A Comparison of Cosmological Parameters Determined from CMB Temperature Power Spectra from the South Pole Telescope and the Planck Satellite

Several of our tests in this work make use of the publicly available Planck 2015 baseline high-ℓ temperature and low-ℓ temperature and polarization bandpowers. These are optimally combined multifrequency bandpowers, and we refer to them as the Planck Full Sky (PlanckFS) bandpowers (Planck Collaboration et al. 2016d). The Planck parameters we use are obtained from the baseline ΛCDM Monte Carlo Markov chain (MCMC, Planck Collaboration et al. 2016a). We also use the designation PlanckFS to refer to the parameter estimates from this chain.

We also use bandpowers created from the SPT-SZ 150 GHz and Planck 143 GHz maps of the 2540 deg² SPT-SZ survey region. We refer to the cross-correlation of two half-depth maps as either 150 × 150 or 143 × 143. The cross-spectrum 150 × 143 is the correlation of the full-depth SPT and full-depth Planck maps. A detailed description of the creation of these bandpowers is provided by H17.

We obtain parameter estimates by searching the space defined by the likelihood of the cosmological and nuisance parameters ${\boldsymbol{\Theta }}$ given the data ${{\boldsymbol{D}}}^{i\times j}$, a set of temperature bandpowers where i and j both run over the two frequency bands (143 GHz for Planck and 150 GHz for SPT). We assume the likelihood to take the following form

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

where ${{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{i\times j}$ is the bandpower covariance, and the model temperature bandpowers are expressed as

$$\begin{eqnarray}&&{M}_{b}^{i\times j}={Y}^{i}{Y}^{j}{W}_{b{\ell }}^{i\times j}({D}_{{\ell }}^{\mathrm{th}}+{F}_{{\ell }}^{i\times j}){a}_{{\ell }}={W}_{b{\ell }}^{i\times j}{M}_{{\ell }}^{i\times j}\end{eqnarray} \tag{ 2 }$$

for $i,j\in [143,150]$, and we have ignored the normalization constant in the likelihood. The ${Y}^{i},{Y}^{j}$ terms are temperature calibration parameters, ${W}_{b{\ell }}^{i\times j}$ is the bandpower window function (e.g., Knox 1999), and ${F}_{{\ell }}^{i\times j}$ is the foreground model from Story et al. (2013) with frequency dependence included (George et al. 2015). The term a_ℓ includes aberration effects due to our proper motion with respect to the CMB as

$$\begin{eqnarray}&&{a}_{{\ell }}=1-\displaystyle \frac{d\mathrm{ln}{C}_{{\ell }}}{d\mathrm{ln}{\ell }}\beta \langle \cos \theta \rangle ,\end{eqnarray} \tag{ 3 }$$

based on Jeong et al. (2014) and with $\beta =1.23\times {10}^{-3}$ and $\langle \cos \theta \rangle =-0.26$.

The calibration parameters are based on work from H17, where the three in-patch bandpowers are simultaneously calibrated to each other over the common multipole range. The calibration parameters are fixed to 1 for Y¹⁴³ and have a Gaussian prior of ${Y}^{150}=0.9914\pm 0.0017$.³⁵

We use the 2016 May version of CAMB (Lewis & Bridle 2002) to calculate the theoretical prediction for the lensed temperature power spectrum (${D}_{{\ell }}^{\mathrm{th}}$) assuming the standard ΛCDM model. Our free parameters are the approximation to the angular scale of the sound horizon, ${\theta }_{\mathrm{MC}}$, the baryon and total matter densities, ${{\rm{\Omega }}}_{b}{h}^{2}$ and ${{\rm{\Omega }}}_{m}{h}^{2}$, the scalar amplitude, ${A}_{s}{e}^{-2\tau }$, and the scalar spectral index, n_s. While the Hubble constant H₀ is derived from these five parameters, we discuss it throughout this work because the discrepancy between the CMB-determined value of H₀ and local measurements is of particular interest. We place a Gaussian prior on the optical depth τ of 0.07 ± 0.02.

Our parameter vector also includes six nuisance parameters. The ${Y}^{i}{Y}^{j}$ term is treated as a single parameter, and we include five parameters for foregrounds (three for the template amplitudes and two for the frequency dependence), giving ${\boldsymbol{\Theta }}$ a total of 12 elements. All other parameters are fixed to the baseline model values from Planck Collaboration et al. (2016a), which are the default settings for the 2016 May version of CAMB.

