A MEASUREMENT OF SECONDARY COSMIC MICROWAVE BACKGROUND ANISOTROPIES WITH TWO YEARS OF SOUTH POLE TELESCOPE OBSERVATIONS

The 800 deg² of sky analyzed in this work were observed by SPT during the 2008 and 2009 Austral winters. The specific field locations and extents can be found in Table 1 of Keisler et al. (2011, K11). SPT is an off-axis Gregorian telescope with a 10 m diameter primary mirror located at the South Pole. The receiver is equipped with 960 horn-coupled spiderweb bolometers with superconducting transition edge sensors. The telescope and receiver are discussed in more detail in Ruhl et al. (2004), Padin et al. (2008), and Carlstrom et al. (2011).

Each field was observed to an approximate depth of 18 μK arcmin at 150 GHz. The noise levels at 95 and 220 GHz vary significantly between the two years as the focal plane was refurbished to maximize sensitivity to the tSZ effect before the 2009 observing season. This refurbishment added science-quality 95 GHz detectors and removed half the 220 GHz detectors. As a result, the 2009 data are comparatively shallower at 220 GHz, but much deeper at 95 GHz. The 2009 sky coverage accounts for approximately 75% of the total sky used in this analysis. When calculating the power spectra, the 2009 data have a statistical weight of 100% at 95 GHz, 73% at 150 GHz, and 43% at 220 GHz.

The SPT beams are measured for each year using a combination of bright point sources in each field, Venus, and Jupiter as described in S11. The main lobes of the SPT beam at 95, 150, and 220 GHz are well represented by 17, 12, and 10 FWHM Gaussians. As in S11, we calculate the principal components of the full covariance matrix of this measurement and find that three eigenvectors are adequate to represent >99% of the covariance. We treat the eigenvectors from different years as independent. We have tested the opposite extreme, treating them as completely correlated, and have found minimal impact on the science results. The beam uncertainty is included in the bandpower covariance matrix as described in Section 2.3.4.

The observation-to-observation relative calibrations of the time-ordered data (TOD) are determined from repeated measurements of a galactic H ii region, RCW38. As in K11, the absolute calibration at each frequency is determined by comparing the SPT and WMAP7 bandpowers over the multipole range ℓ ∈ [650, 1000]. We use the same ℓ-bins for each experiment: seven bins with Δℓ = 50. For the absolute calibration, we do not weight individual modes within each bin (as we will do in Section 2.3). A uniform weighting is used because these large angular scales are dominated by sample variance rather than instrumental and atmospheric noise. This calibration method is model-independent, requiring only that the CMB power in the SPT fields is statistically representative of the all-sky power. We estimate the uncertainty in the SPT power calibration at 95, 150, and 220 GHz to be 3.5%, 3.2%, and 4.8%, respectively. These uncertainties are highly correlated because the main sources of error, WMAP7 bandpower errors, and SPT sample variance are nearly identical between bands.

We use a pseudo-C_ℓ method to estimate the bandpowers (Hivon et al. 2002). In pseudo-C_ℓ methods, bandpowers are estimated directly from the spherical harmonic 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) to eliminate noise bias. Our scan strategy produces of the order of 100 complete but noisy maps of each field and so is ideally suited to a cross-spectrum analysis. We use a flat-sky approximation. Before Fourier transforming, each map is apodized by a window designed to mask point sources detected at greater than >5σ (6.4 mJy) at 150 GHz, and to avoid sharp edges at the map borders. 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

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

Details of the bandpower estimator have been presented in previous SPT papers (L10; S11; K11). In the following sections, we will only note differences between the current analysis and what was done by S11.

2.3.1. Simulations

2.3.2. Covariance Estimation and Conditioning

The bandpower covariance matrix includes two terms: sample variance and instrumental noise variance. The sample variance is estimated with the signal-only Monte Carlo bandpowers. 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, and frequencies ν_i and ν_j , as described in L10. We expect some statistical uncertainty of the form

$$\begin{equation} \langle(\textbf {C}_{bb^\prime }-\langle\textbf {C}_{bb^\prime }\rangle)^2\rangle=\frac{\textbf {C}_{bb^\prime }^2+\textbf {C}_{bb}\textbf {C}_{b^\prime b^\prime }}{n_{{\rm obs}}} \end{equation} \tag{ 2 }$$

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 expected to be 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 36 square blocks, with the six on-diagonal blocks corresponding to the covariances of 95 × 95, 95 × 150, ... , 220 × 220 GHz spectra. The bandpowers reported in Table 1 are obtained by first computing power spectra and covariance matrices for bins of width Δℓ = 200 with a total of 37 initial ℓ-bins, thus each of these blocks is a 37 × 37 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 six on-diagonal blocks and average all off-diagonal elements at a fixed separation from the diagonal in each block,

$$\begin{equation} \textbf {c}^\prime _{kk^\prime }=\frac{ \sum _{k_1-k_2=k-k^\prime } \frac{\widehat{\textbf {c}}_{k_1k_2}}{\sqrt{\widehat{\textbf {c}}_{k_1k_1}\widehat{\textbf {c}}_{k_2k_2}}} }{\sum _{k_1-k_2=k-k^\prime } 1}. \end{equation} \tag{ 3 }$$

We set the correlation to zero at distances ℓ > 400 where we find no evidence for correlations. We have tested that varying this condition by a factor of two in either direction does not affect the cosmological constraints. This averaged correlation matrix is then applied to all the blocks.

The covariance matrix includes an estimate of the signal and, if both bandpowers share a common map, the noise variance. To illustrate the latter condition, instrumental noise is included in the blocks corresponding to (150 × 150, 150 × 150) and (95 × 150, 150 × 220) covariances, but not the covariance between the 150 × 150 and 220 × 220 bandpowers. Given the map filtering, we neither expect nor observe correlated noise between different frequencies in the signal band.

Table 1. Bandpowers

ℓ Range	ℓeff	95 GHz		150 GHz		220 GHz
		$\hat{D}$ (μK2)	σ (μK2)	$\hat{D}$ (μK2)	σ (μK2)	$\hat{D}$ (μK2)	σ (μK2)
2001–2200	2073	217.4	9.1	212.9	4.8	285.2	14.3
2201–2400	2274	138.4	6.4	130.1	3.1	203.0	11.3
2401–2600	2479	107.5	5.8	97.4	2.5	188.3	10.8
2601–2800	2685	75.1	5.7	73.9	2.0	171.7	10.5
2801–3000	2883	51.9	4.5	55.1	1.7	175.6	10.5
3001–3400	3190	47.5	4.3	41.5	1.2	171.2	9.1
3401–3800	3594	32.8	5.3	32.5	1.1	179.8	10.3
3801–4200	3996	35.5	6.3	32.5	1.3	208.2	12.4
4201–4600	4400	37.2	8.9	35.9	1.5	224.0	13.9
4601–5000	4803	14.8	12.1	37.5	1.8	273.3	17.0
5001–5800	5406	31.3	13.4	45.4	1.8	341.8	18.3
5801–6600	6209	38.6	22.4	57.0	2.5	422.0	23.3
6601–7400	7012	26.8	41.7	70.6	3.6	455.1	27.9
7401–8400	7917	−16.3	78.3	87.7	5.0	614.0	39.8
8401–9400	8924	34.7	156.0	96.2	7.3	792.5	62.1
		95 × 150 GHz		95 × 220 GHz		150 × 220 GHz
2001–2200	2073	207.6	5.8	207.2	9.8	226.0	7.1
2201–2400	2274	129.6	3.8	131.7	7.1	148.6	4.8
2401–2600	2479	94.0	3.1	97.7	6.3	112.8	3.9
2601–2800	2685	72.1	2.5	80.7	6.3	97.4	3.5
2801–3000	2883	49.9	2.1	59.5	5.6	78.6	3.2
3001–3400	3190	37.9	1.6	39.3	4.5	69.2	2.4
3401–3800	3594	22.9	1.6	22.4	5.5	64.8	2.4
3801–4200	3996	25.0	1.9	38.3	6.6	72.4	2.8
4201–4600	4400	25.5	2.2	23.3	8.2	77.6	3.2
4601–5000	4803	24.1	2.9	42.0	10.7	94.5	4.0
5001–5800	5406	31.4	3.2	53.6	10.8	111.4	4.2
5801–6600	6209	38.5	4.9	51.9	17.0	139.3	5.4
6601–7400	7012	48.1	7.8	40.0	26.3	169.9	7.4
7401–8400	7917	57.0	13.1	122.5	39.9	211.2	9.6
8401–9400	8924	83.9	24.4	109.3	79.8	282.0	15.6

