GALAXY CLUSTERS SELECTED WITH THE SUNYAEV–ZEL'DOVICH EFFECT FROM 2008 SOUTH POLE TELESCOPE OBSERVATIONS

The vast majority of known galaxy clusters have been identified by their optical properties or from their X-ray emission. Clusters of galaxies contain anywhere from several tens to many hundreds of galaxies, but these galaxies account for a small fraction of the total baryonic mass in a cluster. Most of the baryons in clusters are contained in the intra-cluster medium (ICM), the hot (∼10⁷–10⁸ K) X-ray-emitting plasma that pervades cluster environments. Sunyaev & Zel'dovich (1972) noted that this same plasma should also interact with cosmic microwave background (CMB) photons via inverse Compton scattering, causing a small spectral distortion of the CMB along the line of sight to a cluster.

The thermal SZ effect has been observed in dozens of known clusters (clusters previously identified in the optical or X-ray) over the last few decades (Birkinshaw 1999; Carlstrom et al. 2002). However, it was not until very recently that the first previously unknown clusters were identified through their thermal SZ effect (Staniszewski et al. 2009). This is mostly due to the small amplitude of the effect. The magnitude of the temperature distortion at a given position on the sky is proportional to the integrated electron pressure along the line of sight. At the position of a massive galaxy cluster, this fluctuation is only on the order of a part in 10⁴, or a few hundred μK.²³ It is only with the current generation of large (∼ kilopixel) detector arrays on 6–12 m telescopes (Fowler et al. 2007; Carlstrom et al. 2009) that large areas of sky are being surveyed to depths sufficient to detect signals of this amplitude.

A key feature of the SZ effect is that the SZ surface brightness is insensitive to the redshift of the cluster. As a spectral distortion of the CMB (rather than an intrinsic emission feature), SZ signals redshift along with the CMB. A given parcel of gas will imprint the same spectral distortion on the CMB regardless of its cosmological redshift, depending only on the electron density n_e and temperature T_e. This makes SZ surveys an excellent technique for discovering clusters over a wide redshift range.

Another aspect of the thermal SZ effect that makes it especially attractive for cluster surveys is that the integrated thermal SZ flux is a direct measure of the total thermal energy of the ICM. The SZ flux is thus expected to be a robust proxy for total cluster mass (Barbosa et al. 1996; Holder & Carlstrom 2001; Motl et al. 2005).

A mass-limited cluster survey across a wide redshift range provides a growth-based test of dark energy to complement the distance based tests provided by supernovae (SNe; Perlmutter et al. 1999; Schmidt et al. 1998). Recent results (e.g., Vikhlinin et al. 2009; Mantz et al. 2010) have demonstrated the power of such tests to constrain cosmological models and parameters.

The South Pole Telescope (SPT; Carlstrom et al. 2009) is a 10 m off-axis telescope optimized for arcminute-resolution studies of the microwave sky. It is currently conducting a survey of a large fraction of the southern sky with the principal aim of detecting galaxy clusters via the SZ effect. In 2008, the SPT surveyed ∼200 deg² of the microwave sky with an array of 960 bolometers operating at 95, 150, and 220 GHz. Using 40 deg² of these data (and a small amount of overlapping data from 2007), Staniszewski et al. (2009, hereafter S09) presented the first discovery of previously unknown clusters by their SZ signature. Lueker et al. (2010, hereafter L10) used ∼100 deg² of the 2008 survey to measure the power spectrum of small-scale temperature anisotropies in the CMB, including the first significant detection of the contribution from the SZ secondary anisotropy.

In this paper, we expand upon the results in S09 and present an SZ-detection-significance-limited catalog of galaxy clusters identified in the 2008 SPT survey. Redshifts for 21 of these objects have been obtained from follow-up optical imaging, the details of which are discussed in a companion paper (High et al. 2010). Using simulated observations, we characterize the SPT cluster selection function—the detectability of galaxy clusters in the survey as a function of mass and redshift—for the 2008 fields. A simulation-based mass scaling relation allows us to compare the catalog to theoretical predictions and place constraints on the normalization of the matter power spectrum on small scales, σ₈, and the dark energy equation of state parameter w.

This paper is organized as follows: Section 2 discusses the observations, including data reduction, mapmaking, filtering, cluster-finding, optical follow-up, and cluster redshift estimation; Section 3 presents the resulting cluster catalog; Section 4 provides a description of our estimate of the selection function; Section 5 investigates the sample in the context of our current cosmological understanding and derives parameter constraints; we discuss limitations and possible contaminants in Section 6, and we close with a discussion in Section 7.

For our fiducial cosmology, we assume a spatially flat ΛCDM model (parameterized by Ω_bh², Ω_ch², H₀, n_s, τ, and A₀₀₂) with parameters consistent with the Wilkinson Microwave Anisotropy Probe (WMAP) five-year ΛCDM best-fit results (Dunkley et al. 2009),²⁴ namely, Ω_M = 0.264, Ω_b = 0.044, h = 0.71, and σ₈ = 0.80. All references to cluster mass refer to M₂₀₀, the mass enclosed within a spherical region of mean overdensity 200 × ρ_mean, where ρ_mean is the mean matter density on large scales at the redshift of the cluster.

