FIRST SEASON QUIET OBSERVATIONS: MEASUREMENTS OF COSMIC MICROWAVE BACKGROUND POLARIZATION POWER SPECTRA AT 43 GHz IN THE MULTIPOLE RANGE 25 ⩽ $\ell$ ⩽ 475

We observe four CMB fields, referred to henceforth as "patches." Table 1 lists their center positions and total integration times, while Figure 2 indicates their positions on the sky.²⁹ The number of patches is determined by the requirement to always have one patch above the lower elevation limit of the mount (43°). The specific positions of each patch were chosen to minimize foreground emission using WMAP three-year data. The area of each patch is ≈250 deg². In addition to the four CMB patches, we observe two Galactic patches. These allow us to constrain the spectral properties of the polarized low-frequency foregrounds with a high-signal-to-noise ratio. The results from the Galactic observations will be presented in a future publication.

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Scanning the telescope modulates the signal from the sky, converting CMB angular scales into frequencies in the polarimeter time streams. Since QUIET targets large angular scales, fast scanning (≈5° s⁻¹ in azimuth) is critical to ensure that the polarization modes of interest appear at higher frequencies than the atmospheric and instrumental 1/f knee frequencies.

So that each module sees a roughly constant atmospheric signal, QUIET scans are periodic motions solely in azimuth with both the elevation and deck-rotation axes fixed. Each scan has an amplitude of 75 on the sky, with period 10–22 s. These azimuth scans are repeated for 40–90 minutes; each set of scans at fixed elevation is denoted a "constant-elevation scan" (CES). We repoint the telescope and begin a new CES when the patch center has moved by 15° in order to build up data over an area of ≈15° × 15° for each patch. Note that a central region ≃ 8° across is observed by all polarimeters since the instrument's field of view has a diameter of ≃ 7°. Diurnal sky rotation and weekly deck rotations provide uniform parallactic-angle coverage of the patch, and ensure that its peripheral regions are also observed by multiple polarimeters.

The polarized flux from Tau A provides a 5 mK signal which we observe at four parallactic angles. The sinusoidal modulation of the signal induced by the changing parallactic angles is fitted to yield responsivity coefficients for each detector. Figure 3 shows the response of the four polarization channels from the central feed horn to Tau A. A typical responsivity is 2.3 mV K⁻¹, with a precision from a single set of observations of 6%. The absolute responsivity from Tau A was measured most frequently for the central feed horn. We choose its +Q diode detector to provide the fiducial absolute responsivity.

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The responsivities of other detectors relative to the fiducial detector are determined with the sky dips as described below. We have three independent means of assessing the relative responsivities among polarimeters: from nearly simultaneous measurements of the Moon, from simultaneous measurements of responses to the rotating sparse wire grid in post-season tests, and from Tau A measurements. The errors from these methods are 4%, 2%, and 6%, respectively, while the error from the sky-dip method is 4%. All the methods agree within errors.

Sky dips generate temperature signals of several 100 mK and thus permit measurement of the TP responsivities. The signals vary slightly with PWV. We estimate the slope from the data as 4% mm⁻¹ and correct for it. This slope is consistent with the atmospheric model of Pardo et al. (2001). Because the ratios of the responsivities for the TP and polarized signals from each detector diode are stable quantities within a few percent of unity, we use sky dips performed immediately before each CES to correct short-term variations in the polarimeter responsivities. The responsivities vary by ≲ 10% over the course of a day due to changing thermal conditions for the bias electronics. Further post-season tests provide a physical model: the relevant temperatures are varied intentionally while the responsivities are measured with sky dips. We confirm the results with the polarized broadband source.

We bound the uncertainty in the absolute responsivity of the polarimeter array at 6%. The largest contributions to this estimate are uncertainties in (1) the beam solid angle (4%, see below), (2) the response difference between polarized and TP signals for each diode detector (3%), and (3) the Tau A flux (3%; Weiland et al. 2011). The first enters in converting the flux of Tau A into μK, while the second enters because, although one fiducial diode detector is calibrated directly from Tau A, for the rest we find relative responsivities from sky dips and normalize by the fiducial diode's responsivity.

For the differential-temperature modules, all detectors observe the signal from Jupiter simultaneously, providing the absolute responsivity for all channels upon comparison with the Jupiter flux from Weiland et al. (2011). Observations of Venus (Hafez et al. 2008) and RCW38 agree with the Jupiter measurements within errors, and sky dips track short-term variations. We calibrate the absolute responsivity with 5% accuracy.

The global pointing solution derives from a physical model of the three-axis mount and telescope tied to observations of the Moon with the central feed horn in the array, as well as Jupiter and Venus with the differential-temperature feed horns. Optical observations are taken regularly with a co-aligned star camera and used to monitor the time evolution of the pointing model.

During the first two months in the season, a mechanical problem with the deck-angle encoder resulted in pointing shifts. The problem was subsequently repaired. Based on pointing observations of the Moon and other astronomical sources, we verify that these encoder shifts are less than 2°. Systematic uncertainties induced by this problem are discussed in Section 8.1.

After the deck-angle problem is fixed, no significant evolution of the pointing model is found. The difference in the mean pointing solution between the start and the end of the season is smaller than 1'. Observations of the Moon and Jupiter also provide the relative pointing among the feed horns. The rms pointing error in the maps is 35.

