The Atacama Cosmology Telescope (ACT): Beam Profiles and First SZ Cluster Maps

Abstract

The Atacama Cosmology Telescope (ACT) is currently observing the cosmic microwave background with arcminute resolution at 148 GHz, 218 GHz, and 277 GHz. In this paper, we present ACT's first results. Data have been analyzed using a maximum-likelihood map-making method which uses B-splines to model and remove the atmospheric signal. It has been used to make high-precision beam maps from which we determine the experiment's window functions. This beam information directly impacts all subsequent analyses of the data. We also used the method to map a sample of galaxy clusters via the Sunyaev-Zel'dovich (SZ) effect and show five clusters previously detected with X-ray or SZ observations. We provide integrated Compton-y measurements for each cluster. Of particular interest is our detection of the z = 0.44 component of A3128 and our current non-detection of the low-redshift part, providing strong evidence that the further cluster is more massive as suggested by X-ray measurements. This is a compelling example of the redshift-independent mass selection of the SZ effect.

Full text

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arXiv:0907.0461v2 [astro-ph.CO] 31 Aug 2009
SUBMITTED TO APJ
Preprint typeset using LATEX style emulateapj v. 10/09/06
THE ATACAMA COSMOLOGY TELESCOPE (ACT): BEAM PROFILES AND FIRST SZ CLUSTER MAPS
A. D. HINCKS1, V. ACQUAVIVA2,3, P. ADE4, P. AGUIRRE5, M. AMIRI6, J. W. APPEL1, L. F. BARRIENTOS5, E. S. BATTISTELLI7,6,
J. R. BOND8, B. BROWN9, B. BURGER6, J. CHERVENAK10, S. DAS1,2, M. J. DEVLIN3, S. DICKER3, W. B. DORIESE11,
J. DUNKLEY12,1,2, R. DÜNNER5, T. ESSINGER-HILEMAN1, R. P. FISHER1, J. W. FOWLER1, A. HAJIAN2,1, M. HALPERN6,
M. HASSELFIELD6, C. HERNÁNDEZ-MONTEAGUDO13, G. C. HILTON11, M. HILTON14,15,16, R. HLOZEK12, K. HUFFENBERGER17,
D. H. HUGHES18, J. P. HUGHES19, L. INFANTE5, K. D. IRWIN11, R. JIMENEZ20, J. B. JUIN5, M. KAUL3, J. KLEIN3, A. KOSOWSKY9,
J. M. LAU21,22,1, M. LIMON23,3,1, Y.-T. LIN24,2,5, R. H. LUPTON2, T. MARRIAGE2, D. MARSDEN3, K. MARTOCCI25,1, P. MAUSKOPF4,
F. MENANTEAU19, K. MOODLEY14,15, H. MOSELEY10, C. B. NETTERFIELD26, M. D. NIEMACK11,1, M. R. NOLTA8, L. A. PAGE1,
L. PARKER1, B. PARTRIDGE27, H. QUINTANA5, B. REID20,1, N. SEHGAL21,19, J. SIEVERS8, D. N. SPERGEL2, S. T. STAGGS1,
O. STRYZAK1, D. SWETZ3, E. SWITZER25,1, R. THORNTON3,28, H. TRAC29,2, C. TUCKER4, L. VERDE20, R. WARNE14, G. WILSON30,
E. WOLLACK10, Y. ZHAO1
Submitted to ApJ
ABSTRACT
The Atacama Cosmology Telescope (ACT) is currently observing the cosmic microwave background with
arcminute resolution at 148GHz, 218GHz, and 277GHz. In this paper, we present ACT’s first results. Data
have been analyzed using a maximum-likelihood map-making method which uses B-splines to model and
remove the atmospheric signal. It has been used to make high-precision beam maps from which we determine
the experiment’s window functions. This beam information directly impacts all subsequent analyses of the
data. We also used the method to map a sample of galaxy clusters via the Sunyaev-Zel’dovich (SZ) effect,
and show eight clusters previously detected with X-ray or SZ observations. We provide integrated Compton-y
measurements for each cluster. Of particular interest is our detection of the z = 0.44 component of Abell 3128
and our currentnon-detectionof the low-redshiftpart, providingstrong evidencethat the furthercluster is more
massive as suggested by X-ray measurements. This is a compelling example of the redshift-independent mass
selection of the SZ effect.
Subject headings: cosmic microwave background – cosmology: observations – galaxies: clusters: general –
methods: data analysis
1Joseph Henry Laboratories of Physics, Jadwin Hall, Princeton Univer-
sity, Princeton, NJ, USA 08544
2Department of Astrophysical Sciences, Peyton Hall, Princeton Univer-
sity, Princeton, NJ USA 08544
3Department of Physics and Astronomy, University of Pennsylvania, 209
South 33rd Street, Philadelphia, PA, USA 19104
4Department of Physics & Astronomy, Cardiff University, 5 The Parade,
Cardiff, Wales, UK CF24 3AA
5Departamento de Astronomía y Astrofísica, Facultad de Física, Pontif-
icía Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
6Department of Physics and Astronomy, University of British Columbia,
Vancouver, BC, Canada V6T 1Z4
7Department of Physics, University of Rome “La Sapienza”, Piazzale
Aldo Moro 5, I-00185 Rome, Italy
8Canadian Institute for Theoretical Astrophysics, University of Toronto,
Toronto, ON, Canada M5S 3H8
9Department of Physics and Astronomy, University of Pittsburgh, Pitts-
burgh, PA, USA 15260
10Code 553/665, NASA/Goddard Space Flight Center, Greenbelt, MD,
USA 20771
11NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boul-
der, CO, USA 80305
12Department of Astrophysics, Oxford University, Oxford, UK OX1 3RH
13Max Planck Institut für Astrophysik, Postfach 1317, D-85741 Garching
bei München, Germany
14Astrophysics and Cosmology Research Unit, School of Mathematical
Sciences, University of KwaZulu-Natal, Durban, 4041, South Africa
15Centre for High Performance Computing, CSIR Campus, 15 Lower
Hope St. Rosebank, Cape Town, South Africa
16Southern African Astronomical Observatory, Observatory Road, Obser-
vatory 7925, South Africa
17Department of Physics, University of Miami, Coral Gables, FL, USA
33124
18Instituto Nacional de Astrofísica, Óptica y Electrónica (INAOE), To-
nantzintla, Puebla, Mexico
19Deptartment of Physics and Astronomy, Rutgers, The State University
of New Jersey, Piscataway, NJ 08854-8019
1. INTRODUCTION
A new generation of experiments is measuring the cos-
mic microwave background (CMB) at arcminute resolu-
tions.Within the past year alone, results from the
South Pole Telescope (Staniszewski et al. 2008), ACBAR
(Reichardt et al. 2009a), AMiBA (Umetsu et al. 2009),
APEX-SZ (Reichardt et al. 2009b), the Cosmic Background
Imager (Sievers et al. 2009), the Sunyaev-Zel’dovich Array
(Sharp et al.2009), andQUaD(Friedman et al.2009)havere-
vealed the ∼arcminute structure of the CMB with higher pre-
cision than ever. The angular power spectrum of temperature
fluctuations at these scales (ℓ ? 1000) will further constrain
20ICREA & Institut de Ciencies del Cosmos (ICC), University of
Barcelona, Barcelona 08028, Spain
21Kavli Institute for Particle Astrophysics and Cosmology, Stanford Uni-
versity, Stanford, CA, USA 94305-4085
22Department of Physics, Stanford University, Stanford, CA, USA 94305-
4085
23Columbia Astrophysics Laboratory, 550 W. 120th St. Mail Code 5247,
New York, NY 10027
24Institute for the Physics and Mathematics of the Universe, The Univer-
sity of Tokyo, Kashiwa, Chiba 277-8568, Japan
25Kavli Institute for Cosmological Physics, Laboratory for Astrophysics
and Space Research, 5620 South Ellis Ave., Chicago, IL, USA 60637
26Department of Physics, University of Toronto, 60 St. George Street,
Toronto, ON, Canada M5S 1A7
27Department of Physics and Astronomy, Haverford College, Haverford,
PA, USA 19041
28Department of Physics , West Chester University of Pennsylvania, West
Chester, PA, USA 19383
29Harvard-Smithsonian Center for Astrophysics, Harvard University,
Cambridge, MA 02138
30Department of Astronomy, University of Massachusetts, Amherst, MA,
USA 01003