Notes. ℓ-band multipole range, weighted multipole value ℓ_eff, bandpower $\hat{D}$, and bandpower uncertainty σ for the six auto and cross-spectra of the 95 GHz, 150 GHz, and 220 GHz maps. The quoted uncertainties are based on the diagonal elements of the covariance matrix, which includes the instrumental noise, Gaussian sample variance, and beam uncertainties. The sample variance of the tSZ effect and calibration uncertainty are not included. Calibration uncertainties are quoted in Section 2.2. Point sources with flux >6.4 mJy at 150 GHz have been masked out at all frequencies 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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We impose one final condition on the estimated covariances of off-diagonal blocks for which the instrumental noise term is included, e.g., the (95 × 150, 150 × 220) covariance. In these blocks, the noise on the covariance estimate for an ℓ-bin can be large compared to the true covariance on small angular scales. As a result, we fit an approximate template based on the measured beam shapes from ℓ = 2000 to 4400 and extrapolate this template to estimate the diagonal elements of the covariance matrix at ℓ > 4400. We have tested zeroing these elements of the covariance matrix instead and found that it does not affect the cosmological constraints.

2.3.3. Field Weighting

We construct a diagonal weight matrix for each field i, wⁱ _bb = 1/Cⁱ _bb . We smooth the weights with Δℓ = 1000 at ℓ < 3000 and Δℓ = 2600 on smaller scales to deal with uncertainty in the covariance estimate since we expect the optimal weights to vary slowly with angular scale.

The combined bandpowers are calculated from the individual-field bandpowers according to

$$\begin{equation} D_b = \sum _{i}D_{b}^{i}w^{i}_{bb} \end{equation} \tag{ 4 }$$

where

$$\begin{equation} {\sum _{i}w^{i}_{b b}} = 1. \end{equation} \tag{ 5 }$$

The covariance matrix is combined likewise according to

$$\begin{equation} \textbf {C}_{bb^\prime } = \sum _{i}w^{i}_{bb}\textbf {C}_{bb^\prime }^{i}w^{i}_{b^\prime b^\prime }. \end{equation} \tag{ 6 }$$

2.3.4. Beam Uncertainties

The procedure for estimating the beams and beam uncertainties is described in Section 2.2; here we discuss how the beam uncertainties are included in the cosmological analysis. In S11, three parameters per frequency band describing the beam uncertainties were marginalized over in the Monte Carlo Markov chain (MCMC) with a Gaussian prior. In this work, we instead follow the treatment laid out in K11 and include the beam uncertainties in the bandpower covariance matrix.

Following K11, we estimate the "beam correlation" matrix for each year:

$$\begin{equation} \pmb \rho ^{\rm beam}_{bb^\prime } = \left(\frac{\delta D_b}{D_b}\right) \left(\frac{\delta D_{b^\prime }}{D_{b^\prime }}\right), \end{equation} \tag{ 7 }$$

where

$$\begin{equation} \frac{\delta D_b}{D_b} = 1-\left(1+\frac{\delta B_b}{B_b}\right)^{-2}, \end{equation} \tag{ 8 }$$

and B_b is the Fourier transform of the beam map averaged across bin b.

We combine the correlation matrix for each year according to the yearly weights. The correlation matrix is translated into a covariance matrix by multiplying by the measured bandpowers:

$$\begin{equation} \textbf {C}^{\rm beam}_{bb^\prime } = \pmb \rho ^{\rm beam}_{bb^\prime }D_{b}D_{b^\prime }. \end{equation} \tag{ 9 }$$

This beam covariance is added to the bandpower covariance matrix due to sample variance and instrumental noise.

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_b h², the density of cold dark matter Ω_c h², the optical depth to recombination τ, the angular scale of the sound horizon at last scattering Θ_s , the amplitude of the primordial density fluctuations ln [10¹⁰ A_s ], and the scalar spectral index n_s .

We calculate the primary CMB power spectrum in the MCMC chains using PICO (Fendt & Wandelt 2007a, 2007b) instead of CAMB (Lewis et al. 2000) for speed. With lensing on, we find chains run with PICO (and the fast WMAP likelihood code (Dvorkin & Hu 2010) modified for the latest WMAP7 likelihood) to be roughly 2000 times faster than ones run with CAMB. PICO allows us to replace the unlensed spectra plus "lensing template" approximation used in L10 and S11 with the full lensing treatment. We have trained PICO using the 2011 January version of CAMB with lensing including nonlinear corrections turned on. Note that we observe no change to the foreground or secondary anisotropy constraints from the current data when using the former lensing template approximation. However, reducing the lensing power at ℓ = 3000 (by turning off lensing or neglecting nonlinear effects) would increase the inferred kSZ power by an approximately equal amount. For instance, neglecting nonlinear effects would increase the kSZ power by ∼1 μK².

We adopt two different models for the tSZ power spectrum. S11 considered four tSZ models: a model by Sehgal et al. (2010) which predicts high levels of tSZ power and three models which all predicted lower tSZ powers (Battaglia et al. 2010; Shaw et al. 2010; Trac et al. 2011). Here, we have limited the model set to one from each group: the high power model by Sehgal et al. (2010) and the model by Shaw et al. (2010) from the lower set.

We use the tSZ power spectrum predicted by Shaw et al. (2010) as the baseline model. 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. These astrophysical effects—in particular, the inclusion of non-thermal pressure support and AGN feedback—tend to suppress the tSZ power (see also Battaglia et al. 2011b; Trac et al. 2011). We use the baseline model from that work (hereafter the Shaw model), which predicts D^tSZ ₃₀₀₀ = 4.3 μK² at ℓ = 3000 and 152.9 GHz. 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). We use this model in all MCMC chains where another model is not explicitly specified.

We also consider the tSZ power spectrum model presented by Sehgal et al. (2010, hereafter the Sehgal model). Sehgal et al. (2010) combined the semi-analytic model for the intracluster medium (ICM) of Bode et al. (2009) with a cosmological N-body simulation to produce simulated tSZ and kSZ maps from which the template power spectra were measured. The assumed cosmological parameters are the same as with the Shaw model. At ℓ = 3000, this model predicts D^tSZ ₃₀₀₀ = 7.4 μK² of tSZ power at 152.9 Ghz.