The results presented in this work are based on observations performed by the SPT in 2008. Carlstrom et al. (2009) and S09 describe the details of these observations; we briefly summarize them here. Two fields were mapped to the nominal survey depth in 2008: one centered at right ascension (R.A.) 5^h30^m, declination (decl.) −55° (J2000), hereafter the 5^h field; and one centered at R.A. 23^h30^m, decl. −55°, hereafter the 23^h field. Results in this paper are based on roughly 1500 hr of observing time split between the two fields. The areas mapped with near uniform coverage were 91 deg² in the 5^h field and 105 deg² in the 23^h field.

This work considers only the 150 GHz data from the uniformly covered portions of the 2008 fields. The detector noise for the 95 GHz detectors was very high for the 2008 season, and the 220 GHz observations were contaminated by the atmosphere at large scales where they would be useful for removing CMB fluctuations. Including these bands did not significantly improve the efficiency of cluster detections and they were not used in the analysis presented here. The final depth of the 150 GHz maps of the two fields is very similar, with the white noise level in each map equal to 18 μK arcmin.

The two fields were observed using slightly different scan strategies. For the 5^h field, the telescope was swept in azimuth at a constant velocity (∼025 s⁻¹ on the sky at the field center) across the entire field then stepped in elevation, with this pattern continuing until the whole field was covered. The 23^h field was observed using a similar strategy, except that the azimuth scans covered only one half of the field at any one time, switching halves each time one was completed. One consequence of this observing strategy was that a narrow strip in the middle of the 23^h field received twice as much detector time as the rest of the map. The effect of this strip on our catalog is minimal and is discussed in Section 4.6.

A single observation of either field lasted ∼2 hr. Between individual observations, several short calibration measurements were performed, including measurements of a chopped thermal source, 2° elevation nods, and scans across the galactic H ii regions RCW38 and MAT5a. This series of regular calibration measurements was used to identify detectors with good performance, assess relative detector gains, monitor atmospheric opacity and beam parameters, and model pointing variations.

The data reduction pipeline applied to SPT data in this work is very similar to that used in S09. Broadly, the pipeline for both fields consists of filtering the time-ordered data from each individual detector, reconstructing the pointing for each detector, and combining data from all detectors in a given observing band into a map by simple inverse-variance-weighted binning and averaging.

The small differences between the data reduction used in this work and that of S09 are as follows.

• 1.  
  In S09, a 19th-order polynomial was fit and removed from each detector's timestream on each scan across the field. Samples in the timestream which mapped to positions on the sky near bright point sources were excluded from the fit. A similar subtraction was performed here, except that a first-order polynomial was removed, supplemented by sines and cosines (Fourier modes). Frequencies for the Fourier modes were evenly spaced from 0.025 Hz to 0.25 Hz. This acts approximately as a high-pass filter in the R.A. direction with a characteristic scale of ∼1° on the sky.
• 2.  
  In S09, a mean across functioning detectors was calculated at each snapshot in time and subtracted from each sample. Here, both a mean and a slope across the two-dimensional array were calculated at each time and subtracted. This acts as a roughly isotropic spatial high-pass filter, with a characteristic scale of ∼05.
• 3.  
  As in Vieira et al. (2010), a small pointing correction (∼5'' on the sky) was applied, based on comparisons of radio source positions derived from SPT maps and positions of those sources in the AT20G catalog (Murphy et al. 2010).

The relative and absolute calibrations of detector response were performed as in L10. The relative gains of the detectors and their gain variations over time were estimated using measurements of their response to a chopped thermal source. These relative calibrations were then tied to an absolute scale through direct comparison of WMAP five-year maps (Hinshaw et al. 2009) to dedicated large-area SPT scans. This calibration is discussed in detail in L10, and is accurate to 3.6% in the 150 GHz data.

The cluster extraction procedure used in this work for both fields is identical to the procedure used in S09, where more details can be found.

The SPT maps were filtered to optimize detection of objects with morphologies similar to the SZ signatures expected from galaxy clusters, through the application of spatial matched filters (Haehnelt & Tegmark 1996; Herranz et al. 2002a, 2002b; Melin et al. 2006). In the spatial Fourier domain, the map was multiplied by

$$\begin{equation*} \psi (k_x,k_y) = \frac{B(k_x,k_y) S(|\vec{k}|)}{B(k_x,k_y)^2 N_{\rm astro}(|\vec{k}|) + N_{\rm noise}(k_x,k_y)}, \end{equation*}$$

where ψ is the matched filter, B is the response of the SPT instrument after timestream processing to signals on the sky, S is the assumed source template, and the noise power has been broken into astrophysical (N_astro) and noise (N_noise) components. For the source template, a projected spherical β-model, with β fixed to 1, was used:

$$\begin{equation*} \Delta T = \Delta T_0 \big(1+\theta ^2/\theta _c^2\big)^{-1}, \end{equation*}$$

where the normalization ΔT₀ and the core radius θ_c are free parameters.

The noise power spectrum N_noise includes contributions from atmospheric and instrumental noise, while N_astro includes power from primary and lensed CMB fluctuations, an SZ background, and point sources. The atmospheric and instrumental noise were estimated from jackknife maps as in S09, the CMB power spectrum was updated to the lensed CMB spectrum from the WMAP5 best-fit ΛCDM cosmology (Dunkley et al. 2009), the SZ background level was assumed to be flat in ℓ(ℓ + 1)C_ℓ with the amplitude taken from L10, and the point-source power was assumed to be flat in C_ℓ at the level given in Hall et al. (2010).