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2The ACT Collaboration
modelsofthe earlyuniverse. Furthermore,secondaryfeatures
such as the Sunyaev-Zel’dovich (SZ) effect and gravitational
lensing probe the growth of structure.
With its first science release, the Atacama Cosmology
Telescope (ACT) now adds to these endeavors.
meter, off-axis Gregorian telescope, it was commissioned
on Cerro Toco in northern Chile in October of 2007. Its
current receiver is the Millimeter Bolometer Array Cam-
era (MBAC), containing three 32×32 arrays of transition
edge sensor (TES) bolometers observing at central frequen-
cies of 148GHz, 218GHz, and 277GHz, with beam full-
widths at half-maxima (FWHM) of 1.37′, 1.01′, and 0.91′,
respectively.It has operated for two seasons and is cur-
rently in its third season.In 2007 one month of science
observations were made using only the 148GHz array. The
other two frequencies were added for the 2008 season, which
lasted about 3.5 months.The telescope optical design is
described in Fowler et al. (2007). Hincks et al. (2008) and
Switzer et al. (2008) report on the telescope performance
and provide an overview of hardware and software systems.
The MBAC design and details of TES detector properties
and readout are in Niemack (2006), Marriage et al. (2006),
Battistelli et al. (2008), Niemack et al. (2008), Swetz et al.
(2008), Thornton et al. (2008), and Zhao et al. (2008).1
ACT is located at one of the premier sites for millimeter
astronomy because of the high altitude (5200m) and the dry
atmosphere. The precipitable water vapor (PWV) had a me-
dian value of 0.56mm during the nights of our 2008 season.
Nevertheless, atmospheric emission remains the largest sig-
nal external to the receiver in our raw data, a reality for any
ground-based millimeter-wave telescope. The atmospheric
power dominates only at low temporal frequencies and this
is the main reason we observe while scanning our telescope
in azimuth. Though much of the atmospheric power is below
the frequency of our 0.0978Hz scans, on typical nights the
atmosphere dominates the detector noise up to about 2Hz.
In this paper we present a map-making method designed to
model and remove the atmospheric signal in a manner which
is unbiased with respect to the celestial signal. The method
currently produces its best results on small scales (? 1◦), so
it is well-suited to making maps of objects with small angular
sizes. One of the most useful applications has been the study
of our instrumental beam with high signal-to-noise maps of
planets. The beam profile affects all aspects of data analysis,
includingcalibration,andwe providethe beamcharacteristics
in this paper. Additionally, we present new SZ measurements
of eight known clusters.
We proceedas follows: in §2 we introducethe map-making
method, showing both the theory and some qualitative prop-
erties; §3 describes how we analyzed our beams, and presents
the key measured parameters along with beam maps and ra-
dial profiles; window functions are derived in §4; §5 shows
a selection of clusters imaged with the mapper; and we con-
clude in §6.
A 6-
2. THE COTTINGHAM MAPPING METHOD
In this section, we present a technique for removing the
atmospheric power first described by Cottingham (1987)
and used by Meyer et al. (1991), Boughn et al. (1992), and
Ganga et al. (1993). The temporal variations in atmospheric
signalsaremodelledusingB-splines,aclassoffunctionsideal
1Reprints of all the references in this paragraph may freely be downloaded
from: http://www.physics.princeton.edu/act/papers.html.
for interpolation, discussed more below. The technique com-
putes maximum-likelihoodestimates of both the celestial and
the atmospheric signals, using all available detectors in a sin-
gle frequencyband. We refer to it hereafteras the Cottingham
Method.
In the following subsections, we give a mathematical de-
scription of the CottinghamMethod (§2.1), followed by a dis-
cussion of its benefits and a comparison to the “destriping”
method developed for Planck, which has close similarities
(§2.2). Our approach for including the effects of spatial vari-
abilityacross the detectorarraysis in §2.3. We discuss the use
of B-splines in §2.4, and finish by outlining our implementa-
tion of the method (§2.5) and map-making steps (§2.6).
2.1. The Algorithm
The measuredtimestream d is modelledas a celestial signal
plus an atmospheric component:
d = m+Bα+n,
(1)
where the pointing matrix P projects the celestial map m into
the timestream, B is a matrix of basis functions with ampli-
tudes α which model the temporal variation of atmospheric
power, and n is the noise. The timestream of measurements
d may be a concatenation of multiple detectors if they have
been properly treated for relative gain differences. Through-
out this paper, this is the case: all working detectors from one
frequency band are processed simultaneously.
We seek ? α and ? m, estimates of the atmospheric amplitudes
d−B? α. The maximum likelihood estimator is then given by
? m =?PTN−1P?−1PTN−1d′= Π?d−B? α?,
Π ≡?PTN−1P?−1PTN−1
the projection matrix and N ≡ ?nnT? is the noise covariance.
The projection matrix Π is designed in such a way that the
map estimate is not biased, in the sense that the error, ? m−m,
minimizesthevarianceofthe mappixelresidualswith respect
totheamplitudes ? α. Theresidualsarethedifferencesbetween
∆d = d′−P? m = d−B? α−PΠ?d−B? α?
where 1 is the identity matrix. We differentiate χ2:
and the celestial map, respectively. Eq. 1 prescribes that we
subtract the atmosphericterm to obtain the map estimate: d′=
the standard map-making equation (e.g., Tegmark 1997):
(2)
where we call
(3)
does not depend on m.
Given a set of basis functions B, the Cottingham method
the celestial signals measured in the timestream and the map
estimate:
= (1−PΠ)?d−B? α?,
(4)
∂χ2
∂? α=
∂
∂? α
?∆dTN−1∆d?
= −2BTN−1(1−PΠ)?d−B? α?.
stituent elements (c.f. Eq. 3) and simplifying. If we define the
following:
= −2BT(1−PΠ)TN−1(1−PΠ)?d−B? α?
The last equality can be obtained by expanding Π to its con-
(5)

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ACT: Beam Profiles and First SZ Clusters3
Ξ ≡ BTN−1(1−PΠ),
then when we set the derivative in Eq. 5 to zero, we have the
simple expression:
Θ ≡ ΞB,
φ ≡ Ξd,
(6)
Θ? α = φ.
(7)
This is a linear equation which is straightforward to solve for
the atmospheric basis function amplitude estimates ? α. These
sphere and celestial map, respectively, for a given set of basis
functions B. We show this explicitly in Appendix A.
There is an arbitrary overall offset to the computed B? α
2.2. Discussion
The chief strength of the Cottingham Method is that it es-
timates the atmospheric power in a way that is unbiased with
respect to the map estimate itself. This important but subtle
point is encapsulated in the term (1−PΠ) in Eq. 5, whose
effect is to project out the map estimate from the data. There-
fore, the solution to Eq. 7 is not sensitive to the estimated ce-
lestial temperature, but only to a time-varying term which is
represented by the atmospheric estimate B? α. This should be
Such approaches require masking of high signal-to-noise ce-
lestial objects (like planets or clusters) and/or multiple iter-
ations to prevent corruption of the maps. They also remove
low-frequency power without regard to its origin, the major-
ity of it atmospheric, but also inevitably containing celestial
signal, and therefore require extensive simulations to under-
stand the effects of the filters on the final maps.
The Cottingham Method has close similarities to the “de-
striping” technique developed in particular for Planck anal-
ysis (Delabrouille 1998; Burigana et al. 1999; Maino et al.
2002). In fact, the linear algebra presented in §2.1 is identi-
cal to some versionsof destriping(e.g.,Keihänen et al. 2004).
The destriping techniques are intended primarily to remove
1/f instrumental noise—thus, for example, Keihänen et al.
(2005) impose a prior on the estimate B? α based on detec-
we use the Cottingham Method to remove atmospheric power
with a flat prior. A distinct feature of our method is that
we process multiple detectors simultaneously since the at-
mospheric signal is common across detectors (see, however,
§2.3). Further, our approach differs in that it uses B-splines
as the basis for modelling the atmosphere (§2.4).
can then be used in Eq. 2 to estimate the map. In fact, both
? α and ? m are the maximum likelihood estimators of the atmo-
which must be estimated to remove the background from
maps. We return to this point in §2.6.
contrasted with high-pass filtering or fitting a slowly-varying
function to the timestream to remove low-frequency power.
tor noise. Sutton et al. (2009) also consider the effects of im-
posing a prior on the atmospheric noise. On the other hand,
2.3. Spatial Structure in the Atmosphere
The Cottingham Method as presented thus far assumes that
the atmospheric signal Bα is common among all the detec-
tors. In fact, we know that there is also spatial structure in the
atmosphere, meaning that in principle, each detector might
see a different atmospheric signal. In practice, the finite tele-
scopebeamsets a lowerlimit onthe spatial scale. We find that
theatmosphericsignalis coherentacrossaquartertoa thirdof
the array, or about 5–7′. For reference, our 148GHz channel,
which has a 1.37′FWHM in the far-field (§3.2), is sensitive
to an angular size of approximately 10′at a 1km distance,
roughly the distance to a typical turbulence layer in the atmo-
sphere when pointed at 50◦in altitude (Pérez-Beaupuits et al.
2005).
Toaccountforthis, wedividethe32×32detectorarrayinto
9 square sub-arrays of roughly equal size and fit for nine sep-
arate temporal atmospheric signals Bsαs, with the subscript s
denoting the sub-array. These can all be done simultaneously
if we adapt Eq. 1:
d = Pm+S




B1
0 B2 ... 0
...
00 ... B9
0 ... 0
...
...
...








α1
α2
...
α9



+N
= Pm+SB′α′+n,
(8)
where S is a book-keepingmatrix that remembers from which
sub-array each measurement in d came.
Method proceeds exactly as before, except that we change
B → SB′and α → α′.
2.4. The B-Spline as a Model of Atmospheric Signal
We follow Cottingham (1987) in choosing cubic basis B-
splines for the basis functions B. B-splines are widely used
in the field of geometrical modeling, and numerous text-
books cover them (e.g., Bojanov et al. 1993; de Boor 2001;
Schumaker2007); here we summarize basic properties. Basis
B-splines are a basis of functions whose linear combinationis
called a B-spline. The basis B-splines are fully determinedby
a knot spacing τkand a polynomialorder p; a B-spline is flex-
ible on scales larger than τk, while on smaller scales it is rela-
tively rigid. The basis B-splines bj,p(t) of order p are readily
evaluatedusing the Cox–de Boor recursionon the polynomial
order. For m knots {tj} with j = 0 to m−1:
?1
bj,p(t)=
tj+p−tjbj,p−1(t)+
with j values restricted so that j+ p+1 < m−1. For m knot
times, m+ p−1 basis B-splines cover the interval between the
first and last knot time. The individual basis B-splines bj,p(t)
are compact functions,such that the B-spline receives support
from no more than p of its bases at any point. For model-
ingtheatmosphericsignal, we always chooseknotsuniformly
spaced in time and use p = 3 (cubic).
Due to their flexibility on large scales, B-splines are ideal
for modelling the slowly varyingatmospheric signal. The fre-
quency fkbelow which power will be removed is determined
by the knot spacing τk. Empirically, we find:
The Cottingham
bj,0(t)=
if tj≤t <tj+1
otherwise
t −tj
0
,
tj+p+1−t
tj+p+1−tj+1bj+1,p−1(t),
(9)
fk≈ 1/2τk.
(10)
Figure 1 shows an example of atmospheric estimation us-
ing the Cottingham Method. The B-spline knot-spacing is
τk= 0.25s, chosen for this example because it has fk= 2Hz,
the approximate frequency at which the atmospheric power
meets the detector noise level. Longer knot spacings produce
qualitatively similar results, except that they cut off at lower
frequencies, as per Eq. 10.
The Cottingham Method is effective at suppressing atmo-
spheric contamination, but some covariance between atmo-
spheric and celestial map estimates remains. This is typically

