IMPROVED MEASUREMENTS OF THE TEMPERATURE AND POLARIZATION OF THE COSMIC MICROWAVE BACKGROUND FROM QUaD

QUaD observed a ∼100 square degree area of sky, centered on R.A. 55, decl. −50°. The field is fully contained within the shallow field observed by the 2003 flight of the Boomerang experiment (hereafter referred to as B03, Masi et al. 2006). The QUaD field also partially overlaps with B03's deep field. The observations employed a lead–trail scheme, whereby each hour of observations was split equally between two adjoining subfields, separated in R.A. by 05—the lead field, centered on R.A. 525, and the trail field, centered on R.A. 575.

The scanning strategy consisted of constant-elevation scans back and forth over a 75 throw in azimuth, applied as a modulation on top of sidereal tracking of the field center. Each hour of observation was equally split between the lead and trail fields. These half-hour sessions were further divided into four "scan-sets," consisting of 10 "half-scans" each, and the telescope was stepped in elevation by 002 between scan-sets. After a half-hour scanning of the lead field, the telescope pointing returned to its starting position in azimuth and elevation, and repeated the same scan pattern with respect to the ground, but now scanning the trail CMB field. The trail field's scan pattern was thus a replica (in azimuth/elevation coordinates) of the lead field's. After an hour the pointing moved on to a fresh part of sky and the process repeated. This scan pattern was designed to facilitate the lead–trail differencing analysis presented in Paper II, whereby each pair of lead–trail partner scans is differenced point by point. Any ground signal, which is stable in time over the half hour which separates the lead and trail observations, will be completely removed by this differencing at the expense of a reduction in the effective sky area by a factor of 2.¹⁷

In addition to the lead–trail scheme, a further redundancy is present in the scan strategy due to the movement of the CMB field across the sky during each scan-set. The time elapsed from when a given sky pixel is first visited on the first half scan of a set to when it is last visited on the tenth half-scan is ∼6 minutes. During this time the sky rotates by 15. Using only data from a single scan-set, there is therefore scope for separating signals originating on the ground from those originating on the sky, on scales smaller than 15, corresponding to ℓ ∼ 250 in multipole space. One can achieve further separation of ground from sky by combining the data from lead and trail partner observations. For this work, we have made use of both the lead–trail and sky rotation redundancies to separate the ground and sky signals, and to reconstruct the CMB fields over the full sky area.

Field differencing is a suboptimal use of the redundancy in the scan strategy to mitigate against ground pickup. One can retain more of the sky information by constructing and removing estimates of the ground signal. To facilitate the removal of the ground signal, we can model the TOD as

$$\begin{equation} d_i = S_i(\theta) + g(\alpha) + n_i + o_{\rm scan} \,, \end{equation} \tag{ 1 }$$

where S_i(θ) is the sky signal,

$$\begin{equation} S_i(\theta) = \case{1}{2} \left[I (\theta) + Q(\theta)\cos (2\phi _i) +U(\theta)\sin (2\phi _i)\right] \,. \end{equation} \tag{ 2 }$$

Here, I, Q, and U are the Stokes parameters in the direction θ on the sky, and ϕ_i is the polarization sensitivity angle (a combination of detector orientation on the focal plane and boresight rotation) of each detector.

In Equation (1), g(α) represents the ground signal as a function of azimuth, α. In what follows, we will be constructing and removing estimates of g(α) for each QUaD detector and for each pair of lead and trail scan-sets independently. Within these subsets of the data, the elevation is constant, and so the ground signal estimates (which we refer to as ground "templates") are constructed as a function of azimuth only. However, since we allow the templates to differ between detectors and between lead–trail scan-set pairs, in practice, the ground signal, which we remove from the data, does depend on elevation also. Moreover, since templates are constructed for each detector individually, our model also allows the ground signal to depend on frequency.

The resolution with which to construct the ground templates, in general, needs to be determined through trial and error. In practice, we find that the effect of changes in this parameter on the resulting maps is imperceptible. This suggests that the ground signal varies smoothly in azimuth and that a fairly coarse resolution is sufficient to characterize it. For this analysis, we have used a resolution of Δα = 01, midway between the minimum and maximum resolutions we have investigated.

In Equation (1), we have split the noise component into a random part (n_i, which we model as a Gaussian random variable; see Section 6.2) and an offset (o_scan) which we model as a constant for each half-scan. We have found that it is essential to remove these offsets before constructing ground templates from the data. Otherwise the resulting templates tend to be dominated by the long-timescale part of the 1/f atmospheric noise (i.e., the offsets) rather than the ground signal which we are attempting to characterize.

Regardless of the technique employed to reconstruct maps of the Stokes parameters (naive, maximum likelihood etc.), one can recast the well-known map-making equation (e.g., Stompor et al. 2002 and references therein),

$$\begin{equation} \mathbf {A}^T \mathbf {N}^{-1}\mathbf {A}\mathbf {m}= \mathbf {A}^T \mathbf {N}^{-1} \mathbf {d}\,, \end{equation} \tag{ 3 }$$

to reconstruct both the sky and the ground signals simply by making the substitutions

$$\begin{eqnarray} \mathbf {m}&\rightarrow & \mathbf {m}+ \mathbf {g}\nonumber \\ \mathbf {A}&\rightarrow & (\mathbf {A}^{{\rm CMB}}, \mathbf {A}^G) \,, \end{eqnarray} \tag{ 4 }$$

where m is the reconstructed CMB map, g is the reconstructed ground signal, and A^CMB and A^G are the "pointing matrices" associated with the CMB and ground signals, respectively. For a highly redundant scan strategy, Equation (3) should be soluble exactly, and the CMB and ground signals should be completely separable. However, for a scan strategy such as that used for QUaD with limited revisiting of the same sky pixels at different azimuths, this complete separation between sky and ground is not possible. In this case, one can still separate the ground and sky signals on smaller scales, but one loses all information on the largest scales where the separation is degenerate. For QUaD, we find that below ℓ ∼ 200 our template removal procedure offers essentially no improvement over field-differencing and that the separation is near perfect by ℓ ∼ 1000.

Attempting to simultaneously solve for the sky and ground signals in the QUaD data by applying Equation (3), we find that large-scale (ground signal) gradients are introduced to the resulting CMB maps due to the degeneracy between the CMB and the ground signal on the largest scales. One could certainly modify the procedure (e.g., by marginalizing over the large-scale CMB and ground-signal modes) to solve this problem. For the analysis presented here, we adopt a simpler approach and simply solve for the ground signal independently, subtract this from the TOD, and then construct the CMB maps from the ground-cleaned TOD. We account for the resulting filtering of the CMB signal in our Monte Carlo analysis. Our analysis assumes that the ground signal does not change between the start of the lead scan set and the end of the corresponding trail scan set (∼36 minutes). This is only a slight relaxation of the assumption that was made for our previous analysis where we assumed that the ground signal was constant over a 30 minute timescale.

To apply the template removal, we proceed as follows. First, we estimate and remove the atmospheric 1/f offsets, o_scan, from each half-scan. To estimate the offsets, we simply take the mean of the data within each scan. Note however that we restrict the azimuth range over which we calculate the offsets to the central azimuth range where all the scans within a scan-set overlap. This ensures that our estimated ground templates will be unbiased over the full azimuth range.

