A Comparison of Maps and Power Spectra Determined from South Pole Telescope and Planck Data

The SPT is a 10 m telescope located at the Amundsen–Scott South Pole station. From 2008 to 2011, the first camera on the SPT, a three-band bolometer array known as the SPT-SZ camera, was used to conduct a survey of ∼2500 deg² of the southern sky with low Galactic dust contamination, referred to as the SPT-SZ survey. As shown in Figure 1, the survey area is a contiguous region extending from ${20}^{{\rm{h}}}$ to ${7}^{{\rm{h}}}$ in right ascension (R.A.) and from $-65^\circ$ to $-40^\circ$ in declination. The survey was conducted by observing a series of 19 areas, ranging in area from roughly 100 to 300 deg², which together form the full survey region (see, e.g., S13, Figure 2). If not specified, in this paper we use "field" to refer to individual observing areas of SPT-SZ.

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There were approximately 200 observations of each field, with each individual observation taking roughly 2 hr. Estimates of the primary CMB temperature power spectrum from this survey, as well as the resulting cosmological interpretation, are discussed in Keisler et al. (2011, hereafter K11), S13, and Hou et al. (2014). Our analysis is identical to that of S13 up to the single-observation map step. We briefly review that part of the analysis here and refer the reader to K11 and S13 for more details.

To process time-ordered data (TOD) into maps, the TOD from each SPT detector are first filtered and multiplied by a calibration factor. The filtering steps important for this analysis are a high-pass filter and the subtraction of a common mode across all detectors in each of the six 160-element modules in the focal plane. The high-pass filter cuts off signals below a temporal frequency corresponding to an angular frequency along the scan direction of ${\ell }=270$. The common-mode subtraction acts as an isotropic high-pass filter with a cutoff at roughly ${\ell }=500$. In both of these filtering steps, bright point sources ($S\gt 50$ mJy at 150 GHz) are masked to avoid large filtering artifacts.

The TOD from individual SPT detectors are then binned into maps using inverse-variance weighting, i.e., TOD samples corresponding to times when an individual detector was pointed at a particular pixel are averaged together using inverse-variance weighting and then assigned to that pixel. Maps are made using the oblique Lambert equal-area azimuthal projection (Snyder 1987) with 1 arcmin square pixels.

The SPT maps are simply binned and averaged maps of filtered data and thus are biased representations of the sky. The signal in these maps is the true sky signal convolved with the instrument beam, the effect of the TOD filtering, and the effect of binning. Beams are discussed below, and the filter transfer function is estimated through simulations, as discussed in Section 3.3.

We use the S13 estimates of the SPT 150 GHz beam transfer function ${B}_{{\ell }}$ and its uncertainty, and we refer the reader to S13 for more details. Briefly, the main lobe is measured using bright point sources in the survey fields, while the sidelobes are measured using observations of Jupiter. Venus observations are used to stitch the two together. The main lobe of the beam is well approximated by a 1.2 arcmin FWHM Gaussian. The beam uncertainty arises from several statistical and systematic effects, including residual atmospheric noise in the maps of Venus and Jupiter, and the weak dependence of ${B}_{{\ell }}$ on the choice of radius used to stitch the inner and outer beam maps.

The TOD from each detector in each observation is calibrated using the response to an internal thermal source, which is in turn tied to the brightness of the Galactic H ii region RCW38. For details of this calibration, see Schaffer et al. (2011). The full-depth maps are then compared to CMB satellite data to provide the overall absolute calibration. In K11 and S13, the power spectrum of the full-depth maps was compared to the full-sky WMAP estimate of the CMB power spectrum in the multipole range $650\leqslant {\ell }\leqslant 1000$. For this work, we initially use the calibration determined in George et al. (2015), using the comparison of SPT power spectra to the full-sky Planck power spectrum in the multipole range $670\leqslant {\ell }\leqslant 1170;$ however, the map-based comparison to Planck undertaken here ends up providing a significantly more precise absolute calibration, as detailed in Section 4.2.

The Planck satellite (Planck Collaboration et al. 2014a) was launched in 2009 by the European Space Agency, with the goal of measuring the CMB temperature and polarization anisotropy with significantly better sensitivity, angular resolution, and wavelength coverage than was achieved by its space predecessor, the WMAP mission (Bennett et al. 2013). Planck mapped the full sky in nine bands, ranging in frequency from 30 to 857 GHz. In this work, we use HFI data from the 2015 data release. Specifically, we use the 143 GHz full-mission, halfmission-1, and halfmission-2 maps downloaded from the NASA/IPAC Infrared Science Archive (IRSA).³⁵

In this work, we use a cross-spectrum pipeline similar to that used in K11 and S13 to compare SPT and Planck data on the SPT-SZ sky patch. To use this pipeline for Planck data, the Planck maps must have the same pixelization, projection, and field definitions as the SPT maps. To achieve this, we create mock TOD using the Planck 143 GHz full-mission or half-mission maps and the SPT pointing and detector weight information from individual SPT observations. The mock TOD are then binned into maps in the same manner as the real SPT TOD, using the same projection and pixelization. Instead of using the SPT bandpass filtering, a much simpler high-pass filter is applied to the Planck mock TOD to simply remove the signals on scales with ${\ell }\lt 100$. Our simulations show that applying this simple high-pass filter greatly improves the numerical stability of our unbiased power spectrum calculations. In the power spectrum analysis described below, we use the projected Planck maps generated from the mapmaking pipeline with this simple high-pass filter applied to the Planck mock TOD.