Our primary measurement of the polarization angle for each detector comes from observing the radial polarization of the Moon, as illustrated in Figure 4. The measurement has a high signal-to-noise ratio and its inaccuracy is dominated by systematic error due to the temperature gradient of the Moon surface. One can see the effect in the different amplitudes of the two positive envelopes in Figure 4. The fluctuations of the detector polarization-angle measurements over many observations with different phases of the Moon and telescope orientations are typically 1° in rms. Although simulations suggest these fluctuations can be due to the failure to account in analysis for the temperature gradient, we conservatively assign them as upper limits on the fluctuations of the polarization angles during the season. Based on this conservative limit, we estimate the systematic error in the CMB-power-spectra measurement in Section 8.2, resulting in a negligible contribution.

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Two other less precise methods also give estimates of the detector angles: fits to the Tau A data, and the determination of the phases of the sinusoidal responses of all the detectors to rotation of the sparse wire grid. In each case, the differences between the detector angles determined by the secondary method and the Moon are described by a standard deviation of ≈3°. However, we find a mean shift between the Tau A derived and Moon-derived angles of 17. To estimate the errors in the angles in light of this shift, we use an empirical approach: in Section 8.2 we estimate the impact on the power spectra from using the Tau A results instead of the Moon results, and find it to be small.

The polarization and differential-temperature beams are obtained from maps created using the full data sets of Tau A and Jupiter observations, respectively, with square pixels of 18 on a side. For polarization, this process produces the main and leakage beam maps simultaneously, with the latter describing the instrumental polarization. The average FWHM for the beams across the array is 273, measured with 01 precision for the central feed horn and for the differential-temperature feed horns at the edge of the focal plane. The non-central–polarization-horn FWHMs are measured less frequently and thus are less precisely known, with an uncertainty of 15. The beam elongation is typically small (1%), and its effect is further reduced by the diurnal sky rotation and weekly deck rotations which result in a symmetrized effective beam in the CMB maps. We compute one-dimensional symmetrized beam profiles, with a resolution of 06. These profiles are modeled as a sum of six even Gauss–Hermite terms (Monsalve 2010). The main-beam solid angles are computed by integrating these models out to 54' (roughly −28 dB), yielding 78.0 ± 0.4 μsr for the differential-temperature horns and 74.3 ± 0.7 μsr for the central horn. An average gives 76 μsr for all horns in the array. We also examine alternative estimates such as integrating the raw beam map instead of the analytical fit. We assign a systematic uncertainty of 4% based on the differences among these different estimates. The systematic error includes possible contributions from sidelobes, which we constrain to 0.7 ± 0.4 μsr with antenna range measurements carried out before the observation season.

The window functions, encoding the effect of the finite resolution of the instrument on the power spectra, are computed from the central-horn and the temperature-horn–profile models. The central-horn beam profile and window function are shown in Figure 5. The uncertainty accounts for statistical error and differences between polarization and differential-temperature beams, as described in Section 8.1.

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Instrumental imperfections can lead to a spurious polarization signal proportional to the unpolarized CMB temperature anisotropy. We call this the I to Q (or U) leakage term. In our instrument, a fraction of the power input on one port of the correlation module is reflected because of a bandpass mismatch to the septum polarizer, and a fraction of the reflected power re-enters the other port. The dominant monopole term comes from this effect. We measure the monopole term from the polarimeter responses to temperature changes, using sky dips; Moon, Tau A, and Galactic signals; as well as variations from the weather. The average magnitude is 1.0% (0.2%) for the Q (U) diodes. Note that the discrepancy in the Q and U averages was predicted from measurements of the properties of the septum polarizers and confirmed in the field. We do not correct for this effect but assign systematic errors as described in Section 8.3.

To prepare the TOD for map making, we execute three steps: pre-processing, noise modeling, and filtering. Of these steps, only the filtering is significantly different between the two pipelines.

5.1.1. Pre-processing

5.1.2. Noise Model

5.1.3. Filtering

The fundamental unit of data used for analysis is the double-demodulated output of one detector diode for a single CES, referred to as a "CES-diode." Selecting only those CES-diodes that correspond to good detector performance and observing conditions is a critical aspect of the data analysis. The data-selection criteria began with a nominal set of cuts and evolved into several distinct configurations, as many as 33 in the case of pipeline A. For each configuration, analysis validation (see Section 6) was performed yielding statistics quantifying the lack of contamination in the data set. The final data set was chosen when these statistics showed negligible contamination and were little affected by changes to the cuts.

Cut efficiencies, defined as the fractions of CES-diodes accepted for the analysis, are given for both pipelines in Table 3. While each pipeline applies its own cuts uniformly to all four patches, the efficiencies among patches are non-uniform because of differences in weather quality. Over the course of the eight-month observing season, patch CMB-1 is primarily visible at night, when the atmosphere tends to be more stable; patch CMB-3 is mostly observed during the day.

The first step of the data selection is simply to remove known bad data: data from six non-functional detector diodes, data during periods of mount malfunctions, and CESs lasting less than 1000 s. Furthermore, we cut individual CES-diodes that show deviation from the expected linear relationship between the demodulated and TP signals. This cut removes data with poor thermal regulation of the electronics or cryostat, or residual ADC nonlinearity.

