COSMOS: THREE-DIMENSIONAL WEAK LENSING AND THE GROWTH OF STRUCTURE
1
Richard Massey,
2
Jason Rhodes,
2, 3
Alexie Leauthaud,
4
Peter Capak,
2
Richard Ellis,
2
Anton Koekemoer,
5
Alexandre Re
´
fre
´
gier,
6
Nick Scoville,
2
James E. Taylor,
2, 7
Justin Albert,
2
Joel Berge
´
,
6
Catherine Heymans,
8
David Johnston,
3
Jean-Paul Kneib,
4
Yannick Mellier,
9,10
Bahram Mobasher,
5
Elisabetta Semboloni,
9,11
Patrick Shopbell,
2
Lidia Tasca,
4
and Ludovic Van Waerbeke
8
Received 2006 September 22; accepted 2007 January 24
ABSTRACT
We present a three-dimensional cosmic shear analysis of the Hubble Space Telescope COSMOS survey, the largest
ever optical imaging program performed in space. We have measured the shapes of galaxies for the telltale distortions
caused by weak gravitational lensing and traced the growth of that signal as a function of redshift. Using both 2D and
3D analyses, we measure cosmological parameters
m
, the density of matter in the universe, and
8
, the normali-
zation of the matter power spectrum. The introduction of redshift information tightens the constraints by a factor of
3 and also reduces the relative sampling (or ‘‘cosmic’’) variance compared to recent surveys that may be larger but
are only two-dimensional. From the 3D analysis, we find that
8
(
m
/0:3)
0:44
¼ 0:866
þ0:085
0:068
at 68% confidence limits,
including both statistical and potential systematic sources of error in the total budget. Indeed, the absolute calibration
of shear measurement methods is now the dominant source of uncertainty. Assuming instead a baseline cosmology to
fix the geometry of the universe, we have measured the growth of structure on both linear and nonlinear physical
scales. Our results thus demonstrate a proof of concept for tomographic analysis techniques that have been proposed
for future weak-lensing surveys by a dedicated wide-field telescope in space.
Subject heading gs: cosmology: observations — gravitational lensing — la rge-scale structure of universe
Online m aterial: color figures
1. INTRODUCTION
The observed shapes of distant galaxies become slightly
distorted as light from them passes through foreground mass
structures. Such ‘‘cosmic shear ’’ is induced by the (differential)
gravitational deflection of a light bundle, and happens regardless
of the nature and state of the foreground mass. It is therefore a
uniquely powerful probe of the dark matter distribution, directly
and simply linked to theories of structure formation that may be
ill-equipped to predict the distribution of light (for reviews, see
Bartelmann & Schneider 2001; Wittman 2002; Refregier 2003).
Furthermore, the main difficulties in this technique lie within the
optics of a telescope that has been built on Earth and can be thor-
oughly tested. It is not limited by systematic biases from un-
known physics such as astrophysical bias (Dekel & Lahav 1999;
Hoekstra et al. 2002; Smith et al. 2003a; Weinberg et al. 2004) or
the mass-temperature relation for X-ray-selected galaxy clusters
(Huterer & White 2002; Pierpaoli et al. 2001; Viana et al. 2002).
The study of cosmic shear has rapidly progressed since the
simultaneous detection of a coherent signal by four independent
groups (Bacon et al. 2000; Kaiser et al. 2000; Wittman et al.
2000; Van Waerbeke et al. 2000). Large, dedicated surveys with
ground-based telescopes have recently measured the projected
two-dimensional power spectrum of the large-scale mass distri-
bution and drawn competitive constraints on cosmological param-
eters (Brown et al. 2003; Bacon et al. 2003; Hamana et al. 2003;
Jarvis et al. 2003; Van Waerbeke et al. 2005; Massey et al. 2005;
Hoekstra et al. 2006). The addition of photometric redshift esti-
mation for large numbers of galaxies has led to the first measure-
ments of a changing lensing signal as a function of redshift ( Bacon
et al. 2004; Wittman 2005; Semboloni et al. 2006).