After removing the offsets, templates of the ground signal are constructed by simple binning of the timestream data in azimuth. That is, for each ground template "pixel," we construct

$$\begin{equation} \hat{g}(\alpha) = \frac{1}{N_{\rm hits}} \sum _{i \in \Delta \alpha } d_i \,, \end{equation} \tag{ 5 }$$

where the sum is over all data from the lead scan-set, and its partner trail scan-set which falls in the azimuth range, Δα.

Although they will be unbiased estimates of the ground signal, the templates constructed using Equation (5) will also contain CMB signal and noise. The expectation value of the constructed templates is

$$\begin{equation} \langle \hat{g}(\alpha) \rangle = g(\alpha) + \frac{1}{N_{\rm hits}} \left[ \sum _{i \in \Delta \alpha } S_i(\theta) + \sum _{i \in \Delta \alpha } n_i \right] \,, \end{equation} \tag{ 6 }$$

where S_i(θ), g(α), and n_i are the quantities defined in Equation (1). In the limit of a highly redundant scan strategy and uncorrelated noise, the terms in brackets will average to zero, and the templates will contain only the ground signal.¹⁸ For a realistic scan strategy and correlated noise, these terms will be nonzero, and so removing the templates will have the effect of filtering both large-scale CMB signal and long-timescale noise. We correct for this filtering by including the template removal procedure in our Monte Carlo simulations (Section 6.2).

Finally, to obtain estimates of the TOD which are free of ground pickup, the templates are subtracted from the original TOD:

$$\begin{equation} \mathbf {d}^{\rm clean} = \mathbf {d}- \mathbf {A}^G \hat{\mathbf {g}} \,, \end{equation} \tag{ 7 }$$

where A^G is the pointing matrix associated with the ground signal, and $\hat{\mathbf {g}}$ is the estimated ground signal constructed using Equation (5). This will result in TOD which is, in principle, free of ground contamination and can thus be modeled as in Equation (1) but now without the g(α) term. Once the ground contamination has been removed, our map-making proceeds as described in Paper II. Explicitly, we perform the following operations. For each azimuth scan the best-fit third-order polynomial is subtracted to remove the long-timescale part of the atmospheric 1/f noise. For each PSB pair, the data are then summed and differenced to yield temperature (s_i) and polarization (d_i) TOD. Maps of the temperature (or Stokes I) CMB field are then constructed from the summed data using a simple weighted average

$$\begin{equation} T = \frac{1}{\sum _i w_i} \sum _i w_i s_i \,, \end{equation} \tag{ 8 }$$

where the weights are given by w_i = W(x)/v_scan. Here, v_scan is the variance of the data across the parent half-scan—noisy data (e.g., due to bad weather) is thus downweighted. W(x), where x denotes the fractional position within the scan, is an apodization (the same for each scan) which we use to downweight the scan ends. We apply this apodization to reduce the tiling effects seen in our previous analysis (see Section 6.3 and Figure 15 in Paper II) whereby the interaction of the polynomial filtering with different sky coverage for different detectors produced visible step features in the final maps. The exact form used for the apodization is not important.

Zoom In Zoom Out Reset image size

We construct maps of the Stokes polarization parameters, Q and U, as

$$\begin{eqnarray} &&\left(\begin{array}{@{}c} Q \\ U\end{array} \right) &=& \left(\begin{array}{@{}cc@{}} \langle \cos ^2(2\phi _i)\rangle & \langle \cos (2\phi _i)\sin (2\phi _i)\rangle \\ \langle \cos (2\phi _i)\sin (2\phi _i)\rangle & \langle \sin ^2(2\phi _i)\rangle \\ \end{array} \right)^{-1} \nonumber\\ &&\times\left(\begin{array}{@{}c@{}} \langle \cos (2\phi _i)d_i\rangle\\ \langle \sin (2\phi _i)d_i\rangle \\ \end{array} \right), \qquad \qquad\qquad\qquad \quad \end{eqnarray} \tag{ 9 }$$

where the angled brackets denote an average taken over all data falling within each map pixel and the angle, ϕ_i is a combination of the polarization sensitivity direction of each detector on the focal plane and the "deck angle" (rotation about the telescope boresight) of the observation. Note that to construct the averages (e.g., 〈cos(2ϕ_i)d_i〉) on the right-hand side of Equation (9), we also use inverse-variance weights as in Equation (8).

Note that the only difference between the approaches of our two pipelines in applying the above operations is during the final co-addition of the template-subtracted data into CMB maps: Pipeline 1 uses HEALPix¹⁹ to pixelize the sky at a resolution of ∼1.7 arcmin (N_side = 2048). Pipeline 2 works under the flat sky approximation and pixelizes the sky into a two-dimensional (2D) Cartesian grid with a spacing of 1.2 arcmin. Figure 1 shows an example of the performance of the template removal procedure (for Pipeline 1) for the 150 GHz Stokes U polarization map. Note that in order to highlight the success of the template removal, for this demonstration, we have not applied the third-order polynomial removal mentioned above to the TOD and have only removed the mean from each scan.²⁰

In Figure 2, we present the full set of maps (T, Q, and U at 100 and 150 GHz, again for Pipeline 1) over the full sky area as estimated using the template removal procedure (now including the third-order polynomial removal). For the purposes of visual illustration only, we have smoothed each of the maps with a 5 arcmin Gaussian kernel in order to bring out the CMB structure.

Zoom In Zoom Out Reset image size

We can quickly (and crudely) assess the relative amounts of E- and B-mode power in the polarization maps by decomposing the Q and U maps into E- and B-modes. We do this under the flat sky approximation. To minimize the impact of the noisy edge regions of the maps, and to reduce the effects of E/B mixing due to the finite survey geometry, we apply an apodization to our maps before Fourier transforming. (E/B mixing is fully accounted for during our power spectrum estimation described in Section 6.) The resulting E and B maps at 150 GHz are shown in Figure 3. We clearly detect significantly more E-mode than B-mode structure. The reconstructed B-mode map shows similar levels of fluctuations to our polarization jackknife maps (see Section 6.5) and is consistent with noise.

Zoom In Zoom Out Reset image size

Both of our pipelines broadly follow the so-called pseudo-C_ℓ technique (Hivon et al. 2002), extended to polarization (Brown et al. 2005). Note that, for both pipelines, in order to minimize edge effects, the T, Q, and U maps are first apodized with an inverse-variance mask as described in Section 6.1 of Paper II.

Pipeline 1 works on the curved sky and uses fast spherical harmonic transforms to estimate the pseudo-C_ℓ spectra. These spectra are then corrected for the effects of the sky cut, noise and filtering, and binned into band powers according to

$$\begin{equation} {\bf P}_b=\sum _{b^{\prime }}{\bf K}^{-1}_{bb^{\prime }} \, \sum _\ell P_{b^{\prime }\ell } \, (\widetilde{\bf C}_\ell -\langle \widetilde{\bf N}_\ell \rangle _{\rm MC}) \,. \end{equation} \tag{ 11 }$$

In this equation and throughout this section, we use bold face to denote six-spectrum quantities, e.g., P_b is a 6 n_band vector, P_b = {P^TT_b, P^EE_b, P^BB_b, P^TE_b, P^TB_b, P^EB_b}, and ${\bf K}_{bb^{\prime }}$ is a 6 n_band × 6 n_band matrix, given by

$$\begin{equation} {\bf K}_{bb^{\prime }} = \sum _\ell P_{b \ell } \sum _{\ell ^{\prime }}{\bf M}_{{\ell \ell ^{\prime }}} F_{\ell ^{\prime }}Q_{{\ell ^{\prime }}b^{\prime }} \,. \end{equation} \tag{ 12 }$$

In Equation (11), $\widetilde{\bf C}_\ell$ are the pseudo-C_ℓ spectra estimated from the data maps, and $\langle \widetilde{\bf N}_\ell \rangle _{\rm MC}$ are the noise power spectra as measured from simulations.

