SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP^*) OBSERVATIONS: SKY MAPS, SYSTEMATIC ERRORS, AND BASIC RESULTS

The sixth and seventh years of WMAP's operation span the interval from 2006 August 9, 00:00:00 UT (day number 222) to 2008 August 10, 00:00:00 UT (day number 222) and include nine short periods when observations were interrupted. These periods include eight scheduled events: six station-keeping maneuvers and one maneuver to avoid flying through the Earth's shadow (2007 November 11), followed by a small orbital correction 19 days later. The other interruption to observations occurred as a result of the failure of the primary transmitter used to telemeter data to Earth. WMAP was subsequently reconfigured to use its backup transmitter and normal operations resumed with no performance degradation. On 2008 August 1 (day number 141) WMAP flew through the Moon's shadow, causing a ≈4% decrease in the incident solar flux lasting ≈6 hr. WMAP remained in normal observing mode throughout this event, but data were excluded from sky map processing due to minor instrument thermal perturbations as described in Section 2.3.

The algorithm used to calibrate the time-ordered data (TOD) is the same as was used for the five-year processing (Hinshaw et al. 2009). Calibration occurs in two steps. First, an hourly absolute gain and baseline are determined. The absolute calibration is based on the CMB monopole temperature (Mather et al. 1999) and the velocity-dependent dipole resulting from WMAP's orbit about the solar system barycenter. The calibration is performed iteratively, since removing the fixed sky signals arising from the barycentric CMB signal and foregrounds requires values of both the gain and baseline solutions. The only change made to this step of the calibration procedure is that the initial value for the barycentric CMB dipole signal has been updated to agree with the value determined from the five-year analysis.

The second step in the calibration procedure consists of fitting the hourly radiometer gain values to a model which relates the gain to measured values of radiometer parameters. The functional form of the gain model is the same as for the five-year data release, but the fitting procedure now utilizes all seven years of observational data. The seven-year mean changes in the calibration are presented in Table 1. Note that the rms of the fractional calibrations change, ΔG/G, is only 0.05%. The absolute calibration accuracy is estimated to be 0.2%, unchanged from the previously published value.

For the K, Ka, and Q bands, there is a small but significant year-to-year variation in the WMAP calibration that can be seen as variations of the Galactic plane brightness in yearly sky maps. This has been measured by correlating each yearly map against the seven-year map for pixels at |b| < 10°. A slope and offset is fit to each correlation and the slope values are adopted as yearly calibration variations. These are shown in Figure 1 and are consistent with the 0.2% absolute calibration uncertainty.

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The WMAP instrument provides the highest quality data when the observatory is scanning the sky in its nominal observing mode and the instrument thermal environment is stable. Periodically, conditions arise that result in one or both of these conditions not being met. These situations arise from scheduled station-keeping maneuvers, and unscheduled events, such as solar flares or on-board equipment malfunctions. Data taken during these periods are excluded from the sky map processing to ensure the highest quality data products. Previously, potentially corrupted data segments were identified by manually inspecting the event logs and instrument thermal trend plots. The current data release uses a more objective automated procedure to identify the unusable data segments. The automated procedure was designed to approximate the manual procedure, but small differences will occur in the sky coverage between the first five years of the current data release and the previous five-year data release. Using the new procedure WMAP still achieves an overall observing efficiency of ≈98.4%, down slightly from its previous value of ≈99%.

The automated procedure identifies suspect events based on the time derivative of the focal plane assembly (FPA) temperature. The measured FPA temperature is averaged over one hour intervals, and the time derivative is formed by differencing successive values of these averages. Suspect thermal events are delineated by the times at which the magnitude of this derivative initially exceeds then finally falls below a threshold value. The threshold has been chosen to be five times the rms deviation of the temperature derivative signal occurring during normal observing periods, and corresponds to a value of ≈0.75 mK hr⁻¹. In situations where the observatory was taken out of normal observing mode (e.g., during a scheduled station-keeping maneuver), the duration of the event is lengthened if needed to encompass the entire time the observatory was not in normal observing mode. For each event, data from 1.2 hr before the event began to 7.25 hr after the event ended are excluded from sky map processing.

In the one-, three-, and five-year data analyses, observations when the boresight of each telescope beam fell within 15 of Mars, Jupiter, Saturn, Uranus, or Neptune were excluded from sky map processing, preventing contamination of the sky maps by emission from the planets. Subsequent analysis has shown that even with these exclusion criteria microwave emission from Jupiter could generate as much as a 75 μK (in the K band) errant signal in a narrow annulus surrounding the region of excluded observations. Although there is no indication that this influenced the cosmological analysis, the radii used to exclude observations have been increased for the seven-year analysis, chosen to limit planetary leakage to less than 1 μK. However, given the uncertainty in the determination of the beam profiles at these levels, we adopt a conservative upper bound on possible planetary signal leakage into the TOD used for sky map production of 5  μK. The new values are displayed in Table 2.

Table 2. Data Exclusion Radii for the Planets

Planet	Frequency Band
	K	Ka	Q	V	W
Mars	20	15	15	15	15
Jupiter	30	25	25	22	20
Saturn	20	15	15	15	15
Uranus	20	15	15	15	15
Neptune	20	15	15	15	15

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The mask for diffuse Galactic emission has been revised by including areas near the plane where the diffuse foreground cleaning algorithm appears to be less efficient than for the sky at large. These areas are found by performing a pixel-by-pixel χ² test comparing null maps to cleaned Q–V and V–W maps with resolution degraded from r9¹⁵ to r5. All pixels with χ² greater than four times that of the χ² in the polar caps regions are cut. Small islands of cut pixels are eliminated from the cut if they contain 4 or fewer contiguous pixels. The two resulting masks based on Q–V and V–W analyses, respectively, are combined and promoted back to r9. The edges of the cut are smoothed by convolving the mask with a Gaussian of 3° FWHM and cutting the result at a value of 0.5. As the final step, the smoothed cut is combined with the five-year KQ85 or KQ75 cut (Gold et al. 2011). The resulting masks, termed KQ85y7 and KQ75y7, admit 78.3% and 70.6% of the sky, respectively, as compared to 81.7% and 71.6% for the five-year versions—a decrease of the admitted sky area by 3.4% of the full sky for KQ85 and 1.0% for KQ75.

