CLUSTER MORPHOLOGIES AND MODEL-INDEPENDENT Y_SZ ESTIMATES FROM BOLOCAM SUNYAEV–ZEL'DOVICH IMAGES

Bright quasars located near the clusters were observed for 10 minutes once every ≃90 minutes in order to determine the offset of our focal plane relative to the telescope pointing coordinates. These observations were used to construct a model of the pointing offset as a function of local coordinates (az, el), with a single model for each cluster. The uncertainty in the pointing models is ≲5 arcsec. This pointing uncertainty is quasi-negligible for Bolocam's 58 arcsec FWHM beams, especially for extended objects such as clusters. We made two 20 minute long observations each night of Uranus, Neptune, or a source in Sandell (1994) for flux calibration. Using the quiescent detector resistance as a proxy for detector responsivity and atmospheric transmission, we then fit a single flux-calibration curve to the entire data set. We estimate the uncertainty in our flux calibration to be 4.3%, with the following breakdown: 1.7% from the Rudy temperature model of Mars scaled to measured WMAP values (Halverson et al. 2009; Wright 1976; Griffin et al. 1986; Rudy et al. 1987; Muhleman & Berge 1991; Hill et al. 2009), 1.5% in the Uranus/Neptune model referenced to Mars (Griffin & Orton 1993), 1.4% due to variations in atmospheric opacity (S09), 3.1% due to uncertainties in the solid angle of our point-spread function (PSF) (S09), and 1.5% due to measurement uncertainties (S09).

The raw Bolocam timestreams are dominated by noise sourced by fluctuations in the water vapor in the atmosphere, which have a power spectrum that rises sharply at low frequencies. In order to optimally subtract the atmospheric noise, we have used a slightly modified version of the average subtraction algorithm described in Sayers et al. (2010, hereafter S10). We have modified the S10 algorithm because these cluster data contain additional atmospheric noise caused by the Lissajous scan pattern. Since we are scanning the telescope parallel to R.A. and decl., the airmass we are looking through is constantly changing. As a result, our data contain a large amount of atmospheric signal in narrow bands centered on the two fundamental scan frequencies.

Following the algorithm in S10, we first create a template of the atmosphere by averaging the signal from all of our detectors at each time sample (i.e., the average signal over the FOV). In S10, this template is subtracted from each detector's timestream after weighting it by the relative gain of that detector, which is determined from the correlation coefficient between the timestream and the template. We use a single correlation coefficient for each detector for each 10 minute long observation. However, a significant fraction of the atmospheric noise at the fundamental scan frequencies remains in the data after application of the S10 algorithm, indicating that we have slightly misestimated the correlation coefficients. Therefore, we modified the S10 algorithm to compute the correlation coefficients for the template based only on the data within a narrow band centered on the two fundamental scan frequencies. The atmospheric noise power in these narrow frequency bands is roughly an order of magnitude above the broadband atmospheric noise at nearby frequencies; consequently, the data in these narrow bands provide a high signal-to-noise estimate of each detector's response to atmospheric signal. The narrowband atmospheric noise features are completely removed using this modified S10 algorithm, and the amount of residual broadband atmospheric noise is slightly reduced compared to the results from the original S10 algorithm.

After applying this average subtraction algorithm to the timestream data, we then high-pass filter the data according to

$$\begin{eqnarray*} &&\mathcal {F} = 1 - \frac{1}{1 + (10^{f/f_0 - 1})^\kappa } \end{eqnarray*}$$

with f₀ = 250 mHz and κ = 8. The value of f₀ was chosen to maximize the spatially extended signal-to-noise ratio (S/N) for the typical cluster in our sample based on tests with f₀ varying from 0 to 400 mHz, and the value of κ was chosen to produce a sharp cutoff with minimal ringing. Figure 2 shows a typical pre- and post-subtraction timestream noise power spectral density (PSD).

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In addition to subtracting atmospheric noise, the FOV-average subtraction and timestream high-pass filter also remove some cluster signal. Since we use the data timestreams to both determine the atmospheric fluctuation template and the correlation coefficient of each detector's data timestream with the template, the FOV-average subtraction acts on the data in a nonlinear way. Consequently, its impact on the data depends on the cluster signal. As described below, we quantify the effects of the FOV-average subtraction and the timestream high-pass filter via simulation by processing a known cluster image through our data-reduction pipeline. These simulations are computation-time intensive, and we find, in practice (see Section 6.1), that the filtering is only mildly dependent on the cluster signal. Thus, in the end, we determine the effects of the filtering for a particular cluster's data set using the cluster model that best fits those data. We use the term transfer function to describe the effect of the filtering, although this terminology is not rigorously correct because the filtering depends on the cluster signal.

