An Improved Measurement of the Secondary Cosmic Microwave Background Anisotropies from the SPT-SZ + SPTpol Surveys

Before Fourier-transforming the maps, we apply a window to each map that smoothly goes to zero at the map edges. The window also masks point sources above 6.4 mJy at 150 GHz from the source catalog in Everett et al. (2020). The mask for each point source has a 2'-radius disk for sources detected with S_{150 GHz} ∈ [6.4, 50] mJy and a 5'-radius disk for sources above 50 mJy. In both cases, a Gaussian taper with ${\sigma }_{\mathrm{taper}}=5^{\prime}$ is applied outside the radius of the disk. For the combined SPTpol and SPT-SZ field that has anisotropic noise due to variations in the amount of integration time, this window also preferentially weights the lower noise regions.

After Fourier-transforming the windowed maps, we take the weighted average of the two-dimensional power spectrum within an ℓ-bin b,

$$\begin{eqnarray}&&{\widehat{D}}_{b}^{{\nu }_{i}\times {\nu }_{j},{AB}}\equiv {\left\langle \displaystyle \frac{{\ell }({\ell }+1)}{2\pi }\mathrm{Re}\left[{\tilde{m}}_{{\ell }}^{{\nu }_{i},A}{\tilde{m}}_{{\ell }}^{{\nu }_{j},B* }\right]\right\rangle }_{{\ell }\in b},\end{eqnarray} \tag{ 2 }$$

where ${\tilde{m}}^{{\nu }_{i},A}$ is the Fourier-transformed map. Here A, B are the observation indices, while ν_i , ν_j are the observation frequencies (e.g., 150 GHz). We average all cross-spectra ${\widehat{D}}_{b}^{{AB}}$ that have A ≠ B to get the binned power spectrum ${\widehat{D}}_{b}$. As in R12, we eliminate the noisier modes along the scan direction by excluding modes with ℓ_x < 1200. We refer to the binned power, ${\widehat{D}}_{b}$, as a "bandpower."

The transfer function and sample variance for the combined SPTpol and SPT-SZ field are calculated from a suite of 200 signal-only simulations. We convolve the simulated skies by the measured beam for each frequency and observing year before sampling the realizations based on the pointing information. The simulated TOD are filtered and binned into maps in the same way as the real data.

The simulated skies include Gaussian realizations of the best-fit lensed Planck 2013 ΛCDM primary CMB model, SZ models, and extragalactic source contributions. Following G15, the kSZ power spectrum is based on the Sehgal et al. (2010) simulations with an amplitude of 2.0 μK² at ℓ = 3000. The tSZ power spectrum is taken from the Shaw et al. (2010) simulations, normalized to have an amplitude of 4.4 μK² at ℓ = 3000 at 153 GHz. The extragalactic source term can be split into three components: spatially clustered and Poisson-distributed DSFGs, and Poisson-distributed radio galaxies. Motivated by the predictions of the De Zotti et al. (2005) model for a 6.4 mJy flux cut at 150 GHz, the radio power is set to ${D}_{3000}^{r}=1.28\,\mu {{\rm{K}}}^{2}$ at 150 GHz. We assume a radio spectral index of α_r = −0.53 ⁴⁵ and 1σ scatter on the spectral index of 0.1. The DSFG Poisson power is set to 7.54 μK² at 154 GHz with a modified blackbody spectrum ⁴⁶ with T_dust = 12 K and β = 2. The clustered DSFG component is modeled by a D_ℓ ∝ ℓ^0.8 term normalized to ${D}_{3000}^{c}=6.25\,\mu {{\rm{K}}}^{2}$ and the same spectral dependence as the Poisson DSFG. Each of the frequency-dependent terms (tSZ, radio, CIB) is estimated at the effective frequency for the nominal band noted in Section 5.1, e.g., 96.9 GHz for the 95 GHz band CIB map. These simulations do not include non-Gaussianity in the tSZ, kSZ, and extragalactic source signals and therefore slightly underestimate the sample variance on these terms. While this would increase the uncertainties on these terms in Section 6, the effect is very minor. In particular, the non-Gaussian variance of the tSZ term on this survey size is <10% (Millea et al. 2012), which is small compared to the 30% model uncertainty assumed when interpreting the tSZ power. We also estimate the non-Gaussian variance of the kSZ term on a 2500 deg² survey using the Flender (ModI) kSZ map (Flender et al. 2016), finding the power level to vary by 2.3%. This variance is very small compared to the kSZ measurement uncertainties in Section 6.

