The Astrophysical Journal, 790:77 (16pp), 2014 July 20 doi:10.1088/0004-637X/790/1/77
C
2014. The American Astronomical Society. All rights reserved. Printed in the U.S.A.
THE GISMO TWO-MILLIMETER DEEP FIELD IN GOODS-N
Johannes G. Staguhn
1,2
, Attila Kov
´
acs
3,4
, Richard G. Arendt
2,5
, Dominic J. Benford
2
, Roberto Decarli
6
, Eli Dwek
2
,
Dale J. Fixsen
2,7
, Gene C. Hilton
8
,KentD.Irwin
8,9
, Christine A. Jhabvala
2
, Alexander Karim
10,11
,
Samuel Leclercq
12
, Stephen F. Maher
2,13
, Timothy M. Miller
2
, S. Harvey Moseley
2
,
Elmer H. Sharp
2,14
, Fabian Walter
6
, and Edward J. Wollack
2
1
The Henry A. Rowland Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218, USA
2
Observational Cosmology Lab, Code 665, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA
3
California Institute of Technology 301-17, 1200 East California Boulevard, Pasadena, CA 91125, USA
4
Institute for Astrophysics, University of Minnesota, 116 Church St SE, Minneapolis, MN 55455, USA
5
CRESST, University of Maryland, Baltimore County, Baltimore, MD 21250, USA
6
Max-Planck-Institute f
¨
ur Astronomie, K
¨
onigstuhl 17, D-69117 Heidelberg, Germany
7
CRESST, University of Maryland, College Park, College Park, MD 20742, USA
8
NIST Quantum Devices Group, 325 Broadway Mailcode 817.03, Boulder, CO 80305, USA
9
Department of Physics, Stanford University, Stanford, CA 94305, USA
10
Department of Physics, Durham University, South Road, Durham DH1 3LE, UK
11
Argelander-Institut f
¨
ur Astronomie, University of Bonn, Auf dem H
¨
ugel 71, D-53121 Bonn, Germany
12
Institut de Radio Astronomie Millim
´
etrique, 300 Rue de la Piscine, F-38406 Saint Martin d‘Heres, France
13
Science Systems and Applications, Inc., 10210 Greenbelt Road, Suite 600, Lanham, MD 20706, USA
14
Global Science and Technology, Inc., 7855 Walker Drive, Suite 200, Greenbelt, MD 20770, USA
Received 2013 February 22; accepted 2014 June 7; published 2014 July 7
ABSTRACT
We present deep continuum observations using the GISMO camera at a wavelength of 2 mm centered on the Hubble
Deep Field in the GOODS-N field. These are the first deep field observations ever obtained at this wavelength.
The 1σ sensitivity in the innermost ∼4
of the 7
diameter map is ∼135 μJy beam
−1
, a factor of three higher in
flux/beam sensitivity than the deepest available SCUBA 850 μm observations, and almost a factor of four higher
in flux/beam sensitivity than the combined MAMBO/AzTEC 1.2 mm observations of this region. Our source
extraction algorithm identifies 12 sources directly, and another 3 through correlation with known sources at 1.2 mm
and 850 μm. Five of the directly detected GISMO sources have counterparts in the MAMBO/AzTEC catalog,
and four of those also have SCUBA counterparts. HDF850.1, one of the first blank-field detected submillimeter
galaxies, is now detected at 2 mm. The median redshift of all sources with counterparts of known redshifts is
˜z = 2.91 ±0.94. Statistically, the detections are most likely real for five of the seven 2 mm sources without shorter
wavelength counterparts, while the probability for none of them being real is negligible.
Key words: galaxies: evolution – galaxies: high-redshift – galaxies: luminosity function, mass function –
galaxies: photometry – galaxies: starburst – infrared: galaxies
Online-only material: color figures
1. INTRODUCTION
Submillimeter and millimeter observations have revealed the
existence of a population of previously unknown high-redshift
dust-enshrouded starburst galaxies. Virtually all their stellar
UV and optical radiation is absorbed and reradiated by the
dust at infrared (IR) wavelengths. They are among the most
luminous galaxies in the universe, and their relative contribution
to the galaxy number counts and co-moving luminosity density
increases with redshift (e.g., Sargent et al. 2012).