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4The ACT Collaboration
-100
-50
0
50
100
4050 607080
Response (mK)
Time (s)
1
10
100
1000
0.001 0.010.1
Frequency (Hz)
110100
Spectral Density (mK s1/2)
FIG. 1.— An example of the Cottingham Method. The fit is done using
300s of data from 605 148GHz detectors. The PWV was 0.8mm, about
0.25mm higher than the median in 2008. The knot spacing is 0.25s and
the order is cubic. In both plots, the original signal is plotted with a solid,
light line, the B-spline atmosphere model with a dashed line and the signal
minus the model with a solid, dark line. Each plot has been smoothed with
a 5-sample boxcar filter for readability. The temperature units are with re-
spect to a Rayleigh-Jeans spectrum. Top: A portion of one of the detectors’
timestreams. Bottom: The spectral densities for the same single detector. A
Welch window was applied before computing the Fourier transform.
at harmonics of the scan frequency (≈ 0.1Hz), as exempli-
fied in the bottom panel of Fig. 1. For this work we have
used the white noise approximation for the detectors (N = 1),
which results in maps of bright sources which are clean down
to the −40dB level in most cases (§3).
ual atmospheric-celestial covariance is manifested as striping
along lines of constant altitude since with our 7◦peak-to-
peak, 1.5◦s−1azimuthal scans (or 4.47◦at 0.958◦s−1when
projected on the sky at our observing altitude of 50.3◦), the
knot spacing (τk= 1.0s for beam maps (§3) and 0.5s for clus-
ter maps (§5)) corresponds to an angular scale smaller than
the scan width. When noted, we fit straight lines to rows of
pixels in the map, after masking out any bright source, and
subtract them. We call this process “stripe removal”. In both
our beam analysis and our cluster studies, we have done tests
which show that the bias introduced by this process is not
significant—see §3.2 and §5.3. Nevertheless, future exten-
sions of the Cottingham Method would benefit from the full
treatment of the noise covariance.
The small resid-
2.5. Implementation
Before making maps with the Cottingham Method, some
preprocessing must be done. The data, which are sampled
at 400Hz, are divided into fifteen-minute time-ordered data
(TOD) files and the preprocessing is performed on each in-
dividual TOD—a future paper will describe the steps which
we only summarize here. The data acquisition electronics’
digital anti-aliasing filter as well as measured detector time
constants are deconvolved from the raw data. Low frequency
signal due to cryogenic temperature drifts are measured with
dark detectors (i.e., detectors uncoupled to sky signal) and re-
moved from signal detectors; a sine wave with period 10.23s
is also fit to each timestream and removed to reduce scan-
synchronous contamination. Calibration to units of power
uses nightly load curves obtained by sweeping through de-
tector bias voltages and measuring the response. Relative
gainimprecisions are removedby using the large atmospheric
signal itself to flat-field the detectors (e.g., Kuo et al. 2004);
this is done independently in each of the nine sub-arrays (see
§2.3). Finally, calibration to temperature units uses measure-
ments of Uranus, which we estimate to have a net error of
6%. (The beam maps require no calibration to temperature—
in fact, the temperature calibration is obtained from them.)
The timestreams require no further preprocessing.
To improve the speed of the Cottingham algorithm, we ex-
ploit the fact that the map pixelization used for calculating
the atmospheric signal (Eqs. 5–7) need not be the same as
the map-making pixelization. In general we only use a se-
lection of the possible pixels on the map; additionally, we
down-sample the number of hits in each pixel. We call the
former “pixel down-sampling” and denote the fraction of re-
tained pixels np; the latter we term “hit down-sampling” and
denotethe fraction of retainedhits nh. Consequently,the frac-
tion of total available data used is np×nh. Each of these
down-samplings is done in an even manner such that there
are no large gaps in the remaining timestream.
We have specified four parameters for the Cottingham
Method: the knot-spacing τk, the pixel size ξ, the pixel down-
sampling fraction, np, and the hit down-samplingfraction, nh.
Of these, we always choose ξ = 18′′(about 1/3 the 277GHz
beam size).
1
10
100
0.01 0.1 1
Time (min)
Fraction of Total Data (npnh)
2
3
5
7
103
104
105
L
Approx. Num. Data Points Per Second
0.25s0.5s1.0s1.5s2.0s
FIG. 2.— Top panel: The ratio L of low-frequency power to white noise
after removing the atmospheric signal, as defined by Eq. 11. All data points
are from the same TOD using all working detectors; only the parameters τk,
npand nh(see text) were varied to obtain each point. The x-axis is a product
of the resamplings npand nh, and the colors are different knot spacings τkas
indicated in the plot key. Bottom panel: The computation time required for
the data shown in the top panel. (The segmented lines are an artifact of how
the data were recorded.) As the fraction npnhof total data used increases,
the efficacy L of power removal flattens out and adding more data does not
significantly improve the fit but only takes more computation time. In this
example, there were ≈ 103000 possible map pixels with an average of 287
hits per pixel.
To evaluate the effect of varying the other three variables,
we define a figure of merit which compares the average de-

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ACT: Beam Profiles and First SZ Clusters5
tector spectral density below 1Hz to the white noise level,
calculated in the range 5–25Hz:
L ≡
1
Nd
Nd
?
i
??1Hz
0HzGi(f)df
?1Hz
0Hzdf
??25Hz
5HzGi(f)df
?25Hz
5Hzdf
?
,
(11)
wherethesum runsovertheNddetectorsusedfortheCotting-
ham calculation and Giis the spectral density of the ithdetec-
tor after removing the estimated atmosphere signal. Fig. 2
shows a plot of measured values of L for a selection of knot-
spacings and pixel down-samplings. As expected, shorter
knot spacings remove more power: note however that only
the τk= 0.5s and τk= 0.25s are capable of removing power
up to the 1Hz for which L is defined (c.f Eq. 10). Because the
curves flatten out as npnhincreases, at a certain point adding
more data does not substantially improve the fit. This sup-
ports our conclusion that we only need to use a fraction of the
data to estimate the atmosphere.
The timing data in the lower panel of Fig. 2 were measured
on a 64-bit Intel Xeon
time is dominated by the calculation of the variables in Eq. 6.
In general, these go linearly with the number of data points
d and quadratically with the number of basis functions B; in
the case of B-splines, the compact support of the bases can
be exploited so that the quadratic rate is subdominant to the
linear, as the plot shows.
The number of basis B-splines is small enough that it is ac-
tuallyfeasibletosolveEq.7exactly. Formostcases, however,
the conjugategradientmethod (e.g.,Press et al. 1992, p. 83ff)
is much faster and yields indistinguishable results. Therefore,
we use the latter in our implementation.
R ?2.5GHz processor. Computation
2.6. Map-Making
Once the atmospheric model Bα has been calculated for
a TOD with the Cottingham Method, we create its celestial
map. Maps are made in (∆a, ∆Acos(a)) coordinates, where
∆a and ∆A are the distances from the altitude and azimuth of
the map center.2We use a pixel size of 10.6′′per side, about
20% the size of the 277GHz beam; note that this is differ-
ent from the pixel size ξ used for calculating the atmosphere
(§2.5).
We make the white noise approximation for each detector
andweightitbytheinverseofits varianceinthemapestimate.
The detector variances are obtained iteratively: we make a
map with equal detector weights and measure the variances
of individual detector maps against the total map, remake the
map with the new variances and repeat until the total map
variance converges. The atmosphere estimates returned by
the Cottingham Method have arbitrary offsets, which can be
different for the nine sub-arrays we use (see §2.3). Thus, in
the same iterative process, we also fit for the sub-array offsets
and remove them when coadding detectors.
Coaddition of TOD maps is done after all of the steps de-
scribed above. The maps are weighted by their inverse vari-
ances (calculated after masking bright sources or clusters).
Finally, we mention that the software used for the results
in this paper has a completely independent pipeline from our
main map-making software which solves for the full survey
area coverage. It has been especially useful for studying and
optimizingthesignalextractioninsmall, targetedregions,and
has provided important double-checks for our other pipeline.
2This is a very good approximation to the Gnomonic projection for the
small map sizes we use.
TABLE 1
SUMMARY OF BEAM PARAMETERS
148GHz 218GHz277GHz
Map Properties (§3.1)
# TODs
Stripe-removal?
Map Radius (′)
Mask Radius (′)a
16
no
21
18
15
no
15
11
yes
15
9,11,136,8,10,12
Beam Centers (§3.2)
Major FWHM (′)
Minor FWHM (′)
Axis Angle (◦)
1.406±0.003
1.344±0.002
62±2
1.006±0.01
1.001±0.003
137±9
0.94±0.02
0.88±0.02
98±13
θWWing Fits (§3.2)
Fit Start, θ1(′)
Fit End, θ2(′)b
Best-fit θW(′)
7
13
5 4.5
6–107–11
0.526±0.0020.397±0.010.46±0.04
Solid Angles (§3.2)
Solid Angle (nsr)
% interpolated
218.2±4
2.8
118.2±3
4.3
104.2±6
7.2
Beam Fits (§4.1)
θ0(′) 0.2137 0.15620.1367
NOTE. — See text for definitions of these parameters and how they are
measured. Values for the 148GHz and 218GHz bands are obtained from
the coadded, unprocessed maps whereas the 277GHz values are average
values from stripe-removed maps with the mask sizes indicated in the
table.
aThe 218GHz and 277GHz beam properties are averaged from the re-
sults at these mask radii—see text.bThe fit ranges for the 218GHz and
277GHz band are varied along with the mask radii so that θ2is never
larger than the mask—see text.
3. BEAM MAPS AND PROPERTIES
Understanding the telescope beams, or point-spread func-
tions (PSF’s), is of primary importance for the interpretation
of our maps since they determine the relative response of the
instrumentto differentscales on the sky and are central to cal-
ibration. For ACT’s measurement of angular power spectra,
the Legendre transform of our measured beam profile, called
a window function, determines the response of the instrument
as a function of angular scale.
Planets are excellent sources for measuring the telescope’s
beam because they are nearly point sources and are brighter
than almost any other celestial object. The best candidates
for the ACT are Saturn and Mars; of the rest, Jupiter is too
bright and saturates the detectors, Venus is available too near
to sunrise or sunset when the telescope is thermally settling,
and the others are too dim for exploring the far sidelobes of
the beam. (However, Uranus is useful as a calibrator.) The
beam maps presented in this section are from observations
of Saturn, which was available from early November through
December of 2008.
3.1. Data Reduction
Maps were made for each night-time TOD of Saturn, using
the Cottingham Method with τk=1s, np=0.32 and nh= 0.36.
We had more TODs than were needed to make low-noise
beam maps, and we excluded about 1/3 of the maps which
hadhigherresidualbackgroundcontamination,manifestedby
beam profiles that significantly diverged from those of the
cleaner maps. The map sizes and number of TODs per fre-
quency band are shown in Table 1.
In the analysis of §3.2, below, we sometimes compare un-
processedmaps to stripe-removedmaps. Stripe-removedmap