The covariances for the bandpowers 150 × 150, 143 × 143, and 150 × 143 contain sample and noise variance along with beam uncertainty. A correlation matrix is formed for each source of beam uncertainty as

$$\begin{eqnarray}&&{\rho }_{{\ell }{{\ell }}^{{\prime} }}^{B,i\times j}=\left({\displaystyle \frac{\delta {D}_{{\ell }}}{{D}_{{\ell }}}}^{B,i\times j}\right)\left({\displaystyle \frac{\delta {D}_{{{\ell }}^{{\prime} }}}{{D}_{{{\ell }}^{{\prime} }}}}^{B,i\times j}\right)\end{eqnarray} \tag{ 4 }$$

where

$$\begin{eqnarray}&&{\displaystyle \frac{\delta {D}_{{\ell }}}{{D}_{{\ell }}}}^{B,i\times j}={\left(1+{\displaystyle \frac{\delta {B}_{{\ell }}}{{B}_{{\ell }}}}^{i}\right)}^{-1}{\left(1+{\displaystyle \frac{\delta {B}_{{\ell }}}{{B}_{{\ell }}}}^{j}\right)}^{-1}-1,\end{eqnarray} \tag{ 5 }$$

and ${\tfrac{\delta {B}_{{\ell }}}{{B}_{{\ell }}}}^{i}$ is either the Planck 143 GHz or the SPT-SZ 150 GHz fractional beam uncertainty template. The SPT beam functions (B_ℓ) and their uncertainty ($\delta {B}_{{\ell }}$) are determined using a combination of maps from Jupiter, Venus, and the 18 brightest point sources in the SPT patch. We refer the reader to S13 and Schaffer et al. (2011) for more details on the creation of the beam templates. The correlation matrix ${\rho }_{{\ell }{{\ell }}^{{\prime} }}^{B,i\times j}$ is then added into the full covariance as

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{i\times j}={{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{S,i\times j}+{{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{N,i\times j}+{{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{B,i\times j}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{B,i\times j}={W}_{b{\ell }}^{i\times j}{W}_{{b}^{{\prime} }{{\ell }}^{{\prime} }}^{i\times j}{M}_{{\ell }}^{i\times j}{M}_{{{\ell }}^{{\prime} }}^{i\times j}{\rho }_{{\ell }{{\ell }}^{{\prime} }}^{B,i\times j},\end{eqnarray} \tag{ 7 }$$

where S, N, and B superscripts signify the sample, noise, and beam covariances, respectively.

Planck Collaboration et al. (2016d) show that the uncertainty in the Planck beams has an effect weaker than a 0.2% on 143 × 143 bandpowers, and the resulting impact on parameter estimation is also extremely small (Planck Collaboration et al. 2016a). We thus make the simplifying assumption that ${\tfrac{\delta {B}_{{\ell }}}{{B}_{{\ell }}}}^{143}=0$.

In 2013, S13 presented the SPT-SZ 150 GHz bandpowers and resulting ΛCDM parameter constraints. There are some differences between the parameter estimation for S13 and for this analysis. First, we here assume massive neutrinos with a total mass of 0.06 eV and a Planck-based τ prior. Calabrese et al. (2017) point out the importance of these assumptions when comparing parameters estimated from the Planck data with other CMB results.

We handle calibration uncertainty by accounting for it in our model instead of including it in the bandpower covariance, and we use a different calibration prior than was used in S13. The method outlined above for handling beam uncertainty differs from S13 in that the beam covariance is now formed in a model-dependent way based on ${M}_{{\ell }}^{i\times j}$, the cosmology of each MCMC sample. The previous methods for handling calibration and beam uncertainty used in S13 produced biased parameter constraints, lowering ${A}_{s}{e}^{-2\tau }$ and n_s (see the Appendix for further detail). The new, much tighter calibration prior is based on the in-patch bandpower comparisons in H17.

In S13, aberration effects due to our proper motion with respect to the CMB were not included in the parameter estimation. Based on Jeong et al. (2014), we include aberration in Equation (2). Accounting for aberration leads to a shift of approximately 0.3σ in the S13 ${{\rm{\Theta }}}_{\mathrm{MC}}$ value toward the PlanckFS value.

In Figure 1 we show the differences in parameter estimates that are due to the above changes by comparing the parameters from S13 to parameters estimated from the same bandpowers, but with the updated likelihood (S13*). The decrease in ${\theta }_{\mathrm{MC}}$ is primarily the result of including aberration effects, ${A}_{s}{e}^{-2\tau }$ is increased by the new calibration prior, and n_s is mostly increased by the new method of handling beam uncertainties. The changes to the likelihood relative to S13 lead to greater consistency between Planck and SPT.

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In H17 and this work, we use the 150 × 150 bandpowers generated from half-power SPT maps instead of the bandpowers from S13, which were generated from cross-spectra of hundreds of single-observation maps. This choice makes the data easier to simulate and simplifies the 143 GHz cross-spectrum analysis, since that data were created in a similar manner. No significant difference was found between the 150 × 150 and S13 bandpowers—for more details see the Appendix of H17. In Figure 1 we compare the differences between parameters estimated from S13, S13* (the S13 bandpowers with the updated likelihood), and 150 × 150 (H17).

To obtain parameter estimates for our in-patch bandpowers, we use the Metropolis–Hastings algorithm to produce a chain from which we generate the posterior for ${\boldsymbol{\Theta }}$ and then marginalize over the nuisance parameters (foreground and calibration parameters). We generate the chains using the likelihood sampler Cosmoslik (Millea 2017).