The principal difference between the two models is the inclusion by Shaw et al. (2010) of non-thermal pressure support from random gas motions. Bode et al. (2009) calibrated their intracluster gas model parameters by matching their model predictions to observed low-redshift X-ray scaling relations and radial profiles; Shaw et al. (2010) updated this analysis having included a radially dependent non-thermal pressure component (and its effect on hydrostatic mass estimates), calibrated from the results of hydrodynamical simulations (Lau et al. 2009). The result was a significantly suppressed thermal pressure fraction—particularly outside of the central regions of clusters—compared to the Bode et al. (2009) model, and thus a lower thermal SZ power spectrum amplitude than that produced by the Sehgal et al. (2010) simulations.

As described in S11, the model of Shaw et al. (2010) is also used to rescale both tSZ model templates as a function of cosmological parameters. Around the fiducial cosmological model, the tSZ amplitude roughly scales as

$$\begin{equation} D^{\rm tSZ} \propto \left(\frac{h}{0.71}\right)^{1.7} \left(\frac{\sigma _8}{0.80}\right)^{8.3} \left(\frac{\Omega _b}{0.044}\right)^{2.8}. \end{equation} \tag{ 10 }$$

Both tSZ models exhibit a similar angular scale dependence over the range of multipoles to which SPT is sensitive. When the normalization of each model is allowed to vary, we detect similar tSZ power for both (see Table 3). However, as we discuss in Section 7.2, the predicted model amplitude is crucial in interpreting the detected tSZ power as a constraint on σ₈ and also leads to significant changes in the fit quality when the tSZ power is fixed to cosmologically scaled model expectations.

Thomson scattering between CMB photons and clouds of free electrons that have a coherent bulk velocity produces fluctuations in the observed brightness temperature of the CMB. This effect, known as the kinetic Sunyaev–Zel'dovich (kSZ) effect, leads to hot or cold spots, depending on whether the ionized gas is moving toward or away from the observer. The shift is identical to a change in the CMB blackbody temperature, except for tiny relativistic corrections.

KSZ temperature anisotropy requires perturbations in the free electron density. This can be due to local perturbations in the baryon density, $\delta _{\rm b} =\rho _{\rm b}/\bar{\rho }_{\rm b}-1$ (where $\bar{\rho _b}$ is the mean baryon density in the universe), or ionization fraction, $\delta _{x} = x_e / \bar{x}_e-1$. Perturbations that are correlated with the large-scale velocity field will produce an observable change in temperature of the CMB blackbody. The total contribution to the temperature anisotropy is given by the integral over conformal time,

$$\begin{equation} \frac{\Delta T_{\rm kSZ}}{T_{{\rm CMB}}} ({\bf \hat{n}}) = \sigma _T\overline{n}_{e,0} \int d \eta [a^{-2} e^{-\tau (\eta)} \bar{x}_e(\eta)] {\bf \hat{n}}\,{\bm \cdot}\, {\bf q}, \end{equation} \tag{ 11 }$$

where σ_T is Thomson scattering cross-section, a is the scale factor, τ(η) is the optical depth from the observer to conformal time η, $\bar{n}_{e,0}$ is the mean electron density of the universe at the present day, and ${\bf \hat{n}}$ is the line-of-sight unit vector. Finally,

$$\begin{equation} {\bf q}=(1+\delta _x) (1+\delta _{\rm b}) \mathbf {v} \,, \end{equation} \tag{ 12 }$$

where v represents the peculiar velocity of free electrons at η.

The kSZ power spectrum can be broken down loosely into two components. The first is the kSZ signal from the post-reionization epoch, z < z_end, where we define z_end as the redshift at which hydrogen reionization is complete. The second component comes from the epoch of reionization itself. Models of inhomogeneous (or patchy) reionization predict bubbles of free electrons around UV-emitting sources embedded in an otherwise neutral medium. Any large-scale bulk velocity of these bubbles will impart a temperature anisotropy onto the CMB. The kSZ signal from reionization is thus sensitive to fluctuations in both the gas density and ionization fraction. Note that in the case of instantaneous reionization, the post-reionization signal is the only source of kSZ power.

We henceforth refer to the post-reionization component as the "homogeneous kSZ" signal and that from the reionization epoch as the "patchy kSZ" signal. We now discuss our modeling of each in more detail.

5.3.1. Homogeneous Kinetic Sunyaev–Zel'dovich Anisotropy

For the homogeneous (or post-reionization) kSZ signal, we adopt the non-radiative (NR) and cooling plus star formation (CSF) models presented by Shaw et al. (2011). These models are constructed by calibrating an analytic model for the homogeneous kSZ power spectrum with two hydrodynamical simulations. The first simulation was run in the non-radiative regime, while the second included metallicity-dependent radiative cooling and star formation. Shaw et al. (2011) measured the power spectrum of gas density fluctuations in both simulations over a range of redshifts and used this to calculate the kSZ power spectrum for each case. They argue that these two cases are likely to bracket the true homogeneous kSZ power. For our fiducial cosmology (detailed in Section 5.2) at ℓ = 3000, the NR template predicts 2.4 μK², whereas the CSF simulation predicts 1.6 μK². The scaling of these models with cosmological parameters is given approximately by

$$\begin{eqnarray} D^{\rm kSZ} &\propto & \left(\frac{h}{0.71}\right)^{1.7} \left(\frac{\sigma _8}{0.80}\right)^{4.7} \left(\frac{\Omega _b}{0.044}\right)^{2.1} \nonumber \\ &&\times \left(\frac{\Omega _m}{0.264}\right)^{-0.4} \left(\frac{n_s}{0.96}\right)^{-0.2}. \end{eqnarray} \tag{ 13 }$$

Note that, for both models, we assume that reionization occurs instantaneously at z_end = 8. Both models also assume that helium remains neutral; if helium is singly ionized at the same time as hydrogen, the predicted power would increase by 16%.

5.3.2. Kinetic Sunyaev–Zel'dovich Anisotropy from Patchy Reionization Scenarios

The CIB is produced by thermal emission from DSFGs over a very broad range in redshift (Lagache et al. 2005; Marsden et al. 2009). The dust grains, ranging in size from a few molecules to 0.1 mm, absorb light at wavelengths smaller than their size, and re-radiate it at longer wavelengths. The result of this absorption is roughly equal amounts of energy in the CIB as in the unprocessed starlight that makes up the optical/UV background (Dwek & Arendt 1998; Fixsen et al. 1998).

The frequency dependence of the CIB (and radio galaxies in Section 5.5) is naturally discussed and modeled in units of flux density (Jy) rather than CMB temperature units. However, as the power spectrum is calibrated in CMB temperature units, we need to determine the ratio of powers between different frequency bands in CMB temperature units. To calculate the ratio of power in the ν_i × 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

$$\begin{equation} \epsilon _{\nu _1,\nu _2} \equiv \frac{\frac{dB}{dT}|_{\nu _0} \frac{dB}{dT}|_{\nu _0}}{\frac{dB}{dT}|_{\nu _i} \frac{dB}{dT}|_{\nu _j}}, \end{equation} \tag{ 14 }$$

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 of the SPT bandpasses for each foreground are presented in Section 5.7.

5.4.1. Poisson Term

Statistical fluctuations in the number of DSFGs lead to a Poisson distribution on the sky. DSFGs are effectively point sources at these angular scales, so they contribute a constant C_ℓ (thus D_ℓ∝ℓ²) to the SPT bandpowers. The bulk of the millimeter-wavelength CIB anisotropy power is expected to come from faint (≲ 1 mJy) DSFGs—so the residual CIB power is largely independent of the source mask flux threshold. This is a key assumption in combining the Planck/HFI and SPT data as the flux cuts differ by a factor of eight at 220 GHz. The 217 GHz Planck/HFI cut is at 160 mJy while the 150 GHz SPT flux cut of 6.4 mJy scales to approximately 22 mJy at 220 GHz. We test the impact of the flux cut with the modeled source counts of Béthermin et al. (2011) and Negrello et al. (2007), and find the two models predict a difference in the Poisson power of 4% or 16%, respectively. Hence in the model fitting, we scale the modeled DSFG Poisson power up by 10% when comparing the model to the Planck/HFI CIB bandpowers. We have tested changing this assumption by tens of percent and find the cosmological constraints are insensitive to this—the Poisson term is subdominant at all multipoles (ℓ ∈ [80, 2240]) in the Planck/HFI bandpowers.