The Planck 143 GHz beam has been measured using the Planck TOD around planets, such as Mars, Jupiter, and Saturn (Planck Collaboration et al. 2014b, 2016f ). The beam window function $B({\ell })$ for individual frequency detector sets has been released by the Planck Collaboration as part of the "Reduced Instrument Model" (RIMO).³⁶ From the 2013 Planck data release to the 2015 data release, there was a marked improvement in the beam characterization, such that the uncertainty on the 2015 beam window functions for 143 GHz data is at the 0.1% level for the ℓ range of interest to this work.

We use an estimator similar to that used in K11 and S13 to calculate the various cross-spectra in this work. Each cross-spectrum is calculated by correlating maps made from different observations of the same field—either correlating full-depth SPT maps with full-depth Planck maps or correlating two half-depth maps from the same experiment. In the latter case, there is a noise penalty relative to K11 or S13, who used ${ \mathcal O }(100)$ independent maps depending on the field. However, the noise penalty is largely insignificant for the SPT maps (less than 2% in power) on the angular scales of interest, and it is unavoidable for the Planck maps because a larger number of independent splits are not publicly available. We mask and zero-pad each map before calculating its two-dimensional Fourier transform, ${\tilde{m}}_{{\boldsymbol{\ell }}}$.³⁷ The mask is a product of an apodization window and a point-source mask.³⁸ The maps and masks are zero-padded before Fourier-transforming such that each Fourier-space pixel has a width of ${\delta }_{{\ell }}=5$. The raw bandpowers are the binned average of the cross-spectra between two maps A and B within a multipole bin b:

$$\begin{eqnarray}&&{\hat{D}}_{b}^{A\times B}={\left\langle \displaystyle \frac{{\ell }({\ell }+1)}{2\pi }{H}_{{\boldsymbol{\ell }}}\mathrm{Re}[{\tilde{m}}_{{\boldsymbol{\ell }}}^{A}{\tilde{m}}_{{\boldsymbol{\ell }}}^{B* }]\right\rangle }_{{\ell }\in b}.\end{eqnarray} \tag{ 1 }$$

For the SPT-only power spectrum, A and B are the two SPT half-survey maps; for the SPT and Planck cross-spectrum, A and B are the full-mission maps of each experiments; and for the Planck-only 143 power spectrum, A and B are the two Planck half-mission maps. ${H}_{{\boldsymbol{\ell }}}$ in the above equation represents the two-dimensional weighting of the Fourier modes that was used by S13 to handle the anisotropic noise in the SPT maps. While this weighting is suboptimal for Planck data, we still use the same S13-derived weighting for all bandpowers to minimize the differential sample variance between cross-spectra.

Since the input maps are biased estimates of the true sky, due to effects such as the application of a mask to the maps, the raw bandpowers ${\hat{D}}_{b}$ are a biased estimate of the true bandpowers, D_b. The biased and unbiased estimates are related by

$$\begin{eqnarray}&&{\hat{D}}_{{b}^{{\prime} }}\equiv {K}_{{b}^{{\prime} }b}{D}_{b},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{K}_{{{bb}}^{{\prime} }}={P}_{b{\ell }}({M}_{{\ell }{{\ell }}^{{\prime} }}\,{F}_{{{\ell }}^{{\prime} }}{B}_{{{\ell }}^{{\prime} }}^{2}){Q}_{{{\ell }}^{{\prime} }{b}^{{\prime} }},\end{eqnarray} \tag{ 3 }$$

${Q}_{{{\ell }}^{{\prime} }{b}^{{\prime} }}$ is the binning operator and P_ℓ is its reciprocal, ${M}_{{\ell }{\ell }^{\prime} }$ is the mode-coupling matrix, ${F}_{{\ell }}$ is the filter transfer function that accounts for the signal suppressed by TOD filtering, and ${B}_{{\ell }}$ is the beam function. For details on the unbiasing procedure, see K11, S13, and Hivon et al. (2002).

The unbiased bandpowers are calculated on a field-by-field basis and then combined. We combine the bandpowers obtained from individual fields using the effective area of single fields as the weighting, as in K11 and S13.

We use simulations both in the calculation of the bandpowers and to characterize the degree of consistency between the SPT and Planck cross-spectra over the SPT-SZ survey region. In this section, we turn our attention to how these simulations are created and used. The final product of this procedure is 400 sets of simulated unbiased bandpowers for each of the three combinations of data (150 × 150, 150 × 143, and 143 × 143).

3.2.1. Sky Signals

3.2.2. Planck Noise Simulations

3.2.3. SPT Noise Realizations

The noise-free simulated observations are used to calculate F_ℓ, the filtering transfer function, and ${H}_{{\boldsymbol{\ell }}}$, the two-dimensional weight in the power spectrum calculation. Given the input power spectrum of the simulations, the effective transfer function can be derived by comparing the power spectrum of the simulations with the known input spectrum using an iterative scheme (Hivon et al. 2002). This method was used in K11 and S13; in this analysis, we make adjustments to Equations (6) and (7) in Keisler et al. (2011) to include the Planck beam and pixel window function for the SPT-Planck cross-spectrum and Planck-only spectrum.