The most significant change in the current processing is the use of asymmetric masking in the iterative reconstruction of the sky maps. WMAP's differential design means that the TOD represents differences between the intensities of pairs of points on the sky observed by the two telescope beams. Reconstructing sky maps from differential data requires solving a set of linear equations that describe the relation between the differential TOD and the sky signal. Solution of this set of equations is performed iteratively, using sky maps from earlier iterations to alternately remove estimated sky signals for each beam from the TOD, leaving the value associated with the opposite beam which is then used to generate the next iteration of the sky map. This procedure becomes problematic when one of the beams is in a region of high intensity and the other is in a region of low intensity. Small errors arising from pixelization, residual pointing errors, beam ellipticity, or radiometer gain errors limit the degree to which the signal associated with the beam in the high emission region can be estimated. Such errors result in errant signals being introduced into sky map pixels associated with the beam in the low-intensity region. This potential problem is circumvented by masking portions of the data when such errors are likely to be introduced.

In the WMAP first-year results, an asymmetric masking procedure was used (Hinshaw et al. 2003). Asymmetric masking means that when one beam is in a high Galactic emission region (as determined by a processing mask; Limon et al. 2010) and the other beam is in a low Galactic emission region, only the pixel in the high emission region is iteratively updated. This allows for the reconstruction of full-sky maps while avoiding the situation that could produce an errant signal. The maps, t₁, were obtained through a simple iterative solution of the linear equation

$$\begin{equation} {\bf t}_{\rm 1} = {\bf W} {\bf d}_{\rm 1} \end{equation} \tag{ 1 }$$

where d₁ is the pre-whitened TOD and W = (M^TM)⁻¹ · M^T. The matrix M is the mapping function, which has N_p columns and N_t rows, corresponding to the number of sky map pixels and TOD points, respectively. When the map processing includes polarization degrees of freedom N_p is four times the number of map pixels, corresponding to the Stokes I, Q, and U components and a spurious component, S, used to absorb effects arising from bandpass differences between the two radiometers comprising each differencing assembly (DA). See Jarosik et al. (2007) for a description of the implementation of the spurious mode, S. Multiplying a sky map vector by M effectively generates the TOD that would be obtained if WMAP observed a sky corresponding to the map vector. The sky maps obtained by solving Equation (1) are an unbiased representation of the true sky signal, but do not treat the noise terms optimally.

The WMAP three-year (Jarosik et al. 2007) and five-year (Hinshaw et al. 2009) analyses generated maximum likelihood map solutions using a conjugate gradient iterative technique. The maximum likelihood estimate of the sky map, ${\bf {\tilde{t}}}$, is

$$\begin{equation} {\bf { \tilde{t}}} = ({\bf M}^{\rm T} {\bf N}^{-1} {\bf M})^{-1} \cdot ({\bf M}^{\rm T} {\bf N}^{-1} {\bf d}) \end{equation} \tag{ 2 }$$

where d is the calibrated but unfiltered TOD and N⁻¹ is the inverse of the radiometer noise covariance matrix. Solution of this equation involves a direct evaluation of the last term of Equation (2) once, resulting in a map t₀ = M^TN⁻¹d, followed by the iterative solution of the equation

$$\begin{equation} {\bf { \tilde{t}}} = ({\bf M}^{\rm T} {\bf N}^{-1} {\bf M})^{-1} \cdot {\bf t_0}. \end{equation} \tag{ 3 }$$

Solving this equation with the conjugate gradient method requires the use of symmetric masking since this method can only be used when the matrix multiplying t₀ (in this case (M^TN⁻¹M)⁻¹) is symmetric. Symmetric masking means that if either beam is in a region of high Galactic emission, neither pixel is updated. Symmetric masking is implemented by simply replacing all occurrences of the full-sky mapping matrix, M, with a masked version of the matrix in Equations (2) and (3). Since symmetric masking produces sky maps with no information in the high Galactic emission regions, in previous analyses two sets of sky maps were generated, symmetrically masked maps, and full-sky maps incorporating no masking. Data from full-sky maps were used to fill in regions of the symmetrically masked maps which contained no data. This procedure was a significant advancement over the iterative procedure used in the WMAP first-year results, in that it produced maximum likelihood maps and allowed for direct determination of the level of convergence of the solution. Its major disadvantages are that two sets of maps have to be generated then pieced together, and there is no simple method of calculating a pixel–pixel noise matrix which describes the noise covariance between pixels obtained from the two different input maps. The pixel–pixel noise covariance matrix delivered with the data release only applied to the pixels in the low-Galactic emission regions, and only diagonal weights are used to describe the noise properties in the regions of high Galactic emission.

Incorporating asymmetric masking into the maximum likelihood sky maps solution requires solving the equation

$$\begin{equation} {\bf { \tilde{t}}} = \big({\bf M_{\rm am}^T} {\bf N}^{-1} {\bf M}\big)^{-1} \cdot \big({\bf M_{\rm am}^T} {\bf N}^{-1} {\bf d}\big) \end{equation} \tag{ 4 }$$

where M^T_am is the mapping matrix incorporating asymmetric masking and M is the unmasked mapping matrix. Again, this equation is solved by direct evaluation of the last term,

$$\begin{equation} {\bf t}_0 ={\bf M_{\rm am}^{\rm T}} {\bf N}^{-1} {\bf d} \end{equation} \tag{ 5 }$$

once, followed by an iterative solution of the equation

$$\begin{equation} {\bf { \tilde{t}}} = \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1} \cdot {\bf t_0}. \end{equation} \tag{ 6 }$$

Note that the matrix M^T_amN⁻¹M is not symmetric, and therefore this equation cannot be solved using a simple conjugate gradient algorithm as was done previously. A bi-conjugate gradient stabilized method (Barrett et al. 1994) was chosen to solve this equation since it offered good convergence properties and was straightforward to implement, requiring relatively minor changes to the existing procedures. This technique was verified by reconstructing input maps to numerical precision from simulated noise-free TOD. Utilizing asymmetric masking eliminates both of the aforementioned shortcomings of the simple conjugate gradient method—each DA only requires a single map solution, and a pixel–pixel inverse noise covariance matrix can be generated which describes the noise correlation between all the pixels in the map.

