The WiggleZ Dark Energy Survey: small‐scale clustering of Lyman‐break galaxies at z < 1
ABSTRACT The WiggleZ Dark Energy Survey is a large-scale structure survey of intermediate-redshift ultraviolet-selected (UV-selected) emission-line galaxies scheduled to cover 1000 deg2, spanning a broad redshift range 0.2 < z < 1.0. The main scientific goal of the survey is the measurement of baryon acoustic oscillations (BAO) in the galaxy clustering pattern at a significantly higher redshift than previous studies. The BAO may be applied as a standard cosmological ruler to constrain dark energy models. Based on the first 20 per cent of the data set, we present initial results concerning the small-scale clustering of the WiggleZ targets, together with survey forecasts. The WiggleZ galaxy population possesses a clustering length r0= 4.40 ± 0.12 h−1 Mpc, which is significantly larger than z= 0 UV-selected samples, with a slope γ= 1.92 ± 0.08. This clustering length is comparable to z= 3 Lyman-break galaxies with similar UV luminosities. The clustering strength of the sample increases with optical luminosity, UV luminosity and reddening rest-frame colour. The full survey, scheduled for completion in 2010, will map an effective volume Veff≈ 1 Gpc3 (evaluated at a scale k= 0.15 h Mpc−1) and will measure the angular diameter distance and Hubble expansion rates in three redshift bins with accuracies of ≈5 per cent. We will determine the value of a constant dark energy equation-of-state parameter, wcons, with a higher precision than existing supernovae observations using an entirely independent technique. The WiggleZ and supernova measurements lie in highly complementary directions in the plane of wcons and the matter density Ωm. The forecast using the full combination of WiggleZ, supernova and cosmic microwave background (CMB) data sets is a marginalized error Δwcons= 0.07, providing a robust and precise measurement of the properties of dark energy including cross-checking of systematic errors.
- Michael J. Drinkwater, Russell J. Jurek, Chris Blake, David Woods, Kevin A. Pimbblet, Karl Glazebrook, Rob Sharp, Michael B. Pracy, Sarah Brough, Matthew Colless, [......], Ben Jelliffe, Nick Jones, I-hui Li, Barry Madore, D. Christopher Martin, Gregory B. Poole, Todd Small, Emily Wisnioski, Ted Wyder, H. K. C. Yee[Show abstract] [Hide abstract]
ABSTRACT: The WiggleZ Dark Energy Survey is a survey of 240 000 emission-line galaxies in the distant Universe, measured with the AAOmega spectrograph on the 3.9-m Anglo-Australian Telescope (AAT). The primary aim of the survey is to precisely measure the scale of baryon acoustic oscillations (BAO) imprinted on the spatial distribution of these galaxies at look-back times of 4–8 Gyr.The target galaxies are selected using ultraviolet (UV) photometry from the Galaxy Evolution Explorer satellite, with a flux limit of NUV < 22.8 mag. We also require that the targets are detected at optical wavelengths, specifically in the range 20.0 < r < 22.5 mag. We use the Lyman break method applied to the UV colours, with additional optical colour limits, to select high-redshift galaxies. The galaxies generally have strong emission lines, permitting reliable redshift measurements in relatively short exposure times on the AAT. The median redshift of the galaxies is zmed= 0.6. The redshift range containing 90 per cent of the galaxies is 0.2 < z < 1.0.The survey will sample a volume of ∼1 Gpc3 over a projected area on the sky of 1000 deg2, with an average target density of 350 deg−2. Detailed forecasts indicate that the survey will measure the BAO scale to better than 2 per cent and the tangential and radial acoustic wave scales to approximately 3 and 5 per cent, respectively. Combining the WiggleZ constraints with existing cosmic microwave background measurements and the latest supernova data, the marginalized uncertainties in the cosmological model are expected to be σ(Ωm) = 0.02 and σ(w) = 0.07 (for a constant w model). The WiggleZ measurement of w will constitute a robust, precise and independent test of dark energy models.This paper provides a detailed description of the survey and its design, as well as the spectroscopic observations, data reduction and redshift measurement techniques employed. It also presents an analysis of the properties of the target galaxies, including emission-line diagnostics which show that they are mostly extreme starburst galaxies, and Hubble Space Telescope images, which show that they contain a high fraction of interacting or distorted systems. In conjunction with this paper, we make a public data release of data for the first 100 000 galaxies measured for the project.Monthly Notices of the Royal Astronomical Society 11/2009; 401(3):1429 - 1452. · 5.52 Impact Factor - [Show abstract] [Hide abstract]