The shear measurement methods used for these ground-based
surveys have been precisely calibrated on simulated images con-
taining a known shear signal by the Shear Testing Program (STEP;
Heymans et al. 2006; Massey et al. 2007). This program has also
sped the development of a next generation of even more accu-
rate shear measurement methods (Bridle et al. 2002; Refregier
& Bacon 2003; Bernstein & Jarvis 2002; Massey & Refregier
2005; Mandelbaum et al. 2005; Kuijken 2006; Nakajima &
Bernstein 2007; Massey et al. 2006). With several ambitious plans
for dedicated telescopes both on the ground (e.g., the CTIO Dark
1
Based on observations with the NASA / ESA Hubble Space Telescope, ob-
tained at the Space Telescope Science Institute, which is operated by the Asso-
ciation of Universities for Research in Astronomy (AURA), Inc. under NASA
contract NAS5-26555; also based on data collected at the Subaru Telescope, which
is operated by the National Astronomical Observatory of Japan; the European
Southern Observatory, Chile; Kitt Peak National Observatory, Cerro Tololo Inter-
American Observatory, and the National Optical Astronomy Observatory, all of
which are operated by AURA under cooperative agreement with the National
Science Foundation; the National Radio Astronomy Observatory, which is a facil-
ity of the American National Science Foundation operated under cooperative agree-
ment by Associated Universities, Inc.; and the Canada-France-Hawaii Telescope
operated by the National Resear ch Council of Canada, the Cen tre National de la
Recherche Scientifique de France, and the University of Hawaii.
2
California Institute of Technology, 1200 East California Boulevard, Pasa-
dena, CA 91125; rjm@astro.caltech.edu.
3
Jet Propulsion Laboratory, Pasadena, CA 91109.
4
Laboratoire d’Astrophysique de Marseille, BP 8, Traverse du Siphon,
F-13376 Marseille Cedex 12, France.
5
Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD
21218.
6
Service d’Astrophysique, CEA /Saclay, F-91191 Gif-sur-Yvette, France.
7
Department of Physics and Astronomy, University of Waterloo, 200 Uni-
versity Avenue West, Waterloo, ON N2L 3G1, Canada.
8
Department of Physics and Astronomy, University of British Columbia,
6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada.
9
Institut d’Astrophysique de Paris, UMR7095 CNRS, Universi te
´
Pierre &
Marie Curie-Paris, 98 bis Boulevard Arago, F-75014 Paris, France.
10
Observatoire de Paris-LERMA, 61 avenue de l’Observatoire, F-75014
Paris, France.
11
Argelander-Institut f u
¨
r Astronomie, Auf dem Hu
¨
gel 71, D-53121 Bonn,
Germany.
A
239
The Astrophysical Journal Supplement Series, 172:239Y 253, 2007 September
# 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
Energy Survey [CTIO-DES], the Panoramic Survey Telescope
and Rapid Response System [Pan-STARRS], the VISTA/VLA
Survey Telescope Kilo-Degree Survey [VISTA/VST-KIDS], the
Large Synoptic Survey Telescope [LSST]) and in space (e.g., the
Dark Universe Explorer [DUNE], the Supernova/Acceleration
Probe [SNAP], and other possible Joint Dark Energy Mission
[JDEM] incarnations), the importance of weak lensing in future
cosmological and astrophysical contexts seems assured.
In this paper, we present statistical results from the first space-
based survey comparable to those from dedicated ground-based
observations. The Cosmic Evolution Survey (COSMOS; Scoville
et al. 2007a) combines the largest contiguous expanse of deep
imaging from space with extensive, multicolor follow-up from
the ground. High-resolution imaging is particularly needed for
weak lensing because the shapes of galaxies that would also be
detected from the ground are much less affected by the telescope’s
point-spread function (PSF), and a much higher density of new
galaxy shapes are resolved. This allows the signal to be measured
on smaller physical scales for the first time. Parameter constraints
from our survey still carry a fair deal of statistical uncertainty due
to cosmic variance in the finite survey size, but to a far lesser ex-
tent than previous space-based surveys (Rhodes et al. 2001, 2004;
Refregier et al. 2002; Heymans et al. 2005). More importantly, the
potential level of observational systematics is much lower from
space than from the ground, where the presence of the atmosphere
fundamentally limits all weak-lensing measurements.