To compute the transfer function, we first insert a simulated, beam-smoothed cluster profile into our data timestreams by reverse mapping it using our pointing information. These data are then processed in an identical way to the original data, and an output image, or map, is produced. When processing the data-plus-simulated-cluster timestreams we use the FOV-average subtraction correlation coefficients that were determined for the original data. This ensures that the simulated cluster is processed in an identical way to the real cluster in our data. In the limit that the best-fit cluster model is an accurate description of the data, this process is rigorously correct. The original data map is then subtracted from this data-plus-simulated-cluster map to produce a noise-free image of the processed cluster. In Figure 3, we show an example cluster image, along with the noise-free processed image of the same cluster. The Fourier transform of this processed cluster image is divided by the Fourier transform of the input cluster to determine how the cluster is filtered as a function of two-dimensional Fourier mode (i.e., what we term the transfer function, see Figure 4).

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At small angular scales, there is very little signal in the beam-smoothed input cluster, and numerical noise prevents us from accurately characterizing the transfer function at these scales. The transfer function is expected to be unity at small angular scales, and is asymptoting to this value at the larger scales where we can accurately characterize it. Therefore, we set the transfer function to a value of 1 for u>0.75 arcmin⁻¹. Deconvolving the processed cluster image using the transfer function (see Section 5), which has been approximated as 1 at small angular scales, produces an image that is slightly biased compared to the input cluster. The residuals between these two images are approximately white, with an rms of ≲0.1 μK_CMB. This transfer-function-induced bias is negligible compared to our noise, which has an rms of ≃10 μK_CMB.

As noted above, the transfer function (weakly) depends on the profile of the cluster; larger clusters are more heavily filtered than smaller clusters. Therefore, we determine a unique transfer function for each cluster using the best-fit elliptical Nagai model for that cluster (Nagai et al. 2007, hereafter N07). The details of this fit are given in Section 4. Since the transfer function depends on the best-fit model, and vice versa, we determine the best-fit model and transfer function in an iterative way. Starting with a generic cluster profile, we first determine a transfer function, and then fit an elliptical Nagai model using this transfer function (i.e., the Nagai model parameters are varied while the transfer function is held fixed). This process is repeated, using the best-fit model from the previous iteration to calculate the transfer function, until the best-fit model parameters stabilize. This process converges fairly quickly, usually after a single iteration for the clusters in our sample. The model dependence of the resulting transfer function is quantified in Section 6.1.

In order to accurately characterize the sensitivity of our images, we compute our map-space noise directly from the data via 1000 jackknife realizations of our cluster images. In each realization, random subsets of half of the ≃100 observations are multiplied by −1 prior to adding them into the map. Each jackknife preserves the noise properties of the map while removing all of the astronomical signal, along with any possible fixed-pattern or scan-synchronous noise due to the telescope scanning motion.³ Since these jackknife realizations remove all astronomical signal, we estimate the amount of astronomical noise in our images separately, as described below. After normalizing the noise estimate of each map pixel in each jackknife by the square root of the integration time in that pixel, we construct a sensitivity histogram, in μK_CMB s^1/2, from the ensemble of map pixels in all 1000 jackknifes. The width of this histogram provides an accurate estimate of our map-space sensitivity (see Figure 5). We then assume that the noise covariance matrix is diagonal⁴ and divide by the square root of the integration time in each map pixel to determine the noise rms in that pixel. This method is analogous to the one used in S09, where it is described in more detail.

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There is a non-negligible amount of noise in our maps from two main types of astronomical sources: anisotropies in the CMB and unresolved point sources. The SPT has recently published power spectra for both of these sources at 150 GHz over the range of angular scales probed by our maps, so we use their measurements to estimate the astronomical noise in our maps (Lueker et al. 2010; Hall et al. 2010). Note that these measurements inherently include all astrophysical effects, such as lensing and clustering of point sources. Using the SPT power spectra, we generate simulated maps of our cluster fields assuming Gaussian fluctuations. Note that this is a poor assumption for both the SZ-sourced CMB fluctuations and the signal sourced by background galaxies that are lensed by the cluster. However, since the noise power from both of these sources is quasi-negligible compared to the total noise in our images, any failure of our Gaussian assumption will have a minimal impact on our results. The good match of the SDS1 data to our noise model validates our Gaussian assumption (see Figure 5). These simulated images of the CMB plus point sources are then reverse mapped into our timestream data and processed to estimate how they will appear in our cluster maps. We then add these processed astronomical realizations to our jackknife realizations to provide a complete estimate of the noise in our images. Sensitivity estimates from our signal-free map of SDS1 agree well with the sensitivity estimates from our noise maps, indicating that there are no additional noise sources that have not been included in our estimate (see Figure 5). The properties of this noise are described in more detail in Section 6.2. In general, the noise from astronomical sources is quasi-negligible compared to the other noise in our images. However, the large-scale correlations from CMB fluctuations are non-negligible in our deconvolved images (see Table 5).