To avoid spurious decrements from the wings of bright point sources, all positive sources above a given flux (roughly 7 mJy, or 5σ in a version of the map filtered to optimize point-source signal to noise) were masked to a radius of 4' before the matched filter was applied. Roughly 150 sources were masked in each field, of which 90%–95% are radio sources. The final sky areas considered after source masking were 82.4 and 95.1 deg² for the 5^h and 23^h fields, respectively.

The maps were filtered for 12 different cluster scales, constructed using source templates with core radii θ_c evenly spaced between 025 and 30. Each filtered map M_ij, where i refers to the filter scale and j to the field, was then divided into strips corresponding to distinct 90' ranges in elevation. The noise was estimated independently in each strip in order to account for the weak elevation dependence of the survey depth. The noise in the kth strip of map M_ij, σ_ijk, was estimated as the standard deviation of the map within that strip.

Signal-to-noise maps $\widetilde{M}_{ij}$ were then constructed by dividing each strip k in map M_ij by σ_ijk. SZ cluster decrements were identified in each map $\widetilde{M}_{ij}$ by a simple (negative) peak detection algorithm similar to SExtractor (Bertin & Arnouts 1996). The highest signal-to-noise value associated with a decrement, across all filter scales, was defined as ξ, and taken as the significance of a detection. Candidate clusters were identified in the data down to ξ of 3.5, though this work considers only the subset with ξ⩾5.

These detection significances are robust against choice of source template: the use of Nagai (Nagai et al. 2007), Arnaud (Arnaud et al. 2010), or Gaussian templates in place of β-models was found to be free of bias and to introduce negligible (∼2%) scatter on recovered ξ.

Optical imaging was used for confirmation of candidates, for photometric redshift estimation, and for cluster richness characterization. A detailed description of the coordinated optical effort is presented in High et al. (2010) and is summarized here.

The 5^h and 23^h fields were selected in part for overlap with the Blanco Cosmology Survey (BCS; see Ngeow et al. 2009), which consists of deep optical images using g, r, i, and z filters. These images, obtained from the Blanco 4 m telescope at CTIO with the MOSAIC-II imager in 2005–2007, were used when available. The co-added BCS images have 5σ galaxy detection thresholds of 24.75, 24.65, 24.35, and 23.05 mag in griz, respectively.

For clusters that fell outside the BCS coverage region, as well as for five that fell within, and for the unconfirmed candidate, images were obtained using the twin 6.5 m Magellan telescopes at Las Campanas, Chile. The imaging data on Magellan were obtained by taking successively deeper images until a detection of the early-type cluster galaxies was achieved, complete to between L* and 0.4L* in at least one band. The Magellan images were obtained under a variety of conditions, and the Stellar Locus Regression technique (High et al. 2009) was used to obtain precise colors and magnitudes.

Spectroscopic data were obtained for a subset of the sample using the Low Dispersion Survey Spectrograph on Magellan (LDSS3; see Osip et al. 2008) in longslit mode. Typical exposures were 20–60 minutes, with slit orientations that contained the brightest cluster galaxy (BCG) and as many additional red cluster members as possible.

Photometric redshifts were estimated using standard red sequence techniques and verified using the spectroscopic subsample. A red sequence model was derived from the work of Bruzual & Charlot (2003), and local overdensities of red galaxies were searched for near the cluster candidate positions. By comparing the resulting photo-z's to spectroscopic redshifts within the subsample of 10 clusters for which spectroscopic data are available, H10 estimates photo-z uncertainty σ_z as given in Table 1, of order 3% in (1 + z).

Completeness in red sequence cluster-finding was estimated from mock catalogs. At BCS depths, galaxy cluster completeness for the masses relevant for this sample is nearly unity up to a redshift of one, above which the completeness falls rapidly, reaching 50% at about redshift 1.2, and 0% at redshift 1.25. At depths about a magnitude brighter (corresponding to the depth of the Magellan observations of the unconfirmed candidate, Section 3.1), the completeness deviates from unity at redshift ∼0.8.

SPT-CL 2259-5617.The SZ signal from this cluster is anomalously compact for such a low-redshift object. The cosmological analysis (Section 5) explicitly excludes all z < 0.3 clusters, so this cluster not used in parameter estimation.

SPT-CL J2331−5051. This cluster appears to be one of a pair of clusters at comparable redshift, likely undergoing a merger. It will be discussed in detail in a future publication. The fainter partner is not included in this catalog as its significance (ξ = 4.81) falls below the detection threshold.

SPT-CL J2332−5358. This cluster is coincident with a bright dusty point source which we identify in the 23h 220 GHz data. Although the 150 GHz flux from this source could be removed with the aid of the 220 GHz map, a multi-frequency analysis is outside the scope of the present work. The impact of point sources on the resulting cluster catalog is discussed in Section 6.2.

SPT-CL J2343−5521. No optical counterpart was found for this candidate. The field was imaged with both BCS and Magellan, and no cluster of galaxies was found to a 5σ point-source detection depth. The simulated optical completeness suggests that this candidate is either a false positive in the SPT catalog or a cluster at high redshift (z ≳ 1.2). While the relatively high ξ = 5.74 indicates a ∼7% chance of a false detection in the SPT survey area (see discussion of contamination below, Section 4.7), the signal to noise of this detection exhibits peculiar behavior with θ_c (see Figure 8), preferring significantly larger scales than any other candidate, consistent with a CMB decrement. Further multi-wavelength follow-up observations are underway on this candidate, and preliminary results indicate it is likely a false detection.