The beam sidelobes, described in Section 2, introduce contamination to the data if the telescope scanning motion causes them to pass over the ground or the Sun. Ground pickup is dealt with by filtering as described in Section 5.1.3. The less frequent cases of Sun contamination are handled by cutting those CES-diodes for which the Sun's position overlaps with the measured sidelobe regions for each diode.

Additional cuts are specific to each pipeline. Pipeline A removes data taken during bad weather using a statistic calculated from fluctuations of the TP data during 10 s periods, averaged across the array. This cut removes entire CESs. Several more cuts remove individual CES-diodes. While these additional cuts are derived from the noise modeling statistics, they also target residual bad weather. During such marginal weather conditions only some channels need to be cut, since the sensitivity for a given detector diode to atmospheric fluctuations depends on its level of instrumental polarization. Next, we reject CES-diodes with poor agreement between the filtered data and the noise model in three frequency ranges: a narrow range (only 40 Fourier modes) about the scan frequency, from twice the scan frequency to 1 Hz, and from 1 Hz to 4.6 Hz. We also cut CES-diodes that have higher than usual 1/f knee frequencies, or large variations during the CES in the azimuthal slopes of the double-demodulated time streams; both these cuts help to eliminate bad weather periods. Finally, we also remove any CES-diodes with an outlier greater than 6σ in the time domain on three timescales (20 ms, 100 ms, and 1 s).

For pipeline B, the weather cut rejects CESs based on a statistic computed from fluctuations of the double-demodulated signals from the polarization modules on 10 s and 30 s timescales. Three cuts are applied to remove individual CES-diodes. The first is a cut on the 1/f knee frequency, similar to that of pipeline A. Second, a cut is made on the noise model χ² in the frequency range passed by the filter, and third, we reject CES-diodes having a large χ² in the azimuth-binned TOD. This cut rejects data with possible time variation in the ground signal. Finally, an entire CES is removed if more than 40% of its detectors have already been rejected.

After filtering, the TOD for all diodes are combined to produce Q and U maps for each of the QUIET patches. The maps use a HEALPix N_side = 256 pixelization (Gorski et al. 2005; http://healpix.jpl.nasa.gov/). This section describes the map making and power-spectra estimation from the maps for each of the pipelines.

5.3.1. Pipeline-A Map Making

5.3.2. Power-spectra Estimation in Pipeline A

The MASTER (Monte Carlo Apodized Spherical Transform Estimator) method is used in pipeline A (Hivon et al. 2002; Hansen & Gorski 2003); it is based on a pseudo-$C_{\ell }$ technique and takes account of effects induced by the data processing using Monte Carlo (MC) simulations. The pseudo-$C_{\ell }$ method allows estimation of the underlying $C_{\ell }$ using spherical-harmonics transformations when the observations do not cover the full sky uniformly (Wandelt et al. 2001). The pseudo-$C_{\ell }$ spectrum, designated by $\tilde{C}_\ell$, is related to the true spectrum $C_{\ell }$ by

$$\begin{equation} \langle \tilde{C}_\ell \rangle = \sum _{\ell ^\prime } M_{\ell \ell ^\prime } F_{\ell ^\prime } B_{\ell ^\prime }^2 \langle C_{\ell ^\prime } \rangle. \end{equation} \tag{ 1 }$$

There is no term corresponding to noise bias, which would arise if we did not employ a cross-correlation technique. Here, $B_\ell$ is the beam window function, described in Section 4.4, and $M_{\ell \ell ^\prime }$ is a mode-mode-coupling kernel describing the effect of observing only a small fraction of the sky with non-uniform coverage. It is calculable from the pixel weights, which are chosen to maximize the signal-to-noise ratio (Feldman et al. 1994). We bin in $\ell$ and recover $C_{\ell }$ in nine band powers, C_b, and $F_\ell$ is the transfer function (displayed in Section 7) due to filtering of the data; its binned estimate, F_b, is found by processing noiseless CMB simulations through pipeline A and used to obtain C_b. For the polarization power spectra, Equation (1) is generalized for the case where $\tilde{C}_\ell$ contains both $\tilde{C}_\ell ^{{\rm EE}}$ and $\tilde{C}_\ell ^{{\rm BB}}$.

In the power-spectra estimates, we include only the cross-correlations among pointing-division maps, excluding the autocorrelations. Because the noise is uncorrelated for different pointing divisions, the cross-correlation technique allows us to eliminate the noise-bias term and thus the possible residual bias due to its misestimate. Cross-correlation between different pointing divisions also suppresses possible effects of ground contamination and/or time-varying effects. Dropping the autocorrelations creates only a small increase in the statistical errors (≈3%) on the power spectra.

The errors estimated for the pipeline-A power spectra are frequentist two-sided 68% confidence intervals. A likelihood function used to compute the confidence intervals is modeled following Hamimeche & Lewis (2008) and calibrated using the MC simulation ensemble of more than 2000 realizations with and without CMB signal. We also use the likelihood function to put constraints on r and calculate the consistency to ΛCDM.