We have three sets of unbiased bandpowers derived from SPT and Planck data, ${{ \mathcal D }}_{b}^{150\times 150}$, ${{ \mathcal D }}_{b}^{150\times 143}$, and ${{ \mathcal D }}_{b}^{143\times 143}$. In this analysis, we use the difference between these bandpowers, in particular the residual and the ratio between the bandpower sets, to quantitatively characterize the consistency between the SPT and Planck data sets. This section presents the metrics we use for these consistency tests.

3.4.1. Covariance Estimation

We estimate the bandpower sample variance and noise variance using 400 sets of simulated signal+noise maps. The final bandpower covariance matrix will also include a contribution due to beam uncertainties; the calculation of the beam covariance is detailed in the next section. For a single field and data combination, the bandpower covariance, ${{\rm{\Xi }}}^{X\times Y}$, is calculated simply as

$$\begin{eqnarray}&&{{\rm{\Xi }}}^{X\times Y}=\langle ({{\boldsymbol{D}}}_{b,\mathrm{sim}}^{X\times Y}-{\bar{{\boldsymbol{D}}}}_{b,\mathrm{sim}}^{X\times Y})({{\boldsymbol{D}}}_{b^{\prime} ,\mathrm{sim}}^{X\times Y}-{\bar{{\boldsymbol{D}}}}_{b^{\prime} ,\mathrm{sim}}^{X\times Y})\rangle ,\end{eqnarray} \tag{ 4 }$$

where ${\bar{{\boldsymbol{D}}}}_{b,\mathrm{sim}}^{X\times Y}$ is the mean over all 400 simulations for cross-bandpowers $X\times Y\in [150\times 150,150\times 143,143\times 143]$. The simulated 150 × 143 cross-bandpowers ${{ \mathcal D }}_{b,\mathrm{sim}}^{150\times 143}$ are derived from cross-spectra of simulated SPT maps including realizations of SPT full-observation noise and simulated Planck maps of the same underlying sky signal including Planck FFP8 noise simulations. For ${{ \mathcal D }}_{b,\mathrm{sim}}^{150\times 150}$, the simulated bandpowers are calculated by the cross-correlation of the two sets of simulated SPT maps with half-observation noise realizations. Similarly, the simulated ${{ \mathcal D }}_{b,\mathrm{sim}}^{143\times 143}$ bandpowers are obtained using two sets of simulated Planck maps including Planck half-mission white-noise realizations. The full bandpower covariance matrix is then obtained by combining the individual-field covariance matrices with the square of the area weighting used to combine the bandpowers themselves.

As we will evaluate consistency by looking at the differences and ratios among sets of bandpowers, we also need the covariance matrix for these quantities. For the differences or residual bandpowers, ${\rm{\Delta }}{{\boldsymbol{D}}}_{{\rm{b}}}\equiv {{\boldsymbol{D}}}_{b}^{i\times j}-{{\boldsymbol{D}}}_{b}^{m\times n}$ (where $i,j,m,n\,\in \{150,143\}$), the noise-plus-sample-variance part of the covariance is easily estimated from the 400 simulations:

$$\begin{eqnarray}&&{{\rm{\Xi }}}_{\mathrm{resid}}=\langle {\rm{\Delta }}{{\boldsymbol{D}}}_{b,\mathrm{sim}}{\rm{\Delta }}{{\boldsymbol{D}}}_{b^{\prime} ,\mathrm{sim}}\rangle .\end{eqnarray} \tag{ 5 }$$

For the ratios relative to the 150 × 150 bandpowers, the covariance can be expressed as

$$\begin{eqnarray}&&{{\rm{\Xi }}}_{\mathrm{ratio}}=\left\langle \left(\displaystyle \frac{{{\boldsymbol{D}}}_{b,\mathrm{sim}}^{i\times j}}{{{\boldsymbol{D}}}_{b,\mathrm{sim}}^{150\times 150}}-1\right)\left(\displaystyle \frac{{{\boldsymbol{D}}}_{b^{\prime} ,\mathrm{sim}}^{i\times j}}{{{\boldsymbol{D}}}_{b^{\prime} ,\mathrm{sim}}^{150\times 150}}-1\right)\right\rangle .\end{eqnarray} \tag{ 6 }$$

3.4.2. Beam Uncertainty

In this section, we present how beam uncertainties are handled for the bandpower comparison. Beam uncertainties appear as a second covariance term, in addition to the noise-plus-sample-variance term presented above, because the simulations did not include beam uncertainties.

We begin by estimating the eigenmodes of the Planck and SPT beam covariance matrices. The Planck beam eigenmodes can be obtained from RIMO in the 2015 data release, and the SPT beam eigenmodes can be derived from the beam correlation matrix calculated in S13. For each eigenmode, we can calculate the fractional beam uncertainty as a function of multipole, $\delta {b}_{{\ell }}/{b}_{{\ell }}$, and propagate this linearly to the bandpower space according to

$$\begin{eqnarray}&&\delta {D}_{b,\mathrm{beam}}^{i\times j}=-\displaystyle \sum _{{\ell }}{W}_{b{\ell }}^{i\times j}{D}_{{\ell }}^{i\times j,\mathrm{fid}}\left(\displaystyle \frac{\delta {b}_{{\ell }}^{i}}{{b}_{{\ell }}^{i}}+\displaystyle \frac{\delta {b}_{{\ell }}^{j}}{{b}_{{\ell }}^{j}}\right),\end{eqnarray} \tag{ 7 }$$

where $i,j\in \{150,143\}$, ${W}_{b{\ell }}^{i\times j}$ is the bandpower window function, and ${D}_{{\ell }}^{i\times j,\mathrm{fid}}$ is the fiducial power spectrum including CMB and extragalactic foregrounds for that frequency combination.

