A Comparison of Maps and Power Spectra Determined from South Pole Telescope and Planck Data
Abstract
We study the consistency of 150 GHz data from the South Pole Telescope (SPT) and 143 GHz data from the Planck satellite over the patch of sky covered by the SPTSZ survey. We first visually compare the maps and find that the residuals appear consistent with noise after accounting for differences in angular resolution and filtering. We then calculate (1) the crossspectrum between two independent halves of SPT data, (2) the crossspectrum between two independent halves of Planck data, and (3) the crossspectrum between SPT and Planck data. We find the three crossspectra are wellfit (PTE = 0.30) by the null hypothesis in which both experiments have measured the same sky map up to a single free calibration parameter—i.e., we find no evidence for systematic errors in either data set. As a byproduct, we improve the precision of the SPT calibration by nearly an order of magnitude, from 2.6% to 0.3% in power. Finally, we compare all three crossspectra to the fullsky Planck power spectrum and find marginal evidence for differences between the power spectra from the SPTSZ footprint and the full sky. We model these differences as a power law in spherical harmonic multipole number. The bestfit value of this tilt is consistent among the three crossspectra in the SPTSZ footprint, implying that the source of this tilt is a sample variance fluctuation in the SPTSZ region relative to the full sky. The consistency of cosmological parameters derived from these datasets is discussed in a companion paper.
Subject headings:
cosmic background radiation1. Introduction
One of the most remarkable results of modern cosmology is that a simple sixparameter model, usually referred to as the Lambda Cold Dark Matter () model, can fit the full range of cosmological observations. With the precision of cosmological observables now reaching the level of a few percent, however, several small discrepancies in the inferred parameter values are attracting attention.
These discrepancies show up in three places. The first is in the inferred parameter values from the CMB compared to some observations of the local universe. For example, the amplitude of local density fluctuations, , that is measured from observations of large scale structure appears lower than the value implied by cosmic microwave background (CMB) measurements (Planck Collaboration et al., 2016f). There is also some tension between direct measurements of the Hubble parameter and the value derived from CMB measurements (Planck Collaboration et al., 2014c, 2016e; Riess et al., 2011, 2016).
Second, there are mild (12) discrepancies between parameter values derived from observations of the CMB by different experiments. In particular, the bestfit parameters of the model given Planck satellite measurements of CMB temperature and polarization power spectra (Planck Collaboration et al., 2016c) are somewhat different from those derived from earlier CMB data, whether from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite (Hinshaw et al., 2013), from the combination of WMAP data and data from the South Pole Telescope (SPT, Hou et al. 2014), or from WMAP + SPT plus data from the Atacama Cosmology Telescope (ACT, Calabrese et al. 2013).
Finally, work has been done on the internal consistency of CDM model parameter values from different subsets of the Planck data. Addison et al. (2016) recently pointed out that the matter density inferred from Planck data at is 2.5 discrepant from that inferred from Planck data at . In response, Planck Collaboration et al. (2017) show that after correcting for certain approximations in the Addison et al. (2016) analysis, and taking into account the fact that matter density had been singled out as the most discrepant parameter, the global discrepancy is only 1.6 .
Taken together, these lowlevel discrepancies have led some to speculate that we are seeing evidence of a potential failure of the model (Wyman et al., 2014; Battye & Moss, 2014), systematic errors in the analysis of the lowredshift probes (Efstathiou et al., 2014), or systematic errors in the Planck data (Planck Collaboration et al., 2017). The analysis presented in this paper is motivated by the fact that it is the high temperature data from Planck that are driving the parameter shifts of interest (Addison et al., 2016; Planck Collaboration et al., 2017). In terms of the measurement uncertainties, data from the 2540squaredegree SPTSZ survey yield constraints within a factor of 2 of the Planck constraints at and better than Planck at (Story et al., 2013, hereafter S13). The SPTSZ data are thus a logical choice for consistency checks of the high Planck data.
In this work, we present the first comparison between Planck and SPT data over the same patch of sky. Checks have previously been performed at the powerspectrum and cosmologicalparameter level, for instance in Planck Collaboration et al. (2014c) and Planck Collaboration et al. (2016e), and the two data sets have been shown to be roughly consistent, but again with some lowlevel discrepancies. The strength of powerspectrum and parameter comparisons using the full datasets are limited by the sample variance of the SPT data at lower and Planck noise at higher . By limiting the comparison to CMB modes measured by both experiments, we can greatly reduce the sample variance to sharpen the consistency tests between the two data sets.