The primary statistic we use to infer the compatibility between various parameter distributions is

$$\begin{eqnarray}&&{\chi }^{2}={\rm{\Delta }}{{\boldsymbol{\theta }}}^{T}{{\boldsymbol{C}}}^{-1}{\rm{\Delta }}{\boldsymbol{\theta }},\end{eqnarray} \tag{ 8 }$$

where ${\boldsymbol{C}}$ is the parameter difference covariance and

$$\begin{eqnarray}&&{\rm{\Delta }}{\boldsymbol{\theta }}={{\boldsymbol{p}}}_{1}-{{\boldsymbol{p}}}_{2}.\end{eqnarray} \tag{ 9 }$$

The ${{\boldsymbol{p}}}_{\alpha }$ are either the means of the parameter posteriors or obtained through minimization of the negative log-likelihood. The latter method is used when simulations are required as minimizing the negative log-likelihood requires significantly less computation time than running an MCMC. The ${{\boldsymbol{p}}}_{\alpha }$ are composed of the five non-τ cosmological parameters: ${\theta }_{\mathrm{MC}}$, ${{\rm{\Omega }}}_{m}{h}^{2}$, ${{\rm{\Omega }}}_{b}{h}^{2}$, ${A}_{s}{e}^{-2\tau }$, and n_s.

When comparing parameters from the in-patch bandpowers to PlanckFS, the parameter difference covariance is approximated as ${\boldsymbol{C}}={{\boldsymbol{C}}}^{i\times j,{{\ell }}_{\max }}+{{\boldsymbol{C}}}^{\mathrm{FS}}$ with $i\times j\in \{143\times 143,150\times 143,150\times 150\}$ and $2000\leqslant {{\ell }}_{\max }\leqslant 3000$. The small correlations between the in-patch and full-sky parameter sets are ignored.

The parameter difference covariance for comparisons between the in-patch bandpowers cannot be calculated as simply. The parameters are obtained from bandpowers in the same sky cut, and therefore a large portion of the sample variance is common between all three sets and must be accounted for. It is necessary to estimate the covariance from the fluctuations across a set of simulations.

To calculate the in-patch parameter difference covariance matrices, we generate 400 bandpower simulations for each of the in-patch spectra. The creation of the simulations is described in H17. To simulate the calibration uncertainty, we multiply each simulation by a random draw from the appropriate calibration prior. The simulated bandpowers are then substituted into Equation (1), and we calculate a set of parameter estimates through minimization of the negative log-likelihood. The minimization is made using the scipy minimize module with the Nelder–Mead method (Jones et al. 2001). Running the minimizer on the 150 × 150 bandpowers returns parameter values similar to the results obtained from the MCMC procedure.

With the 400 sets of parameters for each of the three in-patch bandpowers, we obtain the parameter difference covariances for the three in-patch comparisons. The stability of our covariances was tested by splitting the simulations into two groups of 200 and recalculating ${\chi }^{2}$ with each half. We find the results from the two halves to be consistent, and, as we show below, the simulated parameter differences follow a ${\chi }^{2}$ distribution with five degrees of freedom. After calculating the ${\chi }^{2}$ for a set of parameter differences, we convert it into a PTE, which we use to infer compatibility between the sets of parameters.

Comparing the parameter differences derived from the SPT 150 × 150 and PlanckFS data sets, we find

$$\begin{eqnarray}&&\mathrm{PTE}=0.032\,({\rm{PlanckFS\; versus}}\,150\times 150).\end{eqnarray} \tag{ 10 }$$

When considering parameters estimated from the full multipole range for 150 × 150, the parameters that differ the most (in terms of standard deviations) are ${\theta }_{\mathrm{MC}}$ and ${{\rm{\Omega }}}_{m}{h}^{2}$, and subsequently, H₀. As noted in previous publications, the higher value of H₀ preferred by the SPT is closer to the value reported by Riess et al. (2016). While a PTE of 0.032 might be due to a statistical fluctuation, it could be an indication of potential systematic error or a breakdown of ΛCDM.

In this section, we compare parameters derived from modes that are well measured by both Planck and SPT. We consider data within the SPT-SZ sky region, which covers 2540 deg², or about 6% of the sky, and within the multipole range $650\leqslant {\ell }\leqslant 2000$. Specifically, we compare parameters estimated in this multipole range from the in-patch cross-spectrum bandpowers presented in H17. This comparison provides a sensitive test of unaccounted-for systematic errors in either experiment.

