IMPROVED CONSTRAINTS ON COSMIC MICROWAVE BACKGROUND SECONDARY ANISOTROPIES FROM THE COMPLETE 2008 SOUTH POLE TELESCOPE DATA

The power spectrum analysis presented here depends on an accurate measurement of the beam function, which is the azimuthally averaged Fourier transform of the beam map. Due to the limited dynamic range of the detectors, the SPT beams for the 2008 observing season were measured by combining maps of three sources: Jupiter, Venus, and the brightest point source in each field. Maps of Venus are used to stitch together the outer and inner beam maps from Jupiter and the point source, respectively. Maps of Jupiter at radii >4' are used to constrain a diffuse, low-level sidelobe that accounts for roughly 15% of the total beam solid angle. The beam within a radius of 4' is measured on the brightest point source in the field. The effective beam is slightly enlarged by the effect of random errors in the pointing reconstruction. We include this effect by measuring the main beam from a source in the final map. The main-lobe beam is approximately fit by two-dimensional Gaussians with full width at half-maximum (FWHM) equal to 115 at 150 GHz and 105 at 220 GHz. We have verified that the measured beam is independent of the point source used, although the signal-to-noise drops for the other (dimmer) sources in either field. The beam measurement is similar to that described in L10, although the radii over which each source contributes to the beam map have changed. However, the estimation of beam uncertainties has changed. We now determine the top three eigenvectors of the covariance matrix for the beam function at each frequency, and include these as beam errors in the parameter fitting. These three modes account for >99% of the beam covariance power. The beam functions and the quadrature sum of the three beam uncertainty parameters are shown in Figure 1.

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The absolute calibration of the SPT data is based on a comparison of the CMB power at degree scales in the WMAP 5 year maps with dedicated SPT calibration maps. These calibration maps cover four large fields totaling 1250 deg² which were observed to shallow depth during the 2008 season. Details of the cross-calibration with WMAP are given in L10 and remain essentially unchanged in this work. Although the beam functions changed slightly at ℓ > 1000 as a result of the improved beam measurement described in Section 2.1, this change had a negligible effect at the angular scales used in the WMAP calibration. The calibration factors are therefore unchanged from L10; however as discussed above, we have slightly changed the treatment of beam and calibration errors and are no longer folding part of the beam uncertainty into a calibration uncertainty.

We estimate the uncertainty of the temperature calibration factor to be 3.5%, which is slightly smaller than the value of 3.6% presented in L10. We applied this calibration to the 220 GHz band by comparing 150 GHz and 220 GHz estimates of CMB anisotropy in the survey regions. This internal cross-calibration for 220 GHz is more precise than, but consistent with, a direct absolute calibration using observations of RCW38. We estimate the 220 GHz temperature calibration uncertainty to be 7.1%. Because the 220 GHz calibration is derived from the 150 GHz calibration to WMAP, the calibration uncertainties in the two bands are correlated with a correlation coefficient of approximately 0.5.

As discussed in Section 7.1, our Markov Chain Monte Carlo (MCMC) chains also include two parameters to adjust the overall temperature calibration in each SPT band. The resulting calibration factors are listed in the Table 1 notes.

Table 1. Single-frequency Bandpowers

ℓ Range	ℓeff	150 GHz		150 × 220 GHz		220 GHz
		$\hat{D}$ (μK2)	σ (μK2)	$\hat{D}$ (μK2)	σ (μK2)	$\hat{D}$ (μK2)	σ (μK2)
2001–2200	2056	242.1	6.7	248.8	8.3	295.9	14.9
2201–2400	2273	143.2	4.2	154.7	5.6	201.5	11.7
2401–2600	2471	109.3	3.2	122.1	4.5	193.5	11.0
2601–2800	2673	75.9	2.6	102.8	4.1	172.1	10.4
2801–3000	2892	60.2	2.3	80.4	3.7	179.3	11.1
3001–3400	3184	47.5	1.5	73.5	2.5	169.7	8.2
3401–3800	3580	36.9	1.6	69.5	2.7	180.7	9.2
3801–4200	3993	36.7	1.8	81.0	3.3	240.1	11.6
4201–4600	4401	38.5	2.2	81.5	3.9	226.6	12.6
4601–5000	4789	39.3	2.7	96.9	4.7	276.9	15.2
5001–5900	5448	49.2	2.5	122.3	4.2	361.3	13.3
5901–6800	6359	60.8	3.9	158.7	6.2	457.4	20.0
6801–7700	7256	89.1	6.0	173.9	9.5	488.8	28.4
7701–8600	8159	81.7	9.5	229.2	14.2	653.0	42.3
8601–9500	9061	131.9	14.4	309.1	21.9	895.2	66.0

Notes. Band multipole range and weighted value ℓ_eff, bandpower $\hat{D}$, and uncertainty σ for the 150 GHz auto-spectrum, cross-spectrum, and 220 GHz auto-spectrum of the SPT fields. The quoted uncertainties include instrumental noise and the Gaussian sample variance of the primary CMB and the point-source foregrounds. The sample variance of the SZ effect, beam uncertainty, and calibration uncertainty are not included. To include the preferred calibration from the Markov Chain Monte Carlo (MCMC) chains, these bandpowers should be multiplied by 0.92, 0.95, and 0.98 at 150 GHz, 150 × 220 GHz, and 220 GHz, respectively (see Section 7.1). Beam uncertainties are shown in Figure 1 and calibration uncertainties are quoted in Section 2.2. Point sources above 6.4 mJy at 150 GHz have been masked out in this analysis. This flux cut substantially reduces the contribution of radio sources to the bandpowers, although DSFGs below this threshold contribute significantly to the bandpowers.

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Each detector in the focal plane measures the CMB brightness temperature plus noise, and records this measurement as the TOD. Noise in the TOD contains contributions from the instrument and atmosphere. The instrumental noise is largely uncorrelated between detectors. Conversely, the atmospheric contribution is highly correlated across the focal plane, because the beams of individual detectors overlap significantly as they pass through turbulent layers in the atmosphere.

We bandpass filter the TOD to remove noise outside the signal band. The TOD are low-pass filtered at 12.5 Hz (ℓ ∼ 18000) to remove noise above the Nyquist frequency of the map pixelation. The TOD are effectively high-pass filtered by the removal of a Legendre polynomial from the TOD of each detector on a by-scan basis. The order of the polynomial is chosen to have the same number of degrees of freedom (dof) per unit angular distance (∼1.7 dof per degree). The polynomial fit removes low frequency noise contributions from the instrument and atmosphere.

Correlated atmospheric signals remain in the TOD after the bandpass filtering. We remove the correlated signal by subtracting the mean signal over each bolometer wedge²⁶ at each time sample. This filtering scheme is slightly different from that used by L10, although nearly identical in effect at 150 GHz. L10 subtracted a plane from all wedges at a given observing frequency at each time sample. Since SPT had two wedges of 220 GHz bolometers and three wedges of 150 GHz bolometers in 2008, the L10 scheme filters the 220 GHz data more strongly than the 150 GHz data. The wedge-based common mode removal implemented in this work results in more consistent filtering between the two observing bands.

The data from each bolometer are inverse noise weighted based on the calibrated, pre-filtering detector power spectral density. We bin the data into map pixels based on pointing information, and in the case of the 23^h30^m field, combine two pairs of lead and trail maps to form individual observations as discussed in Section 2. A total of 300 and 240 individual observation units are used in the subsequent analysis of the 5^h30^m and 23^h30^m fields, respectively.

We use a pseudo-C_ℓ method to estimate the bandpowers. In pseudo-C_ℓ methods, bandpowers are estimated directly from the Fourier transform of the map after correcting for effects such as TOD filtering, beams, and finite sky coverage. We process the data using a cross spectrum based analysis (Polenta et al. 2005; Tristram et al. 2005) in order to eliminate noise bias. Beam and filtering effects are corrected for according to the formalism in the MASTER algorithm (Hivon et al. 2002). We report the bandpowers in terms of $\mathcal {D}_\ell$, where

$$\begin{equation} \mathcal {D}_\ell =\frac{\ell \left(\ell +1\right)}{2\pi } C_\ell \;. \end{equation} \tag{ 1 }$$

The first step in the analysis is to calculate the Fourier transform $\tilde{m}^{(\nu _i,A)}$ of the map for each frequency ν_i and observation A. All maps of the same field are apodized by a single window that is chosen to mask out detected point sources and avoid sharp edges at the map borders. After windowing, the effective sky area used in this analysis is 210 deg². Cross-spectra are then calculated for each map pair on the same field; a total of 44,850 and 28,680 pairs are used in the 5^h30^m and 23^h30^m fields, respectively. We take a weighted average of the cross-spectra within an ℓ-bin, b,

$$\begin{equation} D^{\nu _i\times \nu _j, AB}_b\equiv \left\langle \frac{{\rm k}({\rm k}+1)}{2\pi }\tilde{m}^{(\nu _i,A)}_\textbf {k} \tilde{m}^{(\nu _j,B)*}_\textbf {k} \right\rangle_{k \in b}. \end{equation} \tag{ 2 }$$

As in L10, we find that a simple, uniform selection of modes at k_x > 1200 is close to the optimal mode weighting. With the adopted flat-sky approximation, ℓ = k.