Note that we have chosen to model the signal from point sources as an additional noise term in our maps rather than attempting to subtract any individual point sources. Part of the motivation for this approach is the fact that we do not detect any point sources in our cluster or blank-field images. For reference, P10 mapped 15 clusters to a similar depth using the SPT and only detected two point sources; the combined area of the SPT images is approximately 20 times the combined area of our six images. Although there are several known submillimeter galaxies and active galactic nuclei in our images (Zemcov et al. 2007; Ivison et al. 2000; Cooray et al. 1998), even the brightest sources will have a flux of ≃10 μK_CMB in our observing band, rendering them undetectable given our noise rms of ≃10 μK_CMB. Additionally, it is not possible to reliably estimate the flux of these sources in our observing band since most have only been detected at one or two wavelengths, generally separated from our observing band by more than a factor of two in wavelength.

As mentioned above, the SZ effect involves CMB photons inverse Compton scattering off of hot electrons in the ICM. Since the electrons are many orders of magnitude hotter than the CMB photons, there is, on average, a net increase in the energy of the photons. The classical distortion relative to the blackbody spectrum of the CMB is given by

$$\begin{eqnarray*} &&f(x) = x \frac{e^x + 1}{e^x - 1} - 4, \end{eqnarray*}$$

where x = hν/k_BT_CMB. The magnitude of the distortion is proportional to the product of the density and temperature of the electrons in the ICM projected along the line of sight. The frequency-dependent temperature change of the CMB is given by

$$\begin{eqnarray*} &&T_{{\rm SZ}} = f(x) y T_{\rm CMB}, \end{eqnarray*}$$

where

$$\begin{eqnarray*} &&y = \int n_e \sigma _T \frac{k_B T_e}{m_e c^2}, \end{eqnarray*}$$

and n_e and T_e are the density and temperature of the ICM electrons. Relativistic corrections can be included by multiplying f(x) by (1 + δ(x, T_e)) (Itoh et al. 1998). Note that the X-ray brightness of the ICM is proportional to n²_eT^1/2_e; the differing density and temperature dependence of the SZ and X-ray signals makes them highly complementary probes of the ICM (e.g., Bonamente et al. 2006, hereafter B06).

We have fit our data to two types of models: an isothermal beta model (Cavaliere & Fusco-Femiano 1976, 1978) and the pressure profile proposed by N07, hereafter the Nagai profile. The isothermal beta model has been used extensively to describe X-ray and SZ measurements of the ICM, and our beta model fit parameters can be directly compared to these previous results. However, the beta model provides a poor description of deep X-ray data, which generally have a cuspier core and steeper outer profile (e.g., Vikhlinin et al. 2006, N07, and Arnaud et al. 2010). In contrast, the Nagai pressure profile, which is a generalization of the Navarro–Frenk–White dark matter profile, is able to describe deep X-ray observations over a wide range of scales (N07; Arnaud et al. 2010). Therefore, we have also fit our data to the Nagai model. In practice, the beta and Nagai models are highly degenerate over the range of angular scales to which our data are sensitive (1–12 arcmin).

Specifically, the isothermal beta model is described by

$$\begin{eqnarray*} &&p = \frac{p_0}{(1 + r^2/r_c^2)^{3 \beta /2}}, \end{eqnarray*}$$

where p is the pressure profile, p₀ is the pressure normalization, r_c is the core radius, and β is the power-law slope. The beta model can be analytically integrated along the line of sight to give the observed SZ signal, with

$$\begin{eqnarray*} &&T_{{\rm SZ}} = \frac{f(x) y_0 T_{\rm CMB}}{(1 + r^2/r_c^2)^{(3\beta -1)/2}} + \delta T, \end{eqnarray*}$$

where T_SZ is the SZ signal in our map (in μK_CMB) and y₀ is the central Comptonization. We fix the value of β at 0.86 for all of our fits; this is the best-fit value found in P10 for SZ data.⁵ Since our data timestreams are high-pass filtered, we are not sensitive to the DC signal level in our images. Note that because the timestream data, rather than the map, are high-pass filtered, the DC signal level of the map is in general not equal to 0. However, the DC signal of the map is not physically meaningful due to the filtering and must therefore be included as a free parameter, δT, in the model fits. We have also generalized the beta model to be elliptical in the plane of the sky with

$$\begin{eqnarray*} &&T_{{\rm SZ}} = \frac{f(x) y_0 T_{\rm CMB}}{(1 + \frac{r_1^2}{r_c^2} + \frac{r_2^2}{(1-\epsilon)^2r_c^2})^{(3\beta -1)/2}} + \delta T, \end{eqnarray*}$$

where r₁ is oriented along the major axis, described by a position angle of θ(in degrees east of north), r₂ is orthogonal to the major axis, and is the ellipticity.

