The SPTpol Extended Cluster Survey

The survey was conducted using the SPTpol receiver that was installed on the 10 m SPT (Carlstrom et al. 2011) from 2012 to 2016. As detailed in Austermann et al. (2012), the receiver is composed of 768 feedhorn-coupled polarization-sensitive pixels split between the two channels with 588 pixels at 150 GHz and 180 pixels at 95 GHz; each pixel contains two transition-edge-sensor bolometers, resulting in 1536 detectors in total. The primary mirror is slightly underilluminated, resulting in beams well approximated by Gaussians with FWHM of 12 and 17 at 150 and 95 GHz, respectively.

The survey is composed of 10 separate ∼250–270 deg² "fields," each imaged to noise levels of ∼30–40 $\mu {\rm{K}}$ arcmin at 150 GHz; see Table 1. The fields were observed by scanning the telescope at fixed elevation back and forth in azimuth at ∼055 s⁻¹, stepping 10' in elevation, and then scanning in azimuth again. This process is repeated until the full field is covered in a complete "observation." Each field was observed >80 times, and 20 different dithered elevation starting points were used to provide uniform coverage in the final co-added maps.

Table 1.  Summary Information for the 10 Fields That Compose the 2770 deg² SPTpol Extended Cluster Survey

Name	R.A.	δ	Area	σ95	σ150	γfield
	(deg)	(deg)	(deg2)	($\mu {\rm{K}}$ arcmin)	($\mu {\rm{K}}$ arcmin)
ra23hdec-25	345.0	−25.0	276.0	61.3	30.5	0.84
ra23hdec-35	345.0	−35.0	250.2	59.4	36.6	0.80
ra1hdec-25	15.0	−25.0	275.2	80.4	39.2	0.69
ra1hdec-35	15.0	−35.0	251.8	61.5	36.6	0.79
ra3hdec-25	45.0	−25.0	272.9	54.6	28.6	0.90
ra3hdec-35	45.0	−35.0	248.8	43.8	25.3	1.04
ra5hdec-25	75.0	−25.0	277.0	57.0	31.4	0.85
ra5hdec-35	75.0	−35.0	250.3	54.8	31.6	0.88
ra11hdec-25	165.0	−25.0	274.3	77.6	40.0	0.68
ra13hdec-25	195.0	−25.0	270.8	50.7	30.0	0.90

Note. Listed are the field name, center, source-masked effective area, and noise levels at both 95 and 150 GHz, as well as the "field renormalization" factors discussed in Section 5.1.1. The survey contains an additional 122 deg² that are masked in the cluster analysis owing to the presence of millimeter-bright sources. Following Schaffer et al. (2011), the noise levels are measured from 4000 < ℓ < 5000 using a Gaussian beam approximation with FWHM of 1.7 (1.2) arcmin at 95 (150) GHz. The field renormalization factors are normalized with respect to the values from Reichardt et al. (2013) and de Haan et al. (2016) for the SPT-SZ survey.

Download table as:  ASCIITypeset image

The data processing and mapmaking procedures in this work follow closely those in previous SPT-SZ and SPTpol publications (see, e.g., Schaffer et al. 2011; Bleem et al. 2015b; Crites et al. 2015; Henning et al. 2018). First, for each observation, the time-ordered bolometer data (TOD) is corrected for electrical cross talk between detectors, and a small amount of bandwidth (∼1.4 Hz and harmonics) is notch-filtered to remove spurious signals from the pulse tube coolers that cool the optics and receiver cryostats. Next, using the cut criteria detailed in Crites et al. (2015), detectors with poor noise performance, poor responsivity to optical sources, and/or anomalous jumps in TOD are removed. As this work is focused on temperature-based science, we relax the requirement that both bolometers in a pixel polarization pair be active for an observation. Relative gains across the array are then normalized using a combination of regular observations of both an internal calibrator source and the galactic H ii region RCW 38. For the first field observed in the survey—ra23hdec−35⁸⁵ —the internal calibrator was inadvertently disabled during summer maintenance for ∼50% of the observations, and so these data were relatively calibrated only with RCW 38 observations.

The TOD is then processed on a per-azimuth scan basis by fitting and subtracting a seventh-order Legendre polynomial, applying an isotropic common mode filter that removes the mean of all detectors in a given frequency, high-passing the data at angular multipole ℓ = 300 and low-passing the data at ℓ = 20,000. Sources detected in preliminary mapmaking runs at ≥5σ (∼9–15 mJy depending on field depth) at 150 GHz and bright radio sources detected in the Australia Telescope 20-GHz Survey (AT20G; Murphy et al. 2010) at the edges of the field are masked with a 4'  radius during these filtering steps. The SPT-ECS  also contains a small number of sources with extended millimeter-wave emission (see Section 3.2) and more conservative masks around these sources are applied in the filtering steps.⁸⁶ Following filtering, the TOD for each detector is then weighted based on the inverse noise variance in the 1–3 Hz signal band and binned into 025 pixels in maps in the Sanson-Flamsteed projection (Calabretta & Greisen 2002) using reconstructed telescope pointing. We have extended the characterization of the SPT pointing model to incorporate position information from all millimeter-wave-bright AT20G sources (typically 45–60 sources per field detected at S/N > 10 were used, compared to the two to three bright sources that proved sufficient in previous SPT analyses) to better constrain boom flexure and other mechanical aspects of the telescope at the elevations of these fields. With this extension we achieve reconstructed pointing performance of ∼3''–4'' rms when comparing SPT source locations to AT20G positions.

The single-observation maps for each field are then characterized based on both noise properties and coverage; maps with significant outliers from the median of these distributions are flagged and excluded from the co-addition step. The remaining maps are combined in a weighted sum based on their total pixel weights from the previous binning step; final maps consist of 78–150 observations per field.

The SPT-ECS  fields were taken at significantly higher levels of atmospheric loading compared to other SPTpol survey data.⁸⁷ We found it necessary to augment our standard absolute calibration process (see, e.g., Staniszewski et al. 2009) with two additional steps that make use of the 143 GHz full- and half-mission temperature maps from the 2015 Planck data release (Planck Collaboration et al. 2015, 2016a). The first step follows a similar method to the absolute temperature calibration conducted in previous SPT power spectrum analyses (e.g., Henning et al. 2018; Hou et al. 2018). We derive normalization factors to rescale each co-added map by first convolving the Planck maps with the SPT beams and transfer functions (the latter resulting from the TOD filtering process described above) and the SPT maps with the Planck beam and window function. Then, masking bright point sources in the field, we set the normalization as the ratio from 900 ≤ ℓ ≤ 1600 of the cross spectrum of the Planck half-mission maps to the cross spectrum of the Planck full-mission map with the SPT maps. The 95 GHz data required an additional calibration step, as we found—especially in the fields centered at δ = −25°—that the responsivity of the detectors decreased with increasing air mass. This trend is well represented as a linear decline in sensitivity as a function of decl., and we used the Planck data to fit for and correct this variation across the fields.

The thermal SZ signal is produced by the inverse Compton scattering of cosmic microwave background (CMB) photons off high-energy electrons, such as those that reside in the intracluster medium of galaxy clusters. This produces a spectral distortion of the observed CMB temperature at the location x of clusters given by the line-of-sight integral (Sunyaev & Zel'dovich 1972)

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Delta }}T({\boldsymbol{x}},\nu ) & = & {T}_{\mathrm{CMB}}\ {f}_{\mathrm{SZ}}(\nu )\displaystyle \int {n}_{{\rm{e}}}({\boldsymbol{r}})\displaystyle \frac{{k}_{{\rm{B}}}{T}_{{\rm{e}}}({\boldsymbol{r}})}{{m}_{{\rm{e}}}{c}^{2}}{\sigma }_{{\rm{T}}}{dl}\\ & \equiv & {T}_{\mathrm{CMB}}\ {f}_{\mathrm{SZ}}(\nu )\ {y}_{\mathrm{SZ}}({\boldsymbol{x}}),\end{array}\end{eqnarray} \tag{ 1 }$$

where ${T}_{\mathrm{CMB}}=2.7260\pm 0.0013$ K is the mean CMB temperature (Fixsen 2009), ${f}_{\mathrm{SZ}}(\nu )$ is the frequency (ν) dependence of the thermal SZ effect (Sunyaev & Zel'dovich 1980), n_e is the electron density, T_e is the electron temperature, k_B is the Boltzmann constant, m_ec² is the electron rest-mass energy, σ_T is the Thomson cross section, and ${y}_{\mathrm{SZ}}$ is the Compton y-parameter. This effect results in a decrement at the two channels measured by the SPTpol receiver; for a nonrelativistic thermal SZ spectrum the effective band centers are 95.9 and 148.5 GHz.⁸⁸

To identify candidate galaxy clusters, we use a spatial-spectral filter designed to optimally extract thermal SZ cluster signals (Melin et al. 2006). This "matched-filter" approach has been widely used both in previous SPT publications and in analyses by other experiments (see, e.g., Planck Collaboration et al. 2016b; Hilton et al. 2018). We model the cluster profile as a projected spherical β-model with β fixed to 1 (Cavaliere & Fusco-Femiano 1976):

$$\begin{eqnarray}&&{\rm{\Delta }}T={\rm{\Delta }}{T}_{0}{(1+{\theta }^{2}/{\theta }_{{\rm{c}}}^{2})}^{-(3\beta -1)/2},\end{eqnarray} \tag{ 2 }$$

where the normalization ΔT₀ is a free parameter and the core radius, θ_c, is allowed to vary in 12 equally spaced steps from 025 to 3'.