The bandpowers above are band-averaged pseudo-C_ℓs that can be related to the true astronomical power spectrum $\widehat{D}$ by

$$\begin{equation} \widehat{D}^{\nu _i\times \nu _j,AB}_b\equiv (K^{-1})_{bb^\prime }D^{\nu _i\times \nu _j,AB}_{b^\prime }. \end{equation} \tag{ 3 }$$

The K matrix accounts for the effects of timestream filtering, windowing, and band-averaging. This matrix can be expanded as

$$\begin{equation} K^{\nu _i\times \nu _j}_{bb^\prime }=P_{bk}\big(M_{kk^\prime }[\textbf {W}]\,F^{\nu _i\times \nu _j}_{k^\prime }B^{\nu _i}_{k^\prime }B^{\nu _j}_{k^\prime }\big)Q_{k^\prime b^\prime }. \end{equation} \tag{ 4 }$$

Here Q_kb and P_bk are the binning and re-binning operators (Hivon et al. 2002). $B^{\nu _i}_{k}$ is the beam function for frequency ν_i, and $F^{\nu _i\times \nu _j}_{k}$ is the k-dependent transfer function which accounts for the filtering and map-making procedure. The mode mixing due to observing a finite portion of the sky is represented by $M_{kk^\prime }[\textbf {W}]$, which is calculated analytically from the known window $\textbf {W}$.

3.2.1. Transfer Function

We calculate a transfer function for each field and observing frequency. The transfer functions are calculated from simulated observations of 300 sky realizations that have been smoothed by the appropriate beam. Each sky realization is a Gaussian realization of the best-fit lensed WMAP7 ΛCDM model plus a Poisson point-source contribution. The Poisson contribution is selected to be consistent with the results of H10 and includes radio source and DSFG populations. The amplitude of the radio source term is set by the de Zotti et al. (2005) model source counts at 150 GHz with an assumed spectral index of α_r = −0.5. The DSFG term has an assumed spectral index of 3.8. The effective point-source powers are C^150x150_ℓ = 7.5 × 10⁻⁶ μK², C^150x220_ℓ = 2.3  ×  10⁻⁵ μK², and C^220x220_ℓ = 7.8  ×  10⁻⁵ μK². The sky realizations are sampled using the SPT pointing information, filtered identically to the real data, and processed into maps. The power spectrum of the simulated maps is compared to the known input spectrum, C^{ν, theory}, to calculate the effective transfer function (Hivon et al. 2002) in an iterative scheme,

$$\begin{equation} F^{\nu, (i+1)}_{k}=F^{\nu,(i)}_{k}+\frac{\big\langle D^{\nu }_{k}\big\rangle _{\rm MC} - M_{kk^\prime } F_{k^\prime }^{\nu,(i)} {B^\nu _{k^\prime }}^2 C^{\nu,{\rm theory}}_{k^\prime }}{{B^\nu _{k}}^2 C^{\nu,{\rm theory}}_{k} w_2}. \end{equation} \tag{ 5 }$$

Here $w_2= \int d\!x \textbf {W}^2$ is a normalization factor for the area of the window. We find that the transfer function estimate converges after the first iteration and use the fifth iteration. For both fields and both bands, the result is a slowly varying function of ℓ with values that range from a minimum of 0.6 at ℓ = 2000 to a broad peak near ℓ = 5000 of 0.8.

3.2.2. Bandpower Covariance Matrix

The bandpower covariance matrix includes two terms: variance of the signal in the field (cosmic variance) and instrumental noise variance. The signal-only Monte Carlo bandpowers are used to estimate the cosmic variance contribution. The instrumental noise variance is calculated from the distribution of the cross-spectrum bandpowers $D^{\nu _i\times \nu _j,AB}_b$ between observations A and B, as described in L10. We expect some statistical uncertainty of the form

$$\begin{equation} \langle(\textbf {c}_{ij}-\langle\textbf {c}_{ij}\rangle)^2\rangle=\frac{\textbf {c}_{ij}^2+\textbf {c}_{ii}\textbf {c}_{jj}}{n_{{\rm obs}}} \end{equation} \tag{ 6 }$$

in the estimated bandpower covariance matrix. Here, n_obs is the number of observations that go into the covariance estimate. This uncertainty on the covariance is significantly higher than the true covariance for almost all off-diagonal terms due to the dependence of the uncertainty on the (large) diagonal covariances. We reduce the impact of this uncertainty by "conditioning" the covariance matrix.

How we condition the covariance matrix is determined by the form we expect it to assume. The covariance matrix can be viewed as a set of nine square blocks, with the three on-diagonal blocks corresponding to the covariances of a 150 × 150, 150 × 220, or 220 × 220 GHz spectrum. Since the bandpowers reported in Tables 1 and 4 are obtained by first computing power spectra and covariance matrices for bins of width Δℓ = 100 with a total of 80 initial ℓ-bins, each of these blocks is an 80 × 80 matrix. The shape of the correlation matrix in each of these blocks is expected to be the same, as it is set by the apodization window. As a first step to conditioning the covariance matrix, we calculate the correlation matrices for the three on-diagonal blocks and average all off-diagonal elements at a fixed separation from the diagonal in each block:

$$\begin{equation} \textbf {c}^\prime _{kk^\prime }=\frac{ \sum _{k_1-k_2=k-k^\prime } \frac{\widehat{\textbf {c}}_{k_1k_2}}{\sqrt{\widehat{\textbf {c}}_{k_1k_1}\widehat{\textbf {c}}_{k_2k_2}}} }{\sum _{k_1-k_2=k-k^\prime } 1}. \end{equation} \tag{ 7 }$$

This averaged correlation matrix is then applied to all nine blocks.

The covariance matrix generally includes an estimate of both signal and noise variance. However, the covariance between the 150 × 150 and 220 × 220 bandpowers is a special case: since we neither expect nor observe correlated noise between the two frequencies in the signal band, we include only the signal variance in the two blocks describing their covariance.

3.2.3. Combining the Fields

We have two sets of bandpowers and covariances—one per field—which must be combined to find the best estimate of the true power spectrum. We calculate near-optimal weightings for this combination at each frequency and ℓ-bin using the diagonal elements of the covariance matrix:

$$\begin{equation} w^{\nu _i\times \nu _j}_b \propto 1\big/\big(C^{(\nu _i \times \nu _j) \times (\nu _i \times \nu _j)}_{bb}\big). \end{equation} \tag{ 8 }$$

This produces a diagonal weight matrix, w, for which the bandpowers can be calculated:

$$\begin{equation} \widehat{D}^{\nu _i\times \nu _j}_b = \sum _{\zeta } \big(w_\zeta ^{\nu _i\times \nu _j} \widehat{D}_\zeta ^{\nu _i\times \nu _j}\big)_b, \end{equation} \tag{ 9 }$$

where ζ denotes the field. The covariance will be

$$\begin{equation} \textbf {c}^{(\nu _i \times \nu _j) \times (\nu _m \times \nu _n) }_{bb^\prime } =\sum _{\zeta } \big(w_\zeta ^{\nu _i\times \nu _j} \textbf {c}_\zeta ^{(\nu _i \times \nu _j) \times (\nu _m \times \nu _n)} w_\zeta ^{\nu _m\times \nu _n}\big)_{bb^\prime }. \end{equation} \tag{ 10 }$$

After combining the two fields, we average the bandpowers and covariance matrix into the final ℓ-bins. Each initial Δℓ = 100 sub-bin gets equal weight.

We use the standard, six-parameter, spatially flat, lensed ΛCDM cosmological model to predict the primary CMB temperature anisotropy. The six parameters are the baryon density Ω_bh², the density of cold dark matter Ω_ch², the optical depth to recombination τ, the angular scale of the peaks Θ, the amplitude of the primordial density fluctuations ln [10¹⁰A_s], and the scalar spectral index n_s.

Gravitational lensing of CMB anisotropy by large-scale structure tends to increase the power at small angular scales, with the potential to influence the high-ℓ model fits. The calculation of lensed CMB spectra out to ℓ = 10, 000 is prohibitively expensive in computational time. Instead, we approximate the impact of lensing by adding a fixed lensing template calculated for the best-fit WMAP7 cosmology to unlensed CAMB³⁰ spectra. The error due to this lensing approximation is less than 0.5 μK² for the allowed parameter space at ℓ > 3000. This is a factor of 20 (260) times smaller than the secondary and foreground power at 150 (220) GHz and should have an insignificant impact on the likelihood calculation. We tested this assumption by including the full lensing treatment in a single chain with the baseline model described above. As anticipated, we found no change in the resulting parameter constraints.