Extensive ground-based follow-up in multiple filters has also
provided photometric redshift estimates for each galaxy. Lensing
requires a purely geometric measurement, so knowledge of the
distances in a lens system as well as the angles through w hich
light has been deflected are essential. We have extended cosmic
shear analysis into the information-rich three-dimensional shear
field. Our constraints on cosmological parameters are tightened
by observing independent galaxies at multiple redshifts, and the
separate volume in each redshift slice reduces the cosmic vari-
ance. Furthermore, we can directly trace the growth of large-scale
structure on both linear and nonlinear physical scales. Although
these results are still limited by the finite size of the COSMOS
survey, they provide a ‘‘proof of concept’’ for tomographic tech-
niques suggested (by e.g., Taylor 2002; Bernstein & Jain 2004;
Heavens 2006; Taylor et al. 2006) for future missions dedicated
to weak lensing. Throughout this paper, we have assumed a flat
universe, with Hubble parameter h ¼ 0:7.
This paper is organized as follows. In x 2, we describe the
data and analysis techniques. In x 3, we present a traditional 2D
‘‘cosmic shear’’ analysis of the two-point correlation functions,
demonstrating the level to which systematic effects have been
eliminated from the COSMOS data. In x 4, we extend the anal-
ysis into three dimensions via redshift tomography. We show
how the signal grows as a function of redshift, and directly trace
the growth of structure over cosmic time, on a range of physical
scales. In x 5, we use the measured statistics from both the 2D
and 3D analyses to derive constraints on cosmological parame-
ters. We conclude in x 6.
2. DATA ANALYSIS METHODS
2.1. Ima
ge Acquisition
The COSMOS field is a contiguous square, covering 1.64 deg
2
and centered at R:A: ¼ 10
h
00
m
28:6
s
,decl: ¼þ02
12
0
21:0
00
(J2000.0) (Scoville et al. 2007b; Koekemoer et al. 2007). Between
2003 October and 2005 June, the region was completely tiled
by 575 slightly overlapping pointings of the Advanced Camera
for Surveys (ACS) Wide Field Camera ( WFC) with the F814W
(approximately I-band) filter. Four slightly dithered, 507 s expo-
sures were taken at each pointing. Compact objects can be de-
tected on the stacked images in a 0.15
00
diameter aperture at 5
down to F814W
AB
¼ 26:6 (Scoville et al. 2007a).
The individual images were reduced using the standard STScI
ACS pipeline and combined using the program MultiDrizzle
(Fruchter & Hook 2002; Koekemoer et al. 2007). We took care
to optimize various MultiDrizzle parameters for precise galaxy-
shape measurement in the stacked images ( Rhodes et al. 2007).
We use a finer pixel scale of 0.03
00
for the stacked images. Pixel-
ization acts as a convolution, followed by a resampling and, al-
though current algorithms can successfully correct for convolution,
the formalism to properly treat resampling is still under devel-
opment for the next generation of methods.
We use a Gaussian drizzle kernel that is isotropic and with
pixfrac = 0.8, small enough to avoid smearing the object un-
necessarily while large enough to guarantee that the convolution
dominates the resampling. This process is then properly cor-
rected by existing shear measurement methods.
2.2. Shear Measurement
The detection of objects and measurement of their shapes is
fully described in Leauthaud et al. (2007). Modeling of the ACS
PSF is discussed in Rhodes et al. (2007). Here we provide only a
brief summary of the important results.
Objects were detected in the reduced ACS images using
SExtractor ( Bertin & Arnouts 1996). To avoid biasing our result,
the detection threshold was set intentionally low, far beneath the
final thresholds that we adopt. The catalog was finally separated
into stars and galaxies by noting their positions on the magnitude
versus peak surface brightness plane. Objects near bright stars or
any saturated pixels were masked using an automatic algorithm,
to avoid shape biases due t o any background gradient. The im-
ages were then all visually inspected, to mask other defects by
hand (including ghosting, reflected light, and asteroid /satellite
trails).
The size and the ellipticity of the ACS PSF varies over time,
due to the thermal ‘‘breathing’’ of the spacecraft. The long period
of time during which the COSMOS data were collected forces us
to consider this effect. Although other strategies have been dem-
onstrated successfully for observations conducted on a shorter
time span, it would be inappropriate for us to assume, like Lombardi
et al. (2005), that the PSF is constant or even, like Heymans et al.