The Nagai model is described by

$$\begin{equation} p = \frac{p_0}{(r/r_s)^{\mathcal {C}} [1 + (r/r_s)^\mathcal {C}]^{(\mathcal {B} -\mathcal {C})/\mathcal {A}}}, \end{equation} \tag{ 1 }$$

where p is the pressure, p₀ is the pressure normalization, r_s is the scale radius (r_s = r₅₀₀/c, with c ≃ 1.2; Arnaud et al. 2010), and $\mathcal {A}$, $\mathcal {B}$, and $\mathcal {C}$ are the power-law slopes at intermediate, large, and small radii compared to r_s. We have fixed the values of $\mathcal {A}$, $\mathcal {B}$, and $\mathcal {C}$ to the best-fit values found in Arnaud et al. (2010; 1.05, 5.49, 0.31). The Nagai model is not analytically integrable, so we numerically integrate p along the line of sight to determine T_SZ. We have also generalized the Nagai model to be elliptical in the plane of the sky, using the same notation as our elliptical generalization of the beta model. Note that, in all of our fits, we have corrected for relativistic effects via the approximations given in Itoh et al. (1998) using gas temperature estimates from X-ray observations of these clusters (Cavagnolo et al. 2008; Jeltema et al. 2001). The relativistic corrections are ≃5% for the typical temperatures in these massive clusters (≃10 keV).

We find that both the beta model and the Nagai model adequately describe our data, with the exception of A697, and neither model is preferred with any significance. However, the elliptical models (χ²/dof =5605/5413) provide a much better fit to the full ensemble of our cluster data compared to the spherical models (χ²/dof =5813/5423, see Tables 2 and 3). The F-ratio for these two fits is 20.1 for ν₁ = 10 and ν₂ = 5413 degrees of freedom (dof; Bevington & Robinson 1992), corresponding to a probability of <10⁻³⁶ that we would obtain data with equal or greater preference for elliptical models if the clusters were spherical. If we neglect A697, which is not well described by either model, then the F-ratio is 12.2 with a corresponding probability of <10⁻¹⁶. The weighted mean ellipticity of the five clusters is = 0.27 ± 0.03, consistent with results from X-ray data (e.g., Maughan et al. 2008; De Filippis et al. 2005). Our inability to distinguish between the Nagai and beta models is likely due to the limited spatial dynamic range of our images (≃1–12 arcmin). For the clusters in our sample, we are insensitive to the core, r ≲ 0.15r₅₀₀ ≃ 0.5 arcmin, and the outskirts of the cluster r ≳ 2r₅₀₀ ≃ 10 arcmin, which means we are only sensitive to a single power-law exponent in the Nagai model, $\mathcal {A}$. For this reason, the Nagai model is highly degenerate with the beta model for the angular scales probed by our data.

Table 2. Nagai Model Fit Parameters

Cluster	p0 ($10^{-11} \frac{\hbox{erg}}{\hbox{cm}^3}$)	rs (arcmin)	c500		θ (deg)	χ2/dof	PTE$_{\chi ^2}$	PTEsim
Elliptical Nagai model
A697	9.3 ± 1.3	6.9 ± 1.0	0.93 ± 0.14	0.37 ± 0.05	−24 ± 4	1289/1117	0.00	0.00
A1835	8.1 ± 1.7	6.7 ± 1.5	0.94 ± 0.21	0.27 ± 0.07	−16 ± 10	966/945	0.31	0.22
MS 0015.9+1609	6.7 ± 1.4	5.5 ± 1.1	0.62 ± 0.13	0.24 ± 0.08	68 ± 12	1079/1117	0.79	0.73
MS 0451.6−0305	8.7 ± 1.6	4.7 ± 0.7	0.80 ± 0.14	0.26 ± 0.06	85 ± 7	1188/1117	0.07	0.08
MS 1054.4−0321	6.5 ± 1.9	3.6 ± 1.0	0.64 ± 0.19	0.09 ± 0.07	−1 ± 32	1084/1117	0.75	0.72
Spherical Nagai model
A697	11.0 ± 1.8	4.6 ± 0.8	1.40 ± 0.22	...	...	1399/1119	0.00	0.00
A1835	8.9 ± 2.1	5.1 ± 1.2	1.24 ± 0.27	...	...	1007/947	0.08	0.07
MS 0015.9+1609	6.0 ± 1.1	5.4 ± 1.1	0.63 ± 0.12	...	...	1100/1119	0.66	0.61
MS 0451.6−0305	10.6 ± 2.1	3.5 ± 0.5	1.08 ± 0.15	...	...	1220/1119	0.02	0.02
MS 1054.4−0321	6.0 ± 1.8	3.6 ± 1.0	0.64 ± 0.16	...	...	1087/1119	0.75	0.75