We find that the data from SPT, WMAP7, ACBAR, and QUaD considered here prefer gravitational lensing at $\Delta {\rm ln} \mathcal { L}=-5.9 (3.4\sigma)$. Most of this detection significance is the result of the low-ℓ data and not the high-ℓ SPT data presented here; without including SPT data in fits, lensing is preferred at 3.0σ. In estimating the preference for gravitational lensing, we used the true lensing calculation instead of the lensing approximation described above.

We parameterize the secondary CMB anisotropy with two terms describing the amplitude of the tSZ and kSZ power spectra: D^tSZ₃₀₀₀ and D^kSZ₃₀₀₀. The model for the SZ terms can be written as

$$\begin{equation} D^{{\rm SZ}}_{\ell,\nu _1,\nu _2} = D_{3000}^{{\rm tSZ}} \frac{f_{\nu _1} f_{\nu _2} }{ f_{\nu _0}^2} \frac{\Phi _\ell ^{{\rm tSZ}}}{\Phi _{3000}^{{\rm tSZ}}} + D_{3000}^{{\rm kSZ}} \frac{ \Phi _\ell ^{{\rm kSZ}}}{\Phi _{3000}^{{\rm kSZ}}}. \end{equation} \tag{ 11 }$$

Here, Φ^X_ℓ denotes the theoretical model template for component X at frequency ν₀. The frequency dependence of the tSZ effect is encoded in f_ν; at the base frequency ν₀, $D^{{\rm tSZ}}_{3000,\nu _0,\nu _0} = D_{3000}^{{\rm tSZ}}$. The kSZ effect has the same spectrum as the primary CMB anisotropy, so its amplitude is independent of frequency. In this work, we set ν₀ to be the effective frequency of the SPT 150 GHz band (see Section 6.4).

6.2.1. tSZ Power Spectrum

We adopt four different models for the tSZ power spectrum. Following L10, we use the power spectrum predicted by S10 as the baseline model. S10 combined the semi-analytic model for the intracluster medium (ICM) of Bode et al. (2009) with a cosmological N-body simulation to produce simulated thermal and kinetic SZ maps from which the template power spectra were measured. The assumed cosmological parameters are (Ω_b, Ω_m, $\Omega _\Lambda$, h, n_s, σ₈) = (0.044, 0.264, 0.736, 0.71, 0.96, 0.80). At ℓ = 3000, this model predicts D^tSZ₃₀₀₀ = 7.5 μK² of tSZ power in the SPT 150 GHz band. We use this model in all chains where another model is not explicitly specified.

We also consider tSZ power spectrum models reported by Trac et al. (2011), Battaglia et al. (2010), and Shaw et al. (2010). Trac et al. (2011) followed a procedure similar to that of S10, exploring the thermal and kinetic SZ power spectra produced for different input parameters of the Bode et al. (2009) gas model. We adopt the nonthermal20 model (hereafter the Trac model) presented in that work, which differs from the S10 simulations by having increased star formation, lower energy feedback, and the inclusion of 20% non-thermal pressure support. It predicts a significantly smaller value of D^tSZ₃₀₀₀ = 4.5 μK² when scaled to the SPT 150 GHz band. The second template we consider is that produced by Battaglia et al. (2010) from their smoothed particle hydrodynamics simulations including radiative cooling, star formation, and active galactic nucleus (AGN) feedback (hereafter the Battaglia model). This model predicts D^tSZ₃₀₀₀ = 5.6 μK², intermediate between the baseline model and the Trac model, and peaks at slightly higher ℓ than either of those models. Shaw et al. (2010) investigate the impact of cluster astrophysics on the tSZ power spectrum using halo model calculations in combination with an analytic model for the ICM. We use the baseline model from that work (hereafter the Shaw model), which predicts D^tSZ₃₀₀₀ = 4.7 μK² in the 150 GHz band. The model of Shaw et al. (2010) is also used to rescale all the model templates as a function of cosmological parameters, as described in Section 8.

All four tSZ models exhibit a similar angular scale dependence over the range of multipoles to which SPT is sensitive (see Figure 5). We allow the normalization of each model to vary in all chains and detect similar tSZ power in all cases (see Table 6). However, as we discuss in Section 8, the difference between models is critical in interpreting the detected tSZ power as a constraint on cluster physics or σ₈.

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6.2.2. kSZ Power Spectrum

At ℓ > 3000, the measured anisotropy is dominated by astrophysical foregrounds, particularly bright radio point sources. We mask all point sources detected at >5σ (6.4 mJy at 150 GHz) when estimating the power spectrum. In order of importance after masking the brightest point sources, we expect contributions from DSFGs (including their clustering), radio sources (unclustered), and galactic cirrus. We parameterize these galactic and extragalactic foregrounds with an 11-parameter model. Five parameters describe the component amplitudes at ℓ = 3000: three point-source amplitudes in the 150 GHz band (clustering and Poisson for DSFGs and Poisson for radio) and a galactic cirrus amplitude in each of the two observation bands. One parameter describes the correlation between the tSZ and clustered DSFG terms. Three parameters describe the effective mean spectral indices of each point-source component, and the final two parameters describe the distribution about the mean spectral index for the Poisson radio source and DSFG populations.

The frequency dependencies of many of these foregrounds are naturally discussed and modeled in units of flux density (Jy) rather than in CMB temperature units. This is because the flux density of these foregrounds can be modeled as a power law in frequency, (S_ν∝ν^α). However, in order to model these foregrounds in the power spectrum, we need to determine the ratio of powers in CMB temperature units. To calculate the ratio of power in the ν_i cross ν_j cross-spectrum to the power at (the arbitrary frequency) ν₀ in units of CMB temperature squared, we multiply the ratio of flux densities by

$$\begin{equation} \epsilon _{\nu _1,\nu _2} \equiv \frac{\frac{dB}{dT}\big |_{\nu _0} \frac{dB}{dT}\big |_{\nu _0}}{\frac{dB}{dT}\big |_{\nu _1} \frac{dB}{dT}\big |_{\nu _2}}, \end{equation} \tag{ 12 }$$

where B is the CMB blackbody specific intensity evaluated at T_CMB, and ν_i and ν_j are the effective frequencies of the SPT bands. Note that ν_i may equal ν_j. The effective frequencies for each foreground calculated for the measured SPT bandpass (assuming a nominal frequency dependence) are presented in Section 6.4.

6.3.1. Poisson DSFGs

Poisson distributed point sources produce a constant C_ℓ (thus, D_ℓ∝ℓ²). A population of dim (≲ 1 mJy) DSFGs is expected to contribute the bulk of the millimeter wavelength point-source flux. This expectation is consistent with recent observations (Lagache et al. 2007; Viero et al. 2009; H10; Dunkley et al. 2011). Thermal emission from dust grains heated to tens of Kelvin by star light leads to a large effective spectral index (S_ν∝ν^α), where α > 2 between 150 and 220 GHz, for two reasons (e.g., Dunne et al. 2000). First, these frequencies are in the Rayleigh–Jeans tail of the blackbody emission for the majority of the emitting DSFGs. Second, the emissivity of the dust grains is expected to increase with frequency for dust grains smaller than the photon wavelength (2 mm for 150 GHz; Draine & Lee 1984; Gordon 1995).

Assuming that each galaxy has a spectral index α drawn from a normal distribution with mean α_p and variance σ²_p, then following Millea et al. (2011), the Poisson power spectrum of the DSFGs can be written as

$$\begin{equation} D^{p}_{\ell,\nu _1,\nu _2} = D_{3000}^{p} \epsilon _{\nu _1,\nu _2} \eta _{\nu _1,\nu _2} ^{\alpha _{p} + 0.5 {\rm ln}(\eta _{\nu _1,\nu _2}) \sigma _{p}^2} \left(\frac{\ell }{3000}\right)^2, \end{equation} \tag{ 13 }$$

where $\eta _{\nu _1,\nu _2} = (\nu _1 \nu _2 / \nu _0^2)$ is the ratio of the frequencies of the spectrum to the base frequency. D^p₃₀₀₀ is the amplitude of the Poisson DSFG power spectrum at ℓ = 3000 and frequency ν₀.