ABSTRACT: We investigate the use of simple colour cuts applied to the Sloan Digital Sky Survey (SDSS) optical imaging to perform photometric selections of emission-line galaxies (ELGs) out to z < 1. Our selection is aimed at discerning three separate redshift ranges: 0.2 ≲z≲ 0.4, 0.4 ≲z≲ 0.6 and 0.6 ≲z≲ 1.0, which we calibrate using data taken by the COMBO-17 survey in a single field (S11). We thus perform colour cuts using the SDSS g, r and i bands and obtain mean photometric redshifts of and . We further calibrate our high-redshift selection using spectroscopic observations with the AAOmega spectrograph on the 4-m Anglo-Australian Telescope, observing ≈50–200 galaxy candidates in four separate fields. With just 1 h of integration time and seeing of ≈ 1.6 arcsec, we successfully determined redshifts for ≈65 per cent of the targeted candidates. We compare our spectroscopic redshifts to the photometric redshifts from the COMBO-17 survey and find reasonable agreement between the two. We calculate the angular correlation functions of these samples and find correlation lengths of r0= 2.78 ± 0.08, 3.71 ± 0.11 and 5.50 ± 0.13 h−1 Mpc for the low-, mid- and high-redshift samples, respectively. Comparing these results with predicted dark matter clustering, we estimate the bias parameter for each sample to be b= 0.72 ± 0.02, b= 0.93 ± 0.03 and b= 1.43 ± 0.03. We calculate the two-point redshift-space autocorrelation function at z≈ 0.6 and find a clustering amplitude of so= 6.4 ± 0.8 h−1 Mpc. Finally, we use our photometric sample to search for the integrated Sachs–Wolfe signal in the Wilkinson Microwave Anisotropy Probe (WMAP) 5-yr data. We cross-correlate our three redshift samples with the WMAP W, V, Q and K bands and find an overall trend for a positive signal similar to that expected from models. However, the signal in each is relatively weak, with the results in the WMAP W band being wTg(<100 arcmin) = 0.25 ± 0.27, 0.17 ± 0.20 and 0.17 ± 0.16 μK for the low-, mid- and high-redshift samples, respectively. Combining all three galaxy samples, we find a signal of wTg(<100 arcmin) = 0.20 ± 0.12 μK in the WMAP W band, a significance of 1.7σ. However, in testing for systematics where the WMAP data are rotated with respect to the ELG sample, we found similar results at several different rotation angles, implying the apparent signal may be produced by systematic effects.Monthly Notices of the Royal Astronomical Society 04/2010; 403(3):1261 - 1273. · 5.52 Impact Factor - SourceAvailable from: Rob SharpChris Blake, Sarah Brough, Matthew Colless, Carlos Contreras, Warrick Couch, Scott Croom, Tamara Davis, Michael J. Drinkwater, Karl Forster, David Gilbank, [......], Barry Madore, Chris Martin, Kevin Pimbblet, Gregory B. Poole, Michael Pracy, Rob Sharp, Emily Wisnioski, David Woods, Ted K. Wyder, Howard Yee[Show abstract] [Hide abstract]
ABSTRACT: We present precise measurements of the growth rate of cosmic structure for the redshift range 0.1 < z < 0.9, using redshift-space distortions in the galaxy power spectrum of the WiggleZ Dark Energy Survey. Our results, which have a precision of around 10% in four independent redshift bins, are well-fit by a flat LCDM cosmological model with matter density parameter Omega_m = 0.27. Our analysis hence indicates that this model provides a self-consistent description of the growth of cosmic structure through large-scale perturbations and the homogeneous cosmic expansion mapped by supernovae and baryon acoustic oscillations. We achieve robust results by systematically comparing our data with several different models of the quasi-linear growth of structure including empirical models, fitting formulae calibrated to N-body simulations, and perturbation theory techniques. We extract the first measurements of the power spectrum of the velocity divergence field, P_vv(k), as a function of redshift (under the assumption that P_gv(k) = -sqrt[P_gg(k) P_vv(k)] where g is the galaxy overdensity field), and demonstrate that the WiggleZ galaxy-mass cross-correlation is consistent with a deterministic (rather than stochastic) scale-independent bias model for WiggleZ galaxies for scales k < 0.3 h/Mpc. Measurements of the cosmic growth rate from the WiggleZ Survey and other current and future observations offer a powerful test of the physical nature of dark energy that is complementary to distance-redshift measures such as supernovae and baryon acoustic oscillations.Monthly Notices of the Royal Astronomical Society 04/2011; 415(3). · 5.52 Impact Factor
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arXiv:0901.2587v2 [astro-ph.CO] 25 Feb 2009
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 25 February 2009(MN LATEX style file v2.2)
The WiggleZ Dark Energy Survey: small-scale clustering of
Lyman Break Galaxies at z < 1
Chris Blake1⋆, Russell J. Jurek2, Sarah Brough1, Matthew Colless3, Warrick
Couch1, Scott Croom4, Tamara Davis2,5, Michael J. Drinkwater2, Duncan Forbes1,
Karl Glazebrook1, Barry Madore6, Chris Martin7, Kevin Pimbblet2, Gregory B.