The partial sky coverage of QUIET generates a small amount of E/B mixing (Challinor & Chon 2005), which contributes an additional variance to the BB power spectrum. We incorporate it as part of the statistical error. This mixing can be corrected (Smith & Zaldarriaga 2007) in future experiments where the effect is not negligible compared to instrumental noise.

5.3.3. Pipeline-B Map Making

In pipeline B, the pixel-space sky map $\hat{\mathbf {m}}$ (N_side = 256) is given by

$$\begin{equation} \hat{\mathbf {m}} = (\mathbf {P}^{T}\mathbf {N}^{-1}\mathbf {F}\mathbf {P})^{-1} \mathbf {P}^{T}\mathbf {N}^{-1} \mathbf {F} \mathbf {d}, \end{equation} \tag{ 2 }$$

where $\mathbf {P}$ is the pointing matrix, $\mathbf {N}$ is the TOD-noise-covariance matrix, $\mathbf {F}$ corresponds to the apodized bandpass filter discussed in Section 5.1.3, and $\mathbf {d}$ denotes the TOD. This map is unbiased, and for the case $\mathbf {F}=\mathbf {1}$ it is additionally the maximum-likelihood map, maximizing

$$\begin{equation} \mathcal {L}({\mathbf {m}}|\mathbf {d}) =e^{-\frac{1}{2}\left(\mathbf {d}-\mathbf {Pm}\right)^T\mathbf {N}^{-1}\left(\mathbf {d}-\mathbf {Pm}\right)}. \end{equation} \tag{ 3 }$$

The corresponding map–noise-covariance matrix (e.g., Tegmark 1997; Keskitalo et al. 2010) is

$$\begin{equation} \mathbf {N_{\hat{m}}} = (\mathbf {P}^{T}\mathbf {N}^{-1}\mathbf {F}\mathbf {P})^{-1} (\mathbf {P}^{T}\mathbf {F}^{T}\mathbf {N}^{-1}\mathbf {F}\mathbf {P})(\mathbf {P}^{T}\mathbf {N}^{-1}\mathbf {F}\mathbf {P})^{-1}. \end{equation} \tag{ 4 }$$

Note that one often encounters the simplified expression $\mathbf {N_{\hat{m}}} = (\mathbf {P}^{T}\mathbf {N}^{-1}\mathbf {F}\mathbf {P})^{-1}$ in the literature. This corresponds effectively to assuming that $\mathbf {F} = \mathbf {F}^2$ in the Fourier domain, and is strictly valid for top-hat–filter functions only. For our filters, we find that the simplified expression biases the map-domain $\chi ^2 (\equiv \hat{\mathbf {n}}^T \mathbf {N_{\hat{m}}^{-1}}\hat{\mathbf {n}}$, where $\hat{\mathbf {n}}$ is a noise-only map) by ≈3σ, and we therefore use the full expression, which does lead to an unbiased χ².

Equations (2)–(4) apply to both polarization and temperature analysis. The only significant difference lies in the definition of the pointing matrix, $\mathbf {P}$. For polarization, $\mathbf {P}$ encodes the detector orientation, while for temperature it contains two entries per time sample, +1 and −1, corresponding to the two horns in the differential-temperature assembly.

After map making, the maps are post-processed by removing unwanted pixels (i.e., compact sources and low-signal-to-noise edge pixels). All 54 compact sources in the seven-year WMAP point source catalog (Gold et al. 2011) present in our four patches are masked out, for a total of 4% of the observed area. We also marginalize over large-scale and unobserved modes by projecting out all modes with $\ell$ ⩽ 5 ($\ell$ ⩽ 25 for temperature) from the noise-covariance matrix using the Woodbury formula, assigning infinite variance to these modes.

5.3.4. Power-spectra Estimation in Pipeline B

Given the unbiased map estimate, $\hat{\mathbf {m}}$, and its noise-covariance matrix, $\mathbf {N_{\hat{m}}}$, we estimate the binned CMB power spectra, C_b, using the Newton–Raphson optimization algorithm described by Bond et al. (1998), generalized to include polarization. In this algorithm, one iterates toward the maximum-likelihood spectra by means of a local quadratic approximation to the full likelihood. The iteration scheme in its simplest form is

$$\begin{equation} \delta C_b = \frac{1}{2} \sum _{b^{\prime }} \mathcal {F}^{-1}_{bb^{\prime }} {\rm Tr}[(\hat{\mathbf {m}}\hat{\mathbf {m}}^T - \mathbf {C})(\mathbf {C}^{-1}\mathbf {C}_{,b^{\prime }}\mathbf {C}^{-1})], \end{equation} \tag{ 5 }$$

where b denotes a multipole bin, $\mathbf {C}$ is the signal-plus-noise pixel-space covariance matrix, and $\mathbf {C}_{,b}$ is the derivative of $\mathbf {C}$ with respect to C_b. The signal component of $\mathbf {C}$ is computed from the binned power spectra, C_b, and the noise component is based on the noise model described in Section 5.1.2, including diode–diode correlations. Finally,

$$\begin{equation} \mathcal {F}_{bb^{\prime }} = \frac{1}{2} {\rm Tr}(\mathbf {C}^{-1}\mathbf {C}_{,b}\mathbf {C}^{-1}\mathbf {C}_{,b^{\prime }}) \end{equation} \tag{ 6 }$$

is the Fisher matrix. Additionally, we introduce a step length multiplier, α, such that the actual step taken at iteration i is α δC_b, where 0 < α ⩽ 1 guarantees that $\mathbf {C}$ is positive definite. We adopt the diagonal elements of the Fisher matrix as the uncertainties on the band powers.