In principle, the frequency scaling of the Poisson power depends on both α_p and σ_p. H10 combine the dust spectral energy distributions (SEDs) from Silva et al. (1998) with an assumed redshift distribution of galaxies to infer σ_p ∼ 0.3 (although they conservatively adopt a prior range of σ_p ∈ [0.2, 0.7]). However, for α = 3.6, the ratio of Poisson power at 220 to 150 GHz only increases by 2% for σ_p = 0.3 versus σ_p = 0, which is small compared to the constraint provided by the current data. The complete model includes six parameters describing three measured Poisson powers in two bands. In order to reduce this to a reasonable number of parameters, we set σ_p = 0 in the baseline model. The equivalent Poisson radio galaxy spectral variance, σ_r, has a negligible effect on the radio power at 220 GHz and is also set to zero.

6.3.2. Clustered DSFGs

Because DSFGs trace the mass distribution, they are spatially correlated. In addition to the Poisson term, this leads to a "clustered" term in the power spectrum of these sources,

$$\begin{equation} D^{c}_{\ell,\nu _1,\nu _2} = D_{3000}^{c} \epsilon _{\nu _1,\nu _2} \eta _{\nu _1,\nu _2} ^{\alpha _{c}} \frac{\Phi _\ell ^{c}}{\Phi _{3000}^{c}}. \end{equation} \tag{ 14 }$$

D^c₃₀₀₀ is the amplitude of the clustering contribution to the DSFG power spectrum at ℓ = 3000 at frequency ν₀. The model for the clustering contribution to the DSFG power spectrum is Φ^c_ℓ, which we have divided by Φ^c₃₀₀₀ to make a unitless template which is normalized at ℓ = 3000.

We consider two different shapes for the angular power spectrum due to the clustering of the DSFGs shown in Figure 5. The first is the shape of the fiducial model used in Hall et al. (2010). This spectrum was calculated assuming that light is a biased tracer of mass fluctuations calculated in linear perturbation theory. Further, the model assumes that all galaxies have the same SED, that of a gray body, and the redshift distribution of the luminosity density was given by a parameterized form, with parameters adjusted to fit SPT, BLAST, and Spitzer power spectrum measurements. Here we refer to this power spectrum shape as the "linear theory" template.

On the physical scales of relevance, galaxy correlation functions are surprisingly well approximated by a power law. The second shape we consider is thus a phenomenologically motivated power law with index chosen to match the correlation properties observed for Lyman break galaxies at z ∼ 3 (Giavalisco et al. 1998; Scott & White 1999). We adopt this "power-law" model in the baseline model and use a template of the form Φ^c_ℓ∝ℓ^0.8 in all cases except as noted.

We also consider a linear combination of the linear-theory shape and the power-law shape. Such a combination can be thought of as a rough approximation to the two-halo and one-halo terms of a halo model. The present data set has limited power to constrain the ratio of the two terms; however, we do find a slight preference for the pure power-law model over the pure linear-theory model, as discussed in Sections 7.3.2 and 7.3.4.

The spectrum of the clustered component should be closely related to the spectrum of the Poisson component since they arise from the same sources (H10). However, the clustered DSFG term is expected to be more significant at higher redshifts, where the rest-frame frequency of the SPT observing bands is shifted to higher frequencies and out of the Raleigh–Jeans approximation for a larger fraction of the galaxies. One might therefore anticipate a slightly lower spectral index for the clustered term.

We assume a single spectral index for the Poisson and clustered DSFG components in the baseline model (α_c = α_p), and explore the impact of allowing the two spectral indices to vary independently in Section 7.3.1. In general, we require α_c = α_p, except where otherwise noted.

6.3.3. tSZ–DSFG Correlation

The tSZ signal from galaxy clusters and the clustered DSFG signal are both biased tracers of the same dark matter distribution, although the populations have different redshift distributions. Thus, it is natural to expect these two signals to be partially correlated and this correlation may evolve with redshift and halo mass. At frequencies below the tSZ null, this should result in spatially anti-correlated power. At 150 GHz, some models that attempt to associate emission with individual cluster member galaxies predict anti-correlations of up to tens of percent (S10). L10 argue against this being significant in the SPT 150 GHz band, because cluster members observed at low redshift have significantly less dust emission than field galaxies. It is, of course, possible for this emission to be significant at higher redshifts where much of the tSZ power originates. Current observations place only weak constraints on this correlation. (As we show in Section 7.4, the data presented here are consistent with an anti-correlation coefficient ranging from zero to tens of percent.) To the extent that the Poisson and clustered power spectra have the same spectral index, the differenced band powers discussed in Section 7.5 (as well as in L10) are insensitive to this correlation.

A correlation, γ(ℓ), between the tSZ and clustered DSFG signals introduces an additional term in the model,

$$\begin{eqnarray} &&D^{\hbox{\scriptsize tSZ-DSFG}}_{\ell,\nu _1,\nu _2} = \gamma (\ell) \big(\sqrt{D^{c}_{\ell,\nu _1,\nu _1} D^{{\rm tSZ}}_{\ell,\nu _2,\nu _2}} + \sqrt{D^{c}_{\ell,\nu _2,\nu _2} D^{{\rm tSZ}}_{\ell,\nu _1,\nu _1} } \big).\quad\ \ \ \end{eqnarray} \tag{ 15 }$$

We will assume no tSZ–DSFG correlation in all fits unless otherwise noted. In Section 7.4, we will investigate constraints on the degree of correlation and the impact on other parameters from allowing a DSFG–tSZ correlation.

6.3.4. Radio Galaxies

The brightest point sources in the SPT maps are coincident with known radio sources and have spectral indices consistent with synchrotron emission (Vieira et al. 2010, hereafter V10). Much like the Poisson DSFG term, the model for the Poisson radio term can be written as

$$\begin{equation} D^{r}_{\ell,\nu _1,\nu _2} = D_{3000}^{r} \epsilon _{\nu _1,\nu _2} \eta _{\nu _1,\nu _2} ^{\alpha _{r} + 0.5 {\rm ln}(\eta _{\nu _1,\nu _2}) \sigma _{r}^2} \left(\frac{\ell }{3000}\right)^2. \end{equation} \tag{ 16 }$$

The variables have the same meanings as for the treatment of the DSFG Poisson term. Clustering of the radio galaxies is negligible.

In contrast to DSFGs, for which the number counts climb steeply toward lower flux, S²dN/dS for synchrotron sources is fairly constant. As a result, the residual radio source power (D^r₃₀₀₀) in the map depends approximately linearly on the flux above which discrete sources are masked. For instance, if all radio sources above a signal-to-noise threshold of 5σ (6.4 mJy) at 150 GHz are masked, the de Zotti et al. (2005) source count model predicts a residual radio power of D^r₃₀₀₀ = 1.28 μK². For a 10σ threshold, the residual radio power would be D^r₃₀₀₀ = 2.59 μK².

The de Zotti model is based on radio surveys such as the NRAO VLA Sky Survey extrapolated to 150 GHz using a set spectral index for each subpopulation of radio sources. Extrapolating over such a large frequency range obviously introduces significant uncertainties; however, we do have an important cross-check on the modeling from the number counts of bright radio sources detected in the SPT or ACT maps (V10; Marriage et al. 2011). The de Zotti number counts appear to overpredict the number of very high flux radio sources reported by both experiments. We find that the de Zotti model should be multiplied by 0.67 ± 0.14 to match the number of >20 mJy radio sources in the SPT source catalog (V10). There are relatively few radio sources at these fluxes, and V10 and Marriage et al. (2011) have partially overlapping sky coverage. The counting statistics on these relatively rare objects will be improved significantly with the full SPT survey. The discrepancy disappears for lower flux radio sources and we find that, for 6.4 mJy < S₁₅₀ < 20 mJy 150 GHz, the best-fit normalization of the de Zotti model to fit the radio sources in the V10 catalog is 1.05 ± 0.14. Throughout this work, we use the de Zotti model prediction with a 15% uncertainty as a prior for residual power from sources below 6.4 mJy. Relaxing this radio prior by expanding the 1σ amplitude prior about the de Zotti et al. (2005) model from 15% to 50% has no significant impact on resulting parameter constraints.

We analyze the two-frequency V10 catalog to determine the radio population mean spectral index (α_r) and scatter in spectral indices (σ_r) at these frequencies. For each source, we take the full (non-Gaussian) likelihood distribution for the source spectral indices from V10. We combine all sources above 5σ at 150 GHz with a probability of having a synchrotron-like spectrum of at least 50% in order to calculate the likelihoods of a given α_r and σ_r. (We note that the results are unaffected by the specific choice of threshold probability for being a synchrotron source.) The best-fit mean spectral index is α_r = −0.53 ± 0.09 with a 2σ upper limit on the intrinsic scatter of σ_r < 0.3. With the present power spectrum data, both this uncertainty and intrinsic scatter are undetectable. We have checked this assumption by allowing the spectral index to vary with a uniform prior between α_r ∈ [ − 0.66, −0.21] instead of fixing it to the best-fit value of −0.53 for several specific cases and found no significant difference in the resulting parameter constraints. We therefore assume α_r = −0.53 and σ_r = 0 for all cases considered.