The lower cutoff of ${\ell }=650$ is set by the SPT analysis in S13, which did not report data at larger angular scales (lower multipoles) because of the increasing noise from the atmosphere on these scales. The upper cutoff of ${\ell }=2000$ is set by high-ℓ noise in the Planck 143 GHz data resulting from the larger Planck beam (roughly 7 arcmin FWHM, compared to 1 arcmin for SPT 150 GHz) and the slightly higher noise per pixel in the Planck maps (∼25 μK arcmin for Planck 143 GHz compared to ∼18 for SPT). The variance of the 143 × 143 bandpowers (the set with the largest noise variance) is dominated by sample variance to approximately ${\ell }=1500;$ as a result, the three sets of bandpowers have similar uncertainty in this range. The 143 × 143 error bars begin to grow significantly larger than those for 150 × 150 around ${\ell }=1800$, and 150 × 143 begins to show the same behavior around ${\ell }=2200$. We choose ${{\ell }}_{\max }=2000$ to maximize the signal-to-noise ratio of the comparison between 150 × 150 and 150 × 143. When restricted to the SPT-SZ sky area and this range of angular scales, both experiments are measuring a very similar set of modes on the sky with a similar signal-to-noise ratio per mode. Given our null hypothesis, the expected covariance of parameter differences for these modes is thus greatly reduced, making it easier for us to see the impact of any systematic errors.

The parameter estimates for the in-patch bandpowers over this multipole range are in Figure 2, and Figure 3 shows the ratio of the in-patch bandpowers and models to the best-fit PlanckFS model. The in-patch parameters are more similar to each other than to the Planck full-sky values, and the features apparent by eye in the bandpower and best-fit-model ratios to PlanckFS are similar among the three in-patch sets. These plots still include sample variance in the in-patch error bars, however, so it is difficult to assess the statistical consistency of the three data sets. Figure 4 shows the distribution of ${\chi }^{2}$ values for differences in simulation parameters calculated in this comparison (as expected, the histogram closely follows a ${\chi }^{2}$ distribution for five degrees of freedom). This statistic accounts for the large decrease in sample variance in the parameter difference covariance and provides a quantitative assessment of the consistency among the three in-patch parameter sets. The ${\chi }^{2}$ values of the data differences are shown by vertical red lines, and none of these values lie notably outside the main distribution. The PTEs from this test are included in Table 1 and confirm that all three sets of in-patch bandpowers are consistent with each other in the multipole range $650\leqslant {\ell }\leqslant 2000.$

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Table 1.  PTEs between Parameters in the SPT Sky Patch

	${{\ell }}_{\max }$
	2000	2500	3000
150 × 150–150 × 143	0.74	0.66	0.57
150 × 150–143 × 143	0.32	0.38	0.20
150 × 143–143 × 143	0.62	0.73	⋯

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Some parameter differences among the three in-patch sets are visible in Figure 2. Between the 143 × 143 and 150 × 143 data sets, the largest differences are the slightly lower preferred values of ${{\rm{\Omega }}}_{m}{h}^{2}$ and n_s in 150 × 143. This trend continues in 150 × 150, but as shown in Figure 4 and Table 1, these differences are consistent with our null hypothesis.

We pay special attention to the comparison between the parameters derived from 150 × 150 with ${{\ell }}_{\max }=2000$ and those from 150 × 143 with ${{\ell }}_{\max }=2000$ because this comparison provides the most stringent test of our null hypothesis. Figure 5 shows that the expected covariance of parameter differences, indicated by the contours in the upper triangle, are quite small, comparable to the covariance of parameter uncertainties in the PlanckFS posterior. In this regard, examining these parameter differences provides us with a much more powerful test than the comparison between the PlanckFS parameters and the 150 × 150 full ℓ-range parameters.

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In addition to the visual impression of this increase in precision of the test given by Figure 5, we also provide a quantitative description of the increase in precision. We do so by simultaneously diagonalizing the covariances for the 150 × 150 and PlanckFS parameter differences at ${{\ell }}_{\max }\,=3000$ and the 150 × 150 and 150 × 143 parameter differences at ${{\ell }}_{\max }=2000$. We then multiply the square root of the eigenvalue ratios to calculate the reduction in the $1\sigma$ volume for the five-dimensional parameter space. Comparing the ratio of the volumes, we find a ratio of 0.003, i.e., the volume in the parameter difference space containing 68% of the probability is 300 times smaller for the ${{\ell }}_{\max }=2000$ 150 × 150 versus 150 × 143 parameter differences than for the full ℓ-range 150 × 150 versus PlanckFS parameter differences. Despite the precision of this test, we find a perfectly acceptable PTE for this comparison:

$$\begin{eqnarray}&&\mathrm{PTE}=0.74(150\times 150\ \mathrm{versus}\ 150\times 143).\end{eqnarray} \tag{ 11 }$$

In conclusion, when Planck and SPT data are restricted to modes on the sky that are measured well in both experiments, we find that the best-fit cosmological parameters are fully consistent. The observed consistency between the two data sets in this stringent test provides strong evidence against instrumental systematics affecting either data set on these angular scales, on this part of the sky.

Next, we reintroduce smaller angular scales measured within the SPT-SZ survey region. Given the ΛCDM model and a spectrum measured in the range $650\leqslant {\ell }\leqslant 2000$, one can predict the spectrum at other angular scales. Given the consistency found in the previous section, finding significant discrepancy from extending the multipole range could indicate either a systematic affecting SPT data at high ℓ or a failure of the ΛCDM model. Parameter estimates for the in-patch bandpowers at various values of ${{\ell }}_{\max }$ are shown in Figure 2, and PTEs are reported in Table 1.