For the consistency tests, we need the beam covariance for bandpower differences and bandpower ratios. For the bandpower differences, the above equation leads straightforwardly to

$$\begin{eqnarray}&&{{\rm{\Xi }}}_{{bb}^{\prime} ,\mathrm{beam}}^{(i\times j)(m\times n)}=\displaystyle \sum _{{\rm{e}} \mbox{-} \mathrm{modes}}(\delta {D}_{b,\mathrm{beam}}^{i\times j}-\delta {D}_{b,\mathrm{beam}}^{150\times 150})\\ &&\quad \times (\delta {D}_{b^{\prime} ,\mathrm{beam}}^{m\times n}-\delta {D}_{b^{\prime} ,\mathrm{beam}}^{150\times 150}),\end{eqnarray} \tag{ 8 }$$

where $i,j,m,n\in \{150,143\}$ and the sum is taken over all eigenmodes of the beam covariance matrices.

For the bandpower ratios, the beam covariance can be written as

$$\begin{eqnarray}&&{{\rm{\Xi }}}_{{bb}^{\prime} ,\mathrm{beam}}^{(i\times j)(m\times n)}=\displaystyle \sum _{{\rm{e}} \mbox{-} \mathrm{modes}}\left(\displaystyle \frac{\delta {D}_{b,\mathrm{beam}}^{i\times j}-\delta {D}_{b,\mathrm{beam}}^{150\times 150}}{{D}_{b}^{150\times 150}}\right)\\ &&\quad \times \,\left(\displaystyle \frac{\delta {D}_{b^{\prime} ,\mathrm{beam}}^{m\times n}-\delta {D}_{b^{\prime} ,\mathrm{beam}}^{150\times 150}}{{D}_{b^{\prime} }^{150\times 150}}\right).\end{eqnarray} \tag{ 9 }$$

The total bandpower covariance for a consistency test is then the sum of the noise-plus-sample-variance term from the previous section and the appropriate beam covariance term above.

3.4.3. Bandpower Window Function Correction

To characterize the level of consistency between the three sets of bandpowers (150 × 150, 150 × 143, and 143 × 143), we choose one set, ${D}_{b}^{150\times 150}$, as the fiducial that is subtracted from or divided into the other two sets (see Section 4.2). However, the residual bandpowers and bandpower ratios are biased owing to the difference of bandpower window functions (e.g., Knox 1999). Due to the different filtering and the beam transfer functions of the two experiments, the bandpower window functions are different between bandpower sets. In Figure 3, the window functions of the three sets of bandpowers are illustrated for two bins with effective centers at ${\ell }=675$ and ${\ell }=2475$. In both of these bins, relative to the 150 × 150 window functions, the 143 × 143 and 150 × 143 bandpowers receive more weight from multipoles lower than the bin center and less from multipoles higher than the bin center. These differences, if left unaddressed, lead to a bias in the bandpower differences that is dependent, to some degree, on the assumed cosmological model.

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Before comparing the bandpower sets, we need to correct for this bias. The correction is calculated as follows:

$$\begin{eqnarray}&&\delta {D}_{b}^{i}=\displaystyle \sum _{{\ell }}({W}_{{\ell }}^{i}-{W}_{{\ell }}^{150\times 150}){D}_{{\ell }}^{\mathrm{fid}},\end{eqnarray} \tag{ 10 }$$

where $i=150\times 143$, or 143 × 143. The Planck 2015 best-fit cosmology plus the best-fit extragalactic foregrounds are used as the fiducial model. We subtract this correction term from the 150 × 143 and 143 × 143 bandpowers to remove the bias caused by the bandpower window function differences. In Figure 4, the top panel shows the shape of the correction $\delta {D}_{b}$ and the bottom panel shows the ratio between $\delta {D}_{b}$ and the corresponding D_b. For 150 × 143, $\delta {D}_{b}$ yields a roughly flat 1%–2% correction to the bandpower, while for 143 × 143, the correction is more important at higher multipoles, up to $\sim 15 \%$ at ${\ell }=2500$. The window function correction is clearly critical for the consistency analysis between these three sets of bandpowers.

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We now investigate the model dependence of the window function corrections. To characterize this dependence, we calculate the window function corrections for a distribution of input ${D}_{{\ell }}$. The distribution is obtained by randomly sampling the cosmology and foreground models from a Markov chain based on WMAP9+SPT data. The variation of $\delta {D}_{b}$ in Equation (10) is illustrated in the top panel of Figure 4, with the error bars indicating the $3\sigma$ variation of the corrections. Within the band range of interest, this variation level is less than 1% of the bandpower standard error; therefore, we ignore this very small uncertainty in our analysis.