(2005), that the focus is piecewise constant. Fortunately, most of
the PSF variations can be ascribed to a single physical parame-
ter: the distance between the primary and secondary mirrors, or
‘‘effective focus.’’ Variations of order 10 m create ellipticity
variations of up to 5% at the edges of the field, which is over-
whelming in terms of a weak-lensing signal. Jee et al. (2005) built
a PSF model for individual exposures by linearly interpolating
between two PSF patterns, observed above and below nominal
focus. We have used the TinyTim (Krist 2003) ray-tracing pack-
age to continuously model the PSF as a function of effective focus
and CCD position. By matching the dozen or so stars brighter than
F814W
AB
¼ 23 on each typical COSMOS image (Leauthaud
et al. 2007) to TinyTim models, we can robustly estimate the
offset from nominal focus with an rms error of less than 1 m
(Rhodes et al. 2007). We then return to the entire observational
data set, an d fi t a 3 ; 2 ; 2 order polynomial for each parameter
of the PSF model, as a function of x, y, and focus. Using the entire
COSMOS data set strengthens the fit, especially at the extremes
of focus values used, where few stars have been observed. The
final PSF model for each exposure is then extracted from the 3D
fit, at the appropriate focus value.
MASSEY ET AL.240 Vol. 172
We use the shear measurement method developed for space-
based imaging by Rhodes et al. (2000, hereafter RRG). It is a
‘‘passive’’ method that measures the Gaussian-weighted second
order moments I
ij
¼
P
wIx
i
x
j
/
P
wI of each galaxy and corrects
them using the Gaussian-weighted moments of the PSF model.
The RRG method is well suited to the small, diffraction-limited
PSF obtained from space, because it corrects each moment in-
dividually and only divides them to form an ellipticity at the final
stage.
In an advance from previous implementations of the Kaiser-
Squires-Broadhurst method, and spurred by the findings of STEP
(Massey et al. 2007), we allow the shear responsivity factor G
to vary as a function of magnitude. The shear responsivity is the
conversion factor between measured galaxy ellipticity e
i
and the
cosmologically interesting quantity shear
i
. As described in
Leauthaud et al. (2007), we have tested our pipeline on simulated
images created with the same Massey et al. (2004a) package
used for STEP, but tailored specifically to the image character-
istics of the COSMOS data. We found it necessary to multiply
our shears by a mean calibration factor of (0:86)
1
, but then found
the shear calibration hmi accurate to 0.3%, w ith a residua l shear
offset of hci¼0:2 4 ; 10
4
, with no significant variation as a
function of simulated galaxy size or flux. This is particularly
important in t he measurement of a shear signal as a function of
redshift. See Heymans et al. (2006) or Massey et al. (2007) for the
definitions of the multiplicative hmi and additive hci shear errors.
2.3. Char
ge Transfer Effects
As discussed further in Rhodes et al. (2007), the ACS WFC
CCDs also suffer from imperfect charge transfer efficiency (CTE)
during readout. This causes flux to be trailed behind objects,
spuriously elongating them in a coherent direction that mimics
a lensing signal. Furthermore, since this effect is produced by a
fixed number of charge traps in the silicon substrate, it affects
faint sources (with a larger fraction of their flux being affected)
more than bright ones. Thus, it is an insidious effect that also
mimics an increase in shear signal as a function of redshift. CTE
trailing is a nonlinear transformation of the image, and prevents
traditional tests of a weak-lensing analysis that look at bright
stars. As such, it is the most significant hurdle to overcome in
weak-lensing analysis from space.
We are developing a method to remove CTE trailing at the
pixel level. Following the work of Bristow & Alexov (2002) on
the Space Telescope Imaging Spectrograph (STIS), this method
will push charge back to where it belongs, as the very first stage
in data reduction. Because an ACS version of this algorithm is
still under development, in this paper we correct most of the CTE
effect via a parametric model acting at the catalog level. We as-
sume that the spurious change in an object’s apparent ellipticity "
is an additive amount that depends only on t he object’s flux, dis-
tance from the CCD readout register, and date of observation. In
fact, we also allowed variation with object size, although this had
little effect. As shown in Rhodes et al. (2007), this correction is
sufficient for the full catalog of more than 70 galaxies per arcmin
2
when considering mass reconstruction or circularly averaged sta-
tistics o n small scales, where the signa l is strong. However, it is
not adequate for the faintest galaxies when considering statistics
on large scales, as we would like to do in this paper. Fortunately,
the galaxy flux level at which the C TE correction successfully
removes the CTE signal ( leaving a residual signal 1 ord er o f
magnitude below the expected cosmological signal) appears to
coincide with that for which reliable photometric redshifts can
be obtained for almost all objects.