P_bℓ is a binning operator which bins the raw pseudo-C_ℓ into band powers, and Q_ℓb is the inverse operator which "unfolds" a band power into individual C_ℓs. For this analysis, we use "flat" band powers for which the quantity ℓ(ℓ + 1)C_ℓ/2π is constant within each band. That is, the binning operator we use is

$$\begin{equation} P_{b \ell } = \left\lbrace \begin{array}{@{}ll} \frac{1}{2 \pi } \frac{\ell (\ell +1)}{\ell ^{(b+1)}_{\rm low} - \ell ^{(b)}_{\rm low}}, & \mbox{ if } 2 \le \ell ^{(b)}_{\rm low} \le \ell < \ell ^{(b+1)}_{\rm low} \\ 0, & \mbox{ otherwise, }\end{array} \right. \end{equation} \tag{ 13 }$$

with the corresponding inverse operator given by

$$\begin{equation} Q_{\ell b}= \left\lbrace \begin{array}{@{}ll} \frac{2 \pi }{\ell (\ell +1)}, & \mbox{ if } 2 \le \ell ^{(b)}_{\rm low} \le \ell < \ell ^{(b+1)}_{\rm low} \\ 0, & \mbox{ otherwise, }\end{array} \right. \end{equation} \tag{ 14 }$$

where ℓ^(b)_low denotes the nominal lower edge of band b.

The coupling matrix, ${\bf M}_{{\ell \ell ^{\prime }}}$, describes the mode-mixing effects of the apodization mask and sky cut, and is given in Brown et al. (2005) for the full set of six possible CMB spectra. Note that the ${\bf M}_{{\ell \ell ^{\prime }}}$ matrix fully encodes E/B leakage effects due to the finite survey geometry and so our Pipeline 1 estimator explicitly corrects the EE and BB spectra for this leakage in the mean.

F_ℓ is a transfer function which we use to describe the combined effects of timestream filtering, beam suppression, and filtering of the sky signal due to the removal of the ground templates. This function will also encode any other signal suppression effects which are present in our simulations (e.g., pixelization effects). We estimate F_ℓ from our signal-only simulations as described in Section 6.3.

Pipeline 2 works in the flat sky approximation and uses 2D FFTs to estimate power spectra. This pipeline is described in detail in Paper II. The power spectrum estimator for Pipeline 2 can effectively be written as

$$\begin{equation} {\bf P}_b= \, F_b^{-1} \sum _\ell P_{b \ell } \, (\widetilde{\bf C}_\ell -\langle \widetilde{\bf N}_\ell \rangle _{\rm MC}) \,, \end{equation} \tag{ 15 }$$

where F_b is the binned equivalent of the per-multipole transfer function, F_ℓ, and we have implicitly made the connection between the flat sky and curved sky power spectra, C_ℓ ≈ P(k) for ℓ ≈ 2πk.

Note that (in addition to the flat sky approximation), the primary difference between the two pipelines is that Pipeline 1 performs the correction for the mode-mixing effects induced by the sky cut, whereas Pipeline 2 does not perform this correction. Because of this difference, neither the recovered band powers nor their uncertainties are directly comparable between the two pipelines. A proper comparison of the two analyses requires the use of the associated band power window functions (BPWFs; Section 6.3) which fully encode the relation between underlying true sky power and observed power for both pipelines.

In both analyses, we estimate the covariance matrix of our power spectrum estimates from the scatter found in the power spectra measured from simulations containing both signal and noise:

$$\begin{equation} \langle \Delta {\bf P}_b \Delta {\bf P}_{b^{\prime }}\rangle = \langle ({\bf P}_b - \overline{\bf P}_b)({\bf P}_{b^{\prime }} - \overline{\bf P}_{b^{\prime }}) \rangle _{\rm MC} \,, \end{equation} \tag{ 16 }$$

where $\overline{\bf P}_b$ denotes the average of each band power over all simulations. Note finally that the covariance properties of the power spectra estimates are dependent on whether the correction for mode-mixing induced by the sky cut is applied or not. We return to this issue in Section 6.6 where we compare the results from our two analyses.

In simulating QUaD, we follow the procedure described in Section 5 of Paper II with some important differences, which we now discuss. Algorithmically, both of our analysis pipelines adopt the same approach to creating simulated timestreams and only differ in the final map-making stage as described in Section 3.2.

6.2.1. Signal Simulations

6.2.2. Noise Simulations

We estimate the transfer function from our suite of signal-only simulations. In the absence of noise, the mean of the recovered pseudo-C_ℓ spectra will equal their expectation values:

$$\begin{equation} \langle \widetilde{\bf C}_\ell \rangle _{\rm MC} = \sum _{\ell ^{\prime }}{\bf M}_{\ell \ell ^{\prime }}F_{\ell ^{\prime }}{\bf C}_{\ell ^{\prime }}\,, \end{equation} \tag{ 17 }$$

where C_ℓ are the input model spectra used to create the simulations. For a small-area survey such as QUaD, the unbinned coupling matrix, ${\bf M}_{\ell \ell ^{\prime }}$, is singular, and so Equation (17) cannot be solved directly. In Pipeline 1, we iteratively solve this equation to provide an estimate of F_ℓ. With a reasonable starting guess, convergence is typically reached in just a few iterations. For Pipeline 1, the BPWFs, W_bℓ, defined by (Knox 1999),

$$\begin{equation} \langle {\bf P}_b \rangle = \sum _\ell \frac{{\bf W}_{b \ell }}{\ell } \frac{\ell (\ell + 1)}{2 \pi } \langle {\bf C}_\ell \rangle \,, \end{equation} \tag{ 18 }$$

are given by

$$\begin{equation} \frac{{\bf W}_{b \ell }}{\ell } = \frac{2 \pi }{\ell (\ell + 1)} F_\ell \, \sum _{b^{\prime }} {\bf K}^{-1}_{bb^{\prime }} \sum _{\ell ^{\prime }}P_{b^{\prime } {\ell ^{\prime }}} {\bf M}_{\ell ^{\prime } \ell }\,. \end{equation} \tag{ 19 }$$

Pipeline 2 calculates its BPWFs numerically as described in Section 6.6 of Paper II. In order to calculate the transfer functions, Pipeline 2 simply takes the ratio of the mean band powers recovered from signal-only simulations and the expectation values for each band power:

$$\begin{equation} F_b = \frac{\sum _\ell P_{b \ell } \langle \widetilde{\bf C}_\ell \rangle _{\rm MC}}{ \langle {\bf P}_b \rangle } \,. \end{equation} \tag{ 20 }$$

Figure 5 shows the derived transfer function from Pipeline 1. (The transfer function from Pipeline 2 is similar.) As mentioned earlier, this function encapsulates all effects due to timestream filtering, beam suppression and the filtering due to removal of the ground templates. To demonstrate the relative size of these effects, we also plot the transfer function derived from special simulations with these three effects included in isolation. Of particular interest is the transfer function describing the ground-removal procedure—this curve in effect encapsulates the lossiness of the technique. On scales where this curve is ≳0.5, we gain an improvement over the field-differencing technique. We see that below ℓ ∼ 200, there is a little gain from our template-removal technique over field-differencing. The amount of the signal retained then climbs rapidly and is effectively unity by ℓ = 1000.