In order to compare the measured bandpowers to theory, we need to estimate a covariance matrix including both sample variance and instrumental noise variance. As in R12 and G15, the sample variance is estimated from signal-only simulations (Section 2.3.2), and the noise variance is empirically determined from the distribution of the cross-spectrum bandpowers ${D}_{b}^{{\nu }_{i}\times {\nu }_{j},{AB}}$ between observations A and B and frequencies ν_i and ν_j . A noisy estimate of the bandpower covariance matrix could degrade parameter constraints (see, e.g., Dodelson & Schneider 2013). Thus, we follow G15 and "condition" the covariance matrix to minimize the noise on the covariance estimate and largely avoid this degradation.

The covariance matrix depends on the signal power and, if both bandpowers share a common map, the noise power. As the errors on the off-diagonal elements include terms proportional to the (potentially much larger) diagonal elements, the uncertainty on the off-diagonal elements can be large compared to the true covariance. As a result, we estimate these values analytically from the diagonal elements using the equations in Appendix A of L10.

We follow G15 and weight each field and frequency cross-spectrum based on the average of the inverse of the diagonal of the covariance matrix over the bins 2500 < ℓ < 3500. These weights adjust for the differences in noise and sample variance between fields; beam and calibration errors are deliberately not included. As argued by G15, the angular range, 2500 < ℓ < 3500, is where the data have the most sensitivity to SZ signals.

We calculate the combined bandpowers, D_b , as

$$\begin{eqnarray}&&{D}_{b}=\displaystyle \sum _{i}{D}_{b}^{i}{w}^{i},\end{eqnarray} \tag{ 3 }$$

where D_b ⁱ is the bandpower of field i and w_i is the weight. The covariance matrix likewise can be expressed as

$$\begin{eqnarray}&&{C}_{{{bb}}^{{\prime} }}=\displaystyle \sum _{i}{w}^{i}{C}_{{{bb}}^{{\prime} }}^{i}{w}^{i}.\end{eqnarray} \tag{ 4 }$$

The sum of the weights is normalized to unity.

To handle the calibration uncertainties, we include three calibration factors in the parameter fitting, one per frequency. We marginalize over these three factors, with a prior based on the measured calibration uncertainty for all parameter fits.

We follow Aylor et al. (2017) for the treatment of beam uncertainties. The beam correlation matrix, ${\rho }_{{{bb}}^{{\prime} }}^{\mathrm{beam}}$, is calculated as described by G15, using the fractional beam errors for each year and the relative weights of each year of data over the SPT-SZ and SPTpol surveys. At each step in the chain, we use the predicted theory bandpowers (${D}_{b}^{\mathrm{theory}}$) to convert this beam correlation matrix into a beam covariance according to

$$\begin{eqnarray}&&{C}_{{{bb}}^{{\prime} }}^{\mathrm{beam}}={{\boldsymbol{\rho }}}_{{{bb}}^{{\prime} }}^{\mathrm{beam}}{D}_{b}^{\mathrm{theory}}{D}_{{b}^{{\prime} }}^{\mathrm{theory}}.\end{eqnarray} \tag{ 5 }$$

We add this beam covariance to the bandpower covariance matrix that contains the effects of sample variance and instrumental noise. The likelihood for that specific theoretical model is then evaluated using this combined covariance matrix.

The CIB is detected at very high significance and is especially important at 220 GHz. As highlighted in Table 2, adding the CIB terms to the model improves the fit quality by Δχ² ∼ 77,000. With the flux cut of ∼6.4 mJy at 150 GHz in this work, the Poisson CIB power is larger than the radio galaxy power by a factor of seven at 150 GHz and by a factor of 60 at 220 GHz. The radio galaxy power is larger than the CIB power at 95 GHz.