6.3.5. Galactic Cirrus

In addition to the foregrounds discussed above, we include a fixed galactic cirrus term parameterized as

$$\begin{equation} D^{{\rm cir}}_{\ell,\nu _1,\nu _2} = D_{3000}^{{\rm cir}, \nu _1,\nu _2} \left(\frac{\ell }{3000}\right)^{-1.5}. \end{equation} \tag{ 17 }$$

H10 measured the Galactic cirrus contribution in the 5^h30^m field by cross-correlating the SPT map and model eight of Finkbeiner et al. (1999, hereafter FDS) for the same field. We extend that analysis to the combined data set by scaling the 5^h30^m cirrus estimate by the ratio of the auto-correlation power in the FDS map of each field. We find the 23^h30^m field has 15% of the cirrus power measured in the 5^h30^m field. As the fields are combined with equal weight, we expect the effective cirrus amplitude in the combined bandpowers to be 0.575 times the 5^h30^m amplitudes reported above. Based on this analysis, we fix the amplitude of the cirrus term, D_ℓ∝ℓ^−1.5, in all parameter fits to be (D^{cir, 150, 150}₃₀₀₀, D₃₀₀₀^{cir, 150, 220}, D^{cir, 220, 220}₃₀₀₀) = (0.35, 0.86, 2.11) μK², respectively. Note that the cirrus power has a dust-like spectrum and is effectively removed in the differenced bandpowers discussed in Section 7.5. We assume there is zero residual cirrus power in the DSFG-subtracted bandpower analyses. Although we include this cirrus prior in the multi-frequency fitting, it has no significant impact on results; marginalizing over the amplitudes in the two bands or setting them to either extreme (zero or the high value in the 5^h30^m field) does not change the resulting parameter values.

Throughout this work, we refer to the SPT band center frequencies as 150 and 220 GHz. The data are calibrated to CMB temperature units, hence the effective frequency is irrelevant for a CMB-like source. However, the effective band center will depend (weakly) on the source spectrum for other sources. The actual SPT spectral bandpasses are measured by Fourier transform spectroscopy, and the effective band center frequency is calculated for each source, assuming a nominal frequency dependence. For an α = −0.5 (radio-like) source spectrum, we find band centers of 151.7 and 219.3 GHz. For an α = 3.8 (dust-like) source spectrum, we find band centers of 154.2 and 221.4 GHz. The band centers in both cases drop by ∼0.2 GHz if we reduce α by 0.3. We use these radio-like and dust-like band centers to calculate the frequency scaling between the SPT bands for the radio and DSFG terms. For a tSZ spectrum, we find band centers of 153.0 and 220.2 GHz. The 220 band center is effectively at the tSZ null; the ratio of tSZ power at 220 to 150 GHz is 0.2%.

As discussed in Section 6, the baseline secondary anisotropy and foreground model contains four free parameters (beyond the six ΛCDM parameters): the tSZ amplitude, the Poisson DSFG amplitude, the clustered DSFG amplitude, and the DSFG spectral index. For the tSZ component, we use the S10 tSZ model scaled by a single amplitude parameter. For the clustered DSFG component, we use the power-law model presented in Section 6.3.2, again scaled by an amplitude parameter. As discussed in Section 7.3.4, the data show a slight preference for the power-law clustered point-source shape model compared to the linear-theory model that was adopted in L10. We require that both the clustered and Poisson DSFG components have the same spectral index, which we model as the free parameter α_p, with no intrinsic scatter (σ_p = 0). Table 2 shows the improvement in the quality of the fits with the sequential introduction of free parameters to the original ΛCDM primary CMB model. Beyond this baseline model, adding additional free parameters does not improve the quality of the fits.

Table 2. Δχ² with the Addition of Model Components

Variable	dof	$-2\Delta {\rm ln} \mathcal {L}$
Point sources (Poisson)	2	−7821
Point sources (including clustering)	1	−104
tSZ	1	−25
kSZ	0	−1

Notes. The addition of DSFG and SZ terms in the model significantly improves the quality of the maximum likelihood fit. From top to bottom, the improvement in the fit χ² as each term is added to a model including the primary CMB anisotropy and all the terms above it, e.g., the tSZ row shows the improvement from adding the tSZ effect with a free amplitude to a model including the CMB, and point-source power, including the contribution from clustering of the DSFGs. The second column (dof) lists the number of degrees of freedom in each component. The free Poisson point-source parameters are the amplitude and spectral index of the DSFGs. Note that the third Poisson component (the radio power) is constrained by an external prior rather than the data and we have marginalized over the allowed radio power. The free clustered point-source parameter is the amplitude of the power-law clustered template. The spectral index of the DSFG clustering term is fixed to that of the Poisson term. The last two rows mark the addition of the S10 tSZ template with a free amplitude, and the addition of the S10 homogeneous kSZ template with a fixed amplitude. We do not find a significant improvement in the likelihood with any extension to this model, including allowing the kSZ amplitude to vary, or allowing independent spectral indices for the two DSFG components.

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The additional parameters of the model discussed in Section 6 are all fixed or constrained by priors. The kSZ power is fixed to the D^kSZ₃₀₀₀ = 2.05 μK² expected for the homogeneous reionization model of Sehgal et al. (2010). Radio galaxies are included with a fixed spectral index α_r = −0.53 and zero intrinsic scatter and an amplitude D^r₃₀₀₀ = 1.28 μK² with a 15% uncertainty. Here, and throughout this work, we also include a constant Galactic cirrus power as described in Section 6.3.5. In addition to astrophysical model components, for each band we include a term to adjust the overall temperature calibration and three terms to parameterize the uncertainty in the measurement of the beam function. This baseline model provides the most concise interpretation of the data. As will be demonstrated in subsequent sections, relaxing priors on the fixed or tightly constrained parameters in this model has minimal impact on the measured tSZ power.

Parameter constraints from this baseline model are listed in Table 3 and Figures 6 and 7. For the tSZ power, we find D^tSZ₃₀₀₀ = 3.5 ± 1.0 μK², which is consistent with previous measurements (L10; Dunkley et al. 2011) and significantly lower than predicted by the S10 model. We detect both the Poisson and clustered DSFG power with high significance, with amplitudes of D^p₃₀₀₀ = 7.4 ± 0.6 μK² and D^c₃₀₀₀ = 6.1 ± 0.8 μK², respectively. The shared spectral index of these components is found to be α = 3.58 ± 0.09, in line with both theoretical expectations and previous work (H10; Dunkley et al. 2011). The amplitude of the Poisson term is significantly lower than what was measured in H10; however, as we show in the following sections, this discrepancy is eliminated by adopting the linear-theory clustered DSFG template used in that work.

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Table 3. Parameters from Multi-frequency Fits

Model	DtSZ3000	DkSZ3000	DP3000	DC3000	αP	αC	DP, 2203000	DC, 2203000
Baseline model	3.5 ± 1.0	[2.0]	7.4 ± 0.6	6.1 ± 0.8	3.58 ± 0.09	⋅⋅⋅	70 ± 4	57 ± 8
Patchy kSZ	2.9 ± 1.0	[3.3]	7.4 ± 0.6	5.7 ± 0.8	3.62 ± 0.09	⋅⋅⋅	72 ± 4	55 ± 8
Free kSZ	3.2 ± 1.3	2.4 ± 2.0	7.4 ± 0.6	5.9 ± 1.0	3.59 ± 0.11	⋅⋅⋅	71 ± 5	57 ± 9
Free αc	3.4 ± 1.0	[2.0]	7.9 ± 1.0	5.4 ± 1.4	3.46 ± 0.21	3.79 ± 0.37	69 ± 5	59 ± 8
Linear-theory clustering	3.5 ± 1.0	[2.0]	9.3 ± 0.5	4.9 ± 0.7	3.59 ± 0.09	⋅⋅⋅	89 ± 5	47 ± 7
Free αc and linear theory	3.4 ± 1.0	[2.0]	9.2 ± 0.7	5.3 ± 1.4	3.62 ± 0.14	3.47 ± 0.34	90 ± 5	46 ± 7
tSZ–DSFG correlation	3.4 ± 1.0	[2.0]	9.3 ± 0.7	4.9 ± 0.8	3.60 ± 0.13	⋅⋅⋅	89 ± 5	46 ± 7
tSZ–DSFG cor, free kSZ	1.8 ± 1.4	5.5 ± 3.0	9.1 ± 0.7	4.4 ± 0.9	3.62 ± 0.14	⋅⋅⋅	89 ± 5	43 ± 8
Priors and Templates
Model	Npars	Free αc	Free DkSZ3000	tSZ Model	kSZ Model	DG Clustered	$-2\Delta {\rm ln} \mathcal {L}$
						Model
Baseline model	10	No	No	S10	S10	power law	⋅⋅⋅
Patchy kSZ	10	No	No	S10	Patchy	power law	1
Free kSZ	11	No	Yes	S10	S10	power law	0
Free αc	11	Yes	No	S10	S10	power law	0
Linear-theory clustering	10	No	No	S10	S10	linear theory	4
Free αc and linear theory	11	Yes	No	S10	S10	linear theory	4
tSZ–DSFG cor	11	No	No	S10	S10	linear theory	4
tSZ–DSFG cor, free kSZ	12	No	Yes	S10	S10	linear theory	4