Zoom In Zoom Out Reset image size

Figure 6 shows the BPWFs, plotted as W_bℓ/ℓ, for C^BB_ℓ from both our pipelines for the 150 GHz channel. These functions describe the response of our band power measurements to the true sky signal at each multipole. The negative wings in the BPWFs of Pipeline 1 are a direct result of the application of the correction for mode-mixing effects as described in Section 6.1.

Zoom In Zoom Out Reset image size

The expectation values for our EE band powers assuming the best-fit ΛCDM model to the WMAP 5-year data are shown in Figure 7. Note that the apparent improvement in going from Pipeline 2 to Pipeline 1 in terms of the agreement between the true sky power and the band power expectation values does not come without a price—as a result of applying the mode-mixing correction, the error bars for Pipeline 1 are enhanced with respect to the Pipeline 2 errors. The covariance properties also change between the two analyses such that the total information content is preserved.

Zoom In Zoom Out Reset image size

We emphasize that, in terms of either the accuracy or the precision of the recovery of true sky power, neither analysis is superior. This is clear from the fact that one can transform between the band power estimates and covariances of the two analyses via a simple and exact matrix operation. Whether to apply the mode-mixing correction is simply a matter of preference and is only relevant for visual interpretation of the results—one can choose to have smaller error bars or one can choose to have band powers which better trace the underlying true sky power but one cannot have both.

We apply the procedures described in Section 6.1 to estimate the six possible CMB power spectra (TT, TE, EE, BB, TB, EB) from the QUaD maps. In addition to the 100 and 150 GHz auto-spectra, we also estimate the 100–150 GHz cross spectra as described in Paper II. In the case where the noise is uncorrelated between the two frequency channels, the cross spectra do not require the noise-debiasing step. Although in practice we do apply this correction, the correction is modest for the cross spectra, so these measurements will be much less sensitive to the details of our noise model. The power spectrum results from Pipeline 1 are presented in Figure 8. We make strong detections of the TT, TE, and EE spectra which show good agreement with those predicted by the best-fitting ΛCDM model to the 5-year WMAP results. The BB, TB, and EB spectra are consistent with null within the noise. The results from Pipeline 2 show a similar agreement with the ΛCDM model.

Zoom In Zoom Out Reset image size

We have subjected our analysis to the same set of systematic tests as performed in Paper II. These tests involve splitting the data into two roughly equal parts. Maps made from the two data subsets are subtracted, and the power spectra of the residual maps are calculated. Any deviation of these "jackknife" spectra from null would indicate systematic contamination in the data. For a detailed description of the various ways in which we split the data, we refer the reader to Section 7 of Paper II.

The strongest test is the so-called "deck jackknife" test where we split the data according to the boresight rotation angle (deck angle) of the observations. This test, in particular, will be strongly sensitive to any residual ground contamination remaining in the data after applying the procedure described in Section 3. In Figure 8, the deck jackknife spectra are plotted alongside the spectra measured from the undifferenced maps. It is clear from this figure that the power measured in the deck-differenced maps is small compared to the measured signals in TT, TE, and/or EE. The other data splits which we consider are splitting the data according to orientation of the PSB pairs on the focal plane (see Section 2), a split between the forward and backward scans and a split in observation dates which roughly separates the data into the 2006 and 2007 observing seasons. We also take the difference of the 100 and 150 GHz maps. This "frequency jackknife" test is not strictly a test for systematic effects in the data. Rather, it is a strong test for "foreground" (i.e., non-CMB) astrophysical emission.

The cancellation of the signal apparent in Figure 8 is visually impressive. To investigate whether the differenced spectra are formally consistent with null, as in Paper II, we have performed χ² tests against the null model. Note that we compare this statistic with the χ² distribution as measured from our simulations rather than against the theoretical χ² curve, calculating the "probability to exceed" (PTE) the observed value by random chance. Low numbers therefore indicate a problem. The PTE values for each of our measured spectra (from Pipeline 1) are presented in Table 1.

Examining the table, the PTE values for all the spectra bar TT reveal no significant problems, the numbers being consistent with a uniform distribution between 0 and 1. In contrast, many of our TT jackknife spectra are clearly inconsistent with a null signal. The failure is perhaps excusable in the case of the frequency difference (since there are astrophysical reasons why this test might fail) but taken as a whole, the statistics for TT (and to a lesser extent, some of the TE numbers) suggest that there is some degree of residual systematics present in our temperature maps. The PTE statistics for Pipeline 2, although not identical, show the same general pattern of jackknife failures for the TT spectra. Comparing to our previous results, there were hints of a problem with the TT jackknife tests in Paper II but to a lesser extent than is apparent now. This is possibly due to the fact that with our increased sensitivity we can measure both the signal and systematics to greater precision. Alternatively, it could indicate that the template removal procedure is not as effective as field-differencing in removing the ground.

These jackknife failures indicate that residual systematic effects in our temperature maps are significant with respect to the errors on the jackknife spectra. However, the jackknife errors contain only noise, whereas the errors on our measured TT and TE CMB power spectra also contain considerable sample variance. To assess how significant the residual systematics are, in Figure 9, we plot the measured TT and TE jackknife band powers alongside the signal band powers for the 150 GHz channel. Clearly, when compared to the sample variance-dominated signal spectra, the degree of residuals is much less significant. Repeating the χ² analysis using the signal covariance matrices in place of the jackknife covariance matrices, we find that the residual contamination is negligible. In summary, although our TT jackknife tests indicate the presence of residual systematics, they also clearly demonstrate that these residuals are irrelevant compared to both the measured sky signal and its associated sample variance. Note this is also true for the levels of foreground contamination in our maps since our frequency difference test is sensitive to foregrounds.

Zoom In Zoom Out Reset image size

To produce a set of final power spectra, we combine the single frequency and 100–150 GHz cross spectra shown in Figure 8 following the procedure described in Section 9 of Paper II. The final estimates of the power spectra are shown in Figure 10 for both of our analysis pipelines. Once again, the model curves plotted are the best-fitting ΛCDM model to the WMAP 5-year results. We have performed χ² tests of all spectra, and both analyses show perfectly acceptable agreement with this model and with each other. Pipeline 1 χ² and PTE values are given in Table 2.

Zoom In Zoom Out Reset image size

Comparing the two sets of results presented in Figure 10, we see that the nominal error bars are smaller for Pipeline 2 and that Pipeline 2's points appear to trace a slightly smoothed version of Pipeline 1's points. Both of these effects are simply a result of the differing BPWFs as discussed in Section 6.3. Note that neighboring band powers in Pipeline 1 are ∼10% anti-correlated, whereas they are ∼10% positively correlated in Pipeline 2. Correlations between non-adjacent band powers are negligible for both analyses.

Figure 11 shows a comparison with our previous results from Paper II. Note that we perform this comparison using Pipeline 2, since this pipeline is an extension of the analysis presented in Paper II and so is directly comparable. Two effects are apparent in this figure. First, the uncertainties on all of our power spectra have been reduced by ∼30% as a result of the increase in sky area afforded by our template-based ground removal technique. Second, implementing the improved beam models described in Section 4 has resulted in a slight increase in the amplitude of our power spectrum measurements for multipoles, $\ell \mathrel {\rlap{\lower4pt\hbox{$\sim $}} \raise1pt\hbox{$>$}}700$. This impacts mostly on the high signal-to-noise measurements of the TT spectrum on small scales.

