ARRAY FOR MICROWAVE BACKGROUND ANISOTROPY: OBSERVATIONS, DATA ANALYSIS, AND RESULTS FOR SUNYAEV–ZEL'DOVICH EFFECTS

Before the 60 cm antennas were mounted onto the platform, their beam patterns were individually measured in the far field (∼100 m) by scanning a wide-band white source with a system retrofitted from a commercial equatorial mount (Koch et al. 2006). Figure 1(a) shows an example. An azimuthal average of the beam pattern (Figure 1(b)) reveals a full width at half-maximum (FWHM) of about 23 arcmin. The first side lobe is about 20 db below the primary peak and the beam profile is as per designed. The index of asymmetry (Wu et al. 2001) is calculated showing that the discrepancy of the beam pattern from its azimuthal average is below 10% within the FWHM. All beam patterns of the antennas are well fitted by a Gaussian beam of 23 arcmin within the 2σ region for <10% error.

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In 2007, AMiBA was used with the seven 57.6 cm diameter antennas close-packed on the platform, providing 21 baselines of three different lengths L^b = 60.6, 105.0, and 121.2 cm. Each baseline operates at two frequency channels centered at f = 90 and 98 GHz, each with a bandwidth of 8 GHz. Thus, the three baseline lengths correspond to six multipole bands $\ell =2\pi \sqrt{u^2+v^2}=2\pi L^b f/c=[1092,1193]$, [1193, 1295], [1891, 2067], [2067, 2243], [2184, 2387], [2387, 2590]. These baselines have a sixfold symmetry (Ho et al. 2009). To achieve better u–v coverage and thus a better imaging capability, we observe each target at eight different polarization angles of the platform with an angular step of 75, uniformly covering azimuthal angles in the u–v plane with 48 discrete samples for each of the three different baseline lengths at each of the two frequency bands. This makes a total of 288 discrete u–v samples. Figure 2 shows such typical u–v coverage with the corresponding noise-weighted point-spread function from a typical observation, the so-called dirty beam. The FWHM of the dirty beam is about 6 arcmin, defining the synthesized resolution of the seven-element AMiBA. When using an equal-weighted (rather than the current noise-weighted) scheme to construct the dirty beam, we still obtain a similar resolution of about 6 arcmin because the noise level for each baseline is about the same.

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To minimize, if not completely remove, the effects from ground pickup and from a DC component in the electronics that leaks into the data, we adopt a "two-patch" observing strategy that alternates between the target and a region of nominally blank sky. This strategy is similar to that used by CBI for the measurement of CMB polarization (Readhead et al. 2004).

In this mode, a leading patch that contains our target is tracked for 3 minutes, and then a trailing patch of the presumably blank sky is tracked for the same period of time. The trailing patch is located at an R.A. 3^m10^s greater than the R.A. of the target. Since it takes 10 s for the telescope to slew between the fields, the two patches are observed over identical azimuth and elevation tracks. During both tracks the platform polarization angle is controlled so that each baseline corresponds to a fixed u–v mode. The recorded data during the two tracks should then contain the same contribution from ground pickup, which is removed by a subsequent differencing of the two tracks. This strategy requires that both the electronic offset and the ground emission are stable within 6 minutes, which is far shorter as compared with the measured time dependence (Lin et al. 2009). The penalty for adopting such a two-patch strategy is the loss of efficiency by a factor of 4, where a factor of 2 comes from doubling the observing time and the other factor of 2 from doubling the noise variance when differencing the two patches. We also note that there is a CMB component as well as possible combinations from Galactic foreground and extragalactic point sources in the differencing map though at a level much lower than the expected SZE of massive clusters (Liu et al. 2009). Nevertheless before observations we did check our fields for such contaminations.

Figure 3 demonstrates the application of the two-patch observing strategy for SZE mapping. The left panel is an image constructed from the leading patch, containing cluster Abell 2142 (see Section 4 for the formalism and procedure of data analysis). The middle panel shows the corresponding image from the trailing patch of blank sky. It is clear that both patches are contaminated by signals much stronger than the SZE (mainly from the ground pickup), and that the images are closely similar. The difference of the two patches reveals the cluster signal (right panel), whose signal strength is only about 1% of the contamination signal in individual patches.