The optimal filter described in Section 2.3 provides an estimate of the β-model normalization, ΔT₀, and core size for each cluster, based on the filter scale θ_c at which the significance ξ was maximized. Assuming prior knowledge of the ratio θ₂₀₀/θ_c (where θ₂₀₀ is the angle subtending the physical radius R₂₀₀ at the redshift of the cluster), one can integrate the β-profile to obtain an estimate of the integrated SZ flux, Y. Basic physical arguments and hydrodynamical simulations of clusters have demonstrated Y to be a tight (low intrinsic scatter) proxy for cluster mass (Barbosa et al. 1996; Holder & Carlstrom 2001; Motl et al. 2005; Nagai et al. 2007; Stanek et al. 2010).

In single-frequency SZ surveys, the primary CMB temperature anisotropies provide a source of astrophysical contamination that greatly inhibits an accurate measure of θ_c (Melin et al. 2006). The modes at which the primary CMB dominates must be filtered out from the map, significantly reducing the range of angular scales that can be used by the optimal filter to constrain θ_c. This range is already limited by the ∼1' instrument beam, which only resolves θ_c for the larger clusters. Any integrated quantity will thus be poorly measured. Melin et al. (2006) demonstrated that if the value of θ_c can be provided by external observations, e.g., X-ray, Y can be accurately measured.

The inability to constrain θ_c can be seen in Figure 8, where the highest signal to noise associated with a peak is plotted against each filter scale. For several clusters (for example, SPT-CL J0516−5430, SPT-CL J0551−5709, and SPT-CL J2332−5358), the peak in signal to noise associated with a cluster is very broad in θ_c. Because of this confusion, we do not report Y in this work. Instead, as described below (Section 4.8), we use detection significance as a proxy for mass.

Multi-frequency surveys are not in principle subject to this limitation as the different frequencies can be combined to eliminate sources of noise that are correlated between bands, thus increasing the range of angular scales available for constraining cluster profiles.

Simulated SZ maps were generated using the method of Shaw et al. (2009), where a detailed description of the procedure can be found. In brief, the semianalytic gas model of Bode et al. (2007) was applied to halos identified in the output of a large dark matter light cone simulation. The cosmological parameters for this simulation were chosen to be consistent with those measured from the WMAP five-year data combined with large-scale structure observations (Dunkley et al. 2009), namely, Ω_M = 0.264, Ω_b = 0.044, and σ₈ = 0.8. The simulated volume was a periodic box of size 1 h⁻¹ Gpc. The matter distribution in 421 time slices was arranged into a light cone covering a single octant of the sky from 0 < z ⩽ 3.

Dark matter halos were identified and gas distributions were calculated for each halo using the semianalytic model of Bode et al. (2007). This model assumes that intra-cluster gas resides in hydrostatic equilibrium in the gravitational potential of the host dark matter halo with a polytropic equation of state. As discussed in Bode et al. (2007), the most important free parameter is the energy input into the cluster gas via non-thermal feedback processes, such as SNe and outflows from active galactic nuclei (AGNs). This is set through the parameter _f such that the feedback energy is E_f = _fM_*c², where M_* is the total stellar mass in the cluster. Bode et al. (2007) calibrate _f by comparing the model against observed X-ray scaling relations for low-redshift (z < 0.25) group and cluster mass objects. We note that the redshift range in which the model has been calibrated and that encompassed by the cluster sample presented here barely overlaps; comparison of the model to the SPT sample (as, for example, in Section 5) thus provides a test of the predicted cluster and SZ signal evolution at high redshift.

For our fiducial model, we adopt _f = 5 × 10⁻⁵, however for comparison we also generate maps using the "standard" and "star formation only" versions of this model described in Bode et al. (2009). There are two principal differences between these models. First, the stellar-mass fraction M_*/M_gas is constant with total cluster mass in the fiducial model, but mass dependent in the "standard" and "star formation" model. Second, the amount of energy feedback is significantly lower in the "standard" than model than in the fiducial, and zero in the "star formation" model.

From the output of each model, a two-dimensional image of SZ intensity for each cluster with mass M>5 × 10¹³ M_☉ h⁻¹ was produced by summing up the electron pressure along the line of sight. SZ cluster sky maps were constructed by projecting down the light cone, summing up the contribution of all the clusters along the line of sight. Individual SZ sky maps were 10 × 10 degrees in size, resulting in a total of 40 independent maps. For each map, the mass, redshift, and position of each cluster was recorded.

From SPT pointed observations of X-ray-selected clusters, Plagge et al. (2010) have demonstrated that cluster radial SZ profiles match the form of the "universal" electron pressure profile measured by Arnaud et al. (2010) from X-ray observations of massive, low-redshift clusters. To complement the set of maps generated using the semianalytic gas model, SZ sky maps were generated in which the projected form of the Arnaud et al. (2010) pressure profile was used to generate the individual cluster SZ signals.

At 150 GHz and at the flux levels of interest to this analysis (∼1 to ∼10 mJy), the extragalactic source population is expected to be primarily composed of two broad classes: sources dominated by thermal emission from dust heated by a burst of star formation, and sources dominated by synchrotron emission from AGN. We refer to these two families as "dusty sources" and "radio sources" and include models for both in our simulated observations.