Page 6

6The ACT Collaboration
havebeentreated as outlinedin §2.4, while unprocessedmaps
have not been altered after map-making except for the sub-
traction of a background level. The planet is masked out
before calculating the mean map value which estimates this
level. The mask sizes for the three arrays, whether used for
stripe removalor backgroundestimation, are listed in Table 1.
In each frequencyband, the selected TOD beam maps were
coadded. Weights were determined from the rms of the mean
background level, calculated outside the mask radius. Rel-
ative pointing of individual detectors was measured to sub-
arcsecond precision using the ensemble of Saturn observa-
tions. The overall telescope pointing was determined from
each planet observationprior to map-makingand used to cen-
ter each TOD map, so recentering of the maps was unneces-
sary before coaddition.
3.2. Beam Measurements
Fig. 3 shows coadded beam maps for the three arrays using
a color scale which highlights the features in the sidelobes.
The 148GHz and 218GHz maps have striking similarities,
most notably along the altitude (or vertical) direction where
both exhibit more power near the top of the map. This is
due to the off-axis design of the telescope (see Fowler et al.
2007): since the 148GHz and 218GHz arrays sit at about the
same vertical offset from the center of the focal plane, their
resemblance along this axis is expected. Note that we recover
structure in these map at a < −40dB level. The 277GHz map
is clearly inferior, showing residual striping in the scan direc-
tion, although this occurs below −20dB. We believe this is
from a combination of the brighter atmosphere at 277GHz,
as well as detector noise correlation induced by large opti-
cal loads (such as Saturn), which we are still investigating.
Nonetheless we are still able to measure the 277GHz solid
angle to about 6% (see below), and work is underway to im-
prove it.
Forthe 148GHz and218GHz arrays, we do ourbeamanal-
ysis on maps which have not had stripe-removal because this
process removes the real vertical gradient from the maps (see
Fig. 3). Nonetheless, the solid angles (see below) from stripe-
removed maps are within 1σ of the values from unprocessed
maps. On the other hand, the larger residual striping in the
277GHz maps necessitates the use of stripe-removed maps.
Thebeamcenteris characterizedbyfittinganellipticalAiry
pattern—the function describing the beam of an optical sys-
tem with a perfect aperture—to the top ∼3dB of the beam
map. Thisprovidesameasurementofthelocationofthebeam
center, its FWHM along the major and minor axes of the el-
lipse, and its orientation, which we define as the angle of the
major axis from the line of zero altitude relative to the beam
center. The uncertainties in these parameters are determined
using the bootstrap method (Press et al. 1992, pp. 691ff) and
give errors consistent with the standard deviation of values
measured from the individual TOD maps. The FWHM and
angles are listed in Table 1. They are included for reference
but are not used in any analysis.
We denote the beam map by B(θ,φ), where we use coordi-
nates with radial distance θ from the beam center and polar
angle φ. By definition, B(0,φ) = 1. The symmetrized beam is
averaged around the polar angle:
bS(θ) ≡
?dφ′B(θ,φ′)
?dφ′
.
(12)
Another quantity of interest is the accumulated solid angle,
which measures the total normalized power within a given ra-
dius:
Ω(θ) =
2π
?
0
dφ′
θ
?
0
θ′dθ′B(θ′,φ′).
(13)
The beam solid angle is ΩA≡ Ω(θ = π).
Fig.4shows measuredbeamprofilesandaccumulatedsolid
angles for the three arrays. We measure the beam profiles
down to about −45dB. If the beams exactly followed an Airy
pattern, these data would account for 98% of the solid angle.
Since systematic effects could corrupt our maps at the largest
radii, we seek a way to robustly estimate the last few percent
of the solid angle on each beam. The method is to extrapolate
the data with a fit to the asymptotic expression for the Airy
Pattern:
bS(θ ≫ θF) =
?θW
θ
?3
,
(14)
where θFis the beam FWHM and θWdefines the wing scale.
Eq.14 is goodto better than1% beyondabout5θF(Schroeder
2000, §10.2b).3
Knowledge of θW allows us to infer the
amount of unaccounted solid angle beyond the map bound-
ary. A simple integration shows that the solid angle beyond a
radius θbis:
ΩW(θ > θb) = 2πθ3
W
θb.
(15)
We can also use this expression to estimate the amount
of true beam power which was “mistakenly” included in the
measurementof the backgroundlevel outside the mask radius
and subtracted from the map. In our analysis of the beam
profiles and solid angles (includingthose displayed in Fig. 4),
we use the fits of θWto calculate this missing power and add
it back into the map. A new θW is then calculated from the
corrected map; after two such iterations the θWfit converges.
Fig. 4 includes over-plots of the wing estimates from the
best-fit values of θW on unprocessed maps. We denote the
radii between which the fits were performed as θ1–θ2, and
choose θ1≈5θFfor each array. For 148GHz, we obtain good
fits for any choice of θ2up to 13′, or about the −40dB level
in the profile. Thus, we use θ2= 13′, for which we fit with χ2
of 40 for 35 degrees of freedom. The fits to the other profiles
are not as robust: 218GHz has a reduced-χ2of 2.8 for θ2=
7′and 277GHz has reduced-χ2of 25 for θ2= 6′. Larger θ2
gave poorer fits. Consequently, for these profiles we calculate
θwat different mask sizes, as indicated in Table 1. At each
mask size we varied θ2in 2′increments, always keeping it
lower than the mask size. The average value from the whole
ensemble of fits gives us θwand we take its standard error as
the uncertainty. Although Eq. 14 may be too simple a model
fortheseprofiles, contributionsto thesolid angleat these radii
are only a few percent of the total solid angle, which has an
uncertaintydominatedby the contributionof the beam at radii
less than θ2—see below. The values of θwfor all three beam
profiles are listed in Table 1.
3In full generality Eq. 14 is also proportional to cos(πDθ/λ − 3π/4),
where D is the telescope aperture diameter and λ the wavelength; we have
smoothed over cosine cycles.

Page 7

ACT: Beam Profiles and First SZ Clusters7
148GHz218GHz277GHz
Alt
Azcos(Alt)
FIG. 3.— Beam maps for the three frequency bands: from left to right, 148GHz, 218GHz and 277GHz. The maps are from coadded observations on 11–15
nights (see Table 1) and have radii of 21′(148GHz) and 15′(218GHz and 277GHz). Maps are normalized to unity and contours are in decrements of −10dB.
The color scale has been chosen to highlight the fact that we have made < −40dB beam measurements of our 148GHz and 218GHz bands. Even in the inferior
277GHz map, striping is still below −20dB. The circles show the sizes of the beam FWHM for each band (see Table 1). A Gaussian smoothing kernel with
σ = 0.54′has been applied to highlight large-scale structure; smoothing is not otherwise performed in the analysis. No stripe removal has been done on these
maps.
-50
-40
-30
-20
-10
0
0510 1520
Beam Profile (dB)
148GHz
05
Radius (arcmin)
10 15
218GHz
05 1015
0
50
100
150
200
Accumulated Solid Angle (nsr)
277GHz
Destriped Profile
Un-Destriped Profile
Best-fit θw
Accum. Solid Angle
FIG. 4.— The beam profiles (Eq. 12) and accumulated solid angles (Eq. 13) for the three arrays, calculated from coadded maps (see text). The beam profiles are
shown for both unprocessed maps, with dark errorbars, and stripe-removed maps, with light errorbars. Over-plotted on each profile is the best fit of θW(Eq. 14) to
the unprocessed beam profiles. The error on the profiles are standard errors from the azimuthal average. The accumulated solid angles are from the unprocessed
maps (without any solid angle extrapolation via Eq. 15) for 148GHz and 218GHz, and from the stripe-removed map for 277GHz. Saturn is bright enough that
the rms power from the CMB falls below all the points in these plots.
Our θwfits allow us to calculate precise solid angles. At
radiismaller thanθ2, we integratethe normalizedpowerin the
map (c.f. Eq. 13). Beyond θ2, we use Eq. 15 to extrapolate
the remaining solid angle. (In the case of the 218GHz and
277GHz solid angles, we use the smallest θ2and the largest
mask size in the ranges shown in Table 1. Other choices from
these ranges do not significantly alter the results.) Finally, in
the approximationthat Saturn is a solid disk, it adds half of its
solid angle ΩSto the measured instrument solid angle—this
is shown in Appendix B. Thus, the total solid angle is:
ΩA = Ω(θ ≤ θ2) + ΩW(θ > θ2) − ΩS/2.
During the period of our observations, Saturn subtended solid
angles from 5.2 to 6.0 nanosteradians (nsr). We use the mean
value of 5.6nsr.
(16)
Determining the rest of the uncertainty in the solid angle is
not straightforward since systematic errors dominate. For our
total error, we add the estimated uncertainties of each of the
terms on the RHS of Eq. 16 in quadrature. The uncertainty
from Saturn’s solid angle we take to be 1nsr, both because of
its varying angular size and to account for any systematic er-
ror due to the disk approximation.4The uncertainty of ΩWis
derived from the error of the fitted θW. For Ω(θ < θ2), which
dominates, we estimate the error by looking at the distribu-
tion of values from the individual TOD maps which comprise
the coadded map. We did this in two ways. First, we cal-
4The rings of Saturn add a layer of complication to its solid angle cal-
culation, particularly since they have a different temperature than the disc.
The ring inclination was low during our observations (< 6◦) and we have
estimated that their contribution is negligible within the error budget.