Notes. Best-fit parameters and 1σ uncertainties for our Nagai model fits with the power-law exponents fixed to the best-fit values found in Arnaud et al. (2010; 1.05, 5.49, 0.31). From left to right the columns give the normalization of the Nagai profile, p₀, the scale radius of the major axis, r_s, the concentration parameter, c₅₀₀, the ellipticity, , the position angle of the major axis in degrees east of north, θ, the χ² and dof for the fit, and the goodness of fit quantified by the probability to exceed the given χ²/dof based on the standard χ² probability distribution function and also empirically by the fraction of our 1000 noise realizations that produce a larger χ² value when a model cluster is added to them (see Section 6.2). The values of c₅₀₀ for the spherical fits can be compared to the nominal value of 1.17 that is given in Arnaud et al. (2010) based on spherical fits of X-ray data. See Section 6.2 for a description of how the parameter uncertainties are calculated. Note that we have included the 4.3% uncertainty in our flux calibration in the error estimates for p₀.

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Table 3. Beta Model Fit Parameters

Cluster	y0 (10−4)	rc (arcsec)	rc/r500		θ (deg)	χ2/dof	PTE$_{\chi ^2}$	PTEsim
Elliptical beta model
A697	2.93 ± 0.26	98 ± 10	0.26 ± 0.03	0.38 ± 0.04	−10 ± 2	1288/1117	0.00	0.00
A1835	2.49 ± 0.31	79 ± 11	0.21 ± 0.03	0.28 ± 0.07	−9 ± 10	970/945	0.28	0.20
MS 0015.9+1609	2.59 ± 0.27	85 ± 12	0.43 ± 0.07	0.24 ± 0.07	70 ± 12	1078/1117	0.79	0.73
MS 0451.6−0305	2.89 ± 0.22	68 ± 8	0.31 ± 0.04	0.26 ± 0.06	80 ± 8	1193/1117	0.06	0.06
MS 1054.4−0321	2.13 ± 0.22	58 ± 12	0.41 ± 0.10	0.09 ± 0.07	7 ± 33	1083/1117	0.76	0.73
Spherical beta model
A697	2.74 ± 0.24	72 ± 8	0.19 ± 0.02	...	...	1397/1119	0.00	0.00
A1835	2.46 ± 0.31	63 ± 8	0.17 ± 0.02	...	...	1008/947	0.08	0.08
MS 0015.9+1609	2.54 ± 0.29	80 ± 11	0.40 ± 0.05	...	...	1099/1119	0.66	0.61
MS 0451.6−0305	2.92 ± 0.24	53 ± 7	0.24 ± 0.03	...	...	1222/1119	0.02	0.02
MS 1054.4−0321	2.13 ± 0.22	56 ± 10	0.40 ± 0.07	...	...	1085/1119	0.76	0.76

Notes. Best-fit parameters and 1σ uncertainties for our beta model fits with the power-law exponent β set to the best-fit value found in P10 (0.86). From left to right the columns give the normalization of the beta profile, y₀, the core radius of the major axis, r_c, the relative value of the core radius, r_c/r₅₀₀, the ellipticity, , the position angle of the major axis in degrees east of north, θ, the χ² and dof for the fit, and the goodness of fit quantified by the probability to exceed the given χ²/dof based on the standard χ² probability distribution function and also empirically by the fraction of our 1000 noise realizations that produce a larger χ² value when a model cluster is added to them (see Section 6.2). The values of r_c/r₅₀₀ for the spherical fits can be compared to the nominal value of 0.20 given in P10 based on spherical fits of SPT SZ data. See Section 6.2 for a description of how the parameter uncertainties are calculated. Note that we have included the 4.3% uncertainty in our flux calibration in the error estimates for y₀.