The Poisson power spectrum of the DSFGs can be written as

$$\begin{equation} D^{p}_{\ell ,\nu _1,\nu _2} = D_{3000}^{p} \epsilon _{\nu _1,\nu _2} \frac{\eta _{\nu _1} \eta _{\nu _2} }{\eta _{\nu _0} \eta _{\nu _0}} \left(\frac{\ell }{3000}\right)^2, \end{equation} \tag{ 15 }$$

where η_ν encodes how the CIB brightness scales with observing frequency (see Section 5.4.3). D^p ₃₀₀₀ is the amplitude of the Poisson DSFG power spectrum at ℓ = 3000 and frequency ν₀. Note that we neglect the decohering effect of frequency dependence varying from source to source. Hall et al. (2010) argued that the spectral index variation in the DSFG population was less than σ_α = 0.7. This level of variation has a negligible effect on the inferred Poisson powers across the SPT frequency bands.

5.4.2. Clustered Term

5.4.3. Frequency Dependence

Unless otherwise noted, all results presented in this work assume the modified blackbody (modified BB) model for the frequency dependence of the CIB:

$$\begin{equation} \eta _{\nu } = \nu ^\beta B_\nu (T), \end{equation} \tag{ 16 }$$

where B_ν(T) is the blackbody spectrum for temperature T, and β is an effective dust emissivity index. T and β are free parameters with uniform priors T ∈ [5, 35 K] and β ∈ [0, 2]. The upper limit on the temperature prior is unimportant because the SPT frequencies are in the Rayleigh–Jeans tail of >15 K objects. Therefore, there would be a perfect degeneracy between 30 K and 100 K dust. We have tested instead fixing the temperature to 9.7 K with a wider β range as was done in Addison et al. (2012) and found little effect on the results reported in this work.

For some results, we also examine the single-SED model presented by Hall et al. (2010) (hereafter the single-SED model); however, we do not find substantive differences in the results from the modified BB model. This model assumes a single SED for all DSFGs. This SED is a modified blackbody with T = 35 K and β = 2. The DSFGs have a density as a function of redshift parameterized by a Gaussian with mean z_c and variance σ² _z . The number counts are set to zero at z < 0. We fix all but one parameter in the model; we allow z_c to vary freely. As in Hall et al. (2010), we set σ_z = 2. The frequency dependence predicted by the single-SED model has a slight ℓ-dependence; we ignore this and use the frequency scaling at ℓ = 3000. The single-SED model has one fewer parameter and is more easily interpretable physically than the modified BB model. However, the data are in mild tension with the model—the preferred z_c is near the prior cutoff at 0. This does not degrade the quality of the fit to the data, but it leads to a one-sided distribution of CIB spectral indices, which results in a slight artificial reduction in some parameter uncertainties.

A simplifying assumption we make for the analyses presented here is that the various components of the CIB (Poisson and clustering terms) have the same frequency dependence. Even in the context of the single-SED model, we expect their spectral indices to differ to some degree, because the weighting of sources as a function of redshift will be different for the two terms. Our assumption here is, effectively, that the redshift distributions of the source populations giving rise to the Poisson and clustered components are not sufficiently different as to cause a significant difference in frequency dependence. We have tested allowing separate frequency dependencies for the Poisson and clustered CIB terms and find minimal impact on the SZ constraints.

We have also checked whether the higher frequency, 217 and 353 GHz, Planck/HFI CIB bandpowers (Planck Collaboration et al. 2011) are consistent with the assumption of a single frequency dependence for the Poisson and clustered terms. If the frequency dependence of each term was different, then we would expect to see variations in the CIB power ratio between 217 GHz and 353 GHz as a function of angular scale since the relative power in each term is ℓ-dependent. We find that the CIB power ratio is consistent with a single spectral index with α_217/353 = 3.22 ± 0.08. With (9-1) degrees of freedom (dof), the null hypothesis of a single spectral index has χ² = 8.0—the probability to exceed this χ² is 0.44. This spectral index is flatter than preferred by the lower frequency SPT data; however, we expect the CIB frequency dependence to flatten toward higher frequencies as the rest-frame emission frequency approaches the peak of the graybody dust emission.

5.4.4. tSZ–CIB Correlations

As tracers of the same dark matter distribution, we expect some degree of correlation between the tSZ and CIB power spectra. In previous papers, we have argued that the correlation is likely to be small since most of the DSFGs contributing to the CIB are in the field rather than in the massive galaxy clusters that contribute most of the tSZ power. Unfortunately, observational constraints on the IR flux (and therefore tSZ–CIB correlation) are largely limited to low-redshift, high-mass clusters. The observations are also at frequencies close to the peak of the CIB emission, introducing a significant scaling uncertainty in translating the results to the longer wavelengths important for tSZ surveys. The simulations of Sehgal et al. (2010) show that a model with significant correlations at 148 GHz can be constructed without violating current observational constraints by concentrating the CIB galaxies in the lower-mass, higher-redshift galaxy clusters.

We adopt a simplified treatment of tSZ–CIB correlations in this work. We assume that the correlation, ξ, is independent of angular multipole, and that both the Poisson and clustering CIB components correlate with tSZ fluctuations. Note that the choice of which CIB terms correlate does not significantly change the resulting tSZ and kSZ constraints, although it does rescale the amplitude of the inferred correlation.

The tSZ–CIB correlation introduces a term to the model of the form

$$\begin{equation} D^{{\rm tSZ-CIB}}_{\ell ,\nu _1,\nu _2} = \xi \big(\sqrt{\big(}D^{\rm tSZ}_{\ell ,\nu _1,\nu _1} D^{\rm CIB}_{\ell ,\nu _2,\nu _2}\big) + \sqrt{\big(}D^{\rm tSZ}_{\ell ,\nu _2,\nu _2} D^{\rm CIB}_{\ell ,\nu _1,\nu _1}\big) \big). \end{equation} \tag{ 17 }$$

Here $D^{{\rm tSZ-CIB}}_{\ell ,\nu _1,\nu _2}$ is the power due to correlations (which should be negative below the tSZ null of 217 GHz, i.e., ξ < 0), $D^{\rm tSZ}_{\ell ,\nu _1,\nu _1}$ is the tSZ power spectrum, and $D^{\rm CIB}_{\ell ,\nu _1,\nu _1}$ is the sum of the Poisson and clustered CIB components.

High-flux point sources in the SPT maps are coincident with known radio sources from long-wavelength catalogs (e.g., SUMSS; Mauch et al. 2003) and have spectral indices consistent with synchrotron emission (Vieira et al. 2010). The Poisson radio source term can be expressed as

$$\begin{equation} D^{r}_{\ell ,\nu _1,\nu _2} = D_{3000}^{r} \epsilon _{\nu _1,\nu _2} \eta _{\nu _1,\nu _2} ^{\alpha _{r} + 0.5 {\rm ln}(\eta _{\nu _1,\nu _2}) \sigma _{r}^2} \left(\frac{\ell }{3000}\right)^2, \end{equation} \tag{ 18 }$$

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^r ₃₀₀₀ is the amplitude of the Poisson radio source power spectrum at ℓ = 3000 and frequency ν₀. The mean spectral index of the radio source population is α_r with scatter σ_r .