For dusty sources, the source count model of Negrello et al. (2007) at 350 GHz was used. These counts are based on the physical model of Granato et al. (2004) for high-redshift SCUBA-like sources and on a more phenomenological approach for late-type galaxies (starburst plus normal spirals). Source counts were estimated at 150 GHz by assuming a scaling for the flux densities of S_ν ∝ ν^α, with α = 3 for high-redshift protospheroidal galaxies and α = 2 for late-type galaxies. For radio sources, the de Zotti et al. (2005) model for counts at 150 GHz was used. This model is consistent with the measurements of Vieira et al. (2010) for the radio source population at 150 GHz.

Realizations of source populations were generated by sampling from Poisson distributions for each population in bins with fluxes from 0.01 mJy to 1000 mJy. Sources were distributed in a random way across the map. Correlations between sources or with galaxy clusters were not modeled, and we discuss this potential contamination in Section 6.2.

Simulated CMB anisotropies were produced by generating sky realizations based on the power in the gravitationally lensed WMAP five-year ΛCDM CMB power spectrum. Non-Gaussianity in the lensed power was not modeled.

The effects of the transfer function, i.e., the effects of the instrument beam and the data processing on sky signal, were emulated by producing synthetic SPT timestreams from simulated skies sampled using the same scans employed in the observations. The sky signal was convolved with the measured SPT 150 GHz beam, timestream samples were convolved with detector time constants, and the SPT data processing (Section 2.2) was performed on the simulated timestreams to produce maps.

Full emulation of the transfer function is a computationally intensive process; to make a large number of simulated observations, the transfer function was modeled as a two-dimensional Fourier filter. The accuracy of this approximation was measured by comparing recovered ξ of simulated clusters in skies passed through the full transfer function against the ξ of the same clusters when the transfer function was approximated as a Fourier filter applied to the map. Systematic differences were found to be less than 1% and on an object-by-object basis the two methods produced measured ξ that agreed to better than 3%.

Noise maps were created from SPT data by subtracting one half of each observation from the other half. Within each observation, one direction (azimuth either increasing or decreasing) was chosen at random, and all data when the telescope was moving in that direction were multiplied by −1. The data were then processed and combined as usual to produce a "jackknife" map which contained the full noise properties of the final field map, but with all sky signal removed.

Due to the observing strategy employed on the 23^h field, a ∼15 strip in the middle of that map contains significantly lower atmospheric and instrumental noise than the rest of the map. The jackknife noise maps (Section 4.5) used in simulated observations naturally include this deep strip, so any effects due to this feature are taken into account in the simulation-based estimation of the average selection function (Section 4.7) and scaling relation (Section 4.8) across the whole survey region. The cosmological analysis (Section 5) uses these averaged quantities; simulated observations performed with and without a deep strip demonstrated that any bias or additional scatter from using the averaged quantities is negligible compared to the statistical errors.

Forty realizations of the 2008 SPT survey (two fields each) were simulated, from which clusters were extracted and matched against input catalogs.

Figure 1 shows the completeness of the simulated SPT sample, the fraction of clusters in simulated SPT maps that were detected with ξ ⩾ 5, as a function of mass and redshift. The exact shape and location of the curves in this figure depend on the detailed modeling of intra-cluster physics, which remain uncertain. The increase in SZ brightness (and cluster detectability) with increasing redshift at fixed mass is due to the increased density and temperature of high-redshift clusters, and is in keeping with self-similar evolution. At low redshifts (z ≲ 0.3), CMB confusion suppresses cluster detection significances and drives a strong low-redshift evolution in the selection function. These completeness curves were not used in the cosmological analysis (Section 5), where uncertainties on the mass scaling relation (Section 4.8) account for uncertainties in the modeling of intra-cluster physics.

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The SZ sky was removed from simulations to estimate the rate of false positives in the SPT sample. Figure 2 shows this contamination rate as a function of lower ξ threshold, averaged across the survey area. A ξ ⩾ 5 threshold leads to approximately one false detection within the survey area.

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To test for biases introduced by an SZ background composed of low-mass systems, a simulation was run including only SZ sources well below the SPT threshold, with masses M < 10¹⁴ M_☉ h⁻¹. This background was found to have negligible effect on the detection rate as compared to the SZ-free false detection simulation.

As discussed in Section 3.2, the integrated SZ flux Y is poorly estimated in this analysis and so is not used as a mass proxy. However, the noise σ_ijk measured in each elevation strip is relatively even across the SPT maps, so it is possible to work in the native space of the SPT selection function and use detection significance ξ as proxy for mass. Additional uncertainty and bias introduced by use of such a relation (in place of, for example, a Y-based scaling relation) are small compared to the Poisson noise of the sample and the uncertainties in modeling intra-cluster physics.

The steepness of the cluster mass function in the presence of noise will result in a number of detections that have boosted significance. Explicitly, ξ is a biased estimator for 〈ξ〉, the average detection significance of a given cluster across many noise realizations. An additional bias on ξ comes from the choice to maximize signal to noise across three free parameters, R.A., decl., and θ_c. These biases make the relation between ξ and mass complex and difficult to characterize.

In order to produce a mass scaling relation with a simple form, the unbiased significance ζ is introduced. It is defined as the average detection signal to noise of a simulated cluster, measured across many noise realizations, evaluated at the preferred position and filter scale of that cluster as determined by fitting the cluster in the absence of noise.