Here we compare the SPTSZ data in the 150 GHz band and Planck fullmission data in the 143 GHz band, restricted to the SPTSZ observing region. We calculate angular crossspectra of the SPT and Planck maps (), and, for comparison, the crossspectrum of one half of SPT data with the other half (), and the crossspectrum of one half of Planck data with the other half (). We calculate the difference between the SPT spectrum and the other two, as well as the ratio of the SPT spectrum to the others. Using simulated observations of mock skies (including realistic noise for both experiments), we calculate the expected uncertainty in these differences and ratios, and we use this to calculate the and probability to exceed this under the null hypothesis that there are no systematic biases between the experiments. This investigation is similar to that performed by Louis et al. (2014) on ACT and Planck data over 592 deg of sky and two observing bands (143/148 GHz and 217/218 GHz), the conclusion of which was that ACT and Planck measured statistically consistent CMB fluctuations over that patch of sky.
This paper is organized as follows. In Section 2, we discuss the SPT 150 GHz and Planck 143 GHz temperature maps in the 2540 patch of sky that constitutes the whole of the SPTSZ survey region. In Section 3 we present how power spectra are calculated from the maps and the simulations generated to debias the data and build the covariance matrix. In Section 4 we compare the power spectra and use simulations to test the null hypothesis that the two experiments are measuring the same sky, and we discuss these results in the context of comparisons between SPT and fullsky Planck data. We present our conclusions in Section 5.
2. Data
The main goal of this work is to compare maps and power spectra from the SPT 150 GHz and Planck 143 GHz data sets within the SPTSZ survey area. As an overview of the two datasets, Figure 1 shows a sky map of Planck HFI 143 GHz fullmission data, with the SPTSZ survey area outlined by a solid black curve. In this section, we discuss details of the mapmaking and instrumental characteristics of each experiment.
2.1. Spt
The SPT is a 10meter telescope located at the AmundsenScott South Pole station. From 2008 to 2011, the first camera on the SPT, a threeband bolometer array known as the SPTSZ camera, was used to conduct a survey of 2500 deg of the Southern sky with low Galactic dust contamination, referred to as the SPTSZ survey. As shown in Figure 1, the survey area is a contiguous region extending from to in right ascension (R.A.) and from to in declination. The survey was conducted by observing a series of 19 areas, ranging in area from roughly 100 to 300 square degrees, 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 SPTSZ.
There were approximately 200 observations of each field, with each individual observation taking roughly 2 hours. Estimates of the primary CMB temperature power spectrum from this survey, and 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 singleobservation map step. We briefly review that part of the analysis here and refer the reader to K11 and S13 for more details.
To process timeordered 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 highpass filter and the subtraction of a common mode across all detectors in each of the six 160element module in the focal plane. The highpass filter cuts off signals below a temporal frequency corresponding to an angular frequency along the scan direction of . The commonmode subtraction acts as an isotropic highpass filter with a cutoff at roughly . In both of these filtering steps, bright point sources ( mJy at 150 GHz) are masked to avoid large filtering artifacts.
The TOD from individual SPT detectors are then binned into maps using inversevariance weighting, i.e., TOD samples corresponding to times when an individual detector was pointed at a particular pixel are averaged together using using inversevariance weighting, then assigned to that pixel. Maps are made using the oblique Lambert equalarea azimuthal projection (Snyder, 1987) with 1arcmin square pixels.
The SPT maps are simply binnedandaveraged 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 and its uncertainty, and 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 wellapproximated by a 1.2arcmin 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 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 region RCW38. For details of this calibration, see Schaffer et al. (2011). The fulldepth maps are then compared to CMB satellite data to provide the overall absolute calibration. In K11 and S13, the power spectrum of the fulldepth maps was compared to the fullsky WMAP estimate of the CMB power spectrum in the multipole range . For this work, we initially use the calibration determined in George et al. (2015), using the comparison of SPT power spectra to the fullsky Planck power spectrum in the multipole range ; however, the mapbased comparison to Planck undertaken here ends up providing a significantly more precise absolute calibration, as detailed in Section 4.2.