For 143 × 143, increasing ${{\ell }}_{\max }$ from 2000 to 2500 adds little information to the parameter estimates. This is consistent with Figure 3, where the error bars of the 143 × 143 bandpowers have become significantly larger by ${\ell }=1800$. Some parameters in the 150 × 143 measurement do shift when we expand the ℓ range: ${\theta }_{\mathrm{MC}}$, ${{\rm{\Omega }}}_{b}{h}^{2}$, and n_s shift away from the 143 × 143 values with increasing ${{\ell }}_{\max }$. Nevertheless, at ${{\ell }}_{\max }=2500$, we still find that the 150 × 143 and 143 × 143 measurements are consistent with a PTE of 0.62.³⁶

For 150 × 150, when ${{\ell }}_{\max }$ is increased from 2000 to 2500, ${\theta }_{\mathrm{MC}}$ increases in a manner similar to what we saw with 150 × 143, the baryon density increases, and the matter density decreases. These shifts correspond to an increase in H₀. At ${{\ell }}_{\max }=2500$, the 150 × 150 measurement remains consistent with 150 × 143 and 143 × 143, with PTEs of 0.66 and 0.38, respectively. The trend in parameter shifts continues when we increase ${{\ell }}_{\max }$ to 3000 to include the full range of the 150 × 150 data, yet the PTEs remain moderate. We also plot in Figure 2 parameter results for ${{\ell }}_{\max }$ = 1800, and we see the trend toward the PlanckFS values continues. Uncertainties rapidly grow for ${{\ell }}_{\max }$ < 1800. Finally, we calculate the ${\chi }^{2}$ and PTEs for the comparison of parameters from 150 × 150 data at ${{\ell }}_{\max }=2000$ to parameters from 150 × 150 data with ${{\ell }}_{\max }=2500$ and 3000. These PTEs are 0.88 and 0.75, respectively. Thus, while the parameter shifts with increasing ${{\ell }}_{\max }$ are suggestive of a potentially interesting trend, they are consistent with our expectations under the null hypothesis.

In the previous sections, we have found that the ΛCDM parameters estimated from the Planck and SPT in-patch bandpowers are consistent for all ℓ ranges considered. In this section, we relax the restrictions on sky coverage and compare the in-patch parameters to the PlanckFS parameters in different fixed ℓ ranges. This effectively tests ΛCDM and the assumptions of statistical isotropy in the CMB. The PTEs between the in-patch parameters and PlanckFS parameters in all ℓ ranges tested are listed in Table 2 and the parameter constraints are shown in Figure 2.

Table 2.  PTEs between PlanckFS and In-patch Parameters

	${{\ell }}_{\max }$
	2000	2500	3000
150 × 150	0.24	0.094	0.032
150 × 143	0.19	0.18
143 × 143	0.29	0.31

Note. The entries for 150 × 143 and 143 × 143 at ${{\ell }}_{\max }=3000$ are blank since these spectra have negligible signal-to-noise ratio above ${\ell }=2500$.

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We first compare PlanckFS to 143 × 143 at ${{\ell }}_{\max }=2000$ and 2500. Because of the rapidly increasing noise at high ℓ in the 143 × 143 bandpowers, we do not consider ${{\ell }}_{\max }=3000$ for this comparison, and we see very little difference in the parameters and comparison PTEs for ${{\ell }}_{\max }=2000$ and 2500. These PTEs are 0.29 and 0.31, respectively. Although the PTE values indicate no discepancy between the data sets, we do see small differences in ${\theta }_{\mathrm{MC}}$, ${A}_{s}{e}^{-2\tau }$, and n_s between PlanckFS and 143 × 143 at ${{\ell }}_{\max }=2000$ and 2500. The two main differences between the sets of bandpowers that could drive parameter differences are the Planck low ℓ data (which are not included in the in-patch bandpowers) and the sky outside of the patch. However, we can rule out the low-ℓ data as an explanation based on the results of Planck Collaboration et al. (2016c), who found only marginal parameter shifts when cutting the low-ℓ (${\ell }\leqslant 650$) data. Therefore, the majority of the parameter differences between PlanckFS and 143 × 143 can be attributed to differences in the Planck data from the SPT-SZ patch and the Planck data from the rest of the sky.

As noted in H17, all three sets of in-patch bandpowers have more power than the PlanckFS model at moderate ℓ and less power at high ℓ, creating a tilt. In Figure 3 we show the ratios of the in-patch bandpowers to the PlanckFS best-fit model. We also show the ratios of best-fit in-patch models to the PlanckFS best-fit model. The tilt is clearly visible by eye in all bandpowers and best-fit models, and this tilt drives the differences in ${A}_{s}{e}^{-2\tau }$ and n_s. We discuss this tilt further in Section 3.5. There is also an oscillatory pattern in the best-fit model ratios, consistent with the difference in best-fit ${\theta }_{\mathrm{MC}}$, although this feature is not as obviously discernible directly in the bandpower ratios as is the tilt.