Page 8

8The ACT Collaboration
culated the mean and standard deviation of the solid angles
measured in each individual map. This also reassures us that
the coaddition step does not introduce any systematic error
through, for example, pointing misalignments or changes in
telescope focus from night to night. Second, we used the
bootstrap method to generate 1000 coadded maps with ran-
dom subsets of individual maps and used this ensemble to es-
timate the 68th percentile(i.e., 1σ) of solid angles. These two
error estimates were consistent with each other.
The solid angles and their uncertainties are reported in Ta-
ble 1. The formal uncertainties have been doubled and we
quote them as 1σ, in case there are systematic effects for
which we have not accounted. In particular,the maps used for
power spectrum estimation will come from an independent
pipeline and will treat the instrumental response in slightly
different ways—for example, by weighting detectors differ-
ently. We expect the beam uncertainties to decrease as our
analysis evolves.
4. WINDOW FUNCTIONS
The statistics of the CMB are frequently characterized by
an angular power spectrumCℓ:
∆T(ˆ n) =
?
ℓ,m
aℓmYℓm(ˆ n); ?a∗
ℓ′m′aℓm? = δℓ′ℓδm′mCℓ,
(17)
where ∆T(ˆ n) is the CMB temperatureat position ˆ n andYℓmis
a spherical harmonic. In spherical harmonic space, the beam
is encoded in a window function wℓdescribing the response
of the experiment to different multipoles ℓ, such that the total
variance of a noiseless power spectrum is:
Var =
?
ℓ
2ℓ+1
4π
Cℓwℓ.
(18)
Inthe case ofa symmetricbeam, thewindowfunctionis the
square of the Legendre transform of the beam radial profile
(White & Srednicki 1995; Bond 1996):
?
4.1. Basis Functions
For calculation of the window function and its covariance
we model each beam with a set of basis functions which is
complete but not necessarily orthogonal:
wℓ= b2
ℓ; bℓ≡2π
ΩA
bS(θ)Pℓ(θ)d(cosθ).
(19)
bS(θ) =
nmax
?
n=0
anbn(θ).
(20)
Because the beam is truncated by a cold Lyot stop
(Fowler et al. 2007), its Fourier transform is compact on a
disk, which suggests that a natural basis with which to de-
compose the Fourier transform of the beam image is the set
of Zernike polynomials that form an orthonormalbasis on the
unit disk (Born & Wolf 1999). The Zernike polynomials, ex-
pressed in polar coordinates ρ and ϕ on the aperture plane,
are:
Vm
n(ρ,ϕ) = Rm
n(ρ)eimϕ,
(21)
wherem and n are integerssuch that n≥0, n>|m| andn−|m|
is even. In the case of an azimuthally symmetric beam, we
need only consider the m = 0 radial polynomials, which can
be expressed in terms of Legendre polynomials, Pn(x) as:
R0
2n(ρ) = Pn(2ρ2−1).
(22)
The radial Zernike polynomials have a convenient analytic
form for their Fourier transform:
˜R0
2n(θ) =
?
ρdρe−iρθR0
2n(ρ) = (−1)nJ2n+1(θ)/θ,
(23)
where Jnis a Bessel function of the first kind. Motivated by
this, we adopt
bn(θ) =
?θ
θ0
?−1
J2n+1
?θ
θ0
?
(24)
as our set of basis functions to fit the radial beam profile.5
Here, we have introduced a fitting parameter, θ0, to control
the scale of the basis functions.
4.2. Fitting Basis Functions to the Beam Profile
Below θ1(c.f. Table 1), we fit the bases bnof Eq. 24 to the
measured beam profile, and beyond θ1, we use the power law
defined in Eq. 14 with the parameters θwlisted in Table 1. We
assume vanishing covariance between the power law and the
basis functions as they are fitted to independent sets of data
points.
We employ a nonlinear, least-squares method to solve for
the coefficients anand their covariancematrixCaa′
rithm uses a singular value decomposition to determine if the
basis functions accurately characterize the data and also com-
putes a goodness-of-fit statistic (Press et al. 1992, §15.4). As
inputs to the fitting procedure we are required to specify the
scale parameter, θ0, and polynomial order, nmax. We searched
the {θ0,nmax} parameter space until a reasonable fit was ob-
tained that kept nmaxas small as possible. For all three bands,
nmax= 13 gives a reduced χ2≈ 1. No singular values is found
for any of the fits. The parameters θ0we use for each fre-
quency band are listed in Table 1.
mn. The algo-
4.3. Window Functions and their Covariances
Given the amplitudes anof the radial beam profile fitted to
thebasisfunctionsandthe covariancematrixCaa′
amplitudes amand an, the beam Legendre transform is:
mnbetweenthe
bℓ=
nmax
?
n=0
anbℓn.
(25)
and the covariance matrix of the beam Legendre transforms
bℓand bℓ′ is:
Σb
ℓℓ′ =
nmax
?
m,n=0
∂bℓ
∂amCaa′
mn
∂bℓ′
∂an
(26)
and therefore the covariance for the window function is:
Σw
ℓℓ′ = 4wℓwℓ′Σb
ℓℓ′.
(27)
In Fig. 5 we show the window functions for each of the
three frequency bands with diagonal error bars taken from
5It may be asked why the Airy pattern, which was somewhat suitable
for the high-θ fit in §3.2, is not used here. We find that at low θ, it is a
poor fit since the optics are more complicated than the perfect-aperture model
assumed by the Airy pattern.

Page 9

ACT: Beam Profiles and First SZ Clusters9
0
0.25
0.5
0.75
1
wℓ
0
0.025
0.05
200040006000800010000
∆wℓ/wℓ
Multipole ℓ
148GHz
218GHz
277GHz
FIG. 5.— Normalized window functions (top) and diagonal errors (bottom)
computed from the basis functions for each of the three frequency bands.
The window functions have been normalized to unity at ℓ = 0. In practice, the
normalization will take place over the range of multipoles corresponding to
the best calibration. Only statistical errors are shown.
the covariance matrix, Σw
function for each of the frequency bands has fallen to less
than 15% of its maximum value at ℓ = 10000. The statis-
tical diagonal errors are at the 1.5%, 1.5%, and 6% levels
for the 148GHz, 218GHz, and 277GHz bands respectively,
as shown in Fig. 5. They are computed following Eq. 17 of
Page et al. (2003). The off-diagonal terms in the beam co-
variance matrix are comparable in magnitude to the diagonal
terms. Singular value decompositions of Σb
handful of modes with singular values larger than 10−3of the
maximum values: 5 modes for 148GHz, 4 for 218GHz, and
2 for 277GHz. Thus, the window function covariances can
be expressed in a compact form which will be convenient for
power spectrum analyses.
For 277GHz we estimate a 10% systematic uncertainty
from destriping. Another source of systematic error in the
window functions arises from the beams not being perfectly
symmetric. The symmetrized beam window function gen-
erally underestimates the power in the beam (e.g., Fig. 23,
Hinshaw et al. 2007). In practice, the scans in our surveyfield
are cross-linked (see §5.1) so that the quantity of relevance
to the power spectrum calculation is the cross-linked window
function i.e.,
ℓℓ′. We observe that the window
ℓℓ′ yield only a
wℓ=
1
2π
?2π
0
b(ℓ)b∗(ℓ)dφℓ,
(28)
where φℓis the polar angle in spherical harmonic space and
b(ℓ) is the transform of the beam profile. To study the mag-
nitude of this effect on the window function we computed
thefractionaldifferencebetweenthewindowfunctionderived
from the Legendre transform of the symmetrized beam and
the cross-linked window function, derived for a typical cross-
linking angle of 60◦. The difference between the two was
found to be at the 1%, level for the 148GHz and 218GHz
arrays, and at the 4% level for the 277GHz array.
5. SZ GALAXY CLUSTERS
In addition to beam maps, the Cottingham Method mapper
has been used for making maps of SZ clusters. The maps and
analysispresentedinthissectionarethefirstresultsfromACT
on SZ science. For this first overview,we present results from
only the 148GHz band, the most sensitive during our 2008
season. We focus on known clusters, including four new SZ
measurements.
5.1. Data
Table 2 lists the clusters studied in this paper, including in-
formation on the maps and a summary of the results of our
analysis (§5.2, §5.3). Fig. 6 shows the cluster maps and com-
panion difference maps (see below).
Apart from planets, ACT has done no targeted observations
of specific objects, so the cluster maps come from our reg-
ular survey data, which were taken at two different central
azimuth pointings, one on the rising sky and the other on the
setting sky. Therefore, the maps presented here are “cross-
linked”, i.e., they consist of data taken with two distinct an-
gles between the azimuthal scan direction and the hour angle
axis. The integration time is short, ranging from about 3 to 11
minutes—see Table 2.
The clusters were discovered in a full-survey 148GHz map
produced by our main map-maker. (Full maps will be shown
and discussed in a future publication.) A Wiener filter was
constructed using the polytropic model of Komatsu & Seljak
(2001) as an SZ template, and included detector noise, CMB
power and point source contributions in the noise model.
Clusters were thenidentified fromthe filtered maps. We make
two points about the clusters presented in this paper: first, al-
though our template-based detection method has some built-
in bias, the detections presented here are significant (≥ 3σ in
the filtered survey map); and second, we have only included a
sample of our significant detections.
Cluster maps are made using the procedures outlined in
§2.5 and §2.6. The knot spacing was τk= 0.5s and the down-
sampling fractions were np≈ 0.42 and nh≈ 0.40. All maps
are 0.4◦in diameter. Straight-line stripe removal (§2.6) has
beenperformed,using a 6′radiusmask overthe cluster decre-
ment.
We have made companion “difference” maps for each clus-
ter from the same data. For each of the rising and setting
observations, a map made from the first half of the nights’
data is subtracted from the second half. The rising and setting
difference maps are then coadded to produce the full, cross-
linked coadded difference map—the same procedure used for
the signal maps.
The map noise, listed in Table 2, is the rms of the map com-
puted outside a 6′mask and converted to an effective pixel
size of one square arcminute. By examining the power spec-
tra of the maps we found that the rms values we quote are
dominated by the white noise level and do not have signifi-
cant contributions from residual low-frequencypower.
5.2. Characterizing the Detections
The cluster center positions are determined by finding the
coldest point in the map smoothed with a 2′FWHM Gaus-
sian kernel. The one exceptionis ACT-CL J0509−5345(SPT-
CL 0509−5342), which has a complex structure. In this case,
we choose a center which gives a maximal signal-to-noise
(see below). As a rough guide, we also quote the cluster
depths, ∆TSZ, from these smoothed maps in Table 2, but we
stress that these values should not be used for quantitative
analysis. All other cluster properties are measured from un-
smoothed maps.
We quantify the significance of the detections by a signal-
to-noise (SNR) measurement. Denoting the signal map mxy
andthe differencemap dxy, we define the SNR within an aper-
ture size θ as:

Page 10

10The ACT Collaboration
TABLE 2
SELECTION OF SZ CLUSTERS DETECTED BY ACT
ACT DescriptorCatalog NameJ2000 Coordinatesa
rmsb
[µK]
tintc
[min]
SNR (θ)d
∆TSZe
[µK]
1010×Y(θ)f
RADec.
θ ≤ 2′
(±0.2)
θ ≤ 4′
(±0.6)
θ ≤ 6′
(±1.2)
ACT-CL J0245−5301
ACT-CL J0330−5228
ACT-CL J0509−5345
ACT-CL J0516−5432
ACT-CL J0546−5346
ACT-CL J0638−5358
ACT-CL J0645−5413
ACT-CL J0658−5556
Abell S0295
Abell 3128 (NE)
SPT-CL 0509−5342
Abell S0520
SPT-CL 0547−5345
Abell S0592
Abell 3404
1E 0657−56 (Bullet)
02h45m28s
03h30m50s
05h09m20s
05h16m31s
05h46m35s
06h38m46s
06h45m29s
06h58m33s
−53◦01′36′′
−52◦28′38′′
−53◦45′00′′
−54◦32′42′′
−53◦46′04′′
−53◦58′40′′
−54◦13′52′′
−55◦56′49′′
44
49
47
55
46
55
59
80
10.1
10.3
10.1
6.8
9.5
7.5
9.3
3.4
15.2 (6.8′)
12.8 (4.3′)
7.7 (5.2′)
4.2 (4.1′)
13.9 (5.8′)
8.1 (3.1′)
2.8 (2.0′)
12.1 (2.7′)
−250
−260
−70
−110
−250
−230
−120
−510
0.89
0.94
0.33
0.19
0.91
0.70
0.12
1.60
2.36
2.69
1.07
-0.11
2.36
1.40
-0.18
2.95
3.91
4.34
1.50
-0.55
3.67
2.07
-0.69
3.56
aPosition of the deepest point in 2′FWHM Gaussian smoothed map, except for ACT-CL J0509−5345 which has a position which gives a maximal
SNR (see text).bMap rms measured outside a 6′mask and reported for a one square arcminute area.cIntegration time, defined as the approximate
total time (in minutes) that the telescope was pointed in the map region.dMaximum signal-to-noise ratio (Eq. 29) and the radius θ at which it was
obtained.eCluster depth, as measured in a 2′FWHM Gaussian smoothed map at the listed coordinates; intended as a guide to the magnitude of the
decrement.fSee Eq. 32 and following discussion.
SNR(θ) ≡ min







????????
?
N(θ)?σ2
√
x2+y2<θ
?
?mxy±dxy
m+σ2
?
d
?
????????







,
(29)
where the sum is over pixels which are within a radius θ from
the cluster center, N(θ) is the numberof pixels in the sum, and
σ2
respectively, calculated outside a 6′mask. Because the sign
of a difference map is arbitrary (we subtracted one half of the
data from the other half), we take the smaller of the possible
two SNR. The purpose for including the subtraction is that
coincident flux in the difference map indicates that the signal
map probably contains spurious flux.
In Table 2, we quote SNR(θi) at the value of θiwhich max-
imized the SNR.
mand σ2
dare the variances of the signal and differencemaps,
5.3. Integrated Compton-Y Values
The SZ effect occurs when CMB photons inverse
Compton-scatter off hot electrons in clusters of galaxies
(Sunyaev & Zeldovich 1970). The imprint on the CMB is
proportional to the integrated electron gas pressure:
∆T
TCMB
= y f(x);
y ≡kBσT
mec2
?
dlneTe,
(30)
where the integral is along the line of sight, me, ne, and Te
are the electron mass, number density, and temperature, re-
spectively, σT is the Thomson scattering cross-section, and
the variable y is the Compton-y parameter. The function f(x)
encodes the dependence on frequency:
f(x) =?xcoth(x/2)−4?[1+δSZ(x,Te)],
with x ≡ hν/kBTCMB. For hot clusters, a relativistic correc-
tion δSZ(x,Te) may be required (Rephaeli 1995), in which
case f must be included in the integral of Eq. 30 in the non-
isothermal case.However, in this paper we assume non-
relativistic f for simplicity.
A robust measure of the SZ signal is the integrated
Compton-yparameter, since it is model-independentand sim-
ply sums pixels in the maps (Benson et al. 2004):
(31)
Y(θ) =
??
|θ′|<θ
dΩθ′y(θ′),
(32)
where θ is the angular distance from the cluster center. We
use steradians as the unit of solid angle, so Y is dimension-
less. As an example, it is plotted for ACT-CL J0638−5358
in the lower panel of Fig. 7; the upper panels show its az-
imuthally symmetrized temperature profile. The values of Y
at 2′, 4′, and 6′are shown for each cluster in Table 2. The er-
rors were determined by calculating the standard deviation of
Y values from maps of random patches of the CMB, made in
the same way as the cluster maps. We find 1σ uncertainties of
0.2×10−10, 0.6×10−10and 1.2×10−10forY at 2′, 4′, and 6′, re-
spectively. These values are dominated by systematic errors,
including uncertainties in the background level, contributions
from CMB in the map, and residual contamination from the
stripe removal. Accordingly,they should not be interpreted as
the significance of the detection; the SNR values (see Table 2)
serve that purpose.
Note thatin twoclusters—ACT-CL
(Abell S0520) and ACT-CL J0645−5413 (Abell 3404)—we
measure negative values ofY(4′) andY(6′). Their maps show
that the measured SZ signal is compact and the negative
values are consistent with noise.
As a check that the choice of knot spacing (τk= 0.5) is
not creating a significant bias via covariance of the celes-
tial signal with the low-frequency atmospheric estimate (see
§2.4), we created maps with τkfrom 0.15s to 1.5s for ACT-
CL J0245−5301 and ACT-CL J0638−5358. The temperature
profiles for the latter are plotted in the middle panel of Fig. 7.
Even the shortest spacing does not produce a profile which
is significantly different from the others. Its knot spacing,
τk= 0.15s, is the only one from the ensemble with a corre-
sponding angular scale smaller than the map size. We con-
clude that the results are not biased by having knots of too
high a frequency.
J0516−5432
5.4. Comparisons to Previous Measurements
The clusters shown in this paper are previously known X-
ray, optical, and/or SZ clusters; all are massive systems. For
four of the sources (Abell S0295, Abell 3128 (NE), Abell

Page 11

ACT: Beam Profiles and First SZ Clusters11
Abell S0295
Abell S0295
-53.2
-53.1
-53
-52.9
02h45m
02h46m
Difference
Difference
02h45m
02h46m
-200
-150
-100
-50
0
50
100
150
200
Abell 3128 (NE)Abell 3128 (NE)
-52.6
-52.5
-52.4
-52.3
03h30m
03h31m
03h32m
Difference
Difference
03h30m
03h31m
-200
-100
0
100
200
SPT-CL 0509-5342
SPT-CL 0509-5342
-53.9
-53.8
-53.7
-53.6
05h08m
05h09m
05h10m
Difference
Difference
05h08m
05h09m
-100
-50
0
50
100
Abell S0520
Abell S0520
-54.7
-54.6
-54.5
-54.4
05h16m
05h17m
Difference
Difference
05h16m
05h17m
-100
-50
0
50
100
SPT-CL 0547-5345
SPT-CL 0547-5345
-53.9
-53.8
-53.7
-53.6
05h46m
05h47m
Difference
Difference
05h46m
05h47m
-200
-100
0
100
200
Abell S0592
Abell S0592
-54.1
-54
-53.9
-53.8
06h38m
06h39m
06h40m
Difference
Difference
06h38m
06h39m
-200
-150
-100
-50
0
50
100
150
200
Abell 3404Abell 3404
-54.4
-54.3
-54.2
-54.1
06h45m
06h46m
Difference
Difference
06h45m
06h46m
-100
-50
0
50
100
1E 0657-56 (Bullet)
1E 0657-56 (Bullet)
-56.1
-56
-55.9
-55.8
06h58m
06h59m
07h00m
Difference
Difference
06h58m
06h59m
-400
-200
0
200
400
FIG. 6.— Cluster maps made using the Cottingham Method at 148GHz, paired with their difference maps (see §5.1). The coordinates are J2000 right ascension
(hours) and declination (degrees). The color bars are µK (CMB); note that the scale is different for each cluster. The gray disc in the top corner of the the signal
plots is 2.43′in diameter, the FWHM size of the beam convolved with the Gaussian smoothing kernel which was applied to these images. In each difference plot,
a cross shows the coordinates of the darkest spot in its corresponding signal map (except for SPT-CL 0509−5342 (ACT-CL J0509−5345)—see text).