Relating ζ and ξ is a two step process. The expected relation between ζ and 〈ξ〉 is derived and compared to simulated observations in Appendix B, and found to be $\zeta = \sqrt{\langle \xi \rangle ^2-3}$. Given a known 〈ξ〉, the expected distribution in ξ is derived by convolution with a Gaussian of unit width. The relation between ζ and 〈ξ〉 is taken to be exact, and was verified through simulations to introduce negligible additional scatter; i.e., the scatter in the ζ–ξ relation is the same as the scatter in the 〈ξ〉–ξ relation, namely, a Gaussian of unit width.

The scaling between ζ and M is assumed to take the form of power-law relations with both mass and redshift:

$$\begin{equation} \zeta = A \left(\frac{M}{5\times 10^{14}\,M_\odot \,h^{-1}}\right)^B \left(\frac{1+z}{1.6}\right)^C, \end{equation} \tag{ 1 }$$

parameterized by the normalization A, the slope B, and the redshift evolution C. Appendix C presents a physical argument for the form of this relation, along with the expected ranges in which the values of the parameters B and C are expected to reside based on self-similar scaling arguments.

Values for the parameters A, B, and C were determined by fitting Equation (1) to a catalog of ζ>1 clusters detected in simulated maps, using clusters with mass M>2 × 10¹⁴ M_☉ h⁻¹ and in the redshift range 0.3 ⩽ z ⩽ 1.2. This redshift range was chosen to match the SPT sample, while the mass limit was chosen to be as low as possible without the sample being significantly cut off by the ζ>1 threshold. The best fit was defined as the combination of parameters that minimized the intrinsic fractional scatter around the mean relation.

Figure 3 shows the best-fit scaling relation obtained for our fiducial simulated SZ maps, where A = 6.01, B = 1.31, and C = 1.6. The intrinsic scatter was measured to be 21% (0.21 in ln(ζ)) and the relation was found to adhere to a power law well below the limiting mass threshold. Over the three gas model realizations (Section 4.1), the best-fit value of A, B, and the intrinsic scatter were all found to vary by less than 10%, while the values of C predicted by the "standard" and "star formation" models drop to ∼1.2. For maps generated using the electron pressure profile of Arnaud et al. (2010), best-fit values of A = 6.89, B = 1.38, and C = 0.6 were found, with a 19% intrinsic scatter. The values of A and B remain within 15% of the fiducial model, although C is significantly lower. Arnaud et al. (2010) measured the pressure profile using a low-redshift (z < 0.2) cluster sample and assume that the profile normalization will evolve in a self-similar fashion. The mass dependence of their pressure profile was determined using cluster mass estimates derived from the equation of hydrostatic equilibrium; simulations suggest that this method may underestimate the true mass by 10%–20% (Rasia et al. 2004; Meneghetti et al. 2010; Lau et al. 2009). We do not take this effect into account in our simulations—doing so would reduce the value of A by approximately 10%. The Bode et al. (2009) gas model is calibrated against X-ray scaling relations measured from low-redshift cluster samples (Vikhlinin et al. 2006; Sun et al. 2009), but assumes an evolving stellar-mass fraction which may drive the stronger redshift evolution.

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Based on these simulations, priors on the scaling relation parameters (A, B, C, scatter) were adopted, with conservative 1σ Gaussian uncertainties of (30%, 20%, 50%, and 20%) about mean values measured from the fiducial simulation model. These large uncertainties in scaling relation parameters are the dominant source of uncertainty in the cosmological analysis (Section 5) and mass estimation (Appendix C). Furthermore, although the weakest prior is on the redshift evolution, C, it is the uncertainty on the amplitude A that dominates the error budget on the measurement of σ₈ (see Section 5.4).

It should be noted that at low redshift (z ≲ 0.3), such a power-law scaling relation fails to fully capture the behavior of the CMB-confused selection function. The cosmological analysis below therefore excludes this region during likelihood calculation. The mass estimates presented in Appendix D may be biased low for low-redshift objects, although this effect is expected to be small compared to existing systematic errors.

Evaluation of cosmological models in the context of the SPT catalog requires a theoretical model that is capable of predicting the number density of dark matter halos as a function of both redshift and input cosmology. For a given set of cosmological parameters, the simulation-based mass function of Tinker et al. (2008) was used in conjunction with matter power spectra computed by CAMB (Lewis et al. 2000) to construct a grid of cluster number densities in the native ξ–z space of the SPT catalog.

• 1.  
  A two-dimensional grid of the number of clusters as a function of redshift and mass was constructed by multiplying the Tinker et al. (2008) mass function by the comoving volume element. The gridding was set to be very fine in both mass and redshift, with Δz = 0.01 and the mass binning set so that Δζ = 0.0025 (see below). The grids were constructed to extend beyond the sensitivity range of SPT, 0.1 < z < 2.6 and 1.8 < ζ < 23. Extending the upper limits was found not to impact cosmological results, as predicted number counts have dropped to negligible levels above those thresholds.
• 2.  
  The parameterized scaling relation (Section 4.8) was used to convert the mass for each bin to unbiased significance ζ for assumed values of A, B, and C.
• 3.  
  This grid of number counts (in ζ–z space) was convolved with a Gaussian in ln(ζ) with width set by the assumed intrinsic scatter in the scaling relation (21% in the fiducial relation).
• 4.  
  The unbiased significance ζ of each bin was converted to an ensemble-averaged significance 〈ξ〉.
• 5.  
  This grid was convolved with a unit-width Gaussian in ξ to account for noise, with the resulting grid in the native SPT catalog space, ξ–z.
• 6.  
  Each row (fixed ξ) of the ξ–z grid which contained a cluster was convolved with a Gaussian with width set as the redshift uncertainty for that cluster. Photometric redshift uncertainties are given in Table 1, and are described briefly in Section 2.4 and in detail in High et al. (2009); spectroscopic redshifts were taken to be exact.
• 7.  
  A hard cut in ξ was applied, corresponding to the catalog selection threshold of ξ ⩾ 5.
• 8.  
  An additional cut was applied, requiring z ⩾ 0.3, to avoid low-redshift regions where the power-law scaling relation fails to capture the behavior of the CMB-confused selection function. This cut excludes three low-redshift clusters from the cosmological analysis, leaving 18 clusters plus the unconfirmed candidate, whose treatment is described below.