2.2. Planck
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 fullmission, halfmission1,
and halfmission2 maps downloaded from the NASA/IPAC Infrared Science Archive
(IRSA).
In this work, we use a crossspectrum pipeline similar to that used in K11 and S13 to compare SPT and Planck data on the SPTSZ 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 fullmission or halfmission 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 highpass filter is applied to the Planck mock TOD to simply remove the signals on scales with . Our simulations show that applying this simple highpass 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 highpass 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, 2016a). The beam window function for individual frequency detector sets has been released by the Planck collaboration as part of the “Reduced Instrument Model” (RIMO).
3. Power Spectrum
We now turn to the methodology for calculating unbiased crossspectra of SPT 150 GHz maps and Planck 143 GHz maps. We introduce the power spectrum estimator in Section 3.1. Simulations are used in several places while estimating the power spectra; we describe how these simulations are generated in Section 3.2. We discuss how these simulations are used to estimate the SPT filter transfer function in Section 3.3 and to calculate the noise and samplevariance parts of the power spectrum covariance in Section 3.4.1. We describe how beam uncertainty is incorporated into the analysis in Section 3.4.2. Finally, we discuss how to calculate the window functions needed to compare the binned crossspectrum estimates—or “bandpowers”—to a theory spectrum in Section 3.4.3.
3.1. Power Spectrum Estimator
We use an estimator similar to that used in K11 and S13 to calculate the various crossspectra in this work.
Each crossspectrum is calculated by correlating maps made from different observations of the same field—either correlating fulldepth SPT maps with fulldepth Planck maps or correlating two halfdepth maps from the same experiment.
In the latter case, there is a noise penalty relative to K11 or S13 who used 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 zeropad each map before calculating its 2dimensional Fourier transform, .
(1) 
For the SPTonly power spectrum, and are the two SPT halfsurvey maps; for the SPT and Planck crossspectrum, and are the full mission maps of each experiments; and for the Planckonly 143 power spectrum, and are the two Planck halfmission maps. in the above equation represents the 2dimensional 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 S13derived weighting for all bandpowers to minimize the differential sample variance between crossspectra.
Since the input maps are biased estimates of the true sky, due to effects as the application of a mask to the maps, the raw bandpowers are a biased estimate of the true bandpowers, . The biased and unbiased estimates are related by
(2) 
where
(3) 
is the binning operator and is its reciprocal, is the modecoupling matrix, is the filter transfer function which accounts for the signal suppressed by TOD filtering, and 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 fieldbyfield 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.
3.2. Simulations
We use simulations both in the calculation of the bandpowers and to characterize the degree of consistency between the SPT and Planck crossspectra over the SPTSZ 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 (, , and ).
Sky Signals
Our simulations include the following components: 1) a gravitationally lensed CMB signal, 2) thermal and kinematic SunyaevZel’dovich (SZ) signals, 3) the cosmic infrared background (CIB) signal, and 4) emission from radio galaxies. We generate 400 realizations of the lensed CMB with LensPix (Lewis, 2005), based on the bestfit model from Planck 2015 (Planck Collaboration et al., 2016e). Modes are generated out to , well above the angular multipoles, , used in this comparison. The maps are stored in HEALPix (Górski et al., 2005) format with resolution parameter .
Unlike S13 which used Gaussian realizations for all extragalactic foregrounds, here we generate realizations of individual sources for the bright CIB galaxies () and all radio galaxies (up to the flux cut at 50 mJy). The bright CIB galaxies are drawn from the modeled of Cai et al. (2013), while the radio galaxy is taken from De Zotti et al. (2005). In both cases, the amplitudes of the distribution are calibrated by the actual observations of Mocanu et al. (2013) at 150 GHz.
The other extragalactic foregrounds, the thermal and kinematic SZ signals and the lowflux CIB, are treated as Gaussian. The shapes of the thermal and kinematic SZ angular power spectra are taken from the Shaw et al. (2010) and Shaw et al. (2012) models respectively, with the amplitudes set to the median values from George et al. (2015), and . We similarly draw upon the median values and templates from George et al. (2015) for the CIB terms. The clustered CIB spectrum is taken to follow with an amplitude . The shotnoise or “Poisson” CIB power from galaxies dimmer than is taken to be .