The radio source power (D^r ₃₀₀₀) in the map depends approximately linearly on the flux above which discrete sources are masked. This reflects the fact that S² dN/dS for synchrotron sources is nearly independent of S (e.g., de Zotti et al. 2005). As a result, the Poisson radio source power in the K11 bandpowers (which had a higher flux cutoff) is expected to be 9.2 ± 2.4 μK² higher than in the 150 GHz bandpowers presented here. We correct for this by subtracting a Poisson term with amplitude D₃₀₀₀ = 9.2 μK² from the K11 bandpowers in the parameter fits. Note that the restriction to ℓ < 2000 that we impose on the K11 bandpowers renders the uncertainty on the Poisson power insignificant. We have tested varying the subtracted Poisson power and found no impact on the results.

Following the treatment in S11, we apply a strong prior to the Poisson radio source term due to sources below the SPT detection threshold. First, we fix the mean spectral index (α_r = −0.53) and scatter (σ_r = 0). Second, based on the de Zotti et al. (2005) source count model, we apply a Gaussian prior to the 150 GHz amplitude of 1.28 ± 0.19 μK². We have tested the impact of removing the radio priors. The data prefer parameter values consistent with the priors, and there are no significant changes to the CIB and SZ constraints. The strong radio priors are included in all cases presented here.

As discussed in Hall et al. (2010), the clustering of radio galaxies at SPT frequencies is likely to be negligible. Clustering on scales of a few arcminutes is a ∼5% modulation of the mean, and the intensity of the radio source mean is quite low. Furthermore in the unified model of active galactic nuclei, the dominant sources at >90 GHz (blazars and flat spectrum radio quasars) are active galactic nuclei with their jets aligned toward the observer. The effect of random alignment would serve to further dilute the clustering signal. Supporting this argument is conservative extrapolations of upper limits on arcminute scale power at 30 GHz (Sharp et al. 2010). If entirely due to radio point-source power, the 30 GHz limit translates to an upper limit of 0.1 μK² at 150 GHz. We do not include a clustered radio component in any results in this work.

Galactic cirrus is the final (and smallest) foreground we include in our modeling. Following the cirrus treatment in Hall et al. (2010) and K11, we cross-correlate the SPT maps with the model 8 galactic dust predictions of Finkbeiner et al. (1999). Galactic cirrus is detected in all frequency bands at a significance of ∼3σ with an angular dependence of

$$\begin{equation} D^{\rm cir}_{\ell ,\nu _1,\nu _2} = D_{3000}^{{\rm cir}, \nu _1,\nu _2} \left(\frac{\ell }{3000}\right)^{-1.2}. \end{equation} \tag{ 19 }$$

The measured powers are 0.16, 0.21, and 2.19 μK² at 95, 150, and 220 GHz. Note that the 150 GHz amplitude is different from the value quoted in K11 because of the different field weights in the two analyses. We fix the amplitude of the galactic cirrus in each frequency band; allowing it to vary with a prior based on the measurement uncertainty has no impact on any parameter constraints.

Throughout this work, we refer to the SPT frequencies as 95, 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 beam-filling source with a nominal frequency dependence. The bandpasses differ slightly from 2008 to 2009 due to the focal plane refurbishment. We calculate the effective frequency for each year's spectral bandpass and average them according to the yearly weights presented in Section 2.1. For an α = −0.5 (radio-like) source spectrum, we find band centers of 95.3, 150.2, and 214.1 GHz. For an α = 3.5 (dust-like) source spectrum, we find band centers of 97.9, 153.8, and 219.6 GHz. The band centers in both cases drop by ∼0.2 GHz if we reduce α by 0.5. 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 97.6, 152.9, and 218.1 GHz. The ratio of tSZ power in the 95 GHz band to that in the 150 GHz band is 2.78. The 220 GHz band center is effectively at the tSZ null; the ratio of tSZ power in the 220 GHz band to that in the 150 GHz band is 0.00004 for a non-relativistic tSZ spectrum.

We adopt a baseline model with the six ΛCDM parameters and six additional parameters for the secondary CMB anisotropies and foregrounds described in Section 5. Four of these extra parameters represent the power in the tSZ effect, kSZ effect, DSFG Poisson term, and DSFG clustering term. The final two parameters describe the frequency dependence of the DSFG anisotropy. All DSFG terms are assumed to have the same frequency scaling, based on the modified BB model. As templates for the angular shape of tSZ and kSZ power spectra, we use the Shaw et al. (2010) tSZ model and Shaw et al. (2011) CSF kSZ model. The DSFG Poisson power (like all unresolved Poisson power) has the shape D_ℓ∝ℓ², while the DSFG clustering term is described by the power-law D_ℓ∝ℓ^0.8 template presented in Section 5.4.2. Figure 3 plots the best-fit values for each component on top of the SPT bandpowers.

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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. Adding additional free parameters beyond the tSZ amplitude does not significantly improve the quality of the fits. However, small improvements are seen with the introduction of tSZ–CIB correlations or treating the clustered and Poisson frequency scalings as independent.

The additional parameters discussed in Section 5 are either fixed or constrained by strong priors. A Poisson population of radio galaxies is included with a fixed spectral index α_r = −0.53, zero intrinsic scatter, and an amplitude of D^r ₃₀₀₀ = 1.28 ± 0.19 μK². Weakening these radio priors—removing the amplitude prior and setting a uniform prior on the radio spectral index of α_r ∈ [ − 1.5, 0]—does not change the inferred SZ levels. We always include the constant Galactic cirrus power described in Section 5.6. We assume no correlation between the tSZ effect and CIB in the baseline model. This assumption is relaxed in Section 6.2.1. The χ² for the 90 SPT bandpowers (84 dof since the ΛCDM parameters are essentially fixed by external data and there are six model parameters beyond ΛCDM) is 73.3; the PTE for this χ² is 79%. We have tested fixing the ΛCDM parameters to the best-fit values and find the derived constraints on the six extended model parameters are unchanged. This baseline model fits the SPT data well and provides the most concise interpretation of the data.

SZ parameter constraints for the baseline model and extensions are listed in Table 3 and are presented in Figure 4. For the tSZ power, we find D^tSZ ₃₀₀₀ = 3.65 ± 0.69 μK², which is consistent with the predictions of the Shaw model. We find a strong 95% confidence level (CL) upperlimit on kSZ power at D^kSZ ₃₀₀₀ < 2.8 μK², although we stress that this limit is dependent on our DSFG modeling assumptions. We have applied a positivity prior to the kSZ power; without this prior, the median kSZ power is negative by 1.2σ. The SZ constraints are independent of the assumed template shape; we find similar power constraints for all tSZ and kSZ model templates considered.

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The current tSZ and kSZ constraints are consistent with earlier measurements of the SZ power. S11 measured D^tSZ ₃₀₀₀ + 0.5D₃₀₀₀ ^kSZ = 4.5 ± 1.0 μK². The equivalent constraint from these data is (1.034)D^tSZ ₃₀₀₀ + 0.5D₃₀₀₀ ^kSZ = 4.27 ± 0.58 μK². The multiplicative factor in parentheses before the tSZ term corrects for the different effective bandcenters in the two data sets. Dunkley et al. (2011) measured D^tSZ ₃₀₀₀ + D₃₀₀₀ ^kSZ = 6.8 ± 2.9 μK², while these data prefer (1.17)D^tSZ ₃₀₀₀ + D₃₀₀₀ ^kSZ = 5.27 ± 0.76 μK². We note that the two analyses differ in that Dunkley et al. (2011) fixed the ratio of thermal and kinetic SZ power, while this ratio is free in this work. The three data sets show excellent consistency.