Zoom In Zoom Out Reset image size

Figure 12 shows our measured power spectra from Pipeline 1 in comparison with the published results from a number of other CMB experiments.

Zoom In Zoom Out Reset image size

The QUaD power spectra data, along with the associated band power covariance matrices and BPWFs (for both of our analysis pipelines) are available for download at http://quad.uchicago.edu/quad.

We have assessed the significance with which we detect the acoustic oscillations in the EE power spectrum by repeating the analysis of Section 9.3 in Paper II. For this test, we compare our EE measurements against both the ΛCDM model and against a heavily smoothed version of the ΛCDM curve. The results of this test are shown in Figure 13. The QUaD detection of acoustic oscillations in the E-mode power spectrum is now beyond question—the probability that the true E-mode spectrum is a smooth curve has dropped from 0.001 with our previous analysis to <10⁻¹⁴. We have also repeated our Paper II analysis where we used "toy models" of the E-mode spectrum to fit the peak spacing, phase and amplitude of the acoustic oscillations. With our previous measurements, we constrained the peak spacing to be Δℓ_s = 306 ± 10, the phase to be ϕ = 13° ± 33° and the amplitude to be a = 0.86 ± 0.17. Repeating the analysis with our new measurements, we find Δℓ_s = 308 ± 7, ϕ = 6° ± 22°, and a = 0.96 ± 0.10. For comparison, when we perform the analysis with the QUaD band power values replaced by their expectation values in the ΛCDM model, we find Δℓ_s = 310, ϕ = 13°, and a = 0.99. These results confirm that the polarization peak spacing and the phase relationship between the temperature and polarization acoustic oscillations are as expected in the ΛCDM model. Passing this (non-trivial) test further strengthens the foundations of this model.

Zoom In Zoom Out Reset image size

To obtain our constraints, we perform a Monte Carlo Markov Chain (MCMC) sampling of the cosmological parameter space. To do this, we use the publicly available CosmoMC package (Lewis & Bridle 2002), which in turn uses the CAMB code (Lewis et al. 2000) to generate the CMB and matter power spectra.²²

We make use of the publicly available WMAP likelihood software from the LAMBDA²³ Web site. We marginalize over a possible contribution to the temperature power spectrum from the Sunyaev–Zel'dovich (SZ) effect following the WMAP analysis (Dunkley et al. 2009). To do this, we use the SZ templates from Komatsu & Seljak (2002; also available from the LAMBDA Web site) and the known frequency dependence of the SZ effect. In order to avoid possible contamination from residual point sources, we exclude the ACBAR band powers above ℓ = 2000. For the same reason, we do not include our own TT measurements at ℓ>2000 (Friedman et al. 2009).

Marginalization over the WMAP beam uncertainty is included in the WMAP likelihood code, and we also marginalize over the quoted ACBAR calibration and beam uncertainties. We take the latter to be a 2% error on a 5 arcmin (FWHM) Gaussian beam as assumed in the ACBAR CosmoMC data file. For QUaD, we marginalize over our 3.5% calibration uncertainty and over the uncertainty in our beam. As described in Appendix A, our beam uncertainties are dominated by uncertainties in the level of our sidelobes rather than in the effective FWHM of our main lobe beams. We therefore marginalize over the full ℓ-dependent beam uncertainty shown in Figure 19. Where we include the SDSS LRG data, we marginalize over both the amplitude of the matter power spectrum and over a correction for scale-dependent nonlinear density evolution using the methods described in Tegmark et al. (2006).

We model the likelihood functions for the QUaD auto-spectra as offset log-normal distributions (Bond et al. 2000). The required noise offsets are derived from our signal-only and noise-only simulations. (We model the TE likelihood as a Gaussian distribution.) We include all covariances apparent (above the numerical noise) in our simulation-derived covariance matrix (Equation (16)). In addition to same-spectrum covariances, this includes nonzero TT–TE, EE–TE, TT–EE, and EE–BB correlations.

Note that, for our main MCMC analysis, we do not include our measurements of the parity-violating spectra, TB and EB, since these spectra are expected to vanish in standard ΛCDM models and its usual variants. However, in Section 7.7, we will use these spectra to constrain possible parity-violating interactions to the surface of last scattering (see e.g., Lue et al. 1999) following our previous work (Wu et al. 2009).

Finally, in Section 7.8, we use our polarization measurements to place a formal upper limit on the strength of the lensing B-mode signal.

Our basic ΛCDM cosmological model is characterized by the following six parameters (where h = H₀/(100 km s⁻¹Mpc⁻¹), with H₀ being Hubble's constant in units, km s⁻¹Mpc⁻¹): the physical baryon density, Ω_bh²; the physical cold dark matter (CDM) density, Ω_ch²; the ratio of the sound horizon to the angular diameter distance at last scattering, θ = r_s/D_A; the optical depth to last scattering, τ; the scalar spectral index, n_s; and the scalar amplitude, $\mathcal {A}_s = \ln \left[ 10^{10} A_s \right]$. Here, A_s is the amplitude of the power spectrum of primordial scalar perturbations, parameterized by $\mathcal {P}_s(k) = A_s(k/k^s_\star)^{n_s-1}$. We discuss the choice of pivot-point, k^s_⋆, in the following section. Other parameters which we quote and which are derived from this basic set, are the dark energy density, Ω_Λ (assumed here to be a simple cosmological constant), the age of the universe, the total matter density, Ω_m, the amplitude of matter fluctuations in 8 h⁻¹Mpc spheres, σ₈, the redshift to reionization, z_re and the value of the present day Hubble constant, H₀. For all our analyses, we assume a flat universe and include the effects of weak gravitational lensing. We impose the following broad priors on our base MCMC parameters: 0.005 < Ω_bh² < 0.100; 0.01 < Ω_ch² < 0.99; 0.5 < θ < 10.0; $2.7 < \mathcal {A}_s < 4.0$; 0.5 < n_s < 1.5; 0.01 < τ < 0.80. There is also a prior imposed on the age of the universe (10 < Age(Gyr) < 20) and on the Hubble constant (40 < H₀(km s⁻¹Mpc⁻¹) < 100).

We also investigate models extended to include both a running in the scalar spectral index, n_run = dn_s/dln k, and/or a possible tensor contribution. Assuming a power law for the tensor modes, $\mathcal {P}_t \propto k^{n_t}$, we parameterize their amplitude by the tensor-to-scalar ratio, $r = \mathcal {P}_t / \mathcal {P}_s$. We adopt a uniform prior measure for r between 0 and 1. For the running spectral index model, we adopt a prior of −0.5 < n_run < 0.5 on the running.