We start the Newton–Raphson search at $C_{\ell }$ = 0, and iterate until the change in the likelihood value is lower than 0.01 times the number of free parameters, corresponding roughly to a 0.01σ uncertainty in the position of the multivariate peak. Typically, we find that 3–10 iterations are required for convergence.

Estimation of cosmological parameters, θ, is done by brute-force grid evaluation of the pixel-space likelihood,

$$\begin{equation} \mathcal {L}(\theta) \propto \frac{-({1}/{2})\mathbf {d}^{T}\mathbf {C}^{-1}(\theta)\mathbf {d}}{\sqrt{|\mathbf {C}(\theta)|}}. \end{equation} \tag{ 7 }$$

Here, $\mathbf {C}(\theta)$ is the covariance matrix evaluated with a smooth spectrum, $C_{\ell }$, parameterized by θ. In this paper, we only consider one-dimensional likelihoods with a parameterized spectrum of the form $C_{\ell }$ = a $C^{\rm fid}_{\ell}$, a being a scale factor and $C^{\rm fid}_{\ell}$ a reference spectrum; the computational expense is therefore not a limiting factor. Two different cases are considered, with a being either the tensor-to-scalar ratio, r, or the amplitude of the EE spectrum, q, relative to the ΛCDM model.

As described in Section 2, we dedicate one pair of modules to differential-temperature measurements. While these modules are useful for calibration purposes, when combined with our polarization data they also enable us to make self-contained measurements of the TE and TB power spectra.

For temperature, both pipelines adopt the pipeline-A data-selection criteria used for polarization analysis (see Section 5.2). The temperature-sensitive modules, however, are far more susceptible to atmospheric contamination than the polarization modules. Thus, these cuts result in reduced efficiencies: 12.4%, 6.9%, and 6.8% for patches CMB-1, CMB-2, and CMB-3, respectively.³⁴ More tailoring of the cuts for these modules would improve efficiencies.

In pipeline A, the analysis proceeds as described in Sections 5.1.3, 5.3.1, and 5.3.2 except for two aspects. First, in the TOD processing a second-order polynomial is fit and removed from each telescope half-scan instead of a linear function. This suppresses the increased contamination from atmospheric fluctuations in the temperature data. Second, we employ an iterative map maker based on the algorithm described by Wright et al. (1996). Map making for differential receivers requires that each pixel is measured at multiple array pointings or cross-linked. In order to improve cross-linking, we divide the temperature data into only four maps by azimuth and deck angle, rather than the 60 divisions used for polarization analysis. To calculate TE and TB power spectra, polarization maps are made for these four divisions, plus one additional map that contains all polarization data with pointings not represented in the temperature data.

For pipeline B, the algorithms for making temperature maps and estimating power spectra are identical to the polarization case, as described in Sections 5.3.3 and 5.3.4.

A smaller number of null tests is used for the temperature analysis. Several are not applicable and others are discarded due to lack of data with sufficient cross-linking. Even so, we are able to run suites of 29, 27, and 23 TT null tests on patches CMB-1, CMB-2, and CMB-3, respectively. We calculate the sums of χ²_null statistics, yielding PTEs of 0.26 and 0.11 for patches CMB-1 and CMB-2, respectively. No significant outliers are found for these patches. However, a 5σ outlier in a single test³⁷ is found in patch CMB-3, implying contamination in its temperature map. CMB-3 is therefore excluded from further analysis. We confirm consistency between the patches CMB-1 and CMB-2 with a PTE of 0.26.

With no significant contamination in TT, EE, or BB spectra, one may be confident that the TE and TB spectra are similarly clean. For confirmation, we calculate TE and TB null spectra for the five null tests that are common to the temperature and polarization analyses. These yield PTEs of 0.61 and 0.82 for TE, and 0.16 and 0.55 for TB, for patches CMB-1 and CMB-2, respectively, with no significant outliers. Patch consistency checks give PTEs of 0.48 for TE and 0.26 for TB. Thus, the TE and TB power spectra, as well as the TT, pass all validation tests that are performed.

The CMB power spectra are reported in nine equally spaced bands with Δ$\ell$ = 50, beginning at $\ell$_min = 25. Given the patch size, modes with $\ell$ < $\ell$_min cannot be measured reliably. The correlation between neighboring bins is typically −0.1; it becomes negligible for bins further apart.

The EE, BB, and EB polarization power spectra estimated by both pipelines are shown in Figure 9. The agreement between the results obtained by the two pipelines is excellent, and both are consistent with the ΛCDM concordance cosmology. Our findings and conclusions are thus fully supported by both pipelines. Only the statistical uncertainties are shown here; we treat systematic errors in Section 8. Because the systematic error analysis was only done for pipeline A, we adopt its power-spectra results (tabulated in Table 5) as the official QUIET results.