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To calibrate these data, we observe planets roughly every 3  hr using the same two-patch observing strategy. This time the two patches are each observed for 4 minutes and separated by 4^m10^s in R.A. Such a separation in R.A. is equivalent to an angular separation of at least 57 arcmin, assuring that leakage of the planetary signal into the second patch is small, less than 0.1% of the planetary flux density according to the primary beam pattern measured in Section 3.1. The detailed formalism on how the data are converted to calibrated visibilities is presented in Section 4.3, and the determination of the planetary flux densities is discussed by Lin et al. (2009). Long-duration observations of the planets show that the system is stable at night (between about 8 p.m. and 8 a.m.) so that a planetary calibration every 3  hr controls the calibration error to be within 5% in gain and 0.1 radian in phase (Lin et al. 2009).

To test the overall performance of our observing, calibration, and analysis strategy, we first used the Sun to successfully phase calibrate an observation of Jupiter, and then Jupiter to phase and amplitude calibrate an observation of Saturn. The left plot of Figure 4 shows a dirty image of Saturn calibrated by an observation of Jupiter made 4 hr later. The FWHM of this image is identical to that of the dirty beam shown in Figure 2, and the index of asymmetry (Wu et al. 2001) of the cleaned image is well below 5%.

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After first light of the seven-element AMiBA in 2006 September, which was a drift scan on Jupiter and whose image was quickly constructed, several months of start-up tests were used to study the instrumental properties, to fine tune our system, to study the ground pickup, and to find an optimal observing strategy. During this period, Saturn, Mars, Venus, and the Crab Nebula were also observed for system tests. In 2007 February, when testing the two-patch observing strategy, we detected a first extragalactic object, quasar J0325 + 2224 with a total (unresolved) flux density of 550 mJy (the right plot of Figure 4).

After further fine tuning of the system, in 2007 April we detected our first SZE cluster, Abell 2142 with a central SZE decrement of around −320 mJy beam⁻¹. During 2007, we observed a total of six S–Z clusters with 0.091 ⩽ z ⩽ 0.322: A1689, A1995, A2142, A2163, A2261, and A2390.

The observations were always at night time, when the system is stable. Due to the weather, system engineering, and observability of the targets, a total of roughly 150 observing hours was spent in observing the clusters or their trailing fields over a span of five months, from April to August in 2007. During this period a lot of efforts were dedicated to the system tests and tuning (Lin et al. 2009; Koch et al. 2009a). Table 1 summarizes the pointing directions, the observing and integration times, and the synthesized beam sizes, for each of the six clusters. Due to the two-patch observing strategy, which modulates between on- and off-source positions, the total time spent during cluster observations is double the on-cluster time listed in the table. The integration time on source (t_int) is less than the actual observing time (t_obs) because of the data trimming that will be discussed in Section 4.1.

Table 1. Observing Log for AMiBA SZE Clusters

	Pointing Directions (J2000)
Cluster	R.A.	Decl.	tobs (hr)	tint (hr)	Synthesized Resolution
A1689	13h11.49m	−1°2047	8.75	7.11	63
A1995	14h52.84m	58°0280	15.90	5.56	66
A2142	15h58.34m	27°1361	7.05	5.18	65
A2163	16h15.57m	−6°0743	7.80	6.49	66
A2261	17h22.46m	32°0762	19.15	8.87	64
A2390	21h53.61m	17°4171	16.05	11.02	64

Notes. Observing log for the AMiBA observations of six clusters of galaxies, including the pointing directions, the observing (t_obs) and integration (t_int) times per baseline, and the synthesized resolution (syn. res.; the FWHM of the azimuthally averaged dirty beam).