At 150 GHz and ℓ = 3000, we find that the Poisson DSFG component has power ${D}_{3000}^{p}=7.24\pm 0.63\,\mu {{\rm{K}}}^{2}$ while the one- and two-halo DSFG clustering terms are ${D}_{3000}^{1 \mbox{-} \mathrm{halo}}=2.21\pm 0.88\,\,\mu {{\rm{K}}}^{2}$ and ${D}_{3000}^{2 \mbox{-} \mathrm{halo}}=1.82\pm 0.31\,\,\mu {{\rm{K}}}^{2}$, respectively. At 220 GHz, this scales to ${D}_{3000}^{p,220\,\mathrm{GHz}}=61.4\pm 9.0\,\mu {{\rm{K}}}^{2}$, ${D}_{3000}^{1 \mbox{-} \mathrm{halo},220\,\mathrm{GHz}}=32.4\pm 11.2\,\,\mu {{\rm{K}}}^{2}$, and ${D}_{3000}^{2 \mbox{-} \mathrm{halo},220\,\mathrm{GHz}}=27.5\pm 4.6\,\,\mu {{\rm{K}}}^{2}$. The β in the modified blackbody functional form of ν^β B_ν (T) rises from 1.48 ± 0.13 for the Poisson term to 2.23 ± 0.18 for the clustered terms. Cast as effective spectral indices from 150 to 220 GHz, these values of β translate to spectral indices of 3.29 ± 0.13 for the Poisson power and 4.04 ± 0.18 for the clustered power. The higher spectral index for the clustered power could indicate a different redshift weighting (the same rest-frame modified blackbody spectrum would appear steeper at low redshift), a dependence of the galaxy spectral energy distributions on the galaxy mass, or contributions from different source populations, such as are considered to explain the CIB-CXB correlation (e.g., Yue et al. 2013).

The constraints from the baseline model are close to both theoretical expectations and previous work (e.g., Dunkley et al. 2011, G15). When considering a similar foreground model (except for the angular dependence of the tSZ-CIB correlation), G15 found Poisson power levels of 7.59 ± 0.69 μK² and 63.4 ± 9.5 μK² at 150 and 220 GHz, respectively. Note that since G15 reported powers at the effective frequency band centers instead of 150 and 220 GHz, to facilitate a comparison we have rescaled the reported numbers to 150 and 220 GHz using the median spectral index in this work. These two sets of constraints agree very closely (0.3σ or 3%–5%). It should be remembered that there is a large overlap between the underlying data, especially at 220 GHz, where only the relative weighting of the data has changed. For the clustered terms, the G15 numbers are 1.6 ± 0.9 μK² and 1.7 ± 0.3 μK² for the one- and two-halo terms, respectively. The agreement is still good: the one-halo term has increased by 0.7σ, while the two-halo term dropped by a smaller amount. The same trends continue at 220 GHz: the one-halo term increases by 0.5σ, while the two-halo term falls slightly. We note that the recovered CIB clustering power in G15 was slightly lower than previous measurements. Thus, the shifts here move toward those earlier measurements. The inferred spectral indices of this work are also within 1σ of the values in G15.

Radio power is detected at high significance at 95 and 150 GHz, with the addition of radio power to the model leading to a large improvement in the fit quality, namely, Δχ² = 5141 for two parameters. As in G15, the data prefer slightly less radio galaxy power than predicted by the De Zotti et al. (2005) model for a 6.4 mJy flux cut at 150 GHz. The preferred radio power at ℓ = 3000 is ${D}_{3000}^{r-150\times 150}=1.01\pm 0.17\,\,\mu {{\rm{K}}}^{2}$, about 25% lower than the 1.28 μK² predicted. The population spectral index for the radio power is constrained to be −0.76 ± 0.15. This is 1σ lower than the median spectral index of −0.60 for synchrotron-classified sources reported by Mocanu et al. (2013). This could be due to random chance, the 150 GHz only selection criteria for masking point sources in this work, or a tendency for the spectral index to flatten for the brightest 150 GHz radio sources, as argued by Mocanu et al. (2013).

As shown in Figure 3, we detect both tSZ and kSZ power. We measure ${D}_{3000}^{\mathrm{tSZ}}=3.42\pm 0.54\,\,\mu {{\rm{K}}}^{2}$ and ${D}_{3000}^{\mathrm{kSZ}}=3.0\pm 1.0\,\,\mu {{\rm{K}}}^{2}$ for the tSZ and kSZ power, respectively, at ℓ = 3000 and 143 GHz.