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Table 5. CMB-Spectra Band Powers from QUIET Q-band Data

$\ell$ bin	EE	BB	EB
25–75	0.33+0.16−0.11a	−0.01+0.06−0.04	0.00+0.07−0.07
76–125	0.82+0.23−0.20	0.04+0.14−0.12	−0.10+0.11−0.12
126–175	0.93+0.34−0.31	0.24+0.28−0.25	0.71+0.22−0.20
176–225	1.11+0.58−0.52	0.64+0.53−0.46	0.18+0.38−0.38
226–275	2.46+1.10−0.99	1.07+0.98−0.86	−0.52+0.68−0.69
276–325	8.2+2.1−1.9	0.8+1.6−1.4	0.9+1.3−1.3
326–375	11.5+3.6−3.3	−2.2+2.7−2.4	0.0+2.0−2.0
376–425	15.0+6.2−5.8	−4.9+5.3−4.9	3.2+3.9−3.9
426–475	21+13−11	2+11−10	4.5+8.3−8.2

Notes. Units are thermodynamic temperatures, μK², scaled as $C_{\ell }$$\ell$($\ell$ + 1)/2π. ^aPatch CMB-1 has significant foreground contamination in the first EE bin.

Download table as:  ASCIITypeset image

The bottom sub-panels in Figure 9 show the window and transfer functions for each bin computed by pipeline A. Figure 10 shows the maps for patch CMB-1 computed by pipeline B, and Figure 11 shows the QUIET power spectra in comparison with the most relevant experiments in our multipole range.³⁸ Additional plots and data files are available at the QUIET Web site at the University of Chicago.³⁹

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View extended figure (2 panels)

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Fitting only a free amplitude, q, to the EE spectrum⁴⁰ relative to the seven-year best-fit WMAP ΛCDM spectrum (Larson et al. 2011), we find q = 0.87 ± 0.10 for pipeline A and q = 0.94 ± 0.09 for pipeline B. Taking into account the full non-Gaussian shapes of the likelihood functions, both results correspond to more than a 10σ detection of EE power. In particular, in the region of the first peak, 76 ⩽ $\ell$ ⩽ 175, we detect EE polarization with more than 6σ significance, confirming the only other detection of this peak made by BICEP at higher frequencies. The χ² relative to the ΛCDM model, with $C^{\rm EB}_{\ell}$ = $C_{\ell }$^BB = 0, is 31.6 (24.3) with 24 degrees of freedom, corresponding to a PTE of 14% (45%) for pipeline A (B).

In order to minimize possible foreground contamination, QUIET's four CMB patches were chosen to be far from the Galactic plane and known Galactic synchrotron spurs. In these regions, contributions from thermal dust emission are negligible in Q band. Spinning dust is expected to be polarized at no more than a few percent in Q band (Battistelli et al. 2006; Lopez-Caraballo et al. 2011), so we expect the contribution to polarized foreground emission in our patches to be small. We therefore consider only two dominant sources of possible foreground contamination, namely, compact radio sources and Galactic diffuse synchrotron emission.

To limit the effect of compact radio sources, we apply a compact-source mask to our maps before computing the power spectra, as described in Section 5. We also evaluate the CMB spectra both with and without the full WMAP temperature compact-source mask (Gold et al. 2011), and find no statistically significant changes. The possible contribution from compact radio sources with fluxes below the WMAP detection level (1 Jy) is small: 0.003 μK² at $\ell$ = 50 and 0.01 μK² at $\ell$ = 100 (Battye et al. 2011).⁴¹ We therefore conclude that our results are robust with respect to contamination from compact radio sources and that the dominant foreground contribution comes from diffuse synchrotron emission.

In Figure 12, we show the power spectra measured from each patch. The CMB-1 EE band power for the first bin is 0.55 ± 0.14 μK², a 3σ outlier relative to the expected ΛCDM band power of 0.13 μK²; while not significant enough to spoil the overall agreement among the patches as shown in Section 6, this is a candidate for a bin with foreground contamination.

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To estimate the Q-band polarized synchrotron contamination in our CMB patches, we process the WMAP7 K-band (23 GHz) map through pipeline A and estimate its band power, $\hat{C}^\mathrm{KK}_b$, as well as the cross-spectra with the QUIET Q-band data, $\hat{C}^\mathrm{QK}_b$. These results are shown for the first bin (25 ⩽ $\ell$ ⩽ 75; b = 1) in Table 6, together with the corresponding QUIET band powers, $\hat{C}^\mathrm{QQ}_b$. Since foregrounds do not contribute to the sample variance, the uncertainties for $\hat{C}^\mathrm{KK}_{b=1}$ and $\hat{C}^\mathrm{QK}_{b=1}$ are given by instrumental noise only, including contributions from both WMAP and QUIET. For $\hat{C}^\mathrm{QQ}_{b=1}$, sample variance as predicted by the ΛCDM model is also included.