For the primordial power spectrum parameterization which we have chosen, we need also to choose a scalar pivot point, k^s_⋆, the wavenumber at which n_s and A_s are evaluated. Within standard ΛCDM, n_s is modeled as independent of scale, and we can map constraints on A_s obtained at one pivot point to an arbitrary new pivot point, k^s'_⋆, using

$$\begin{equation} A_s({k^s_\star }^{\prime }) = A_s(k^s_\star) \,\,\, ({k^s_\star }^{\prime } / k^s_\star)^{n_s-1}. \end{equation} \tag{ 21 }$$

For models including a running in the spectral index, both n_s and A_s are dependent on scale. For these models, we can map constraints from an old to a new pivot point using

$$\begin{equation} A_s({k^s_\star }^{\prime }) = A_s(k^s_\star) \,\,\, ({k^s_\star }^{\prime } / k^s_\star)^{n_s(k^s_\star)-1 + \frac{1}{2} n_{\rm run} \ln ({k^s_\star }^{\prime }/k^s_\star)}, \end{equation} \tag{ 22 }$$

$$\begin{equation} n_s({k^s_\star }^{\prime }) = n_s(k^s_\star) + n_{\rm run} \ln ({k^s_\star }^{\prime } / k^s_\star). \end{equation} \tag{ 23 }$$

Correlations between the two parameters, n_s and n_run, are dependent on the pivot point at which one chooses to present results. In particular, there is a scale at which the uncertainties on these two parameters become uncorrelated (Copeland et al. 1998). Choosing to present results at this "decorrelation scale" has the attractive feature that the marginalized one-dimensional (1D) constraint on n_s is not degraded by allowing the running to be nonzero. Finelli et al. (2006) presented parameter constraints from CMB and large-scale structure data using a pivot point of k^s_⋆ = 0.01 Mpc⁻¹, whereas Peiris & Easther (2006) identified a decorrelation scale of k^s_⋆ ≈ 0.02 Mpc⁻¹ using the WMAP 3-yr data.

In order to find the decorrelation pivot point, we have followed the analysis of Cortês et al. (2007) who describe a technique to fit MCMC chains for the decorrelation scale. They found a decorrelation scale of k^s_⋆ = 0.017 Mpc⁻¹ using the WMAP 3-yr data set. Repeating their analysis using the WMAP 5-yr data, we find k^s_⋆ = 0.013 Mpc⁻¹. For simplicity, we choose to present our main results on $\mathcal {A}_s$ and n_s at this decorrelation scale for all of the models and data set combinations which we have investigated.

For models including a possible tensor component, we still quote our constraints on $\mathcal {A}_s$ and n_s at k^s_⋆ = 0.013 Mpc⁻¹, but for the tensor-to-scalar ratio, r, we use a tensor pivot point of k^t_⋆ = 0.002 Mpc⁻¹. We do not attempt to remap our constraints on r to a more optimal pivot point, since the only meaningful data contributing to a constraint on r are the WMAP temperature power spectrum on very large scales (for which, a tensor pivot point of k^t_⋆ = 0.002 Mpc⁻¹ is appropriate). Note also that for these models, we enforce both the first and second inflation consistency equations (e.g., Lidsey et al. 1997): r = −8n_t and $\frac{d n_t}{d \ln k} = n_t \left[ n_t - (n_s - 1) \right]$. Additionally, enforcing the second equation ensures that the first consistency equation holds to linear order in Δln k on all scales (Cortês et al. 2007).

In Table 3, for each data set combination, we list the mean recovered values for each parameter, along with their associated 68% confidence limits, marginalized over all other parameters. In Figure 14, we plot the marginalized 1D constraints for the WMAP, WMAP + QUaD and WMAP + ACBAR combinations. Clearly, the WMAP data dominate when we add either the ACBAR or the QUaD data, as was found in our previous analysis (Castro et al. 2009). However, the addition of either of these data sets does provide additional information on Ω_bh², Ω_ch², and θ. The biggest improvement in constraints is in θ where the WMAP + ACBAR + QUaD combination tightens the limits by ∼25% compared to WMAP alone. This additional constraining power comes mostly from the QUaD data.

Zoom In Zoom Out Reset image size

Table 3. Basic Six Parameter Constraints

Parameter	WMAP	WMAP+ACBAR	WMAP+QUaD	WMAP+ACBAR+QUaD	WMAP+ACBAR+QUaD+SDSS
Ωbh2	0.0228+0.0006−0.0006	0.0229+0.0006−0.0006	0.0227+0.0005−0.0005	0.0227+0.0005−0.0005	0.0227+0.0005−0.0005
Ωch2	0.109+0.006−0.006	0.111+0.006−0.006	0.108+0.006−0.006	0.109+0.005−0.005	0.108+0.004−0.004
θ	1.0406+0.0030−0.0031	1.0422+0.0027−0.0027	1.0403+0.0024−0.0025	1.0415+0.0022−0.0023	1.0414+0.0022−0.0022
τ	0.090+0.017−0.017	0.090+0.017−0.017	0.090+0.017−0.017	0.089+0.017−0.017	0.088+0.017−0.016
ns	0.965+0.014−0.014	0.966+0.013−0.013	0.962+0.013−0.013	0.962+0.013−0.013	0.962+0.012−0.012
${\cal A}_s$	3.11+0.04−0.04	3.12+0.04−0.04	3.11+0.04−0.04	3.11+0.04−0.04	3.11+0.03−0.03
ΩΛ	0.74+0.03−0.03	0.74+0.03−0.03	0.75+0.03−0.03	0.75+0.03−0.03	0.75+0.02−0.02
Age	13.68+0.14−0.14	13.63+0.12−0.12	13.69+0.12−0.12	13.66+0.11−0.11	13.66+0.10−0.10
Ωm	0.26+0.03−0.03	0.26+0.03−0.03	0.25+0.03−0.03	0.25+0.03−0.03	0.25+0.02−0.02
σ8	0.80+0.04−0.04	0.80+0.03−0.03	0.79+0.03−0.03	0.79+0.03−0.03	0.79+0.02−0.02
zre	10.5+1.4−1.4	10.5+1.4−1.3	10.5+1.3−1.3	10.5+1.3−1.3	10.4+1.4−1.3
H0	72.1+2.6−2.6	72.3+2.5−2.5	72.5+2.5−2.5	72.4+2.4−2.4	72.7+1.7−1.7

Note. We quote the scalar amplitude as $\mathcal {A}_s \equiv \ln \left[10^{10} A_s\right]$ for a pivot point of k^s_⋆ = 0.013 Mpc⁻¹.

Download table as:  ASCIITypeset image

The mean values of these parameters also shift a little, but the only significant discrepancy is perhaps in the recovered value of θ. Here, we find the WMAP + ACBAR combination prefers a somewhat higher value (in agreement with ACBAR's own analysis; Reichardt et al. 2009), whereas the addition of QUaD data does not change the WMAP-only preferred mean value but simply tightens the constraint.

Note that in comparison to the WMAP team's own analysis (Dunkley et al. 2009), we recover slightly different mean values for τ and more significantly different values for z_re. This is due to the different reionization model used in the later version of the CAMB software which we have used. We note in passing that the majority of the constraining power in the QUaD data comes from the measurements of the polarization power spectra as found with our previous analysis (Castro et al. 2009).

The 1D and 2D marginalized constraints on our base parameters for the running spectral index model, as obtained from our WMAP + QUaD runs are shown in Figure 15 along with the constraints using only the WMAP data. The recovered parameter values and their uncertainties are listed in Table 4.