We detect both the DSFG Poisson and clustering power with high significance, with powers of D^p ₃₀₀₀ = 7.54 ± 0.38 μK² and D^c ₃₀₀₀ = 6.25 ± 0.52 μK², respectively, at 150 GHz. The effective spectral index from 150 to 220 GHz of the DSFG power is found to be α = 3.56 ± 0.07. The DSFG constraints are discussed in more detail in Section 9, and are in line with both theoretical expectations and previous work (H10; S11; Dunkley et al. 2011).

Here, we explore modifications to the baseline model and the impact on SZ and CIB constraints. First, we consider allowing a correlation between the tSZ effect and CIB, and show the impact on the derived SZ power. The SZ interpretation is robust to the other CIB model variants we examined. Second, we examine model extensions to the ΛCDM cosmology, and show that they do not affect the SZ and CIB parameters.

6.2.1. tSZ–CIB Correlations

We first consider the effect of a correlation between the tSZ and CIB as prescribed by Equation (17). Note that DSFG overdensities in galaxy clusters would lead to an anti-correlation since the tSZ effect causes a flux decrement at the SPT frequencies. The correlated power is highly degenerate with the kSZ and tSZ power as shown in Figure 5. We see some dependence on the kSZ template shape (not shown); the patchy template, which is falling instead of rising slightly across the angular multipoles of interest, is less degenerate with the correlation coefficient than the CSF homogeneous kSZ model. Increasing anti-correlation increases the allowed kSZ power, while reducing the tSZ power, until ξ ∼ −0.4. The slope of the tradeoff between ξ and kSZ power approaches zero for very negative values of ξ. This is because the correlated power is proportional to the tSZ power, and the tSZ power approaches zero in this limit.

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Decreasing tSZ power with increasing anti-correlation contradicts our naive expectations. In single-frequency bandpowers (e.g., only 150 GHz), increasing tSZ–CIB anti-correlations would increase the allowed tSZ contribution for any observed power level. However, this picture changes when we consider the other SPT frequency combinations. A tSZ–CIB anti-correlation reduces the power at 150 × 150 (which could be compensated by increasing tSZ power) but it also has ∼50% larger effect at 150 × 220 (where there is effectively zero tSZ power). Conversely, a tSZ–CIB anti-correlation reduces the power at 95 × 150 by a similar amount to 150 × 150, whereas there is more tSZ power in the 95 GHz band. Adding kSZ power more effectively cancels the tSZ–CIB correlation term than adding tSZ power in the most sensitive combinations of the three frequencies.

Due to this degeneracy, introducing a tSZ–CIB correlation substantially degrades the constraints on both the tSZ and kSZ power. In a model with ξ as a free parameter, the measured tSZ power is D^tSZ ₃₀₀₀ = 3.26 ± 1.06 μK². The 95% confidence upper limit on the kSZ power is now D^kSZ ₃₀₀₀ < 6.7 μK². The SPT data better constrain the linear combination of D^tSZ ₃₀₀₀ + 0.5D₃₀₀₀ ^kSZ = 4.60 ± 0.63 μK², although the limiting case with zero tSZ and high kSZ power is disfavored. The expanded parameter space resembles the constraint in previous SPT work (L10, S11); the additional frequency information in this analysis is constraining the CIB model instead of breaking the degeneracy between tSZ and kSZ power. To facilitate comparison we express the constraint in the same basis as S11; we find a constraint of (1.034)D^tSZ ₃₀₀₀ + 0.5D₃₀₀₀ ^kSZ = 4.71 ± 0.64 μK². The multiplicative factor in parentheses before the tSZ term corrects for the different effective bandcenters in the two data sets. S11 found D^tSZ ₃₀₀₀ + 0.5D₃₀₀₀ ^kSZ = 4.5 ± 1.0 μK². The two measurements are completely consistent with each other. We will discuss the inferred correlation and its impact on CIB constraints in Section 9.

6.2.2. Beyond ΛCDM

The models for the tSZ power spectrum are based on simulations with a fixed cosmology. Therefore, to correctly compare the measured tSZ power with theoretical predictions, we must rescale the theoretical power spectra so that they are consistent with the cosmological parameters at each point in the MCMC chains. Recall that the cosmological parameters are derived solely from constraints on BAO, H₀, and the primary CMB power spectrum and so are not directly influenced by the amplitude of the measured tSZ power.

Following S11, we define a dimensionless tSZ scaling parameter, A_tSZ, where

$$\begin{equation} A_{\rm tSZ}(\kappa _i) = \frac{D_{3000}^{\rm tSZ}}{\Phi _{3000}^{\rm tSZ}(\kappa _i)} \;. \end{equation} \tag{ 20 }$$

D^tSZ ₃₀₀₀ is the measured tSZ power at ℓ = 3000 and Φ^tSZ ₃₀₀₀(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^tSZ ₃₀₀₀ with the cosmological parameter constraints provided by the primary CMB measurements. As described in Section 5.2, we consider two template models for the tSZ power spectrum, the "Shaw" and "Sehgal" models. These two templates bracket the range of tSZ models considered by S11.

To determine the cosmological scaling of Φ^tSZ ₃₀₀₀(κ_i ), we run the Shaw model at each step in the chain. The approximate cosmological scaling of the Shaw model around the fiducial model cosmological parameters is given by Equation (10).

In the right panel of Figure 6, we show the constraints obtained on A_tSZ. The solid black line in each plot shows the results for the baseline model using the Shaw tSZ template, the CSF kSZ template, and the modified BB CIB frequency scaling, while the green line shows those obtained when a free tSZ–CIB correlation is allowed in our modeling. The blue dashed line shows the results obtained using the Sehgal tSZ model instead. The numerical constraints are also given in Table 4. Recall that a value of A_tSZ ≃ 1 implies that the measured tSZ power is consistent with the tSZ template predictions, while A_tSZ less (greater) than 1 indicates that the model overestimates (underestimates) the amplitude. Note that sample variance will cause deviations from unity on the order of 0.04.

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The preferred value of A_tSZ for the Shaw template is 1.2σ below unity (from the constraints given on ln A_tSZ), while the preferred value of A_tSZ for the Sehgal template is 2.9 σ below unity. Although both templates predict more tSZ power than is measured, the Sehgal template is in significant tension with the data. As an aside, we note that using the single-SED CIB frequency scaling instead of the modified BB has little effect on the tSZ amplitude. Including a tSZ–CIB spatial correlation shifts the preferred value of A_tSZ to a slightly lower value, but principally increases the error bars (by roughly 33%).

In Table 5, we evaluate the goodness of fit of our SZ models by running new MCMC chains in which the tSZ and kSZ terms have been fixed to their predicted, although cosmologically rescaled, values. All four combinations of the Sehgal and Shaw tSZ models and the NR and CSF kSZ models are considered. We show the change in the best-fit χ² for each case relative to our baseline model (Shaw tSZ, CSF kSZ). Models that predict more SZ power are a slightly worse fit to the data. Table 4 shows that the measured tSZ power is a factor of 0.41 ± 0.13 times that predicted by the Sehgal template. This template predicts a tSZ amplitude that is poorly matched to the data, resulting in a Δχ² = +8.4 relative to the baseline model. Likewise, the NR kSZ template predicts 1 μK² more power than the CSF kSZ template and thus provides a worse fit to the measured powers, increasing χ² by 2.1 (for Shaw tSZ) and 2.3 (Sehgal tSZ). Adding a kSZ contribution from patchy reionization would further increase the model tension.