The SPT and Planck maps within the same patch are presented in Figure 5 for visual comparison. The top panels show the filtered SPT map described in Section 2.1 on the left and the projected Planck full-mission 143 GHz map on the right. Some bright point sources can be identified by eye in both the SPT and Planck maps, but the maps do not resemble each other, because the Planck map is dominated by the degree-scale CMB anisotropy filtered out of the SPT data, and the SPT map shows more small-scale structure owing to its higher angular resolution. In the bottom panels of Figure 5, we show SPT and Planck maps of the same spatial modes. The bottom left panel shows the SPT map from the top left panel convolved with the difference between the SPT 150 GHz and Planck 143 GHz beams; the bottom right panel shows a map made by observing a Planck sky with the SPT 150 GHz scan strategy and TOD filtering. Though the Planck map has a visibly higher noise level (as expected), the signals in the two maps appear nearly identical. Figure 6 shows the difference between the bottom left and bottom right maps from Figure 5, on the same color scale, along with a simulated difference map for comparison. The feature at the location of one of the bright point sources is most likely caused by temporal variability of the source, as the SPT and Planck data were not taken simultaneously.

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To make this result more quantitative, we could match the beam and filtering between these two data sets over the full SPT-SZ survey region, mask point sources, calculate the power in the difference map, and compare that power to the expected power from noise alone. In the next section we do a nearly equivalent but somewhat simpler calculation. Using the cross-spectrum formalism outlined in Section 3, we calculate the Planck 143 × 143 power spectrum, the SPT 150 × 150 power spectrum, and the SPT-Planck 150 × 143 cross-spectrum, and we calculate the ${\chi }^{2}$ of the null hypothesis that these three spectra are measuring the same power. This calculation does not completely eliminate sample variance, as the perfectly mode-matched difference-map power spectrum would, but it strongly reduces it (see Figure 8 below).

In this section, we present the comparison between the three sets of window-function-corrected bandpowers, which we denote as ${{ \mathcal D }}_{b}^{150\times 150}$, ${{ \mathcal D }}_{b}^{150\times 143}$, and ${{ \mathcal D }}_{b}^{143\times 143}$. We first show all three sets of bandpowers in the top panel of Figure 7. The error bars contain contributions from sample variance, noise variance, and beam uncertainties. By eye, the three sets of bandpowers look very consistent. At low ℓ, the scatter among the three sets of points is much smaller than the errors on any one set; this is because the errors on all three are dominated by sample variance in this regime, and sample variance is highly correlated among the three data sets. This is also why the three sets of error bars are nearly the same size at low ℓ (though the 143 × 143 bandpowers have slightly smaller error bars in this ℓ range because the modes lost in the SPT TOD filtering process result in slightly increased sample variance). Noise variance begins to dominate the 143 × 143 error bars at ${\ell }\gtrsim 1700$ and the 150 × 143 error bars at ${\ell }\gtrsim 2000$, while the 150 × 150 error bars are sample variance dominated over the entire range plotted.

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The remaining panels of Figure 7 show various comparisons among the three sets of bandpowers. Two comparison schemes are applied: differences and ratios. The former is good for diagnosing additive effects, while the latter is more sensitive to multiplicative systematics. In both schemes, ${{ \mathcal D }}_{b}^{150\times 150}$ is chosen as the fiducial bandpower set. These comparison plots and the ${\chi }^{2}$ and the probability to exceed (PTE) statistics calculated below are thus testing the following set of null hypotheses: (1) two sets of bandpower residuals

$$\begin{eqnarray*}\langle {\rm{\Delta }}{{ \mathcal D }}_{b,c}\rangle & \equiv & \langle {{ \mathcal D }}_{b}^{150\times 143}-{{ \mathcal D }}_{b}^{150\times 150}\rangle =0\\ \langle {\rm{\Delta }}{{ \mathcal D }}_{b,a}\rangle & \equiv & \langle {{ \mathcal D }}_{b}^{143\times 143}-{{ \mathcal D }}_{b}^{150\times 150}\rangle =0\end{eqnarray*}$$

and (2) two sets of bandpower ratios

$$\begin{eqnarray*}\langle {\rm{\Delta }}{{ \mathcal D }}_{b,c}/{{ \mathcal D }}_{b}^{150\times 150}\rangle & \equiv & \langle {{ \mathcal D }}_{b}^{150\times 143}/{{ \mathcal D }}_{b}^{150\times 150}\rangle -1=0\\ \langle {\rm{\Delta }}{{ \mathcal D }}_{b,a}/{{ \mathcal D }}_{b}^{150\times 150}\rangle & \equiv & \langle {{ \mathcal D }}_{b}^{143\times 143}/{{ \mathcal D }}_{b}^{150\times 150}\rangle -1=0.\end{eqnarray*}$$