The most massive galaxies are predicted to be at the center of
galaxy clusters that reside in the most massive dark matter halos.
Surveys that map their distribution with redshift will therefore
reveal the epochs of cluster formation in the early universe. For
example, follow-up observations of two submillimeter galaxies
(SMGs) at optical and near-IR wavelengths have shown that they
are members of protoclusters that formed at z ≈ 5 (AzTEC-3:
Capak et al. 2011; HDF850.1: Walter et al. 2012). A survey of
dusty starbursts is also essential for determining the obscured
cosmic star formation rate at high redshift, and for understanding
the formation and evolution of dust in these objects.
The advantage of using (sub)millimeter wavelength observa-
tions to search for these objects stems from the fact that starburst
galaxies have typical dust temperatures of 35 K, so that their IR
spectrum peaks at ∼90 μm. Submillimeter–millimeter observa-
tions therefore trace the Rayleigh–Jeans part of their spectrum,
and benefit from the fact that the decrease in flux from high-
redshift objects is largely offset by the negative K-correction.
Figure 1 depicts the flux of a typical dusty SMG versus red-
shift (solid lines). This starburst galaxy is characterized by an
IR luminosity of 10
12
L
and a dust mass of 10
8
M
. The dust
was assumed to have a κ(λ) ∝ λ
−β
mass absorption coefficient
with a spectral index of β = 1.5, and a temperature of 35 K. An
interesting effect at high redshifts is the fact that dust heating
by the cosmic microwave background (CMB) becomes compa-
rable to the heating by ambient starlight. When accounting for
both sources of heating, the actual dust temperature can be ex-
pressed as T
d
= (T
4+β
0
+T
4+β
CMB
)
1/(4+β)
, where T
CMB
= 2.73(1+z)
is the CMB temperature at redshift z, and T
0
is the dust tem-
perature when heated by starlight alone. The Goddard IRAM
2 Millimeter Observer (GISMO) observations are a differential
measurement of the flux from a galaxy against the CMB. The
observed galaxy spectrum in such measurement is thus given by
F
ν
(λ) = 4πM
d
κ(λ)
[
B
ν
(λ, T
d
) − B
ν
(λ, T
CMB
]
, (1)
which cannot be characterized by a single blackbody with a
simple λ
−β
emissivity law. The dotted lines in the figure show
1
The Astrophysical Journal, 790:77 (16pp), 2014 July 20 Staguhn et al.
S
ν
(mJy)
Redshift
200 μm
350 μm
500 μm
850 μm
1.2 mm
2.0 mm
0.1
1
10
2 4 6 8 10 12
Figure 1. Redshift-dependent flux density, measured against the CMB, of a
galaxy with fixed dust mass M
d
= 10
8
M
and a dust emissivity index of
β = 1.5, i.e., with an FIR luminosity of L ∼ 10
12
L
, shown for a variety of
wavelengths, based on the multi-temperature empirical dust models of Kov
´
acs
et al. (2010). To radiate the same luminosity against an increasingly warmer
CMB in earlier epochs, the cold dust temperature (T
0
= 35 K) must rise as
T
4+β
eff
d
= T
4+β
eff
CMB
+ T
4+β
eff
0
(the effective dust emissivity, β
eff
,isdefinedin
Kov
´
acs et al. 2010). For comparison, the dashed lines show the same if the CMB
heating is ignored. Note, that the observed flux density at 2 mm wavelengths
increases monotonically and steeply as a function of redshift for z>1.
(A color version of this figure is available in the online journal.)
the fluxes that would be measured if the CMB radiation were
not present.
The figure shows that the 2 mm fluxes tend to be lower than
those at the shorter wavelengths. However, the rising 2 mm flux
with redshift provides the strongest bias toward the detection of
high-redshift galaxies. Furthermore, the atmospheric transmis-
sion is higher, and the atmospheric background noise is lower
at 2 mm than at shorter wavelengths.
We have developed the GISMO instrument that utilizes a
near background-limited detector to fully exploit the advantages
of the 2 mm window. Here we report the first deep survey
conducted with GISMO centered on the Hubble Deep Field
North (HDF-N). The HDF-N is one of the best studied regions
in the sky. Its sky coverage is one Hubble Space Telescope
WFPC2 pointing, i.e., ∼2.