Poole1, Michael Pracy1,8, Rob Sharp3, Todd Small7and David Woods9,10
1Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia
2Department of Physics, University of Queensland, Brisbane, QLD 4072, Australia
3Anglo-Australian Observatory, P.O. Box 296, Epping, NSW 2121, Australia
4School of Physics, University of Sydney, NSW 2006, Australia
5Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark
6Observatories of the Carnegie Institute of Washington, 813 Santa Barbara St., Pasadena, CA 91101, United States
7California Institute of Technology, MC 405-47, 1200 East California Boulevard, Pasadena, CA 91125, United States
8Research School of Astronomy and Astrophysics, Australian National University, Weston Creek, ACT 2600, Australia
9School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
10Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C., V6T 1Z1, Canada
25 February 2009
ABSTRACT
The WiggleZ Dark Energy Survey is a large-scale structure survey of intermediate-
redshift UV-selected emission-line galaxies scheduled to cover 1000 deg2, spanning a
broad redshift range 0.2 < z < 1.0. The main scientific goal of the survey is the
measurement of baryon acoustic oscillations (BAO) in the galaxy clustering pattern
at a significantly higher redshift than previous studies. The BAO may be applied as
a standard cosmological ruler to constrain dark energy models. Based on the first
20% of the dataset, we present initial results concerning the small-scale clustering of
the WiggleZ targets, together with survey forecasts. The WiggleZ galaxy population
possesses a clustering length r0= 4.40 ± 0.12h−1Mpc, which is significantly larger
than z = 0 UV-selected samples, with a slope γ = 1.92 ± 0.08. This clustering length
is comparable to z = 3 Lyman Break Galaxies with similar UV luminosities. The
clustering strength of the sample increases with optical luminosity, UV luminosity and
reddening rest-frame colour. The full survey, scheduled for completion in 2010, will
map an effective volume Veff≈ 1 Gpc3(evaluated at a scale k = 0.15h Mpc−1) and will
measure the angular-diameter distance and Hubble expansion rates in three redshift
bins with accuracies ≈ 5%. We will determine the value of a constant dark energy
equation-of-state parameter, wcons, with a higher precision than existing supernovae
observations using an entirely independent technique. The WiggleZ and supernovae
measurements lie in highly complementary directions in the plane of wcons and the
matter density Ωm. The forecast using the full combination of WiggleZ, supernovae and
CMB datasets is a marginalized error ∆wcons= 0.07, providing a robust and precise
measurement of the properties of dark energy including cross-checking of systematic
errors.
Key words:
galaxies: starburst
surveys, cosmology: observations, large-scale structure of Universe,
1 INTRODUCTION
⋆
E-mail: cblake@astro.swin.edu.au
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Blake et al.
The large-scale structure of the Universe is one of the pil-
lars of our modern understanding of cosmology, encoding
information about the contents and evolution of the Uni-
verse, the physics of the growth of density fluctuations with
time, and the formation and evolution of galaxies within
the underlying network of dark matter haloes. In particular,
the large-scale clustering pattern of galaxies is sensitive to
the properties of the cosmic dark energy component which
is currently poorly understood. Dark energy influences both
the rate of growth of structure and the geometrical distance-
redshift relations. One of the cleanest probes of dark energy
is to delineate as a function of redshift the apparent tangen-
tial and radial size of the baryon acoustic oscillation scale, a
known “standard ruler” preferred separation imprinted into
the galaxy distribution (Cooray et al. 2001; Eisenstein 2002;
Blake & Glazebrook 2003; Seo & Eisenstein 2003; Hu &
Haiman 2003; Linder 2003; Glazebrook & Blake 2005). This
cosmological probe is helping to motivate a new generation
of massive spectroscopic galaxy surveys.
Cosmic structure has been mapped out by a succes-
sion of galaxy redshift surveys of increasing size and depth.
The local Universe (redshifts z < 0.2) has been studied
in exquisite detail by the 2-degree Galaxy Redshift Sur-
vey (2dFGRS; Colless et al. 2001) and the Sloan Digital
Sky Survey (SDSS; York et al. 2000). The SDSS Luminous
Red Galaxy component extended this programme to a mean
redshift z ≈ 0.35 using a specific type of tracer galaxy
(Eisenstein et al. 2001). Indeed, the cosmological conclu-
sions reached should be independent of the galaxy type used,
given that the “bias” with which galaxies trace the underly-
ing dark matter fluctuations is expected to be a simple linear
function on large scales (Coles 1993; Scherrer & Weinberg
1998). In this sense, the choice of the “tracer population”
of galaxies can be determined by observational considera-
tions such as telescope exposure times, the availability of
input imaging data for target selection, and secondary sci-
ence goals.
The WiggleZ Dark Energy Survey, using the AAOmega
multi-object spectrograph at the 3.9m Anglo-Australian
Telescope (AAT), is designed as the next leap forwards in
redshift coverage, targetting the range 0.2 < z < 1.0. The
survey is scheduled to cover a sky area of 1000 deg2, mapping
a cosmic volume V ∼ 1 Gpc3sufficient to measure the im-
print of baryon oscillations in the clustering pattern at a sig-
nificantly higher redshift than has been previously achieved
by 2dFGRS (Cole et al. 2005; Percival et al. 2007) and SDSS
(Eisenstein et al, 2005; Huetsi 2006; Percival et al. 2007;
Gaztanaga et al. 2008). The survey redshift range is moti-
vated by the optimal redshift location for testing a cosmolog-
ical constant model in a spatially-flat Universe (Parkinson
et al. 2007), which is the sensible initial hypothesis to re-
ject in the dark energy parameter space. The target galaxy
population is bright emission-line galaxies selected from UV
imaging by the Galaxy Evolution Explorer (GALEX) satel-
lite (Martin et al. 2005). This choice is motivated by the
short (1-hr) exposure times required to obtain redshifts at
the AAT. The survey commenced in August 2006 and is
scheduled to finish in July 2010, using the equivalent of 165
clear nights of telescope time (220 awarded nights). Sec-
ondary science goals involve the study of star formation and
galaxy evolution as a function of redshift and environment.