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The six clusters were chosen based on observational convenience, the expected flux density based on SZE observations at other frequencies, and the existence of extensive published X-ray data. The observational convenience requires that the targets can be seen at the Mauna-Loa latitude N1954, at the night time when the system is stable (between about 8 p.m. and 8 a.m.), and during the main observing period between April and August in 2007. Several cluster catalogs are compiled and studied for the expected SZE flux at AMiBA frequency, including OVRO (30  GHz; Mason et al. 2001), OVRO/BIMA (30 GHz; Grego et al. 2001; Reese et al. 2002), VSA (34 GHz; Lancaster et al. 2005), and SuZIE II (145, 221, and 355 GHz; Benson et al. 2004). The redshift range of z ≲ 0.3 is so chosen that massive clusters of a typical virial radius of few Mpc is resolvable by AMiBA, whose resolution (6') corresponds to a physical scale of 1.6  Mpc at z = 0.3 in a flat ΛCDM cosmology with matter density parameter Ω_m = 0.3, cosmological constant density parameter Ω_Λ = 0.7, and Hubble constant H₀ = 70 km s⁻¹ Mpc⁻¹. Among these six clusters, only A1689 and A2390 are known to have relaxed with a nearly isothermal profile. A1689 has a regular morphology in X-ray while A2390 is elongated. A2163, by contrast, is a merger with shocks and high-temperature regions, and A2142 is a merger with a strong frontal structure.

Due to the missing flux problem, most studies for the scientific implications of our results require the assistance of X-ray-derived properties of the clusters. In Table 2, we summarize the critical X-ray-derived parameters for the AMiBA clusters, which will be used by our companion science papers (Huang et al. 2009; Koch et al. 2009b; Liu et al. 2009; Umetsu et al. 2009). The X-ray temperatures are from Reese et al. (2002) for A2163, A2261, A1689, and A1995, from Markevitch et al. (1998) for A2142, and from Allen (2000) for A2390. The power-law index β in the isothermal β-model and the core radius θ_c are from Reese et al. (2002) for A2163, A2261, A1689, and A1995, from Sanderson & Ponman (2003) for A2142, and from Böhringer et al. (1998) for A2390. The X-ray core radii are mostly below 1 arcmin except for A2142 and A2163. These parameters are mostly X-emission weighted but not corrected for any cooling flows, which we further discuss in some science analyses such as the estimation of the Hubble constant H₀ (Koch et al. 2009b).

Initially, the cluster data must be processed to remove baselines for which the system is malfunctioning. In addition to checking the instrumental logs of receivers and correlators, nightly drift-scan data for Jupiter or Saturn are used to identify obvious problems. We then apply Kolmogorov–Smirnov (K–S) test to filter out data sets for which the noise appears non-Gaussian (Nishioka et al. 2009). In the K–S test, a 5% significance level is used throughout and shown to be efficient in identifying data sets with known hardware problems. Finally, we remove periods of data where occasional mount control problems appear.

After this flagging, we difference the tracking data c^b(τ_m; t) in the two-patch observation in order to remove ground pickup and the electronic DC offset. 4σ outliers in the temporal domain are clipped before the differenced tracking data are integrated over the 3 minutes (4 minutes for the calibration) tracking period to yield the lag data c^b(τ_m) ≡ 〈c^b(τ_m; t)〉_t. Thus, each two-patch observation yields a set of four lags c^b(τ_m) (m = 1–4). We carefully verified that the noise of the lag data is white within the timescale of 10 minutes, assuring the scaling of the noise level with the two-patch integration time (Nishioka et al. 2009).

Further flagging of 4σ outliers is applied in the visibility space (see Section 4.3). Table 3 shows the percentages of data that are flagged out by each step. On average the fraction of good data is about 60%.

We must invert Equation (5) to convert the four lags c^b(τ_m) to two visibilities v^b(f). Ideally, if τ_m ≡ τ is continuous, then we need only perform an inverse Fourier transform from the τ domain to the f domain. However, τ here is sampled discretely, and conservation of the degrees of freedom provided by four-lag measurements implies that we can obtain at best two uncorrelated complex visibilities, with no redundancy.

We perform the inversion first by dividing the 16 GHz frequency band into two, [f₁, f_d] and [f_d, f₂], each associated with one "band visibility", and define a "visibility vector" as

$$\begin{equation} {\bf v}^b =\left[ \begin{array}{c} \Re \lbrace v^b([f_1,f_d])\rbrace \\[4pt] \Im \lbrace v^b([f_1,f_d])\rbrace \\[4pt] \Re \lbrace v^b([f_d,f_2])\rbrace \\[4pt] \Im \lbrace v^b([f_d,f_2])\rbrace \end{array} \right]. \end{equation} \tag{ 6 }$$

If we also write the four lags in column-vector form c^b ≡ c^b(τ_m) (m = 1–4), then the visibility-to-lag transform resulted from Equation (5) is simply c^b = U^bv^b, where U^b is a 4 × 4 matrix. The four columns in U^b can easily be obtained from Equation (5) as the four vectors c^b when setting each component in v^b to unity in turn while zeroing the other three components. Subsequently, the inversion is

$$\begin{equation} {\bf v}^b=[{\bf U}^b]^{-1} {\bf c}^b = {\bf V}^b {\bf c}^b. \end{equation} \tag{ 7 }$$

Once given any lag data c^b, we can use Equation (7) to construct the band visibilities v^b([f₁, f_d]) and v^b([f_d, f₂]).

In principle, if R^b(f) and ϕ^b(f) are accurately known, then V^b can be accurately determined so that there is no need for further calibration. However, in reality R^b(f) and ϕ^b(f) are time dependent so that V^b is also time dependent and thus needs to be calibrated for every observation. Because the transform (Equation (7)) is linear, any reasonable choice of f_d, R^b(f) and ϕ^b(f) should yield the same v^b after calibration. Hence, we make the simplest choice f_d = (f₁ + f₂)/2 = 94 GHz, R^b(f) = 1, and ϕ^b(f) = 0, and then calibrate the data using the formalism described in Section 4.3. This choice makes the inversion (Equation (7)) exactly a discrete inverse Fourier transform that accounts for the band-smearing effect. An alternative, and equally natural, choice might be to have equal power in the two frequency bands, with f_d so chosen that $\int _{f_1}^{f_d} R^b(f) df = \int _{f_d}^{f_2} R^b(f) df$, but this is difficult to implement since R^b(f) is time dependent.

For calibration, we observe planets using the same two-patch observing strategy (Section 3.4), the same time-domain preprocessing (Section 4.1), and the same lag-to-visibility transform (Equation (7), Section 4.2). The underlying band visibilities of these calibrators have the form

$$\begin{equation} {\bf v}^{b({\rm thy})}_\ast = \left[\begin{array}{cccc} M_1\\[4pt] 0\\[4pt] M_2\\[4pt] 0 \end{array} \right], \end{equation} \tag{ 8 }$$

where M₁ and M₂ are real constants, corresponding to the fluxes at different frequencies. However, the band visibilities constructed from observational data using Equation (7), where we have chosen R^b(f) = 1 and ϕ^b(f) = 0, will take the form

$$\begin{equation} {\bf v}^{b({\rm obs})}_\ast =\left[\begin{array}{c} M_1^{b({\rm obs})} \cos \phi _1^{b({\rm obs})} \\[4pt] M_1^{b({\rm obs})} \sin \phi _1^{b({\rm obs})} \\[4pt] M_2^{b({\rm obs})} \cos \phi _2^{b({\rm obs})} \\[4pt] M_2^{b({\rm obs})} \sin \phi _2^{b({\rm obs})} \end{array} \right]. \end{equation} \tag{ 9 }$$

Since the involved operations are linear, there must exist a calibration matrix C^b that corrects this discrepancy, i.e.,

$$\begin{equation} {\bf v}^{b({\rm thy})}_{\ast } = {\bf C}^b {\bf v}^{b({\rm obs})}_{\ast }. \end{equation} \tag{ 10 }$$

We may model C^b as

$$\begin{equation} {\bf C}^b = \left[\begin{array}{cccc} a^b_{1} \cos \phi ^b_{1} & -a^b_{1} \sin \phi ^b_{1} & 0& 0\\[4pt] a^b_{1} \sin \phi ^b_{1} & a^b_{1} \cos \phi ^b_{1} & 0& 0\\[4pt] 0& 0& a^b_{2} \cos \phi ^b_{2} & -a^b_{2} \sin \phi ^b_{2} \\[4pt] 0& 0& a^b_{2} \sin \phi ^b_{2} & a^b_{2} \cos \phi ^b_{2} \end{array} \right], \end{equation} \tag{ 11 }$$