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Our cluster images are filtered by both the atmospheric noise subtraction and the Bolocam PSF. Therefore, prior to fitting a model to our data, we need to filter the cluster model in an identical way. First, we generate a two-dimensional image using the cluster model, computed directly from the isothermal beta model and via a line-of-sight integration of the Nagai pressure model. Next, the model image is convolved with the Bolocam PSF and the measured transfer function. In practice, the transfer function convolution is performed via multiplication in Fourier space. This filtered model is then compared to our data map, using all of the map pixels contained within a radius where the minimum coverage is greater than 25% of the peak coverage. This radius is typically between 6 and 7 arcmin. We use an iterative least-squares technique to determine the best-fit parameters for each model; we estimate the uncertainty on each parameter via the standard deviation of the best-fit parameter values estimated from noise realizations with model clusters added to them (see Section 6.2).

For each model, we fit both a scale radius (r_c for the beta model and r_s for the Nagai model) and a normalization (y₀ for the beta model and p₀ for the Nagai model). Additionally, as described above, we fit for the observationally unconstrained DC signal level of the map, δT. We also fit for a centroid offset relative to the X-ray pointing center. The offsets for the five clusters range from ≃(0–20) ± 5 arcsec, indicating there are no major differences in the X-ray and SZ centroids. Finally, when we allow the model to be elliptical in the image plane, we fit for the ellipticity and position angle θ. A complete list of the best-fit parameters for spherical and elliptical versions of both models is given in Tables 2 and 3.

All of these clusters have been studied extensively at a wide range of wavelengths, providing us with a large number of published results to compare our model fit parameters. In general, our results agree well with those found from previous studies. In particular, when we fit a spherical beta model to our data using the values of β given in B06, which were determined using Chandra X-ray data and 30 GHz interferometric SZ data, our best-fit values for y₀ agree quite well, with the exception of A697 (see Table 4). The best-fit values we find for r_c are also consistent with those determined in B06, with the exception of A1835 and MS 0451.6−0305. Additionally, the ellipticity and orientation of our elliptical fits are in general consistent with previously published results (Maughan et al. 2008; Donahue et al. 2003; De Filippis et al. 2005; Piffaretti et al. 2003; Girardi et al. 2006; McNamara et al. 2006; Schmidt et al. 2001; Neumann & Arnaud 2000). Finally, the best-fit concentration parameters from our Nagai model fits to three of the five clusters are consistent with those found from X-ray data (Arnaud et al. 2010); MS 0015.9+1609 and MS 1054.4−0321 have significantly lower values of c₅₀₀. We describe each cluster in detail below.

• A697. We find that A697 has a significant ellipticity, and it is not described well by either a beta model or a Nagai model. The poor model fits result from the cluster appearing significantly extended in the SW direction, and extremely compact in the NE direction. A697 is the only cluster in our sample that is not adequately described by the models. The ellipticity we find for A697, = 0.37 ± 0.05, is roughly consistent with the ellipticity of ≃0.25 found from X-ray data (De Filippis et al. 2005; Maughan et al. 2008; Girardi et al. 2006). We find a position angle of −24 deg for A697, which is similar to the value of −16 deg found by Girardi et al. (2006), but somewhat misaligned to the position angle of 16 deg found by De Filippis et al. (2005).
• A1835. We detect an ellipticity in our image of A1835, and we find that it is described well by either an elliptical beta or Nagai model. Our best-fit spherical Nagai model parameters with $\mathcal {A} = 0.9$, $\mathcal {B} = 5.0$, and $\mathcal {C} = 0.4$ (p₀ = 11.1 ± 2.6 × 10⁻¹¹erg cm⁻³, r_s = 4.6 ± 1.1 arcmin) are consistent with the values found in Mroczkowski et al. (2009; p₀ = 13.6 × 10⁻¹¹ erg cm⁻³, r_s = 4.3 arcmin) using a combination of SZA SZ data and Chandra X-ray data. X-ray measurements find an ellipticity of 0.1–0.2 for A1835 (De Filippis et al. 2005; McNamara et al. 2006; Schmidt et al. 2001), consistent with the value of = 0.27 ± 0.07 we find with Bolocam. We find a position angle of −16 deg for A697, which is similar to the values of 7, −20, and −30 deg found by De Filippis et al. (2005), McNamara et al. (2006), and Schmidt et al. (2001).
• MS 0015.9+1609. MS 0015.9+1609 appears to be elliptical in our image, and it is described well by either a spherical or elliptical model. As with A697 and A1835, X-ray data favor slightly lower ellipticities, ≲ 0.20, compared to what we find with Bolocam, = 0.24 ± 0.08 (De Filippis et al. 2005; Maughan et al. 2008; Piffaretti et al. 2003). We find a position angle of 68 deg for MS 0015.9+1609, fairly close to the value of 47 deg found by Piffaretti et al. (2003), but almost orthogonal to the value of −49 deg found by De Filippis et al. (2005).
• MS 0451.6−0305. MS 0451.6−0305 also appears to be elliptical in our image and is adequately described by either an elliptical beta or Nagai model. We find an ellipticity of = 0.26 ± 0.06 for MS 0451.6−0305, in excellent agreement with ellipticities determined using X-ray data (De Filippis et al. 2005; Donahue et al. 2003). Additionally, our best-fit position angle of 85 deg agrees well with the value of 84 deg found by De Filippis et al. (2005) and the value of −75 deg found by Donahue et al. (2003).
• MS 1054.4−0321. Our image of MS 1054.4−0321 shows no evidence for ellipticity, and it is described well by either an elliptical or spherical model. Although MS 1054.4−321 does not appear to be elliptical in our data, X-ray data show a clear ellipticity oriented along the east–west direction (Neumann & Arnaud 2000; Jeltema et al. 2001). However, our non-detection of an ellipticity, = 0.09 ± 0.07, is only marginally inconsistent with the X-ray value determined by Neumann & Arnaud (2000), = 0.29.
• SDS1. We have attempted to fit cluster models to the SDS1 map, but we find best-fit amplitudes that are consistent with 0 and best-fit scale radii that are large compared to the size of our images (i.e., scale radii that are large enough to produce profiles that are approximately constant over the entire image).