In this initial study we focus on the small-scale cluster-
ing properties of the first 20% of the WiggleZ sample. The
clustering strength is an important parameter in the survey
design and cosmological parameter forecasts: the signal-to-
noise with which we can recover the galaxy power spectrum
depends on the bias of the galaxies with respect to the dark
matter fluctuations, which affects the balance between sam-
ple variance and shot noise in the power spectrum error
budget. These initial clustering measurements allow us to
determine the bias parameter and complete the survey fore-
cast.
Furthermore, the joint UV-optical selection in the red-
shift interval 0.2 < z < 1 places the WiggleZ survey in an
interesting location in the parameter space of galaxy evo-
lution. In this context, the clustering strength of a set of
galaxies provides a direct indication of the density of the
environment or (equivalently) the typical mass of the dark
matter haloes hosting the galaxies. The clustering strength
of UV-selected samples has been studied at low redshift
z ≈ 0 (Milliard et al. 2007; Heinis et al. 2007) and the cor-
responding rest-frame samples have been selected at much
higher redshift z ≈ 3 through studies of the clustering of
Lyman Break Galaxies (LBGs; e.g. Giavalisco & Dickinson
2001; Ouchi et al. 2001; Arnouts et al. 2002; Foucaud et
al. 2003; Adelberger et al. 2005; Allen et al. 2005; Ouchi
et al. 2005; Lee et al. 2006; Yoshida et al. 2008). The Wig-
gleZ survey samples a redshift range which is intermediate
to these previous studies. Moreover, the clustering strength
of optically-selected star-forming galaxies at high redshift
has been studied over small areas by the Deep Extragalac-
tic Evolutionary Probe (DEEP2) project (Coil et al. 2008)
and the VIMOS VLT Deep Survey (VVDS; Meneux et al.
2006). WiggleZ is mapping an area ∼ 100 times larger, and is
therefore able to measure accurately the clustering strength
of the most luminous star-forming galaxies, for which these
smaller surveys are limited by small-number statistics and
sample variance.
The backdrop to these studies is the recent concept of
“down-sizing” (Cowie et al. 1996; Glazebrook et al. 2004;
van Dokkum et al. 2004) whereby the stars in more massive
galaxies appear to have formed earlier, and the typical mass
of the most actively star-forming galaxies is expected to de-
crease with time. A recent study of LBGs (Yoshida et al.
2008) has emphasized the importance of studying the clus-
tering segregration with both UV and optical luminosities,
which crudely trace ongoing star formation rate and stellar
mass, respectively. WiggleZ is well-suited for undertaking
such studies over the redshift range 0.2 < z < 1.0.
The plan of this paper is as follows: in Section 2 we
introduce the WiggleZ survey strategy and target selection
and describe the data sample used in this study. In Section 3
we describe the methodology used to produce the small-scale
clustering measurement including the generation of random
(unclustered) realizations of the dataset with the correct se-
lection functions and redshift completeness map. We also
explain how we derive the statistical error in the clustering
measurement. We present the clustering results in Section 4
(split by redshift, absolute magnitude and rest-frame colour)
together with initial comparisons to other studies. Section
5 contains the cosmological parameter forecasts for the full
WiggleZ survey, and Section 6 summarizes our conclusions.
When converting redshifts to co-moving co-ordinates we as-
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WiggleZ survey: small-scale clustering
3
Figure 1. The sky distribution of the seven WiggleZ survey regions compared to the coverage of the SDSS, RCS2 and GALEX Medium
Imaging Survey at the end of 2008.
sume a spatially-flat Universe with cosmological parameters
Ωm = 0.3 and ΩΛ = 0.7.
2 DATA
The design and implementation of the WiggleZ Dark En-
ergy Survey will be fully described in a forthcoming “survey
paper” (Drinkwater et al. 2009, in preparation) which will
accompany our mid-term Data Release. We include a brief
outline here both for ease of reference and to emphasize the
key points relevant to the small-scale clustering analysis.