Notes. Parameter constraints from multi-frequency fits to the models described in Section 6. The amplitudes are $\mathcal {D}_\ell$ at ℓ = 3000 for the 150 GHz band, in units of μK². In all cases, the error specified is one-half of the 68% probability width for a given parameter after marginalizing over other parameters as described in the text. Where α_c is unspecified, both the clustered and Poisson DSFG components share an identical spectral index given by α_p. The majority of the chains use a fixed kSZ spectrum; in these cases, the kSZ value appears in brackets. In addition to the model parameters described in Section 6, we include two derived parameters, D^{P, 220}₃₀₀₀ and D^{C, 220}₃₀₀₀, which are the amplitude of the Poisson and clustered DSFG component in the SPT 220 GHz band, respectively. We also summarize the free parameters, templates, and relative goodness of fit for each case.

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We explore the effect of altering this baseline model by replacing the tSZ template with the three alternatives discussed in Section 6.2.1: the Shaw, Trac, and Battaglia models. As anticipated, we find no significant difference between the models for any of the measured parameters. The resulting tSZ powers are listed in Table 6; they are statistically identical. The other model parameters also vary by less than 1% for all four tSZ model templates.

The baseline kSZ model is the homogeneous reionization model presented in S10. In this section, two alternative kSZ cases are considered. First, for the same set of model parameters, we change the (fixed) kSZ power spectrum from the baseline homogeneous reionization model to a template with additional power from patchy reionization. In the second case, we allow the kSZ amplitude to vary in order to explore the tSZ–kSZ degeneracy and to set upper limits on the SZ terms.

We explore the effect of replacing the homogeneous kSZ template with the patchy kSZ template described in Section 6.2.2. The patchy kSZ model increases the power at ℓ = 3000 by 50% to D^kSZ₃₀₀₀ = 3.25 μK² and has a slightly modified angular scale dependence with additional large-scale power. The likelihood of the best-fit model is nearly unchanged from the homogeneous kSZ case. The increase in kSZ power by 1.2 μK² leads to a decrease in tSZ power by 0.6 μK², as one would expect given the near degeneracy along the tSZ + 0.5 kSZ axis which we discuss below. The kSZ spectrum shape difference has no impact on the results.

As an alternative to the fixed kSZ models discussed above, we also allow the kSZ amplitude to vary, adding one additional parameter to MCMC chains. With two frequencies, the current SPT data does not allow the separation of the tSZ, kSZ, and clustered point-source components. The chains show a strong degeneracy between the components, as shown in Figure 8. The best-constrained eigenvector in the kSZ/tSZ plane is approximately tSZ + 0.5 kSZ, very similar to the proportionality used in the differenced bandpowers in Section 7.5. For this linear combination, we find D^tSZ₃₀₀₀ + 0.5 D^kSZ₃₀₀₀ = 4.5 ± 1.0 μK². We note that the data show some preference for a positive tSZ power, at just under 2σ. We tested whether this preference is due to the detailed shapes of the two templates by swapping the tSZ and kSZ templates. This model swap had no significant impact on the two-dimensional likelihood surface, indicating that the preference for positive tSZ power is driven by the unique frequency signature of the tSZ. When marginalizing over the tSZ amplitude and other parameters, we find the upper limit on the kSZ power to be D^kSZ₃₀₀₀ < 6.5 μK² at 95% confidence. The corresponding upper limit on tSZ power is D^tSZ₃₀₀₀ < 5.3 μK² at 95% confidence.

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The baseline model contains DSFG sources that generate both Poisson and clustering contributions to the power spectrum. The spectral index of the Poisson power and clustering power is fixed to be the same. In this section, we examine two changes to this baseline model. First, we allow the clustered DSFG spectral index to vary independently and second, we change the clustered template from a power-law to linear-theory model. We explore all four permutations of these two Boolean cases (one of the four is the baseline case). One-dimensional likelihood curves for each case are shown in Figure 9 and discussed below. We also consider a mixture of the power-law and linear-theory templates, each with its own variable amplitude, again for a total of five free parameters. As we will discuss in Section 7.3.4, the SPT data slightly prefer the power-law template.

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7.3.1. Free α_c

7.3.2. Linear-theory Model

7.3.3. Free α_c & Linear-theory Model

7.3.4. Discriminating between the Power-law and Linear-theory Models

In principle, the likelihood of a fit to the data can be used to distinguish between the power-law and linear-theory clustered DSFG templates described above. We explore this in two ways. First we consider the improvement in the best-fit log-likelihood between the models which use each template. The power-law model produces a marginally better fit to the data, with $\Delta {\rm ln}\, \mathcal {L} = -2$ when compared to the linear-theory model.

In addition to the choice of one template or the other, we extend the baseline model to include a second DSFG clustering component. The two DSFG clustered amplitudes, corresponding to the power-law and the linear-theory templates, are allowed to vary independently. The spectral index for all three (Poisson and two clustered) DSFG components are tied. The two-dimensional likelihood surface for the amplitudes of each template is shown in Figure 10. The data show a slight preference for the power-law template over the linear-theory template; however, we can not rule out the linear-theory model or a linear combination of the two with any confidence. We can expect this constraint on DSFG clustering shape to improve with future data.

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As discussed in Section 6.3.3, there are reasons to expect that the signal from the tSZ and the clustered DSFGs may be correlated. In order to study that correlation, we extend the model to include the correlation coefficient of these components, γ(ℓ), detailed in Section 6.3.3. We assume that the correlation coefficient is constant in ℓ, thus reducing γ(ℓ) to γ. We would expect an anticorrelation at frequencies below the SZ null; however, we do not require that the correlation coefficient is negative. Instead, we set a flat prior on the correlation γ and allow it to range from −1 to 1.

We consider the impact of introducing correlations between the tSZ and DSFGs on two chains—both allow the correlation factor to vary and one also allows the kSZ amplitude to vary. In both chains, we assume the linear-theory clustered DSFG template, because its shape is similar to the S10 tSZ template.

The results for this model without the possibility of a tSZ–DSFG correlation have been presented in Section 7.3.2. The marginalized one-dimensional likelihood function for the tSZ–DSFG correlation factor is shown in Figure 11. In the fixed kSZ case, the data are consistent with zero correlation: γ = 0.0  ±  0.2. When the kSZ power is allowed to vary, the data remain consistent with zero correlation, however, large anti-correlations are also possible. The 95% CL upper limit on D^tSZ₃₀₀₀ allowing for a tSZ–DSFG correlation is unchanged from the equivalent chain with the correlation coefficient set to zero; marginalizing over a non-zero tSZ–DSFG correlation factor does not allow additional tSZ power. As Figure 12 shows, this is because kSZ power increases for larger negative correlation in order to preserve the observed power in the 150 × 220 bandpowers. Naively, one might expect the best-fit tSZ power to increase when the possibility of a tSZ canceling correlation is taken into account; however, in practice, a negative correlation leads to less tSZ power in the best-fit model. The anti-correlation reduces the predicted power in the 150  ×  220 spectrum, and kSZ power is increased to compensate. The sum of kSZ and tSZ is well constrained by the data, so more kSZ power leads to less tSZ power. When marginalizing over the tSZ–DSFG correlation, the 95% CL on tSZ decreases slightly to D^tSZ₃₀₀₀ < 4.5 μK², while the kSZ upper limit increases to D^kSZ₃₀₀₀ < 9.7 μK². From this, as well as the agreement between the multi-frequency fits and the DSFG-subtracted result presented in Section 7.5, we conclude that the addition of a tSZ–DSFG correlation cannot explain the discrepancy between the observed tSZ power and models that predict a significantly larger tSZ contribution.

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These results depend on the Poisson and clustered DSFG terms having similar spectral indices. We expect the upper limits to be substantially weakened if the two spectral indices are allowed to separate, but note that the required spectral index separation would need to be much larger than the α_p–α_c ∼ 0.2 difference expected from models (H10).