The absolute calibration of the SPT data from George et al. (2015) has a statistical uncertainty of ∼1% at 150 GHz. We expect the bandpower comparison to be significantly more precise than this, so before plotting the bandpower residuals and ratios and before testing the null hypotheses above, we apply a recalibration parameter r_c to bandpowers containing SPT data:

$$\begin{eqnarray}&&{\rm{\Delta }}{{ \mathcal D }}_{b,c}\equiv {r}_{c}{{ \mathcal D }}_{b}^{150\,\times \,143}-{r}_{c}^{2}{{ \mathcal D }}_{b}^{150\,\times \,150}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{{ \mathcal D }}_{b,a}\equiv {{ \mathcal D }}_{b}^{143\,\times \,143}-{r}_{c}^{2}{{ \mathcal D }}_{b}^{150\,\times \,150},\end{eqnarray} \tag{ 12 }$$

and similarly for the bandpower ratios. Because of the precision of the George et al. (2015) calibration, we expect r_c to be very close to 1. We then calculate and minimize ${\chi }^{2}$ as a function of r_c:

$$\begin{eqnarray}&&{\chi }^{2}({r}_{c})={\rm{\Delta }}{{\boldsymbol{D}}}_{{\rm{b}}}^{T}({r}_{c}){{\rm{\Xi }}}^{-1}{\rm{\Delta }}{{\boldsymbol{D}}}_{{\rm{b}}}({r}_{c}),\end{eqnarray} \tag{ 13 }$$

and similarly for the bandpower ratios, where ${\rm{\Delta }}{{\boldsymbol{D}}}_{{\rm{b}}}$ denotes the vector that includes both ${\rm{\Delta }}{{ \mathcal D }}_{b,c}$ and ${\rm{\Delta }}{{ \mathcal D }}_{b,a}$. We fit the two sets of residuals or ratios simultaneously. In principle, we should include the recalibration parameter in an adjustment to the noise contribution to the covariance matrix, but we neglect it for simplicity, with the justification that the correction is very small at less than 1%. In Figure 7, we have included the best-fit recalibration parameter for SPT, r_c, in all three panels.

In the middle panel of Figure 7 we show the bandpower residuals ${\rm{\Delta }}{{ \mathcal D }}_{b,c}$ and ${\rm{\Delta }}{{ \mathcal D }}_{b,a}$ with error bars given by the square root of the diagonal elements of the full covariance matrix. This figure shows that the residual bandpowers are consistent with zero given the errors. In the bottom panels of the same figure we show the bandpower ratios. Similar to the residuals, the error bars of the bandpower ratios are the square root of the diagonal elements of the full covariance matrix. Qualitatively, these results appear to be consistent with our null hypotheses.

The quantitative characterization comes from the ${\chi }^{2}$ value with the best-fit recalibration parameter. The best-fit values of r_c, minimum ${\chi }^{2}$, and probabilities to exceed that ${\chi }^{2}$ are listed in Table 1 for several combinations of data and covariance. The primary results are from the combined residual bandpowers. Using the full 2 × 2 block covariance matrix for the recalibration fit, these results give a best fit of ${r}_{c}\,=1.0087\pm 0.0015$ with ${\chi }^{2}=78.7$. There are 37 ℓ bins in our analysis, so we have 74 data points among the two residual bandpowers and one free parameter. The PTE for ${\chi }^{2}=78.7$ and 73 degrees of freedom is 0.30. We find very similar results from the combined-ratio fit: ${r}_{c}=1.0092\pm 0.0015$, ${\chi }^{2}\,=81.02$, $\mathrm{PTE}=0.24$. Put another way, given the noise properties and the beam uncertainties of the two experiments, 30% (24%) of our simulations have a higher ${\chi }^{2}$ for the bandpower differences (ratios) than we find with the real data. We thus find as our primary result that the SPT and Planck data in the SPT-SZ sky patch are quite consistent with the null hypothesis that there is no systematic offset in the two experiments' measurement of the sky. A by-product of this analysis is that we recalibrate the SPT data with a statistical precision of 0.30% in power (0.15% in temperature) relative to Planck 143 GHz. This is comparable with the absolute calibration uncertainty of the Planck 143 GHz, 0.14% in power (0.07% in temperature; Planck Collaboration et al. 2016a), so we add this uncertainty in quadrature for a final SPT 150 GHz calibration uncertainty of 0.33% in power.

Table 1.  Best-fit Recalibration Parameter, ${\chi }^{2}$, and PTE for the Various Bandpower Comparisons

Comparison	Best rc	${\chi }^{2}$	PTE
Residual, combined, full covariance	1.0087 ± 0.0015	78.7	0.30
Residual, combined, no beam error	1.0117 ± 0.0009	89.8	0.09
$150\times 143\mbox{--}150\times 150$, full covariance	1.0087 ± 0.0016	42.1	0.22
$150\times 143\mbox{--}150\times 150$, no beam error	1.0115 ± 0.0010	50.4	0.06
$143\times 143\mbox{--}150\times 150$, full covariance	1.0076 ± 0.0022	36.8	0.43
$143\times 143\mbox{--}150\times 150$, no beam error	1.0110 ± 0.0013	41.8	0.23
Ratio, combined, full covariance	1.0092 ± 0.0015	81.0	0.24
Ratio, combined, no beam error	1.0120 ± 0.0009	91.0	0.08
$(150\times 143)/(150\times 150)$, full covariance	1.0090 ± 0.0016	43.3	0.19
$(150\times 143)/(150\times 150)$, no beam error	1.0118 ± 0.0010	50.9	0.05
$(143\times 143)/(150\times 150)$, full covariance	1.0082 ± 0.0022	38.1	0.37
$(143\times 143)/(150\times 150)$, no beam error	1.0113 ± 0.0013	42.4	0.21

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In Table 1, we also report the quantities of consistency from the single pair of bandpowers. For example, the PTE is 0.22 for the residual between ${{ \mathcal D }}_{b}^{150\times 143}$ and ${{ \mathcal D }}_{b}^{150\times 150}$ with the full covariance matrix. For the other pair of residual bandpowers, ${{ \mathcal D }}_{b}^{143\times 143}$ and ${{ \mathcal D }}_{b}^{150\times 150}$, the PTE is 0.43 with the full covariance matrix. The results without the beam uncertainties are also listed in Table 1.