2.1 WiggleZ survey strategy
The WiggleZ survey strategy is to harvest low signal-to-
noise spectra of a large number of UV-selected emission-line
galaxies in relatively short exposure times (1-hr integrations
at the AAT). The survey tolerates a relatively low redshift
completeness of 70% but generates a large statistical sample
of galaxy redshifts. The survey goal is to cover 1,000 deg2
of the equatorial sky, gathering ∼ 350,000 spectra of which
∼ 245,000 are expected to yield successful redshifts. The
survey was designed such that the average galaxy number
density n is related to the amplitude of the galaxy clustering
power spectrum Pgalon the relevant baryon oscillation scales
by n ∼ 1/Pgal, implying that the contributions of sample
variance and shot noise to the clustering error are equal. This
is the optimal survey strategy for fixed number of galaxies.
The WiggleZ survey area, illustrated in Figure 1, is split
into seven equatorial regions to facilitate year-round observ-
ing. We require that each region should possess a minimum
angular dimension of ∼ 10 deg, corresponding to a spatial
co-moving scale that exceeds by at least a factor of two the
standard ruler preferred scale [which projects to (8.5, 4.6,
3.2, 2.6) deg at z = (0.25,0.5,0.75,1.0)]. The survey cover-
age within individual regions should also be highly (> 70%)
contiguous, otherwise the significance of the detection of the
acoustic features is degraded by convolution with the survey
window function. The survey duration is forecast to be ∼ 165
clear nights between August 2006 and July 2010, using the
multi-object capability of the 2dF positioner system coupled
to the AAOmega spectrographs (Saunders et al. 2004; Sharp
et al. 2006).
Galaxy redshifts are obtained from the bright emission
lines associated with star-forming galaxies, in particular red-
shifted [OII] 3727˚ A, Hβ 4861˚ A and [OIII] 4959˚ A, 5007˚ A.
Low-resolution (5˚ A FWHM) spectra are obtained spanning
the (observed-frame) wavelength range 5500−9500˚ A, hence
the majority of successful redshifts in the range z < 0.95 are
confirmed by multiple emission lines. Single-line redshifts
are almost invariably [OII], for which we usually resolve the
doublet in the range z > 0.8, increasing our confidence in
the line identification. Redshifts are obtained by visual in-
spection of each spectrum using the interactive software tool
“runz”, and are classified by a quality flag 1 ≤ Q ≤ 5, where
the range Q ≥ 3 denotes a “reliable” redshift (see Colless et
al. 2001). The fraction of stellar contamination is very small
(< 1%) and we find a similarly low fraction of high-redshift
quasar interlopers. The galaxy continuum is typically de-
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Blake et al.
tected with low signal-to-noise (an average of S/N ∼ 1 per
resolution element).
2.2 WiggleZ survey target selection
WiggleZ targets are chosen by a joint UV-optical selection.
The primary selection dataset is the Medium Imaging Sur-
vey undertaken by the Galaxy Evolution Explorer (GALEX)
UV satellite, which provides typical exposure times of 1500
seconds in two filter bands, FUV (1350−1750˚ A) and NUV
(1750 − 2750˚ A). The GALEX point-spread function is too
broad to allow for accurate placement of the spectrograph
optical fibres, therefore the UV imaging is cross-matched
with optical data. For our NGP regions, the SDSS imaging
data are used. For our SGP regions, the SDSS 2.5◦stripes
are too narrow compared to the preferred baryon oscilla-
tion scale hence we use imaging data from the second Red
Cluster Sequence (RCS2) project instead (Yee et al. 2007).
Sources are cross-matched between the GALEX and optical
catalogues with a matching tolerance of 2.5 arcsec (which
produces a negligible fraction of incorrect matches). In each
imaging dataset, the majority of galaxies possess relatively
low signal-to-noise (S/N = 3−5) but their detection in both
datasets ensures a robust sample. We note that acquisition
of our eventual requirement of ∼ 1250 GALEX orbits of data
is still ongoing. About 70% of this total had been obtained
at the end of 2008.
Targets are chosen from the UV-optical matched sample
using a series of magnitude and colour cuts. These cuts are
tuned to optimize the fraction of targets lying at high red-
shift z > 0.5. Firstly the galaxy magnitudes are de-reddened
using standard dust corrections based on the local value of
E(B − V ) measured from the Schlegel, Finkbeiner & Davis
(1998) dust maps. The primary GALEX selection criterion
is a red FUV − NUV colour (FUV − NUV > 1 or FUV
drop-out), motivated by the Lyman Break passing through
the FUV filter for z > 0.5, and tuned by looking at galaxy
templates. At the depth of the Medium Imaging Survey this
colour is noisy, resulting in a significant amount of contami-
nation by low-redshift (z < 0.5) galaxies which are partially
removed by the additional cuts described below. We also im-
pose a faint UV magnitude limit NUV < 22.8 and an addi-
tional signal-to-noise requirement S/N > 3 for the detected
NUV flux (which becomes relevant for fields with unusu-
ally high dust content or low exposure time). The GALEX
field-of-view is circular with radius ∼ 0.6 deg; we only select
sources within the central 0.55 deg because of concerns over
the photometry at the edge of the field.
Our primary optical selection cuts are derived from
SDSS r-band imaging. We require a UV-optical colour in the
range −0.5 < NUV − r < 2 based on the expected model
tracks of star-forming galaxies. We impose a bright r-band
limit 20 < r < 22.5; the UV-optical colour cut implies that
the median optical magnitude of our targets is r ∼ 21.5.