The likelihood ratio for the SPT catalog was then constructed, as outlined in Cash (1979), using the Poisson probability,

$$$ \mathcal {L}=\prod _{i=1}^N P_i=\prod _{i=1}^N \frac{e_i^{n_i}e^{-e_i}}{n_i !}, $$$

where the product is across bins in ξ–z space, N is the number of bins, P_i is the Poisson probability in bin i, and e_i and n_i are the fractional expected and integer observed number counts for that bin, respectively.

The unconfirmed candidate was accounted for by simultaneously allowing it to either be at high redshift or a false detection. Its contribution to the total likelihood was calculated as the union of the likelihoods for n = 0 and n = 1 within a large z>1.0 bin, the redshift range corresponding to where the optical completeness for this candidate's follow-up deviates from unity.

Ultimately, two sources of mass-observable scatter—the intrinsic scatter in the scaling relation, and the 1σ measurement noise—were included in this analysis, along with redshift errors and systematic uncertainties on scaling relation parameters. Other sources of bias and noise (such as point-source contamination, Section 6.2, and the mass function normalization described below) are thought to be subdominant to these and were disregarded.

While Tinker et al. (2008) claim a very small (<5%) uncertainty in the mass function normalization, Stanek et al. (2010) have demonstrated that the inclusion of non-gravitational baryon physics in cosmological simulations can modify cluster masses in the range of the SPT sample by ∼±10% relative to gravity-only hydrodynamical simulations. The large 30% uncertainty on the amplitude A of the scaling relation effectively subsumes such uncertainties in the mass function normalization.

This analysis does not account for the effects of sample variance (Hu & Kravtsov 2003); for the mass and redshift range of the SPT sample, this is not expected to be a problem (Hu & Cohn 2006). The SPT survey fields span of order 100 h⁻¹ Mpc, where the galaxy cluster correlation function would be expected to be a few percent or less (Bahcall et al. 2003; Estrada et al. 2009). This leads to clustering corrections to the uncertainty on number counts on the order of a few percent or less of the Poisson variance.

The present SPT sample only meaningfully constrains a subset of cosmological parameters, so to explore the cosmological implications of the SPT catalog it is necessary to include information from other experiments. Existing analyses of other cosmological data in the form of Markov Chain Monte Carlo (MCMC) chains provide fully informative priors. These were importance sampled by weighting each set of cosmological parameters in the MCMC chain by the likelihood of the SPT cluster catalog given that set of parameters.

In this analysis, four MCMC chains were used to explore constraints on parameters: the first two use only the seven-year data set from the WMAP experiment to explore the standard spatially flat ΛCDM and wCDM cosmologies, while the third explores wCDM while adding data from baryon acoustic oscillations (BAO; Percival et al. 2010) and SNe (Hicken et al. 2009). These three chains were taken from the official WMAP analysis²⁵ (Komatsu et al. 2010). The fourth chain was computed by L10 and allows for a direct comparison with that work. It explores a spatially flat ΛCDM parameter space based on the "CMBall" data set: WMAP five-year + QUaD (Brown et al. 2009) + ACBAR (Reichardt et al. 2009) + SPT (power spectrum measurements with A_SZ as a free parameter; L10).

Figure 4 shows the number density of clusters in the SPT catalog, plotted over theoretical predictions calculated using the method described in Section 5.1, for 100 random positions in the WMAP7-only MCMC chain. The SPT data are adequately described by many cosmological models that are allowed by this data set, and the MCMC chains are well sampled within the region of high probability.

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Uncertainties in the scaling relation parameters were accounted for by marginalizing over them: at each step in the chain, the likelihood was maximized across A, B, C, and scatter, subject to the priors applied to each parameter, using a Newton–Raphson method. The parameter values selected in this way at the highest likelihood point in each MCMC chain are given in Table 3 in Appendix C. The fiducial values of B and the scatter appear consistent with those preferred by the chains, while the preferred values of the normalization A and redshift evolution C are both approximately 10% lower than their fiducial values.

This weaker-than-fiducial redshift evolution could come from a variety of sources, and is consistent with other simulated models, e.g., the "standard" and "star formation" models, see Section 4.8. Uncertainties in the redshift evolution are not a significant source of error in this analysis: recovered parameter values and uncertainties (Section 5.3) are found to be insignificantly affected by widely varying priors on C.

The resulting constraints on σ₈ and w are given for all chains in Table 2. The parameter best constrained by the SPT cluster catalog is σ₈. CMB power spectrum measurements alone have a large degeneracy between the dark energy equation of state, w, and σ₈. Figure 5 shows this degeneracy, along with the added constraints from the SPT cluster catalog. Including the cluster results tightens the σ₈ contours and leads to an improved constraint on w. This is a growth-based determination of the dark energy equation of state, and is therefore complementary to dark energy measurements based on distances, such as those based on SNe and BAO.