As discussed in Section 3.2.2, the white-noise assumption in simulations for ${{ \mathcal D }}_{b}^{143\times 143}$ results in an overestimate of the noise contribution to the covariance matrix, at roughly the 10% level in ${{ \mathcal D }}_{b}$ errors, or the 20% level in variance. If we assumed that the Planck noise was the dominant contribution to the residual and ratio covariance in the 150 × 150 versus 143 × 143 comparisons, we would expect roughly a 20% increase in ${\chi }^{2}$ if we were able to use more realistic noise simulations. The resulting ${\chi }^{2}$ would still correspond to a reasonable PTE ($\gtrsim 0.16$) for the null hypothesis. This would also be true of the combined constraints, particularly because the main constraining power comes from ${{ \mathcal D }}_{b}^{150\times 143}$, which is unaffected by the assumption of white noise in the Planck-only bandpowers.

Because of the largely reduced sample variance contribution to the difference between bandpowers within the same sky coverage (roughly a factor of six in $150\times 143-150\times 150$ compared to 150 × 150), the comparison of ${{ \mathcal D }}_{b}^{150\times 150}$ with the ${{ \mathcal D }}_{b}^{150\times 143}$ and ${{ \mathcal D }}_{b}^{143\times 143}$ bandpowers derived from within the SPT patch provides tighter tests, over a wide range of angular scales, than can be achieved by the comparison of ${{ \mathcal D }}_{b}^{150\times 150}$ with the more precise Planck spectra derived from nearly the full sky. In Figure 8, we compare the uncertainties on the two sets of bandpower residuals ${{ \mathcal D }}_{b}^{143\times 143}-{{ \mathcal D }}_{b}^{150\times 150}$ (blue) and ${{ \mathcal D }}_{b}^{150\times 143}-{{ \mathcal D }}_{b}^{150\times 150}$ (green) to the uncertainties on the SPT-only bandpowers ${{ \mathcal D }}_{b}^{150\times 150}$ (red). In the lower-ℓ region (${\ell }\lt 1800$), the green curve has the lowest error because the sample variance has been greatly reduced by subtracting the SPT bandpowers from the cross-bandpowers, while at higher ℓ the green curve rises owing to the Planck noise contribution. In the most constraining case (${{ \mathcal D }}_{b}^{150\times 143}-{{ \mathcal D }}_{b}^{150\times 150}$), the errors for almost all bins are $\leqslant 10\,\mu {{\rm{K}}}^{2}$, comparable to the uncertainty in the PlanckFS bandpowers. Both sets of bandpower residuals yield very stringent tests on the consistency of the two data sets, and our results show that these data sets are formally consistent in this patch.

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While restricting the bandpower comparison to overlapping sky substantially reduces differential sample variance, it does increase the Planck covariance. In this section, we instead investigate differences between the "in-patch" bandpowers (i.e., the 150 × 150, 150 × 143, and 143 × 143 bandpowers from the 2540 deg² SPT-SZ survey region) and the full-sky Planck 2015 TT high-ℓ unbinned frequency combined bandpowers (Planck Collaboration et al. 2016b), which we refer to as "PlanckFS." In this case, the null hypothesis is that any differences between PlanckFS and the in-patch bandpowers are consistent with expectations given Gaussian statistics and statistical isotropy.

In Figure 9 we show the residuals and ratios for the in-patch to PlanckFS bandpowers. The residual plots show that all three sets of the in-patch bandpowers prefer greater power at $650\lt {\ell }\lt 1200$, indicating that the SPT-SZ patch has greater power at these multipoles than the full-sky average. The ratio plots also suggest a tilt with respect to PlanckFS, although the significance is modest. The slope of the tilt is consistent to 0.5σ between the three cross-spectra, being slightly larger in 150 × 150 and smaller in 143 × 143.

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To quantify the statistical significance of this tilt, we model the ratio of the in-patch and PlanckFS bandpowers as a power law:

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{{\ell }}^{{th},i\times j}}{{D}_{{\ell }}^{\mathrm{PlanckFS}}}=A{\left(\displaystyle \frac{{\ell }}{4000}\right)}^{n}.\end{eqnarray} \tag{ 14 }$$

We assume a Gaussian likelihood with

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }(A,n| {D}^{i\times j},{D}^{\mathrm{PlanckFS}})={{\rm{\Delta }}}_{b}{{\rm{\Sigma }}}_{{bb}^{\prime} }^{-1}{{\rm{\Delta }}}_{b}^{\prime} ,\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{b}=\displaystyle \frac{{D}_{b}^{i\times j}}{{W}_{b{\ell }}^{i\times j}{D}_{{\ell }}^{\mathrm{PlanckFS}}}-{R}_{b}^{i\times j}(A,n),\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{R}_{b}^{i\times j}(A,n)={W}_{b{\ell }}^{i\times j}\left(A{\left(\displaystyle \frac{{\ell }}{4000}\right)}^{n}+{F}_{{\ell }}^{i\times j}\right).\end{eqnarray} \tag{ 17 }$$