5 ×2.
5, which is less than the 2
×4
instantaneous field of view of the GISMO array. The HDF
is located in the greater GOODS-N region, which has also
been studied in exquisite detail over the last decade at many
wavelengths, X-rays: Brandt (2008), UV: Teplitz et al. (2006),
optical: Giavalisco et al. (2004), optical spectroscopy: Cowie
et al. (2004), near-infrared: Yan et al. (2006), mid-infrared:
Rodighiero et al. (2006), far-infrared: Borys et al. (2003),
Frayer et al. (2006), Perera et al. (2008), radio: Morrison et al.
(2010). In the (sub-)millimeter regime, the HDF and GOODS-N
have been studied by the (sub-)millimeter cameras SCUBA,
MAMBO, and AzTEC in the past (Hughes et al. 1998; Pope
et al. 2005; Greve et al. 2008; Penner et al. 2011). The currently
available data of the full HDF at 1 mm reach 1σ sensitivities of
0.5 mJy beam
−1
(MAMBO observations combined with AzTEC
observations; Penner et al. 2011); the SCUBA “super” map
(Pope et al. 2005) reaches a peak depth of 0.4 mJy beam
−1
at
850 μm, however the sensitivity varies significantly over the
observed area in the field.
The paper is organized as follows. We first describe the
instrument and its characteristics in Section 2. In Section 3
we describe observations and the data reduction. In Section 4
we describe the source extraction and its results, and present
simulations used to characterize the data and to evaluate the
completeness and reliability of the extracted sources. Section 5
presents the 2 mm number counts and the analysis of the
properties of select individual sources.
2. THE GISMO 2 mm CAMERA
Continuum observations in the 2 mm atmospheric window
have not been astronomically explored from the ground to the
same degree as has been done at shorter wavelengths (1 mm
or less), except for Sunyaev–Zel’dovich observations with dedi-
cated 6–10 m class telescopes (Swetz et al. 2011; Carlstrom et al.
2011; Dobbs et al. 2006). The reason for this is predominantly of
technical nature, in particular the very demanding requirements
on the noise performance of a background-limited camera oper-
ating in this low opacity atmospheric window. In order to provide
background-limited observations in the 2 mm window at a good
mountain site such as the IRAM 30 m telescope on Pico Veleta,
the required sensitivity, expressed in Noise Equivalent Power,
for the detectors is ∼5 ×10
−17
W
√
s (Staguhn et al. 2006), a re-
quirement that is met by our “high” temperature (T
c
= 450 mK)
Transition Edge Sensor (TES) detectors. Consequently we have
proposed and built a 2 mm wavelength bolometer camera, the
GISMO (Staguhn et al. 2008), for astronomical observations at
the IRAM 30 m telescope on Pico Veleta, Spain (Baars et al.
1987). GISMO uses a compact optical design (Sharp et al. 2008)
and uses an 8 × 16 array of close-packed, high sensitivity TES
bolometers with a pixel size of 2 × 2mm
2
(Benford et al.
2008), which was built in the Detector Development Labora-
tory at NASA/GSFC (Allen et al. 2008). The array architecture
is based on the Backshort Under Grid design (Allen et al. 2006).
GISMO’s bandpass is centered on 150 GHz and has a fractional
bandwidth of 20%. The superconducting bolometers are read
out by SQUID time domain multiplexers from NIST/Boulder
(Irwin et al. 2002). This design is scalable to kilopixel size
arrays for future ground-based, suborbital and space-based
X-ray and far-infrared through millimeter cameras (e.g.,
Staguhn et al. 2012).
3. OBSERVATIONS AND REDUCTION
The GISMO Deep Field (GDF) observations of the HDF-N
were obtained between 2011 April 13 and 18, and on 2012 April
11, 12, and 23. The total integration time was t ∼ 39 hr; however
2/3 of those observations were obtained with GISMO’s lower
sensitivity during the Spring 2011 run (see Section 3.4). The
FWHM of GISMO’s beam is 17.
5.
3.1. Data Reduction
The data were reduced, using CRUSH
15
(Kov
´
acs 2008),
which is the standard reduction software for the GISMO camera.