where the parameters a^b_i and ϕ^b_i account for the gain and phase corrections relevant to the effects from R^b(f) and ϕ^b(f) in Equation (5), respectively. By comparing the moduli and phases of v^b(thy)_* and v^b(obs)_*, we obtain

$$\begin{eqnarray} a^b_{1}&=&{M_1}/{M^{b({\rm obs})}_1}, \quad a^b_{2}={M_2}/{M^{b({\rm obs})}_2}, \nonumber\\&&[-6.5pt]\\[-6.5pt] \phi ^b_{1}&=&-\phi ^{b({\rm obs})}_1,\quad \phi ^b_{2}=-\phi ^{b({\rm obs})}_2. \nonumber \end{eqnarray} \tag{ 12 }$$

Finally, for a set of observed lag data c^b, the calibrated band-visibility vector is

$$\begin{equation} {\bf v}^b = {\bf C}^b {\bf V}^b {\bf c}^b. \end{equation} \tag{ 13 }$$

We emphasize that this is a linear transform, so different initial choices for R^b(f) and ϕ^b(f) that generate different V^b should lead to different C^b but the same final calibrated band visibilities v^b. For each two-patch observation, we obtain one v^b for each cross-polarization mode and each baseline. For each baseline and each day, 4σ outliers in |v^b| are flagged.

In further analyses such as image making and model fitting for the cluster profile, a reliable estimate for the error in the band visibilities derived above will be needed. First, the two-patch differenced lag data c^b(t) ≡ c^b(τ_m; t) are modeled as the linear sum of the signal and noise, c^b(t) = s^b_c(t) + n^b_c(t). Since such data for clusters are dominated by electronic noise (i.e., n^b_c(t) ≫ s^b_c(t)) and the noise is white (Nishioka et al. 2009), the noise variance associated with each time-integrated c^b ≡ 〈c^b(t)〉_t can be directly estimated as

$$\begin{equation} \big({\bf \sigma }^b_{c}\big)^2 \equiv \big\langle \left[{\bf n}^b_{c}\right]^2 \big\rangle \equiv \big\langle \big\langle {\bf n}^b_{c}(t) \big\rangle _t^2 \big\rangle \approx \frac{\langle [{\bf c}^b (t)]^2 \rangle _t}{\#_t}, \end{equation} \tag{ 14 }$$

where \#_t is the number of temporal data in the time integration. Then (σ^b_c)² constitutes the diagonal elements of the lag–lag noise correlation matrix N^b_c ≡ 〈n^b_cn^{b T}_c〉. The off-diagonal elements are found to be less than 10% of the diagonal elements (Nishioka et al. 2009) so we approximate them as zeros.

In a similar fashion, the band visibility obtained from Equation (13) can be modeled as a linear sum of the signal and noise, v^b(t) = s^b_v(t) + n^b_v(t). Finally, the noise correlation matrix for the calibrated band visibilities is

$$\begin{equation} {\bf N}^b_v \equiv \big\langle {\bf n}^b_v {{\bf n}^b_v}^{\rm T} \big\rangle \approx {\bf C}^b {\bf V}^b {\bf N}^b_c {{\bf V}^b}^{\rm T} {{\bf C}^b}^{\rm T}. \end{equation} \tag{ 15 }$$

We construct the "dirty" image of a cluster by making a continuous inverse Fourier transform of all the band visibilities from all days and all baselines, with noise weighting based on the noise variance. The instantaneous u–v coverage is improved in AMiBA operation by rotating the platform to eight polarization angles, giving a uniform angular interval of 75 between sampled u–v modes k (see Section 3.2). Figure 6 shows the dirty images of these clusters. The flux decrement at the center of each target is evident, as expected for the SZE signal at the AMiBA center frequency of 94 GHz (see Section 2). The left part of Table 4 summarizes the angular sizes (azimuthally averaged FWHM) and the peak fluxes directly measured from these images.