The consistency between the SZ models and measured bandpowers can be tested further by including additional external cosmological constraints. Both the tSZ and kSZ power spectra scale most sensitively with σ₈. We therefore importance sample each of the MCMC chains presented in Table 5 imposing a prior of σ₈(Ω_M /0.25)^0.47 = 0.813 ± 0.027, based on the X-ray cluster abundance measurements of Vikhlinin et al. (2009). The subsequent change in χ² is given in the third column of the table. The prior from Vikhlinin et al. (2009) increases the significance of the discrepancy with models that predict more tSZ power. This result is particularly interesting since it eliminates one possible solution for the high-power tSZ models—massive neutrinos. The CMB data alone are consistent with high neutrino masses which slow the growth of structure (see Section 7.3). This reduces the tSZ power predicted by all models. As a result there are cosmological parameter sets allowed at 1σ by the CMB data, such as ∑m_ν = 0.4 eV and σ₈ = 0.7, for which the Sehgal model is a closer match to the observed power than the Shaw model. The X-ray constraint, σ₈ ≃ 0.8, which is independent of the tSZ modeling, rules out this explanation.

To obtain joint constraints on σ₈ from both the primary CMB anisotropy and the tSZ power spectrum, we must determine the probability of observing the measured value of A_tSZ for a given set of cosmological parameters. In the absence of sample 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. In practice, the expected sample variance of the tSZ signal is non-negligible. Based on the 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 eight times that of L10 and thus we reduce the width of this distribution by $\sqrt{8}$. For each of the tSZ templates, we construct new MCMC chains by importance sampling to include this prior.

The one-dimensional σ₈ constraints for the two tSZ templates are shown in Figure 7 and Table 4. Using the Shaw template results in σ₈ = 0.797 ± 0.011, while the Sehgal model prefers σ₈ = 0.767 ± 0.009. The constraint without the tSZ amplitude information (i.e., from the primary CMB + BAO + H₀ alone) is σ₈ = 0.812 ± 0.018. Adding the tSZ information therefore both reduces the size of the statistical error by 40% and lowers the preferred value, depending on the amplitude of the tSZ template.

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The difference in the preferred value of σ₈ provided by the Shaw & Sehgal templates is significantly larger than the statistical errors. This implies that the systematic error due to the theoretical uncertainty in the tSZ modeling significantly exceeds the statistical error. To produce a constraint on σ₈ that accounts for this, we must add an additional theory uncertainty to the sample variance when setting the prior on A_tSZ.

It is difficult to determine the level of theory uncertainty on the predicted tSZ amplitude for a fixed set of cosmological parameters. A significant fraction of the total tSZ power is contributed by groups and clusters of mass M₅₀₀ < 2 × 10¹⁴  h⁻¹  M_☉ and redshift z > 0.65 (Trac et al. 2011; Battaglia et al. 2011b; Shaw et al. 2010). To date, such objects have not been studied in detail with SZ and X-ray observations. From a theoretical perspective, models and simulations have demonstrated that astrophysical processes such as feedback from active galactic nuclei and turbulence-driven non-thermal pressure support can significantly alter predicted cluster thermal pressure profiles, and thus their SZ signal (Battaglia et al. 2011a; Trac et al. 2011; Parrish et al. 2012; Shaw et al. 2010).

Following S11, we choose to add a 50% theory uncertainty on ln (A_tSZ). This uncertainty encompasses all four models considered by S11. As the sample variance is insignificant compared to the theory uncertainty, the new prior on A_tSZ is effectively a lognormal distribution of width σ(ln A_tSZ) = 0.5. With this assumption, we obtain constraints of σ₈ = 0.807 ± 0.016 for the Shaw model.

We estimate the contribution of the systematic theory uncertainty to the total uncertainty quoted above as follows. We form a new prior based on the median value of A_tSZ = 0.7 with the sample-variance-based width of σ(ln (A_tSZ)) = 0.042. By sampling around the preferred value of A_tSZ instead of unity, we avoid introducing artificial tension if the modeled amplitude is incorrect. Importance sampling the MCMC chain with this prior leads to parameter constraints that include only statistical noise and sample variance. The resulting σ₈ constraint is σ(σ₈) = 0.010. Assuming the errors add in quadrature, this result implies a systematic modeling uncertainty of σ(σ₈) = 0.012.

Given the current modeling uncertainties, the measured amplitude of tSZ power in this work only marginally improves our knowledge of σ₈ over existing constraints. A detailed and systematic study of the thermal pressure profiles of low-mass (≈10¹⁴  h⁻¹  M_☉) clusters at z ≈ 0.65 is required in order to reduce the theory uncertainty and thus increase the precision of σ₈ constraints derived from the tSZ power spectrum.

We can apply the methodology outlined above to other cosmological parameters, most notably the sum of the neutrino masses, ∑m_ν. For CMB + H₀ + BAO, neutrino masses are highly degenerate with σ₈ as shown in Figure 8. Higher neutrino masses lead to lower σ₈ since massive neutrinos slow the growth of structure below the neutrino-free-streaming length. Introducing massive neutrinos weakens the CMB + H₀ + BAO constraint on σ₈ to σ₈ = 0.756 ± 0.044. The sum of the neutrino mass is constrained to be less than 0.52 eV at 95% CL.

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The tSZ power spectrum presents an independent probe of σ₈ that breaks the ∑m_ν degeneracy and thereby improves the neutrino mass determination. The preference for positive neutrino masses increases with models that predict more tSZ power. Models that predict more tSZ power require a lower σ₈ to match the observed tSZ power. As can be seen in Figure 8, lower σ₈ values favor larger neutrino masses.

With massive neutrinos and no modeling uncertainty (but including a 4% sample variance), adding the tSZ information reduces the uncertainty on σ₈ by a factor of two to three. The median σ₈ moves up slightly to σ₈ = 0.776 ± 0.019 with the Shaw model and down slightly to σ₈ = 0.732 ± 0.017 with the Sehgal model. In this optimistic scenario, we would find ∑m_ν = 0.15 ± 0.09 or 0.29 ± 0.10 eV for the Shaw and Sehgal models, respectively. Note that the X-ray σ₈ measurement mentioned earlier is in tension with the preferred σ₈ with the Sehgal model. With a more realistic 50% modeling uncertainty on the Shaw model, the constraint is σ₈ = 0.768 ± 0.031. With this modeling uncertainty, the Sehgal and Shaw model results are consistent at ∼0.5σ. The upper limit on ∑m_ν is reduced from 0.52 eV to 0.40 eV at 95% CL with the addition of tSZ information. Even with a large theory prior, the SPT tSZ measurements improve the neutrino mass constraints by disfavoring regions of parameter space with very low σ₈ values.

We consider how the uncertainties on σ₈ and ∑m_ν break down between systematic and statistical uncertainties using the method presented in Section 7.2. The prior is constructed as before and centered the preferred value of A_tSZ = 1.14 in the ∑m_ν > 0 eV chain. We find that the uncertainty on σ₈ can be allocated as σ_sys(σ₈) = 0.024 (for the modeling uncertainty) and σ_stat(σ₈) = 0.020 (for statistical noise plus sample variance). Similarly, the uncertainty on ∑m_ν can be divided into σ_sys(∑m_ν) = 0.10 eV and σ_stat(∑m_ν) = 0.10 eV. The 50% theory uncertainty is dominant for both parameters.