The pixel–pixel noise covariance matrix is given by ${\bf \Sigma } = \langle {\bf {\tilde{t}}_{\rm n}} {\bf {\tilde{t}}}^{\rm T}_{\rm n}\rangle$, where ${\bf \tilde{t}_{\rm n}}$ is the noise component of a reconstructed sky map and the brackets denote an ensemble average. The TOD, d, used to generate the map may be written as the sum of a signal term and a noise term

$$\begin{equation} {\bf d } = {\bf Mt } + {\bf n} \end{equation} \tag{ 7 }$$

where M is the full-sky (unmasked) mapping matrix, t is a vector representing the input sky signal, and n is a vector corresponding to the radiometer noise, its covariance being described as

$$\begin{equation} {\bf N} = \langle {\bf n}{\bf n}^{\rm T}\rangle. \end{equation} \tag{ 8 }$$

The noise component of the maximum likelihood asymmetrically masked map solution may be written as

$$\begin{equation} {\bf { \tilde{t}_{\rm n}}} = \big({\bf M_{\rm am}^T} {\bf N}^{-1} {\bf M}\big)^{-1} \cdot \big({\bf M_{\rm am}^T} {\bf N}^{-1} {\bf n}\big). \end{equation} \tag{ 9 }$$

The noise covariance matrix becomes

$$\begin{eqnarray} {\bf \Sigma }&=& \big\langle \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1} \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf n}\big) \cdot \big[\big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1}\qquad\nonumber\\ &&\times \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf n}\big)\big ]^{\rm T} \big\rangle \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} \hspace{-5pt}&=& \big\langle \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1} \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf n}\big) \,{\cdot}\, \big({\bf n}^{\rm T} {\bf N}^{-1} {\bf M}_{\rm am}\big)\qquad\hspace{5pt}\nonumber\\ &&\times ({\bf M}^{\rm T} {\bf N}^{-1} {\bf M}_{\rm am})^{-1} \big\rangle \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \hspace{-1.7pt}&=&\big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1} \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1}\big)\,{\cdot}\, (\langle {\bf n} {\bf n}^{\rm T}\rangle {\bf N}^{-1}){\bf M}_{\rm am}\qquad\hspace{1.7pt}\nonumber\\ &&\times ({\bf M}^{\rm T} {\bf N}^{-1} {\bf M}_{\rm am})^{-1} \end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray} \hspace{1.5pt}&=&\big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big)^{-1} \cdot \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}_{\rm am}\big)\cdot ({\bf M}^{\rm T} {\bf N}^{-1} {\bf M}_{\rm am})^{-1}.\hspace{-1.5pt}\nonumber\\ \end{eqnarray} \tag{ 13 }$$

In practice, the terms M^T_amN⁻¹M and M^T_amN⁻¹M_am are evaluated at r4 with the N⁻¹ normalized to unity at zero lag, and the inverse pixel–pixel noise covariance matrix is generated by forming the product

$$\begin{equation} {\bf \Sigma }^{-1} =({\bf M}^{\rm T} {\bf N}^{-1} {\bf M}_{\rm am}) \cdot \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}_{\rm am}\big)^{-1}\cdot \big({\bf M}^{\rm T}_{\rm am} {\bf N}^{-1} {\bf M}\big). \end{equation} \tag{ 14 }$$

Regions of the Σ⁻¹ matrix corresponding to combinations of Stokes I, Q, and U are then converted to noise units by dividing by the appropriate combinations of σ₀(I) and σ₀(Q, U) given in Table 1.

The term M^T_amN⁻¹M_am is numerically inverted and is used as the source for the off-diagonal terms of the preconditioner for the bi-conjugate gradient stabilized algorithm.

The asymmetric masking used to generate sky maps requires a new procedure for calculating the effective number of observations, N_obs, for each map pixel. In previous data releases, the effective number of observations was calculated by accumulating the number of TOD points falling within each pixel, weighted by the appropriate transmission imbalance coefficients, x_im, and polarization projection factors (sin 2γ, cos 2γ). These values corresponded to the diagonal elements of the inverse pixel–pixel noise covariance matrix Σ⁻¹ = M^TN⁻¹M (Jarosik et al. 2007, Equation (26)), evaluated assuming white radiometer noise, i.e., N⁻¹ = I. The simple procedure for evaluating these terms is not applicable to the form of Σ⁻¹ for the asymmetrically masked maps, Equation (14). The N_obs fields of the r9 and r10 sky maps were generated by evaluating the diagonal elements of Equation (14) with N⁻¹ = I using sparse matrix techniques. The N_obs fields of the r4 sky maps are described in Section 2.10.

As described in Jarosik et al. (2007), errors in the determination of the transmission imbalance parameters, x_im, are a potential source of systematic artifacts in the reconstructed sky maps. These time-independent parameters, which specify the difference between the transmission between the A-side and B-side optical systems for each radiometer, are measured from the flight data, and are presented in Table 3. To prevent biasing the cosmological analyses, sky map modes that can be excited by these measurement errors are identified and projected from the Σ⁻¹ matrices.