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Table 4. Properties of AMiBA SZE Images

	Dirty Images		Cleaned Images
Cluster	Size	Flux (mJy)	Size	Flux (mJy)	S/N Ratio	Scale (Mpc)
A1689	(61)	−217	(57)	−168	6.0	(1.05)
A1995	(68)	−167	(68)	−161	6.4	(1.91)
A2142	77	−320	90	−316	13.7	0.92
A2163	78	−347	112	−346	11.7	2.24
A2261	(62)	−115	(58)	−90	5.2	(1.25)
A2390	74	−180	80	−158	6.6	1.78

Notes. Basic properties measured from the dirty and cleaned SZE images, including the angular sizes (azimuthally averaged FWHM), the peak flux, and the S/N ratios. For cleaned images, the peak flux is corrected for the attenuation by the primary beam due to an offset to the pointing center. The last column "scale" indicates the physical scale at z (see Table 2) corresponding to the angular size shown in the fourth column. Here, we have assumed a flat ΛCDM cosmology with Ω_m = 0.3, Ω_Λ = 0.7, and H₀ = 70 km s⁻¹ Mpc⁻¹. The brackets indicate that the cluster appears unresolved in the SZE image.

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A dirty beam for each cluster data set is constructed using the same method, and the dirty image and dirty beam are then processed by a CLEAN procedure (Hogbom 1974) in MIRIAD (MIRIAD-ATNF 2008) to yield a cleaned image. Figure 7 shows the cleaned SZE images of the six AMiBA clusters, where the cleaned regions are indicated by the white circles, which are the FWHM contour of the primary beam. This process significantly reduces the convolution effects from the finite u–v coverage. The basic properties measured from the cleaned images are summarized in the right part of Table 4. The apparent angular sizes, as compared with the synthesized resolution of about 6', indicate that we have partially resolved clusters A2142, A2163, and A2390, while A1689, A1995, and A2261 appear unresolved in these images. The SZE signals observed here are at the level of few hundred mJy, indeed consistent with the theoretical expectation ΔI_SZE ∼ −200 mJy as discussed in Section 2.

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In Table 4, the signal-to-noise (S/N) ratios are computed as the peak flux of the cleaned model (uncorrected for the primary beam attenuation) divided by the rms of the noise residual map. We note that the noise level in these results are dominated by the instrumental noise, leading to the fact that the scaling among the peak flux, S/N ratios, and integration time is consistent with the noise equivalent flux (point-source sensitivity) of 63 mJy hr^−1/2 for the on-source integration time in a two-patch observation (Lin et al. 2009), i.e., for a 2-hr observation, 1 hr per patch, the noise rms is 63 mJy. Simulations also show that our analysis method (described in Section 4.2–4.4) does not bias the SZE amplitude but induces a statistical error at the few-percent level. The calibration error is controlled within 5% (Lin et al. 2009). The systematic effect from CMB anisotropy is estimated to be at a similar level as the instrumental noise, and the point-source contamination causes an underestimate of the SZE amplitude by about 10% (Liu et al. 2009).

It is useful for science purposes to estimate parameters describing the structures of the SZE clusters from the visibilities. We employ a maximum-likelihood analysis to estimate the band visibilities taking the priors that all the visibilities represent the same circularly symmetric source and that the visibility moduli within a frequency band are constant. Thus, there are only six nonredundant band visibilities {v_p;  p = 1–6} of six multipole bands corresponding to the two frequency bands [fⁿ₁, fⁿ₂] (n = 1,  2) of three baselines lengths (see Section 3.2 and Table 5), i.e., the observed band visibilities are modeled as

$$\begin{eqnarray} v^b\big(\big[f^n_1,f^n_2\big]| \lbrace v_p\rbrace, {\bf x}_0\big) & = & B({\bf x}_0-{\bf x}_1) v_{p(\in b,n)}\nonumber \\ && \times\,\frac{1}{k^n_2-k^n_1} \int _{{\bf k}^n_1}^{{\bf k}^n_2} e^{i{\bf k}({\bf x}_0-{\bf x}_1)} d{\bf k},\quad \end{eqnarray} \tag{ 16 }$$

where x₀ is the SZE cluster center, x₁ is the pointing direction, B(x) is the primary beam, and the integration accounts for the frequency band-smearing effect. These {v_p} are equivalent to the band visibilities at x₁ = x₀ and therefore are real numbers. They indicate the cluster profile in the visibility space.