Finally we increase the high-redshift efficiency by introduc-
ing optical colour cuts. Different cuts are used for the SDSS
and RCS2 regions, governed by the available imaging bands
and depths. For the SDSS regions analyzed in this paper,
we apply cuts for those (brighter) galaxies with good detec-
tions in the SDSS g- and i-bands. Specifically, for targets
with g < 22.5 and i < 21.5 we reject galaxies in the colour
space defined by r − i < g − r − 0.1 and r − i < 0.4 which
is occupied by low-redshift galaxies both theoretically and
empirically (more details will be given in Drinkwater et al.
2009, in preparation). The final fraction of z > 0.5 galaxies
obtained is ≈ 70%. The redshift distribution is displayed in
Figure 2.
An average of 3-4 pointings of the 2dF spectrograph per
patch of sky is required in order to achieve the required tar-
get density of 350 deg−2. For any observing run the optimal
placement of field centres (based on the current availabil-
ity of targets) is achieved using the “Metropolis” (simulated
annealing) algorithm (Campbell, Saunders & Colless 2004).
Galaxies are prioritized for spectroscopic follow-up on the
basis of optical r-band magnitude, in the sense that fainter
targets are observed first. The motivation for this strategy is
to combat the potential inefficiency of “mopping up” resid-
ual galaxies in the final pointing for any patch of sky: the
brighter remaining galaxies can be observed in a shorter ex-
posure time by configuring fewer fibres.
2.3WiggleZ July 2008 data sample
In this paper we analyze the subset of the WiggleZ sample
assembled from our first observations in August 2006 up un-
til the end of the 08A semester (July 2008). At this point we
had utilized 108 of our allocated nights, of which the equiv-
alent of 70 nights were clear. The available galaxy database
included ≈ 97,000 reliable (Q ≥ 3) WiggleZ unique galaxy
redshifts.
In this analysis we only use those galaxies lying in the
SDSS regions of our optical imaging because work is still
on-going on the RCS2 portion of the angular selection func-
tion. Specifically, we include the WiggleZ 9-hr, 11-hr, 15-hr
and 0-hr (SDSS) regions illustrated in Figure 1. The num-
ber of existing AAOmega pointings in these regions is (42,
98, 140, 48). The numbers of galaxies in each region with
reliable redshifts satisfying the final survey selection criteria
are (5782, 14873, 21629, 4383), constituting a total sample
of N = 46,667 for this initial analysis (about 20% of the
final sample).
3ANALYSIS
3.1 Correlation function estimator
We quantify the small-scale clustering of the galaxy distri-
bution using a standard set of techniques based on the 2-
point correlation function. This statistic compares the num-
ber of observed close galaxy pairs to that expected by ran-
dom chance, as a function of spatial separation. The key
requirement is an ensemble of random (unclustered) realiza-
tions of the survey possessing the same selection function
as the observed galaxy distribution. With this in place we
convert the data (D) and random (R) galaxy angle-redshift
distributions into a grid of co-moving co-ordinates (x,y,z)
using an assumed cosmological model (we use a flat model
with Ωm = 0.3). We then bin the number of data-data (DD),
data-random (DR) and random-random (RR) pairs in the
two-dimensional space of separation perpendicular to the
line-of-sight (denoted by σ) and parallel to the line-of-sight
(denoted by π). This decomposition is motivated by the in-
fluence of galaxy peculiar velocities (redshift-space distor-
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WiggleZ survey: small-scale clustering
5
tions) which shift galaxies in π, but not in σ. Each of our
random realizations contains the same number of targets as
the data sample, and is generated by a method described
below. The pair counts DR and RR are determined by av-
eraging over 10 random realizations.
The 2D redshift-space correlation function ξz(σ,π) is
derived using the estimator proposed by Landy & Szalay
(1993):
ξz(σ,π) =DD(σ,π) − 2DR(σ,π) + RR(σ,π)
RR(σ,π)
(1)
(where this last equation assumes an equal number of data
and random galaxies). We bin galaxy pairs by the absolute
value of the line-of-sight separation, i.e. π ≡ |π|. The “real-
space” correlation function (independent of the redshift-
space distortion) can be obtained by summing equation 1
over π. We first define the projected correlation function
Ξ(σ) (Davis & Peebles 1983):
Ξ(σ) = 2
∞
?
π=0
ξz(σ,π)∆π (2)
where the factor of 2 extrapolates the result to the full range
−∞ < π < ∞. If we assume that the real-space correlation
function ξr is well-described by a power-law ξr(r) = (r0/r)γ,
where r0 is the clustering length, γ is the slope and r =
√σ2+ π2, and if we neglect the coherent infall velocities
described below, we can then derive
ξr(r) =Ξ(r)
rCγ
(3)
where
Cγ =
?∞
−∞
(1 + u2)−γ/2du =Γ(1
2)Γ(γ−1
Γ(γ
2)
2)
(4)
The difficulty with this method is that the measurement of
ξz(σ,π) becomes noisy for large π and therefore the sum-
mation in equation 2 must be truncated at some π = πmax,
invalidating equation 3. We therefore adopted the follow-
ing approach (similar to the methodology of Coil et al.