Next we compare PlanckFS to 150 × 143 at ${{\ell }}_{\max }=2000$ and 2500 (and again refer the reader to Figure 2). Compared to 143 × 143, all parameters except ${A}_{s}{e}^{-2\tau }$ shift away from PlanckFS values when SPT data are included and as ${{\ell }}_{\max }$ is increased. Nevertheless, the PTE of the comparison to PlanckFS still gives PTEs of 0.19 at ${{\ell }}_{\max }=2000$ and 0.18 at ${{\ell }}_{\max }=2500$.

Finally, we compare PlanckFS to 150 × 150 with varying ${{\ell }}_{\max }$. In Figure 2 we see a trend toward the PlanckFS values as ${{\ell }}_{\max }$ is lowered, with near-convergence of Ω_bh², Ω_mh², and H₀ by ${{\ell }}_{\max }=1800$. For ${{\ell }}_{\max }=1800,2000,$ and 2500, we find 150 × 150 and PlanckFS to be at least marginally consistent (minimum PTE of 0.094). The discrepancy between the data sets only approaches the 2σ level when we extend the parameter estimation to the full 150 × 150 multipole range, where we find the PTE of 0.032 with which we began this investigation. As noted in the previous section, the only notable difference between the 150 × 143 and 150 × 150 data at ${{\ell }}_{\max }=2500$ is the marginally lower matter density (and hence higher Hubble constant) that 150 × 150 prefers; these parameters are pushed even farther in this direction when ${{\ell }}_{\max }$ is increased to 3000. From this comparison, we see that the 150 × 150 bandpowers above ${\ell }\gt 1800$ drive some of the discrepancy with PlanckFS.

H17 showed that the bandpowers from the SPT-SZ patch (from both SPT and Planck data) have a tilt relative to the PlanckFS bandpowers. We also see this feature prominently in the ratios of in-patch bandpowers to the PlanckFS best-fit model, as plotted in Figure 3. In this section, we investigate the impact of this tilt on cosmological parameters. To do so, we fit a power law to the ratio of the in-patch bandpowers to the PlanckFS best-fit model and multiply this power law into the theory spectrum in Equation 2 when estimating new parameters.

The power law takes the form

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{{\ell }}^{\mathrm{th},i\times j}}{{D}_{{\ell }}^{\mathrm{PlanckFS}}}=A{\left(\displaystyle \frac{{\ell }}{1000}\right)}^{n}.\end{eqnarray} \tag{ 12 }$$

Using the PlanckFS model instead of the PlanckFS bandpowers allows us to better assess how this tilt drives the best-fit parameters away from the PlanckFS values. We assume Gaussianity and use the likelihood

$$\begin{eqnarray}&&-2\mathrm{ln}{ \mathcal L }(A,n| {D}^{i\times j})={{\rm{\Delta }}}_{b}{{\rm{\Sigma }}}_{{{bb}}^{{\prime} }}^{-1}{{\rm{\Delta }}}_{b}^{{\prime} }\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}{{\rm{\Delta }}}_{b} & = & {D}_{b}^{i\times j}-{Y}^{i}{Y}^{j}{W}_{b{\ell }}^{i\times j}{a}_{{\ell }}({D}_{{\ell }}^{\mathrm{PlanckFS}}A\\ & & \times \left.{\left(\displaystyle \frac{{\ell }}{1000}\right)}^{n}+{F}_{{\ell }}^{i\times j}\right),\end{eqnarray} \tag{ 14 }$$

where ${D}_{{\ell }}^{\mathrm{PlanckFS}}$ is the PlanckFS best-fit model. In Figure 6 we present the best-fit power laws for each spectrum with ${{\ell }}_{\max }=2500$. As the amount of SPT data included is increased (i.e., as we go from 143 × 143 to 150 × 143 to 150 × 150), we see an increase in the tilt, with 150 × 150 having a best-fit value of n that is discrepant with 0 by $2.2\sigma$. We note, however, that the best-fit tilt values from the three bandpower sets are consistent within 1σ.

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To connect this feature with parameters, we remove this tilt from the 150 × 150 bandpowers by multiplying the theory spectrum in Equation (2) by the best-fit power law for 150 × 150 and run a new chain. As shown in Figure 7, removing this tilt from the full range of the 150 × 150 bandpowers results in a decrease in ${A}_{s}{e}^{-2\tau }$ and an increase in n_s to significantly better agreement with PlanckFS. By contrast, we note that removing the tilt hardly affects the two density parameters (and H₀); therefore, the preference for higher H₀ by 150 × 150 is due to high-ℓ information unrelated to the tilt.