2.4. Photometric Redshifts
Reliable photometric redshift estimation is vital to the success
of our 3D shear measurement. For this r eason, the COSMOS
field has been observed from the ground in a comprehensive range
of wavelengths (Capak et al. 2007). Deep imaging is currently
available in the Subaru B
J
, V
J
, g
þ
, r
þ
, i
þ
, z
þ
, NB816, CFHT u
,
i
, CTIO/KPNO K
s
, and SDSS u
0
, g
0
, r
0
, i
0
, and z
0
bands. The
COSMOS photometric redshift code was used as described in
Mobasher et al. (2007). This code contains a luminosity function
prior in order to maximize the global accuracy of photometric
redshifts for the faintest and most distant population. It returns
both a best-fit redshift and a full redshift probability distribution
for each galaxy. The size of 68% confidence limits for each es-
timated redshift are well modeled by 0:03(1 þ z) out to z 1:4
anddowntomagnitudeI
F814W
¼ 24 (Mobasher et al. 2007;
Leauthaud et al. 2007).
Before a large spectroscopic redshift sample becomes avail-
able to calibrate the galaxy redshift distribution, our 3D analysis
will be limited by the reliability of photometric redshifts. We do
not impose a strict magnitude cut in the single I
F814W
band, but
instead using color information from many bands, and select
those galaxies with accurately measured redshifts. This includes
96% of detected galaxies brighter than I
F814W
¼ 24 and an in-
complete sample fainter than that (Leauthaud et al. 2007). The
selection function, and the final redshift distribution, thus depend
on the spectral energy distribution of individual galaxies. How-
ever, since the background galaxies are unrelated to the fore-
ground m ass that is lensing them, such incompleteness has no
detrimental effect on our analysis.
We specifically select galaxies that are observed in the mul-
ticolor ground-based data and that have a 68% confidence limit
in their redshift probability distribution function smaller than
z ¼ 0:5. The latter cut primarily removes galaxies with double
peaks in the photometric redshift PDF due to redshift degenera-
cies. W ithin the range of colors currently observed in the COSMOS
field, one particular degeneracy dominates: between 0:1 < z < 0:3
and 1:5 < z < 3:2, where the 4000 8 break can be confused with
coronal line absorption features. At z > 1:5, the 4000 8 break is
well into the IR, where sufficiently deep data are not yet available
for conclusive identification. To avoid catastrophic errors be-
tween these specific redshifts, we therefore also exclude galax-
ies with any finite probability below z ¼ 0:4 and above z ¼1:0.
After these cuts, we have redshift (and shear) measurements for
40 galaxies arcmin
2
.
3. 2D SHEAR ANALYSIS
3.1. 2D Source Redshift Distribution
The distribution of galaxies with reliably measured shears and
redshifts is shown in Figure 1. The effects of cosmic variance are
quite apparent, with all the spikes below z 1:2 corresponding
to known structures in the field. Beyond that, the photometric
redshifts are limited by the finite number of observed colors for
each galaxy, and the peaks at z ¼1 :3, 1.5, and 2.2 arise artifi-
cially at locations where spectral features move between filters.
The median photometric redshift is z
med
¼1:26. To minimize the
impact of g alaxy shape measur ement noise, we do wnweight
the contribution to the measured signal from faint and therefo re
noisier galaxies. We apply a weight
w ¼
1
(mag) þ 0:1
; ð1Þ
3D WEAK LENSING IN COSMOS 241No. 1, 2007
where the rms dispersion of observed galaxy ellipticities is well
modeled by
(mag) 0:32 þ 0:0014(mag 20)
3
: ð2Þ
The error distribution of the shear estimators is discussed in more
detail in Leauthaud et al. (2007). After this weighting, the me-
dian photometric redshift is z
med
¼ 1:11. In most cosmic shear
analyses to date, an estimate of this value is all that was known
about the redshift distribution. The smooth, dotted curve shows
the distribution that would have been obtained from a Smail et al.
(1994) fitting function
P(z) / z
exp 1:41z=z
med
ðÞ
hi
; ð3Þ
with ¼ 2, ¼ 1:5, z
med
¼ 1:26, and an overall normalization
to ensure the correct projected number density of galaxies. This
would have been a better fit to the high-redshift tail apparent
in Figure 1, had the free parame ter in the model, z
med
,been
1.17.