Zoom In Zoom Out Reset image size

Table 4. Parameter Constraints for the Running Spectral Index Model

Parameter	WMAP	WMAP+ACBAR	WMAP+QUaD	WMAP+ACBAR+QUaD	WMAP+ACBAR+QUaD+SDSS
Ωbh2	0.0221+0.0009−0.0009	0.0221+0.0007−0.0007	0.0219+0.0007−0.0007	0.0219+0.0006−0.0006	0.0223+0.0006−0.0006
Ωch2	0.116+0.009−0.009	0.120+0.008−0.008	0.117+0.008−0.008	0.120+0.008−0.008	0.111+0.004−0.004
θ	1.0399+0.0030−0.0031	1.0414+0.0027−0.0027	1.0399+0.0024−0.0024	1.0409+0.0023−0.0023	1.0413+0.0022−0.0022
τ	0.093+0.018−0.018	0.095+0.019−0.019	0.095+0.019−0.018	0.096+0.019−0.019	0.096+0.019−0.018
ns	0.964+0.014−0.014	0.967+0.014−0.014	0.963+0.013−0.013	0.965+0.013−0.013	0.967+0.013−0.013
${\cal A}_s$	3.16+0.06−0.06	3.18+0.05−0.05	3.17+0.06−0.06	3.19+0.05−0.05	3.15+0.04−0.04
nrun	−0.031+0.028−0.028	−0.040+0.023−0.023	−0.038+0.024−0.024	-0.046+0.021−0.021	−0.028+0.018−0.018
ΩΛ	0.70+0.05−0.05	0.69+0.05−0.05	0.70+0.05−0.05	0.69+0.05−0.05	0.74+0.02−0.02
Age	13.81+0.18−0.18	13.78+0.14−0.15	13.82+0.14−0.14	13.82+0.13−0.13	13.72+0.11−0.11
Ωm	0.30+0.05−0.05	0.31+0.05−0.05	0.30+0.05−0.05	0.31+0.05−0.05	0.26+0.02−0.02
σ8	0.81+0.04−0.04	0.83+0.03−0.03	0.81+0.04−0.04	0.83+0.03−0.03	0.79+0.02−0.02
zre	11.2+1.6−1.6	11.4+1.6−1.6	11.4+1.6−1.6	11.7+1.6−1.6	11.2+1.5−1.5
H0	68.9+4.0−4.0	68.0+3.4−3.4	68.5+3.5−3.6	67.5+3.2−3.2	71.4+1.9−1.9

Note. The pivot point used for $\mathcal {A}_s$ and n_s is k^s_⋆ = 0.013 Mpc⁻¹.

Download table as:  ASCIITypeset image

The impact of QUaD data is greater for this model—the QUaD data add significantly to the constraints on Ω_bh², Ω_ch², θ, and n_run, reducing the marginalized 1D errors on these parameters by up to 20%. Adding both the QUaD and ACBAR data has an even greater impact, reducing the errors on these parameters by up to a third compared to the WMAP-only uncertainties. Of particular interest are the constraints in the n_s–n_run plane since many theories of inflation predict both a deviation from n_s = 1 and/or a small negative running. Constraints from the WMAP + ACBAR + QUaD combination are shown in the left panel of Figure 16, together with the constraints from WMAP alone. Our 1D marginalized constraint on the running from the combined data set is n_run = 0.046 ± 0.021, 2.2σ away from the n_run = 0 model. Adding the LSS data to the mix improves the constraints even further, in our analysis tightening the 1σ error on n_run from 0.021 to 0.018. The significance of a nonzero running is also reduced on addition of the LSS data.

Zoom In Zoom Out Reset image size

Comparing Tables 3 and 4, we see also that the 1D marginalized constraint on the spectral index, n_s, is not weakened by allowing a nonzero running. This is due to our use of the decorrelation pivot scale as described in Section 7.2. The results also show that the constraints on n_s obtained for the standard six-parameter ΛCDM model are robust to marginalization over a possible running. For example, with the WMAP + ACBAR + QUaD combination, the constraint on n_s goes from n_s = 0.962 ± 0.013 to n_s = 0.965 ± 0.013 when we allow for a possible nonzero running. For comparison, if instead, we use the WMAP-preferred pivot point (k^s_⋆ = 0.002 Mpc⁻¹), the marginalized 1D constraint on n_s is degraded to n_s = 0.962 ± 0.019 in the presence of running.

Hints of a negative running in the spectral index have been observed in previous CMB analyses (e.g., Dunkley et al. 2009; Reichardt et al. 2009). With the addition of the new QUaD data, this suggestion of a negative running not only persists, but is strengthened. We do, however, stress that the combined result shown in the left panel of Figure 16 is still consistent with zero running at the 3σ level. Nevertheless, it is worth examining the implications for inflation models if the running was as large and as negative as the best-fit value returned from our analysis of the combined CMB data set. In this respect, Malquarti et al. (2004) have pointed out that an observed running of n_run ≲ −0.02 would effectively rule out large field inflation models. More generally, Easther & Peiris (2006) have demonstrated that for a large negative running, single field slow-roll inflation models will last less than 30 e-folds after entering the horizon. This amount of inflation is insufficient if inflation happened at the GUT scale. Consequently, an observation of n_run ∼ −0.05 would require inflation theory to move beyond the simplest models, e.g., by considering multiple fields and/or modifications to the slow-roll formalism (e.g., Chung et al. 2003; Makarov 2005; Ballesteros et al. 2006).

Our constraints for the ΛCDM model including a possible tensor component are listed in Table 5 in terms of the mean recovered parameter values and their uncertainties. For r, we quote the 95% one-tail upper limit, since, as expected, no detection of tensors is made. For our WMAP-only analysis, we recover a slightly weaker limit (r < 0.48) than that obtained by the WMAP team themselves (r < 0.43; Dunkley et al. 2009).²⁴ Adding either the ACBAR or QUaD data, this is reduced to r < 0.40.

Table 5. Parameter Constraints Including a Possible Tensor Component

Parameter	WMAP	WMAP+ACBAR	WMAP+QUaD	WMAP+ACBAR+QUaD	WMAP+ACBAR+QUaD+SDSS
Ωbh2	0.0235+0.0008−0.0008	0.0234+0.0007−0.0007	0.0232+0.0007−0.0007	0.0231+0.0006−0.0006	0.0229+0.0006−0.0006
Ωch2	0.104+0.007−0.007	0.106+0.007−0.007	0.103+0.007−0.007	0.105+0.006−0.006	0.107+0.004−0.004
θ	1.0421+0.0033−0.0033	1.0433+0.0028−0.0028	1.0412+0.0026−0.0026	1.0423+0.0023−0.0023	1.0419+0.0022−0.0023
τ	0.094+0.018−0.018	0.092+0.018−0.018	0.093+0.018−0.018	0.091+0.017−0.017	0.088+0.017−0.017
ns	0.990+0.023−0.023	0.986+0.021−0.020	0.982+0.020−0.020	0.978+0.018−0.018	0.973+0.015−0.015
${\cal A}_s$	3.07+0.05−0.05	3.08+0.04−0.04	3.08+0.04−0.04	3.09+0.04−0.04	3.09+0.03−0.04
r	<0.48 (95% c.l.)	<0.40 (95% c.l.)	<0.40 (95% c.l.)	<0.33 (95% c.l.)	<0.27 (95% c.l.)
ΩΛ	0.78+0.03−0.03	0.77+0.03−0.03	0.78+0.03−0.03	0.77+0.03−0.03	0.76+0.02−0.02
Age	13.52+0.18−0.18	13.51+0.16−0.16	13.57+0.15−0.15	13.57+0.13−0.13	13.61+0.11−0.11
Ωm	0.22+0.03−0.03	0.23+0.03−0.03	0.22+0.03−0.03	0.23+0.03−0.03	0.24+0.02−0.02
σ8	0.77+0.04−0.04	0.78+0.04−0.04	0.76+0.04−0.04	0.78+0.03−0.03	0.78+0.02−0.02
zre	10.5+1.4−1.4	10.5+1.4−1.4	10.5+1.4−1.3	10.4+1.4−1.3	10.3+1.3−1.3
H0	75.8+3.8−3.8	75.2+3.4−3.4	75.5+3.4−3.4	74.8+3.1−3.1	73.8+1.8−1.9

Note. The pivot point used for $\mathcal {A}_s$ is k^s_⋆ = 0.013 Mpc⁻¹, while the pivot point used for the tensor-to-scalar ratio, r, is k^t_⋆ = 0.002 Mpc⁻¹.