Table 6. Band and Cross Powers for $\ell$ = 25–75

Patch	Spectrum	$\hat{C}^\mathrm{KK}_{b=1}$	$\hat{C}^\mathrm{QK}_{b=1}$	$\hat{C}^\mathrm{QQ}_{b=1}$
CMB-1	EE	$\mathbf { 17.4 \pm 4.7}$	$\mathbf { 3.30 \pm 0.55 }$	$\mathbf { 0.55 \pm 0.14 }$
	BB	4.8 ± 4.5	0.40 ± 0.41	0.06 ± 0.08
	EB	−6.2 ± 3.2	0.27 ± 0.38	0.10 ± 0.08
CMB-2	EE	5.5 ± 3.7	0.01 ± 0.56	0.23 ± 0.19
	BB	4.6 ± 3.4	0.18 ± 0.48	−0.11 ± 0.13
	EB	−5.5 ± 2.8	−0.39 ± 0.41	−0.20 ± 0.12
CMB-3	EE	0.2 ± 1.9	0.64 ± 0.43	0.10 ± 0.18
	BB	−0.3 ± 2.6	0.33 ± 0.35	0.01 ± 0.13
	EB	1.4 ± 1.7	−0.34 ± 0.30	$\mathbf { -0.27 \pm 0.11}$
CMB-4	EE	−5.2 ± 5.1	0.7 ± 1.2	0.65 ± 0.58
	BB	−2.6 ± 5.2	−0.1 ± 1.1	−0.37 ± 0.52
	EB	−1.0 ± 3.9	0.0 ± 0.9	−0.15 ± 0.47

Notes. Power-spectra estimates for the first multipole bin for each patch, computed from the WMAP7 K-band data and the QUIET Q-band data. The units are $\ell$($\ell$ + 1)$C_{\ell }$/2π(μK²) in thermodynamic temperature. Uncertainties for $\hat{C}^\mathrm{KK}_{b=1}$ and $\hat{C}^\mathrm{QK}_{b=1}$ include noise only. For $\hat{C}^\mathrm{QQ}_{b=1}$, they additionally include CMB sample variance as predicted by ΛCDM. Values in bold are more than 2σ away from zero.

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There is significant EE power in patch CMB-1 as measured by $\hat{C}^\mathrm{KK}_{b=1}$. We also find a correspondingly significant cross-correlation between the WMAP K band and the QUIET Q band, confirming that this excess power is not due to systematic effects in either experiment and is very likely a foreground. No significant power is found in any other case. The non-detection of foreground power at $\ell$ > 75 is consistent with the expected foreground dependence: ∝$\ell$^−2.5 (Carretti et al. 2010), and the low power found in $\hat{C}_{b=1}^\mathrm{KK}$.

The excess power observed in the first EE bin of CMB-1 is fully consistent with a typical synchrotron frequency spectrum. To see this, we extrapolate $\hat{C}^\mathrm{KK}_{b=1}$ from K band to Q band, assuming a spectral index of β = −3.1 (Dunkley et al. 2009), and calculate the expected power in C^QK_{b = 1} and C^QQ_{b = 1},

$$\begin{equation} C^\mathrm{QK}_{b=1} = \frac{1.05}{1.01} \left(\frac{43.1}{23} \right)^\beta \hat{C}^\mathrm{KK}_{b=1} = 2.57 \pm 0.69\,\mu {\rm K}^2\:, \end{equation} \tag{ 8 }$$

$$\begin{equation} C^\mathrm{QQ}_{b=1} = \left[\frac{1.05}{1.01} \left(\frac{43.1}{23} \right)^\beta \right]^2 \hat{C}^\mathrm{KK}_{b=1} = 0.38 \pm 0.10\,\mu {\rm K}^2\:, \end{equation} \tag{ 9 }$$

where the prefactor accounts for the fact that β is defined in units of antenna temperature, and the uncertainties are scaled from that of $\hat{C}^\mathrm{KK}_{b=1}$. These predictions are fully consistent with the observed values of $\hat{C}^\mathrm{QK}_{b=1}$ and $\hat{C}^\mathrm{QQ}_{b=1}$, when combined with the ΛCDM-expected power. We conclude that the excess power is indeed due to synchrotron emission.

We constrain the tensor-to-scalar ratio, r, using the QUIET measurement of the BB power spectrum at low multipoles (25 ⩽ $\ell$ ⩽ 175). Here, r is defined as the ratio of the primordial-gravitational-wave amplitude to the curvature-perturbation amplitude at a scale k₀ = 0.002 Mpc⁻¹. We then fit our measurement to a BB-spectrum template computed from the ΛCDM concordance parameters with r allowed to vary. For simplicity, we fix the tensor spectral index at n_t = 0 in computing the template.⁴² This choice makes the BB-power-spectrum amplitude directly proportional to r.

For pipeline A, we find r = 0.35^+1.06_−0.87, corresponding to r < 2.2 at 95% confidence. Pipeline B obtains r = 0.52^+0.97_−0.81. The results are consistent; the lower panel of Figure 11 shows our limits on BB power in comparison with those from BICEP, QUaD, and WMAP. QUIET lies between BICEP and WMAP in significantly limiting r from measurements of CMB-B-mode power in our multipole range. Although we neither expected nor detected any BB foreground power, the detection of an EE foreground in patch CMB-1 suggests that BB foregrounds might be present at a smaller level. We emphasize that the upper limit we report is therefore conservative.

Figure 13 compares the QUIET and WMAP Q-band temperature maps and TT, TE, and TB power spectra. Agreement with the ΛCDM model is good. This is a strong demonstration of the raw sensitivity of the QUIET detectors; the single QUIET differential-temperature assembly produces a high-signal-to-noise map using only 189 hr (after selection) of observations. The high sensitivity of these modules makes them very useful for calibration, pointing estimation, and consistency checks (see Section 4).