CRUSH is open-source and available in the public domain. The
data reduction tool of CRUSH consists of a highly configurable
pipeline, which uses a series of statistical estimators in an
iterated scheme to separate the astronomical signals from
15
http://www.submm.caltech.edu/∼sharc/crush
2
The Astrophysical Journal, 790:77 (16pp), 2014 July 20 Staguhn et al.
0
5
10
15
20
25
30
35
40
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
# Scans
Line-of-sight opacity at 2-mm
Figure 2. Histogram of the 2 mm line-of-sight opacities for the GDF
observations.
the bright and variable atmospheric background and various
correlated instrumental noise signals. It determines proper noise
weights for each sample in the time series, removes glitches,
identifies bad pixels and other unusable data, and determines
the relevant relative gains. It also applies appropriate filters for
1/f -type noise, and other non-white detector noise profiles.
For the detection of point sources, the resulting “deep”-mode
maps are spatially filtered above 50
FWHM to remove spatially
variant atmospheric residuals. The fluxes in each 10 minute scan
are corrected for the line-of-sight atmospheric opacities, based
on the IRAM radiometer measurements. Point-source fluxes
are also corrected scan-wise for the flux-filtering effect of each
and every pipeline step, and for the large-scale structure (LSS)
filtering of the final map. As a result, comparison to point-like
calibrator sources (e.g., planets and quasars) is straightforward,
even if different reduction options are used for these and the
science targets.
3.2. Calibration
Mars, Uranus, and Neptune were observed for primary
flux calibration. Of those, measurements of Mars cover the
widest range of weather conditions. Using the atmospheric
transmission model of the Caltech Submillimeter Observatory
16
and the IRAM 30 m Telescope 225 GHz radiometer readings, we
obtain excellent calibration in effectively all weather conditions:
a 7% rms blind calibration up to τ
225 GHz
∼ 1 is obtained.
Note that any model uncertainties due to the different elevation
of Mauna Kea, the site of the CSO, and Pico Veleta, will be
very small and therefore irrelevant for the accuracy of derived
calibration factors. Figure 2 shows the histogram of the 2 mm
line-of-sight opacities for all data.
3.3. Pointing and Astrometric Accuracy
During the GISMO observing runs in 2011 and 2012, we
obtained a large number of pointing measurements over the
16
http://www.submm.caltech.edu/cso/weather/atplot.shtml
entire sky, from which we derived appropriate pointing models
according to Greve et al. (1996). Our pointing models yield
<3
rms accuracy in both Az and El directions on all pointing
measurements obtained during the two observing runs (424
and 392 individual pointing observations for the 2011 and
2012 observing runs, respectively). Additionally, we frequently
checked pointing on nearby quasars during GDF observations.
Triggered by a reduction flag, CRUSH will automatically
incorporate the measured differential offsets with respect to
the pointing model, to further improve pointing accuracy, and
to remove most systematic pointing errors in the pointing
model, or due to structural deformations of the telescope. The
resulting residual pointing errors are expected to be independent
and random between independent pointing sessions. Thus, a
representative lower bound to the final astrometric accuracy
is given by the instantaneous pointing rms (<4.
2) divided
by the square root of the number of independent pointing
sessions spanning the observations. In our case, approximately
30 independent pointing sessions bracket the GDF observations.
Therefore, the astrometric accuracy of our map (notwithstanding
the inherent positional uncertainties of any detections) could
be as low as 0.
8 rms, or somewhat higher in the presence of
systematics errors, which are not eliminated by the use of nearby
pointing measurements.
3.4. Instrument Performance
The noise equivalent flux density (NEFD) of measurements
during the 2011 run was typically 15–17 mJy
√
s, under most
weather conditions. The obtained sensitivity at that time was
mainly limited by a neutral density filter with 40% transmission.
This filter was needed, since there was a significant amount of
THz light scattered into the GISMO beam by the low pass filters,
which were positioned very close to the entrance window of the
dewar. In early 2012 we mounted a 77 K baffle dewar in front of
the GISMO optical entrance window, which reduces the stray
light significantly and eliminates the need for a neutral density
filter in the instrument (Sharp et al. 2012). As a result of this,
the NEFD obtained during the 2012 observing run was typically
10 mJy
√
s.