Figure 1 also shows the lensing sensitivity function
g() ¼ 2
Z
h
(
0
)
D
A
()D
A
(
0
)
D
A
(
0
)
a
1
() d
0
; ð4Þ
of the observed source redshift distribution, where is a distance
in comoving coordinates (in which the power spectrum is mea-
sured),
h
is the distance to the horizon, D
A
are angular diameter
distances, (with the extra factor of a
1
converting these into
comoving coordinates), and () is the distribution function of
source galaxies in redshift space, normalized so that
Z
h
0
() d ¼ 1: ð5Þ
This represents the sensitivity of a projected lensing analysis to
mass overdensities as a function of their redshift, and peaks at
z 0:4, about halfway to the peak of the source galaxy redshift
distribution in terms of angular diameter distance.
3.2. 2D Shear Correlation Functions
The 2D power spectrum of the projected shear field is given by
C
‘
¼
9
16
H
0
c
4
2
m
Z
h
0
g()
D
A
()
2
Pk;ðÞd; ð6Þ
where is a comoving distance,
h
is the horizon distance, g()
is the lensing weight function, and P(k;) is the underlying 3D
distribution of mass in the universe. The two-point shear corre-
lation functions can be expressed (Schneider et al. 2002) in terms
of the projected power spectrum as
C
1
() ¼
1
4
Z
1
0
C
‘
J
0
(‘) þ J
4
(‘)
‘ d‘; ð7Þ
C
2
() ¼
1
4
Z
1
0
C
‘
J
0
(‘) J
4
(‘)
‘ d‘: ð8Þ
These can be measured by averaging over galaxy pairs, as
C
1
(a) ¼
r
1
(r)
r
1
(r þ a)
; ð9Þ
C
2
(a) ¼
r
2
(r)
r
2
(r þ a)
; ð10Þ
where is the separation between the galaxies and the super-
script r denotes components of shear rotated so that
ˆ
g
r
1
(
ˆ
g
r
2
)in
each galaxy points along (at 45
from) the vector between the
pair. In practice, we compute this measurement in discrete bins of
varying angular scale. However, they will need to be integrated
later, so to keep this task manageable, we use fine bins of 0.1
00
throughout the calculations, and only rebin for the sake of clarity
in the final plots.
A third shear-shear correlation function can be formed,
C
3
(a) ¼
r
1
(r)
r
2
(r þ a)
þ
r
2
(r)
r
1
(r þ a)
; ð11Þ
for which parity invariance of the universe requires a zero signal.
The presence or absence of C
3
(a) can therefore be used as a first
test for the presence of systematic errors in our measurement, al-
though many systematics can still be imagined that would not show
up in this test.
The 2D shear correlation functions measured from the entire
COSMOS survey are shown in Figure 2. Note that the measure-
ments on scales smaller than 1
0
are new. For a given survey size,
these are obtained more easily from space than from the ground
because of the higher number density of resolved galaxies.
The additional, spurious signal that would have been obtained
without correction for CTE trailing is shown as roughly horizon-
tal solid lines in Figure 2. This was calculated by recomputing
the correlation functions, but rather than constructing a shear cat-
alog by subtracting the CTE contamination from each galaxy’s
raw shear measurement, the CTE contamination was used as a
direct replacement. An estimate of the residual CTE contamina-
tion for the galaxy population after correction, according to the
Fig. 1.— Thin solid line: Distribution of the best-fit redshifts returned by the
COSMO S photometric redshift code ( Mobasher et al. 2007) with a luminosity
function prior. Thick solid line: Distri bution after accounting for the different
weights given to galaxies. In both cases, the bin size is z ¼ 0:02. Peaks below
z 1:2 correspond to real structures in the field, but the artificial clustering at
higher redshift is due to limitations in the finite number of observed near-IR
colors. The dashed curve shows the redshift sensitivity function, assuming a
CDM universe with WMAP parameters. The dotted line shows the redshift
distribution that would have been expe cted, wi th knowledge of only the median
photometric redshift and a Sm ail et al. (1994) fitting function.
MASSEY ET AL.242 Vol. 172
performance evaluation in Rhodes et al. (2007) is shown as dot-
ted lines. Although this is now below the signal, the uncorrected
level was more than an order of magnitude larger than the signal
on large scales. Minimizing CTE by careful hardware design to
avoid the need for this level of correction will be a vital aspect of
dedicated space-based weak-lensing missions in the future.