Table 5. SZE Centers and Visibility Profiles for AMiBA Clusters

	SZE Center x0 (J2000)
Cluster	R.A.	Decl.	v1 (mJy)	v2 (mJy)	v3 (mJy)	v4 (mJy)	v5 (mJy)	v6 (mJy)
A1689	13h11.41 ± 0.03m	−1°20.7 ± 04	−123 ± 92	−94 ± 89	−566 ± 109	−130 ± 108	−414 ± 185	85 ± 212
A1995	14h53.12 ± 0.06m	58°02.6 ± 10	−234 ± 86	−106 ± 80	−241 ± 138	−207 ± 126	−53 ± 216	−32 ± 200
A2142	15h58.30 ± 0.03m	27°13.8 ± 05	−508 ± 78	−366 ± 76	−140 ± 113	−205 ± 117	−9 ± 139	−206 ± 162
A2163	16h15.73 ± 0.04m	−6°09.5 ± 05	−652 ± 110	−332 ± 113	93 ± 167	−236 ± 158	−457 ± 298	−31 ± 260
A2261	17h22.46 ± 0.06m	32°08.9 ± 05	−34 ± 68	−108 ± 67	−160 ± 94	−78 ± 99	−412 ± 138	−37 ± 163
A2390	21h53.72 ± 0.05m	17°38.8 ± 07	−148 ± 87	−185 ± 75	−260 ± 107	−273 ± 112	−96 ± 141	−44 ± 166

Notes. The maximum-likelihood results for the coordinates of SZE cluster centers x₀ and band-visibility profiles {v_p}. The values v₁–v₆ correspond to the multipole bands of ℓ = [1092, 1193], [1193, 1295], [1891, 2067], [2067, 2243], [2184, 2387], [2387, 2590], respectively (see Sections 3.2, 4.6). v₁, v₃, v₅ correspond to the frequency band of [86, 94] GHz, and v₂, v₄, v₆ correspond to [94, 102] GHz. The three baseline lengths L^b are 0.606 m (for v₁, v₂), 1.05 m (for v₃, v₄), and 1.212 m (for v₅, v₆).

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The Markov chain Monte Carlo (MCMC) approach with Metropolis–Hastings sampling is used for this eight-dimensional likelihood analysis. For each cluster three MCMC chains of 200,000 samples are used. The results are given in Table 5 and Figure 8. For all clusters the expected SZE decrement in the flux density is evident. We note that although the primary CMB makes a nonnegligible contribution to our observed visibilities (Liu et al. 2009), it appears with random phase in u–v space and thus will not bias our analysis here but rather contribute as part of the error bars.

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We emphasize that these results for {v_p} and x₀ are independent of cluster model. They can be further compared or combined with other experimental results, or fitted with a specific cluster model, for further study. The approach here is different from that in Liu et al. (2009), where an isothermal β-model is directly fitted with the visibilities to estimate the central brightness I₀, with the core radius θ_c and the power index β taken from X-ray analysis.

A commonly used powerful test is the so-called sum-and-difference test, where the temporal data are divided into two subsets of equal size and then processed separately. Figure 9 shows an example for A2142, where the two half-data images (left and middle) show a clear signal while their difference (right) reveals no signal but noise. Their average is very close to the overall dirty image shown in Figure 6. We have verified that such feature for the existence of signal is independent of the scheme for dividing the data into two halves. All data of the six clusters have passed this test.

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We carried out several other tests for systematics. Long-duration (12 hr) observations of planets were used to check the stability of the system; an independent analysis path produced results for A2142 consistent to high accuracy with those presented here; and several two-patch blank-sky observations showed no signal above the expected CMB confusion (Lin et al. 2009). The noise properties of the lag data were investigated in detail indicating no systematics and no significant non-Gaussianity (Nishioka et al. 2009). The asymmetry of the antenna beams (Wu et al. 2001) was shown to be negligible. The pointing error is small enough that it has negligible effects on our results (Koch et al. 2009a), and the radio alignment was tuned to improve efficiency and tested for stability (J.-H. P. Wu et al. 2009, in preparation).