The procedure follows the same method as used in the three- and five-year analyses and consists of generating a one-year span of simulated TOD using the nominal values of the loss imbalance parameters. Sky maps are processed from this archive using the input values of x_im and altered values to simulate errors in the measured values of the coefficients. The differences between the resultant maps are used to identify map modes resulting from processing the data with altered x_im values. Two modifications have been made to this procedure relative to the previous analyses. (1) The simulated sky maps are formed at r4 by direct evaluation of Equations (5) and (6) at r4. This change was adopted to eliminate contamination of the transmission imbalance templates by the poorly measured sky map modes associated with monopoles in the Stokes I and spurious mode S sky maps. It was found that varying the x_im factors produced large changes in the amplitudes of these poorly measured modes in addition to exciting the mode associated with the loss imbalance. Through utilization of a singular value decomposition it was possible to null the poorly measured modes associated with the aforementioned monopoles while preserving the modes associated with the transmission imbalance while evaluating Equation (6). (2) Transmission imbalance modes were evaluated both for the case when the x_im for both radiometers comprising each DA were increased 20% above their measured values, and for the case when the x_im value for radiometer 1 was increased by 10% and that of radiometer 2 decreased by 10%. (Previously the only combination used was that in which both x_im values were increased.) For six of the DAs very similar sky map modes were generated by both sets of x_im, while for the V2, W1, W2, and W4 somewhat different modes were generated for the two different sets. As a result, only one mode was projected from the K1, Ka1, Q1, Q2, and W3 matrices, while two modes were projected out of the V2, W1, W2, and W4 matrices. In each case, the modes were removed from Σ⁻¹ following the method described in Jarosik et al. (2007).

The low-resolution sky maps were generated by performing an inverse noise weighted degradation of the high-resolution maps that takes into account the intra-pixel noise correlations of the high-resolution input maps. The weight matrix for the polarization (Stokes Q and U) and spurious mode (S) of each high-resolution map pixel, p9, is given by

$$\begin{equation} {\bf N}_{\rm obs}({\rm p9}) = \left(\begin{array}{@{}ccc@{}} N_{\rm QQ} & N_{\rm QU} & N_{\rm QS}\\ N_{\rm QU} & N_{\rm UU} & N_{\rm SU}\\ N_{\rm QS} & N_{\rm SU} & N_{\rm SS} \end{array} \right), \end{equation} \tag{ 15 }$$

where each element N_XY is an element of Σ⁻¹ (Equation (14)) evaluated as described in Section 2.8. For each DA year combination, the Q, U, S map sets were generated as

$$\begin{eqnarray} &&\left(\begin{array}{@{}c@{}} Q_{\rm p4} \\ U_{\rm p4} \\ S_{\rm p4} \end{array} \right)& = & \left({\bf N}_{\rm obs}^{\rm tot}({\rm p4}) \right)^{-1} \sum _{{\rm p9} \in {\rm p4}}{\bf N}_{\rm obs}({\rm p9}) \left(\begin{array}{@{}c@{}} Q_{\rm p9} \\ U_{\rm p9} \\ S_{\rm p9} \end{array} \right), \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} {\bf N}_{\rm obs}^{\rm tot}({\rm p4})& = &\sum _{{\rm p9} \in p4} {\bf N}_{\rm obs}({\rm p9}). \end{eqnarray} \tag{ 17 }$$

The N_obs fields of the low-resolution sky maps contain the diagonal elements of the corresponding portions of the Σ⁻¹ matrices as described in Section 2.7 before the scaling by σ₀ is applied.

The single-year single-DA maps were combined to form a seven-year map for each DA and seven-year maps for each frequency band. The individual low-resolution maps were inverse noise weighted using the ${\bf \tilde{\Sigma }}$ matrices (Section 2.9), scaled to reflect the noise level of each DA. The weighted polarization maps were formed as

$$\begin{eqnarray} &&\left(\begin{array}{@{}c@{}} Q \\ U \end{array} \right)& = & {\bf \tilde{\Sigma }}^{\rm tot} \sum _{\rm yr, DA} {\bf \tilde{\Sigma }^{-1}}_{\rm yr, DA} \left(\begin{array}{@{}c@{}} Q \\ U \\ \end{array} \right)_{\rm yr, DA}, \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} {\bf \tilde{\Sigma }}^{\rm tot}& = &\left(\sum _{\rm yr, DA} {\bf \tilde{\Sigma }^{-1}}_{\rm yr, DA} \right)^{-}, \end{eqnarray} \tag{ 19 }$$

where ()⁻ represents a pseudo-inverse of the sum in parenthesis. This summation is performed over the year/DA combinations to be included in the final map. For K and Ka bands, the pseudo-inverse is calculated by performing a singular value decomposition of the sum of the QU × QU ${\bf \tilde{\Sigma }^{-1}}$ matrices and inverting all its eigenvalues except for the smallest, which is set to zero. This procedure de-weights the one nearly singular mode associated with the monopoles in the I and S maps. No modes are removed from the Q-, V-, and W-band maps.

3.1.1. Secondary Tip/Tilt

3.1.2. Fitting Process

The overall fitting of beams progresses much as it did for the five-year data. On each of the A and B sides, the primary mirror is modeled with Fourier modes added to the nominal mirror figure. Similarly, the secondary mirror is modeled with Bessel modes. A complete fit is done using modes with spatial frequency on the mirror surface up to some maximum value. When convergence is achieved, the current best-fit parameters become the starting point for a new fit with a higher spatial frequency limit. The primary mirror spatial frequencies are indexed using an integer k, such that the wavelength of the mirror surface distortion is 280 cm/k. The secondary modes are specified by the Bessel mode indexes n and k. Details are given in Hill et al. (2009, Section 2.2.2). The maximum spatial frequencies fitted are defined by k_max = 24 on the primary and n_max = 2 on the secondary.