2008) to convert a model real-space correlation function
ξr(r) = (r0/r)γinto a projected correlation function which
may be compared with the data. In the linear regime, the
effect of coherent infall velocities on the correlation function
can be modelled by
ξz(σ,π) = ξ0(r)P0(µ) + ξ2(r)P2(µ) + ξ4(r)P4(µ)(5)
where Pℓ(µ) are the Legendre polynomials, µ = cosθ and
θ is the angle between r and π. For a power-law real-space
correlation function,
ξ0(r)=
?
?
8β2
35
1 +2β
3
+β2
5
?
ξr(r)(6)
ξ2(r)=
4β
3
+4β2
7
??
γ
γ − 3
?
ξr(r)(7)
ξ4(r)=
?
γ(2 + γ)
(3 − γ)(5 − γ)
?
ξr(r)(8)
where β ≈ Ωm(z)0.55/b is the redshift-space distortion pa-
rameter (Hamilton 1992; Hawkins et al. 2003) and b is the
linear galaxy bias parameter. We assumed β = 0.6 for this
model, consistent with our measurements (see Section 4.3),
Figure 2. The redshift probability distribution of WiggleZ tar-
gets with reliable redshifts in the four survey regions analyzed in
this study (normalized such that?
the combined regions as the thicker line.
P(z)dz = 1). We plot cubic
spline fits to the redshift distribution. We also show the result for
and for each set of trial values (r0,γ) we employed the above
set of equations to calculate ξz(σ,π). For each value of σ
we then integrated this function in the π direction up to
π = πmax in order to compare with the correlation function
measurements. We assumed πmax = 20h−1Mpc, and we
checked that our results did not depend sensitively on the
value of πmax.
We treated each of the four survey regions indepen-
dently, measuring the correlation function and correspond-
ing error. We then constructed the “combined” correla-
tion function by averaging the measurements in the four
regions with inverse-variance weighting. For convenience,
we plot projected correlation functions in this paper as
Ξ(σ)/(σ Cγ,reduced) ∝ (r0/σ)γ, where
?σ/πmax
Cγ,reduced=
−σ/πmax
(1 + u2)−1/2du (9)
3.2Selection function
We now discuss the generation of the random survey real-
izations that are required for calculation of the correlation
function. This determination of the survey “selection func-
tion” will be described fully in a forthcoming paper (Blake
et al. 2009, in preparation) and we give a brief summary
here.
The calculation begins with the angular selection func-
tion of the “parent” sample of UV-optical matches. This
function is defined firstly by the boundaries of the GALEX
fields and SDSS coverage map. Secondly, because the UV
magnitudes of our sample lie close to the threshold of the
GALEX MIS observations, there is a significant incomplete-
ness in the GALEX imaging that depends on the local dust
extinction and GALEX exposure time. We used the GALEX
number counts as a function of dust and exposure time to
calibrate the relation between these quantities and the par-
ent WiggleZ target density. This angular completeness func-
tion is displayed in Figure 3 for the four survey regions ana-
lyzed in this paper. We used this map to produce a series of
random realizations of the parent catalogue for each region.
Page 6
6
Blake et al.
Figure 3. Greyscale map illustrating the angular completeness of the parent catalogue of SDSS-GALEX matches for the four survey
regions analyzed in this paper. This parent target density varies with dust extinction and GALEX exposure time because the UV
magnitudes of WiggleZ galaxies lie close to the threshold of the Medium Imaging Survey data. The x-axis and y-axis of each panel are
right ascension and declination, respectively.
The next step is to process these random parent cat-
alogues into random realizations of the redshift catalogue.
The spectroscopic follow-up of the parent catalogue com-
prises a network of overlapping AAOmega pointings, with
field centres optimized by the simulated annealing algorithm
and not lying on a regular grid. The fraction of successful
redshifts in each pointing varies considerably depending on
weather conditions. Furthermore, the redshift completeness
within each AAOmega field exhibits a significant radial vari-
ation due to acquisition errors at the plate edges.
In Figure 4 we illustrate how the redshift completeness
varies across these survey regions by simply taking the ratio
of successful redshifts to parent galaxies in each pixel. This is
a useful visualization, but in fact the number of unique sec-
tors defined by the overlapping AAOmega fields is so large
that this determination of the redshift completeness map is
very noisy. Indeed, some unique sectors contain zero parent
galaxies.
One possible approach is to smooth this completeness
map over larger areas to reduce the Poisson noise at the ex-
pense of a diminished sensitivity to small-scale completeness
variations between AAOmega pointings. In this analysis we
use an alternative approach, which is to apply the AAOmega
pointing sequence to each of the random realizations of the
parent catalogue, and thereby create an ensemble of ran-
dom realizations of the redshift catalogue. The AAOmega
pointing sequence is defined by the right ascension and dec-
lination of the field centre together with the number of suc-
cessful and unsuccessful redshifts obtained for that pointing.