Download table as:  ASCIITypeset image

The WMAP + ACBAR + QUaD combination produces a constraint on tensor modes of r < 0.33, the strongest from the CMB alone to date. Note that this constraint does not come from our upper limits on the BB spectrum. It is, in fact, driven by a preference of the small-scale data (particularly the QUaD EE and TE data) for a somewhat lower spectral index compared to that preferred by WMAP alone—a lower n_s allows more of the large-scale TT power observed by WMAP to come from scalar perturbations and therefore the maximum allowed tensor contribution is reduced. Our CMB-only constraints in the r–n_s plane are plotted in the right panel of Figure 16.

When we allow for both a running in the spectral index and a tensor contribution the constraints weaken considerably versus either on their own. In the left panel of Figure 17, for the WMAP + ACBAR + QUaD combination, we plot constraints in the n_run–n_s plane with and without marginalization over a possible tensor component. The right panel of this figure shows the corresponding constraints in the r–n_s plane with and without marginalization over a possible running in the spectral index. For this model, the addition of QUaD and ACBAR data still improves the constraints on the spectral index running (and indeed, still strongly suggests a small negative running), but the constraints in the r–n_s plane in the presence of running do not improve on the WMAP-only result. This degradation in the constraints on r when we allow for a running in the spectral index is further demonstrated in Table 6 where we quote the 1D marginalized constraints on the parameters {r, n_s, n_run} for the tensors-only, running-only, and tensors + running models. In this table, we present the results for the WMAP + ACBAR + QUaD combination and for the case where we add in the SDSS LRG data.

Zoom In Zoom Out Reset image size

In the preceding sections, we used the QUaD TT, EE, TE, and BB spectra to constrain the parameters of standard ΛCDM models and its usual extensions. For that analysis, we did not use our measurements of the cross-polarization spectrum (EB) or the correlation of temperature with B-modes (TB), since these spectra are expected to vanish in a universe which respects parity conservation (which the above models do). In this section, we use these two spectra (along with the TE, EE, and BB measurements) to constrain a possible parity-violation signal on cosmological scales. In the presence of parity-violating interactions, a rotation in the polarization direction of CMB photons will be induced as they propagate from the surface of last scattering. If parity-violating effects are present on cosmological scales, there will therefore be a net local rotation of the observed Stokes parameters, Q and U, in the measured polarization map. This will mix E and B modes resulting in nonzero expectations for the TB and EB power spectra. Parameterizing the parity-violation effect with a rotation angle, Δα, the expectation values for the TB and EB spectra in terms of the cosmological TE and EE spectra are given by

$$\begin{equation} C_\ell ^{TB} = C_\ell ^{TE} \sin \, (2 \Delta \alpha) \end{equation} \tag{ 24 }$$

$$\begin{equation} C_\ell ^{EB} = \case{1}{2} C_\ell ^{EE} \sin \, (4 \Delta \alpha) \,. \end{equation} \tag{ 25 }$$

In addition, assuming that primordial and lensed B-modes are negligible (which is an excellent assumption given our sensitivity), the expectation value for the BB spectrum is

$$\begin{equation} C_\ell ^{\it BB} = C_\ell ^{EE} \sin ^2(2 \Delta \alpha) \,. \end{equation} \tag{ 26 }$$

We used our previous results to place a constraint of Δα = 053 ± 082 ± 050 (Wu et al. 2009), where the two quoted uncertainties are the random and systematic components, respectively.²⁵

Here, we repeat this analysis with our new measurements. The analysis is model independent in the sense that we construct our estimator for Δα in terms of the observed power spectra and do not assume a cosmological model for the EE or TE signals. For details of our estimator and analysis (which has not changed since our previous work), we refer the reader to Wu et al. (2009).

We apply the estimator to the real QUaD data and assign error bars due to random noise and sample variance by processing the suite of simulations containing both signal and noise through the analysis. The result is shown in Figure 18, where we plot both the results from simulations and from the real data. Note that our simulations contain no parity-violating signals and so should scatter about zero, which they do. We take the scatter in the results from the simulations as our random error. Adding in the systematic error, our final result is

$$\begin{equation} \Delta \alpha = 0.64 \pm 0.50 \mbox{ (random)} \pm 0.50 \mbox{ (systematic)} \,. \end{equation} \tag{ 27 }$$

The random error has been reduced by ∼40% with respect to our previous analysis in line with expectations.

Zoom In Zoom Out Reset image size

Our result can be compared to the limits obtained from the WMAP 5-year data (Δα = −1.7 ± 2.1; Komatsu et al. 2009) or to the limits obtained from the combination of the WMAP 5-year data and the B03 results (Δα = −2.6 ± 1.9; Xia et al. 2008). We note that both of these quoted results include random errors only and do not include estimates of the systematic errors on the WMAP and B03 polarization calibration angles. Even when we include this systematic uncertainty for QUaD, our result is clearly a marked improvement over these previous analyses.

Although QUaD has not made any detection of B-modes, it is the most sensitive small-scale CMB polarization experiment to date. We can therefore place the leading upper limit on the presence of a small-scale B-mode signal. The signal expected to dominate on the scales at which QUaD is sensitive is that induced by gravitational lensing of E-modes by intervening large-scale structure. As well as measuring cosmological B-modes from inflation, future polarization experiments will target this lensing signal from which useful information can be gained on dark energy and massive neutrinos (e.g., Kaplinghat et al. 2003; Smith et al. 2006).

Assuming a single flat band power between ℓ = 200 and ℓ = 2000, we find ℓ(ℓ + 1)C^BB_ℓ/2π = 0.17 ± 0.17 μK² with a 95% upper limit of 0.57 μK². The errors quoted are estimated from the scatter in the results obtained from the suite of simulations containing both signal and noise. For comparison, the ΛCDM expectation value for this band power is 0.058 μK². Alternatively, assuming the ΛCDM shape for the lensing signal, and simply fitting for its amplitude between ℓ = 200 and ℓ = 2000, our constraint on the amplitude²⁶ is 2.5 ± 4.5 with a 95% upper limit of 12.5.

Note that although we have used all of our BB band powers to obtain the above constraints, the window function of our estimator is strongly skewed towards lower multipoles where the band power uncertainties are much smaller. The effective range of multipoles to which our upper limits apply is, in fact, 170 < ℓ < 400 rather than the nominal 200 < ℓ < 2000 range.

Although our 2σ upper limits are an order of magnitude larger than the expected ΛCDM signal, they are, in turn, roughly an order of magnitude better than previously reported limits on the amplitude of the B-mode signal in this angular scale range.