3.3. Error Estimation and Verification
The error bars in Figure 2 include statistical errors due to both
intrinsic galaxy shape noise within the survey and the effect of
sample (‘‘cosmic’’) variance due to the finite survey size. The
shape noise dominates on small angular scales, and the cosmic
variance on scales larger than 10
0
. Surveys covering a simi-
lar area but in multiple lines of sight, such as ACS parallel data
(Schrabback et al. 2007; J. Rhodes et a l. 2007, in preparation),
will suffer less from the latter effect.
The statistical shape noise is easy to measure from the galaxy
population. To measure the sample variance, we split the COSMOS
field into four equally sized quadrants and recalculate the cor-
relation functions in each. Of course, large-scale correlations in
the mass distribution mean that the four adjacent quadrants are
not completely independent at large scales, and the measured
variance underestimates the true error. To correct for this effect,
we artificially increased the measured errors on 20
0
Y 40
0
scales
by 15%, in line with initial calculations.
After the fact, we compared our final error bars to independent
predictions from a full ray-tracing analysis through n-body sim-
ulations by Semboloni et al. (2007). Figure 3 shows the predicted
and observed 1 errors on C
þ
() C
1
() þ C
2
()(assuming
40 background galaxies per square arcminute in the simulations,
distributed in redshift with z
med
¼ 1:11 and with
"
¼ 0:32). Av-
eraging across all thirteen angular bins with equal weight, the
mean ratio between our measured error and the predicted non-
Gaussian error is 0.994. Future work may therefore improve the
error estimation, but in the COSMOS field at least, our quadrant
technique reaches a level of precision sufficient for this paper.
We also use the quadrant technique to measure the full co-
variance matrix between each angular bin. As shown in Figure 4,
the off-diagonal elements are nonzero. This is expected even in
an ideal case, because the same source population of galaxies is
used to construct pairs separated by different amounts. Nor are
the upper-left and lower-right quadrants of Figure 4 expected to
be zero: the same pairs go into the calculation of both C
1
() and
C
2
(), and after deconvolution from the PSF,
ˆ
r
1
and
ˆ
r
2
are no
longer formally independent. We will use the full, nondiagonal
covariance matrix during our measurement of cosmological pa-
rameters in x 5.
The final datum in the C
3
() panel of Figure 2 is significantly
(5 ) nonzero. This may be real; a finite region may not be
parity invariant on scales comparable to the field size. But even if
this does indicate a systematic problem, it is not as troubling as it
appears, because on this scale the error bars are large for C
1
()
and C
2
(), so the point carries very little weight. For a possible
explanation, note that the spurious C
3
signal has the same sign as
the uncorrected CTE signal. On scales that span almost the entire
COSMOS survey, one of the galaxies in a pair must lie near the
edge of the survey field that was observed last and that suffers
most from CTE degradation. If the temporal dependence of the
CTE signal is not linear, as we have assumed, the spiral observing
Fig. 2.—Correlation functions of the 2D shear field. The open circles indi-
cate negative values. The inner error bars show statistical erro rs only; the outer
error bars, visi ble only on large scales, also include the contribution of cosmic
variance. The six parallel curves show theoretical predictions for a flat CDM
cosmology with
m
¼ 0:3and
8
varying from 0.7 (bottom) to 1.2 (top). The
roughly horizontal lines indicate the level of the spurious signal due to CTE
trailing before and after correction.
Fig. 3.— Comparison of the error bars that we measured from the data, to ad-
vance predictions from Semboloni et al. (2007) obtained by ray-tracing through
n-body simulations of large-scale structure. The two solid lines show the pre-
dictions assuming a Gaussianized mass distribution (bottom) and with the f ull,
non-Gaussian distribution (top).
3D WEAK LENSING IN COSMOS 243No. 1, 2007
![Fig. 10.—E-B decomposition of the 3D cosmic shear signal, in different redshift bins, colored as in Fig. 8. For clarity, only the E-modes are shown. Open circles depict negative values. The B-modes are as noisy, but are consistent with zero. Note that adjacent data points are highly correlated. [See the electronic edition of the Supplement for a color version of this figure.]](/figures/fig-10-e-b-decomposition-of-the-3d-cosmic-shear-signal-in-2w159r2z.webp)