Each component is scaled appropriately to the effective frequency of the SPT or Planck maps and then coadded together to create the final sky realization.
Planck Noise Simulations
The Planck collaboration released the 8th Full Focal Plane (FFP8) simulation in the 2015 data release (Planck Collaboration et al., 2016d). There are 1000 fullmission FFP8 noise simulations available at NERSC.
The FFP8 noise simulations are only available for the fullmission data, and thus cannot be used for the Planck halfmission crossspectrum. To create unbiased Planck 143 GHz bandpowers within the SPT area, we also generate halfmission noise simulations to characterize the noise contribution to the PlanckPlanck () bandpowers as presented in Section 4.2. The halfmission noise simulations are based on the pixelspace noise variance released by the Planck collaboration along with the halfmission observation maps. We approximate the Planck map noise as Gaussian and uncorrelated between pixels (“white”) for the halfmission noise simulations.
To test this whitenoise approximation, we compare the noise power spectrum over the SPT survey area from 100 of these Gaussian whitenoise realizations to 100 FFP8 fullmission noise simulations. The average power ratio is plotted in Figure 2. Clearly, the whitenoise realizations overestimate the noise power for , i.e. most angular multipoles of interest. This implies that the Planck noise contribution is overestimated in our Planckonly bandpowers, at a maximum level of roughly 15% and a mean level of across the range of interest. We discuss the possible impact of this overestimate on our results in Section 4.2 and find that our main conclusions would likely be unchanged had we used more realistic noise simulations for the Planckonly bandpowers.
SPT Noise Realizations
SPT noise realizations are created directly from the data. For each individual SPT observation of each SPTSZ survey field, we create a residual map which is the difference between a map made from all leftgoing telescope scans and a map made from all rightgoing telescope scans. There are approximately 200 jackknife maps for each field (the number of individual 2hour observations). We use these maps to create the noise part of the simulated SPT observations. For each field, we multiply +1 or 1 randomly to the residual maps and coadd them together to form one noise realization. Using this method, we create 400 noise realizations.
With the same method, we also create SPT noise realizations of the first and second half sets of observations by coadding the residual maps of half of the observations within one field. Similar to the halfmission noise simulations for Planck we use the SPT half noise simulations to characterize the noise contribution to the SPTonly bandpowers.
3.3. Filtering Transfer Function
The noisefree simulated observations are used to calculate , the filtering transfer function, and , the twodimensional weight in the powerspectrum 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 Eq.6 and Eq.7 in Keisler et al. (2011) to include the Planck beam and pixel window function for the SPTPlanck crossspectrum and Planckonly spectrum.
3.4. Consistency Metrics
We have three sets of unbiased bandpowers derived from SPT and Planck data, , , and . 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 datasets. This section presents the metrics we use for these consistency tests.
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, , is calculated simply as
(4) 
where is the mean over all 400 simulations for crossbandpowers . The simulated crossbandpowers are derived from crossspectra of simulated SPT maps including realizations of SPT fullobservation noise and simulated Planck maps of the same underlying sky signal including Planck FFP8 noise simulations. For , the simulated bandpowers are calculated by the cross correlation of the two sets of simulated SPT maps with halfobservation noise realizations. Similarly, the simulated bandpowers are obtained using two sets of simulated Planck maps including Planck halfmission whitenoise realizations. The full bandpower covariance matrix is then obtained by combining the individualfield covariance matrices with the square of the area weighting used to combine the bandpowers themseeves.
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, (where ), the noiseplussamplevariance part of the covariance is easily estimated from the 400 simulations:
(5) 
For the ratios relative to the bandpowers, the covariance can be expressed as:
(6) 
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 noiseplussamplevariance 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 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, , and propagate this linearly to the bandpower space according to:
(7) 
where , is the bandpower window function, and 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:
(8)  
where and the sum is taken over all eigenmodes of the beam covariance matrices.
For the bandpower ratios, the beam covariance can be written as:
(9)  
The total bandpower covariance for a consistency test is then the sum of the noiseplussamplevariance term from the last section and the appropriate beam covariance term above.
Bandpower Window Function Correction
To characterize the level of consistency between the three sets of bandpowers (, , and ), we choose one set, , as the fiducial which is subtracted from or divided into the other two sets (see Section 4.2). However, the residual bandpowers and bandpower ratios are biased due 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 and . In both of these bins, relative to the window functions, the and 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.
Before comparing the bandpower sets, we need to correct for this bias. The correction is calculated as follows
(10) 
where , or . The Planck 2015 bestfit cosmology plus the bestfit extragalactic foregrounds are used as the fiducial model. We subtract this correction term from the and bandpowers to remove the bias caused by the bandpower window function differences. In Figure 4, the upper panel shows the shape of the correction and the lower panel shows the ratio between and the corresponding . For , yields a roughly flat correction to the bandpower; while for , the correction is more important at higher multipoles, up to at . The window function correction is clearly critical for the consistency analysis between these three sets of bandpowers.
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 . The distribution is obtained by randomly sampling the cosmology and foreground models from a Markov chain based on WMAP9+SPT data. The variation of in Eq. 10 is illustrated in the upper panel of Figure 4 with the error bars indicating the variation of the corrections. Within the band range of interest, this variation level is less than 1% of the bandpower standard error, we therefore ignore this very small uncertainty in our analysis.
4. Results
In this section, we present the results of the comparison between SPT 150 GHz data and Planck 143 GHz data. We first present a qualitative maplevel comparison of the two data sets in a common sky area. We then present the major result of the paper: the bandpower comparison among three sets of bandpowers calculated on the 2540 SPTSZ sky patch: , , and . Setting as the fiducial and subtracting it from the other two, we use the residual as the quantity to characterize the level of consistency between SPT 150 GHz and Planck 143 GHz data. As a byproduct, for the first time we recalibrate the SPT data to the Planck data on the same patch of sky.
We also characterize the ratios between these sets of bandpowers. Note that the residual and ratio tests are not intended to be independent checks, as they contain nearly the same information presented in different ways. The reason we use both metrics is that the systematic errors most likely to cause differences between SPT and Planck data (such as unmodeled foreground residuals or beam systematics) fall into two broad categories: additive or multiplicative systematics. Additive systematics will show up most obviously in the residual bandpower test, while multiplicative systematics will show up most obviously in a bandpower ratio test.
Finally, we compare each set of 2540 bandpowers to the fullsky Planck 143 GHz bandpowers. Obviously such bandpower comparisons are not distinct from the map comparison. Rather, the residuals and the ratios of these sets of bandpowers are the quantitative version of the byeye map comparison.
4.1. Maplevel Comparison
The SPT and Planck maps within the same patch are presented in Figure 5 for visual comparison. The upper panels show the filtered SPT map described in Section 2.1 on the left and the projected Planck fullmission 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 degreescale CMB anisotropy filtered out of the SPT data, and the SPT map shows more smallscale structure due to its higher angular resolution. In the lower panels of Figure 5, we show SPT and Planck maps of the same spatial modes. The lowerleft panel shows the SPT map from the upperleft panel convolved with the difference between the SPT 150 GHz and Planck 143 GHz beams; the lowerright 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 lowerleft and lowerright 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.
To make this result more quantitative, we could match the beam and filtering between these two data sets over the full SPTSZ 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 crossspectrum formalism outlined in Section 3, we calculate the Planck power spectrum, the SPT power spectrum, and the SPTPlanck crossspectrum, and we calculate the of the null hypothesis that these three spectra are measuring the same power. This calculation does not completely eliminate sample variance, as the perfectly modematched differencemap power spectrum would, but it strongly reduces it (see Figure 8).
4.2. Bandpower Comparison and Recalibration
In this section, we present the comparison between the three sets of windowfunctioncorrected bandpowers, which we denote as , , and . We first show all three sets of bandpowers in the upper 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 , 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 (though the bandpowers have slightly smaller error bars in this range because the modes lost in the SPT TOD filtering process result in slightly increased sample variance). Noise variance begins to dominate the error bars at and the error bars at , while the error bars are samplevariancedominated over the entire range plotted.
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, is chosen as the fiducial bandpower set. These comparison plots and the and the probability to exceed (PTE) statistics calculated below are thus testing the following set of null hypotheses: 1) two sets of bandpower residuals
and 2) two sets of bandpower ratios
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 to bandpowers containing SPT data:
(11)  
(12) 
and similarly for the bandpower ratios. Because of the precision of the George et al. (2015) calibration, we expect to be very close to 1. We then calculate and minimize as a function of :
(13) 
and similarly for the bandpower ratios, where denotes the vector that includes both and . 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 bestfit recalibration parameter for SPT, , in all three panels.
In the middle panel of Figure 7 we show the bandpower residuals and with error bars given by the square root of the diagonal elements of the full covariance matrix. This figure shows the residual bandpowers are consistent with zero given the errors. In the lower 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. Qualitively, these results appear to be consistent with our null hypotheses.
Comparison  best  PTE  

Residual, combined, full covariance  78.7  0.30  
Residual, combined, no beam error  89.8  0.09  
, full covariance  42.1  0.22  
, no beam error  50.4  0.06  
, full covariance  36.8  0.43  
, no beam error  41.8  0.23  
Ratio, combined, full covariance  81.0  0.24  
Ratio, combined, no beam error  91.0  0.08  
, full covariance  43.3  0.19  
, no beam error  50.9  0.05  
, full covariance  38.1  0.37  
, no beam error  42.4  0.21 
The quantitative characterization comes from the value with the bestfit recalibration parameter. The bestfit values of , minimum , and probabilities to exceed that are listed in Table 1 for several combinations of data and covariance. The primary results are from the combined residual bandpowers. Using the full block covariance matrix for the recalibration fit, these results give a bestfit of with . There are 37 bins in our analysis, so we have 74 data points among the two residual bandpowers and one free parameter. The PTE for and 73 degrees of freedom is 0.30. We find very similar results from the combinedratio fit: , , . Put another way, given the noise properties and the beam uncertainties of the two experiments, 30% (24%) of our simulations have a higher 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 SPTSZ sky patch are quite consistent with the null hypothesis that there is no systematic offset in the two experiments’ measurement of the sky. A byproduct 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. 2016b), so we add this uncertainty in quadrature for a final SPT 150 GHz calibration uncertainty of 0.33% in power.
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 and with the full covariance matrix. For the other pair of residual bandpowers, and , 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 whitenoise assumption in simulations for results in an overestimate of the noise contribution to the covariance matrix, at roughly the 10% level in errors, or the 20% level in variance. If we assumed the Planck noise was the dominant contribution to the residual and ratio covariance in the vs. comparisons, we would expect roughly a 20% increase in if we were able to use more realistic noise simulations. The resulting would still correspond to a reasonable PTE () for the null hypothesis. This would also be true of the combined constraints, particularly because the main constraining power comes from , which is unaffected by the assumption of white noise in the Planckonly 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 compared to ), the comparison of with the and bandpowers derived from within the SPT patch, provide tighter tests, over a wide range of angular scales, than can be achieved by the comparison of 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 (blue), and (green) to the uncertainties on the SPTonly bandpowers (red). In the lower region (), 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 the green curve rises due to the Planck noise contribution. In the most constraining case (), the errors for almost all bins are , comparable to the uncertainty in the PlanckFS bandpowers. Both sets of bandpower residuals yield very stringent tests on the consistency of the two datasets, and our results show that these datasets are formally consistent in this patch.
4.3. Comparison to the Fullsky Planck 2015 TT High Bandpowers
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 “inpatch” bandpowers (i.e., the 150 150, 150 143, and 143 143 bandpowers from the 2540 deg SPTSZ survey region) and the fullsky Planck 2015 TT high unbinned frequency combined bandpowers (Planck Collaboration et al., 2016e), which we refer to as “PlanckFS.” In this case, the null hypothesis is that any differences between PlanckFS and the inpatch bandpowers are consistent with expectations given Gaussian statistics and statistical isotropy.
In Figure 9 we show the residuals and ratios for the inpatch to PlanckFS bandpowers. The residual plots show all three sets of the inpatch bandpowers prefer greater power at , indicating the SPTSZ patch has greater power at these multipoles than the fullsky 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 between the three crossspectra, being slightly larger in and smaller in .
To quantify the statistical significance of this tilt we model the ratio of the inpatch and PlanckFS bandpowers as a power law:
(14) 
We assume a Gaussian likelihood with
(15)  
(16)  
(17) 
Here is the bestfit PlanckFS power spectrum, (only the SPT beam uncertainty is included and crosscorrelations between and the are negligible), and 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 to account for the SZ, Poissonpointsource, and clustered CIB uncertainties. The bestfit recalibration from Section 4.3 has been applied to the inpatch SPTonly and SPTPlanck bandpowers. The bestfit values for and are reported in Figure 9. We find only marginal evidence for a tilt between PlanckFS and any of the three crossspectra; the most significant tilt is for and is 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 .
PTE  

0.33  0.41  
1.79  0.85  
1.18  0.55 

The and PTE for the null hypothesis that the
tilt relative to PlanckFS is the same in the inpatch
, , and bandpowers.
To determine the expected statistical properties of the differences in the bestfit values of and we construct a covariance matrix () from the set of bestfit values of and in each of the 400 simulations of Section 3.2. We then calculate for these differences as
(18) 
where is the vector of parameter differences and 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 crossspectra are completely consistent with each other.
All our tests are consistent with the following explanation for the tilts we observe in the inpatch spectra relative to the bestfit Planck fullsky spectrum: they are driven by a samplevariance fluctuation away from the fullsky average, the magnitude of which is roughly consistent with expectations under the assumption of statistical isotropy and our noise model.
4.4. Pipeline Checks
All published power spectra from the SPT collaboration have been calculated using some variant of the crossspectrum pseudo 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 SPTSZ and Planck data used in this work (because we have mockobserved 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 there was no issue with either data set.
To test this scenario, we have analyzed the inpatch Planck data using an alternate pipeline.
Specifically, we created a HEALPix version of the SPTSZ skypatch and pointsource
mask (stitched together from the individualfield masks), and we
handed this mask and the halfmission Planck 143 GHz maps to the
PolSpice
5. Conclusions
In this paper, we have compared 150 GHz SPT data and 143 GHz Planck data in the same region of the sky, namely the 2540 SPTSZ survey footprint. We have performed a visual comparison of maps constructed from the two data sets and found the difference between the two maps to be visually consistent with noise, once they have been filtered to display the same angular modes.
We then performed a quantitative analysis of the consistency of the maps, relying primarily on a comparison of the crossspectrum of two halves of the SPT data with the SPT Planck cross spectrum. We also compared the SPT SPT spectrum with the crossspectrum of two halves of the Planck data. These comparisons were made using differences between and ratios of two sets of binned power spectra, or bandpowers, at a time, always using the SPT SPT bandpowers as the fiducial set. To test the null hypothesis that the bandpower differences (after recalibrating the SPT data) are consistent with zero—or that the ratios are consistent with unity—we created a suite of 400 simulations of the signal and noise properties of the SPT and Planck maps, including signal contributions from the CMB and extragalactic foregrounds. We found our most stringent test, based on the expected variance of the differences, to be the comparison of the SPT SPT spectrum with the SPTPlanck cross spectrum. Forming a quantity from these bandpower differences and the simulationbased covariance matrix, we have found a value that is exceeded by 22% of the analogous values derived from the simulated data, i.e., corresponding to the PTE value 0.22. When we add the residuals between the SPT SPT and Planck Planck crossspectra, we find a PTE of 30%. All other tests result in similarly unremarkable PTEs. We find no evidence of a failure of our null model; i.e., we see no evidence for systematic errors or under or overestimate of statistical errors.
We have also compared the three sets of bandpowers from the 2540 deg SPTSZ survey region to the fullsky Planck 143 GHz power spectrum. Relative to the Planck fullsky spectrum, we have found a hint for a tilt in the inpatch bandpowers. For all three sets of inpatch bandpowers, the amplitude of the tilts we have obtained are consistent with each other and roughly consistent with expected noise and sample variance fluctuations.
This work shows that the SPT 150 GHz and Planck 143 GHz data are in very good agreement with each other within the 2540 deg SPTSZ survey area. In a companion paper (Aylor et al., 2017), we extend this comparison to the cosmological parameters that can be derived from these bandpowers.
Footnotes
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of California, Davis, CA, USA 95616
 affiliation: Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL 60510
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA 60439
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA 60439
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Enrico Fermi Institute, University of Chicago, Chicago, IL, USA 60637
 affiliation: High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA 60439
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025
 affiliation: Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: California Institute of Technology, Pasadena, CA, USA 91125
 affiliation: Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
 affiliation: Canadian Institute for Advanced Research, CIFAR Program in Cosmology and Gravity, Toronto, ON, M5G 1Z8, Canada
 affiliation: Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309
 affiliation: Department of Physics, University of California, Davis, CA, USA 95616
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: European Southern Observatory, KarlSchwarzschildStraße 2, 85748 Garching, Germany
 affiliation: Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309
 affiliation: Department of Physics, University of Colorado, Boulder, CO, 80309
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
 affiliation: Canadian Institute for Advanced Research, CIFAR Program in Cosmology and Gravity, Toronto, ON, M5G 1Z8, Canada
 affiliation: Astronomy Department, University of Illinois at UrbanaChampaign, 1002 W. Green Street, Urbana, IL 61801, USA
 affiliation: Department of Physics, University of Illinois UrbanaChampaign, 1110 W. Green Street, Urbana, IL 61801, USA
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: University of Chicago, Chicago, IL, USA 60637
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305
 affiliation: Department of Physics, University of California, Davis, CA, USA 95616
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA 94720
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: University of Chicago, Chicago, IL, USA 60637
 affiliation: Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721
 affiliation: Department of Physics, University of Michigan, Ann Arbor, MI, USA 48109
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Enrico Fermi Institute, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of California, Davis, CA, USA 95616
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Faculty of Physics, LudwigMaximiliansUniversität, 81679 München, Germany
 affiliation: Excellence Cluster Universe, 85748 Garching, Germany
 affiliation: MaxPlanckInstitut für extraterrestrische Physik, 85748 Garching, Germany
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
 affiliation: Department of Physics and McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Minnesota, Minneapolis, MN, USA 55455
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
 affiliation: Physics Department, Center for Education and Research in Cosmology and Astrophysics, Case Western Reserve University,Cleveland, OH, USA 44106
 affiliation: Physics Department, Center for Education and Research in Cosmology and Astrophysics, Case Western Reserve University,Cleveland, OH, USA 44106
 affiliation: Center for Astrophysics and Space Astronomy, Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO, 80309
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Enrico Fermi Institute, University of Chicago, Chicago, IL, USA 60637
 affiliation: Liberal Arts Department, School of the Art Institute of Chicago, Chicago, IL, USA 60603
 affiliation: Department of Physics, University of California, Berkeley, CA, USA 94720
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Physics Department, Center for Education and Research in Cosmology and Astrophysics, Case Western Reserve University,Cleveland, OH, USA 44106
 affiliation: Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA
 affiliation: HarvardSmithsonian Center for Astrophysics, Cambridge, MA, USA 02138
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Kavli Institute for Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA 94305
 affiliation: Dept. of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305
 affiliation: Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
 affiliation: Department of Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON, M5S 3H4, Canada
 affiliation: Astronomy Department, University of Illinois at UrbanaChampaign, 1002 W. Green Street, Urbana, IL 61801, USA
 affiliation: Department of Physics, University of Illinois UrbanaChampaign, 1110 W. Green Street, Urbana, IL 61801, USA
 affiliation: Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
 affiliation: Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637
 slugcomment: 2018ApJ…853….3H
 http://irsa.ipac.caltech.edu/data/Planck/release_2/allskymaps/
 http://irsa.ipac.caltech.edu/data/Planck/release_2/ancillarydata/
 When calculating angular power spectra in this work, we use the flatsky approximation, in which we replace spherical harmonic transforms with twodimensional Fourier transforms.
 In this work, we use the same point source masks used in S13 but slightly different apodization masks; we have confirmed using simulations that any effect of the different masking on power spectra or cosmological parameters is negligible for this analysis.
 https://crd.lbl.gov/departments/computationalscience/c3/c3research/cosmicmicrowavebackground/cmbdataatnersc/
 http://www2.iap.fr/users/hivon/software/PolSpice
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