3.5. Noise Properties of the Beam-smoothed Map
To estimate the noise we randomly multiply each of the
individual 10 minute scans by +1 or −1, a method known as
“jackknifing.” This eliminates any stationary noise (including
sources and foreground) but retains random noise, including that
from the atmosphere. The histogram of the signal-to-noise ratio
(S/N) for the jackknifed, beam-smoothed, and filtered GISMO
map is shown in Figure 3. The distribution is well fit by a
Gaussian with σ = 1.00. The S/N histogram for the regular
(not jackknifed) smoothed and filtered map (also Figure 3)
shows distinct excess deviations from the Gaussian distribution
on both extremes of the distribution. When we subtract the
12 detected sources (see Section 4.3) from the image, the
S/N histogram does approach the expected noise distribution
as shown in Figure 4. The subtraction of sources (both real
and false) causes this histogram to be truncated at the 2.99σ
detection limit. Below this limit both the positive and negative
sides of the histogram are more closely Gaussian because the
effective smoothed and filtered point-spread function (PSF)
used for the subtraction has both positive and negative features
(main beam and surrounding filter bowl). There is, however, a
slight (∼5%) symmetric excess remaining in the post-extraction
3
The Astrophysical Journal, 790:77 (16pp), 2014 July 20 Staguhn et al.
Figure 3. Signal-to-noise histogram for the smoothed and filtered GDF data
(black), which was used for the source extraction, and shown in Figure 5,and
for a corresponding time-based random jackknife realization (green). The dotted
line shows the expected Gaussian noise distribution, based on the radiometric
down-integration of the detector timestream noise, with σ = 1.00. The close
agreement of the jackknifed noise distribution and the Gaussian expectation
indicates that our measurement noise is both closely Gaussian in nature, and
radiometric. At the same time, the histogram of the regular (non-jackknifed) map
exhibits distinct deviations from the Gaussian noise. A symmetric widening is
caused by additional noise from unresolved sources (i.e., confusion noise) on top
of the radiometric measurement noise, while resolved emission sources cause
the asymmetric excess on the positive half of the distribution.
(A color version of this figure is available in the online journal.)
residuals, relative to the jackknifed map. This excess would be
consistent with the presence of confusion noise in our map.
We cannot, however, quantify the corresponding level of 2 mm
confusion noise with any precision because the noise estimation
errors are relatively large given the low number of beams in the
field.
4. SOURCE EXTRACTION AND SIMULATIONS
4.1. Extraction—Method and Reliability
The Gaussian nature of noise in our map (Section 3.5)isa
notable feature of the GISMO data and allows us to provide
the strongest possible statistical characterization of our source
candidates relying exclusively on formal Gaussian statistics.
Because we do not detect any deviation from Gaussian noise,
nor see any signs of non-radiometric down-integration in the
jackknifed maps, we need not worry about potential troublesome
statistical biases that could otherwise result from non-Gaussian,
or correlated, noise features.
We extract sources from the beam-smoothed filtered map,
which were produced by CRUSH. Beam-smoothing is math-
ematically equivalent to maximum-likelihood PSF amplitude
fitting at every map position (Kov
´
acs 2006), i.e., the PSF-
smoothed map value and its noise directly provide the amplitude
and uncertainty of a fitted PSF at each map position. The ef-
fective filtered PSF for GISMO maps reduced in “deep” mode
in CRUSH is accurately modeled as a combination of a 17.
5
FWHM Gaussian main beam combined with a negative 50
FWHM Gaussian bowl such that the combined PSF yields zero
integral (signifying that we have no DC sensitivity due to the
sky-noise removal, and other filtering during the reduction). The
Figure 4. Signal-to-noise histogram of the smoothed and filtered GDF data
(Figure 5), after the 12 blindly identified sources are removed and flagged.
The vertical dashed line marks the source extraction threshold, resulting in an
abrupt truncation of the histogram above the detection threshold. The dotted line
shows the expected Gaussian noise distribution from the jackknifed map with
σ = 1.00. The post-extraction residuals are more closely Gaussian, as expected,
albeit still hinting at the presence of further (fainter) resolved and unresolved
sources in the field, which manifest as an asymmetric and a symmetric excess,
respectively.