To prevent spurious decrements from the filtering process, we mask regions around bright emissive sources before applying the matched filters to the maps. These sources are detected in the 150 GHz data using a matched filter designed to optimize the S/N of point sources. Masks of 4'  radius are placed over sources detected at >5σ, and candidates detected within 8'  of these sources are excluded from the final cluster lists. Additionally, as referenced above in Section 2.2, there are three extended sources in these fields (NGC 55, NGC 253, and NGC 7293; Dreyer 1888) and one exceptionally bright quasar (QSO B0521–365; e.g., Planck Collaboration et al. 2018) that require additional masking. Masks of radius 033 are used for the NGC sources and radius 025 for the quasar. Regions around these sources are also inspected following the cluster filtering process, and a small number of spurious candidates are rejected. In total 122 deg² are masked, 4.5% of the full survey area.

Cluster candidates are identified as peaks in the matched-filtered maps. For each location we define our SZ observable, ξ, as the maximum detection significance over the 12 filter scales. As in prior SPT analyses, there is a small decl. dependence in the noise owing to atmospheric loading, detector responsivity, and coverage changes across each field. To capture this in the ξ estimates, each filtered map is split into 90'  strips in decl. and—as in Huang et al. (2019)—noise in each strip is measured by measuring the standard deviation of a Gaussian fit to unmasked pixels. In this work, all candidates ξ ≥ 5 are reported, and for 4 < ξ < 5, where our follow-up is currently highly spatially incomplete, we also report confirmed systems in the DES common region (see Section 4).

We make use of simulations to estimate the contamination of our catalogs by spurious detections and to renormalize the measured SZ detection significances to account for the varying field depths (see Section 5.1.1). Simulations were previously used to this effect in, e.g., Reichardt et al. (2013), de Haan et al. (2016), and Huang et al. (2019); we briefly overview the process here and describe some small changes to the process from the SPT-SZ simulations. For more details on these simulations see Huang et al. (2019).

For each field we construct sets of simulated millimeter-wave skies consisting of the following:

• 1.  
  Primary lensed CMB (Keisler et al. 2011).
• 2.  
  Signals from Poisson and clustered dusty sources that we approximate as Gaussian random fields with amplitude and spectral indices matching George et al. (2015).
• 3.  
  Discrete radio sources below the masking threshold with the source population drawn from the model of De Zotti et al. (2005) and with spectral indices drawn from the results of George et al. (2015) and Mocanu et al. (2013).
• 4.  
  Thermal SZ constructed using a halo light cone from the Outer Rim (Habib et al. 2016; Heitmann et al. 2019) simulation with thermal SZ profiles painted for each halo with ${M}_{200c}\gt {10}^{13}$ following the methodology of Flender et al. (2016) and using the pressure profiles of Battaglia et al. (2012). The thermal SZ power is consistent with the results of George et al. (2015). The SZ signal is omitted in the false detection simulations.
• 5.  
  Atmospheric and instrumental noise from jackknife noise maps constructed via co-adding field observations where half of the observations were randomly multiplied by −1.

Each sky realization is convolved with the SPT beam and transfer function. As in Huang et al. (2019), there are two significant changes compared to simulations used for SPT-SZ cluster studies. First, we use discrete radio sources, as opposed to Gaussian random fields, to account for radio contamination. This change was found to be important for properly capturing the false detection rates of the deeper SPTpol 100d and 500d cluster surveys but has negligible impact at the noise levels of the SPT-ECS  and SPT-SZ surveys. We adopt it for consistency here. Second, we use the measured SPT beams, as opposed to Gaussian approximations, which enables more consistent scalings between the SPT-SZ and SPTpol experiments.

To estimate the number of spurious detections in each field, we run the cluster detection algorithm on the simulated SZ-free maps. As in de Haan et al. (2016), to reduce shot noise in our estimates from our finite number of simulations, we model the false detection rate with the function

$$\begin{eqnarray}&&{N}_{\mathrm{false}}(\gt \xi )={\alpha }_{\mathrm{field}}{e}^{-{\beta }_{\mathrm{field}}\left(\xi -5\right)}\times \mathrm{field}\ \mathrm{area}.\end{eqnarray} \tag{ 3 }$$

All of the fields are well approximated by α ∼ 0.008 and β = 4.3; as each field is approximately 260 deg², this results in ∼2 false detections per field expected above ξ = 5 and 17–18 above ξ = 4.5.

As detailed in de Haan et al. (2016), the field depth rescaling factors, which track changes in "unbiased significance" as a function of mass for the varying field depths, are determined by measuring the S/N of simulated clusters at their known locations and optimal filter scales from the simulated maps (see also Section 5.1.1). We list the field depth rescaling factors γ_field in Table 1. Following previous SPT publications, the absolute normalization is set to correspond to the unit scaling adopted in Reichardt et al. (2013). While in principle the field scaling simulations should be sufficient to properly scale the SPT-ECS  field depths relative to SPT-SZ, the extra calibration steps required for the SPT-ECS  survey make this challenging. To capture any residual uncertainty in this process, we introduce a new parameter, γ_ECS, which rescales all field scalings in the SPT-ECS  survey ${\gamma }_{\mathrm{SPT}-\mathrm{ECS},i}={\gamma }_{\mathrm{ECS}}\times {\gamma }_{\mathrm{field},i}$. With this parameterization, γ_ECS = 1 means that our simulations capture the entirety of the relative difference in effective depth between SPT-SZ and SPT-ECS. We empirically calibrate γ_ECS in Sections 5.1.2 and 6.1.

Galaxy clusters contain an overdensity of galaxies relative to the field, and galaxies emit radiation at millimeter wavelengths. Since the thermal SZ signal from the cluster gas is a decrement in the frequency bands used in this work, any positive emission above the background will act as a negative bias to the SZ signature. We can classify the potential bias from cluster galaxy emission into two regimes, one in which the integrated emission from many cluster members produces an average bias to all clusters in a given mass and redshift range, with little variation from cluster to cluster, and one in which a single bright galaxy (or a very small number of bright galaxies) imparts a significant bias to a random subsample of clusters. We can also separate the contributions to this effect from the two primary classes of millimeter-wave-emissive sources: active galactic nuclei producing synchrotron emission ("radio sources") and star-forming galaxies producing thermal dust emission ("dusty sources").

The contribution to the second type of bias from dusty sources is expected to be negligible, because the dusty source population falls off steeply at high flux (e.g., Mocanu et al. 2013), so that the areal density of dusty sources bright enough to fill in a cluster decrement at a level important for this work is very low. This statement is for the field galaxy population, so if galaxies in clusters were more likely than field galaxies to be dusty and star-forming, the bright population could still be an issue. In fact, the opposite is expected to be true, i.e., compared to the field population, galaxies in clusters are less likely to be dusty and star-forming, at least at z < 1 (e.g., Bai et al. 2007; Brodwin et al. 2013; Alberts et al. 2016). In Vanderlinde et al. (2010), it was argued that the other regime of bias from dusty sources is also negligible for clusters more massive than ∼2 × 10¹⁴ M_⊙, which includes all the clusters in this sample (see also, e.g., Soergel et al. 2017 for an analysis of a sample of low-z optically selected clusters, and Erler et al. 2018; Melin et al. 2018 for explorations of the Planck sample).

To assess the potential contamination from radio sources, we make use of the publicly available maps from the 1.4 GHz National Radio Astronomy Observatory Very Large Array (VLA) Sky Survey (NVSS; Condon et al. 1998).⁸⁹ NVSS covers the full sky north of decl. −40° and thus has nearly 100% overlap with the survey fields in this work. The data for the NVSS were taken between 1993 and 1997, so source variability will limit the fidelity of the estimate of contamination to any individual cluster, but we can make some statements about the average or median contamination across the catalog and the fraction of clusters expected to be strongly affected by radio source contamination.

For each of our survey fields, we download all NVSS postage-stamp maps (each 4° × 4°) that have any overlap with that field and reproject them onto the same pixel grid as used in our cluster analysis. We then make beam- and transfer-function-matched NVSS maps for each of the SPTpol observing frequencies by convolving the NVSS maps with a kernel defined by the Fourier-space ratio of the SPTpol beam and transfer function at that frequency and the effective NVSS beam (a 45'' FWHM Gaussian). We scale the intensity in these maps from 1.4 GHz to SPTpol frequencies assuming a spectral index of −0.7 (roughly the mean value found for radio sources in clusters by Coble et al. 2003), and we convert the result to CMB fluctuation temperature.

We then combine the SPTpol-matched NVSS maps in our two bands using the same band weights as used in the cluster-finding process (Section 3) and filter the output with the same β-model-matched filters as used in the cluster-finding process. For each cluster candidate in the catalog, we take the combined NVSS map filtered with the same β-model profile as the cluster candidate, and we record the value of the combined, filtered NVSS map at the candidate location. We divide that value by the same noise value used in the denominator of the ξ value for the cluster candidate, and we record that value as our best estimate of the contamination to the ξ value of that cluster candidate from radio sources. Since the NVSS maps contain all the radio flux at 1.4 GHz (not just the sources bright enough to be included in the NVSS catalog), this test accounts for both regimes of bias discussed above.

The median contamination calculated in this way is ${\rm{\Delta }}{\xi }_{\mathrm{med}}=-0.05$, or 1% of the threshold value for inclusion in the catalog of ξ = 5. Of the 266 candidates in the catalog, 13 (∼5%) have a predicted contamination of greater than 10% of their measured SZ flux, and 7 (∼2.6%) have a predicted contamination of greater than 20%. One cluster candidate, SPT-CL J2357-3446, has an anomalously large predicted bias of Δξ = −11.1. This candidate is almost certainly the low-redshift (z = 0.048) cluster A3068 (separation 01), and it is within 06 of the NVSS source NVSS J235700−344531, which has a catalog 1.4 GHz flux of 1.28 Jy. This NVSS source is a cross-identification of PKS 2354−35, which lies at a redshift consistent with being a member of A3068 (z = 0.049) and is identified as the central galaxy of this cluster by many authors (e.g., Schwartz et al. 1991). Given the relative redshift dependence of the thermal SZ signature of clusters and the flux density of member emission, it is not surprising that the highest level of radio source contamination occurs in one of the lowest-redshift clusters in the sample. It is somewhat surprising, though, that a cluster with a predicted radio source contamination of Δξ > 10 would be detected at ξ = 5.5, as this one is in our catalog. The apparent answer to this puzzle is source variability. More recent observations of this source with the Australia Compact Telescope Array (ATCA) at 5 GHz (Burgess & Hunstead 2006) resulted in a measured flux density of 99 mJy, which would imply a spectral index of <−2.0 if naively combined with the NVSS measurement at 1.4 GHz. We conclude that during our observations the 150 GHz flux of this source was likely <10 mJy (as implied by the ATCA measurement and a spectral index of −0.7) rather than the ∼50 mJy implied by the NVSS measurement. Finally, we also note that all previous SPT cluster cosmology results have cut clusters below z = 0.3 or 0.25, so this cluster would not normally be included in an SPT cosmology analysis.

These contamination numbers will be diluted somewhat by any false detections. However, removing the 22 unconfirmed candidates at ξ > 5 has a negligible effect. If we extend the sample to all confirmed systems at ξ > 4 (for a total of 448 clusters), we find a similar median contamination (Δξ_med = −0.050) and fraction of systems above a given level of contamination: 17 (∼4%) and 8 (∼2%) above 10% and 20% contamination, respectively. We flag candidates with >10% potential contamination of their measured SZ signal in Tables 8 and 9.

In addition to the potential bias to our sample from the millimeter-wave emission from cluster members, there is a potential bias from the avoidance of millimeter-wave-bright sources in our cluster finding. As discussed in Section 3.2, we discard any cluster detection within 8' of a source detected at 5σ (∼9–15 mJy, field dependent) at 150 GHz. If there were a strong physical association between galaxy clusters and such sources, our measured cluster abundance would be biased low.

The majority of sources with 150 GHz flux density above 9 mJy are flat-spectrum quasars (see, e.g., Mocanu et al. 2013; Gralla et al. 2019), and, based on studies of radio galaxies from lower-frequency surveys (e.g., Lin et al. 2009; Gralla et al. 2011, 2014; Gupta et al. 2017), there is not expected to be a significant SZ selection bias from these sources. However, we can perform several checks on the effects of masking with the data in hand. First, as in B15, we perform a secondary cluster search, this time only masking emissive sources detected at >100 mJy at 150 GHz. Each candidate from this run was visually inspected, and, as expected, this candidate list was dominated by filtering artifacts; no new clusters were identified in this secondary run.

We also check for any statistical association between the flux-limited DES redMaPPer optically selected cluster catalog and associated random locations (discussed in more detail below in Section 4.1.1 and in Rykoff et al. 2016) and SPT-selected emissive sources. To increase the sensitivity of this test, we also include sources from the SPT-SZ survey, which had a 5σ source threshold of lower flux (∼6 mJy; W. Everett et al. 2020, in preparation). We first measure the probability of optical clusters and random locations to be within the 8'  source masks and find marginal differences between the two. Adopting a 3'  radius to reduce the noise from chance associations and further restricting the cluster sample to z > 0.25 where we expect the SPT selection to be well understood, we find an excess probability over random of ≲1% for clusters to fall in the source-masked regions (see Table 2). While the purity of the flux-limited sample is expected to decrease at high redshift (thus limiting our ability to test for trends with redshift), we note that we find no significant difference in the fraction of clusters in masked regions between the full sample and two subsamples constructed by splitting the optical sample at its median redshift of z = 0.755.

Table 2.  Optical Clusters near Millimeter-wave-bright Sources

λ Range	Nclusters	% in Masked Region	${N}_{\mathrm{clusters}}^{\mathrm{SPT} \mbox{-} \mathrm{SZ}}$	%
Randoms	2.1e6	0.9	1.2e6	1.05
20–30	2.3e4	1.11	1.3e4	1.4
30–50	9.3e3	0.95	5.4e3	1.1
50–80	1.9e3	1.22	1.1e3	1.65
>80	3.5e2	1.13	2.0e2	1.5

Note. Percentage of DES redMaPPer clusters at z > 0.25 that fall within 3'  of bright emissive sources; N_clusters corresponds to the total number of clusters in a given richness bin within the SPT-SZ+SPT-ECS  (left) or SPT-SZ only (right) footprint. The top row provides statistics for random sources. Less than 1% of clusters over random fall in the masked areas.

Download table as:  ASCIITypeset image

4.1.1. The Dark Energy Survey and redMaPPer

4.1.2. The Parallel Imager for Southern Cosmology Observations

4.1.3. Spitzer/IRAC and Magellan/Fourstar

4.1.4. Pan-STARRS1

4.1.5. WISE

4.1.6. Literature Search

We adopt two different techniques for confirming candidates in clusters: a probabilistic matching to RM clusters in the common overlap region and, for candidates outside the volume probed by RM cluster finding on DES data, the identification of significant overdensities of red sequence or 1.6 μm rest-frame galaxies at the locations of cluster candidates using the techniques described in B15. We show the distribution of the origins of SPT-ECS  cluster redshifts in Figure 2.

Zoom In Zoom Out Reset image size

4.2.1. Confirmation with redMaPPer in Scanning Mode

The RM catalog makes strict cuts on sky coverage and photometric depth to ensure a well-understood optical selection function. In the case of matching to an SZ-selected sample, we can somewhat relax these criteria to also enable targeted searches for red sequence galaxy counterparts in regions excluded by these cuts. We have run the RM algorithm in "scanning mode" centered on the SPT location, where the likelihood of a red sequence overdensity in apertures of 500 kpc radius is computed as a function of redshift from z = 0.1 to 0.95 in steps of δ_z = 0.005. At each redshift the optical richness is computed at both the SZ location and the most likely optical center; for systems with significant red sequence overdensities the richness and redshift are refined at the highest likelihood redshift using the standard RM radius/richness scaling. Richnesses are recorded for each location where λ ≥ 5. We repeat this scanning procedure for 100 mock SZ samples (at over 100,000 sky locations) to compute the probability over random of finding a cluster of richness λ at an SZ location. We report as "confirmed" clusters for which the probability of random association is less than 5%, which corresponds to λ > 19.3. As this probability distribution is a continuum (with no clear breaks), this choice is somewhat arbitrary; setting this threshold at 0.05 leads to an expectation of <2 false associations in the RM-confirmed sample. In Figure 3 we plot the distribution of matched clusters against the probability of random associations for SPT-SZ (ξ > 4.5) and SPT-ECS  clusters (ξ > 4).

Zoom In Zoom Out Reset image size

For cluster candidates in the common region not confirmed via the RM scanning-mode process we make use of both DES and WISE imaging and photometric catalogs at the cluster locations and, where available, pointed follow-up imaging as described in the next subsection. The confirmation of these clusters follows a similar process to that described below.

4.2.2. Cluster Confirmation from Other Imaging Data Sets

Here we describe the techniques used for confirming cluster candidates not confirmed via the RM algorithm or literature search. We obtained imaging for ∼100 candidates outside of the volume searched by RM, as well as some imaging redundant to the DES imaging (in terms of confirmation), as part of our strong-lensing search program that we use here for redshift comparisons. In total, 173 candidates were imaged with Magellan/PISCO (about 2/3 in common with RM; see Figure 4), 19 with Magellan/Fourstar, and 7 with Spitzer (note that the NIR imaging overlaps areas with optical imaging). Ten candidates are located in the DES footprint but are either at high redshift or missing photometry in filters required for RM, and 22 candidates only fall in the Pan-STARRS footprint.

Zoom In Zoom Out Reset image size

To conduct our targeted search for red sequence galaxies in these areas, we first calibrate our synthetic model for the colors and magnitudes of red sequence galaxies, generated with the GALAXEV package (Bruzual & Charlot 2003) by assuming a passively evolving stellar population with single formation burst at z = 3, to match the relevant survey photometry using samples of known clusters with spectroscopic redshifts. For PISCO, the data set that we most use to confirm clusters outside of DES, we use 58 SPT-SZ galaxy clusters with spectroscopic redshifts that were imaged as part of our broader SPT characterization program. In Figure 4 we plot in red the measured PISCO redshifts versus those from the training sample, as well as additional SPT-ECS  clusters with spectroscopic redshifts reported in the literature. The typical redshift precision is σ_z/(1 + z) ∼ 0.015, with uncertainties increasing toward higher redshifts. We also plot in black the PISCO redshifts compared to those from the DES RM catalog for 318 systems in SPT-SZ and SPT-ECS  and find generally good agreement between the two, though the comparison suggests that the redshifts estimated from PISCO may tend be underestimated at the highest redshifts. More spectroscopic data on high-redshift clusters from ongoing SPT programs will help further validate/improve the PISCO redshift calibration for such systems. Given the excellent redshift precision of the RM algorithm, we adopt RM scanning-mode redshifts by default when clusters are confirmed by both methods. We repeat a similar process with DES photometry, finding σ_z/(1 + z) ∼ 0.015, and with 35 spectroscopic clusters (as identified in the SPT-ECS  literature search) at 0.108 < z < 0.72 in the Pan-STARRS1 footprint, finding σ_z/(1 + z) ∼ 0.03 in these shallower data.

We search the optical/NIR imaging for an excess of red sequence galaxies (or alternatively 1.6 μm rest-frame galaxies in the case of Spitzer and WISE confirmations) in the vicinity (2'–3') of the SPT cluster candidates. We call a cluster "confirmed" when significant excesses of these galaxies over background are identified (see, e.g., B15 for more details on the confirmation procedure). In Song et al. (2012) we estimated that <4% of cluster candidates identified via this procedure would be false associations, and for clusters in common between the PISCO and DES imaging, we can cross-check our assigned confirmations against the statistical process described in Section 4.1.1. We note that this is of course a lower limit to the false association rate, as the DES data are also of finite redshift reach. In this comparison we find that ∼1% (3/318) of candidates with RM cluster matches that were also targeted with PISCO were assigned a different cluster counterpart when using the PISCO data. In two circumstances the PISCO data were insufficiently deep to correctly confirm the higher-redshift (z ∼ 0.9) clusters, while the remaining system was a superposition of two rich clusters (λ = 75 and λ = 65) for which the targeted RM algorithm selected the lower-redshift system as the richer cluster and the PISCO data the higher.

For confirming higher-redshift systems without targeted Spitzer or Magellan/Fourstar data we combine data from "unWISE" with optical source catalogs. Following Gonzalez et al. (2019), we adopt a 15  matching radius to associate WISE sources with optical galaxies and exclude sources with i < 21.3 and W1–W2 < 0.2, as these cuts were found to remove low-redshift (z < 0.8) galaxies. Similar to the analysis of clusters with Spitzer imaging, we search for a local excess of galaxies at 1.6 μm rest frame in the vicinity of the SPT cluster candidates. We validated this search process on clusters from the SPT-SZ sample (B15; Khullar et al. 2019) with spectroscopic redshifts z > 0.85, finding that we were able to robustly confirm 20/23 of these systems. From this spectroscopic sample we were able to quantify the redshift uncertainty in our WISE measurements as σ_z/(1 + z) ∼ 0.1. Improving this redshift precision via more sophisticated catalog cuts and photometric analysis of the WISE data is work in progress.

4.2.3. 2dFLenS

The cluster catalog extracted from SPT-ECS  should be statistically consistent with the catalog extracted from the SPT-SZ survey once the different survey properties such as depth are accounted for. To test this, we use a cluster number count (NC) analysis to calibrate the parameters of the ξ–mass scaling relation assuming a fixed cosmology and compare the results with those obtained for SPT-SZ.

5.1.1. The SZ ξ–Mass Relation

To connect the observed SZ significance, ξ, to cluster mass, we adopt an observable–mass scaling relation of the form

$$\begin{eqnarray}&&\langle \mathrm{ln}\zeta \rangle =\mathrm{ln}\left[{A}_{\mathrm{SZ}}{\left(\displaystyle \frac{{M}_{500c}}{3\times {10}^{14}{M}_{\odot }{h}^{-1}}\right)}^{{B}_{\mathrm{SZ}}}{\left(\displaystyle \frac{H(z)}{H(0.6)}\right)}^{{C}_{\mathrm{SZ}}}\right],\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&P(\mathrm{ln}\zeta | M,z)={ \mathcal N }\left[\langle \mathrm{ln}\zeta \rangle (M,z),{\sigma }_{\mathrm{ln}\zeta }^{2}\right],\end{eqnarray} \tag{ 5 }$$

where A_SZ is the normalization, B_SZ the slope, C_SZ the redshift evolution, ${\sigma }_{\mathrm{ln}\zeta }$ the lognormal scatter on ζ, and H(z) the Hubble parameter. The variable ζ represents the "unbiased" significance that accounts for the maximization of ξ over position and filter scales during cluster detection:

$$\begin{eqnarray}&&P(\xi | \zeta )={ \mathcal N }(\sqrt{{\zeta }^{2}+3},1)\end{eqnarray} \tag{ 6 }$$

for ζ > 2 (Vanderlinde et al. 2010). As in previous SPT publications, we rescale A_SZ on a field-by-field basis to account for the variable depth of the survey: ${A}_{\mathrm{SZ},\mathrm{field}}={\gamma }_{\mathrm{field}}\times {A}_{\mathrm{SZ}}$ (e.g., Reichardt et al. 2013; de Haan et al. 2016). These field renormalization factors, γ_field, are computed using the simulations described in Section 3.4 and are reported in Table 1 on the same reference scale as the analogous factors for the SPT-SZ survey.

The different fields of the SPT-SZ and SPT-ECS  surveys have a small amount of overlap at the field boundaries. We correct the field areas such that the total effective survey area corresponds to the unique sky area that is surveyed. These corrections are between 0.03% and 1.9%. SZ detections in the field overlap regions are matched by keeping the candidate with the larger detection significance ξ. Note that this approach is different from the one adopted in de Haan et al. (2016) and Bocquet et al. (2019), who double-counted clusters in the field overlap regions in SPT-SZ in their NC analyses. The exact treatment of the field boundaries has negligible impact on our results; for example, the change in our total predicted cluster counts due to not correcting for the field overlap area is much smaller than the recovered uncertainty.

5.1.2. γ_ECS Constraints from the Cluster Abundance

We model the cluster sample as independent Poisson draws from the halo mass function. The likelihood function for the vector of cosmological and scaling relation parameters ${\boldsymbol{p}}$ is

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}{ \mathcal L }({\boldsymbol{p}}) & = & \displaystyle \sum _{i}\mathrm{ln}\displaystyle \frac{{dN}({\xi }_{i},{z}_{i}| {\boldsymbol{p}})}{d\xi {dz}}\\ & & -\,{\displaystyle \int }_{{z}_{\mathrm{cut}}}^{\infty }{dz}{\displaystyle \int }_{{\xi }_{\mathrm{cut}}}^{\infty }d\xi \displaystyle \frac{{dN}(\xi ,z| {\boldsymbol{p}})}{d\xi {dz}}.\end{array}\end{eqnarray} \tag{ 7 }$$

The sum runs over all clusters i in our sample, and

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{dN}(\xi ,z| \,{\boldsymbol{p}})}{d\xi {dz}} & = & \iint {dM}\,d\zeta \,\left[P(\xi | \zeta )P(\zeta | M,z,{\boldsymbol{p}})\right.\\ & & \left.\displaystyle \frac{{dN}(M,z| {\boldsymbol{p}})}{{dMdz}}{\rm{\Omega }}(z,{\boldsymbol{p}})\right],\end{array}\end{eqnarray} \tag{ 8 }$$

where ${\rm{\Omega }}(z,{\boldsymbol{p}})$ is the survey volume and dN/dM dz is the halo mass function given by Tinker et al. (2008). The second line in Equation (7) corresponds to the total number of clusters in the survey.

We analyze the SPT-ECS  NC assuming our fixed ΛCDM cosmology. By construction of our scaling relation model, the amplitude A_SZ and the correction factor ${\gamma }_{\mathrm{ECS}}$ (introduced in Section 3.4) are fully degenerate. The constraints on the SZ scaling relation parameters B_SZ, C_SZ, and the scatter ${\sigma }_{\mathrm{ln}\zeta }$ from SPT-SZ and SPT-ECS are consistent at the ≪1σ level. To test the consistency of the relative scaling between the two surveys, we analyze the joint NC from SPT-SZ and SPT-ECS. In this analysis, any residual in the relative calibration between the two surveys is absorbed by γ_ECS. We recover

$$\begin{eqnarray}&&{\gamma }_{\mathrm{ECS}}=1.124\pm 0.045.\end{eqnarray} \tag{ 9 }$$

We provide and discuss an alternate calibration of γ_ECS in Section 6.1.

The ξ–mass relation defined above in Equations (4)–(6) allows us to compute mass estimates for all sample clusters. We adopt A_SZ = 4.08, B_SZ = 1.65, C_SZ = 0.64, and ${\sigma }_{\mathrm{ln}\zeta }=0.20$. These mean scaling relation parameters were determined in Bocquet et al. (2019) for our fixed reference flat ${\rm{\Lambda }}\mathrm{CDM}$ cosmology and using the SPT-SZ sample at ξ > 5 and z > 0.25. As discussed in the previous section, the SZ scaling relation parameters barely shift between an NC analysis using SPT-SZ clusters alone and one using SPT-SZ and SPT-ECS clusters, and we thus use the SPT-SZ-only numbers presented in Bocquet et al. (2019) for consistency with their mass estimates.

There is naturally significant overlap between the Planck and SPT-ECS  cluster samples, as both identify massive clusters by the SZ effect. Here we focus our comparison on the reported masses and redshifts, two quantities critical for cosmological analyses. To directly compare the properties of the two samples for clusters in common, we first associate the catalog from Planck Collaboration et al. (2016b) with the SPT-ECS  catalog using a 4' matching radius and find that 82 SPT candidates (81 confirmed clusters) detected at ξ > 4 match Planck systems within this radius.

Overall we find good agreement between the redshifts for matched clusters, with three outliers for which the estimated redshifts reported in the Planck catalog and this work differ by δ_z > 0.1. These three systems each have redshifts in this work from the RM algorithm. Two of the systems (J0046−3911 and J0516−2236) have photometric redshifts reported in the literature, while the third system, J0348−2144 (PSZ2 G215.19−49.65, separated from the SPT position by 058), was associated in the Planck catalog with ACO 3168 (RXC J0347.4−2149), for which a spectroscopic redshift of z = 0.2399 was derived from five cluster members in Chon & Böhringer (2012). This system is significantly offset (86, 89) from the SPT and Planck detections, respectively. We instead associate this cluster candidate with a closer (12, 17) and richer (λ = 186 vs. 10) system at z = 0.347 ± 0.008.

Beyond the direct redshift comparisons, we also provide here redshifts for 13 PSZ2 systems from Planck Collaboration et al. (2016b); 11 of these clusters were not confirmed by the Planck Collaboration. These systems are listed in Table 3. Two of these clusters have previously reported redshifts in Maturi et al. (2019), and a third we associate with ACO S 1048 (Abell et al. 1989). We find good agreement with the Maturi et al. (2019) redshift estimate for PSZ2 G011.92−63.53 but find δ_z > 0.2 for PSZ2 G011.36−72.93.

We can also compare the reported SZ mass estimates for each of these samples. In Figure 7 we show the Planck and SPT clusters in the SPT-ECS  and SPT-SZ footprints. In the top panel, plotted as small open symbols, are clusters that are only found in one of the catalogs, while the filled diamonds represent clusters that are in both SPT and Planck. The 13 Planck clusters for which we report a redshift from SPT-ECS  in Table 3 are plotted as large open diamonds. Including clusters from the SPT-SZ region brings the joint SPT-Planck sample to a total of 150 clusters with mass estimates for which the reported redshifts differ by δ_z < 0.1,⁹⁷ and 88 such systems at z > 0.25, where SPT masses are expected to be unbiased. In the bottom panel we plot the ratio of SPT to Planck mass as a function of redshift and, in the inset, as a function of the SPT mass estimate. Qualitatively, we notice a trend with redshift where at z < 0.25 the ratio of the SPT to Planck masses is significantly higher than at higher redshifts (1.44${}_{-0.14}^{+0.05}$ vs. ${1.1}_{-0.03}^{+0.055}$). We note that mass estimates for SPT clusters at z < 0.25 are more uncertain—though not expected to be biased high—given increased noise contributions from both the primary CMB and atmosphere, as well as the removal of large-scale sky signal by the map filtering.

Zoom In Zoom Out Reset image size

Comparisons to the Planck SZ masses are often reported in terms of a mass bias, 1 − b, where ${M}_{{Planck}}=(1-b){M}_{\mathrm{True}}$. For purposes of comparison here we treat the SPT masses as the "true" cluster masses and both compute the median mass bias and check for a mass-dependent trend. The latter is achieved via making use of the Bayesian linear regression routines provided by Kelly (2007) and fitting for the power-law index, α:

$$\begin{eqnarray}&&{M}_{500c\mathrm{Planck}}\propto {M}_{500c\,\mathrm{SPT}}^{\alpha }.\end{eqnarray} \tag{ 10 }$$

Here we consider only the statistical errors in the SPT and Planck masses, as we are directly comparing properties of the same clusters.

As discussed in Battaglia et al. (2016), one must take care in such comparisons, as they can be impacted at the level of 3%–15% in (1 − b) by the presence of Eddington bias in the reported Planck masses.⁹⁸ We follow Battaglia et al. (2016) and recompute the SPT masses not accounting for this bias. Restricting to a subset of 69 clusters where the absolute difference between the Planck and SPT S/N was less than 2 (so the bias would be somewhat comparable), we find a median (1 − b)_Eddington = ${0.77}_{-0.025}^{+0.02}$ and α = 1.03 ± 0.14 for the full sample and (1 − b)_Eddington = ${0.80}_{-0.01}^{+0.09}$ and α = 1.3 ± 0.27 for 15 such clusters with 0.25 < z < 0.35 and uncorrected ${M}_{500c\mathrm{SPT}}\gt 5.5\times {10}^{14}$ (where both samples are more complete). To aid comparison with previous studies in the literature, we also compute these values for the entire matched sample with debiased SPT masses, finding (1 − b) = 0.83 ± 0.02 for the full matched sample and (1 − b) = ${0.91}_{-0.05}^{+0.01}$ and α = 0.75 ± 0.06 at z > 0.25.

Comparisons between (Eddington-biased) Planck and (debiased) SPT mass estimates were previously reported by Planck Collaboration et al. (2016b) and Hilton et al. (2018) for the SPT-SZ and PSZ2 samples. Planck Collaboration et al. (2016b) found the SPT-reported masses to be on average 20% higher than the Planck masses—in good agreement with the results derived above with the larger SPT-SZ and SPT-ECS  sample. Hilton et al. (2018) explored the relation between SPT-SZ, ACT, and PSZ2 masses, finding the mean mass ratio of ACT to SPT clusters to be 1.00 ± 0.04 for 18 clusters in common between the samples; the SPT-ECS  sample provides no additional overlapping systems between ACT and SPT to further this comparison. Hilton et al. (2018) additionally noted a mass-dependent trend between the Planck and SPT/ACT masses, finding for the ACT comparison α = 0.55 ± 0.18, a result ∼1σ lower than our value.

A number of studies have also contrasted the estimated Planck masses against masses estimated using other observables, with values of (1 − b) ranging from ∼0.7 to unity (e.g., von der Linden et al. 2014, 0.688 ± 0.072; Hoekstra et al. 2015, 0.76 ± 0.05; Smith et al. 2016, 0.95 ± 0.04; Medezinski et al. 2018, 0.80 ± 0.14). Our recovered values fall within this range. Other works report values for the power-law index, α (e.g., Schellenberger & Reiprich 2017, 0.76 ± 0.08; Mantz et al. 2016, 0.73 ± 0.02), consistent with our measurement when using debiased SPT masses.

We also examine the SPT-ECS  footprint for clusters detected by Planck but not by SPT. Based on the selection function shown in Figure 6, we expect the SPT-ECS  sample to contain essentially all confirmed Planck clusters at z ≥ 0.25 in the common sky area. Including the new confirmations discussed above, there are 117 confirmed Planck clusters that fall within the SPT-ECS footprint, and 82 of these are associated with SPT cluster candidates at ξ > 4. Of the remaining 35 clusters in Planck but not confirmed by SPT, 32 are at redshift z < 0.25—where the SPT filtering reduces both the completeness of the catalog and the fidelity of the mass estimates—and 5 of these 32 confirmed clusters also excluded because they are in regions excluded by the SPT point-source veto. For the three Planck clusters at z > 0.25 but not confirmed by SPT, we find that two of these systems match candidates just below the SPT selection threshold, with PSZ2 G244.74−28.59 (Planck S/N = 5.9, z = 0.33) at ξ = 3.97 and PSZ2 G251.13–78.15 (S/N = 4.6, z = 0.3) at ξ = 3.2. There is also a radio source nearby to PSZ2 G244.74−28.59, which—based on the methodology of Section 3.5—could reduce the ξ value by 0.08–0.7 for a source spectral index of −1 to −0.5. The final unmatched cluster, PSZ2 G282.14+38.29 (S/N = 4.9, z = 0.33 with validation from Pan-STARRS), is flagged as having a nearby point source detected at 857 GHz and is measured at ξ = −0.3 in our sample. We do not detect a large excess of red sequence galaxies in Pan-STARRS at this cluster location. While the SZ flux from one source (PSZ2 G244.74−28.59) may be diminished by the presence of a nearby radio source (which should also influence the Planck detection) and we do not independently confirm G282.14+38.29, we find the SPT selection to be consistent with expectations as relates to the PSZ2 sample with 39/42 of the reported Planck clusters at z ≥ 0.25 and not in a point-source vetoed region also in the SPT-ECS  sample. Further exploration of the differences between the estimated masses for the Planck and SPT samples will require detailed modeling of the selection functions of the two surveys in their jointly accessible mass and redshift ranges and is beyond the scope of this work.

The strong gravitational lensing regime, often identified via the presence of highly magnified and multiply imaged background galaxies lensed by foreground gravitational potentials, provides a unique probe of the cores of massive structures. Galaxy clusters have long been recognized as areas in which to productively search for strong gravitational lenses (see review by Meneghetti et al. 2013 and more recent works by Bayliss et al. 2011; Kneib & Natarajan 2011; Diehl et al. 2017; Lotz et al. 2017; Sharon et al. 2019, among many others). We examine the SPT-ECS  sample for signatures of strong lensing in the Magellan/PISCO and DES imaging data, as well as in archival and dedicated observations from the Hubble Space Telescope (HST), the latter from a snapshot program (PID 15307, PI: Gladders) designed to characterize the central regions of massive clusters from SPT-SZ and SPT-ECS.

We find that 44 of the SPT-ECS  systems exhibit unambiguous signs of strong lensing; we flag all of these systems in Tables 8 and 9. Some of these systems have been previously identified as strong lenses—see Smail et al. (1991), Sand et al. (2005), Covone et al. (2006), Zitrin et al. (2011), Hamilton-Morris et al. (2012), Gruen et al. (2014), Ebeling et al. (2017, 2018), Newman et al. (2018), Repp & Ebeling (2018), Jacobs et al. (2019), Petrillo et al. (2019), and Coe et al. (2019)—and in the online data for Tables 8 and 9 we also link individual previously known strong lenses to these works. In total over 110 systems from SPT-SZ and SPT-ECS  have been identified as strong gravitational lenses; a robust statistical characterization of the PISCO and HST data will be the subject of future work. In Figure 8 we display high-quality PISCO data for three of the SPT-ECS  strong lenses, as well as data from our HST program for the third.

Zoom In Zoom Out Reset image size

We use the optical richness (λ) measurements of SPT clusters matched to the Y3 RM catalog to calibrate the richness–mass relation, taking the SPT selection into account. Assuming our fiducial fixed cosmology, we simultaneously constrain the SZ scaling relation parameters through the NCs of the SPT cluster sample (as discussed in Section 5.1.2) and the parameters of the richness scaling relation. This analysis follows Saro et al. (2015), with the exception that we now also account for the effects of correlated scatter among ζ and richness.

6.1.1. Richness–Mass Relation: Likelihood Function

Along with the ζ–mass relation defined above in Equation (4), we define the richness–mass relation

$$\begin{eqnarray}\begin{array}{rcl}\langle \mathrm{ln}\lambda \rangle & = & \mathrm{ln}{A}_{\lambda }+{B}_{\lambda }\mathrm{ln}\left(\displaystyle \frac{{M}_{500c}}{3\times {10}^{14}\,{M}_{\odot }\,{h}^{-1}}\right)\\ & & +\,{C}_{\lambda }\mathrm{ln}\left(\displaystyle \frac{E(z)}{E(z=0.6)}\right).\end{array}\end{eqnarray} \tag{ 11 }$$

A covariance matrix describes the correlated intrinsic scatter between the two observables ζ and λ:

$$\begin{eqnarray}{{\rm{\Sigma }}}_{\zeta -\lambda }=\left(\begin{array}{cc}{\sigma }_{\mathrm{ln}\zeta }^{2} & {\rho }_{\mathrm{SZ}-\lambda }{\sigma }_{\mathrm{ln}\zeta }{\sigma }_{\mathrm{ln}\lambda }\\ {\rho }_{\mathrm{SZ}-\lambda }{\sigma }_{\mathrm{ln}\zeta }{\sigma }_{\mathrm{ln}\lambda } & {\sigma }_{\mathrm{ln}\lambda }^{2}+{\lambda }^{-1}\end{array}\right).\end{eqnarray} \tag{ 12 }$$

The contribution λ⁻¹ to the intrinsic scatter in richness represents the Poisson noise in the number of member galaxies observed at a fixed cluster mass. We note that we expect positive correlation in the scatter between ζ and λ, as both are projected quantities (see, e.g., Angulo et al. 2012).

The joint scaling relation then reads

$$\begin{eqnarray}&&P\left(\left[\begin{array}{c}\mathrm{ln}\zeta \\ \mathrm{ln}\lambda \end{array}\right]| M,z\right)={ \mathcal N }\left(\left[\begin{array}{c}\langle \mathrm{ln}\zeta \rangle (M,z)\\ \langle \mathrm{ln}\lambda \rangle (M,z)\end{array}\right],{{\rm{\Sigma }}}_{\zeta -\lambda }\right).\end{eqnarray} \tag{ 13 }$$

Following Bocquet et al. (2019), the likelihood function for our NCs and richness calibration analysis is

$$\begin{eqnarray}\begin{array}{rcl}\mathrm{ln}{ \mathcal L }({\boldsymbol{p}}) & = & \displaystyle \sum _{i}\mathrm{ln}\displaystyle \frac{{dN}({\xi }_{i},{z}_{i}| {\boldsymbol{p}})}{d\xi {dz}}\\ & & -\,{\displaystyle \int }_{{z}_{\mathrm{cut}}}^{\infty }{dz}{\displaystyle \int }_{{\xi }_{\mathrm{cut}}}^{\infty }d\xi \displaystyle \frac{{dN}(\xi ,z| {\boldsymbol{p}})}{d\xi {dz}}\\ & & +\,\displaystyle \sum _{j}\mathrm{ln}{P}_{j}(\mathrm{match})P({\lambda }_{\mathrm{obs},j}^{\gt 5}| {\xi }_{j},{z}_{j},{\boldsymbol{p}})\end{array}\end{eqnarray} \tag{ 14 }$$

up to a constant, where the first sum runs over all clusters i in the SPT sample above $\xi \gt 5$ and z > 0.25 and the second sum runs over all SPT clusters j for which an RM richness measurement is available. Note that the first two lines represent the NC likelihood defined earlier in Equation (7). The term $P(\mathrm{match})=1-P(\mathrm{random})$ describes the excess probability of matching an RM cluster to an SPT cluster over random associations P(random). The other term in the last line is computed as

$$\begin{eqnarray}\begin{array}{rcl}P({\lambda }_{\mathrm{obs}}| \xi ,z,{\boldsymbol{p}}) & = & \iiint {dM}\,d\zeta \,d\lambda \left[P({\lambda }_{\mathrm{obs}}| \lambda )P(\xi | \zeta )\right.\\ & & \times P\left.(\zeta ,\lambda | M,z,{\boldsymbol{p}})P(M| z,{\boldsymbol{p}})\right].\end{array}\end{eqnarray} \tag{ 15 }$$

Finally, we account for the richness cut λ_obs > 5 in the volume-limited redMaPPer catalog and evaluate

$$\begin{eqnarray}&&P({\lambda }_{\mathrm{obs}}^{\gt 5}| \xi ,z,{\boldsymbol{p}})=\displaystyle \frac{{\rm{\Theta }}({\lambda }_{\mathrm{obs}}\gt 5)P({\lambda }_{\mathrm{obs}}| \xi ,z,{\boldsymbol{p}})}{{\int }_{5}^{\infty }d{\lambda }_{\mathrm{obs}}{\rm{\Theta }}({\lambda }_{\mathrm{obs}}\gt 5)P({\lambda }_{\mathrm{obs}}| \xi ,z,{\boldsymbol{p}})}\end{eqnarray} \tag{ 16 }$$

with the step function Θ.

6.1.2. Richness–Mass Relation: Results

With this machinery in place we are now ready to explore the mass-richness relation of the SPT-RM sample. Assuming our fiducial cosmology, we evaluate the likelihood presented in Equation (14) of the SPT cluster NCs (which constrains the SZ scaling relation parameters) and the likelihood of the RM richnesses (which constrains the RM richness scaling relation parameters). We only use the SPT-SZ sample for the SPT NCs (to enable an independent constraint on γ_ECS, described below), but we use redMaPPer richnesses for the full SPT-SZ+SPT-ECS  sample.

We present the constraints on the richness–mass relation in Figure 10 and in Table 4. Compared to previous constraints using 19 clusters from SPT-SZ at ξ > 4.5 in the DES Science Verification region (Saro et al. 2015), we find a normalization that is ∼1.2σ higher, with the slope and redshift evolution consistent.

Zoom In Zoom Out Reset image size

Table 4.  Parameters of the Richness–Mass Relation

Parameter	Constraint
Aλ	76.9 ± 8.2
Bλ	1.020 ± 0.080
Cλ	0.29 ± 0.27
σlnλ	0.23 ± 0.16
ρ	>−0.78 (95% CL)

Note. The richness–mass relation is defined in Equation (11). We also quote the constraint on the correlation coefficient between the scatter in the SZ signal and richness ${\rho }_{\mathrm{SZ}-\lambda }$. The constraints are obtained using redMaPPer matches to the ξ > 4.5, z > 0.25 SPT sample.

Download table as:  ASCIITypeset image

We also compare against the DES weak-lensing analysis of the Year 1 RM sample reported in McClintock et al. (2019), which was also analyzed at our fiducial cosmology. Note that the DES weak-lensing analysis constrains $P({M}_{200{\rm{m}}}| \lambda )$—with masses defined with respect to the mean density of the universe—whereas our analysis constrains $P(\lambda | {M}_{500c})$. We convert ${M}_{500c}$ to ${M}_{200{\rm{m}}}$ assuming a Navarro, Frenk, and White (Navarro et al. 1996) profile and the concentration–mass relation from Child et al. (2018).⁹⁹ We invert our relation as

$$\begin{eqnarray}&&P({M}_{200{\rm{m}}}| \lambda )=\int {{dM}}_{200{\rm{m}}}P(\lambda | {M}_{200{\rm{m}}})P({M}_{200{\rm{m}}})\end{eqnarray} \tag{ 17 }$$

with the halo mass function prior $P({M}_{200{\rm{m}}})$.

In Figure 11, we show the mass–lambda relation from our work and examples from the literature. At our scaling relation pivot redshift (z = 0.6; see Equation (11)), the scaling relation normalizations are consistent at λ ≈ 60 or M_{200 m} ≈ 5 × 10¹⁴ M_⊙. However, there are some visible differences in the slope. We approximate¹⁰⁰ the slope F_λ in our $P({M}_{200{\rm{m}}}| \lambda )$ relation as

$$\begin{eqnarray}&&{F}_{\lambda }\equiv 1/{B}_{\lambda }=0.981\pm 0.077.\end{eqnarray} \tag{ 18 }$$

Zoom In Zoom Out Reset image size

We find that our slope is ∼30% shallower than the slope from the DES Y1 analysis (F_λ = 1.356 ± 0.052; McClintock et al. 2019), with a 4σ offset between the two constraints. To reproduce the slope of the McClintock et al. (2019) relation, we would require a significant shift in our assumed cosmology along the Ω_m and σ₈ degeneracy direction (see, e.g., Costanzi et al. 2019); however, a full cosmological interpretation is beyond the scope of this work and would depend on fully accounting for selection effects in the RM sample under study, as well as on degeneracies and covariances in a wider multidimensional parameter space.

A weak-lensing analysis using data from the Sloan Digital Sky Survey (SDSS) finds an amplitude and slope that are consistent with McClintock et al. (2019) at better than 1σ (Simet et al. 2017). Another weak-lensing study using SDSS data finds a much shallower slope centered at λ ∝ M^0.64 using lensing alone; this slope becomes consistent with unity—and thus our measurement—when combining lensing and cluster abundance (Murata et al. 2018). Qualitatively similar results are presented in an analysis of the richness–mass relation using first-year HSC data (Murata et al. 2019). A weak-lensing calibration of an X-ray-selected cluster sample yields constraints on the richness–mass relation that are centered on the results from McClintock et al. (2019), but with large uncertainties (Mantz et al. 2016).

Moving beyond optical weak lensing, two calibrations of the mass–λ relation using lensing of the CMB (Baxter et al. 2018; Raghunathan et al. 2019) recover amplitudes of the mass–λ relation that are compatible with both the calibration from DES Y1 shear measurements and this work. Note, however, that the slope parameters were not constrained by the CMB lensing measurements, where informative priors were applied. The richness–mass relation has also been calibrated using the clustering of clusters (Baxter et al. 2016) and the measurement of pairwise velocity dispersions (Farahi et al. 2016); both methods show consistency at the 2σ level. Finally, a study of the phase space of galaxy dynamics provides a calibration of the richness–mass relation with a slope that is consistent with unity at <1σ (Capasso et al. 2019). However, their relation exhibits strong redshift evolution, which leads to an offset in the relations at our pivot redshift z = 0.6.

6.1.3. Richness–Mass Relation: Constraint on γ_ECS

By using only the SPT-SZ data in the NCs, we can also use this test to independently evaluate our estimated value for γ_ECS presented in Section 5.1.2. We obtain a calibration of the correction factor γ_ECS between SPT-SZ and SPT-ECS  

$$\begin{eqnarray}&&{\gamma }_{\mathrm{ECS}}=1.054\pm 0.075.\end{eqnarray} \tag{ 19 }$$

This determination of γ_ECS is different from the result presented above in Section 5.1.2. In both cases, A_SZ is constrained to yield NCs from SPT-SZ that match our fixed fiducial cosmology. In Section 5.1.2, γ_ECS was calibrated by also demanding that the SPT-ECS  NCs match that cosmology—any relative offset in the amplitude of the ζ–mass relation between SPT-SZ and SPT-ECS  is thus absorbed by γ_ECS. In the calibration presented here, the redMaPPer richnesses serve as the relative anchor between the SPT-SZ and SPT-ECS  surveys. The two determinations of γ_ECS agree at the 0.8σ level. We conclude that our empirical modeling of the full SPT-SZ+SPT-ECS  sample with an overall amplitude offset is adequate, and when reporting cluster masses we adopt the mean recovered constraint from the more precise NC analysis result as our default.

We next explore the distribution of separations between the redMaPPer- and SPT-determined cluster centers. Based on visual inspection of >100 matched X-ray and RM clusters in SDSS, Rozo & Rykoff (2014) found that the gas centers and central galaxies should be well aligned (within 50 kpc) 80% of the time, with it being rare to find a separation of >300 kpc between the two (results consistent with previous findings by, e.g., Lin & Mohr 2004). In the SDSS sample, the RM algorithm selected the visually identified central galaxy in 86% ± 3% of systems and had a long uniform tail to 800 kpc for the remainder of systems. These gas-central galaxy separations were further quantified for RM clusters by Saro et al. (2015) with 19 SPT-RM clusters in DES Science Verification data and Zhang et al. (2019) for 144 (67) systems in SDSS (DES); the latter analysis using archival Chandra X-ray data as analyzed in Hollowood et al. (2019). With differing model parameterizations, these works found ∼63%–84% of all clusters to be well centered.

Following these previous works, we adopt two different models for this offset distribution for the SPT-RM sample, one modeling offsets relative to the cluster mass scale (via R_500c) and the other relative to a cluster extent that scales as a function of RM galaxy richness. Both of these models assume that the offset distribution can be modeled as a central core of well-aligned clusters with small separation combined with a subdominant population of clusters with large offsets. Physically this corresponds to the cluster population being composed of a mixture of relaxed and merging clusters with some additional scatter introduced via possible misidentification of central galaxies by the RM algorithm.

The dynamical state of the cluster population is also traced by the morphology of the cluster gas, which can be measured by X-ray observations (note that the filtering applied to the SPT maps makes it difficult to extract a robust gas morphological measurement from the SZ data). The X-ray morphology has been measured via the A_phot statistic for 50 of the SPT-RM clusters that are also part of a Chandra X-ray Visionary Project (XVP; PI: Benson, Nurgaliev et al. 2017); 38 of the systems in the Zhang et al. (2019) DES Y1 analysis mentioned above are part of the SPT-XVP. The A_phot statistic is a quantification of the amount of azimuthal asymmetry present in the X-ray photon count distribution and has been shown to be a robust morphological measure even when used on X-ray data with a relatively low number of counts (∼2000 counts per cluster) such as the SPT-XVP observations (Nurgaliev et al. 2013). We plot as an inset in Figure 12 the SZ−optical offset distribution of these 50 clusters. The outlier in this inset plot is SPT-CL J2331−5051 (A_phot = 0.14), which may be captured pre-merger with SPT-CL J2332−5053. The RM algorithm has selected what appears to be the central galaxy of the latter cluster and found a smaller structure of λ = 8 at the location of SPT-CL J2331−5051, which is the more massive system inferred from both the SZ and X-ray observations (see further discussion of this system in Andersson et al. 2011; Huang et al. 2019). The A_phot distribution is a continuum, but adopting the somewhat arbitrary choice of McDonald et al. (2017) with A_phot < 0.1 classified as "relaxed" (17 systems) and A_phot > 0.5 as "disturbed" (10 systems), we find the median offset of the relaxed (disturbed) systems to be ${0.067}_{-0.02}^{+0.005}{R}_{500c}$ (${0.23}_{-0.04}^{+0.01}{R}_{500c}$), with the relaxed systems having a closer alignment between the SZ center and the RM most probable central galaxy, as expected.

Zoom In Zoom Out Reset image size

6.2.1. Offset Distribution Relative to R_500c

We first consider the offset distribution relative to the cluster scale R_500c. For this analysis we split the cluster population into two parts: a high-significance subset with ξ ≥ 5 (249 clusters, median λ = 81), which is the threshold used for SPT cosmological analyses (see, e.g., Bocquet et al. 2019), and a lower-significance sample at 4 < ξ < 5 (161 systems, median λ = 55). In Figure 12 we plot the distribution of separations between the SZ centroids and RM central galaxies for these two subsamples.

To characterize this distribution, our model follows that of Saro et al. (2015). We have added a third Gaussian term to account for the long tail to large separations. As noted above, such a tail was also previously seen in analyses of SDSS clusters (and given its small sample size, the absence of a significant tail in Saro et al. 2015 is unsurprising). Examination of clusters with the largest separations revealed systems where the RM algorithm identified a bright galaxy near what was the lesser of two SZ peaks in merging clusters (see, e.g., Figure 13 and the discussion of SPT-CL J2331−5051 above), rich systems split into multiple detections (i.e., "mispercolation"; see discussion in Hollowood et al. 2019), systems with significant masking of the optical data near the SPT position, and—for a few of the lower-significance clusters—systems with higher (but still less than 5%) chance of random association between the SZ candidate and RM cluster.

Zoom In Zoom Out Reset image size

We write the probability distribution as a function of the fractional separation, $x={r}_{\mathrm{offset}}/{R}_{500c}$, as

$$\begin{eqnarray}&&P(x)=2\pi x\left(\displaystyle \frac{{\rho }_{0}}{2\pi {\sigma }_{0}^{2}}{e}^{\tfrac{-{x}^{2}}{2{\sigma }_{0}^{2}}}+\displaystyle \frac{{\rho }_{1}}{2\pi {\sigma }_{1}^{2}}{e}^{\tfrac{-{x}^{2}}{2{\sigma }_{1}^{2}}}+\displaystyle \frac{(1-{\rho }_{0}-{\rho }_{1})}{2\pi {\sigma }_{2}^{2}}{e}^{\tfrac{-{x}^{2}}{2{\sigma }_{2}^{2}}}\right)\end{eqnarray} \tag{ 20 }$$

convolved with the SPT positional uncertainty. The SPT positional uncertainty is given by the cluster detection significance, ξ, and detection scale, θ_c,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{SPT}}=\displaystyle \frac{\sqrt{{\theta }_{\mathrm{beam}}^{2}+{({\kappa }_{\mathrm{SPT}}{\theta }_{c})}^{2}}}{\xi }\end{eqnarray} \tag{ 21 }$$

convolved with a general astrometric uncertainty of 4''–6''  (see Section 2.2; W. Everett et al. 2020, in preparation), where θ_beam = 13 is a combination of the 95 + 150 GHz beams and κ_SPT is a parameter of order unity (Story et al. 2011; Song et al. 2012).

We use the emceepackage in Python (Foreman-Mackey et al. 2013) to conduct a Markov Chain Monte Carlo (MCMC) maximum likelihood analysis adopting priors of

$$\begin{eqnarray*}\begin{array}{rcl}0 & \leqslant & \rho \leqslant 1.0\\ 0 & \leqslant & {\rho }_{0}-{\rho }_{1}\leqslant 1\\ 0 & \leqslant & {\sigma }_{0}\leqslant 0.3\\ {\sigma }_{0} & \lt & {\sigma }_{1}\lt 2\\ {\sigma }_{1} & \lt & {\sigma }_{2}\lt 3\\ 0.5 & \leqslant & {\kappa }_{\mathrm{SPT}}\leqslant 2.\end{array}\end{eqnarray*}$$

Results for this parameterization for both samples are shown in Figure 12 and reported in Table 5. We note that the lower-significance sample does not have the power to constrain κ_SPT, and so we fix it to the best-fit value from the ξ ≥ 5 sample.

Table 5.  Miscentering Model 1 Fits

Parameter	$\xi \geqslant 5$	4 < ξ < 5
ρ0	${0.675}_{-0.08}^{+0.07}$	${0.54}_{-0.20}^{+0.13}$
${\sigma }_{0}$	${0.02}_{-0.01}^{+0.01}$	${0.065}_{-0.035}^{+0.03}$
${\rho }_{1}$	${0.25}_{-0.06}^{+0.08}$	${0.29}_{-0.16}^{+0.14}$
${\sigma }_{1}$	${0.15}_{-0.03}^{+0.03}$	${0.24}_{-0.11}^{+0.14}$
${\sigma }_{2}$	${0.70}_{-0.09}^{+0.125}$	${0.48}_{-0.07}^{+0.15}$
${\kappa }_{\mathrm{SPT}}$	1.0 ± 0.2	⋯

Note. Best-fit miscentering parameters for the SPT-RM volume-limited sample as characterized in Equation (20).

Download table as:  ASCIITypeset image

For the high-significance, ξ ≥ 5 sample we find that the fraction of clusters in the well-centered component in this version of RM is consistent with Saro et al. (2015) (${\rho }_{0}={67}_{-8}^{+6} \%$ vs. ${63}_{-25}^{+15} \%$), with the uncertainty reduced a factor of 2 in this work. The width of this component is slightly smaller (σ₀ = 0.02 ± 0.01 vs. 0.07 ± 0.02), and the width of the second component is also smaller (σ₁ = 0.15 ± 0.03 vs. 0.25 ± 0.07), though we note that some of this spread is absorbed in the third Gaussian term that captures the offsets to high R_500c.

Turning to the lower-significance sample, we find it overall less well centered than the higher-significance sample, but with the parameters also less well constrained. Future studies using SZ clusters from the 500d SPTpol survey or from SPT-3G will significantly increase the number of lower-mass clusters in our SZ-matched sample and will allow us to more robustly explore miscentering trends as a function of mass.

6.2.2. Offset Distribution Relative to R_λ

As a second model of the SZ–central galaxy offset distribution we explore a miscentering model tied to the RM cluster radius, R_λ, where

$$\begin{eqnarray}&&{R}_{\lambda }={\left(\displaystyle \frac{\lambda }{100}\right)}^{0.2}{h}^{-1}\ \mathrm{Mpc}.\end{eqnarray} \tag{ 22 }$$

R_λ is determined by the RM cluster-finding algorithm and corresponds to the maximum separation between the RM central galaxy and cluster members that contribute to the optical richness measurement. Here we focus on the better-constrained SPT clusters at ξ > 5, and we plot this distribution in Figure 14. As can be seen, there are a significant number of systems (14 of 249, 6%) that have offsets greater than R_λ. Examination of these clusters shows, unsurprisingly, that they display similar characteristics to the outliers in the previous subsection (and many are in common). Additionally, it is worth noting that issues that reduce the richness estimate will more adversely affect a fractional offset when the cluster scale is set by the richness measure (e.g., R_λ) as opposed to being set by the SZ mass estimate.

Zoom In Zoom Out Reset image size

Following McClintock et al. (2019) and Zhang et al. (2019), we model the probability distribution for the separation between SZ centroids and RM central galaxies as the combination of an exponential distribution that reflects the well-centered systems and a Gamma distribution Γ(2, τ) that characterizes those clusters with larger separations:

$$\begin{eqnarray}&&P(x)=\rho \displaystyle \frac{1}{{\sigma }_{\lambda }}{e}^{-\tfrac{x}{{\sigma }_{\lambda }}}+(1-\rho )\displaystyle \frac{x}{{\tau }^{2}}{e}^{-\tfrac{x}{\tau }},\end{eqnarray} \tag{ 23 }$$

where now x = r_offset/R_λ, σ_λ characterizes the exponential distribution, τ is the scale parameter of the Gamma distribution function, and, as in the previous model, we also incorporate the SPT positional uncertainties when conducting the fit.

The two-dimensional convolutions required for properly incorporating the SPT positional uncertainty in this model are computationally expensive to repeat many times in an MCMC analysis. Numerical computations of the probability distribution can instead be replaced by relatively inexpensive yet highly precise emulators. For this purpose, we use Gaussian processes (GPs; Rasmussen & Williams 2006), a method that has facilitated robust forward modeling of various astrophysical functions (e.g., Heitmann et al. 2006; Habib et al. 2007 and other applications). We detail the construction and validation of our emulator of the miscentering distribution model in Appendix A.

For this analysis we adopt the priors

$$\begin{eqnarray*}\begin{array}{rcl}0.3 & \leqslant & \rho \leqslant 1.0\\ 0.001 & \leqslant & {\sigma }_{\lambda }\leqslant 0.25\\ 0.05 & \leqslant & \tau \leqslant 1.0.\end{array}\end{eqnarray*}$$

We plot these results in Figure 14 and report the parameter constraints in Table 6. In Figure 14 we also overplot the best-fit model curves from Zhang et al. (2019) convolved with the SPT positional uncertainty.

Table 6.  Miscentering Model 2 Fits

Parameter	$\xi \geqslant 5$
ρ	${0.87}_{-0.03}^{+0.02}$
σ	${0.12}_{-0.01}^{+0.015}$
τ	${0.69}_{-0.09}^{+0.12}$

Note. Best-fit miscentering parameters for the SPT-RM volume-limited sample as characterized in Equation (23).

Download table as:  ASCIITypeset image

While Zhang et al. (2019) explored the separation between X-ray peaks and central galaxies, the analysis here quantifies the central galaxy offset from the gas center averaged over a larger scale via the SPT matched filter. This should generally have a small effect; studies with X-ray centering proxies (see, e.g., Mann & Ebeling 2012) have found an additional 20–60 kpc ($\sim 0.02{R}_{\lambda }\mbox{--}0.06{R}_{\lambda }$) scatter in the BCG and X-ray centroid separation (to which our measurement is most analogous) as compared to the X-ray peak to BCG separation, though there can be notable outliers in the case of merging clusters. With this caveat in mind, we find that our results at ξ > 5, with $\rho ={0.87}_{-0.03}^{+0.02}$ of the clusters within the "well-centered" component of the distribution, agree with previous RM results on DES ($\rho ={0.84}_{-0.07}^{+0.11}$) and are higher than those found in SDSS ($\rho ={0.68}_{-0.05}^{+0.03}$). However, our recovered value of τ is notably higher than previous results ($\tau ={0.69}_{-0.09}^{+0.12}$, vs. ${0.16}_{-0.04}^{+0.11}$), as it is significantly affected by clusters in the long tail. In comparison, Zhang et al. (2019) only found 1 of 67 systems (1.5%) with gas–BCG separations at R > R_λ compared to the 14 (6%) found here. If we reanalyze the cluster sample excluding systems with offsets R > R_λ, we find ρ and σ_λ significantly less well constrained ($\rho ={0.74}_{-0.30}^{+0.22}$, ${\sigma }_{\lambda }\,={0.105}_{-0.07}^{+0.045}$) and τ shifted to lower values consistent with previous work ($\tau ={0.13}_{-0.045}^{+0.075}$). It will be important in future weak-lensing analyses to quantify this tail while incorporating all the cluster selection effects relevant to the analysis at hand, as Zhang et al. (2019) found that shifts in τ at the 0.04 level can lead to systematic shifts in the weak-lensing-derived mass calibrations at the level of $\delta \mathrm{log}{M}_{200}=0.015.$