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The uncertainty in the beam window function is another multiplicative factor, one which increases with multipole. We estimate this uncertainty using the difference of the beam window functions measured for the central module and the modules of the differential-temperature assembly, which are at the edge of the array. The difference is statistically significant, coming from the different locations (with respect to the optics) in the focal plane; it is expected from the pre-season antenna range measurements.

Uncertainties in pointing lead to distortions in polarization maps. E power will be underestimated and spurious B power (if the distortions are nonlinear) generated (Hu et al. 2003). We quantify these effects by using the differences in pointing solutions from two independent models: the fiducial model used for the analysis and an alternative model based on a different set of calibrating observations. We also modeled and included the effects of the deck-angle-encoder shift which occurred for a portion of the season (Section 4.3).

Responsivity shifts, particularly within CESs, lead to distortions in the maps. Full-pipeline simulations quantify the shifts caused by variations in the cryostat or electronics temperatures. Similarly shifts from using responsivities determined from the Moon data, Tau A data, or from the sparse wire grid, rather than those from the sky dips, are determined. We also incorporate the uncertainty in the atmospheric-temperature model used in analyzing the sky-dip data. The largest possible effects on the power spectra are shown in Figure 14.

Uncertainties in the orientation of the polarization axes of the modules can lead to leakage between E and B modes. To quantify this leakage, we use the differences in power spectra where these angles are determined from Moon data, Tau A data, and the sparse-wire-grid data. As expected, the largest effects show up in EB power. We also estimate systematic error due to possible fluctuation of the detector angles over the course of the season. The contribution is negligibly small compared to the overall shift of the angles described above. Both effects are included in the "Polarization Angle" points in Figure 14.

As described in Section 4.5, the I to Q (U) leakage coefficients for the QUIET detector diodes are small: 1% (0.2%). Except in the case of patch CMB-4, our scanning strategy significantly reduces this effect with the combination of sky and deck-angle rotation.

We estimate spurious Q and U in the maps for each CES-diode using the WMAP temperature map and our known leakages. Shown in Figure 14 are the estimates of spurious EE, BB, and EB powers from full-pipeline simulations, where for each realization the spurious Q and U are added to the Q and U from simulated ΛCDM E modes. While this method has an advantage of being able to use the real (not simulated) temperature map, it does not incorporate TE correlation, which only affects the spurious EE power. As a complement, we repeat the study, but using simulated ΛCDM maps for both temperature and polarization; this only changes the estimate of spurious EE power by 30% at most. Because the spurious power is as small as it is, we have treated it as a systematic rather than correcting for it. Doing so would give us a further order of magnitude suppression.

Differing beam ellipticities can also induce higher multipole polarization signals. We measure these leakages from Tau A and Jupiter observations and find that the higher-order multipoles are at most 0.1% of the main-beam peak amplitude. The corresponding effects on the power spectra, which are seen in Figure 14, are of little concern.

While we make cuts to reduce the effects of far sidelobes seeing the Sun (Sections 2 and 5.1.3), small contaminations could remain. We make full-season maps for each diode in Sun-centered coordinates and then use these maps to add contamination to full-pipeline CMB simulations. The excess power found in the simulations is taken as the systematic uncertainty. We do not observe any signature from the Moon, nor do we cut on proximity to it. We estimate the related systematic error and find that it is negligibly small compared to that assigned to the contamination from the Sun.

Here, we discuss a few additional potential sources of systematic uncertainty, which are found to be subdominant.

Ground-synchronous signals. QUIET's far sidelobes do see the ground for some diodes at particular elevations and deck angles. Ground pickup that is constant throughout a CES is removed by our TOD filters; the net effect of this filtering in the full-season maps is a correction of ≈1 μK.

The only concern is ground pickup that changes over the short span of a single CES. We find little evidence for changes even over the entire season, let alone over a single CES. We therefore conservatively place an upper limit on such changes using the statistical errors on the ground-synchronous signal. We start with the CES and module with the largest ground pickup. We then simulate one day's worth of data, inserting a ground-synchronous signal that changes by its statistical error. Given the distribution in the magnitude of the ground-synchronous signal and assuming that changes in this signal are proportional to the size of the signal itself, by considering that the signals from changing pickup add incoherently into the maps made from multiple CES-diodes at a variety of elevations and deck angles, we estimate an upper limit on residual B power from possible changing ground-pickup signals. The result is ≲ 10⁻⁴ μK² at multipoles below 100.

ADC nonlinearities. The possible residual after the correction for the nonlinearity in the ADC system results in effects similar to the I to Q (or U) leakage and the variation of the responsivity during the CES. We estimate such effects based on the uncertainty in the correction parameters, confirming that there is at most a 3% additional effect for the leakage bias, and that the responsivity effect is also small, less than half of the systematic error shown for the responsivity in Figure 14.

Data-selection biases. Cuts can cause biases if they are, for example, too stringent. We expect none but to be sure we apply our selection criteria to 144 CMB + noise simulations. No bias is seen, and in particular we limit any possible spurious B modes from this source to ≲ 10⁻³ μK² at multipoles below 100.