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The tSZ (kSZ) power is detected at approximately 7σ (3σ). While our fiducial results assume the Shaw tSZ template (Shaw et al. 2010) and CSF+patchy kSZ template (Z12, Shaw et al. 2012), the current data offer little information about the specific shape of the SZ spectra. The recovered SZ power levels for four different tSZ templates and three kSZ templates are reported in Table 3—no significant shifts are seen between the different templates considered. The tSZ power spectrum level is a probe of large-scale structure growth and the pressure profiles in galaxy clusters. The total kSZ power has contributions from the EoR and from the bulk flows of large-scale structure at later times; we discuss the implications of the kSZ measurement for reionization in Section 7.

Table 3. SZ Constraints

tSZ Template	kSZ Template	${D}_{3000}^{\mathrm{tSZ}}\,(\mu {{\rm{K}}}^{2})$	${D}_{3000}^{\mathrm{kSZ}}\,(\mu {{\rm{K}}}^{2})$	ξ
Shaw	CSF+patchy	3.42 ± 0.54	3.0 ± 1.0	0.076 ± 0.040
Shaw	CSF	3.39 ± 0.58	3.1 ± 1.3	0.077 ± 0.047
Shaw	Patchy	3.45 ± 0.56	3.5 ± 1.2	0.086 ± 0.050
Battaglia	CSF+patchy	3.74 ± 0.54	2.4 ± 1.0	0.051 ± 0.033
Bhattacharya	CSF+patchy	3.46 ± 0.54	3.0 ± 1.0	0.071 ± 0.036
Sehgal	CSF+patchy	3.59 ± 0.54	2.8 ± 1.0	0.064 ± 0.039
Shaw w. Bispectrum	CSF+patchy	3.53 ± 0.48	2.8 ± 0.9	0.069 ± 0.036

Note. Measured tSZ power, kSZ power, and tSZ-CIB correlation at ℓ = 3000 (and 143 GHz in the case of the tSZ) for different tSZ and kSZ models. The results are robust to the assumed templates. The first two columns indicate which of three templates has been used for the tSZ and kSZ terms. In the case of the kSZ, the three templates are the CSF homogeneous kSZ template (Shaw et al. 2012), the patchy kSZ template (Z12), or the sum of both. In the case of the tSZ, the three templates are taken from the Battaglia (Battaglia et al. 2013b), Shaw (Shaw et al. 2010), Bhattacharya (Bhattacharya et al. 2012), or Sehgal (Sehgal et al. 2010) simulations. The last row shows the results when a prior on the tSZ power based on the bispectrum measurement by Crawford et al. (2014) is added.

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The joint analysis of the SPTpol and SPT-SZ surveys allows the first detection (at 3σ) of kSZ power. The reported kSZ power in this work falls within the 95% CL upper limits on kSZ power reported in previous works (e.g., Dunkley et al. 2013, G15). G15 also report a central value when including a tSZ prior based on the bispectrum of ${D}_{3000}^{\mathrm{kSZ}}=2.9\pm 1.3\,\,\mu {{\rm{K}}}^{2}$, which agrees extremely well (although with 30% larger uncertainties) with the value in this work. If we add the same bispectrum-based tSZ prior to the current results, we find ${D}_{3000}^{\mathrm{kSZ}}=2.8\pm 0.9$ μK², which translates to a 3.1σ detection of kSZ power.

The joint analysis also significantly reduces the measurement uncertainties on the tSZ power. This tSZ measurement is consistent with (<1σ) earlier observations of the tSZ power scaled to 143 GHz: ${D}_{3000}^{\mathrm{tSZ}}={4.38}_{-1.04}^{+0.83}$ μK² (G15), ${D}_{3000}^{\mathrm{tSZ}}=4.20\pm 1.37$ μK² (R12), and ${D}_{3000}^{\mathrm{tSZ}}=3.9\pm 1.7\,\,\mu {{\rm{K}}}^{2}$ (Dunkley et al. 2013). With the same bispectrum-based prior, the preferred tSZ power in this work is ${D}_{3000}^{\mathrm{tSZ}}=3.53\pm 0.48$ μK².

We also consider the cosmological implication of the measured tSZ power, specifically on σ₈ since the tSZ power scales steeply with σ₈. We assume the relationship ${D}_{3000}^{\mathrm{tSZ}}\propto {\sigma }_{8}^{8.34}$ (Shaw et al. 2010). We consider three sets of models: the Shaw et al. (2010) model that forecasts a tSZ power at 143 GHz and ℓ = 3000 of 5.5 μK², the Bhattacharya et al. (2012) model predicting 5.0 μK², and the Battaglia et al. (2012) model predicting 5.9 ± 0.9 μK² (Dunkley et al. 2013). Ignoring the significant modeling uncertainty, the measured tSZ power favors σ₈ = 0.735 ± 0.013, 0.745 ± 0.013, and 0.730 ± 0.013, respectively. More fairly allowing for a 30% modeling uncertainty weakens the σ₈ constraints to σ₈ = 0.735 ± 0.027, 0.744 ± 0.027, and 0.730 ± 0.027, respectively. As noted by G15, the σ₈ levels inferred from the tSZ spectrum are limited by uncertainty in modeling.

