kogut et al. 2007). The dust temperature and polarization model is extrapolated to 100 and 150 GHz, and filtered according to Bicep's scan strategy. The resulting polarized dust emission is ℓ(ℓ + 1)C_ℓ/(2π) = 9.6 × 10⁻⁵μK² and 6.1 × 10⁻⁴μK² at ℓ = 100 for 100 and 150 GHz, respectively. These values are 2 orders of magnitude below the 95% confidence level for upper limits on the BB amplitude from Bicep (discussed in Section 13), as shown in Figure 10.

Zoom In Zoom Out Reset image size

We have also tested for dust contamination in the Bicep maps by studying the cross power spectrum between FDS Model 8 maps and Bicep temperature and polarization maps. The cross power spectrum of the real data is consistent with the distribution from signal-plus-noise simulations, providing additional evidence that the Bicep maps contain no spatial correlation with the FDS dust maps.

To estimate the polarized synchrotron emission in the Bicep field of view, we have used the WMAP Markov Chain Monte Carlo (MCMC) synchrotron maps (Gold et al. 2009), extrapolated to 100 and 150 GHz using the mean spectral index of the Bicep field of view, calculated from the spectral index maps provided in the same analysis. As for the FDS dust maps, we filter the extrapolated 100 and 150 GHz synchrotron maps with the Bicep scan strategy, and find the estimated level of polarized synchrotron emission at ℓ ∼ 100 to be 3 × 10⁻³μK² and 4 × 10⁻⁴μK² for 100 and 150 GHz, respectively, both below the level of Bicep sensitivity. Furthermore, the WMAP MCMC map has poor S/N in regions far from the Galactic plane, and we have found that within the Bicep CMB field, the map is dominated by variance in the Monte Carlo fit (see Gold et al. 2009, for details). The estimated levels of polarized synchrotron emission in Figure 10 should therefore be viewed as conservative estimates or upper limits. We have also derived synchrotron estimates from WMAP K-band polarization maps and the temperature maps of Haslam et al. (1981), assuming a 30% polarization fraction; both methods give synchrotron estimates that are lower than those from the WMAP MCMC maps.

Similarly to our analysis for thermal dust, we have studied the cross power spectrum of the synchrotron maps with the Bicep maps and found there to be no significant spatial correlation of the Bicep data with the synchrotron emission.

At degree-scale resolution, the Bicep maps do not show any obvious point source detections, so we rely on a combination of the 4.85 GHz Parkes–MIT–NRAO (PMN) survey (Wright et al. 1994), the WMAP point source catalog (Wright et al. 2009), and the Acbar catalog (Reichardt et al. 2009) to search for point source contamination. We search for point source contamination by optimally filtering CMB and noise fluctuations out of the Bicep temperature map and determine the significance of the resulting pixel values by repeating the process with simulated maps of CMB with detector noise. Although the resulting maps have a few 2σ detections at the suspected point source locations, there is no statistical evidence for point source contamination above the expected Gaussian distribution of noise. As a further test, we have simulated the effects of masking out the 27 Acbar sources that lie within the Bicep field and have found it has no significant impact on the power spectra.

The CMB and foreground emission have different frequency dependence, so we can test for the presence of foreground contamination in Bicep data by performing a frequency jackknife. We difference the 100 and 150 GHz maps, compute the power spectra, calculate χ², and compare the results to signal-plus-noise simulations, as described in Section 8.1. The probabilities to exceed the χ² values are {0.050, 0.152, 0.732} for EE, BB, and EB, respectively. We find no evidence for foreground contamination in the frequency jackknives.