MEASUREMENTS OF SECONDARY COSMIC MICROWAVE BACKGROUND ANISOTROPIES WITH THE SOUTH POLE TELESCOPE

The SPT beams are measured by combining maps of three sources: Jupiter, Venus, and the brightest point source in the 100 deg² field. Observations of Jupiter are used to measure a diffuse, low-level sidelobe in the range 15' < r < 40', where r is the radius to the beam center. Although this sidelobe has a low amplitude (−50 dB at r = 30'), it contains approximately 15% of the total beam solid angle. A measurement of this sidelobe is necessary for the cross-calibration with the Wilkinson Microwave Anisotropy Probe (WMAP) described in Section 2.2. The observations of Jupiter show signs of potential nonlinearity in the response of the detectors for r <10'. For this reason we only use the observations of Jupiter to map the sidelobe at r >15'. Observations of Venus are used to measure the beam in the region 4'< r <15'. The angular extent of Jupiter or Venus has a negligible effect on the measurement of the relatively smooth beam features present at these large radii. The brightest point source in the map of the 100 deg² field is used to measure the beam within a radius of 4'. In this way, the random error in the pointing reconstruction (7'' rms) and its impact on the effective beam are taken into account. The pointing error has a negligible effect on the relatively smooth outer (r >4') region of the beam.

A composite beam map, B(θ, ϕ), is assembled by merging maps of Jupiter, Venus, and the bright point source. The TOD are filtered prior to making these maps, in order to remove large-scale atmospheric noise. Masks with radii of 40', 25', and 5' are applied around the locations of Jupiter, Venus, and the point source, respectively. These masks ensure that the beam measurements are not affected by the filtering.

Using the flat-sky approximation, we calculate the Fourier transform of the composite beam map, B(ℓ, ϕ_ℓ). From this, we compute the azimuthally averaged beam function,

$$\begin{equation} B_\ell = {\rm {Re}}\bigg \lbrace \frac{1}{2\pi }\int B(\ell,\phi _\ell)d\phi _\ell \bigg \rbrace. \end{equation} \tag{ 1 }$$

We note that averaging |B(ℓ, ϕ_ℓ)|² instead of B(ℓ, ϕ_ℓ) would result in a percent-level noise bias in B_ℓ at very high multipoles due to the presence of noise. The results in this work assume an axially symmetric beam, which is only an approximation for SPT. We simulate the effects of ignoring the asymmetry on the bandpowers and find that the errors introduced by making this assumption are negligible.

Although the measured beam function B_ℓ is used for the bandpower estimation, an empirical fit is used to quantify the errors on B_ℓ. B_ℓ is fit to the empirical model

$$\begin{equation} B_\ell = ae^{-\frac{1}{2}(\sigma _b\ell)^{1.5}} + (1-a)e^{-\frac{1}{2}(0.00292*\ell)^{1.8}}. \end{equation} \tag{ 2 }$$

There are two components: a main lobe (first term) and a diffuse shelf (second term). The form of the model and the numerical values of the slopes of the exponents were constructed to provide a good fit to the measured B_ℓ. We note that B¹⁵⁰_ℓ and B²²⁰_ℓ are measured and fitted separately. The rms difference between the model and the measured B_ℓ's is approximately 1%. Two parameters remain free: σ_b, which describes the width of the main lobe, and a, which sets the relative normalization between the main lobe and the diffuse shelf. These parameters are left free to quantify the uncertainty in B_ℓ. The uncertainty in the values of these parameters directly translates to an uncertainty in B_ℓ.

There are a number of factors that limit the accuracy of the measurement of B_ℓ. These include residual map noise, errors associated with the map-merging process, and spectral differences between the CMB and the sources used to measure the beam. The final uncertainties in the beam model parameters σ_b and a are constructed as the quadrature sum of the estimated uncertainties due to each of these individual sources of error. To a good approximation, the uncertainties on σ_b and a can be taken to be Gaussian and uncorrelated.

In practice, a change in the value of a is equivalent to a change in the overall calibration for ℓ>700. After the beam uncertainties are estimated, the uncertainty in a is folded into the estimated uncertainty on the absolute calibration and the parameter a is fixed to the best-fit value of 0.85. The quoted beam uncertainties in the CosmoMC²⁴ (Lewis & Bridle 2002) data files in Section 6.1 include only the uncertainty on σ_b. Figure 1 shows the measured beam functions for 150 and 220 GHz, along with the 1σ uncertainties in the main lobe beam width σ_b.

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The calibration of the SPT data is tied to the superb WMAP5 absolute calibration through a direct comparison of 150 GHz SPT maps with WMAP5 V- and W-band (61 and 94 GHz) maps (Hinshaw et al. 2009) of the same sky regions. Bright radio sources are masked for this comparison. The WMAP5 maps are resampled according to the SPT pointing information and the resulting TOD are passed through the SPT analysis pipeline to capture the effects of TOD filtering. The ratio of the cross-spectra of the filtered WMAP and SPT maps after correcting for the instrumental beams,

$$\begin{equation} c = \frac{\left<a_{\ell m,{{\it WMAP}_i}}^{*}\,\,a_{\ell m,{{\it WMAP}_j}}\right>}{\left<a_{\ell m,{\it SPT}}^{*} \,\,a_{\ell m,{{\it WMAP}_j}} \left(\frac{B_\ell ^{{\it WMAP}_i} }{B_\ell ^{{\it SPT}}}\right)\right>}, \end{equation} \tag{ 3 }$$

is used to estimate the relative calibration factor between the two experiments. A similar procedure was used to calibrate the Boomerang, ACBAR, and QUaD experiments (Jones et al. 2006; Reichardt et al. 2009a; Brown et al. 2009). The calibration is weighted most heavily to ℓ ∈ [256, 512] and is flat with ℓ ∈ [128, 640]. The largest source of uncertainty is statistical noise in the maps of both experiments. The second largest source of uncertainty is due to the use of a secondary calibration source (RWC38) to bridge the several month gap between the science observations and calibration scans. Dedicated SPT calibration scans of four large fields totaling 1250 deg² of sky were obtained during 2008. The results for these four fields are combined to achieve an absolute temperature calibration uncertainty of 3.6% at 150 GHz.

The 150 GHz calibration is transferred to 220 GHz through the overlapping coverage of SPT's high S/N maps. We calculate the relative calibration by examining the ratio of the cross-spectra between the 150 and 220 GHz maps to the auto-spectra of the 150 GHz map after correcting for the beam and filtering differences. We estimate the relative calibration uncertainty to be 6.2% and the final absolute calibration uncertainty of the 220 GHz temperature map to be 7.2%.

We generate one map per frequency band for each 2 hr observation of the field. The map-making pipeline used for this analysis is very similar to that used by S09, with some small modifications. An overview of the pipeline is provided below, emphasizing the aspects of data processing and map-making that are most important for understanding the power-spectrum analysis.

3.1.1. Data Selection

3.1.2. Time-ordered Data Filtering

Let d_αj be a measurement of the CMB brightness temperature by detector j at time α. Contributions to d_αj are the celestial sky temperature and the atmospheric and instrumental noise. The instrumental noise is largely uncorrelated between detectors. However, the atmospheric signal is highly correlated across the focal plane due to the overlap of individual detector beams as they pass through turbulent layers in the atmosphere.

The signals in the data have been low-pass-filtered by the bolometers' optical time constants. These single-pole bolometer transfer functions are measured by the method described in S09. We deconvolve the transfer functions for each bolometer on a by-scan basis, as the first step in TOD filtering. At the same time, we apply a low-pass filter with a cutoff at 12.5 Hz. In the frequency domain, this filter has the frequency profile:

$$\begin{equation} F(\nu)=e^{-\left(\frac{\nu }{12.5\;{\rm Hz}}\right)^6}. \end{equation} \tag{ 4 }$$

The shape of the filter is chosen to be very sharp, yet not induce a significant amount of ringing. This filter removes noise above the Nyquist frequency associated with the final map pixelation. These operations can be represented as a linear operation ${ \bPi ^{\rm fft}}$ on the TOD,

$$\begin{equation} d^{\prime }_{\alpha j} = \Pi ^{\rm fft}_{\alpha \beta } d_{\beta j}, \end{equation} \tag{ 5 }$$

where a sum is implied over repeated indices.

To remove 1/f noise and atmospheric noise in the scan direction, we project out a 19th-order Legendre polynomial from the TOD for each detector on a scan-by-scan basis. This operation ${\bPi ^{\rm t}}$ can be described by

$$\begin{equation} d^{\prime \prime }_{\alpha j} = \Pi ^{\rm t}_{\alpha \beta } d^{\prime }_{\beta j}. \end{equation} \tag{ 6 }$$

The effect on the signal is similar to having applied a one-dimensional high-pass filter in the scan direction at ℓ ≳ 300.

Significant atmospheric signal remains in the data after this temporal polynomial removal. Because the atmospheric power is primarily common across the entire focal plane, we can exploit the spatial correlations to remove atmosphere without removing fine-scale astronomical signal. The method used in this analysis is to fit for and subtract a plane, a₀ + a_xx + a_yy, across all detectors in the detector array at each time sample, where x and y are set by the angular separation of each pixel from the boresight of the telescope. All detectors are equally weighted in this fit. This operation can be represented as a linear operation on all detectors at each time sample α to produce an atmosphere-cleaned data set, d‴

$$\begin{equation} d^{\prime \prime \prime }_{\alpha j} = \Pi ^{\rm s}_{j k} d^{\prime \prime }_{\alpha k}. \end{equation} \tag{ 7 }$$

We mask the brightest point sources in the map before applying the polynomial subtraction and the spatial-mode filtering. The 92 sources that were detected above 5σ in a preliminary 150 GHz map have been masked. A complete discussion of the point sources detected in this field can be found in Vieira et al. (2009, hereafter V09).

3.1.3. Map Making

The data from each bolometer are inverse noise weighted according to the calibrated, pre-filtering detector power spectral density (PSD) in the range 1–3 Hz, corresponding to 1400 < ℓ < 4300. This multipole range covers the angular scales where we expect the most significant detection of the SZ effect. We represent the mapping between time-ordered bolometer samples and celestial positions with the pointing matrix L, which we apply to the cleaned TOD to produce a map m

$$\begin{equation} {\bf m} = {{\bf L }\bLambda _{\bf w} \bPi ^{\rm s} \bPi ^{\rm t} \bPi ^{\rm fft} {\bf d}}. \end{equation} \tag{ 8 }$$

${ \bLambda _{\bf w}}$ is a diagonal matrix encapsulating the detector weights. Information on the pointing reconstruction can be found in S09 and Carlstrom et al. (2009). This procedure is repeated for all 300 observations used in this analysis. The mean maps for the 150 GHz and 220 GHz data are shown in Figure 2.

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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

$$\begin{equation} \mathcal {D}_\ell =\frac{\ell \left(\ell +1\right)}{2\pi } {\rm C}_\ell \;. \end{equation} \tag{ 9 }$$

As the first step in our analysis, we calculate the Fourier transform, $\tilde{m}_{\bf {k}}$, of the single-observation maps for each frequency. Each map is apodized using the same window $\textbf {W}$, and thus $\tilde{m}^{(\nu,A)}_{\bf {k}}\equiv \hbox{FFT}(\textbf {W}\textbf {m}^{(\nu,A)})$, where the first superscript, ν, indicates the observing frequency, the second superscript, A, indicates the observation number, and the subscript denotes the angular frequency. Cross-spectra are computed for every map pair and averaged within the appropriate ℓ-bin b:

$$\begin{equation} D^{\nu _i\times \nu _j, {\it AB}}_b\equiv \frac{1}{N_b}\sum _{{\bf k} \in b} \frac{{\rm k}({\rm k}+1)}{2\pi }\tilde{m}^{(\nu _i,A)}_{\bf {k}} \tilde{m}^{(\nu _j,B)*}_{\bf k}. \end{equation} \tag{ 10 }$$

Here, N_b is the number of modes in bin b. We use the abbreviation, $D^{\nu _i}\equiv D^{\nu _i\times \nu _i}$, when referring to single-frequency auto-spectra. Recall that |k| = ℓ in the flat-sky approximation.

The FFT of a map, $\widetilde{m}_{\bf k}^{(\nu,A)}$, is linearly related to the FFT of the astronomical sky, a^ν_k. It also includes a noise contribution, n^(ν,Ai)_k, which is uncorrelated between the 300 observations:

$$\begin{equation} \widetilde{m}^{(\nu,A)}_{\bf k}=\widetilde{W}_{\bf {k}-{\bf k}^\prime }G_{\bf {k}^\prime }\left(B_{\bf {k}^\prime }a^{\nu }_{\bf {k}^\prime }+n^{(\nu,A)}_{\bf {k}^\prime }\right). \end{equation} \tag{ 11 }$$

Here G_k is the two-dimensional amplitude transfer function, which accounts for the TOD filtering as well as the map-based filtering described in Section 3.2.2. $\widetilde{\textbf {W}}_{\bf {k}}$ is the Fourier transform of the apodization mask.

Following the treatment by Hivon et al. (2002), we take the raw spectrum, D^AB_b, to be linearly related to the true spectrum by a transformation, $K_{bb^\prime }$. The K transformation combines the power-spectrum transfer function, F_k, which includes the effects of TOD-based and map-based filtering (Section 3.2.3), the beams, B_k (Section 2.1), and the mode-coupling matrix, $M_{kk^\prime }[\textbf {W}]$ (Section 3.2.1). The mode-coupling matrix accounts for the convolution of the spectrum due to the apodization window, $\textbf {W}$

$$\begin{equation} K^{\nu _i\times \nu _j}_{bb^\prime }=P_{bk}\left(M_{kk^\prime }[\textbf {W}]\,F^{\nu _i\times \nu _j}_{k^\prime }B^{\nu _i}_{k^\prime }B^{\nu _j}_{k^\prime }\right)Q_{k^\prime b^\prime }. \end{equation} \tag{ 12 }$$

Here, P_bk is the re-binning operator (Hivon et al. 2002):

$$\begin{eqnarray} P_{bk}&=\left\lbrace \begin{array}{@{}ll} \displaystyle\frac{k(k+1)}{2\pi \Delta k_{(b)}}& k_{(b-1)} < k < k_{(b)}\\ 0 & \hbox{otherwise,}\end{array}\right. \end{eqnarray} \tag{ 13 }$$

while the inverse of the re-binning operator is Q_kb:

$$\begin{equation} Q_{k b}=\left\lbrace \begin{array}{@{}ll}\displaystyle\frac{2\pi }{k(k+1)}&k_{(b-1)}< k <k_{(b)} \\ 0 &\hbox{otherwise}\end{array}\right.. \end{equation} \tag{ 14 }$$

In the rest of the paper, we will often refer to band-averaged quantities, such as C_b = P_bkC_k. Cross-spectra between observations that have been corrected for apodization and processing are denoted as

$$\begin{equation} \widehat{D}^{\nu _i\times \nu _j,{\it AB}}_b\equiv \left(K^{-1}\right)_{bb^\prime }D^{\nu _i\times \nu _j,{\it AB}}_{b^\prime }. \end{equation} \tag{ 15 }$$

The final bandpowers are then computed as the average of all cross-spectra,

$$\begin{equation} q_b^{\nu _i\times \nu _j}=\frac{1}{n_{\rm obs}\left(n_{\rm obs}-1\right)}\sum _A \sum _{B \ne A} \widehat{D}^{\nu _i\times \nu _j,{\it AB}}_b. \end{equation} \tag{ 16 }$$

We make the simplifying assumption that the noise properties of each observation are statistically equivalent; hence, a uniform weighting is chosen here. The data selection criteria in Section 3.1.1 ensure that these observations target the same region of the sky, have roughly the same number of active bolometers, and have similar noise properties. Therefore, this uniform weighting should be unbiased and only slightly sub-optimal. Simulations suggest that properly weighting the observations would reduce the bandpower uncertainties by 14%.

The cross-spectrum bandpowers, $D^{\nu _i\times \nu _j,{\it AB}}_b$, generated from the 300 observations are used in conjunction with signal-only Monte Carlo bandpowers to generate empirical covariance matrices, as described in the Appendix. The variance of the Monte Carlo bandpowers is used to estimate the sample variance contribution to the covariance matrix. Meanwhile, we use the variance in the spectra of the real data to estimate the uncertainty due to noise in the maps.

3.2.1. Apodization Mask and Calculation of the Mode-mixing Kernel

Since we have only mapped a fraction of the full sky, the angular power spectra of the maps are convolutions of the true C_ℓs with an ℓ-space, mode-mixing kernel that depends on the map size, apodization, and point-source masking. We calculate this mode-mixing kernel, $M_{kk^\prime }[\textbf {W}]$, following the derivation in Hivon et al. (2002) for the flat-sky case:

$$\begin{equation} M_{kk^\prime }[\textbf {W}]\equiv \frac{1}{(2\pi)^2}\int d\theta _k\, d\theta _{k^\prime } \left|\widetilde{W}_{\bf {k}-{\bf k}^\prime }\right|^2. \end{equation} \tag{ 17 }$$

At high-k, elements of the mode coupling kernel depend only on the distance from the diagonal. The normalization of f(x) is ∫^∞₀dxf(x) = w₂, where $w_n\equiv \left<\textbf {W}^n\right>$ represents the nth moment of the apodization window. Thus, the coupling kernel also serves the purpose of re-normalizing the power spectrum to account for modes lost due to apodization. As we discuss in the Appendix, the coupling kernel also plays an important role in determining the shape and normalization of the covariance matrix.

We avoid areas of the map with sparse or uneven coverage in any single observation. Thus, the apodization window is conservative in its avoidance of the map edges. We also mask 144 point sources detected in the 150 GHz data above a significance of 5σ, which corresponds to a flux of 6.4 mJy. This source list is from a more refined analysis than the preliminary list used in Section 3.1.2; the differences are in sources near the 5σ detection threshold. Each point source is masked by a 2' radius disk, with a Gaussian taper outside this radius. In other words, pixels located at a distance r from the source location receive the weight:

$$\begin{equation} \begin{array}{@{}cl} 0 & r < 2^{\prime }\\ 1-e^{\frac{1}{2}(\frac{r-2^{\prime }}{5^{\prime }})^2} & r < 15^{\prime }\\ 1 & r > 15^{\prime }. \end{array} \end{equation} \tag{ 18 }$$

Many different tapered mask shapes were tested for both efficacy in removing point-source power and noise performance. Given the relatively small area masked by point sources, varying the shape of the point-source mask has little effect on the final spectrum. The effective sky area of the mask is 78 deg². Simulations have been performed to test whether the application of this mask will bias the inferred power and we see no bias with a 5σ cut. As could be expected, we do observe a mild noise bias when using a more aggressive 3σ point-source mask. The same 5σ mask is used for all maps at both frequencies and for all observations.

3.2.2. Fourier Mode Weighting

3.2.3. Transfer Function Estimation

In order to empirically determine the effect of both the TOD-based filtering and the Fourier mode weighting, we calculate a transfer function, F_k, as defined in Hivon et al. (2002). Note that this power-spectrum transfer function is distinct from the amplitude transfer function G_k. Specifically, the transfer function accounts for all map- and TOD-processing not taken into account by the mode-coupling kernel or the beam functions.

We created 300 Monte Carlo sky realizations at 150 and 220 GHz in order to calculate the transfer function of the filtering. The simulations also serve as an input for the covariance matrix estimation. These simulations contain two components: a CMB component and a point-source component. The CMB component is computed for the best-fit WMAP5 lensed ΛCDM model.

The point-source component includes two different populations of dusty galaxies, a low-z population and a high-z population. For each population we generate sources from a Poisson distribution. Sources are generated in bins of flux, S, ranging from 0.01 to 6.75 mJy. This upper limit in flux is close to the 5σ detection threshold in the 150 GHz maps. In each flux bin, the 150 GHz source density, dN/dS, of each population is taken from the model of Negrello et al. (2007). We relate the flux of each source at 220 GHz to its flux at 150 GHz with a power law in intensity, S_ν ∝ ν^α. The power-law spectral index, α, for each source is drawn from a normal distribution. We use spectral indices of α = 3 ± 0.5 for the high-z protospheroidal galaxies and α = 2 ± 0.3 for the low-z IRAS-like galaxies. As with any Poisson distribution of point sources, the power spectrum of these maps is white (C^ps_ℓ = constant) and related to the flux cutoff, S₀, by

$$\begin{equation} C_\ell ^{\rm ps} = \left(\frac{{\it dB}_\nu }{\it dT}\right)^{-2}_{\rm T_{\rm CMB}}\int _0^{S_0} S^2 \frac{\it dN}{\it dS} {\it dS}. \end{equation} \tag{ 19 }$$

The power spectra of these simulated point-source maps are well represented by a constant C^ps_ℓ = 1.1 × 10⁻⁵ μK² at 150 GHz and C^ps_ℓ = 6.8 × 10⁻⁵ μK² at 220 GHz.

These simulated maps are smoothed by the appropriate beam. From each map realization, we construct simulated TOD using the pointing information. Each realization of the TOD is processed using the low-pass filter, polynomial removal, and the spatial-mode subtraction described in Section 3.1.2. No time-constant deconvolution is applied, since these realizations of the TOD have not been convolved by the bolometer time constants. The filtered, simulated TOD are then converted into maps according to Equation (8).

We apply the same apodization mask and Fourier mode weighting to these map realizations as is used on the actual data. We then compute the Monte Carlo pseudo-power spectrum, (D_k)_MC, for each map. The simulated transfer function is calculated iteratively by comparing the Monte Carlo average, <D_k>_MC, to the input theory spectrum, C^theory_k.

For the single-frequency bandpowers, we start with an initial guess of the transfer function:

$$\begin{equation} F^{\nu,(0)}_{k}=\frac{\left<D^{\nu }_{k}\right>_{\rm MC}}{w_2 {B^{\nu }_k}^2 C^{\nu,{\rm theory}}_k}. \end{equation} \tag{ 20 }$$

The superscript (0), indicates that this is the first iteration in the transfer function estimates. For this initial guess, we approximate the coupling kernel as largely diagonal. Thus, the factor w₂ is the normalization factor required by the apodization window. We then iterate on this estimate using the full mode-coupling kernel:

$$\begin{equation} F^{\nu, (i+1)}_{k}=F^{\nu,(i)}_{k}+\frac{\left<D^{\nu }_{k}\right>_{\rm MC} - M_{kk^\prime } F_{k^\prime }^{\nu,(i)} {B^\nu _{k^\prime }}^2 C^{\nu,{\rm theory}}_{k^\prime }}{{B^\nu _{k}}^2 C^{\nu,{\rm theory}}_{k} w_2}. \end{equation} \tag{ 21 }$$

We iterate on this estimate five times, although the transfer function has largely converged after the first iteration.

This method may misestimate the transfer function if the simulated spectrum is significantly different from the true power spectrum. The primary CMB anisotropy has been adequately constrained by previous experiments; however, the foreground power spectrum is less well known. We repeated the transfer function estimation using an input power spectrum with twice the nominal point-source power. The resulting transfer function was unchanged at the 1% level, giving confidence that the transfer function estimate is robust.

The transfer function for a multifrequency cross-spectrum is taken to be the geometric mean of the two individual transfer functions:

$$\begin{equation} F_k^{\nu _i\times \nu _j}=\sqrt{F_k^{\nu _i}F_k^{\nu _j}}. \end{equation} \tag{ 22 }$$

This treatment is only strictly correct for isotropic filtering. As a cross-check, we have also computed the cross-spectrum transfer function directly. For the angular multipoles reported in this work, the geometric-mean transfer function estimate is in excellent agreement with the estimate obtained using $\left<D^{\nu _1\times \nu _2}_k\right>_{\rm MC}$ in Equations (20) and (21).

3.2.4. Window Function Estimation

The bandpower window functions, ${\mathcal W}^b_\ell / \ell$, are defined by the following equation:

$$\begin{equation} q_b^{\rm th} = ({\mathcal W}^b_\ell / \ell) C_\ell ^{\rm th} \end{equation} \tag{ 23 }$$

in order to facilitate the comparison of experimental bandpowers, q_b, to a theoretical power spectrum, C^th_ℓ. Following the formalism laid out above, we can write this as

$$\begin{equation} q_b^{\rm th} = (K^{-1})_{bb^{\prime }} P_{b^{\prime } k^{\prime }} M_{k^{\prime } k} F_k B_k^2 C_k^{{\rm th}}, \end{equation} \tag{ 24 }$$

which implies that

$$\begin{equation} {\mathcal W}^b_k / k = (K^{-1})_{bb^{\prime }} P_{b^{\prime } k^{\prime }} M_{k^{\prime } k} F_k B_k^2. \end{equation} \tag{ 25 }$$

3.2.5. Frequency-differenced Spectra

We are interested in the power spectra of linear combinations of the 150 and 220 GHz maps designed to remove astronomical foregrounds. One method for obtaining such power spectra would be to directly subtract the maps after correcting the maps for differences in beams or processing. In such a map subtraction, the scaling of the 220 GHz map, x, can be adjusted to optimally remove foregrounds. The differenced maps can then be processed using our standard pipeline to obtain spectra with a reduced foreground contribution.

Equivalently, the differenced spectrum can be generated from the original bandpowers, qⁱ_b, using a linear spectrum transformation, ξ:

$$\begin{equation} q^{{150}-x\times {220}}_b=\sum _{i} \xi ^{i}(x) q_b^i. \end{equation} \tag{ 26 }$$

Here, the index, i, denotes the 150 GHz auto-spectrum, 220 GHz auto-spectrum, or 150 × 220 GHz cross-spectrum. This transformation is computationally fast and takes advantage of the fact that the cross-frequency bandpowers include information on the relative phases of each Fourier component in the map. In this way, the difference spectrum can be represented in terms of the three measured spectra:

$$\begin{eqnarray} q_b^{\nu _i-x\times \nu _j} &=\frac{1}{\left(1-x\right)^2}\sum _{{\bf k}\in b} | a_{\bf k}^{\nu _i}-x a_{\bf k}^{\nu _j}|^2\nonumber\\ &=\frac{1}{\left(1-x\right)^2}\sum _{{\bf k} \in b}|a_{\bf k}^{\nu _i}|^2-2x \hbox{Re}\left(a_{\bf k}^{\nu _i} a_{\bf k}^{\nu _j*}\right)+x^2 |a_{\bf k}^{\nu _j}|^2\nonumber\\ &=\frac{1}{\left(1-x\right)^2}\left(q_b^{\nu _i}-2 x q_b^{\nu _i \times \nu _j}+x^2 q_b^{\nu _j}\right). \end{eqnarray} \tag{ 27 }$$

The overall normalization is chosen such that the CMB power is unchanged in the subtracted spectrum. For clarity, we have momentarily avoided here the complications of beams and filtering and have expressed the bandpowers in terms of the Fourier transform of the celestial sky, a_k. For a given proportionality constant, x, the values of ξ(x) are

$$\begin{eqnarray} \xi ^{{150}}(x)&=&\frac{1}{(1-x)^2}, \nonumber \\ \xi ^{{150}\times {220}}(x)&=&\frac{-2x}{(1-x)^2},\nonumber \\ \xi ^{{220}}(x)&=&\frac{x^2}{(1-x)^2}. \end{eqnarray} \tag{ 28 }$$

In order to compute covariances for the subtracted spectrum, one needs to know the correlations between bandpowers measured at different frequencies. Since both frequencies cover the exact same area of sky, the 150 GHz, 220 GHz, and cross-spectra are nearly completely correlated at low ℓ's where the errors are dominated by sample variance. At higher ℓ-ranges the cross-spectrum bandpowers are also partially correlated with both the 150 GHz and 220 GHz bandpowers, since the instrumental contribution to the cross-spectrum variance originates from the 150 GHz and 220 GHz noise. We represent the correlation between ℓ-bins, b and b', measured at frequencies, i and j as follows:

$$\begin{equation} \textbf {C}^{(i, j)}_{bb^\prime }\equiv \big <q_b^i q_{b^\prime }^j\big >- \big <q_{b}^i\big >\big <q_{b^\prime }^j\big >. \end{equation} \tag{ 29 }$$

As before, the superscripts i and j may stand for 150 GHz, 220 GHz, or the cross-spectrum 150 × 220 GHz. The method for computing the multifrequency covariance matrix is discussed in the Appendix. By combining Equations (26) and (29), one can calculate the covariance matrix for the subtracted bandpowers from the multifrequency covariance matrices:

$$\begin{eqnarray} \textbf {C}^{150-x\times 220}_{bb^\prime }&\equiv \left<q_b^{150-x\times 220}q_{b^\prime }^{150-x\times 220}\right>\nonumber\\ &\quad -\left<q_b^{150-x\times 220}\right>\left<q_{b^\prime }^{150-x\times 220}\right>\\ &=\sum _{i,j} \xi ^{i}(x)\textbf {C}^{(i,j)}_{b b^\prime } \xi ^{j}(x). \end{eqnarray} \tag{ 30 }$$

In this way, we account for correlations between bandpowers of different frequencies when computing the subtracted-spectrum covariance matrix. Ignoring these correlations can lead to a gross overestimate of the uncertainty in the subtracted spectrum. For the SPT bandpowers presented in Section 6.1, neglecting these correlations leads to a 100% overestimate of δC_ℓ at ℓ = 2000 for x = 0.325.

Figure 4 shows the bandpowers we compute by applying the analysis methods described in Section 3 to one 100 deg² field observed by SPT at 150 and 220 GHz. The bandpowers for the two frequencies and their cross-spectrum are tabulated in Table 1. We combine these multifrequency data as described in Section 3 to produce the "DSFG-subtracted" bandpowers listed in Table 2. This power spectrum is compared to the results from WMAP5, ACBAR, and QUaD in Figure 5. The best-fit model to this combined data set including the primary CMB, kSZ, tSZ, and a Poisson point-source contribution is shown for reference. Due to the normalization of the DSFG-subtracted spectrum, the tSZ amplitude is a factor of 2.2 higher in the subtracted spectrum than at 150 GHz. The primary CMB anisotropy is estimated for a spatially flat, ΛCDM model, which includes gravitational lensing. The bandpower uncertainties are derived from the combination of simulations and the measured intrinsic variations within the SPT data as described in Section 3. The bandpowers can be compared to theory using the associated window functions (Knox 1999). The bandpowers, uncertainties, and window functions may be downloaded from the SPT Web site.²⁵

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The SPT bandpowers are dominated by the damping tail of the primary CMB anisotropy on angular multipoles 2000 < ℓ < 3000. At these multipoles, the bandpowers are in excellent agreement with the predictions of a ΛCDM model determined from CMB observations on larger angular scales. On smaller scales, the SPT bandpowers provide new information on secondary CMB anisotropies and foregrounds which dominate the primary CMB anisotropy. The SPT data presented here represent the first highly significant detection of power at these frequencies and angular scales where the primary CMB anisotropy is sub-dominant. After masking bright point sources, the total S/N on power in excess of the primary CMB are 55, 55, and 45 at 150, 150 × 220, and 220 GHz, respectively. The majority of the high-ℓ power can be attributed to a Poisson distribution of point sources (likely DSFGs) on the sky. H10 use the bandpowers presented in this work to measure both Poisson and clustered contributions to the measured powers and investigate the implications for the properties of DSFGs. The largest source of secondary CMB anisotropy at 150 GHz is expected to be the SZ effect and we investigate SZ constraints in the following sections.

Our immediate goal is to measure the amplitude of the tSZ power spectrum. However, several signals in these maps have similar angular power-spectrum shapes and the single-frequency maps only constrain the sum of the power from these sources. For instance, the 150 GHz data effectively constrain the sum of the tSZ, kSZ, and clustered DSFG power.

As discussed earlier, each of these components has a distinct frequency dependence so a linear combination of SPT's two frequency bands can be constructed (following Section 3.2.5) to minimize any one of them. H10 find significant evidence for a clustered DSFG power contribution to the single-frequency bandpowers listed in Table 1 with an amplitude comparable to that of the tSZ effect. We expect the kSZ effect to be smaller than the tSZ effect at 150 GHz on theoretical grounds. Additionally, due to the frequency dependences of the components, removing the kSZ effect would inflate the relative contribution of clustered DSFGs with respect to the tSZ effect. Therefore, we choose to remove DSFGs from the SPT bandpowers.

For a mean DSFG spectral index, α, the proper weighting ratio, x, to apply to the 220 GHz spectrum for DSFG removal would be

$$\begin{equation} x =\frac{S_{150}\big/\big(\frac{{\it dB}_\nu }{{\it dT}_{\rm CMB}}|_{150}\big)}{S_{220}\big /\big (\frac{{\it dB}_\nu }{{\it dT}_{\rm CMB}}|_{220}\big)}= (150/ 220)^\alpha \frac{\frac{{\it dB}_\nu }{{\it dT}_{\rm CMB}}|_{220}}{\frac{{\it dB}_\nu }{{\it dT}_{\rm CMB}}|_{150}}. \end{equation} \tag{ 32 }$$

The spectrum in Table 2 and Figures 5 and 6 is produced with a weighting factor of x = 0.325 corresponding to a mean spectral index of α = 3.6. The contribution from DSFGs can be completely removed only if every galaxy has the same spectral index. However, the comparative closeness of SPT's two frequency bands ensures that power leakage into the difference maps remains small despite some expected scatter in the spectral index of the dusty galaxies. In Section 6.2.1, we motivate this choice of x and discuss predictions for the residual point-source power.

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We assume that there is no tSZ contribution to the 220 GHz data as the 220 GHz band is designed to be centered on the SZ null. Fourier transform spectroscopy measurements of the 220 GHz band pass confirm that the tSZ amplitude in the 220 GHz band will be ≲5% of the 150 GHz amplitude. Any error incurred by subtracting roughly one-third of the 220 GHz amplitude from the 150 GHz data would be less than 3%, far below the present ∼40% statistical uncertainty on A_SZ (see Table 3).

It is important to note that the apparent tSZ power in the DSFG-subtracted bandpowers will be a factor of 1/(1 − x)² = 2.2 higher than at 150 GHz as the differenced bandpowers have been normalized to preserve the amplitude of the primary CMB anisotropy. In this work, we report SZ amplitudes scaled to 153 GHz which is the effective band center of the 150 GHz band for a tSZ spectrum.

6.2.1. Residual Point-source Power

The DSFG-subtracted maps have substantially less power due to both unclustered and clustered point sources as seen in Figure 6. However, we include a Poisson point-source amplitude in all fits since we expect a small fraction of the point-source power to remain in the DSFG-subtracted maps. The best-fit amplitude is consistent with zero and unphysical negative values of the Poisson point-source power are allowed due to noise. To prevent this, we place a prior on the residual Poisson point-source power based on what we know about the observed DSFG population from H10 and radio source population from V09 and de Zotti et al. (2005).

The residual power in the subtracted map due to Poisson-distributed DSFGs, C^DSFG_ℓ, depends on the scatter in spectral indices σ_α, the accuracy to which the mean spectral index $\bar{\alpha }$ is known, and an estimate of the Poisson DSFG power in the 150 GHz band, C^ps,150_ℓ. For a given combination of these parameters, this residual DSFG power will be

$$\begin{eqnarray} C_\ell ^{\rm DSFG} &=& C_\ell ^{{\rm ps},150} \left(\sigma _\alpha ^2 \left[\ln (\nu _{150}/ \nu _{220})\right]^2 + \left(1 - \frac{x}{x_{\rm true}} \right)^2 \right), \nonumber\\ \end{eqnarray} \tag{ 33 }$$

where ν₁₅₀ and ν₂₂₀ are the effective bandcenters of the 150 and 220 GHz bands, and x and x_true are the assumed and true values of map weighting ratio.

We examine the residual Poisson point-source amplitudes for a broad range of weighting ratios to estimate the optimal x value and the error in that estimate. For each x, we estimate the probability that C^ps_ℓ(x) is less than C^ps_ℓ(x = 0.325) using the MCMC chains described in Section 6.3. The resulting probability distribution is taken to be the likelihood function for x_true. As shown in Figure 7, there is a broad maximum for x = 0.25 to 0.4 and we adopt the best fit, x = 0.325, for the following results. In the absence of a direct measurement, we place a conservative uniform prior on the scatter in DSFG spectral indices, 0.2 < σ_α < 0.7, as discussed in H10.

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Our expectation for the residual radio contribution to the DSFG-subtracted bandpowers, C^radio_ℓ, is based on the de Zotti et al. (2005) radio source count model. This model is in excellent agreement on the high flux end with the SPT source counts (V09). As discussed in H10, the residual radio source power after masking is expected to be a small fraction of the DSFG power at 150 GHz. However, this small radio contribution may be comparable to the residual DSFG power in the DSFG-subtracted spectrum. The radio source power can be calculated from the integral of S²dN/dS for the de Zotti et al. (2005) counts model from zero to the flux masking threshold of 6.4 mJy. We compute the power these sources contribute to the optimal DSFG-subtracted spectrum to be 3.9 × 10⁻⁷ μK² by assuming an average spectral index of α = −0.5 based on the detected sources in V09. This power level is nearly identical to that predicted by the Sehgal et al. (2010) simulations. To allow for a variation in spectral index as well as uncertainty in the model normalization when extended to lower flux sources, we assign a conservative uncertainty of 50% on the predicted residual radio source power in the prior.

We combine this information to create a prior on the residual point-source power in the DSFG-subtracted maps C^ps_ℓ = C^DSFG_ℓ + C^radio_ℓ. This prior spans the range C^ps_ℓ∈[3.5, 9.0]×10⁻⁷ μK² at 68% confidence and [1.2, 13.9]×10⁻⁷ μK² at 95% confidence. The best-fit value of the residual Poisson component before applying the prior is C^ps_ℓ(no prior) = (6.2 ± 6.4) × 10⁻⁷ μK² and lies at the middle of our assumed prior range. The upper end of the 95% range is approximately 20% of the best-fit value of the Poisson point-source power in the undifferenced 150 GHz bandpowers. This suggests that we have subtracted over 80% of the point-source power from the 150 GHz spectrum, with the residual point-source power largely from radio sources. Without this prior on the Poisson point-source amplitude, the uncertainty on the A_SZ detection presented in the next section would increase by ∼50%.

6.2.2. Residual Clustered Point-source Power

The DSFG-subtracted bandpowers presented in Section 6.2 detect at high significance a combination of the primary CMB anisotropy, secondary SZ anisotropies, and residual point sources. In this section, we use an MCMC analysis to separate these three components and to produce an unbiased measurement of the tSZ power-spectrum amplitude.

6.3.1. Elements of the MCMC Analysis

6.3.2. Constraints on SZ Amplitude

We fit for the normalization factor of a fixed tSZ template and, in Table 3, report both this template-specific normalization, A_SZ, and the total inferred SZ-power at ℓ = 3000, near the multipole with maximum tSZ detection significance. This estimate of the SZ power includes both tSZ and kSZ terms, and is included to facilitate comparison with other SZ models. We expect both the thermal and the kinetic SZ spectra to vary slowly with angular multipole.

The chains are run for three different assumptions about the kSZ effect: the no kSZ, homogeneous kSZ, and patchy kSZ models described above. In each case, we produce MCMC chains with a fixed kSZ amplitude. The χ² of the 15 SPT bandpowers is between 16.3 and 16.4 for the best fits of the three kSZ cases considered; there is essentially no impact on the quality of the fit. The A_SZ and power constraints for each case are listed in Table 3 and plotted in Figure 8. Shifting the frequency-differencing factor, x, by ±0.05 changes the resultant best-fit A_SZ by less than 0.5σ.

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The bandpower uncertainties in Table 2 do not include the sample variance of the tSZ effect, and we must convolve the A_SZ distribution in the chains with an estimate of the sample variance to find the true A_SZ likelihood function. The sample variance of the tSZ effect is estimated using the simulations of Shaw et al. (2009) re-run with the same intracluster gas model parameters as Sehgal et al. (2010). The simulation consists of 300 map realizations of the size of the SPT sky patch. These are constructed from a base sample of 40 independent maps by separating the components of each map into eight redshift bins (between 0 ⩽ z ⩽ 3) and shuffling these bins between maps to generate a larger sample. We take the spectrum of each realization and find the best-fit A_SZ amplitude after weighting ℓ-bins based on the SPT bandpower uncertainties. We use the distribution of these amplitudes to map out the likelihood function for the tSZ power (see Figure 9). Due to the number of independent realizations, we have limited ability to resolve the tail of the likelihood function. Fortunately, the non-Gaussianity is small for such a large sky area and the distribution of ln(A_SZ) is fit well by a Gaussian with a 12% width. We use this Gaussian fit as an estimate of the full likelihood surface. Small deviations from the true description of the tSZ sample variance will not impact the final results as the sample variance is small compared to both the statistical uncertainties and the assumed 50% model uncertainty. The uncertainties on A_SZ are essentially unchanged by the inclusion of the tSZ sample variance. The sample variance and model uncertainty are shown in Figure 9.

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As discussed earlier, the DSFG-subtracted bandpowers are sensitive to a linear combination of the tSZ and kSZ effects. We expect analysis of 2009 and later SPT data which include 95 GHz data to be able to separate the two SZ effects. The linear combination is not a simple sum, as the frequency differencing used to produce the DSFG-subtracted spectrum suppresses the kSZ relative to the tSZ by a factor of (1 − x)² = 0.46. This factor is uncertain at the 15% level due to the relative calibration uncertainty between the bands. The SPT data detect the combined SZ effect at 2.6σ with tSZ + 0.46 × kSZ = 4.2 ±  1.5 μK² at ℓ = 3000. Using this combined constraint implicitly assumes that the tSZ and kSZ templates are perfectly degenerate, which is a good assumption for the current data quality and templates used in this work.

We can compare the power detected with SPT to that reported by the CBI Collaboration (Sievers et al. 2009). We use the best-fit normalization of a Komatsu & Seljak template for WMAP5 parameters to compare directly the results of the two experiments. There are sub-percent differences in the assumed values of σ₈ and Ω_bh between the template we adopted here and that used in Sievers et al. (2009), which would change the amplitude of the template by ∼1% for an assumed scaling of σ⁷₈(Ω_bh)². This effect is negligible. We find the best-fit normalization of the WMAP5 Komatsu & Seljak model to be 0.37 ± 0.17 for the SPT data under the homogeneous kSZ scenario. This is 2.4σ below the best-fit CBI normalization of 3.5 ± 1.3. The model includes the frequency dependence of the tSZ effect. The smaller SPT bandpowers suggest that the CBI excess power may be produced by foregrounds with a frequency dependence falling more steeply than the SZ effect such as radio sources.

The best-fit normalization for the fiducial tSZ spectrum, A_SZ, is significantly lower than unity. The cosmological parameters assumed when generating this template may be slightly different than the best-fit models, so we scale the template to match the cosmologies explored by the Markov chain. When these scalings are taken into account, the measured values of A_SZ are still low. The tSZ power spectrum depends on the details of how the baryon intracluster gas populates dark matter halos and on cosmology through the number density of these halos. The paucity of tSZ power may reflect an overestimate of the intracluster gas pressure by the fiducial model and so we compare the template to other SZ models. At the same time, this low value of A_SZ favors a shift in the derived cosmological parameters, particularly σ₈.

Even a ∼2.5σ detection of A_SZ will produce cosmologically interesting constraints on σ₈, as we expect A_SZ to scale strongly with σ₈ and less strongly with the baryon density. The scaling is approximately σ^γ₈(Ω_bh)² where 7 < γ < 9, depending on the exact cosmology (Komatsu & Kitayama 1999; Komatsu & Seljak 2002). We explore the σ₈ dependence under the Press–Schechter halo model and find that this relationship steepens with the currently favored lower values of σ₈. Sampling cosmological parameter values from the WMAP5 MCMC chains, we find that the modeled amplitude of the tSZ power spectrum varies roughly as σ^β₈, with β ≈ 11. This sampling reflects the degeneracies between parameters, and so the value of this exponent, β, includes the degeneracy between σ₈ and Ω_mh. Constraining the amplitude of the tSZ effect offers an independent measurement of σ₈ that can be compared to measurements based on primary CMB anisotropy or large-scale structure. Such comparisons test our understanding of the physical processes involved in structure formation.

For each point in the MCMC chain, we calculate the predicted A_SZ value from the six basic cosmological parameters. For this calculation, we use the mass function of Jenkins et al. (2001) to determine the abundance of galaxy clusters of a given mass. We then use the mass–concentration relation of Duffy et al. (2008) to determine the dark matter halo properties, and the gas model used in Komatsu & Seljak (2002) to estimate the tSZ signal for each halo according to its mass. In order to convert this analytic tSZ spectrum into an amplitude, we take an ℓ-weighted average designed to match the relative weights each multipole receives in the real tSZ fits. This amplitude is normalized to unity for the cosmological parameters assumed in the fiducial tSZ model. We also allowed the mass–concentration index to vary with cosmology by appropriate scaling of the characteristic mass M_*, but this was found to be a negligible effect within the explored range in parameters.

At each point in the chain, the measured tSZ amplitude is compared to the predicted tSZ amplitude to construct a tSZ scaling factor, A_SZ/A^theory_SZ. This procedure will account for any correlations between the measured A_SZ parameter and the six ΛCDM parameters in a self-consistent fashion, although we do not see evidence for such correlations in the current data. The distribution of scaling factors versus σ₈ is illustrated by the black contours in Figure 10.

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In general, the tSZ scaling factors are less than unity. These low scaling factors suggest either an overestimate of the tSZ effect or lower values of σ₈. Models predicting larger kSZ or tSZ effects lead to lower scaling factors. However, the results cannot be purely explained by an overestimate of the kSZ effect since this tension persists in the no-kSZ case. Alternatively, the scaling factors may indicate that the explored range in cosmological parameters is systematically overestimating the rms of the mass distribution. For instance, as we see in Figure 10, points of the chain with lower values of σ₈ have scaling factors closer to unity.

The first interpretation of the low tSZ scaling factor is that the Sehgal tSZ template overestimates the tSZ power spectrum. There is currently some degree of uncertainty in the expected shape and amplitude of the tSZ power spectrum as predicted by analytic models or hydrodynamical simulations. One reason for this is that cosmological simulations of the intracluster medium have only recently begun to investigate in detail the impact of radiative cooling, non-gravitational heating sources (such as AGNs), and possible regulatory mechanisms between them. The computational expense of running hydrodynamical simulations with sufficient resolution to resolve small-scale processes (such as star formation) while encompassing a large enough volume to adequately sample the halo mass function is prohibitive to a detailed analysis of the predicted SZ power spectrum. In order to accurately predict the tSZ power spectrum, it is especially important to correctly model the gas temperature and density distribution in low-mass (M < 2 × 10¹⁴h⁻¹M_☉) and high-redshift (z>1) clusters, which contribute significantly to the power spectrum at the angular scales where SPT is most sensitive (Komatsu & Seljak 2002).

In Figure 11, we plot the tSZ power spectrum derived from several different simulations (note that all curves have been normalized to the fiducial cosmology). The thick black solid line shows the base template, obtained from maps generated by Sehgal et al. (2010). The black dot-dashed line shows the power spectrum obtained from the simulations of Shaw et al. (2009), which have a lower energy feedback parameter than the Sehgal et al. (2010) simulations. Increasing the amount of feedback energy has the effect of inflating the gas distribution and suppressing power at small angular scales. The red solid and dot-dashed lines show the power spectrum from maps constructed from a 240 h⁻¹ Mpc box simulation run using the Eulerian hydrodynamics code ART (Kravtsov et al. 2005; D. Rudd 2009, private communication) and from the MareNostrum simulation (O. Zahn et al. 2010, in preparation), respectively. Both of these simulations were run in the adiabatic regime, i.e., no cooling, star formation, or feedback. The gas distribution is more centrally concentrated than in the Shaw or Sehgal models, producing tSZ power spectra with significantly more power at small angular scales and less at larger angular scales. The blue solid line shows the analytic template predicted by the Komatsu & Seljak (2002) halo model calculation. The histogram shows the SPT sensitivity to the thermal SZ signal in each ℓ band (with arbitrary normalization).

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The current data are in tension with even the high-feedback simulations for the CMB-derived best-fit cosmological parameter set. Even with the kSZ effect set to zero, the tSZ scaling factor is only 0.55 ± 0.21 of what is predicted for the fiducial WMAP5 cosmology. Meanwhile, the tSZ scaling factors are 0.42 ± 0.21 for the homogeneous kSZ model and 0.34 ± 0.21 for the patchy kSZ model. The tension grows worse for the medium-feedback simulations by Shaw et al. (2009). Although the Bode et al. (2007) model is calibrated to reproduce observed X-ray scaling relations for high-mass, low-redshift clusters, it may significantly overestimate the contribution of low-mass or high-redshift clusters (for which there are few direct X-ray observations with which to compare).

As mentioned above, an alternate interpretation of the low tSZ scaling factor is that the CMBall parameter chains without SZ constraints overestimate σ₈. In order to constrain σ₈ based on the measurement of A_SZ, we need to construct the likelihood of observing A_SZ for a given cosmological parameter set. At a minimum, this likelihood would reflect the 12% uncertainty due to sample variance of the tSZ effect. However given the large range of tSZ predictions, we add an additional theory uncertainty in quadrature with the sample variance. The gray region in Figure 11 shows the 1σ region encompassed by a lognormal distribution of width σ_Asz = 0.5 around the fiducial model (black solid line). In the range where SPT is most sensitive (2000 ⩽ ℓ ⩽ 6000), all the predicted curves lie within this region. In order to account for the range of predicted power spectra, we adopt a 50% theory uncertainty on the value of ln(A_sz) in the likelihood calculation. The resulting σ₈ constraints are dominated by the theory uncertainty. With this prior, we construct a new chain from the original parameter space through importance sampling. The regions preferred by this prior are shown in Figure 10, as are the results of the new Markov chain.

Constraints on σ₈ with and without including the SPT A_SZ measurements are shown in Table 3 and Figure 12. Under the assumption of the homogeneous kSZ model and a 50% theoretical uncertainty in the amplitude of the tSZ power spectrum, the addition of the SPT data slightly tightens the constraint on σ₈, while reducing the central value from σ₈ = 0.794 ± 0.028 to 0.773 ± 0.025. The uncertainty in the resulting constraint on σ₈ is dominated by the large theoretical uncertainty in the tSZ amplitude. If instead we assume that the fiducial tSZ model is perfectly accurate and do not account for model uncertainty, the uncertainty on σ₈ is reduced by 30% and the preferred value is significantly reduced to σ₈ = 0.746 ± 0.017 for the homogenous kSZ model. Despite the fact that the adopted template is the lowest of the tSZ models shown in Figure 11, the Sehgal et al. (2010) template, the value of σ₈ inferred by the SPT data using this model is lower than that favored by WMAP. Additional SPT data will soon determine if the apparent tension between σ₈ inferred from SZ and primordial CMB measurements is robust. In any case, improving our theoretical understanding of both the kSZ and tSZ power spectra is essential for fully realizing the potential of SZ power-spectrum measurements to constrain cosmological parameters such as σ₈.

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Following Tristram et al. (2005), we represent the expected covariance between two cross-spectra as Ξ:

$$\begin{eqnarray} \hspace{-5.4pc}\Xi _{\ell \ell ^\prime }^{{\it AB},{\it CD}}&\equiv \left<\left(\widehat{D}_\ell ^{\it AB}-\left<\widehat{D}_{\ell }^{\it AB}\right>\right)\left(\widehat{D}_{\ell ^\prime }^{\it CD}-\left<\widehat{D}_{\ell ^\prime }^{\it CD}\right>\right)\right> \end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray} &=\left(K[\textbf {W}]^{-1}\right)_{b^{\prime \prime }b} \left(\left<D^{\it AB}_{b^{\prime \prime }} D^{\it CD}_{b^{\prime \prime \prime }}\right> -\left<D^{\it AB}_{b^{\prime \prime }}\right>\left<D^{\it CD}_{b^{\prime \prime \prime }}\right> \right) \nonumber\\ &\quad\times\left(K[\textbf {W}]^{-1}\right)_{b^{\prime \prime \prime }b^\prime }. \end{eqnarray} \tag{ A2 }$$

Our goal is to express this covariance in terms of the noise and signal in the maps, and to compute the magnitude of the diagonal elements, as well as the correlation between bandpowers. As the first step, the central term can be rewritten as

$$\begin{eqnarray} &\big\langle D^{\it AB}_{b^{}} D^{\it CD}_{b^{\prime }}\big\rangle -\big\langle D^{\it AB}_{b^{}}\big\rangle\big\langle D^{\it CD}_{b^{\prime }}\big\rangle\nonumber\\ &\qquad=P_{bk}P_{b^\prime k^\prime }\frac{1}{(2\pi)^2}\int d\theta _k d\theta _{k^\prime } \Big(\big\langle\widetilde{m}^{A}_{\bf {k}^{}}\widetilde{m}^{B*}_{\bf {k}^{}} \widetilde{m}^{C}_{\bf {k}^{\prime }}\widetilde{m}^{D*}_{\bf {k}^{\prime }}\big\rangle\nonumber\\ &\qquad \qquad-\big\langle\widetilde{m}^{A}_{\bf {k}^{}}\widetilde{m}^{B*}_{\bf {k}^{}}\big\rangle \big\langle\widetilde{m}^C_{\bf {k}^{\prime }}\widetilde{m}^{D*}_{\bf {k}^{\prime }}\big\rangle \Big)\nonumber\\ &\qquad=P_{bk}P_{b^\prime k^\prime }\frac{1}{(2\pi)^2}\int d\theta _k d\theta _{k^\prime } \Big(\big\langle\widetilde{m}^{A}_{\bf {k}^{}}\widetilde{m}^{C}_{\bf {k}^{\prime }}\big\rangle \big\langle\widetilde{m}^{B*}_{\bf {k}^{}}\widetilde{m}^{D*}_{\bf {k}^{\prime }}\big\rangle\nonumber\\ &\qquad\qquad+\big\langle\widetilde{m}^{A}_{\bf {k}^{}}\widetilde{m}^{D*}_{\bf {k}^{\prime }}\big\rangle \big\langle\widetilde{m}^{B*}_{\bf {k}^{}}\widetilde{m}^{C}_{\bf {k}^{\prime }}\big\rangle \Big). \end{eqnarray} \tag{ A3 }$$

To simplify Equation (A3), Tristram et al. (2005) and Polenta et al. (2005) make the following assumptions.

• 1.  
  Fluctuations in the map, i.e., from CMB anisotropies, confusion limited point sources and noise, are well described by a Gaussian random field.
• 2.  
  The beams and filtering applied to the data are isotropic; E_k ≡ G_kB_k depends only on $|\textbf {k}|$.
• 3.  
  The instrumental noise is isotropic.
• 4.  
  The power spectrum, C_k, is smoothly varying with k, and changes little over scales comparable to the width of the mode coupling matrix.

Using the assumption that C_k and |E_k| do not vary much over small changes in $\textbf {k}$, these products become

$$\begin{eqnarray} \left< \widetilde{m}^{A}_{\bf k}\widetilde{m}^{B*}_{\bf {k}^\prime }\right> &=& \sum _{\bf {k}^{\prime \prime }{\bf k}^{\prime \prime \prime }} \widetilde{\textbf {W}}_{\bf {k}^{}-{\bf k}^{\prime \prime }} \widetilde{\textbf {W}}^{*}_{\bf {k}^{\prime }-{\bf k}^{\prime \prime \prime }} E_{\bf {k}^{\prime \prime }}E^{*}_{\bf {k}^{\prime \prime \prime }} \left<a^{A}_{\bf {k}^{\prime \prime }}a^{B*}_{\bf {k}^{\prime \prime \prime }}\right>\nonumber\\ &=&\sum _{\bf {k}^{\prime \prime }} \widetilde{\textbf {W}}_{\bf {k}^{}-{\bf k}^{\prime \prime }} \widetilde{\textbf {W}}_{\bf {k}^{\prime \prime }-{\bf k}^\prime } \left|E_{\bf {k}^{\prime \prime }}\right|^2 C^{\it AB}_{k^{\prime \prime }}\nonumber\\ &\approx& \widetilde{\textbf {W}^2}_{\bf {k}^{}-{\bf k}^\prime }\left|E_{\bf {k}}\right|^2C^{\it AB}_{k}. \end{eqnarray} \tag{ A4 }$$

Here C^AB_k is shorthand for $C_k+\frac{N^{A}_k}{B_k^2}\delta _{\it AB}$, the expected cross-spectrum between two unfiltered, perfectly beam-corrected—though noisy—maps. The additional term is the noise bias that exists in the map auto-spectrum. Assuming isotropic beams and filtering, we can combine Equations (A3) and (A4) to obtain a relatively simple expression:

$$\begin{eqnarray} &\left<D^{\it AB}_{b^{}} D^{\it CD}_{b^{\prime }}\right> -\left<D^{\it AB}_{b^{}}\right>\left<D^{\it CD}_{b^{\prime }}\right>\nonumber\\ &\quad=P_{bk}P_{b^\prime k^\prime }\frac{1}{\left(2\pi \right)^2}\int d\theta _k d\theta _{k^\prime } \left|\widetilde{\textbf {W}^2}\right|^2\left|E_{k}\right|^2\left|E_{k^\prime }\right|^2\nonumber\\ &\qquad\times\left(C_k^{\it AB}C_{k^\prime }^{\it CD}+C_k^{AD}C_{k^\prime }^{BC}\right) \nonumber \\ &\quad=P_{bk}P_{b^\prime k^\prime }E_k^2 E_{k^\prime }^2 M[\textbf {W}^2]_{kk^\prime }\left(C_k^{AC}C_{k^\prime }^{BD}+C^{AD}_{k}C^{BC}_{k^\prime }\right).\qquad \end{eqnarray} \tag{ A5 }$$

Combining Equations (A3) and (A5) yields

$$\begin{eqnarray} \Xi ^{{\it AB},{\it CD}}_{bb^\prime }&=&\left(K[\textbf {W}]^{-1}\right)_{b^{(2)}b}\Big[P_{b^{(2)}k}P_{b^{(3)} k^\prime }M[\textbf {W}^2]_{kk^\prime }E_{k}^2E_{k^\prime }^2\nonumber\\ &&\times\left(C_k^{AC}C_{k^\prime }^{BD}+C^{AD}_{k}C^{BC}_{k^\prime }\right)\Big]\left(K[\textbf {W}]^{-1}\right)_{b^{(3)}b^\prime }.\qquad \end{eqnarray} \tag{ A6 }$$

To obtain a simplified expression for the magnitude of the diagonal elements of the covariance matrix, one typically assumes the mode-coupling matrix is nearly diagonal: $M_{kk^\prime }[\textbf {W}]\approx w_2 \delta _{kk^\prime }$. In this approximation $\left(K[\textbf {W}]^{-1}\right)_{bb^\prime }\approx w_2^{-1}E_b^{-2}\delta _{bb^\prime }$ and $M_{kk^\prime }[\textbf {W}^2]\approx w_4 \delta _{kk^\prime }$ and thus the covariance of any two cross spectra is

$$\begin{eqnarray} \Xi ^{{\it AB}, {\it CD}}_{bb^\prime }&\approx \frac{w_4}{N_b w_2^2}\left(\frac{\ell _{{\rm eff},\, b}(\ell _{{\rm eff},\, b}+1)}{2\pi }\right)^2\nonumber\\ &\quad\times\left(C^{AC}_bC^{BD}_{b}+C^{AD}_bC^{BC}_b\right)\delta _{bb^\prime }\nonumber \\ &=\frac{\left<\widehat{D}^{AC}_b\right>\left<\widehat{D}^{BD}_{b}\right>+\left<\widehat{D}^{AD}_b\right>\left<\widehat{D}^{BC}_b\right>}{\nu _b}\delta _{bb^\prime }, \end{eqnarray} \tag{ A7 }$$

where ν_b is the effective number of independent k-modes in each ℓ-band. For isotropic filtering, $\nu ^{\rm {iso}}_b=\frac{N_b w_2^2}{w_4}$.

One subtlety in estimating the covariance is the fact that although the noise in each map is independent, each map has the same sky coverage. Hence the signal in all maps is correlated. The correct estimator must take this correlation into account. Under the simplifying assumption that all maps are statistically equivalent, the correlation between two cross-spectra depends only on whether the spectra have a map in common (e.g., the cross-spectrum D¹²_b is more strongly correlated to the spectrum D²³_b than D³⁴_b). Comparing the covariance of two cross-spectra taken from four different observations:

$$\begin{eqnarray} \Xi _{bb}^{{\it AB}, {\it CD}}|_{A\ne B\ne C\ne D}&=\frac{2}{\nu _b}C^2_b, \end{eqnarray} \tag{ A8 }$$

to the covariance of a pair of cross-spectra with a common map,

$$\begin{eqnarray} \Xi _{bb}^{{\it AB}, {\it BC}}|_{A\ne B\ne C}\ &=\frac{1}{\nu _b}\left(2C_b^2+\frac{N_b}{B_b^2}C_b\right), \end{eqnarray} \tag{ A9 }$$

to the covariance of a pair of cross-spectra with two maps in common,

$$\begin{eqnarray} \Xi _{bb}^{{\it AB}, {\it AB}}|_{A\ne B} &=\frac{1}{\nu _b}\left(2C_b^2+2\frac{N_b}{B_b^2}C_b+\frac{N^2_b}{B^4_b}\right). \end{eqnarray} \tag{ A10 }$$

The degree of correlation is also ℓ dependent since it depends on the relative signal versus noise power in the maps. In the high-ℓ regime, where noise dominates the power in an individual map, all cross-spectra are nearly independent. Conversely all cross-spectra are nearly completely correlated at low-ℓ, where the primary CMB anisotropy overwhelms the noise.

Given the assumption of statistical equivalence, we can then compute the expected variance of the mean spectrum based on these variance estimates. This is equal to the correlation between any particular cross-spectrum and the mean,

$$\begin{eqnarray} \Xi _{bb}^{{\rm mean},\rm {mean}} &=& \Xi _{bb}^{{\rm mean}, {\it AB}}|_{A\ne B}\nonumber\\ &\approx & \frac{1}{\nu _b}\left(2C_b^2+4C_b \frac{N_b}{n_{\rm obs} B_b^2}+2\frac{N_b^2}{n_{\rm obs}^2 B_b^4}\right).\qquad \end{eqnarray} \tag{ A11 }$$

This estimate of the variation of the mean spectrum agrees with the uncertainty estimates given in Polenta et al. (2005).

It should be noted that the noise and filtering of the SPT data are anisotropic. Equation (A7) can therefore be used only as a guideline. In the case of anisotropic filtering or anisotropic beams, the number of independent modes per bin will be typically smaller than ν^iso_b, since anisotropic filtering will weight different k-space modes unevenly. For example, the k-space mask completely eliminates all modes with k_x < 1200. The variance in each ℓ-bin increases with fewer independent modes. However, even if we account for the effective number of independent modes in an ℓ-bin, Equation (A7) does not account for the anisotropic nature of the atmospheric noise contribution.

The existing analytic treatments are not directly applicable to the SPT data due to anisotropies in the noise and filters. By the nature of SPT's scan strategy, atmospheric fluctuations preferentially contaminate low k_x modes. Likewise the filters intended to remove these fluctuations preferentially remove low k_x modes. Instead, we have designed an empirical estimator which reproduces the analytical results when applied to isotropic data, while accurately accounting for the increased uncertainty due to the noise and filtering anisotropies in the actual data.

The noise covariance matrix estimate is divided into two parts, a signal contribution obtained from the Monte Carlo simulations described in Section 3.2.3 and a noise contribution obtained from real single-observation maps:

$$\begin{equation} \textbf {C}^{}_{bb^\prime }=\textbf {C}^{\rm {MC,s}}_{bb^\prime }+\textbf {C}^{\rm {data}}_{bb^\prime }. \end{equation} \tag{ A12 }$$

The signal contribution is straightforward to estimate with an approach similar to the MASTER power-spectrum error estimator. We use the signal only simulations to obtain an empirical estimate of the sample variance:

$$\begin{equation} \textbf {C}^{\rm MC,s}_{bb^\prime }=\overline{\Delta \widehat{D}^{\rm MC, s}_{b}\Delta \widehat{D}^{\rm MC, s}_{b^{\prime }} }. \end{equation} \tag{ A13 }$$

Note that here $\Delta x\equiv x - \overline{x}$ is defined with respect to the sample mean. Since the simulations include only CMB realizations and point sources in the confusion limit, the simulated signals are essentially Gaussian. Therefore, we expect the usual sample variance contribution:

$$\begin{equation} \left<\textbf {C}^{}_{bb^\prime, (\hbox{MC,s})}\right>=\frac{2 C_b^{\rm theory}}{\nu _b}. \end{equation} \tag{ A14 }$$

As before, ν_b is the effective number of independent Fourier modes in each ℓ-band.

The noise contribution is computed from the cross-spectra of single-observation maps. We use the following estimator for the noise contribution:

$$\begin{eqnarray} \textbf {C}_{bb^\prime }^{\rm data}&\equiv & \frac{2f(n_{\rm obs})}{n^4_{\rm obs}}\sum _{\lambda }\sum _{\alpha \ne \lambda } \Bigg(\Delta \widehat{D}_{b}^{\lambda \alpha }\Delta \widehat{D}_{b^\prime }^{\lambda \alpha }\nonumber\\ &&+ 2\Bigg[\sum _{\beta \ne \lambda, \alpha }\Delta \widehat{D}_{b}^{\lambda \alpha }\Delta \widehat{D}_{b^\prime }^{\lambda \beta } \Bigg] \Bigg). \end{eqnarray} \tag{ A15 }$$

Here f(n_obs) is a correction due to the finite number of realizations. In the limit of many observations this function asymptotes to unity; we use 300 observations so this term can be ignored. The first term can be identified as the sample variance of the cross-spectra. The second term accounts for the additional correlations between cross-spectra with a common map.

We can now calculate the expectation value for the noise component of the covariance estimator defined in Equation (A15):

$$\begin{eqnarray} \left<\textbf {C}^{\rm data}_{bb}\right> &\approx& \frac{2}{n_{\rm obs}^2}\Xi _{bb}^{\lambda \alpha,\lambda \alpha }+\frac{4}{n_{\rm obs}}\Xi _{bb}^{\lambda \alpha,\lambda \beta }-\left(\frac{4}{n_{\rm obs}}+\frac{2}{n_{\rm obs}^2}\right)\Xi ^{\rm {mean},\rm {mean} }\nonumber\\ &=& \frac{1}{\nu _b}\left(4C_b\frac{N_b}{n_{\rm obs} B_b^2}+2\frac{N_b^2}{n_{\rm obs}^2B_b^4}\right).\qquad\quad \end{eqnarray} \tag{ A16 }$$

This is combined with the signal, or cosmic variance, component in Equation (A13) to get the expectation value of the estimator:

$$\begin{equation} \left<\textbf {C}_{bb}\right>\approx \frac{1}{\nu _b}\left[2 {C_b^{\rm theory}}^2+4C_b\frac{N_b}{n_{\rm obs} B_b^2}+2\frac{N_b^2}{n_{\rm obs}^2B_b^4}\right]. \end{equation} \tag{ A17 }$$

This agrees with the analytic estimate (Equation (A11)) for the variance of the mean.

A multifrequency data set requires an estimate of both the covariance of the each individual set of bandpowers (i.e., both the single-frequency bandpowers and the cross-frequency bandpowers) and the cross-covariance between these sets. We naturally expect signal correlations between different sets of bandpower due to the fact that all three sets of bandpowers reported here are derived from the same patch of sky. However we also expect noise correlations between the 150 GHz × 220 GHz cross-spectrum bandpowers and each set of single-frequency bandpowers since the noise uncertainty in the cross-spectrum is entirely due to noise in the 150 GHz and 220 GHz data. Thus, we compute the cross-covariance matrices, ${\textbf {C}_{bb^\prime }}^{(i,j)}$, where i and j denote one of the three sets of bandpowers: 150 GHz, 220 GHz, and 150 GHz × 220 GHz:

$$\begin{eqnarray} &&{\textbf {C}_{bb^\prime }}^{(i,j)} = \overline{{\Delta \widehat{D}^{\rm MC, s}_{b}}^{(i)}{\Delta \widehat{D}^{\rm MC, s}_{b^{\prime }}}^{(j)} }+\frac{2f(n_{\rm obs})}{n^4_m}\nonumber\\ &&\times\sum _{\lambda }\sum _{\alpha \ne \lambda }\left({\Delta \widehat{D}_{b}^{\lambda \alpha }}^{(i)}{\Delta \widehat{D}_{b^\prime }^{\lambda \alpha }}^{(j)}+ 2\left[\sum _{\beta \ne \lambda, \alpha }{\Delta \widehat{D}_{b}^{\lambda \alpha }}^{(i)}{\Delta \widehat{D}_{b^\prime }^{\lambda \beta }}^{(j)} \right] \right).\nonumber\\ \end{eqnarray} \tag{ A18 }$$

Given the finite number of simulations and data maps, we expect some statistical uncertainty in the covariance estimate, particularly in the off-diagonal elements. Such uncertainty is not unique to the estimation technique described here, rather it is expected for any covariance estimate which is computed from a finite number of realizations. We expect the covariance estimates to be Wishart distributed, with n_obs = 300 degrees of freedom. A given covariance element, $\textbf {C}_{ij}$ has a statistical variance of

$$\begin{equation} \left<\left(\textbf {C}_{ij}-\left<\textbf {C}_{ij}\right>\right)^2\right>=\frac{\textbf {C}_{ij}^2+\textbf {C}_{ii}\textbf {C}_{jj}}{n_{\rm obs}}. \end{equation} \tag{ A19 }$$

For diagonal elements we expect a standard deviation of $\sqrt{2/n_{\rm obs}}=1/\sqrt{150}=8.1\%$. In addition, there is a statistical uncertainty on the apparent correlation between two bins. If we assume that the true correlation between bins is small, then the standard deviation of the apparent correlation between two bins is $\sqrt{1/n_{\rm obs}}$ or 5.7%. For the choice of bin size, the statistical error on the correlation of any two bins is much larger than the expected correlation (i.e., the fractional error on the apparent correlation estimates is greater than 100% even for adjacent bins). For bins that are widely separated, these false bin–bin correlations may skew model fitting. Therefore, we "condition" the published covariance matrices in order to reduce this statistical uncertainty on the covariance matrices.

From Equation (A5), we see that the shape of the correlation matrix (i.e., the relative size of the on-diagonal to off-diagonal covariance elements as a function of bin separation) is determined by the apodization window through the quadratic mode-coupling matrix, $M[\textbf {W}]_{kk}$. For the ℓ range considered in this work, the off-diagonal elements of this matrix depend only on the distance from the diagonal, |k − k'|. Therefore, we condition the estimated covariance matrix, $\widehat{\textbf {C}}_{kk^\prime }$ by first computing the corresponding correlation matrix, and then averaging all off-diagonal elements of a fixed separation from the diagonal:

$$\begin{equation} \textbf {C}^\prime _{kk^\prime }=\frac{ \sum _{k_1-k_2=k-k^\prime } \frac{\widehat{\textbf {C}}_{k_1k_2}}{\sqrt{\widehat{\textbf {C}}_{k_1k_1}\widehat{\textbf {C}}_{k_2k_2}}} }{\sum _{k_1-k_2=k-k^\prime } 1}. \end{equation} \tag{ A20 }$$

The bandpowers reported in Tables 1 and 2 are obtained by first computing power spectra and covariance matrices for a bin width of Δℓ = 100 with a total of 80 preliminary bins. This covariance matrix was then conditioned according to Equation (A20) before averaging the bandpowers and covariance matrix into the final bands.

Equation (A5) is based on the assumption that the filtering is isotropic. In order to test the validity of this equation for the anisotropic filtering, we perform 10,000 simple Monte Carlo simulations. In each simulation, a white noise realization is subjected to a simplified, though similarly anisotropic, version of the filtering scheme. The variance of the resultant spectra is computed and compared to Equation (A5). Though the apparent correlations between all bins exhibit the expected 1% scatter, the correlations between neighboring bins are consistent with Equation (A5).