Because the Fourier modes are non-orthogonal over the circular domain of the primary mirror distortions, the fitting is actually done in an orthogonalized space (Hill et al. 2009). A bug in the orthogonalization code was corrected before the seven-year analysis began; this bug only affected speed of convergence, not the definition of χ², so the five-year results remain valid. However, this change made it practical to refit the beam models ab initio from the nominal mirror figures, i.e., with zero for all Fourier and Bessel coefficients, rather than beginning from the five-year solution.

The χ² changes from the five-year to the seven-year beam models are shown in Table 5. The χ² values shown for the five-year models are recomputed using seven-year data. It is clear from Table 5 that the overall improvement in the model for the 10 DAs collectively is driven primarily by the W band on the A side and the V and W bands on the B side.

Table 5. Changes in χ² from Five-year to Seven-year Beam Fits

DA	χ2ν	χ2ν	Δχ2ν
	(Five-year)	(Seven-year)
Side A
All	1.060	1.049	−0.011
K1	1.009	1.006	−0.004
Ka1	1.012	1.016	0.004
Q1	1.078	1.074	−0.003
Q2	1.094	1.102	0.008
V1	1.144	1.155	0.011
V2	1.162	1.171	0.008
W1	1.225	1.171	−0.053
W2	1.239	1.151	−0.088
W3	1.244	1.181	−0.063
W4	1.208	1.156	−0.052
Side B
All	1.065	1.047	−0.018
K1	1.022	1.017	−0.005
Ka1	1.010	1.009	−0.000
Q1	1.093	1.046	−0.046
Q2	1.045	1.052	0.007
V1	1.288	1.171	−0.117
V2	1.178	1.159	−0.019
W1	1.226	1.197	−0.029
W2	1.169	1.155	−0.013
W3	1.223	1.203	−0.021
W4	1.226	1.165	−0.061

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Figure 2 shows model beam profiles from an example DA, W1, from both the five-year and the seven-year analyses. The A-side model has a similar profile in the five-year and seven-year fits, with small tradeoffs in fit quality at various radii. However, the B-side model shows a new feature in the seven-year fitting, namely, an elevated response at large radii and very low signal levels in the V and W bands. This feature is seen at radii larger than 2° in the right-hand panel of Figure 2, where the seven-year profile (red) is very slightly higher than the five-year profile (blue). The five-year and seven-year profiles for the example DA differ by a factor of ∼2–10 in this region, although both are ≳50 dB down from the beam peak.

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Several attempts have been made using jumps in the fitting parameters (as in simulated annealing) to find a region of parameter space in the A-side fit that would lead to the existence of elevated tails similar to those on the B side. So far, such attempts have failed. The best A-side fit lacks the elevated-tail feature, while the best B-side fit has it.

Just as in the five-year analysis, none of the beam models meets a formal criterion for a good fit, since the number of degrees of freedom is of order 10⁵, or 10⁴ per DA, while the overall χ²_ν is ∼1.05. However, the model beams are used only in a restricted role, more than ∼45 dB below the beam peak, so that the primary contribution to the beams and b_ℓ is directly from Jupiter data. As a result, the effect of any particular feature of the models is either omitted or diluted in the final result. However, the B-side model tail mentioned above increases the beam uncertainty slightly at large radii as compared to the five-year analysis.

The process for incorporating information from the models into the Jupiter-based beams is explained by Hill et al. (2009). This process is termed hybridization. Briefly, the beam profiles are integrated from Jupiter TOD. The two-dimensional coordinates of each TOD point within the beam model are determined. If the model gain is below a certain threshold, then the model gain is substituted for the Jupiter measurement. The thresholds are optimized for each DA by minimizing the uncertainty in the solid angle, under the assumption that the beam model is subject to a scaling uncertainty of 100%. Final hybrid profiles combining the A and B sides are shown for the W1 DA, for both the five-year and seven-year analyses, in Figure 3.

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The beam transfer function, b_ℓ, is integrated directly from the hybrid beam profiles, and the error envelope for b_ℓ is computed using a Monte Carlo method. These procedures are the same as in the five-year analysis (Hill et al. 2009). A comparison of five-year and seven-year b_ℓ and the corresponding uncertainties is shown in Figure 4.

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The conversion factor from peak antenna temperature to flux density for a point source is given by (Page et al. 2003a)

$$\begin{equation} \Gamma = c2/2 k_B \Omega _{\rm eff} (\nu _{\rm eff})^2, \end{equation} \tag{ 20 }$$

where Ω_eff is the effective beam solid angle of the A and B sides combined and ν_eff is the effective band center frequency, both of which depend on the source spectrum. New values of these quantities have been calculated that supersede previous results given for point sources.

For a point source with antenna temperature spectrum T_A ∝ ν^β, the effective frequency is determined from

$$\begin{equation} \nu _{\rm eff}^{\beta } = \int f(\nu) G_m(\nu) \nu ^{\beta } d\nu / \int f(\nu) G_m(\nu) d\nu, \end{equation} \tag{ 21 }$$

where f(ν) is the passband response and G_m(ν) is the forward gain. This is consistent with the definition used by Jarosik et al. (2003) for a beam-filling source, except in that case the forward gain is not included. Values of ν_eff(β) are calculated using pre-flight bandpass measurements and pre-flight GEMAC¹⁷ measurements of forward gain. A correction for scattering is applied to the GEMAC measurements,

$$\begin{equation} G_m(\nu) = G_m^{\rm GEMAC}(\nu) e^{(-(4 \pi \sigma / \lambda)^2)}, \end{equation} \tag{ 22 }$$

where σ is an effective rms primary mirror deformation whose value is set separately for each DA based on the degree of scattering estimated from the pass 4 physical optics modeling.