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When combined with the wCDM WMAP7 chain, the SPT data provide roughly a factor of 1.5 improvement in the precision of σ₈ and w, finding 0.81 ± 0.09 and −1.07 ± 0.29, respectively. Including data from BAO and SNe, these constraints tighten to σ₈ = 0.79 ± 0.03 and w = −0.97 ± 0.05.

The dominant sources of uncertainty limiting these constraints are the Poisson error due to the relatively modest size of the current catalog and the uncertainty in the normalization A of the mass scaling relation. With weak-lensing- and X-ray-derived mass estimates of SPT clusters, along with an order of magnitude larger sample expected from the full survey, cosmological constraints from the SPT galaxy cluster survey will markedly improve.

The value of the normalization parameter A (which can be thought of as an "SZ amplitude") preferred by the likelihood analysis was found to be lower than the fiducial value, as shown in Figure 6. The prior assumed on this parameter is sufficiently large that it is not a highly significant shift; however, in light of the recent report by L10 of lower-than-expected SZ flux, it is worth addressing. The SPT cluster catalog results are complementary to the results of the power spectrum analysis, in that the majority of the SZ power at the angular scales probed by L10 comes from clusters below the mass threshold of the cluster catalog.

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Figure 6 shows that the amplitude A is strongly degenerate with σ₈. The constraints provided by the SPT cluster catalog indicate either a value of σ₈ that is at the low end of the CMB-allowed distribution (or equivalently an erroneously high mass function normalization), or an overprediction of SZ flux by the fiducial simulations. If the fiducial amplitude is assumed, the best-fit σ₈ drops from the WMAP5+CMBall value of 0.794 ± 0.029 to 0.775 ± 0.015. This value is anomalously low compared to recent results (e.g., Vikhlinin et al. 2009; Mantz et al. 2010), and in slight tension with the results of the power spectrum analysis of L10, where a still lower value of σ₈ = 0.746 ± 0.017 was obtained for similar simulation models.²⁶ The SZ amplitude parameter used in L10, A_sz, is roughly analogous to A² in the current notation. When including the expected contribution from homogeneous reionization, L10 found A_sz = 0.42 ± 0.21, in mild tension (at the ∼1σ level) with the marginalized value of (A/A_fid)² = 0.79 ± 0.30 found in this analysis.

The fiducial simulations in this work use the semianalytic gas model of Bode et al. (2007, 2009), which is calibrated against low-redshift (z < 0.25) X-ray observations but has not previously been compared to higher-redshift systems. One interpretation of these results is that this model may overpredict the thermal electron pressure in high-redshift (z>0.3) systems; this is not in conflict with the low-redshift calibration of the model and suggests a weaker redshift evolution in the SZ signal than predicted by the model. Alternately, a combination of mass function normalization and point-source contamination could potentially account for the difference.

Theoretical arguments (Barbosa et al. 1996; Holder & Carlstrom 2001; Motl et al. 2005) suggest that the SZ flux of galaxy clusters is relatively well understood. However, there is very little high-precision empirical evidence to confirm these arguments, and there are physical mechanisms that could lead to suppressed SZ flux, such as non-thermal pressure support from turbulence (Lau et al. 2009) or non-equilibrium between protons and electrons (Fox & Loeb 1997; Rudd & Nagai 2009).

Cluster SZ mass proxies (such as Y and y₀, the integrated SZ flux and amplitude of the SZ decrement, respectively) depend linearly on the gas fraction and the gas temperature. There remain theoretical and observational uncertainties in both of these quantities. Estimates of gas fractions for individual clusters can disagree by nearly 20% (e.g., Allen et al. 2008; Vikhlinin et al. 2006), while theoretical and observed estimates of the mass–temperature relation currently agree at the level of 10%–20% (Nagai et al. 2007). Adding these in quadrature leads to uncertainties slightly below our assumed prior uncertainty of 30%.

With the number counts as a function of mass, dN/dln M, scaling as M⁻² or M⁻³ for typical SPT clusters (Shaw et al. 2010a), a 10% offset in mass would lead to a 20%–30% shift in the number of galaxy clusters. With a catalog of 22 clusters, counting statistics lead to an uncertainty of at least 20%. Therefore, systematic offsets in the mass scale of order 10% will have a significant effect on cosmological constraints, and the current 30% prior on A will dominate Poisson errors.

A follow-up campaign using optical and X-ray observations will buttress our current theory/simulation-driven understanding of the SPT SZ-selected galaxy cluster catalog.

The sky density of bright point sources at 150 GHz is low enough—on the order of 1 deg⁻² (Vieira et al. 2010)—that the probability of a galaxy cluster being missed due to a chance superposition with a bright source is negligible. However, sources associated with clusters will preferentially fill in cluster SZ decrements. Characterizing the contamination of cluster SZ measurements by member galaxies will be necessary to realize the full potential of the upcoming much larger SPT cluster catalog, but the systematic uncertainty predicted here and in the literature is well below the statistical precision of the current sample; it is disregarded in the current cosmological analysis (Section 5).

6.2.1. Dusty Source Contamination

6.2.2. Gravitational Lensing

6.2.3. Radio Source Contamination