50
FWHM effective PSF bowl is the direct result of explicit LSS
filtering during the reduction with a 50
FWHM Gaussian pro-
file. We checked the Gaussian main beam assumption on quasars
(detected at high S/N >1000), and confirmed that >98% of the
integrated flux inside a R = 50
circular aperture is recovered
in the 17.
5 FWHM beam-smoothed peak during night-time ob-
servations (i.e., for all of our data). We also checked that the
filtered PSF accurately recovers the quasar fluxes, when these
were LSS filtered the same as our deep field map, and found
no further degradation of photometric accuracy associated with
the LSS filtering. Therefore, we are very confident that we have
sufficient understanding of the effective PSF in our map, and
that the systematic errors of the extracted points source fluxes
are kept below 2%.
The source extraction code we used is part of the CRUSH
software package, and is the same source extraction tool that
was used and described in Weiß et al. (2009). It implements an
iterated false-detection rate algorithm. Apart from peak position
and flux, the algorithm calculates an estimated confidence and
an expected cumulative false-detection rate for each extracted
source. We caution that the confidence levels and false-detection
rates are guiding values only, which represent our best statistical
estimate without prior knowledge of the 2 mm source counts.
A more accurate characterization of confidence levels and/or
cumulative false-detection rates would require accurate prior
information of the true counts of the 2 mm source population.
4.1.1. Overview of the CRUSH Source Extraction Tool
Here, we offer a concise summary of the approach imple-
mented by CRUSH “detect” tool, which we used for the source
extraction.
The expected false-detection rate, i.e., the expected number
of pure noise peaks mistakenly identified as a source, is given
by N
f
(Σ) = NQ(Σ), where N is the number of independent
4
The Astrophysical Journal, 790:77 (16pp), 2014 July 20 Staguhn et al.
Gaussian variables in the map, and Q(Σ) = 1 − P (Σ)isthe
complement cumulative Gaussian probability, i.e., the probabil-
ity of measuring a deviation larger than a chosen significance,
Σ. A smoothed and filtered map with extraction area A contains
N ≈
4.5A
1.13(Δ
2
smooth
)
1 −
Δ
2
smooth
Δ
2
filter
(2)
independent variables in terms of the FWHM widths of the
Gaussian smoothing (Δ
smooth
= Δ
beam
) and the applied large-
scale filtering of the map (Δ
filter
= 50
). The right-hand-side
term in the formula accounts for the lost degrees of freedom due
to the explicit spatial filtering of our map. The approximately 4.5
parameters per smoothing beam were determined empirically
based on the occurrence of significant noise peaks in simulated
noise maps. The formula was verified to yield close to the
expected number of false detections in simulated noise maps
with varying areas and filtering properties, and with N
f
targeted
between 0 to 1000. Thus, the above expression will accurately
predict the actual false-detection rate, as a function of detection
threshold, as long as the map noise is known precisely. For our
map, with ∼321 GISMO beams, a 2.99σ cut yields N
f
(2.99) ≈
2 expected false detections.
Due to the presence of many resolved but undetected sources
in the map (asymmetric confusion), our noise estimates are
bound to be slightly overestimated (even with the median-noise
based estimate used). To our best knowledge, all statistical
estimates of map noise, which are based on the observed map
itself, will result in overstated noise estimates in the presence
of asymmetric confusion (resolved sources below detection).
Neither the jackknifed noise, nor radiometric noise, can help
offer better estimates, as the extraction noise should include
the effect of symmetric confusion (unresolved faint sources)
beyond what these can offer. As a result of an inevitably biased
noise estimation process, the corresponding false-detection rate
estimates are slightly above actual, and represent a useful
conservative upper bound. This is confirmed by the simulations,
presented in Section 4.5, which found that if the 2 mm source
counts were those of, e.g., B
´
ethermin et al. (2011) or Lapi
et al. (2011), then the actual false-detection rate would be 1.34
or 0.55, respectively, versus the expected 2. However, as we
stated earlier, we cannot unbias our noise estimates, or quantify
the true false-detection rate, without prior knowledge of the
true 2 mm source counts, which are not well-constrained at
present. Instead, our estimates offer strong upper bounds for the
unknown actual false-detection rates.