The σ₈ value inferred from the observed tSZ power of approximately 0.735 ± 0.027 is consistent at <1σ with the σ₈ = 0.763 ± 0.037 found by Bocquet et al. (2019) from tSZ-selected galaxy cluster number counts in the SPT-SZ survey. However, as has been noted with earlier cluster results (e.g., Douspis et al. 2019), the σ₈ preferred by the tSZ power spectrum is significantly lower (2.7σ) than the Planck CMB-only result of σ₈ = 0.811 ± 0.006 (Planck Collaboration et al. 2018). It remains unclear whether this discrepancy is related to cluster astrophysics or cosmology.

The different CIB models considered in Table 2 have a modest impact on the inferred SZ power levels. For instance, assuming that the tSZ-CIB correlation is independent of angular multipole increases the uncertainties and shifts a small amount of power from the tSZ to kSZ effect. The tSZ power constraint moves from ${D}_{3000}^{\mathrm{tSZ}}=3.42\pm 0.54$ μK² to 3.30 ± 0.64 μK²; the kSZ power constraint moves from ${D}_{3000}^{\mathrm{kSZ}}=3.0\pm 1.0$ μK² to 3.5 ± 1.2 μK². Assuming a power law for the clustered CIB model leads to minor shifts: ${D}_{3000}^{\mathrm{tSZ}}=3.37\pm 0.55$ μK² and ${D}_{3000}^{\mathrm{tSZ}}=3.3\pm 1.1$ μK². Between the CIB models, the tSZ shifts are less than 0.3σ, while the kSZ shifts are less than 0.5σ.

These SZ constraints presume the nonrelativistic tSZ spectrum, which is an imperfect assumption for real galaxy clusters (Remazeilles et al. 2019). We estimate the potential magnitude of this correction by running a chain with the effective tSZ frequencies of each band calculated for a 5 keV tSZ spectrum. We find small shifts in the SZ power level and no change in the CIB terms. For the 5 keV spectrum without the bispectrum prior, the tSZ power increases by 0.5σ to ${D}_{3000}^{\mathrm{tSZ}}=3.70\pm 0.58$ μK². There is also a minor increase in the kSZ power by 0.2σ to 3.2 ± 1.0 μK². The uncertainties on both terms are essentially unchanged. Such small shifts do not substantially change the EoR results in Section 7.2.

We parameterize the tSZ-CIB correlation with a single parameter ξ that scales the Z12 template for the tSZ-CIB correlation as a function of ℓ. An overdensity of dusty galaxies in galaxy clusters would result in a positive value of ξ. The tSZ-CIB correlation is partially degenerate with the tSZ and kSZ power, as illustrated in Figure 4. Increasing the correlation, ξ, slowly decreases the inferred tSZ power while quickly increasing the inferred kSZ power. We measure the tSZ-CIB correlation to be ξ = 0.076 ± 0.040 at ℓ = 3000. The data prefer positive tSZ-CIB correlation, ruling out ξ < 0 at the 0.983 CL. For easier comparison to past works, we also run a chain with ξ that is constant in ℓ. This does very little to the inferred SZ power levels; the preferred values shift by 0.2σ and 0.3σ for the tSZ and kSZ, respectively. For a constant ξ, the bandpowers in this work favor ${D}_{3000}^{\mathrm{kSZ}}$ = 3.5 ± 1.2 μK² and ${D}_{3000}^{\mathrm{tSZ}}$ = 3.30 ± 0.64 μK². This is somewhat less (1σ) tSZ power than found by G15 in the equivalent case, and slightly more kSZ power (0.4σ). The ξ constraint is ξ = 0.078 ± 0.049. This is well within 1σ of past SPT constraints, $\xi ={0.100}_{-0.055}^{+0.069}$ (G15). It is also within the assumed prior range [0, 0.2] of Dunkley et al. (2013).

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