Page 12

12The ACT Collaboration
-300
-200
-100
0
∆TCMB(µK)
-200
-100
0
∆TCMB(µK)
0
1
2
3
024
Radius (arcmin)
6810
1010Y
Signal
Difference
τk= 0.5s
τk= 0.15s
τk= 0.25s
τk= 1.0s
τk= 1.5s
FIG. 7.— The radial profile (top/middle) and integrated Compton Y(θ)
values (bottom) for the SZ decrement ofACT-CL J0638−5358 (Abell S0592).
The profile data are averages from the maps in 22′′-wide annuli, and Y(θ) is
the sum of the pixels within a radius θ, converted to the unitless Compton-y
parameter (Eqs. 30 and 32). The top panel shows the profile of the signal map
and difference map. The middle panel compares profiles for maps made with
different knot spacings τk, showing that only for very short spacings is the
profile noticeably different from the τ = 0.5s profile used for cluster analysis,
and then not significantly so. In all of the panels, the profile centers were
determined by the minimum of the map after smoothing with a 2′FWHM
Gaussian profile. (The profiles were calculated from the unsmoothed map.)
S0592, and Abell 3404), these are the first reported SZ detec-
tions. In this section, we briefly review measurements from
the literature to provide context, and point out some of the
contributions that our new measurements make to this body
of knowledge.
Relevantparametersfromtheliteraturearelisted inTable3;
references for these values are included below. Typical errors
on LXare small (< 20%), while those on the inferredmass are
more substantial (∼50%). Temperatures are measured values
from X-ray spectra. We use a flat ΛCDM cosmology with
ΩM= 0.3 and H0= 70kms−1Mpc−1. Masses are quoted in
units of M500, defined as the mass within a radius having a
mean mass density ?ρ? 500 times greater than the critical den-
sity, i.e., ?ρ? = 500×3H2/(8πG). In the following we briefly
discuss the clusters in the order in which they appear in Ta-
ble 2, with the exception of the South Pole Telescope (SPT)
clusters which are discussed together near the end of this sec-
tion.
5.4.1. Abell S0295
Abell S0295 first appeared in Abell et al. (1989) in their ta-
ble of supplementarysouthernclusters (i.e., clusters that were
not rich enough or were too distant to satisfy the criteria for
inclusion in the rich nearby cluster catalog). It was also found
to be a significant X-ray source in the ROSAT All Sky Survey
(RASS) Bright Source Catalog (Voges et al. 1999). The spec-
troscopic redshift of Abell S0295 was obtained by Edge et al.
(1994), who also reported the discovery of a giant strong-
lensing arc near the brightest cluster galaxy. Efforts to de-
tect the SZ effect at 1.2-mm and 2-mm with the SEST were
attempted, unsuccessfully, by Andreani et al. (1996). ASCA
observations (Fukazawa et al. 2004) yielded values (see Ta-
ble 3) for average temperature and soft band X-ray flux(0.1–
2.4keV), fromwhich we determinedthe correspondingX-ray
luminosity. The cluster mass M500was then estimated from
the luminosity-mass (specifically LX(0.1-2.4 keV) vs. M500)
relations from Reiprich & Böhringer (2002).
5.4.2. Abell 3128 (NE)
Untilquiterecentlythe north-east(NE)componentofAbell
3128 was believed to be part of the Horologium-Reticulum
superclusteratz=0.06. TheX-raymorphologyis clearlydou-
ble peaked with the two components separated on the sky by
some12′. Rose et al. (2002) estimated the virialmasses ofthe
two componentsassuming the redshift of the supercluster and
obtained a value for each of ∼1.5×1014M⊙. Fig. 8 shows
our SZ measurement with overlaid X-ray contours.
Abell 3128
Abell 3128
-52.6
-52.5
-52.4
03h30m03h31m
Declination (deg) - J2000
Right Ascension - J2000
-200
-200
-100
-100
0
0
100
100
200
200
∆TCMB(µK)
FIG. 8.—ACT-CL J0330-5228 (Abell 3128 (NE)) with overlaid contours of
X-ray emission in black. The SZ detection is associated with the NE feature
of Abell 3128, and confirms that it is due to a more massive, higher redshift
cluster than that at the SW lobe—a compelling example of the redshift in-
dependent mass selection of the SZ effect. The X-ray data come from two
separate XMM-Newton observations (Obs Ids 0400130101 and 0400130201)
with a total exposure time of 104ks. The two observations were mosaicked
into a single image over the 0.2–2.0 keV. Contour values are from 1.25×10−8
to 1.25×10−7photons/cm2/s/arcsec2.
RecentlyWerner et al. (2007)carriedouta detailedstudyof
this cluster using XMM-Newton data, which revealed a more
distant and more massive cluster superposedon the northeast-
ern component of Abell 3128. A significant portion of the
X-ray emission comes from this backgroundcluster. The val-
ues we quote in the table for redshift, X-ray luminosity, gas
temperature, and M500correspond to the background cluster
and come from Werner et al. (2007).
The large SZ decrement (SNR = 12.8) seen in the ACT
maps is clearly associated with the NE component where the
z = 0.44 cluster is. We do not detect a significant decre-
ment fromthe southwesterncomponentwhich lies at z=0.06.
Werner et al. (2007) estimate the temperature of the higher
redshift cluster to be 5.14±0.15 keV, which is significantly
hotter than that of the foreground cluster (kT = 3.36±0.04
keV). This system, therefore, is a compelling illustration of
the mass selection, approximately independent of redshift, of
the SZ effect. Werner et al. (2007) note that the temperature,
luminosity and mass estimates of the z = 0.44 background
cluster are all subject to large systematic errors, as the cluster
properties depend upon the assumed properties of the fore-
ground system. A joint X-ray/SZ/optical analysis should be
able to better constrain the characteristics of both systems and

Page 13

ACT: Beam Profiles and First SZ Clusters13
TABLE 3
SUMMARY OF CLUSTER PROPERTIES FROM X-RAY AND OPTICAL STUDIES
ACT Descriptor Catalog NameRedshiftDA
[Mpc]
LX(0.1–2.4 keV)
[1044erg s−1]
M500
kT
1010×Y2500a
[1015M⊙] [keV]
ACT-CL J0245−5301
ACT-CL J0330−5228
ACT-CL J0509−5345
ACT-CL J0516−5432
ACT-CL J0546−5346
ACT-CL J0638−5358
ACT-CL J0645−5413
ACT-CL J0658−5556
Abell S0295
Abell 3128 (NE)
SPT 0509−5342
Abell S0520
SPT 0547−5345
Abell S0592
Abell 3404
1E 0657−56
0.3006
0.44
0.36 (P)
0.294
0.88 (P)
0.2216
0.167
0.296
920
1172
1037
906
1596
737
589
910
8.3
3.9
2.2
3.5
4.7
10.6
8.2
20.5
0.8
0.3
0.4
0.6
0.6
1.0
0.7
1.4
6.7±0.7
5.1±0.2
—
7.5±0.3
—
8.0±0.4
7.6±0.3
10.6±0.1
0.53+0.35
0.15+0.10
—
0.72+0.47
—
1.31+0.86
1.87+1.23
1.61+1.16
−0.21
−0.06
−0.28
−0.52
−0.74
−0.67
NOTE. — See §5.4 for citations to the literature from which these values were obtained. The marker (P) in the redshift column
indicates a photometric redshift measurement.
aPredicted value ofY within R2500from theY-kT scaling relation of Bonamente et al. (2008). Errors come from the uncertainty
on the scaling relation parameters. Although we do not have R2500values for our clusters, the Y(2′) measurements listed in
Table 2 should be roughly comparable to these—see §5.4.8.
thereby contributeto assessing the mass threshold of the ACT
cluster survey.
5.4.3. Abell S0520
This optically-rich cluster (Abell et al. 1989) lies at a red-
shift of z = 0.294 (Guzzo et al. 1999). It is also an X-ray
cluster, RXC J0516−5430 (Böhringer et al. 2004), and has
been detected by SPT (Staniszewski et al. 2008) as SPT-CL
0517−5430. The X-ray temperature,soft X-rayflux and lumi-
nosity are based on XMM-Newton observations (Zhang et al.
2006) and the mass we quote in Table 3 comes from the X-
ray–derivedgasdensity andtemperatureprofiles(Zhang et al.
2008).
5.4.4. Abell S0592
The galaxy cluster Abell S0592 was originally detected
optically (Abell et al. 1989). ROSAT detected it as a bright
source in the All Sky Survey and its redshift (z = 0.2216)
was reported in de Grandi et al. (1999). The cluster is also
known by its REFLEX designation of RXC J0638.7−5358
(Böhringer et al. 2004). The ROSAT flux and luminosity in
the soft X-ray band (0.1-2.4keV) are 7.5×10−12erg cm−2s−1
and 1.1×1045ergs s−1. The X-ray spectrum of Abell S0592
fromaChandraobservation(Hughes et al. 2009)yieldsanin-
tegratedgastemperatureof kT =8.0±0.4keV.ThesoftX-ray
luminosity implies a cluster mass of M500= 1015M⊙.
5.4.5. Abell 3404
Abell 3404, at z = 0.167 (de Grandi et al. 1999), is the low-
est redshift system in the ACT SZ-detected cluster sample
presented here. It is REFLEX cluster RXC J0645.5−5413.
The X-ray temperature, soft X-ray flux and luminosity are
based on XMM-Newton observations (Zhang et al. 2008).
These authors also provide the total cluster mass based on the
X-ray–derivedgas density and temperature profiles.
5.4.6. 1E 0657−56 (Bullet Cluster)
We detect 1E 0657−56, the famous “Bullet” cluster, at high
significance with a strong central decrement and large in-
tegrated Y. Previous detections of the mm-band SZ signal
fromthis clusterhavebeenreportedbyACBAR (Gomez et al.
2004) and APEX-SZ (Halverson et al. 2008).
The spectroscopic redshift of 1E 0657−56 was obtained by
Tucker et al. (1998), the X-ray flux came from the Einstein
Observatory (Markevitch et al. 2002), the X-ray gas temper-
ature from XMM-Newton (Zhang et al. 2006), and the cluster
mass, M500, from a study by Zhang et al. (2008).
ACT-CL 0658-5557 (1E 0657-56/Bullet)
ACT-CL 0658-5557 (1E 0657-56/Bullet)
-56
-55.95
-55.9
06h58m30s 06h59m00s
Declination (deg) - J2000
Right Ascension - J2000
-400
-400
-200
-200
0
0
200
200
400
400
∆TCMB(µK)
FIG. 9.— ACT-CL J0658-5556 (Bullet Cluster) with overlaid contours
of X-ray emission (black) and dark matter distribution (orange). The X-ray
contours come from an 85ks-long Chandra observation (Obs Id 3184) and
correspond to the 0.5-2.0 keV band. Contour values are 4×10−7to 2×10−9
photons/cm2/s/arcsec2. The lensing data are from Clowe et al. (2007) with
contours running from κ = 0.12 to 0.39.
Fig. 9 shows a zoomed-in plot of our SZ map with
X-ray contours from Chandra and lensing contours from
Clowe et al. (2007). As expected, the SZ decrement follows
the X-ray contours more closely than the lensing contours,
since the collisonless darkmatter is expectedto be offsetfrom
the collisional gas in this merging system.
5.4.7. South Pole Telescope Clusters
Staniszewski et al. (2008) recently reported blind SZ de-
tections of four galaxy clusters. Only one of them (SPT-
CL 0517−5430) is a previously known cluster, Abell S0520
(see §5.4.3).
The physical properties of the SZ clusters—photometric
redshifts, luminosities, and mass estimates—have been re-
ported by Menanteau & Hughes (2009) based on optical and
X-ray data. Here we summarize some of their findings (see
Table3). Allfourclustershavecentralellipticalgalaxiesasso-