Each source identified above the significance cut is removed
from the map with the smoothed and filtered PSF before the
extraction proceeds. Subtraction with the filtered PSF allows
the detection of further nearby peaks, which may have been
previously suppressed by the negative filter bowl surrounding
the previous detections. The circular area (r
2
= Δ
2
beam
+
Δ
2
smooth
) containing the main beam of the detected source
is flagged after the extraction, since it no longer contains
meaningful information after the removal of the source from
within. To ensure that our catalog is based on the most accurate
measure of the map noise and zero levels, CRUSH estimates
the zero level using the mode of the map flux distribution,
and estimates the noise from the median observed deviation
median x
2
≈ 0.454937σ
2
. Both measures are relatively robust
and reasonably unbiased by the presence of relatively bright
sources, or localized features, in the map.
For each source candidate, CRUSH estimates a detection con-
fidence based on the expected false-detection rate N
f
. Accord-
ing to Poisson statistics, the detection confidence C of a single
peak is the probability that no such peak occurs randomly, i.e.,
P
0
(N
f
) = e
−N
f
. This is then further refined to include infor-
mation from other sources already detected in the map. Thus, if
n true sources with apparent significance above Σ are known a
priori to exist in the map, than any given peak at significance Σ
may be one of n sources, or one of the N
f
expected false detec-
tions, hence the probability of false detection for each of n + N
f
peaks is reduced by a factor of N
f
/(n+ N
f
). (In other words, we
should expect only N
f
false detections (noise peaks) for every n
actual sources detectable above a given threshold.) CRUSH uses
the number of sources N(>Σ) that were already extracted above
significance Σ minus the expected false-detection rate N
f
(Σ)
as a self-consistent proxy for n, which is a reasonable assump-
tion when prior knowledge of the actual underlying counts is
not readily available (as in our case). As such, the individual
confidence levels of consecutive detections are estimated as
C(Σ) = 1 −
N
f
(Σ)
N(>Σ)
(1 − e
−N
f
(Σ)
). (3)
4.2. Deboosting
Deboosting is a statistical correction to the observed flux
densities, when source counts fall steeply with increasing
brightness (e.g., Crawford et al. 2010, and references therein).
Thus, in a statistical ensemble of sources, the same observed
flux arises more often from one of many fainter sources than
from the few brighter ones, relative to the measured value. We
assume a measurement with Gaussian noise (validated by the
closely Gaussian jackknife noise distribution) anda2mmsource
count model scaled from observationally constrained 850 μm
counts (e.g., Coppin et al. 2006;Weißetal.2009) assuming
T
d
/(1 + z) 10 K (Kov
´
acs et al. 2006) and dust emissivity
index (β)of1.5(Kov
´
acs et al. 2010). We also deboosted our
data using the physical number-count models of Lapi et al.
(2011) and B
´
ethermin et al. (2011); see Section 5.1.
For deboosting we followed a Bayesian recipe, such as
described in Coppin et al. (2005, 2006):
p(S
i
|S
o
,σ) ∝ p(S
i
)p(S
o
,σ|S
i
)(4)
expressing the probability of intrinsic source flux S
i
in terms of
the observed flux S
o
and its measurement uncertainty σ .
However, we made some important modifications to the
recipe to account for the possibility that the observed flux
arises from multiple overlapping galaxies, and we account
for confusion. Accordingly, we replace the single isolated
source assumption p(S
i
) ∝(dN/dS)(S
i
) of Coppin et al. (2005,
2006) with the compound probability that one or more (up to
m) resolved sources in the beam contribute to an aggregated
intrinsic flux S
i
:
p(S
i
) =
S
i
0
dS
1
...
S
m−1
0
dS
m
π(S
1
)...π(S
m
)δ
S
i
−
m
k=1
S
k
.
(5)
Inside the integrals is the product of the individual component
probability densities π (S
k
), which correspond to S
i
arising from
a specific combination of (S
1
...S
m
) individual components. The
delta function ensures that the component fluxes considered add
up to the total intrinsic flux S
i
when integrated. Each nested
integral for S(k) is performed up to the previous flux S
k−1
,
indicating that each successive component S
k
is no brighter
than the previous one, S
k−1
, and ensuring that each particular
5