The effective solid angle is calculated as

$$\begin{equation} \Omega _{\rm eff} = \Omega (\mbox{Jupiter}) \Omega _{\rm GEMAC}(\beta)/\Omega _{\rm GEMAC}(\mbox{Jupiter}), \end{equation} \tag{ 23 }$$

where Ω(Jupiter) is the pass 4 beam solid angle determined from Jupiter observations and Ω(GEMAC) is the beam solid angle calculated for a given source spectrum using the scattering-corrected GEMAC measurements. From Equation (25) of Page et al. (2003b),

$$\begin{equation} \Omega (\mbox{GEMAC}) = 4 \pi \int f(\nu) \nu ^{\beta } d\nu / \int f(\nu) G_m(\nu) \nu ^{\beta } d\nu. \end{equation} \tag{ 24 }$$

Values of ν_eff, Ω_eff, and Γ for a point source with β = −2.1, typical for the sources in the WMAP point source catalog, are given in Table 2.

4.1.1. The Temperature Dipole and Quadrupole

The five-year CMB dipole value was obtained using a Gibbs sampling method (Hinshaw et al. 2009) to estimate the dipole signal in both the five-year ILC map and foreground reduced maps. The uncertainties on the measured parameters were set to encompass both results as an estimate of the effect of residual foreground signals. The dipole measured from the seven-year data shows no significant changes from that obtained from the five-year data, so the best-fit dipole parameters remain unchanged from the five-year values, and are presented in Table 6.

Table 6. WMAP Seven-year CMB Dipole Parameters

dxa	dy	dz	db	l	b
(mK)	(mK)	(mK)	(mK)	(°)	(°)
−0.233 ± 0.005	−2.222 ± 0.004	2.504 ± 0.003	3.355 ± 0.008	263.99 ± 0.14	48.26 ± 0.03

Notes. The measured values of the CMB dipole signal. These values are unchanged from the five-year values and are reproduced here for completeness. ^aCartesian components are give in Galactic coordinates. The listed uncertainties include the effects of noise, masking and residual foreground contamination. The 0.2% absolute calibration uncertainty should be added to these values in quadrature. ^bThe spherical components of the CMB dipole are given in Galactic coordinates and already include all uncertainty estimates, including the 0.2% absolute calibration uncertainty.

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The maximum likelihood value of magnitude of the CMB quadrupole is $l(l+1)C_l/2\pi = 197^{+2972}_{-155}\,\mbox{$\mu $\rm K}^2 (95\%\mbox{ CL})$ (Larson et al. 2011) based on the analysis of the ILC map using the KQ85 mask. This value is essentially unchanged from the five-year results and lies below the most likely value predicted by the best-fit ΛCDM model. This value, however, is not particularly unlikely given the distribution of values predicted by the model, as described in Bennett et al. (2011).

Figure 6 displays the seven-year band average maps for Stokes Q and U components for all five WMAP frequency bands. Polarized Galactic emission is evident in all frequency bands. The smooth large angular scale features visible in the W-band maps, and to a lesser degree in the V-band maps, are the result of a pair of modes that are poorly constrained by the map-making procedure. While these dominate the appearance of the map, they are properly de-weighted when these maps are analyzed using their corresponding Σ⁻¹ matrices, so useful polarization power spectra may be obtained from these maps. The relatively large amplitudes of these modes limits the utility of using difference plots between the five-year and seven-year map sets to test for consistency.

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4.1.2. Low-ℓ W-band Polarization Spectra

Previous analyses (Hinshaw et al. 2009; Page et al. 2007) exhibited unexplained artifacts in the low-ℓ W-band polarization power spectra demonstrating an incomplete understanding of the signal and/or noise properties of these spectra. Specifically, the value of C^EE_ℓ for ℓ = 7 measured in the W band was found to be significantly higher than could be accommodated by the best-fit power spectra, given the measurement uncertainty. This result, and several other anomalies, caused these data to be excluded from cosmological analyses. Significant effort has been expended trying to understand these spectra with the goal of eventually allowing their use in cosmological analyses.

A set of null spectra was formed based on the latest uncleaned W-band polarization sky maps to test for year-to-year and DA-to-DA consistency. Polarization cross power spectra were calculated for pairs of maps using the Master algorithm (Hivon et al. 2002) utilizing the full Σ⁻¹ covariance matrix to weight the input maps. The polarization analysis mask was applied by marginalizing the Σ⁻¹ over pixels excluded by the mask to minimize foreground contamination. Appropriately weighted null signal combinations of these spectra were formed to determine if any individual years or DAs possessed peculiar characteristics. The uncertainties on the power spectra were evaluated using the Fisher matrix technique (Page et al. 2007) and measured map noise levels. Since the input sky maps contain both signal (mostly of Galactic origin) and noise, an additional term was added to the Fisher matrix noise estimate to account for the signal × noise cross term. The signal component of this term was estimated using the seven-year average power spectra values for the combined W1, W2, and W3 DAs. This term was only added for multipole/polarization combinations (EE or BB) for which the estimated signal was greater than 0.

Table 7 shows the result of this analysis. The reduced χ² combinations in the top panel are evaluated for data combinations of all four W-band DAs that compare individual years to the average of the remaining years. Polarization combinations EE and BB were evaluated for multipoles ranging from 2 to 32. Additional combinations were formed to compare the first three years of data to the latter four, and to compare data taken in odd and even numbered years. All combinations resulted in reasonable χ² and probability to exceed values.

Table 7. Seven-year Spectrum χ² W-band Null Tests

Data Combinations	χ2EE	PTEEE	χ2BB	PTEBB
For DAs W1, W2, W3, and W4
{yr1} – {y2, y3, y4, y5, y6, y7}	0.944	0.56	0.976	0.50
{yr2} – {y1, y3, y4, y5, y6, y7}	0.800	0.78	0.956	0.54
{yr3} – {y1, y2, y4, y5, y6, y7}	0.934	0.57	1.166	0.24
{yr4} – {y1, y2, y3, y5, y6, y7}	0.850	0.70	0.920	0.59
{yr5} – {y1, y2, y3, y4, y6, y7}	0.737	0.85	1.286	0.13
{yr6} – {y1, y2, y3, y4, y5, y7}	0.769	0.82	1.101	0.32
{yr7} – {y1, y2, y3, y4, y5, y6}	1.108	0.31	0.831	0.73
{yr1, yr2, y3} – {y4, y5, y6, y7}	0.608	0.96	0.98	0.49
{yr1, yr3, y5, y7} – {y2, y4, y6}	0.860	0.69	1.06	0.38
For seven years of data
{W1} – {W2, W3, W4}	1.195	0.21	0.982	0.49
{W2} – {W1, W3, W4}	0.802	0.77	0.926	0.58
{W3} – {W1, W2, W4}	0.890	0.64	0.897	0.63
{W4} – {W1, W2, W3}	1.163	0.24	2.061	0.0005
{W1, W2} – {W3, W4}	0.867	0.68	1.361	0.09
{W1, W3} – {W2, W4}	0.866	0.68	1.219	0.19

Notes. χ² tests for various null combinations of low-ℓ Master polarization power spectra obtained from uncleaned W-band sky maps. Reduced χ² values are presented along with the probability to exceed (PTE) values based on 31 degrees of freedom, corresponding to 2 ⩽ ℓ ⩽ 32. Polarization cross power spectra were obtained for each DA year × DA year combination, then appropriately weighted combinations generated to null any sky signal. Predicted uncertainties were obtained using the standard Fisher matrix formalism incorporating the inverse noise covariance matrices and the measured sky map noise levels. Since individual power spectra estimates do include signals (mostly foreground), the uncertainties include a contribution for the signal × noise cross term as explained in the text. The only anomalous point occurs when the seven-year W4 data are compared to the seven-year data from the remaining W-band DAs.

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The lower half of the table contains the results of a similar analysis, but in this case combinations were formed to isolate individual DAs. The W4 is singled out with a reduced χ² value of 1.163 which has only a probability to exceed of only 0.05%. This DA has an unusually large 1/f knee frequency, which makes it particularly susceptible to systematic artifacts. For this reason it is excluded from the analysis below, but continues to be studied in case it might contain clues as to the nature of the low-ℓ polarization anomalies seen in the other W-band DAs.

Figure 7 displays polarization power spectra for the EE, BB, and EB modes for the first three, five, and seven years of template cleaned individual year maps from the W1, W2, and W3 DAs. These spectra were also obtained using the Master algorithm utilizing the mask-marginalized Σ⁻¹ covariance matrix sky map weighting. Only cross power spectra are included, so the measurement noise does not bias the measured values. The uncertainties are based on the Fisher matrix technique, and have not had the signal × noise term added, since any signal remaining in the cleaned maps is expected to be small. (Master algorithm based low-ℓ power spectra are not used in any cosmological analysis.)

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There is generally good agreement between the values for the three different time ranges. Note that the high value seen in the three- and five-year analyses for C^EE₇ has fallen significantly with the additional two years of data. In fact, had the data been taken in reverse order (starting with year seven), the C^EE₇ value would not have been identified as particularly anomalous. However, the C^EE₁₂ value has risen in the seven-year combination. To see if these signals might be artifacts of the foreground cleaning, a similar analysis was performed on the uncleaned sky maps and is displayed in Figure 8. By comparing the two figures, it can be seen that the foreground cleaning mainly affects the multipoles ℓ ⩽ 5 leaving the ℓ = 7, 12 multipoles relatively unchanged.

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The bottom panel of these figures plots the ratio between the variance measured among the individual cross spectra for each multipole/polarization combination, to that predicted by the Fisher matrix technique. For ℓ ⩾ 8 there is good agreement between the measured variance and analytical estimate, but the measured variance slowly grows larger with decreasing ℓ for ℓ ⩽ 7 for both the cleaned and uncleaned sky maps. The Fisher matrix noise estimate utilizes the Σ⁻¹ matrix that describes the noise correlation between the pixels of the Stokes Q and U maps. This matrix incorporates information regarding the sky scanning pattern, through the mapping matrices M and M_am, and correlations in the radiometer noise, through the use of inverse radiometer noise covariance matrix N⁻¹. This method does not, however, include the effect of any noise correlations that might be introduced through the baseline fitting and calibration procedure. A program was undertaken to determine if the baseline fitting and calibration procedure could be affecting the low-ℓ polarization results and is described in the following section.

4.1.3. Calibration-induced Spectral Uncertainties

The WMAP data remain one of the cornerstone data sets used for testing the cosmological models and the precision measurement of their parameters. Figure 9 displays the binned TT and TE angular power spectra measured from the seven-year WMAP data (Larson et al. 2011), along with the predicted spectrum for the best-fit minimal six-parameter flat ΛCDM model. The overall agreement is excellent, supporting the validity of this model. Table 8 tabulates the parameter values for this model using WMAP data alone, and in combination with other data sets. Details of the methodology used to determine these values are described in Larson et al. (2011) and Komatsu et al. (2011).