Here ${D}_{{\ell }}^{\mathrm{PlanckFS}}$ is the best-fit PlanckFS power spectrum, ${{\rm{\Sigma }}}_{{bb}^{\prime} }=\langle {{\rm{\Delta }}}_{b}{{\rm{\Delta }}}_{b^{\prime} }\rangle$ (only the SPT beam uncertainty is included and cross-correlations between ${D}_{b}^{i\times j}$ and the ${D}_{{\ell }}^{\mathrm{PlanckFS}}$ are negligible), and ${F}_{{\ell }}^{i\times j}$ is the foreground model adopted from S13 with the frequency dependence from George et al. (2015) included. There are three foreground amplitude parameters included in ${F}_{{\ell }}^{i\times j}$ to account for the SZ, Poisson-point-source, and clustered CIB uncertainties. The best-fit recalibration from Section 4.3 has been applied to the in-patch SPT-only and SPT-Planck bandpowers. The best-fit values for A and n are reported in Figure 9. We find only marginal evidence for a tilt between PlanckFS and any of the three cross-spectra; the most significant tilt is for 150 × 150 and is $1.5\sigma$ away from zero. We conclude that the tilts we see in the observed spectra are roughly consistent with the expectation based on Gaussian statistics and the assumption of statistical isotropy—i.e., they are roughly consistent with n = 0.

To determine the expected statistical properties of the differences in the best-fit values of A and n, we construct a covariance matrix (${\boldsymbol{C}}$) from the set of best-fit values of A and n in each of the 400 simulations of Section 3.2. We then calculate ${\chi }^{2}$ for these differences as

$$\begin{eqnarray}&&{\chi }^{2}={({\boldsymbol{\Theta }}-\bar{{\boldsymbol{\Theta }}})}^{\top }{{\boldsymbol{C}}}^{-1}({\boldsymbol{\Theta }}-\bar{{\boldsymbol{\Theta }}}),\end{eqnarray} \tag{ 18 }$$

where ${\boldsymbol{\Theta }}$ is the vector of parameter differences and $\bar{{\boldsymbol{\Theta }}}$ is the mean simulation difference, which is consistent with zero. The breakdown of PTEs for the various pair differences is shown in Table 2. As can be seen, the most extreme PTE is 0.85 and the lowest is 0.41. We conclude that the observed tilts in the three cross-spectra are completely consistent with each other.

Table 2.  Power-law Parameter PTEs

	${\chi }^{2}$	PTE
$150\times 150-150\times 143$	0.33	0.41
$150\times 150-143\times 143$	1.79	0.85
$150\times 143-143\times 143$	1.18	0.55

Note. The ${\chi }^{2}$ and PTE for the null hypothesis that the tilt relative to PlanckFS is the same in the in-patch 150 × 150, 150 × 143, and 143 × 143 bandpowers.

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All our tests are consistent with the following explanation for the tilts we observe in the in-patch spectra relative to the best-fit Planck full-sky spectrum: they are driven by a sample variance fluctuation away from the full-sky average, the magnitude of which is roughly consistent with expectations under the assumption of statistical isotropy and our noise model.

All published power spectra from the SPT Collaboration have been calculated using some variant of the cross-spectrum pseudo-${C}_{{\ell }}$ pipeline used in this work. Extensive checks have been performed on this pipeline (see, e.g., Section 4.2 of Story et al. 2013), demonstrating that the correct input spectrum is recovered from simulated data, even when that spectrum differs from the spectrum assumed in calculating the filter transfer function. If, however, some aspect of the pipeline were inducing a bias in the estimated power spectra (through some mechanism that has escaped all pipeline tests), this bias would affect both the SPT-SZ and Planck data used in this work (because we have mock-observed the Planck data and analyzed it with the SPT pipeline). If the bias on the two data sets were comparable, it would then divide or subtract out in the bandpower comparison, and we would (wrongly) conclude that there was no issue with either data set.

To test this scenario, we have analyzed the in-patch Planck data using an alternate pipeline. Specifically, we created a HEALPix version of the SPT-SZ sky-patch and point-source mask (stitched together from the individual-field masks), and we handed this mask and the half-mission Planck 143 GHz maps to the PolSpice⁴⁰ (Szapudi et al. 2001; Chon et al. 2004) code, which is designed to estimate the power spectra of masked full-sky maps and properly account for the masking. We binned the ${\rm{\Delta }}{\ell }=1$ PolSpice output into the ${\rm{\Delta }}{\ell }=50$ bins used in the SPT pipeline using the bandpower window functions calculated for the 143 × 143 "scanned, filtered" Planck bandpowers. We then calculated the ratio of the "unscanned, unfiltered" (PolSpice) bandpowers to the scanned, filtered ones and found that they agree to better than 3% in every individual bin in which there is appreciable signal-to-noise ratio in the 143 × 143 spectrum, with an overall ratio of 1.0028 ± 0.0050 over the range $600\lt {\ell }\lt 1800$ and no evidence of a trend with ℓ. We have also redone the tilt calculation in Section 4.3 using the unscanned, unfiltered bandpowers and found results consistent with what we found with the scanned, filtered bandpowers (within a fraction of a sigma). We thus conclude that our fundamental results are not an artifact of the SPT analysis method.