The main result of this work is the set of multi-frequency bandpowers. However, we also follow the treatment in L10 and consider a linear combination of the two frequency maps designed to eliminate the foreground contribution from DSFGs. Specifically, this is the power spectrum of the map m constructed from the 150 and 220 GHz maps (m₁₅₀ and m₂₂₀, respectively) according to

$$\begin{equation} m = \frac{1}{1-x} (m_{150} - x \,m_{220}). \end{equation} \tag{ 18 }$$

In L10, the subtraction coefficient x was selected by minimizing the Poisson point-source power in the resultant bandpowers. We repeat this analysis for the new bandpowers and find x = 0.312 ± 0.06. This is consistent with the value of x = 0.325 ± 0.08 reported in L10. For complete details on the analysis, we refer the reader to that work. Given the consistency of these results, we have chosen to use x = 0.325 to create the "DSFG-subtracted" bandpowers presented in Table 4 to enable the direct comparison of the two data sets. These two subtraction coefficients correspond to a spectral index for the DSFG population of α_p = 3.7 and 3.6, respectively, consistent with the α_p = 3.57 ± 0.09 determined in multi-frequency fits. We compare the L10 bandpowers with those presented in this work in Figure 13 and find excellent agreement between them. Note that the two data sets are not independent since L10 analyzed one of the two fields used in this work.

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Table 4. DSFG-subtracted Bandpowers

ℓ range	ℓeff	$\hat{D}$ (μK2)	σ (μK2)
2001–2200	2056	245.0	9.6
2201–2400	2273	140.3	6.9
2401–2600	2471	110.5	6.0
2601–2800	2674	59.8	5.1
2801–3000	2892	59.0	5.4
3001–3400	3185	38.9	3.9
3401–3800	3581	23.8	4.3
3801–4200	3994	20.7	5.4
4201–4600	4402	20.9	6.7
4601–5000	4789	12.3	7.9
5001–5900	5449	17.4	7.3
5901–6800	6360	13.0	11.8
6801–7700	7257	60.7	18.4
7701–8600	8161	3.6	28.6
8601–9500	9063	56.2	44.8

Notes. Band multipole range and weighted value ℓ_eff, bandpower $\hat{D}$, and uncertainty σ for the DSFG-subtracted auto-spectrum of the SPT fields. This is the power spectrum of a linear combination of the 150 and 220 GHz maps ((m₁₅₀ − 0.325 m₂₂₀)/(1–0.325)) designed to suppress the DSFG contribution. The quoted uncertainties include instrumental noise and the Gaussian sample variance of the primary CMB and the point-source foregrounds. The sample variance of the SZ effect, beam uncertainty, and calibration uncertainty are not included. Beam uncertainties are shown in Figure 1 and calibration uncertainties are quoted in Section 2.2. Point sources above 6.4 mJy at 150 GHz have been masked out in this analysis, eliminating the majority of radio source power.

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This "DSFG-subtracted" power spectrum is interesting for two reasons. First, it allows a direct comparison with the results of L10. Second, in the DSFG-subtracted bandpowers, the residual correlated DSFG power will only be significant if the frequency spectrum of DSFGs in galaxy clusters is substantially different than the spectrum of DSFGs in the field. Therefore, parameter fits to DSFG-subtracted bandpowers are independent of the assumed DSFG power spectrum shape or tSZ–DSFG correlation, and are a potentially interesting cross-check on the multi-frequency fits. However, in practice, it is worth noting that these fits constrain a similar linear combination of tSZ and kSZ, roughly given by tSZ + 0.5 kSZ. Both methods therefore result in a similarly robust detection of tSZ power independent of tSZ–DSFG correlation.

For this DSFG-subtracted case, we simplify the baseline model to include only one free parameter beyond ΛCDM: the tSZ power spectrum amplitude. We also include a Poisson point-source term with a strict prior and a fixed kSZ power spectrum. The S10 model is used for both kSZ and tSZ. The Poisson prior is identical to the point source prior described in L10 with one exception: the radio source amplitude prior has been tightened to match the 15% prior used in this work instead of the 50% log-normal prior used in L10. However, tests with relaxed priors show the results are insensitive to this radio uncertainty. Except for this small change in the radio prior, and the switch to WMAP7 data, these differenced spectrum fits are identical to the parameter fitting used in L10.

As shown in Figure 6, the resulting DSFG-subtracted tSZ amplitude agrees with both the L10 value and the multi-frequency result presented here. We measure the tSZ power at 150 GHz to be D^tSZ₃₀₀₀ = 3.5 ± 1.2 μK², with a fixed (S10 homogeneous) kSZ component. For the same template choices, L10 report D^tSZ₃₀₀₀ = 3.2 ± 1.6 μK² and the multi-frequency fit to the baseline model finds D^tSZ₃₀₀₀ = 3.5 ± 1.0 μK². This close agreement demonstrates the consistency of the two methods of parameter fitting and confirms the results presented in L10.

L10 found the measured amplitude of the tSZ power spectrum to be a factor of 0.42 ± 0.21 lower than that predicted by the simulations of S10 (at their fiducial cosmology). In this analysis, having doubled the survey area to 210 deg², we find that the amplitude of the tSZ power spectrum at ℓ = 3000 is D^tSZ₃₀₀₀ = 3.5 ± 1.0 μK² at 150 GHz. This is 0.47 ± 0.13 times the S10 template.

The discrepancy suggests that either the S10 model overestimates the tSZ power or the assumed set of cosmological parameters may be wrong. To calculate the tSZ power spectrum, S10 combined a semi-analytic model for intra-group and cluster gas with a cosmological N-body lightcone simulation charting the matter distribution out to high redshift. The gas model assumed by S10 was calibrated by comparing it with X-ray observations of low-redshift (z < 0.2) clusters. However, the majority of tSZ power is contributed by groups or clusters at higher redshift (Komatsu & Seljak 2002; Trac et al. 2011). Hence, the S10 model may substantially overpredict the contribution from higher redshift structures. Note that in Section 7.4, we demonstrated that the lower-than-predicted tSZ power is unlikely to be due to a spatial correlation between DSFGs and the groups and clusters that dominate the tSZ signal.

As described in Section 6.2.1, in addition to the S10 template we investigate results obtained using the templates suggested by Trac et al. (2011), Shaw et al. (2010), and Battaglia et al. (2010) (the Trac, Shaw, and Battaglia models). These three alternative models all predict substantially less power at ℓ = 3000 than the S10 template (assuming the same cosmological parameters). For each template we run a new chain substituting the S10 template with one of the three others. The rest of the baseline model remains as described in Section 6.

Alternatively, the amplitude of the tSZ power spectrum is extremely sensitive to σ₈, scaling approximately as D^tSZ_ℓ∝σ₈^7–9(Ω_bh)². The exact dependence on σ₈ depends on the relative contribution of groups and clusters as a function of mass and redshift to the power spectrum, and thus the astrophysics of the ICM (Komatsu & Seljak 2002; Shaw et al. 2010; Trac et al. 2011). All four of the templates that we consider in this work were determined assuming σ₈ = 0.8. The deficit of tSZ power measured in the SPT maps may therefore indicate a lower value of σ₈.

To correctly compare the measured tSZ power spectrum amplitude with theoretical predictions, we must rescale the template power spectra so that they are consistent with the cosmological parameters at each point in the chains. Recall that the cosmological parameters are derived solely from the joint WMAP7+QUaD+ACBAR+SPT constraints on the primary CMB power spectrum and so are not directly influenced by the amplitude of the measured tSZ power.

We define a dimensionless tSZ scaling parameter, A_tSZ, where

$$\begin{equation} A_{\rm tSZ}(\kappa _i) = \frac{D_{3000}^{{\rm tSZ}}}{\Phi _{3000}^{{\rm tSZ}}(\kappa _i)} \;. \end{equation} \tag{ 19 }$$

D^tSZ₃₀₀₀ is the measured tSZ power at ℓ = 3000 and Φ^tSZ₃₀₀₀(κ_i) is the template power spectrum normalized to be consistent with the set of cosmological parameters "κ_i" at step i in the chain. We obtain the preferred value of A_tSZ by combining the measurement of D^tSZ₃₀₀₀ with the cosmological parameter constraints provided by the primary CMB measurements.

To determine the cosmological scaling of each of the templates, we adopt the analytic model described in Shaw et al. (2010). This model uses the halo mass function of Tinker et al. (2008) to calculate the abundance of dark matter halos as a function of mass and redshift, combined with an analytic prescription (similar to that of Bode et al. 2009) to determine the SZ signal from groups and clusters. Shaw et al. (2010) demonstrated that when the same assumptions are made about the stellar mass content and feedback processes in groups and clusters, their model almost exactly reproduces the S10 template. For the Trac and Battaglia models, we modify the astrophysical parameters of the Shaw model so that it agrees with the other models at their fiducial cosmology. Varying the input cosmological parameters thus allows us to estimate the cosmological scaling of the tSZ power spectrum for any given model.