Since we compute the transfer function for each cluster using the best-fit elliptical Nagai model for that cluster, our deconvolved images necessarily have some model dependence. In order to quantify the amount of model dependence, we have computed transfer functions for a range of elliptical Nagai models for one of the clusters in our sample, MS 0451.6−0305. Relative to the best-fit model, we have varied the scale radius, r_s, the ellipticity, , the position angle, θ, and the centroid location, δR.A. and δdecl., by increasing and decreasing each parameter individually by its 1σ uncertainty.⁸ We then deconvolved our processed map of MS 0451.6−0305 using the transfer function computed from each model and subtracted the resulting map from the one produced using the transfer function for the best-fit model. In each case, the residual map was approximately white, with an rms of 1.5, 0.6, and 0.5 μK_CMB for variations in r_s and , θ, and δR.A. and δdecl., respectively. Since the typical noise rms of our deconvolved maps is ≃10 μK_CMB, the additional rms introduced by our uncertainty in determining the model used for calculating a transfer function is quasi-negligible. Note that the best-fit elliptical Nagai model will not provide an exact description of a real cluster. However, the elliptical Nagai model does provide an adequate description of four of the five clusters we have observed, indicating that the difference between the true cluster profile and the model profile is in general less than our noise. Therefore, the artifacts in our deconvolved map produced by using a model to describe the cluster will be smaller than the artifacts produced by our measurement uncertainty on the best-fit model.

Additionally, we created a deconvolved map of MS 0451.6−0305 using the transfer function for a point-like source. The resulting profile is significantly different from the profile obtained using the transfer function for the best-fit Nagai model, indicating that the naive calculation of a transfer function using a point source is inadequate. Compared to using the transfer function calculated from the best-fit elliptical Nagai model, the peak decrement is reduced by ≃50 μK_CMB, while the magnitude of the SZ signal at the edge of the map is increased by ≃50 μK_CMB (i.e., the deconvolved cluster image from a point-source transfer function is systematically broader).

As described in Section 4, we use the best-fit elliptical Nagai model to determine the DC signal level in our maps since its value is unconstrained by our data. The measurement uncertainty in the value of the DC signal is small, <1 μK_CMB, but there is a large amount of uncertainty in the model at large radii. Based on the results from Borgani et al. (2004), N07, and Piffaretti & Valdarnini (2008) presented in Arnaud et al. (2010), the rms scatter in the pressure profiles from cluster to cluster in simulations is ≲25%. Therefore, we include an additional 25% systematic uncertainty on the DC signal that we add to the deconvolved map. Specifically, we estimate that the model uncertainty in the DC signal level of our map is 25% of the signal level at the edge of the map.