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Table 8. WMAP Seven-year Cosmological Parameter Summary

Description	Symbol	WMAP-only	WMAP+BAO+H0
Parameters for the Standard ΛCDM Modela
Age of universe	t0	13.75 ± 0.13 Gyr	13.75 ± 0.11 Gyr
Hubble constant	H0	71.0 ± 2.5 kms−1Mpc−1	70.4+1.3−1.4 kms−1Mpc−1
Baryon density	Ωb	0.0449 ± 0.0028	0.0456 ± 0.0016
Physical baryon density	Ωbh2	0.02258+0.00057−0.00056	0.02260 ± 0.00053
Dark matter density	Ωc	0.222 ± 0.026	0.227 ± 0.014
Physical dark matter density	Ωch2	0.1109 ± 0.0056	0.1123 ± 0.0035
Dark energy density	ΩΛ	0.734 ± 0.029	0.728+0.015−0.016
Curvature fluctuation amplitude, k0 = 0.002 Mpc−1b	$\Delta _{\cal R}^2$	(2.43 ± 0.11) × 10−9	(2.441+0.088−0.092) × 10−9
Fluctuation amplitude at 8h−1 Mpc	σ8	0.801 ± 0.030	0.809 ± 0.024
Scalar spectral index	ns	0.963 ± 0.014	0.963 ± 0.012
Redshift of matter–radiation equality	zeq	3196+134−133	3232 ± 87
Angular diameter distance to matter–radiation eq.c	dA(zeq)	14281+158−161 Mpc	14238+128−129 Mpc
Redshift of decoupling	z*	1090.79+0.94−0.92	1090.89+0.68−0.69
Age at decoupling	t*	379164+5187−5243 yr	377730+3205−3200 yr
Angular diameter distance to decouplingc,d	dA(z*)	14116+160−163 Mpc	14073+129−130 Mpc
Sound horizon at decouplingd	rs(z*)	146.6+1.5−1.6 Mpc	146.2 ± 1.1 Mpc
Acoustic scale at decouplingd	lA(z*)	302.44 ± 0.80	302.40 ± 0.73
Reionization optical depth	τ	0.088 ± 0.015	0.087 ± 0.014
Redshift of reionization	zreion	10.5 ± 1.2	10.4 ± 1.2
Parameters for Extended Modelse
Total densityf	Ωtot	1.080+0.093−0.071	1.0023+0.0056−0.0054
Equation of stateg	w	−1.12+0.42−0.43	−0.980 ± 0.053
Tensor-to-scalar ratio, k0 = 0.002 Mpc−1 b,h	r	<0.36(95%CL)	<0.24(95%CL)
Running of spectral index, k0 = 0.002 Mpc−1b,i	dns/dln k	−0.034 ± 0.026	−0.022 ± 0.020
Neutrino densityj	Ωνh2	<0.014(95%CL)	<0.0062(95%CL)
Neutrino massj	∑mν	<1.3 eV(95%CL)	<0.58 eV(95%CL)
Number of light neutrino familiesk	Neff	>2.7(95%CL)	4.34+0.86−0.88

Notes. ^aThe parameters reported in the first section assume the six-parameter flat ΛCDM model, first using WMAP data only (Larson et al. 2011) and then using WMAP+BAO+H₀ data (Komatsu et al. 2011). The H₀ data consist of a Gaussian prior on the present-day value of the Hubble constant, H₀ = 74.2 ± 3.6 km s⁻¹ Mpc⁻¹ (Riess et al. 2009), while the BAO priors on the distance ratio r_s(z_d)/D_V(z) at z = 0.2, 0.3 are obtained from the Sloan Digital Sky Survey Data Release 7 (Percival et al. 2010). Uncertainties are 68% CL unless otherwise noted. ^bk = 0.002 Mpc⁻¹ ⟷ l_eff ≈ 30. ^cComoving angular diameter distance. ^dl_A(z_*) ≡ π d_A(z_*) r_s(z_*)⁻¹. ^eThe parameters reported in the second section place limits on deviations from the ΛCDM model, first using WMAP data only (Larson et al. 2011) and then using WMAP+BAO+H₀ data (Komatsu et al. 2011), except as noted otherwise. A complete listing of all parameter values and uncertainties for each of the extended models studied is available on LAMBDA. ^fAllows non-zero curvature, Ω_k ≠ 0. ^gAllows w ≠ −1, but assumes w is constant and Ω_k = 0. The value in the last column is obtained from a combination of WMAP +BAO data and luminosity distance information obtained from the "constitution" SNe data set (Hicken et al. 2009) using the methodology described in Komatsu et al. (2011). ^hAllows tensor modes but no running in the scalar spectral index. ⁱAllows running in the scalar spectral index but no tensor modes. ^jAllows a massive neutrino component, Ω_ν ≠ 0. ^kAllows N_eff number of relativistic species, with the prior 0 < N_eff < 10.

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The seven-year WMAP results significantly reduce the uncertainties for numerous cosmological parameters relative to the five-year results. The uncertainties in the densities of baryonic and dark matter are reduced by 10% and 13%, respectively. When tensor modes are included, the upper bound to their amplitude, determined using WMAP data alone, is nearly 20% lower. By combining WMAP data with the latest distance measurements from Baryon Acoustic Oscillations (BAO) in the distribution of galaxies (Percival et al. 2010) and Hubble constant measurements (Riess et al. 2009), the spectral index of the power spectrum of primordial curvature perturbations is n_s = 0.963 ± 0.012, excluding the Harrison–Zel'dovich–Peebles spectrum by more than 3σ.

The reduced noise obtained by using the seven-year data set yields a better measurement of the third acoustic peak in the temperature power spectrum. This measurement, when combined with external data sets, leads to better determinations of the total mass of neutrinos, ∑m_ν, and the effective number of neutrino species, N_eff, as presented in Table 8. Additionally, when augmented by the small scale CMB anisotropy measurements by ACBAR (Reichardt et al. 2009) and QUaD (Brown et al. 2009), this result yields a greater that 3σ detection of the primordial Helium abundance, Y_He = 0.326 ± 0.075, using CMB data alone. Komatsu et al. (2011) also demonstrate that, with the larger data set, the expected radial and azimuthal polarization patterns around hot and cold peaks in the CMB can now be observed directly in pixel-space by stacking sky map data. In addition, they now detect the Sunyaev–Zel'dovich effect at ≈8σ at the location of known galaxy clusters, as determined by ROSAT.

Finally, Weiland et al. (2011) have measured the brightness temperature of Jupiter, Saturn, Mars, Uranus, and Neptune, and five fixed calibrations objects, in all five frequency bands, allowing their use as millimeter-wave celestial calibration sources traceable to WMAP's precise calibration.