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The picture that emerges from the previous sections is the following: when SPT and Planck data are restricted to modes on the sky that are measured well in both experiments—i.e., to modes in the SPT-SZ survey region in the multipole range $650\leqslant {\ell }\leqslant 2000$—the best-fit parameters from the two data sets are fully consistent. When the comparison is relaxed to either just the same sky or just the same multipole range, parameter constraints from the two experiments are still marginally consistent, although parameter differences begin to emerge. Only when we relax all restrictions on sky coverage and multipole range do we find a difference greater than 2σ between Planck and the SPT. This difference arises in roughly equal parts from the SPT data at high ℓ and from differences in the SPT-SZ patch relative to the whole sky at moderate ℓ. The latter fluctuation accounts for nearly all of the difference in ${A}_{s}{e}^{-2\tau }$, and a sizeable fraction of the differences in ${\theta }_{\mathrm{MC}}$ and n_s, but almost none of the differences in the other parameters. The remainder of the parameter differences arises from the high-ℓ fluctuation.

Of the parameter differences driven by high-ℓ SPT data, the Hubble constant is of particular interest given the discrepancy between the value derived from Planck CMB power spectrum data assuming ΛCDM and the traditional distance ladder measurement of Riess et al. (2016). As can be seen in Figure 2, half of the Hubble constant difference between SPT and Planck arises from the SPT data at ${\ell }\gt 2000$.

In our parameterization, the Hubble constant is a derived parameter that can be calculated from ${{\rm{\Omega }}}_{b}{h}^{2}$, ${{\rm{\Omega }}}_{m}{h}^{2}$, and ${\theta }_{\mathrm{MC}}$. From the perspective of the Hubble constant, the angle ${\theta }_{\mathrm{MC}}$ is essentially fixed—the uncertainties and shifts between data sets are so small that the impact on H₀ is negligible. Thus the observed variation in H₀ between data sets is due to changes in the two density parameters. Changing either the baryon density or the matter density (and enforcing a flat universe) would result in a change in the angular size of the sound horizon at recombination and hence a different observed value of ${\theta }_{\mathrm{MC}}$. The only parameter available to preserve the observed ${\theta }_{\mathrm{MC}}$ is H₀.

Specifically, the baryon density affects the sound speed in the early universe. Increasing the baryon density decreases the sound speed and thus the physical size of the sound horizon at recombination. To preserve the angular size, the angular diameter distance to recombination

$$\begin{eqnarray}{d}_{A}({z}_{* }) & \propto & {\displaystyle \int }_{0}^{{z}_{* }}{dz}/H(z)\\ & \propto & \displaystyle \frac{1}{{H}_{0}}{\displaystyle \int }_{0}^{{z}_{* }}\displaystyle \frac{{dz}}{\sqrt{{{\rm{\Omega }}}_{m,0}{(1+z)}^{3}+{{\rm{\Omega }}}_{{\rm{\Lambda }}}}},\end{eqnarray} \tag{ 15 }$$

where z_* is the redshift of recombination, must be made smaller. At fixed matter density and with flatness enforced, this can only be achieved by increasing H₀.

Changing the matter density, meanwhile, affects the expansion rate, both in the early universe and from recombination to today (as can be seen from Equation (15)). In the early universe, this change would affect the physical size of the sound horizon, although its impact is softened by the contribution of radiation density to the expansion rate. Decreasing the matter density at late times would increase the angular diameter distance to recombination, which, at fixed baryon density, would make the angular size of the sound horizon too small. To preserve the measured angular size, H₀ must increase. Thus both the increase in ${{\rm{\Omega }}}_{b}{h}^{2}$ and the decrease in ${{\rm{\Omega }}}_{m}{h}^{2}$ driven by the high-ℓ SPT data lead to an increase in the inferred value of H₀.

In this section, we investigate the possible impact of foreground and beam misestimation on parameter differences, particularly at high ℓ. To test for beam systematics, we include the amplitudes of the fractional beam uncertainty as parameters in the MCMC, rather than analytically marginalizing over the beam uncertainty. This means in practice that we modify Equation (5) to include parametrized amplitudes of the SPT beam error templates for each source of beam uncertainty and multiply this into Equation (2). The covariance in Equation (1) no longer has Equation (7) included in it. Our model bandpowers now take the form

$$\begin{eqnarray}&&{M}_{b}^{i\times j}={W}_{b{\ell }}^{i\times j}{M}_{{\ell }}^{i\times j}\left(1+{\displaystyle \frac{\delta {D}_{{\ell }}}{{D}_{{\ell }}}}^{B,i\times j}\right).\end{eqnarray} \tag{ 16 }$$

At each step of the new 150 × 150 chain, we calculate the five-parameter ${\chi }^{2}$ from Equation (8) using the difference between the cosmological parameters of the current step and the PlanckFS means. If a beam or foreground parameter were connected to the low PTE found in Section 3, we would expect to find the ${\chi }^{2}$ posterior to be correlated with said parameter. In other words, we treat the five-parameter ${\chi }^{2}$ as a derived parameter and look for correlations or degeneracies between this derived parameter and the beam or foreground parameters. We do not find any significant correlation; the largest correlation found between any foreground or beam parameter and the five-parameter ${\chi }^{2}$ was 0.12, indicating a very weak linear response and no clear direction in the foreground and beam parameter space in which the parameter comparison ${\chi }^{2}$ can be lowered.