As mentioned in Section 3, we make the approximation that the noise covariance matrix is diagonal in our processed maps (i.e., there are no noise correlations between pixels). Although fluctuations in both the atmospheric emission and the CMB are correlated over many pixels, the high-pass filter we apply to our data timestreams eliminates these correlations within our ability to measure them. To test for noise correlations, we added processed cluster images of the best-fit models to the 1000 noise realizations for each cluster (i.e., for a given cluster we separately added each of the four best-fit cluster models from Tables 2 and 3 to each of the 1000 noise realizations). We then fit a model of the same type (e.g., if we added the best-fit elliptical Nagai model to the noise realization, then we fit an elliptical Nagai model to the cluster-model-plus-noise map) to each of these model-cluster-plus-noise maps and examined the distribution of χ² values for these fits. For all five clusters, we found that the distribution of χ² values matched the predicted χ² distribution obtained from the diagonal noise covariance matrix.⁹ As an example, the fit quality of our measured χ² distribution for the elliptical-Nagai-model-plus-noise realizations to the predicted χ² distribution, quantified by a probability to exceed (PTE), is 0.58, 0.02, 0.08, 0.66, and 0.21 for A697, A1835, MS 0015.9+1609, MS 0451.6−0305, and MS 1054.4+321 (see Figure 7). Therefore, we conclude that there is a negligible amount of correlated noise in our processed images and our assumption that the noise covariance matrix is diagonal is valid. Furthermore, we have estimated all of our parameter uncertainties (p₀, r_s, etc.) directly from the distribution of values calculated from our noise realizations. Consequently, any failure of our assumption that the covariance matrix is diagonal for the processed images will only affect the pixel-weighting and χ² values for our model fits.

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However, when we perform the same test of fitting a model to our model-cluster-plus-noise maps using noise realizations that have been deconvolved with our transfer function, the result is significantly different. The distribution of χ² values calculated from our deconvolved noise realizations is significantly broader than the predicted distribution based on uncorrelated noise (see Figure 7). This result is not surprising since the deconvolution enhances the large-scale signals in the images, including residual atmospheric noise and CMB fluctuations. Since the noise in the deconvolved images is significantly spatially correlated, our assumption that the noise covariance matrix is diagonal fails. Therefore, we estimate the uncertainties for the deconvolved images using the spread in values for the noise realizations rather than from the diagonal elements of the noise covariance matrix (e.g., the uncertainties in the radial profiles are determined from the rms spread in the radial profiles of the noise realizations). This is the same technique used by Nord et al. (2009) and Basu et al. (2010) to analyze APEX-SZ data.

The model fits to the model-cluster-plus-noise realizations also provide us with estimates of the uncertainties and biases associated with our model-parameter fitting. Specifically, we obtain 1000 best-fit values for each parameter; the standard deviation of these values then gives the uncertainty on our estimate of that parameter in our actual data map. These uncertainties are given in Tables 2 and 3. In addition, if our parameter estimation algorithm is free from biases, then we should, on average, recover the parameters of the input model that was added to the noise realizations. In practice, we find a small, but measurable, bias in our estimates of the pressure normalization and scale radius in our fits; the bias is typically ≲10% of the uncertainty on each parameter. We find no measurable bias in our estimates of the other fit parameters.

Additionally, we have used our signal-free SDS1 maps to further verify our model-fitting procedure and to search for any components of the noise that have not been included in our noise estimate. First, we inserted model clusters into the SDS1 data timestreams based on the best-fit elliptical Nagai profile for each of the five clusters in our sample. These data were then processed and an elliptical Nagai model was fit to each resulting image. In each fit, there are 6 free parameters (p₀, r_s, , θ, δRA, and δdec), giving us a total of 30 fit parameters for the 5 model clusters. Of these 30 fit parameters, 17 (57%) are within 1σ of the input value, 26 (87%) are within 2σ of the input value, and all 30 are within 3σ of the input value; these results indicate that our model fitting and parameter error estimation are working properly. Additionally, we obtain a reasonable goodness of fit for the models using these best-fit parameters, quantified by a PTE ≃0.8, providing further evidence that there is no significant noise in the data that has not been included in our noise estimate. Note that since the five cluster model profiles are fairly similar, we obtain comparable PTEs for all five profiles.

This appendix includes images and radial profiles of the processed and deconvolved maps, an image of the processed residual map after subtracting the best-fit elliptical Nagai model, and an image of one of the 1000 noise estimates generated via jackknife realizations of the data and a model for the astronomical noise (see Figures A1–A6). We have smoothed all of the images using a Gaussian beam with an FWHM of 58 arcsec. A white dot representing the FWHM of the effective PSF for these beam-smoothed images is given in the lower left of each image. The solid white contour lines in the images represent an S/N of −2, − 4, ..., and the dashed white contour lines represent an S/N of +2, + 4, ... The deconvolved images contain a significant amount of noise that is correlated over large angular scales, along with a model-dependent DC signal offset, and we therefore do not display noise contours on the deconvolved images. The error bars on the radial profiles are estimated from the spread in radial profiles computed from our noise realizations, and therefore do include all of the large-angular-scale noise correlations (although they do not include the uncertainty in the DC signal level of the image). Note that the radial profile bins for the deconvolved images are correlated due to the large-angular-scale noise present in those images.

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