As an indication of the relative cosmological scaling of each model at ℓ = 3000, in Table 5 we give the dependence of Φ_tSZ on σ₈, Ω_b, and h around their fiducial values. We assume that the cosmological scaling takes the form

$$\begin{equation} \Phi ^{\rm tSZ}_{3000} \propto \left(\frac{\sigma _8}{0.8}\right)^\alpha \left(\frac{\Omega _b}{0.044}\right)^\beta \left(\frac{h}{0.71}\right)^\gamma \;. \end{equation} \tag{ 20 }$$

Note that the values of α given in Table 5 for the S10 and Trac models agree well with those given in Trac et al. (2011). We find the templates depend only weakly on other cosmological parameters at this angular scale.

Figure 14 shows the posterior probability of A_tSZ obtained for the S10, Shaw, Trac, and Battaglia templates. The results are also given in Table 6. Recall that a value of A_tSZ = 1 would demonstrate that the model is consistent with the measured tSZ power, while A_tSZ less (greater) than 1 indicates that the model over-(under-)estimates the tSZ power spectrum amplitude. All four models display a non-Gaussian distribution with tails extending to higher values of A_tSZ. Therefore, in Table 6, we give both symmetric 68% confidence limits for both A_tSZ and ln (A_tSZ) as well as the asymmetric 95% confidence limits (in brackets) for A_tSZ.

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For the S10 template, the preferred value is A_tSZ = 0.33 ± 0.15. The measured level of tSZ power is thus significantly less than that predicted by the S10 template. When the kSZ amplitude is also allowed to vary, we find a slight decrease in A_tSZ to 0.30  ±  0.16. Note that the constraint on σ₈ obtained from the primary CMB power spectrum is 0.821 ± 0.025, which is greater than the fiducial value of 0.8 assumed by all the templates.³¹ The values of A_tSZ are therefore less than what we would have obtained had we fixed the templates at their fiducial amplitudes.

We find that the discrepancy remains, but is reduced, for the Shaw, Trac, and Battaglia models. The Trac and Shaw templates provide the best match to the measured amplitude with A_tSZ = 0.59 ± 0.25. It is interesting to note that the total amount of tSZ power measured (D^tSZ₃₀₀₀ in Table 6) is essentially independent of the template used. This demonstrates that the differences in the shapes of the templates do not influence the results.

The templates that are most consistent with the observed SPT power at ℓ = 3000 (when combined with cosmological constraints derived from the primary CMB power spectrum) are those that predict the lowest tSZ signal. The S10, Trac, and Shaw models adopt a similar approach for predicting the tSZ power spectrum. Each assumes that intracluster gas resides in hydrostatic equilibrium in the host dark matter halo and include simple prescriptions to account for star formation and AGN feedback. The Trac and Shaw models—which both predict significantly less tSZ power—differ from the S10 model in two principal ways. First, following the observational constraints of Giodini et al. (2009), they assume a greater stellar mass fraction in groups and clusters. This reduces the gas density and thus the magnitude of the SZ signal from these structures. Second, the Trac and Shaw models include a significant non-thermal contribution to the total gas pressure. Hydrodynamical simulations consistently observe significant levels of nonthermal pressure in the ICM due to bulk gas motions and turbulence (Evrard 1990; Rasia et al. 2004; Kay et al. 2004; Dolag et al. 2005; Lau et al. 2009). In terms of the tSZ power spectrum, including a non-thermal pressure component reduces the thermal pressure required for the ICM to be in hydrostatic equilibrium, therefore reducing the predicted tSZ signal.

The Battaglia template was produced using maps constructed from a hydrodynamical simulation that included star formation and AGN feedback. While direct comparisons with the model parameters of the other three templates are difficult, Battaglia et al. (2010) found that, in simulations in which AGN feedback was not included, the tSZ power spectrum increased by 50% at the angular scales probed by SPT. If we assume the tSZ template produced from their simulations without AGN feedback we find A_tSZ = 0.27 ± 0.10 and thus a significant increase in the discrepancy between the observed and predicted tSZ power. Battaglia et al. (2010) demonstrate that the inclusion of AGN feedback reduces the thermal gas pressure in the central regions of clusters by inhibiting the inflow of gas. This results in flatter radial gas pressure profiles and a suppression of tSZ power on angular scales ℓ > 2000.

To obtain joint constraints on σ₈ from both the primary CMB anisotropies and the tSZ amplitude we must determine the probability of observing the measured value of A_tSZ for a given set of cosmological parameters. In the absence of cosmic variance—and assuming the predicted tSZ template to be perfectly accurate—the constraint on σ₈ would be given by its distribution at A_tSZ = 1 for each of the models. However, in practice, the expected sample variance of the tSZ signal is non-negligible. Based on the hydrodynamical simulations of Shaw et al. (2009), L10 assumed the sample variance at ℓ = 3000 to be a lognormal distribution of mean 0 and width $\sigma _{\ln (A_{\rm tSZ})} = 0.12$. The survey area analyzed here is twice that of L10 and thus we reduce the width of this distribution by a factor of $\sqrt{2}$. For each of the four tSZ templates, we construct new chains to include this prior by importance sampling. We address the issue of theoretical uncertainty in the tSZ templates later in this section.

The black contours in Figure 15 show the 68% and 95% confidence limits in the σ₈–A_tSZ plane for the S10 model before applying the prior on A_tSZ. In this case, the constraint on σ₈ is derived entirely from the primary CMB power and is not influenced by the measured tSZ signal. The combined WMAP7+ACBAR+QUaD+SPT constraints on the primary CMB yield σ₈ = 0.821 ± 0.025. The shapes of the contours reflect the dependence of A_tSZ on σ₈; higher values of σ₈ predict more tSZ power and require a lower tSZ scaling parameter, whereas lower values of σ₈ suggest a tSZ amplitude closer to that predicted by the S10 template.

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The dashed red contours in Figure 15 represent the joint 68% and 95% confidence limits on σ₈ and A_tSZ for the S10 template having resampled the chain to include the prior on A_tSZ. The one-dimensional σ₈ constraints for all four templates are shown in Figure 16 and Table 6. For the S10 template the preferred value of σ₈ is 0.771 ± 0.013.³² The Trac, Shaw, and Battaglia models all prefer higher values of σ₈. The Trac template—the most consistent with the measured tSZ amplitude—gives σ₈ = 0.802 ± 0.013. This represents nearly a factor of two improvement in the statistical constraint on σ₈ compared to that obtained from the primary CMB alone.

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It is clear from Table 6 that the four templates produce a spread in the preferred value of σ₈ that is larger than the statistical uncertainty on any of the individual results. This spread reflects the range in amplitude of the tSZ templates (at a fixed value of σ₈) due to the differences in the thermal pressure profiles of groups and clusters in the underlying models and simulations. The dominant uncertainty on the tSZ-derived σ₈ constraint is thus the astrophysical uncertainty on the predicted tSZ amplitude rather than the statistical error on the measurement of D^tSZ₃₀₀₀. Accounting for this uncertainty will clearly degrade the constraint on σ₈.

To produce a constraint on σ₈ that accounts for the range in amplitude of the tSZ templates we must add an additional theory uncertainty to the prior on A_tSZ. Assuming the Trac template—which is the most consistent with the measured tSZ signal—we find that a 50% theory uncertainty on ln (A_tSZ) is sufficient to encompass the other three templates. As the sample variance is insignificant compared to the theory uncertainty, the new prior on A_tSZ is effectively a lognormal distribution of width $\sigma _{\ln (A_{\rm tSZ})} = 0.5$. With this additional uncertainty, the revised constraint on σ₈ for the Trac template is 0.812 ± 0.022. The astrophysical uncertainty clearly degrades the constraint significantly—it now represents only a small reduction of the statistical uncertainty of the primary CMB-only result.

It is clear that, while the amplitude of the tSZ power spectrum is extremely sensitive to the value of σ₈, the constraints that can be obtained on this parameter are limited by the accuracy with which we can predict the tSZ power spectrum amplitude for a given cosmological model. The principal difficulty in producing precise predictions is that groups and clusters spanning a wide range of mass and redshift contribute to the power spectrum at the angular scales being probed by SPT. For instance, Trac et al. (2011) find that 50% of the power in their nonthermal20 template is due to objects at z ⩾ 0.65. Similarly, they find that 50% of the power comes from objects of mass less than 2 × 10¹⁴ h⁻¹ M_☉. To date, both these regimes have been poorly studied by targeted X-ray and SZ observations. It is therefore difficult to test current models and simulations of the tSZ power spectrum sufficiently well to produce precise predictions. X-ray and SZ observations of groups and high redshift clusters will provide constraints on models and simulations that should improve estimates of the tSZ power spectrum and thus allow tighter constraints on σ₈ from measurements of this signal (Sun et al. 2011). Alternatively, better constraints on cosmological parameters and improved measurements of the tSZ power spectrum will continue to shed new light on the relevant astrophysics in low mass and high redshift clusters.