We find furthermore that the foreground and beam posteriors are not driven significantly from their priors in any chain, as one would expect if any of these components were a poor description. The largest deviation of a parameter's posterior mean from the prior mean was $0.13\sigma$, and most parameters showed shifts of less than $0.1\sigma$. These tests provide more support for our hypothesis that the somewhat low PTE between 150 × 150 and PlanckFS is not due to a systematic error in the beam or foreground treatment.

We expand on the foreground tests by adding free parameters for the kinematic Sunyaev–Zel'dovich (kSZ) effect—the S13 foreground model has a single parameter for the sum of kSZ and tSZ—and a cross-correlation between the tSZ and the cosmic infrared background based on George et al. (2015), since these components were also included in the determination of the PlanckFS parameters. This change has a negligible impact on the parameter posteriors for 150 × 150. This result is expected: the motivation for the simplified foreground parameterization in S13 was that these extra foreground components are expected to have a similar power-spectrum shape to tSZ in the multipole range examined in S13 and here and are thus only distinguishable via their frequency dependence. Thus, the differences between Planck and SPT do not appear to be related to foreground components that are included in the Planck analysis, but not in S13.

Potential systematic errors may also creep in through the optical depth measurement, given the proven challenges in recovering the reionization peak from the midst of the Galactic foregrounds. We test whether the τ prior is contributing to the low PTE by running 150 × 150 chains with a low optical depth, $\tau =0.05\pm 0.02$, and a high optical depth, $\tau =0.10\pm 0.02$. For the τ = 0.05 prior there are small shifts in both density parameters toward PlanckFS values, but the shifts only change the PTE from 0.032 to 0.058. For the τ = 0.1 prior we see the opposite effect and calculate a PTE of 0.015. The small improvement in the PTE as τ is lowered argues against the idea that the parameter differences between PlanckFS and 150 × 150 are significantly connected to the τ prior.

Two of the most discrepant parameters between 150 × 150 and PlanckFS are ${{\rm{\Omega }}}_{m}{h}^{2}$ and ${A}_{s}{e}^{-2\tau }$. Since these parameters both impact the lensing amplitude, we test the hypothesis that the parameter differences are lensing- related. The impact of lensing on parameter estimates is often studied by marginalizing over an artifical lensing-power scaling parameter, ${A}_{{\rm{L}}}$. Here we follow Planck Collaboration et al. (2016c) and instead fix the amount of lensing power. With this choice we avoid some difficulties of interpreting parameter constraints after marginalization over ${A}_{{\rm{L}}}$. With marginalization over ${A}_{{\rm{L}}}$, one projects out lensing information. That is useful, but interpretation is complicated by the fact that one also removes any sensitivity to parameter variations that produce effects that can be mimicked by lensing. In contrast, by fixing the lensing potential we can remove the contribution of lensing variation to our parameter constraints, while keeping the contribution of any non-lensing responses of the power spectrum to parameter variations.

Specifically, we fix the 150 × 150 lensing potential to its best-fit ΛCDM value. In practice, we modify our model of the bandpowers (Equation (2)) so that

$$\begin{eqnarray}{M}_{b}^{i\times j} & = & {Y}^{i}{Y}^{j}{W}_{b{\ell }}^{i\times j}{a}_{{\ell }}({D}_{{\ell }}^{\mathrm{th},\mathrm{UL}}(\theta )\\ & & +{D}_{{\ell }}^{\mathrm{Lens}}({\theta }_{* })+{F}_{{\ell }}^{i\times j}),\end{eqnarray} \tag{ 17 }$$

with

$$\begin{eqnarray}&&{D}_{{\ell }}^{\mathrm{Lens}}({\theta }_{* })={D}_{{\ell }}^{\mathrm{th}}({\theta }_{* })-{D}_{{\ell }}^{\mathrm{th},\mathrm{UL}}({\theta }_{* }),\end{eqnarray} \tag{ 18 }$$

where UL signifies an unlensed spectrum and ${\theta }_{* }$ represents cosmological parameters fixed to the 150 × 150 ΛCDM best-fit values.

We first note from the results of this test, shown in Figure 7, is that there is still some preference in the SPT data for lower matter density even with the lensing information removed, albeit it is weaker. Adding the lensing information strengthens this preference.

We note also that the PTE for comparison with PlanckFS only improves to 0.045 with lensing fixed. We attribute this to the small shift in ${A}_{s}{e}^{-2\tau }$ that also occurs when we fix the lensing potential. Thus, although lensing has an impact, removing the impact of lensing on the SPT parameter estimates does not significantly improve the agreement with the PlanckFS parameter estimates.

Finally, we note that in Planck Collaboration et al. (2016c) the Planck collaboration performed a similar test with the Planck data. Fixing the lensing potential lowers the matter density for Planck and increases it for SPT, bringing the preferred matter density values for these two data sets closer together. However, the shifts are relatively small when compared to the full SPT and Planck parameter differences.