In an identical manner to the tSZ, we define a dimensionless scaling parameter A_kSZ for the homogeneous kSZ power. We use the NR and CSF kSZ models of Shaw et al. (2011). Equation (13) gives the scaling of the CSF kSZ template Φ^kSZ ₃₀₀₀ with cosmological parameters.

In Figure 10, we show the constraints obtained on A_kSZ. The solid black line shows our baseline model using the Shaw tSZ template, the CSF kSZ template, and the modified BB CIB frequency scaling. The blue dashed line shows those obtained when a free tSZ–CIB correlation is included in our modeling. The numerical constraints are also given in Table 4. Recall that a value of A_kSZ = 1 implies that the measured kSZ power is consistent with the kSZ template predictions, while A_kSZ less (greater) than 1 indicates that the model overestimates (underestimates) the amplitude. Note that sample variance will cause deviations away from unity on the order of 0.01.

In this work, we have significantly tightened the upper limits on the kinetic SZ power. These limits are not in significant tension with our fiducial homogeneous kSZ template (the CSF model of Shaw et al. 2011). The 2σ upper limit on A_kSZ exceeds 1 for all of the MCMC chains in Table 4. However, for our baseline model, the upper limit is only modestly in excess of 1. The single-SED CIB model allows for slightly less kSZ power than the modified BB frequency scaling. Therefore, while the data are consistent with the CSF kSZ prediction, there is little room for an additional contribution from patchy reionization. Z11 note that if the duration of reionization spans Δz = 2 we can expect an additional 1 μK² of kSZ power. For Δz = 4 we expect 2 μK². Clearly when combined with the post-reionization signal, the predicted kSZ will significantly exceed that measured by SPT in the baseline model.

However, as discussed in Section 6.2.1, incorporating a tSZ–CIB spatial correlation into our model allows for considerably more kSZ power than our fiducial model. In this case, the upper limit on A_kSZ is 4.13, which allows for substantially more power than predicted by the homogeneous kSZ models alone, and thus a significant contribution from patchy reionization.

We now switch to considering limits on the kSZ signal contributed by patchy reionization. For this we use a patchy kSZ template from Z11 based on a model in which reionization started at z = 11 and ended z = 8. As would be expected, the kSZ signals from the reionization and post-reionization era are nearly perfectly degenerate. Therefore, our constraints on the patchy kSZ power depend on the modeling of the post-reionization homogeneous kSZ power. We choose to bracket the latter with three cases: no homogeneous kSZ, the intermediate CSF homogeneous kSZ model, and the highest NR homogeneous kSZ model.

The constraints on patchy kSZ power are shown in Figure 9. Note that we see less difference in the allowed patchy kSZ power than would be inferred from the power differences between the assumed homogeneous kSZ model. The smaller-than-expected differences are because the likelihood function for the total kSZ power, before any amount of homogeneous kSZ power is subtracted, already peaks at zero. We find 95% CL upper limits on the kSZ power due to patchy reionization of 2.6, 2.0, and 1.9 μK² for the zero, CSF, and NR homogeneous kSZ models, respectively. As with all the kSZ constraints, we find the patchy kSZ constraints are most sensitive to tSZ–CIB correlations. With a free tSZ–CIB correlation and the same set of three homogeneous kSZ models, the 95% CL upper limits on patchy kSZ power are degraded to 5.7, 4.9, and 4.7 μK². Z11 interpret these upper limits on patchy kSZ power to constrain the duration of the epoch of reionization and the ionization history of the universe.

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As noted earlier, we find strong evidence for clustered DSFGs in the data. We have modeled the clustering by a power law with D_ℓ∝ℓ^0.8; here we examine if the data prefer an alternate shape. The SPT data alone can be fit with either a linear-theory clustering template or the power-law baseline template—only the extremely wide angular range probed by the combination of SPT and HFI allows shape discrimination in the clustered template. We parameterize the CIB clustering shape in two ways: fitting for the exponent of the power law, or fitting for the amplitude of a linear-theory term in addition to the power-law term (with fixed exponent). Adding linear theory power is similar in effect to lowering the power-law exponent.

We introduce the power-law exponent, γ, as a new parameter to the MCMC parameter chains. A uniform prior is imposed from 0.1 to 1.4. The upper edge is chosen to prevent overlap with the ℓ² scaling of the Poisson term. The constraints on γ are shown in Figure 13. The data prefer γ = 0.81 ± 0.08, consistent with the nominal value of 0.8. Adding the Planck/HFI 353 GHz CIB bandpowers improves the shape measurement to γ = 0.82 ± 0.05. The likelihood of the best-fit point is not improved by the introduction of this new parameter for either set of data. Both results are consistent with the recent constraint of γ = 0.75 ± 0.06 by Addison et al. (2012) using data from BLAST, Planck/HFI, and ACT. For simplicity and ease of comparison with earlier results, we continue to use γ = 0.8 in all other parameter chains.

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We also test introducing the linear theory model with a free amplitude in addition to the power-law term with γ = 0.8. The constraints on the amplitudes of the linear theory and power-law terms at 220 GHz and ℓ = 3000 are shown in Figure 14. We do not find a significant preference for a non-zero amplitude of the linear theory term; the derived 1σ confidence intervals are [−0.02, 0.08 μK²], [−0.2, 0.7 μK²], and [−1.4, 6.2 μK²] at 95, 150, and 220 GHz, respectively. We have allowed negative values to allow the maximum shape flexibility; the total power in the two clustering terms is always positive. Note that these ranges should be compared to 0.67 ± 0.13, 5.7 ± 1.0, and 51 ± 9 μK² for the corresponding amplitude of the power-law term. The quality of the fit is not improved by adding this free parameter. The preferred values imply that most clustered CIB power is in the power-law shape instead of the linear theory shape at ℓ ≳ 500. Adding the Planck/HFI 353 GHz bandpowers does not significantly change this picture.

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The galaxy clusters that contribute to the tSZ power spectrum have a higher than average density of galaxies, each with some amount of star formation. This will lead to an anti-correlation between the tSZ and CIB power spectra. We introduce this correlation following the treatment in Section 5.4.4. Note that in this treatment, the correlation is defined with respect to all CIB power, although the Poisson and clustered components may correlate differently. As discussed in Z11, some models predict a correlation effectively independent of ℓ at current instrumental sensitivities (e.g., Sehgal et al. 2010), while others predict that the magnitude of the correlation will rise across the relevant multipoles. We use a single correlation coefficient on all angular scales. The data prefer a correlation coefficient of ξ = −0.18 ± 0.12 (i.e., an anti-correlation between tSZ and CIB signals, consistent with a fraction of the CIB emission being spatially coincident with the tSZ decrement from clusters). This constraint depends on the assumption that the correlation is independent of ℓ. We tested a correlation model described by Z11 where the absolute magnitude of the correlation doubles from ℓ = 2000 to 10, 000 and found similar uncertainties but a mean value 1σ closer to zero at ℓ = 3000. In the near future, we expect observations with Herschel and Planck in combination with SPT to place better observational constraints on this correlation.

Introducing a tSZ–CIB correlation does very little to the inferred CIB powers; however, the uncertainties increase by 10%–50%. With tSZ–CIB correlations, the constraint on the Poisson power at 150 GHz is D^p ₃₀₀₀ = 8.04 ± 0.48 μK², and the constraint on the clustered power at 150 GHz is D^c ₃₀₀₀ = 6.71 ± 0.74 μK². The best-fit spectral index is lower by 1σ, and the uncertainty increases by 50% to give α = 3.45 ± 0.11. However as discussed earlier, a free correlation significantly degrades the kSZ constraint.