Page 14

14 The ACT Collaboration
ciated with them whose luminosities are consistent with those
of clusters in the Sloan Digital Sky Survey. Their mass esti-
mates from their optical and X-ray luminosities also suggests
that these are fairly massive systems. In Table 3 we list their
M500estimates from their X-ray luminosities.
We make strong detections of SPT-CL 0547−5345
and SPT-CL0509−5342 and also see Abell S0520 (SPT-
CL 0517−5430) with moderate SNR (c.f., Table 2). There
are visible similarities between the ACT maps presented here
and those from SPT. An interesting feature is the bright spot
to the north-westof SPT-CL 0509−5342which figures promi-
nently in both maps and for which we could find no radio or
infrared catalog source.
SPT-CL 0528-5300
SPT-CL 0528-5300
-53.1
-53
-52.9
-52.8
05h27m
05h28m
05h29m
Difference
Difference
05h27m
05h28m
-60
-40
-20
0
20
40
60
FIG. 10.— A map and corresponding difference map centered on the coor-
dinates of SPT-CL 0528−5300, a cluster candidate detected by the South Pole
Telescope but not yet detected with our instrument. The units of the axes are
right ascension (hours) and declination (degrees), and the color bar is ∆TCMB
(µK). The map rms noise is 45µK.
We are unable to confirm the detection of SPT-CL
0528−5300. Fig. 10 shows a map centered on the cluster
coordinates. There is no measureable SNR (Eq. 29) for a
putative decrement centered at its coordinates.
the map noise, we report a 2σ non-detection at the 90µK
level. Recent weak-lensing mass estimates for the SPT clus-
ters (McInnes et al. 2009) indicate that SPT-CL 0528−5300
has a lower mass (M500∼2×1014M⊙, scaled fromtheir M200
assuming a typical factor of 0.6, e.g., Reiprich & Böhringer
(2002)) than any of the other known clusters that we have de-
tected in the SZ with ACT (see Table 3). Further study of the
X-ray and optical properties of SPT-CL 0528−5300 will be
necessary to more accurately predict its expected SZ signal.
Based on
5.4.8. Comparison with Previous SZ measurements
Although the large masses of the ACT-detected clusters we
report here offer strong support for the reality of our detec-
tions, we also compare the quoted integrated Compton-y pa-
rameters for consistency with expections from previous SZ
cluster studies. For this we use theY-kT scaling relation from
Bonamente et al. (2008) (using values for “all clusters” from
their Table 2). Predicted values are given in the last column
of Table 3. The Y values in the scaling relation were inte-
grated within R2500, the radius where the average cluster mass
density is 2500 times the critical density. We do not have
precise R2500values for our clusters, but estimates of R2500
range from about 1′to 3′, so the predicted values of Y(2500)
should, to first order, be roughly comparable to ourY(2′) val-
ues. With that proviso, the predicted and measured Y values
agree to within 2σ for all but two clusters: Abell 3404, whose
X-ray temperature predicts a much higher Y value than we
measure, and Abell 3128 (NE), where the cluster temperature
predicts a much lower Y value than measured. The latter is
a complex system which could have a larger mass than pre-
viously thought. Additionally, the Bonamente et al. (2008)
scaling relation was measured at 30GHz. At 148GHz, the
point source contamination is different. This might explain
why a measured Y value is lower than the predictionbased on
the 30GHz scaling relation. A larger and better-studied sam-
pleof ACT-detectedclusters will benecessary beforedrawing
conclusions about scaling relations.
6. CONCLUSIONS
We have described a maximum-likelihood mapping algo-
rithm which uses B-splines to model atmospheric signal and
to remove it from the data.
The method has been used to make high precision (<
−40dB) beam maps, with solid angles in the 148GHz,
218GHz, and 277GHz bands of (218.2± 4)nsr, (118.2±
3)nsr, and (105.2±6)nsr, respectively. The beam profiles
and window functions will be important for all subsequent
analyses of ACT’s data.
Additionally,we havemademapsshowingSZ detectionsof
eight previously discovered galaxy clusters. Our high-σ de-
tection of the z = 0.44 component of Abell 3128, and our cur-
rent non-dectection of the low-redshift part, corroborates ex-
isting evidence that the further cluster is more massive. This
is a compelling example of the redshift-independent mass se-
lection of the SZ effect.
The maps presented in this paper will be made
publicby Jan1, 2010
site(http://lambda.gsfc.nasa.gov/)
(http://www.physics.princeton.edu/act/).
Acknowledgements.
The ACT project was proposed in
2000 and funded Jan 1, 2004. Many have contributed to
the project since its inception. We especially wish to thank
Asad Aboobaker, Christine Allen, Dominic Benford, Paul
Bode, Kristen Burgess, Angelica de Oliveria-Costa, Peter
Hargrave,Norm Jarosik, Amber Miller, Carl Reintsema, Uros
Seljak, Martin Spergel, Johannes Staghun, Carl Stahle, Max
Tegmark, Masao Uehara, and Ed Wishnow. It is a pleasure
to acknowledge Bob Margolis, ACT’s project manager. Reed
Plimpton and David Jacobson worked at the telescope during
the 2008 season. ACT is on the Chajnantor Science preserve
which was made possible by CONICYT. We are grateful for
the assistance we received at various times from the ALMA,
APEX, ASTE, CBI/QUIET, and NANTEN2 groups.
PWVdatacomefromthepublicAPEXweathersite. Fieldop-
erations were based at the Don Esteban facility run by Astro-
Norte. This research has made use of the NASA/IPAC Extra-
galactic Database (NED) which is operated by the Jet Propul-
sion Laboratory, California Institute of Technology, under
contractwiththeNationalAeronauticsandSpaceAdministra-
tion. We thank the members of our external advisory board—
Tom Herbig (chair), Charles Alcock, Walter Gear, Cliff Jack-
son, Amy Newbury, and Paul Steinhardt—who helped guide
the project to fruition. This work was supported by the U.S.
National Science Foundation through awards AST-0408698
for the ACT project, and PHY-0355328, AST-0707731 and
PIRE-0507768. Fundingwas also providedby PrincetonUni-
versityand the Universityof Pennsylvania. ADH receivedad-
ditional support from a Natural Science and Engineering Re-
search Council of Canada (NSERC) PGS-D scholarship. AK
and BP were partially supported through NSF AST-0546035
and AST-0606975, respectively, for work on ACT. HQ and
LI acknowledgepartial supportfrom FONDAP Centro de As-
through the LAMBDA
ACTand site
The

Page 15

ACT: Beam Profiles and First SZ Clusters 15
trofisica. ES acknowledges support by NSF Physics Frontier
Center grant PHY-0114422 to the Kavli Institute of Cosmo-
logical Physics. KM, MH and RW received financial sup-
port from the South African National Research Foundation
(NRF), the Meraka Institute via fundingfor the South African
Centre for High Performance Computing (CHPC), and the
SouthAfricanSquareKilometerArray(SKA)Project. RH re-
ceivedfundingfromtheRhodes Trust. LVacknowledgessup-
portfromnsf-ast 0707731and FP7-PEOPLE-2007-4-3IRG n
202182.
APPENDIX
A. THE COTTINGHAM METHOD AS A MAXIMUM LIKELIHOOD ESTIMATOR
Eq. 1 can be written in the matrix form,
d =?P B??m
α
?
+n,
(A1)
which has the maximum likelihood estimator:
?
? m
? α
?
=
??PT
BT
?
N−1?P B??−1?PT
BT
?
N−1d =
?(PP) (PB)
(BP) (BB)
?−1?PTN−1d
BTN−1d
?
,
(A2)
where we use the shorthand notation (XY) ≡ XTN−1Y. The inverted matrix evaluates to:
?
?(PP)−(PB)(BB)−1(BP)?−1
−(PP)−1(PB)?(BB)−(BP)(PP)−1(PB)?−1
−(BB)−1(BP)?(PP)−(PB)(BB)−1(BP)?−1
?(BB)−(BP)(PP)−1(PB)?−1
?
.
(A3)
Since this matrix is symmetric, and (XY)T= (YX), we can rewrite the lower-left component as:
(BB)−1(BP)?(PP)−(PB)(BB)−1(BP)?−1=
?
(PP)−1(PB)?(BB)−(BP)(PP)−1(PB)?−1?T
(A4)
=?(BB)−(BP)(PP)−1(PB)?−1(BP)(PP)−1.
(A5)
Thus, the solution for the atmosphere is:
? α =?(BB)−(BP)(PP)−1(PB)?−1BTN−1d −
?(BB)−(BP)(PP)−1(PB)?−1(BP)(PP)−1PTN−1d.
(A6)
To show that this is equivalent to the solution presented in §2.1, we observe that the definitions in Eqs. 3 and 6 of §2.1 can be
recast:
Θ ≡ BTN−1(1−PΠ)B = (BB)−(BP)(PP)−1(PB),
φ ≡ BTN−1(1−PΠ)d =
?BTN−1−(BP)(PP)−1PTN−1?d.
(A7)
This reduces Eq. A6 to:
? α = Θ−1φ,
(A8)
which is the same as Eq. 7 of §2.1.
B. A PLANET’S SOLID ANGLE CONTRIBUTION TO THE BEAM SOLID ANGLE MEASUREMENT
Denote the instrumentresponse with P(n) andthe poweremitted by the planet with P0Ψ(n), where P0is the peakpower emitted
and Ψ is a normalized distribution describing its shape. The coordinate n is a two dimensional vector describing the position on
the sky, with n = 0 at the planet center. The measured beam map,?B, is the convolution of the true beam, B, with the planet:
P(0)=
? B(n) =P(n)
??dΩn′B(n−n′)Ψ(n′)
??dΩn′B(−n′)Ψ(n′).
???dΩn′Ψ(n′)????dΩnB(n)?
(B1)
The measured solid angle is then (c.f. Eq. 13):
?ΩA=
??
dΩn
??dΩn′B(n−n′)Ψ(n′)
??dΩn′B(−n′)Ψ(n′)
=
??dΩn′B(n′)Ψ(n′)
=
ΩΨΩA
??dΩn′B(n′)Ψ(n′),
(B2)
wherein the secondequalitywe broughtthe denominatoroutsidethe outerintegral,and in the numeratorwe switchedthe orderof
and then shifted the dummy variable for the integral over B. In the last equality we recognized that the integrals in the numerator
evaluate to the solid angles of the planet and the true instrument beam, respectively. If the planet is much smaller than the beam,
we can expand the beam appearing the integrand of the denominator in a Taylor series: