MASS CALIBRATION AND COSMOLOGICAL ANALYSIS OF THE SPT-SZ GALAXY CLUSTER SAMPLE USING VELOCITY DISPERSION σ_v AND X-RAY Y_X MEASUREMENTS

The SPT is a 10 m telescope located within 1 km of the geographical South Pole. From 2007 to 2011, the telescope was configured to observe in three millimeter-wave bands (centered at 95, 150, and 220 GHz). The majority of this period was spent on a survey of a contiguous 2500 deg² area within the boundaries 20 hr ⩽ R.A. ⩽ 7 hr and −65° ⩽ decl. ⩽ −40°, which we term the SPT-SZ survey. The survey was completed in 2011 November and achieved a fiducial depth of 18 μK arcmin in the 150 GHz band. Details of the survey strategy and data processing can be found in Schaffer et al. (2011).

Galaxy clusters are detected via their thermal SZE signature in the 95 and 150 GHz maps. These maps are created using time-ordered data processing and map-making procedures equivalent to those described in Vanderlinde et al. (2010), and clusters are extracted from the multi-band data as in Williamson et al. (2011) and Reichardt et al. (2013). A multi-scale matched-filter approach is used for cluster detection (Melin et al. 2006). The observable of the cluster SZE signal is ξ, the detection significance maximized over all filter scales. Because of the impact of noise biases, a direct scaling relation between ξ and cluster mass is difficult to characterize. Therefore, an unbiased SZE significance ζ is introduced, which is the signal-to-noise ratio at the true, underlying cluster position and filter scale (Vanderlinde et al. 2010). For ζ > 2, the relationship between ξ and ζ is given by

$$\begin{equation} \zeta = \sqrt{\langle \xi \rangle ^2-3}. \end{equation} \tag{ 1 }$$

The unbiased significance ζ is related to mass M_{500, c} by

$$\begin{equation} \zeta = A_{{\rm SZ}}\left(\frac{M_{500,c}}{3\times 10^{14}\,M_{\odot }h^{-1}}\right)^{B_{{\rm SZ}}}\left(\frac{E(z)}{E(0.6)}\right)^{C_{{\rm SZ}}}, \end{equation} \tag{ 2 }$$

where A_SZ is the normalization, B_SZ the mass slope, C_SZ the redshift evolution parameter and E(z) ≡ H(z)/H₀. An additional parameter D_SZ describes the intrinsic scatter in ζ, which is assumed to be lognormal and constant as a function of mass and redshift. The scaling parameters and the priors we adopt are summarized in Table 1, and further discussed in Section 4.3.1.

Table 1. ΛCDM Constraints from SZE Cluster Number Counts N(ξ, z) with Mass Calibration from Y_X and σ_v, CMB and Additional Cosmological Probes

Param.	Prior	N(ξ, z)	N(ξ, z)+BBN+H0+			WMAP9	SPTCL+WMAP9		Planck+WP	SPTCL+Planck+WP
			YX	σv	YX+σv			+BAO+SNe Ia			+BAO+SNe Ia
ASZ	6.24 ± 1.87	$6.49^{+2.08}_{-1.89}$	$5.59^{+1.19}_{-1.69}$	$4.38^{+1.05}_{-1.45}$	$4.70^{+0.82}_{-1.24}$	⋅⋅⋅	$3.79^{+0.57}_{-0.63}$	3.47 ± 0.48	⋅⋅⋅	3.27 ± 0.35	3.22 ± 0.30
BSZ	1.33 ± 0.266	1.54 ± 0.16	1.56 ± 0.13	1.65 ± 0.14	1.58 ± 0.12	⋅⋅⋅	1.47 ± 0.11	1.48 ± 0.11	⋅⋅⋅	1.49 ± 0.11	1.49 ± 0.11
CSZ	0.83 ± 0.415	0.75 ± 0.39	0.82 ± 0.35	0.92 ± 0.37	0.91 ± 0.35	⋅⋅⋅	0.40 ± 0.23	0.44 ± 0.23	⋅⋅⋅	0.44 ± 0.21	0.49 ± 0.22
DSZ	0.24 ± 0.16	0.32 ± 0.16	0.28 ± 0.11	$0.24^{+0.11}_{-0.14}$	0.26 ± 0.10	⋅⋅⋅	0.25 ± 0.10	0.27 ± 0.10	⋅⋅⋅	0.25 ± 0.05	0.26 ± 0.05
AX	5.77 ± 0.56	⋅⋅⋅	5.40 ± 0.56	⋅⋅⋅	5.76 ± 0.50	⋅⋅⋅	5.79 ± 0.43	5.94 ± 0.43	⋅⋅⋅	6.10 ± 0.42	6.13 ± 0.40
BX	0.57 ± 0.03	⋅⋅⋅	0.547 ± 0.030	⋅⋅⋅	0.545 ± 0.030	⋅⋅⋅	0.548 ± 0.029	0.549 ± 0.029	⋅⋅⋅	0.546 ± 0.029	0.546 ± 0.029
CX	−0.40 ± 0.20	⋅⋅⋅	−0.37 ± 0.18	⋅⋅⋅	−0.28 ± 0.17	⋅⋅⋅	−0.24 ± 0.17	−0.21 ± 0.17	⋅⋅⋅	−0.17 ± 0.16	−0.16 ± 0.16
DX	0.12 ± 0.08	⋅⋅⋅	0.15 ± 0.07	⋅⋅⋅	0.15 ± 0.07	⋅⋅⋅	0.14 ± 0.07	0.14 ± 0.07	⋅⋅⋅	0.14 ± 0.07	0.14 ± 0.07
$A_{\sigma _v}$a	939 ± 47	⋅⋅⋅	⋅⋅⋅	$971^{+47}_{-43}$	984 ± 39	⋅⋅⋅	973 ± 35	961 ± 35	⋅⋅⋅	948 ± 34	946 ± 33
$B_{\sigma _v}$	2.91 ± 0.15	⋅⋅⋅	⋅⋅⋅	2.91 ± 0.16	2.92 ± 0.16	⋅⋅⋅	2.92 ± 0.15	2.92 ± 0.16	⋅⋅⋅	2.92 ± 0.16	2.91 ± 0.16
$C_{\sigma _v}$	0.33 ± 0.02	⋅⋅⋅	⋅⋅⋅	0.330 ± 0.021	0.331 ± 0.021	⋅⋅⋅	0.329 ± 0.021	0.329 ± 0.020	⋅⋅⋅	0.327 ± 0.021	0.328 ± 0.020
$D_{\sigma _v0}$	0.2 ± 0.04	⋅⋅⋅	⋅⋅⋅	0.176 ± 0.030	0.176 ± 0.030	⋅⋅⋅	0.176 ± 0.028	0.174 ± 0.030	⋅⋅⋅	0.175 ± 0.029	0.175 ± 0.029
$D_{\sigma _v\, {\rm N}}$	3 ± 0.6	⋅⋅⋅	⋅⋅⋅	2.93 ± 0.56	2.92 ± 0.56	⋅⋅⋅	2.92 ± 0.56	2.93 ± 0.54	⋅⋅⋅	2.93 ± 0.54	2.93 ± 0.54
H0b	⋅⋅⋅ c	73.5 ± 2.4	73.2 ± 2.5	73.4 ± 2.4	73.2 ± 2.6	70.0 ± 2.4	70.1 ± 1.7	68.6 ± 1.0	67.6 ± 1.2	68.6 ± 1.1	68.3 ± 0.8
Ωm	⋅⋅⋅	$0.39^{+0.07}_{-0.13}$	$0.41^{+0.07}_{-0.14}$	$0.45^{+0.09}_{-0.16}$	$0.44^{+0.07}_{-0.15}$	0.281 ± 0.028	0.276 ± 0.018	0.292 ± 0.011	0.310 ± 0.017	0.297 ± 0.014	0.299 ± 0.009
σ8	⋅⋅⋅	0.67 ± 0.07	0.69 ± 0.06	0.72 ± 0.07	0.71 ± 0.06	0.825 ± 0.027	0.812 ± 0.017	0.816 ± 0.016	0.841 ± 0.013	0.828 ± 0.011	0.829 ± 0.011
$\sigma _8\left(\frac{\Omega _{\rm m}}{0.27}\right)^{0.3}$d	⋅⋅⋅	0.741 ± 0.064	0.774 ± 0.040	0.831 ± 0.052	0.809 ± 0.036	0.835 ± 0.051	0.817 ± 0.027	0.835 ± 0.022	0.877 ± 0.024	0.852 ± 0.020	0.855 ± 0.016

Notes. N(ξ, z) denotes the cluster sample without additional mass calibration information; SPT_CL contains the clusters with the mass calibration data from X-ray Y_X and velocity dispersion σ_v. The priors are Gaussian as discussed in Section 4.3. The scalar spectral index n_s, the reionization optical depth τ, the baryon density Ω_b, and the Planck nuisance parameters are not shown in this table but are included in the analysis and marginalized out. We fix τ = 0.089 when no CMB data are included in the fit. ^aThe units of $A_{\sigma _v}$ are km s⁻¹. ^bThe units of the Hubble constant H₀ are km s⁻¹ Mpc⁻¹. ^cWe apply a prior H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ when no CMB data are included in the fit. ^dThe uncertainty on σ₈(Ω_m/0.27)^0.3 reflects the width of the likelihood contour in the direction orthogonal to the cluster degeneracy in the Ω_m–σ₈ plane.

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We use SPT-selected clusters for the cosmological cluster number count and mass calibration analysis, described in Section 4. For the number counts, we use a cluster sample identical to the one used in Reichardt et al. (2013). This sample uses data from the first 720 deg² of the SPT-SZ survey and is restricted to ξ > 5 and redshift z > 0.3; it contains 100 cluster candidates. No optical counterparts were found for six of these SZE detections; we discuss their treatment in the analysis in Section 4.1.3. The SPT-SZ 720 deg² survey comprises five fields with different depths which are accounted for by rescaling the SPT ζ–mass relation normalization A_SZ for each field (Reichardt et al. 2013). Our mass calibration data consists of a sub-sample of 64 SPT clusters with additional X-ray and/or spectroscopic follow-up data, as described in Section 2.3 and 2.4. Twenty-two clusters with velocity dispersion σ_v measurements lie outside the SPT-SZ 720 deg² survey. The depths of these fields and the corresponding scaling factors for A_SZ will be presented elsewhere together with the analysis of the full 2500 deg² survey catalog (T. de Haan et al., in preparation). These scaling factors are all between 1.08–1.27 with a median value of 1.17.

The galaxy clusters analyzed here have been followed up in optical and near infrared in the context of the SPT follow-up program, as described in Song et al. (2012), to which we refer the reader for details of the strategy and data reduction. Briefly, the SPT strategy is to target all galaxy clusters detected at SZE significance ξ > 4.5 for multiband imaging in order to identify counterparts to the SZE signal and obtain photometric redshifts. We also obtain Spitzer/IRAC near-infrared imaging for every cluster with SZE significance ξ > 4.8, and we target those systems at lower ξ that are not optically confirmed or have a redshift above 0.9 with ground-based near-infrared imaging using the NEWFIRM imager on the CTIO Blanco 4 m telescope.

We use follow-up optical spectroscopy to measure the velocity dispersion σ_v of 63 clusters. Of these, 53 were observed by the SPT team (Ruel et al. 2014) and 10 have data taken from the literature (Barrena et al. 2002; Buckley-Geer et al. 2011; Sifón et al. 2013). In Ruel et al. (2014), four additional clusters with spectroscopic data are listed, but we choose not to include them in our analysis as they are all at relatively low redshifts below z < 0.1 where the SZE mass-observable scaling relation we adopt is likely not valid. The lowest redshift cluster entering our mass calibration analysis is SPT-CL J2300-5331 at z = 0.2623.

Our own data come from a total observation time of ∼70 hr on the largest optical telescopes (Gemini South, Magellan, and Very Large Telescope, VLT) in the Southern Hemisphere; we specifically designed these observations to deliver the data needed for this velocity dispersion mass calibration study. We obtained low-resolution (R ≃ 300) spectra using several different instruments: GMOS⁴⁰ on Gemini South, FORS2 (Appenzeller et al. 1998) on VLT Antu, LDSS3 on Magellan Clay and IMACS/Gladders Image-Slicing Multislit Option (GISMO⁴¹) on Magellan Baade.

Apart from early longslit spectroscopy using the Magellan LDSS3 spectrograph on a few SPT clusters, the general strategy is to design two masks per cluster for multi-object spectroscopy to get a final average number of 25 member galaxy redshifts per cluster. We typically obtained deep (m^⋆ + 1) pre-imaging in i'-band for spectroscopic observation to (1) accurately localize galaxies to build masks for multi-object spectroscopy, and (2) identify possible giant arcs around cluster cores. This deep pre-imaging is used together with existing shallower optical imaging and near-infrared photometry, where available, to select galaxy cluster members along the red sequence. We refer the reader to Ruel et al. (2014) for a detailed description of the cluster member selection and the data reduction.

Sixteen clusters of our sample have been observed in X-ray using either Chandra or XMM-Newton. The derived properties of 15 of these clusters are published in Andersson et al. (2011). This sub-sample corresponds to the highest SZE significance clusters in the first 178 deg² of the SPT-SZ survey that lie at z  ≳  0.3. We obtained Chandra observations of SPT-CL J2106-5844 in a separate program whose results are published elsewhere (Foley et al. 2011). All of these observations have >1500 source photons within 0.5 × r₅₀₀ and in the 0.5–7.0 keV energy band. X-ray observations are used to derive the intracluster medium temperature T_X and the gas mass M_g. For a detailed description of the data reduction method, we refer the reader to Andersson et al. (2011). Note that there is a calibration offset between temperature measurements from the two satellites (Schellenberger et al. 2014). For our analysis, we adopt priors on the Y_X-mass relation that come from an analysis of Chandra data. Given that only 2/16 systems in this study rely on XMM-Newton data, and the amplitude of the calibration offset is ∼30% in temperature for these massive clusters, we expect an overall temperature bias of ∼4%, corresponding to a ∼2% bias in our mass scale, assuming that the Chandra-derived temperatures are unbiased. Given that this is much smaller than the systematic uncertainty in our Y_X-mass calibration, we neglect any cross-calibration.

Following Benson et al. (2013), we rely on the X-ray observable Y_X ≡ M_gT_X. For the cosmological analysis performed in this work we need to evaluate Y_X as a function of cosmology and scaling relation parameters. In practice, for a given set of cosmological and scaling relation parameters, we iteratively fit for r₅₀₀ and Y_X(r), which is then used to estimate the cluster mass.

We adopt a calibrated scaling relation derived from hydrostatic masses at low redshifts (Vikhlinin et al. 2009a):

$$\begin{equation} \frac{M_{500,c}}{10^{14}\,M_{\odot }} = A_{\rm X} h^{1/2} \left(\frac{Y_{\rm X}}{3\times 10^{14}\,M_{\odot }\, \mathrm{keV}}\right)^{B_{\rm X}} E(z)^{C_{\rm X}}, \end{equation} \tag{ 3 }$$

where A_X is the normalization, B_X the slope and C_X the redshift evolution parameter. We assume an intrinsic lognormal scatter in Y_X denoted D_X and an observational lognormal uncertainty for each cluster. The fiducial values and priors we adopt for the Y_X parameters are discussed in Section 4.3.2 and shown in Table 1.

In addition to our cluster sample, we include external cosmological data sets such as measurements of the CMB anisotropy power spectrum, the BAO, SNe Ia, the Hubble constant (H₀), and big bang nucleosynthesis (BBN). We use these abbreviations when including the data sets in the analysis. We refer to the SPT SZE cluster sample without the follow-up mass information as N(ξ, z) (which stands for the distribution of the clusters in ξ–z space), and we refer to the full cluster sample with mass measurements from σ_v and Y_X as SPT_CL.

We include measurements of the CMB anisotropy power spectrum from two all-sky surveys. We use data from the Wilkinson Microwave Anisotropy Probe (WMAP, 9yr release; Hinshaw et al. 2013) and data from the Planck satellite (1 yr release, including WMAP polarization data (WP); Planck Collaboration et al. 2014a, 2014b). The BAO constraints are applied as three measurements: D_V(z = 0.106) = 457 ± 27 Mpc (Beutler et al. 2011), D_V(z = 0.35)/r_s = 8.88 ± 0.17 (Padmanabhan et al. 2012), and D_V(z = 0.57)/r_s = 13.67 ± 0.22 (Anderson et al. 2012); r_s is the comoving sound horizon at the baryon drag epoch, $D_{\rm V}(z)\equiv [(1+z)^2D_{\rm A}^2(z)cz/H(z)]^{1/3}$, and D_A is the angular diameter distance. We include distance measurements coming from SNe Ia using the Union2.1 compilation of 580 SNe (Suzuki et al. 2012). We adopt a Gaussian prior on the Hubble constant H₀ = 73.8 ± 2.4 km s⁻¹ Mpc⁻¹ from the low-redshift measurements from the Hubble Space Telescope (Riess et al. 2011). Finally, we use a BBN prior from measurements of the abundance of ⁴He and deuterium, which we include as a Gaussian prior Ω_bh² = 0.022 ± 0.002 (Kirkman et al. 2003). Note that both the BBN and H₀ priors are only applied when analyzing the cluster samples without CMB data.

The cluster number count analysis in the SZE observable ξ can be separated from the additional mass calibration in an unbiased way. This approach allows for an easy comparison and combination of the different mass calibrators as we will discuss in Section 4.2. For a detailed derivation of our likelihood function, see the Appendix.

4.1.1. Cluster Mass Function

At each point in the space of cosmological and scaling-relation parameters we use the Code for Anisotropies in the Microwave Background (CAMB; Lewis et al. 2000) to compute the matter power spectrum at 180 evenly spaced redshift bins between 0.2  <  z  <  2. We then use the fitting function presented in Tinker et al. (2008) to calculate the cluster mass function dN/dM for 500 mass bins evenly distributed in log-space between 10^13.5h⁻¹M_☉  ⩽  M  ⩽  10¹⁶h⁻¹ M_☉. This fitting function is accurate at the 5% level across a mass range 10¹¹h⁻¹ M_☉  ⩽  M  ⩽  10¹⁵h⁻¹ M_☉ and for redshifts z ⩽ 2.5.

We move the mass function from its native mass and redshift space to the observable space in ξ–z:

$$\begin{eqnarray} \frac{dN(\xi,z | \boldsymbol{p})}{d\xi dz} &=& \int dMdz\, \Theta (\xi - 5, z-0.3) \nonumber \\ && \times P(\xi | M, z, \boldsymbol{p}) \otimes \frac{dn(M,z|\boldsymbol{p})}{dM} \frac{dV(z)}{dz},\quad \end{eqnarray} \tag{ 6 }$$

where dV/dz is the comoving volume within each redshift bin, $\boldsymbol{p}$ is a vector containing all scaling relation and cosmological parameters, and Θ is the Heaviside step function describing cluster selection in the SZE observable ξ > 5, and observed redshift z > 0.3. The term $P(\xi | M, z, \boldsymbol{p})$ describes the relationship between mass and the SZE observable from the scaling relation (Equations (1) and (2)), and contains both intrinsic and observational uncertainties. In practice, we convolve the mass function with this probability distribution.

Finally, the logarithm of the likelihood $\mathcal {L}$ for the observed cluster counts is computed following Cash (1979). After dividing up the observable space in small bins, the number of expected clusters in each bin is assumed to follow a Poisson distribution. With this the likelihood function is

$$\begin{equation} \ln \mathcal {L}(\boldsymbol{p}) = \sum _i \ln \frac{dN(\xi _i,z_i | \boldsymbol{p})}{d\xi dz} - \int \frac{dN(\xi,z | \boldsymbol{p})}{d\xi dz} d\xi dz, \end{equation} \tag{ 7 }$$

up to a constant offset, and where i runs over all clusters in the catalog. For clusters without spectroscopic data, we integrate the model over redshift weighting with a Gaussian whose central value and width correspond to the cluster's photometric redshift measurement.

The 720 deg² survey area contains five fields of different depths; see Section 2.1. In practice, we perform the above calculation for each field rescaling A_SZ with the corresponding factor, and sum the resulting log likelihoods.

4.1.2. Mass Calibration

For each cluster in our sample containing additional mass calibration information from X-ray and/or velocity dispersions, we include the Y_X or σ_v measurement as follows: At every point in cosmological and scaling relation parameter space $\boldsymbol{p}$, we calculate the probability distribution $P(M | \xi,z,\boldsymbol{p})$ for each cluster mass, given that the cluster has a measured significance ξ and redshift z:

$$\begin{equation} P(M | \xi,z,\boldsymbol{p}) \propto P(\xi | M,z,\boldsymbol{p}) P(M|z,\boldsymbol{p}). \end{equation} \tag{ 8 }$$

In practice, we calculate the probability distribution $P(\xi | M, z,\boldsymbol{p})$ from the SZE scaling relation (Equations (1) and (2)) taking both intrinsic and observational scatter into account, and weight by the mass function $P(M|z,\boldsymbol{p})$, thereby correcting for Eddington bias. We then calculate the expected probability distribution in the follow-up observable(s), which we here call $\mathcal {O}$ for simplicity:

$$\begin{equation} P(\mathcal {O}|\xi,z,\boldsymbol{p}) = \int dM \,P(\mathcal {O}|M,z,\boldsymbol{p})P(M|\xi,z,\boldsymbol{p}). \end{equation} \tag{ 9 }$$

The term $P(\mathcal {O}|M,z,\boldsymbol{p})$ contains the intrinsic scatter and observational uncertainties in the follow-up observable. We assume the intrinsic scatter in the SZE scaling relation and the follow-up measurements to be uncorrelated. For each cluster in the mass calibration sample, we compare the predicted $P(\mathcal {O}|\xi,z,\boldsymbol{p})$ with the actual measurement and extract the probability of consistency. Finally, we sum the log-likelihoods for all these clusters and add the result to the number count likelihood (Equation (7)).

It is important that any cosmological dependence of the mass calibration observations be accounted for. In the case of a single velocity dispersion σ_v, the measurement comes from the combination of redshift measurements from a sample of cluster galaxies; the cosmological sensitivity, if any, is subtle. On the other hand, the X-ray observable Y_X is calculated from the measured temperature and gas mass within r₅₀₀, and the limiting radius and the gas mass are both cosmology dependent. Therefore, Y_X has to be extracted from the observations for each set of cosmological and scaling relation parameters as described in Section 2.4.

4.1.3. Unconfirmed Cluster Candidates

Out of the 100 cluster candidates in the survey, 6 detections could not be confirmed by the optical follow-up and were assigned lower redshift limits based on the depth of the imaging data (Song et al. 2012). In addition, each of these unconfirmed candidates has some probability of being a noise fluctuation.

Our treatment of these candidates takes into account the false detection rate at the detection signal-to-noise as well as the expected number of clusters exceeding the lower redshift bound of the candidate as predicted by the cluster mass function. We calculate the probability of a candidate i to be a true cluster according to

$$\begin{equation} P_{\rm true}^i = \frac{N_{\rm expected} \left(\xi ^i, z_{\rm low}^i| \boldsymbol{p}\right)}{N_{\rm expected} \left(\xi ^i, z_{\rm low}^i| \boldsymbol{p}\right) + N_{\rm false\ detect}(\xi ^i)}, \end{equation} \tag{ 10 }$$

where the number of clusters N_expected above some lower redshift limit is given by $\int _{z_{\rm low}^i}^\infty N(\xi ^i, z|\boldsymbol{p})dz$. The expected number of false detections as a function of ξ has been estimated from simulations and cross-checked against direct follow-up and is assumed to be redshift independent (Song et al. 2012; Reichardt et al. 2013).

In the cosmological analysis, each of the unconfirmed candidates is treated like an actual cluster but weighted with its $P_{\rm true}^i$. However, the specific treatment of the unconfirmed candidates has little effect on the cosmological and scaling relation parameters; for example, simply removing these candidates from the catalog leads to negligible changes in the results.

In previous SPT cluster cosmology studies, we have used a somewhat different method. In that method the expected number density of clusters as a function of ξ, Y_X, and z is calculated on a three-dimensional grid. The likelihood is evaluated by comparing this prediction to the cluster sample in a way analogous to Equation (7). For clusters without Y_X data the likelihood is integrated over the full range of Y_X (Benson et al. 2013).

As we show in the Appendix, the method we employ in the current analysis is mathematically equivalent to this other method; here we assume uncorrelated scatter. For the current application, where we have σ_v and Y_X follow-up measurements, we do not work in the four-dimensional ξ–Y_X–σ_v–z-space, but rather we treat the number count part of the likelihood in its ξ–z-space, and the mass calibration part of the likelihood $P(\mathcal {O}|\xi,z,\boldsymbol{p})$ separately. The results obtained with this analysis method do not show any sign of biases when tested against different sets of mock data (see Section 4.4). This method is convenient when analyzing a cluster sample with multiple different mass observables where only a fraction of the clusters have those observables. In the limit where every cluster in the survey has the same follow-up mass measurements, the likelihood presented and used in our previous analyses (Benson et al. 2013; Reichardt et al. 2013) would be more computationally efficient.

We present the priors used in our analysis and discuss their motivation. All priors are also listed in the first column of Table 1.

4.3.1. Priors on SZE ξ-mass Scaling Relation Parameters

4.3.2. Priors on Y_X-mass Scaling Relation Parameters

4.3.3. Priors on σ_v-mass Scaling Relation Parameters

4.3.4. Additional Priors on Cosmological Parameters

We validate the analysis method using simulated data. In a first step we test the number count part in SZE significance and redshift space using simulated cluster catalogs that match the SPT data but contain orders of magnitude more clusters; our goal here is to minimize statistical noise so as to resolve possible systematics in the analysis at a level far below the statistical noise in our real sample. Our mock generator produces clusters in mass-redshift space, converts the cluster masses to the SZE observable ξ using Equations (1) and (2) with lognormal and normal scatter, respectively, and then applies the survey selection. The crucial part of the analysis—that is, the conversion from mass to observable—is thereby computed differently than in the likelihood code we use to explore cosmological parameter space.

We generate large catalogs using different sets of input values and obtain samples containing on the order of 10⁴ clusters. We then run our analysis pipeline on the mock data using priors equivalent to the ones listed in Table 1; our tests show that we are able to recover the input values to within 1σ statistical uncertainties, verifying that there are no biases in our codes at a level well below the statistical noise in our real cluster ensemble.

We further analyzed mock catalogs produced using the analysis pipeline used in our previous analyses (Benson et al. 2013; Reichardt et al. 2013), recovering the input parameters at the 1σ statistical level. To test the mass calibration module, we use a subset of 500 clusters drawn from the SZE mock catalog described above and additionally convert the cluster masses to X-ray Y_X and velocity dispersion σ_v measurements. We then run our analysis code on the mass calibration part alone, that is, without using the number count information, and use Y_X and/or σ_v, showing that we are able to recover the input values. Finally, we confirm that the combination of number counts and mass calibration produces unbiased results by combining the SZE mock catalog with the X-ray and spectroscopic cluster mass observables. These tests give us confidence that our code is producing unbiased constraints.

In Table 1, we present the results of the analysis of the SPT-SZ survey cluster sample and its mass calibration assuming a flat ΛCDM model. For now we do not include CMB, BAO, or SNe Ia data, because we first wish to isolate the galaxy cluster constraints and the impact of the mass calibration data. However, we include the BBN and H₀ priors, because not all parameters are well constrained by the cluster data.

We present results using the SPT cluster sample N(ξ, z) only, N(ξ, z) with Y_X data, N(ξ, z) with σ_v data, and N(ξ, z) with both Y_X and σ_v. It is clear that the additional mass information from σ_v or Y_X helps in improving the results obtained from N(ξ, z) only. The constraints on the SZE scaling relation normalization A_SZ, the scatter in that relation D_SZ, and the cosmological parameter combination σ₈(Ω_m/0.27)^0.3 tighten. The uncertainty on this parameter reflects the width of the likelihood distribution in Ω_m–σ₈ space in the direction orthogonal to the cluster degeneracy (see Figure 1).

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There is agreement between the results obtained using the mass calibrators σ_v or Y_X, which provides an indication that both methods are reliable and that systematics are under control. The normalization A_SZ decreases by 22% when replacing the Y_X calibration data set with the σ_v data set. Due to the skewness of the probability distributions with tails toward larger values, the constraints on A_SZ from σ_v and Y_X measurements have significant overlap, with the Y_X-favored value displaced 1.15σ from the result obtained from σ_v (see also Figure 2). The constraints on the slope B_SZ, the redshift evolution parameter C_SZ, as well as the scatter D_SZ are not much affected by the choice of the mass calibrator. We note that the Y_X scaling relation is calibrated by observations at z ∼ 0.3, which is extrapolated to higher redshifts using priors motivated by simulations, whereas the σ_v scaling relation is calibrated to simulations over the full redshift range. In terms of the cosmological results, both follow-up methods perform similarly in constraining the fully marginalized values for Ω_m and σ₈. However, the Y_X calibration does better in constraining σ₈(Ω_m/0.27)^0.3.

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Our constraints using SPT clusters with mass calibration from X-ray Y_X only are comparable with previously published results from nearly the same cluster sample (Reichardt et al. 2013). Note that the X-ray sample used here contains measurements of Y_X for two additional clusters (see Section 2.4). We recover almost identical constraints on the SZE and X-ray scaling relation parameters. However, in the Ω_m–σ₈ plane, the constraints presented here extend further along the degeneracy direction toward higher values of Ω_m. This difference is due to a prior on the power spectrum normalization ln (10⁻¹⁰A_s) = [2.3, 4] that was narrow enough to affect the cosmological constraints in Reichardt et al. (2013); we fit for σ₈ in the range [0.4, 1.2], which is much broader than the recovered probability distribution, and hence our choice of prior does not affect our results.

We estimate the effect of potentially larger galaxy velocity bias (see discussion in Section 3 and 4.3.3) by loosening our prior on $A_{\sigma _v}$ from the 5% recommended by Saro et al. (2013) to 10% when analyzing the N(ξ, z)+σ_v+BBN+H₀ data. There is a broadening of the uncertainty on A_SZ by 25%, and a ∼0.3σ shift to a higher value. The constraint on σ₈(Ω_m/0.27)^0.3 degrades by 14% and shifts only by a negligible amount. In addition, we examine the impact of tightening the prior on $A_{\sigma _v}$ to 1%. In this case, we observe improvements on the constraints on A_SZ (28%) and σ₈(Ω_m/0.27)^0.3 (23%).

Because of the consistency of the two calibration data sets, we combine them into a joint mass calibration analysis. We observe that the SZE normalization A_SZ remains close to the value favored by the σ_v measurements, while its 68% confidence region decreases by roughly 20% compared to the individual results. This impact on A_SZ is the best improvement on the SZE parameters we observe when combining the mass calibrators. The constraints on Ω_m and σ₈ lie between the individual results with similar uncertainties. However, σ₈(Ω_m/0.27)^0.3 clearly benefits from the combined mass information, and its uncertainty is 10% (23%) smaller than when using the individual Y_X (σ_v) calibration data.

We now compare the results from our cluster data with constraints from CMB anisotropies as obtained from WMAP9. The probability distributions of the cluster data sets and WMAP9 overlap, indicating agreement between both sets of constraints (see also Figure 1). Moreover, the parameter degeneracies in the Ω_m–σ₈ space for clusters are nearly orthogonal to the ones of CMB data.

We quantify the agreement between two data sets by testing the degree to which their probability distributions $P(\boldsymbol{x})$ overlap in some parameter space $\boldsymbol{x}$. We measure this by first drawing representative samples of points $\left\lbrace \boldsymbol{x_1}\right\rbrace$ and $\left\lbrace \boldsymbol{x_2}\right\rbrace$ from the two probability distributions $P_1(\boldsymbol{x})$ and $P_2(\boldsymbol{x})$. We then compute the distances between pairs of sampled points $\boldsymbol{\delta }\equiv \boldsymbol{x_1} - \boldsymbol{x_2}$ and estimate the probability distribution P_δ from this ensemble $\left\lbrace \boldsymbol{\delta }\right\rbrace$. We then evaluate the likelihood p that the origin lies within this distribution:

$$\begin{equation} p = \int _{S} d\boldsymbol{y}\, P_\delta (\boldsymbol{y}), \end{equation} \tag{ 11 }$$

where the space S is that where $P_\delta <P_\delta (\boldsymbol{0})$, and $P_\delta (\boldsymbol{0})$ is the probability at the origin. We convert p to a significance assuming a normal distribution. Within the PMC fitting procedure used to obtain the probability distributions P, each sample point $\boldsymbol{x}$ is assigned a weight. We calculate the agreement between two distributions using the method presented above, assigning each point $\boldsymbol{\delta }$ a weight that is the product of the weights of the points $\boldsymbol{x_1}$ and $\boldsymbol{x_2}$.

We apply this method in the two-dimensional Ω_m–σ₈ space. Within our baseline model that assumes massless neutrinos we report good consistency (1.3σ) between the results from our cluster sample and from WMAP9. Changing the baseline assumptions to account for one massive neutrino with mass m_ν = 0.06 eV decreases the tension to 1.0σ. We note that this increase in neutrino mass shifts CMB constraints toward lower values of σ₈ by about Δσ₈ ≈ −0.012 while having negligible impact on the cluster constraints. We fit for the sum of neutrino masses in Section 5.7; this further reduces the tension.

Given the overlap between the probability distributions from our clusters and WMAP9, we combine the data sets to break degeneracies and thereby tighten the constraints. In Table 2, we show how the combination of the N(ξ, z) cluster sample with WMAP9 data benefits from the additional mass calibration from σ_v and/or Y_X. It is clear that, even if the cosmological constraints are dominated by the CMB data, the mass calibration from either observable leads to tighter constraints on all four parameters shown in the table. We also observe that the constraints on the cosmological parameters Ω_m, σ₈, and σ₈(Ω_m/0.27)^0.3 obtained when including Y_X data are systematically lower by about half a σ than results obtained without these data; the constraints on A_SZ are higher. These shifts correspond to lower cluster masses; we will come back to this in Section 5.4.

Table 2. Impact of σ_v and/or Y_X Mass Calibration on Results from SPT Clusters N(ξ, z)+WMAP9

Data Set	ASZ	Ωm	σ8	σ8 (Ωm/0.27)0.3
N(ξ, z)+WMAP9	$3.59^{+0.60}_{-1.04}$	0.284 ± 0.027	0.823 ± 0.026	0.835 ± 0.047
N(ξ, z)+WMAP9+σv	$3.51^{+0.65}_{-0.63}$	0.288 ± 0.022	0.824 ± 0.020	0.840 ± 0.035
N(ξ, z)+WMAP9+YX	$3.85^{+0.62}_{-0.66}$	0.273 ± 0.019	0.811 ± 0.019	0.813 ± 0.032
N(ξ, z)+WMAP9+YX+σv	$3.79^{+0.57}_{-0.63}$	0.276 ± 0.018	0.812 ± 0.017	0.817 ± 0.027

Notes. These are fully marginalized constraints. The results from N(ξ, z)+Y_X+σ_v+WMAP9 are presented in more detail in Table 1.

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When adding the WMAP9 data to our full cluster sample SPT_CL, we observe shifts in the SZE scaling relation parameters, as shown in Table 1. There is a decrease in the SZE normalization A_SZ by 19%, and the uncertainty tightens by 42%. We further observe a notable shift in the redshift evolution C_SZ toward a lower value at the 1σ level. This is due to the degeneracy between C_SZ and Ω_m, as the latter also shifts significantly when the WMAP9 data are added. The remaining scaling relation parameters do not benefit from the additional data. Conversely, the SPT cluster data improve the cosmological constraints from the WMAP9 data by reducing the uncertainty on Ω_m by 36%, on σ₈ by 33%, and on σ₈(Ω_m/0.27)^0.3 by 47%. Figure 1 shows how the combination of the data sets leads to improved constraints due to the nearly orthogonal parameter degeneracies of the individual results (red contours in figure).

Finally, we add data from BAO and SNe Ia, which carry additional information on cosmic distances. As expected, we see a further tightening of the constraints on Ω_m = 0.292 ± 0.011 and H₀ = 68.6 ± 1.0 km s⁻¹ Mpc⁻¹.

In Figure 1, we also show the constraints in the Ω_m–σ₈ plane from Planck+WP and report a mild 1.9σ tension between our cluster sample and this CMB data set. The tension is slightly larger than when comparing the clusters to WMAP9. The Planck+WP data favor a larger value of σ₈ than our cluster sample. Assuming one massive neutrino with mass m_ν = 0.06 eV relaxes the tension to 1.5σ.

We proceed and combine our cluster sample with the CMB data from Planck+WP. This data combination prefers a value for σ₈ that is about 1σ lower than suggested by the CMB data. Adding our cluster sample to Planck+WP leads to improvements on the constraints on Ω_m, σ₈ and σ₈(Ω_m/0.27)^0.3, all on the order of 15% (see Table 1, and black/cyan contours in Figure 1).

We add BAO and SNe Ia data to further improve the cosmological constraints, and measure Ω_m = 0.297 ± 0.009, σ₈ = 0.829 ± 0.011, σ₈(Ω_m/0.27)^0.3=0.855 ± 0.016, and H₀ = 68.3 ± 0.8 km s⁻¹ Mpc⁻¹. These represent improvements of 18% (Ω_m), 8% (σ₈), and 11% (σ₈(Ω_m/0.27)^0.3) over the constraints from Planck+WP+BAO+SNe Ia without SPT_CL. In addition, these represent improvements of 18% (Ω_m), 31% (σ₈), 20% (σ₈(Ω_m/0.27)^0.3), and 20% (H₀) over the corresponding parameter uncertainties when using WMAP9 instead of Planck+WP.

Combining the mass calibration from Y_X with σ_v data and further with CMB data leads to shifts in the SZE scaling relation parameters which ultimately shift the mass estimates of the clusters. As shown in Figure 2, there is a systematic increase of the cluster mass scale as we move from X-ray to dispersion only calibration, further on to Y_X+σ_v and finally on to analyses of our SPT_CL data set in combination with external data sets (remember that a decrease in A_SZ corresponds to an increase in cluster mass; see Equation (2)). Also, it is clear that the constraints on the SZE normalization A_SZ obtained when including CMB data are much stronger than the constraints from the cluster data alone. The Gaussian prior on A_SZ is in some tension with the A_SZ constraints after including the CMB data. In this case, we note that the recovered values of A_SZ do not significantly change when removing the prior, because it is much broader than the recovered constraints.

We quantify the agreement between these distributions in the space of A_SZ in a way equivalent to the one presented in Section 5.2. We find that the results from both Y_X and σ_v mass calibration are consistent at the 0.6σ level. There is a mild tension (1.9σ) between mass calibration from Y_X and SPT_CL+Planck+WP+BAO+SNe Ia, while the mass calibration from σ_v is consistent with the multi-probe data set at the 0.8σ level. These shifts would approximately correspond to an increase in the preferred cluster mass scale by 44% and 23%, respectively, when using the multi-probe data set. Note that there are shifts in B_SZ and C_SZ when adding CMB data to our cluster sample, which add a slight ξ (or equivalently mass) and redshift dependence to this comparison of cluster masses.

On average, our cluster mass estimates are higher by 32% than our previous results in Reichardt et al. (2013), primarily driven by using new CMB and BAO data sets. Relative to Reichardt et al. (2013), we have updated the CMB data set from WMAP7 and SPT (Komatsu et al. 2011; Keisler et al. 2011) to Planck+WP (Planck Collaboration et al. 2014a, 2014b), and also updated the BAO data set from Percival et al. (2010) to a combination of three measurements (Beutler et al. 2011; Anderson et al. 2012; Padmanabhan et al. 2012). The new data sets have led to more precise constraints on the cosmological parameters, in particular σ₈(Ω_m/0.27)^0.3, and drive shifts in the preferred cluster mass scale through A_SZ, to improve consistency between the cluster data set and the cosmological constraints. For example, using WMAP9 data instead of Planck+WP+BAO+SNe Ia leads to an average 11% decrease of the cluster masses. Finally, we observe an increase in the slope B_SZ as compared to Reichardt et al. (2013), which reduces the mass change to only ∼15% on the high-mass end of the sample.

Our analysis to this point has focused on extracting parameter confidence regions that emerge from different combinations of our cluster sample with external data sets. We observed shifts especially in the SZE scaling relation parameters when switching among the different data combinations. In the following, we investigate whether the adopted SZE mass-observable scaling relation parameterization is adequate for describing the cluster sample. We execute two tests: (1) we evaluate the goodness of fit of the SZE selected clusters in the ξ–z plane, and (2) we compare the predicted values for the follow-up observables Y_X and σ_v to their actual measurements. Both tests are performed adopting parameter values at the best-fit location in cosmological and scaling relation parameter space from the SPT_CL+Planck+WP+BAO+SNe Ia analysis.

We compare the distribution of the SZE clusters in the observable ξ–z plane with its prediction. This is done using a two-dimensional Kolmogorov–Smirnov (K-S) test as described in Press et al. (1992): At the location of each cluster in ξ and z space, we split the observational space into four quadrants, and calculate the absolute difference between the number of clusters and the number predicted by the model within that area. The largest of these 4 × N_cl values is taken as the maximum difference D between the data and the model. We characterize this difference measure by calculating it for 10,000 independent catalogs that we produce using the best-fit cosmology and scaling relation parameters. Figure 3 contains a histogram of the distribution of differences D from the set of catalogs, and the red line marks the difference for the real sample. This test indicates that there is a 90% chance of obtaining a larger difference D than observed in our real data set. We conclude that there is no tension between our SPT cluster sample and the way we model it through the SZE scaling relation parameterization.

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We now go one step further and ask whether there is tension between the predicted values for the follow-up observables Y_X and σ_v and their actual measurements. Remember that the predicted probability distributions are obtained from the observed SZE signal ξ according to Equation (9). For each cluster, we calculate the percentile of the observed value in its predicted distribution. We get a distribution of percentiles, which we convert to a distribution of pulls (Eadie 1983; Lyons 1989) using the inverse error function:

$$\begin{equation} \mathrm{pull} = \sqrt{2}\times \mathrm{erf}^{-1}(2\times \mathrm{percentile}-1). \end{equation} \tag{ 12 }$$

This distribution is finally compared to a normal distribution of unit width centered at zero using the K-S test. In Figure 4 we show the distribution of pulls for the Y_X and σ_v measurements. For each observable, we show the distribution for two different sets of cosmological and scaling relation parameters: (1) the results obtained from clusters with mass calibration only, and (2) the results from clusters with mass calibration combined with the external cosmological probes. In all four cases, the K-S test provides p-values in the range 0.1 < p < 0.8, indicating no tension between the predicted follow-up mass observables and their measurements. This is an interesting observation given the shifts we observe in the scaling relation and cosmological parameters when adding CMB data to the cluster sample. It shows that the adopted form of the SZE mass-observable scaling relation has enough freedom to compensate for the shifts in cosmological parameters. With a larger cluster and mass calibration data set we could expect to make a more precise consistency test of the data and our adopted scaling relation parameterization.

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The first extension of the ΛCDM model we analyze is the flat wCDM cosmology which includes the dark energy equation of state parameter w. As the dark energy becomes relevant only in the late universe and affects the cluster mass function through its impact on the cosmological growth rate and volume, we expect our cluster sample to provide an important contribution in constraining its nature.

Analyzing our cluster sample using priors on H₀ and BBN, we obtain w = −1.5 ± 0.5. This measurement is compatible with external constraints from WMAP9+H₀ (w = −1.13 ± 0.11) and Planck+WP+BAO (w = −1.13 ± 0.25, 95% confidence limits; Planck Collaboration et al. 2014b), and consistent with the ΛCDM value w = −1. Remember that the results obtained from clusters might in principle be subject to systematics in the mass estimates, while, on the other hand, the CMB anisotropy measurements are most sensitive to the characteristics of the universe at z  ∼  1100, and the distance measurements are subject to their own systematics.

Combining data sets breaks degeneracies and leads to tighter constraints. When adding our SPT_CL sample to the WMAP9+H₀ data, we measure w = −1.07 ± 0.09, or an 18% improvement over the constraint without clusters. Combining our cluster sample with Planck+WP+BAO+SNe Ia (w = −1.051 ± 0.072) data leads to an even tighter constraint, and we measure w = −0.995 ± 0.063 (12% improvement; see also Table 3).

Table 3. Constraints on Extensions of Flat ΛCDM Cosmology from the SPT_CL+Planck+WP+BAO+SNe Ia Data Combination

Parameter	wCDM	νCDM	γ+ΛCDM	γ+νCDM	γ+wCDM
Ωm	0.301 ± 0.014	0.309 ± 0.011	0.302 ± 0.010	0.309 ± 0.012	0.301 ± 0.014
σ8	0.827 ± 0.024	0.799 ± 0.021	$0.793^{+0.046}_{-0.075}$	$0.796^{+0.057}_{-0.080}$	$0.794^{+0.054}_{-0.078}$
H0 (km s−1 Mpc−1)	68.1 ± 1.6	67.5 ± 0.9	68.2 ± 0.8	67.5 ± 0.9	68.3 ± 1.6
w	−0.995 ± 0.063	(− 1)	(− 1)	(− 1)	−1.007 ± 0.065
∑mν(eV)	(0)	0.148 ± 0.081	(0)	$0.143^{+0.066}_{-0.100}$	(0)
∑mν(eV), 95% CL	(0)	<0.270	(0)	<0.277	(0)
γ	(0.55)	(0.55)	0.72 ± 0.24	0.63 ± 0.25	0.73 ± 0.28

Note. These are fully marginalized constraints.

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We now extend the ΛCDM model and include the sum of neutrino masses ∑m_ν as a free parameter. We will refer to this model as νCDM in the following, and we assume three degenerate mass neutrino species.

Massive neutrinos are still relativistic at the epoch of recombination and hence do not significantly affect the structure of CMB anisotropies (as long as m_ν < 0.6 eV for each species, Komatsu et al. 2009). In the late universe, massive neutrinos contribute to Ω_m but do not cluster in structures smaller than their free streaming length, leading to a lower σ₈. Therefore, results from CMB anisotropy data exhibit a strong degeneracy between ∑m_ν and σ₈. Using the Planck+WP+BAO+SNe Ia data combination, we measure ∑m_ν = 0.092 ± 0.058 eV and an upper limit ∑m_ν < 0.182 eV (95% confidence limit, hereafter CL).

Galaxy clusters are ideal probes for measuring σ₈ and therefore represent a valuable piece of information when constraining the νCDM model. When adding our SPT_CL sample to the data set, we observe that the mean of the recovered ∑m_ν increases significantly; we measure ∑m_ν = 0.148 ± 0.081 eV, and an upper limit ∑m_ν < 0.270 eV (95% CL). As discussed earlier, our cluster sample prefers lower values for σ₈ than the CMB data, which here leads to increased neutrino masses due to their degeneracy with σ₈. The results on νCDM from the full data combination are also shown in Table 3.

We recalculate the difference between results from SPT_CL and CMB data as in Section 5.2, but we now adopt our best-fit sum of neutrino masses ∑m_ν = 0.148 eV. This decreases the tension to 0.7σ for WMAP9, and 1.1σ for Planck+WP.

Our constraints on the dark energy equation-of-state parameter confirm once more that the flat ΛCDM model provides an excellent fit to the best currently available cosmological data. However, it still remains unclear what exactly is causing the accelerating expansion in the present epoch. Possible explanations include a new energy component or a modification of gravity on large scales. While measurements of CMB anisotropies and cosmic distances (BAO and SNe Ia) have proven extremely useful for probing the expansion history of the universe, galaxy clusters provide a unique probe for testing its growth history. Combining these tests allows for an interesting consistency test of general relativity (GR) on large scales (e.g., Rapetti et al. 2013).

5.8.1. Parameterized Growth of Structure

We parameterize the linear growth rate of density perturbations f(a) at late times as a power law of the matter density (e.g., Peebles 1980; Wang & Steinhardt 1998)

$$\begin{equation} f(a) \equiv \frac{d \ln \delta }{d\ln a} = \Omega _m(a)^\gamma, \end{equation} \tag{ 13 }$$

where γ is the cosmic growth index and δ ≡ δρ_m/〈ρ_m〉 is the ratio of the comoving matter density fluctuations and the mean matter density. Solving for γ and assuming GR, one obtains

$$\begin{equation} \gamma _{\rm GR} \approx \frac{6-3(1+w)}{11-6(1+w)}, \end{equation} \tag{ 14 }$$

where the leading correction depends on the dark energy equation-of-state parameter w and so γ_GR = 0.55 for a cosmological constant with w = −1. Normalizing the parameterized cosmic growth factor D(z)∝δ(z) at some high redshift z_ini, we can express it as

$$\begin{equation} D_{\rm ini}(z) = \frac{\delta (z)}{\delta (z_{\rm ini})} = \delta (z_{\rm ini}) ^{-1} \exp \int d\ln a \; \Omega _{\rm m}(a)^\gamma \end{equation} \tag{ 15 }$$

and the parameterized matter power spectrum becomes

$$\begin{equation} \mathrm{P}(k,z) = P(k,z_{\rm ini}) D_{\rm ini}^2(z). \end{equation} \tag{ 16 }$$

Note that the complete wavenumber-dependence is contained in P(k, z_ini) while the growth factor D_ini(z), which now depends on γ, evolves with redshift only.

In our analysis, we choose an initial redshift of z_ini = 10 as a starting point for the parameterized growth which corresponds to an era well within matter domination when f(a) = 1 is a very good approximation. We modify the likelihood code presented in Section 4.1.1 so that the matter power spectrum at redshift z_ini is provided by CAMB and then evolves depending on the growth index γ according to Equations (15) and (16).

We note that this parameterization is in principle degenerate with a cosmological model containing neutrino mass as a free parameter; given a particular power spectrum constrained by the CMB anisotropies at very high redshift, variations in both neutrino mass and γ modify the low-redshift power spectrum. However, the SPT sample spans a broad redshift range, which should ultimately allow one to differentiate between the two effects.

5.8.2. Constraints on the Cosmic Growth Index

We fit for a spatially flat ΛCDM model with the additional degree of freedom γ (we will refer to this model as γ+ΛCDM). Using our SPT_CL sample with BBN and H₀ priors, we get results that are consistent with the prediction of GR, γ_GR = 0.55. However, the uncertainty on γ is large, and the 68% confidence interval is [ − 0.2, 0.7]. We tighten the constraints by including the CMB data set which serves as a high-redshift "anchor" of cosmic evolution. To isolate the constraining power clusters have on growth of structure, we choose not to use the constraints on γ that come from the Integrated Sachs–Wolfe (ISW) effect, which has an impact on the low l CMB temperature anisotropy. Regardless, we would expect the additional constraints on γ from the ISW to be less constraining than the cluster-based constraints presented here (see, e.g., Rapetti et al. 2010). We further use distance information from BAO and SNe Ia. As presented in Table 3, we find γ = 0.72  ±  0.24, which agrees with the prediction of GR. In Figure 5, we show the two-dimensional likelihood contours for γ and the most relevant cosmological parameters Ω_m and σ₈. The degeneracy between γ and Ω_m is weak. We see a strong degeneracy with σ₈, as would be expected given the dependence of σ₈ on growth history.

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Our constraints are weaker than those obtained from an X-ray cluster sample (Rapetti et al. 2013). Using 238 clusters from different X-ray catalogs together with CMB anisotropy data from the 5-year WMAP release, these authors obtain γ = 0.415 ± 0.127.

We also consider a γ+νCDM cosmological model, where we additionally allow a non-zero sum of the neutrino masses. There is only a mild degeneracy between γ and ∑m_ν, which does not significantly degrade our constraints on cosmic growth or neutrino masses (see upper panel of Figure 5 and Table 3). However, the best-fit value for γ shifts by ∼0.5σ closer to the GR value.

Finally, we consider a γ+wCDM cosmological model, where we fix ∑m_ν = 0 eV, and allow a varying dark energy equation-of-state parameter w. In doing so we can simultaneously account for possible departures from the standard cosmic growth history as well as departures from the expansion history as described by the ΛCDM model. As presented in Table 3, the results show consistency with the fiducial values γ_GR = 0.55 and w_Λ CDM = −1. Joint parameter constraints are shown in the bottom panel of Figure 6. This combined test confirms that the standard cosmological model accurately describes the evolution of the cosmic expansion and structure formation throughout a wide redshift and distance range.

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