Within each field centre parent galaxies are chosen randomly
to create the synthetic redshift catalogue. It is also necessary
to track the sky coverage of the GALEX data which was con-
temporaneous with each AAOmega pointing. Because the
acquisition of the GALEX imaging data is ongoing with the
spectroscopic follow-up, the boundaries of the angular mask
must be modulated in step with the redshift follow-up. In
addition we impose the radial redshift completeness varia-
tion across each AAOmega field, measured independently
for each observing run.
The redshift distribution N(z) of observed galaxies
varies with position in the sky. This is due to the magni-
tude prioritization described in Section 2.2. Because galax-
ies with fainter r-band magnitudes are targeted first, the
N(z) will be skewed toward higher redshifts for areas of the
survey which have been targeted by fewer AAOmega ob-
servations. This dependence is accounted for in our random
catalogues by measuring the magnitude distribution of tar-
geted galaxies as a function of sky position and drawing a
random redshift from the correctly weighted N(z).
3.3Fibre collision correction
The optical fibres of the 2dF spectrograph cannot be placed
closer together than 0.5 arcmin, and there is a diminishing
probability of observing in a single pointing both members
of a close pair of parent galaxies separated by an angular dis-
tance of less than 2 arcmin [a projected spatial distance of
(0.4,0.8,1.1,1.4)h−1Mpc at z = (0.25,0.5,0.75,1.0)]. This
Page 7
WiggleZ survey: small-scale clustering
7
Figure 4. Greyscale map illustrating the completeness of the spectroscopic follow-up of the WiggleZ targets shown in Figure 3 for the
four survey regions analyzed in this paper. This Figure is generated by taking the ratio of the galaxy densities in the redshift and parent
catalogues in small cells. In our clustering analysis a more accurate approach is adopted in which the full AAOmega pointing sequence
is applied to random realizations of the parent catalogue. The x-axis and y-axis of each panel are right ascension and declination,
respectively.
restriction will eventually be ameliorated by the requirement
of observing each patch of sky with 3-4 AAOmega pointings
to build up the number density of the redshift catalogue.
At present, however, there is a deficit of close angular pairs
in the redshift catalogue, which artificially suppresses the
measured value of the galaxy correlation function on small
scales. The close angular pair deficit is illustrated in Figure
5 by plotting the ratio (1 + wt)/(1 + wp) as a function of
angular separation θ, where wt and wp are the angular cor-
relation functions of the targeted catalogue and the parent
catalogue, respectively. This ratio provides the fraction of
surviving close pairs. In order to correct the galaxy correla-
tion function for the missing close pairs we increased the con-
tribution of each galaxy pair to the data-data pair count as
a function of angular separation by a factor (1+wp)/(1+wt)
(the inverse of the quantity plotted in Figure 5) using a 2-
parameter model {1 + erf[(log10θ − µ)/σ]}/2 fitted to the
data in Figure 5.
We note that for a survey with a redshift-dependent
galaxy number density n(z), the minimum-variance corre-
lation function measurement for separation s is achieved if
galaxies are assigned a redshift-dependent weight w(z) =
[1+4πn(z)J3(s)]−1where J3(s) =?s
density is sufficiently low that w(z) ≈ 1 and this weighting
makes a negligible difference to the results and we do not
use it.
0ξ(x)x2dx (Efstathiou
1988; Loveday et al. 1995). In our case the galaxy number
3.4Redshift blunder correction
The low signal-to-noise spectra obtained by the WiggleZ sur-
vey imply that a small but significant fraction of galaxies
are assigned a “reliable” (Q ≥ 3) redshift which proves to
be incorrect owing to emission-line mis-identification. This
is monitored in the survey by allocating a small number of
fibres (typically 3 to 5 out of 400 per pointing) to re-observe
galaxies with existing Q ≥ 3 redshifts. The fraction of repeat
observations producing a discrepant redshift may be used to
estimate the redshift “blunder” rate.
There is a significant difference in the reliability of
Q = 3 redshifts and Q ≥ 4 redshifts. Q = 3 redshifts (which
represent a fraction 32% of reliable redshifts) are typically
based either on noisy spectra or on a single emission line with
no confirming spectral features, whereas Q ≥ 4 redshifts are
based on multiple detected emission lines. Comparing repeat
observations consisting of a Q = 3 redshift and a Q ≥ 4 red-
shift, assuming that the latter provides the correct redshift
identification, we conclude that ≈ 17% of Q = 3 redshifts
are blunders. This agrees with the internal discrepancy rate
amongst repeated pairs of Q = 3 redshifts (which is 31%,
which must be divided by two to obtain the blunder rate per
object). Comparing repeat observations consisting of Q ≥ 4
redshifts we find that only ≈ 1% of these redshifts are blun-
ders.
The blunder rate for Q = 3 spectra varies significantly
with the true galaxy redshift, which determines how many
emission lines appear in the observed wavelength range. The
Page 8
8
Blake et al.
Figure 5. The ratio of the angular correlation functions of the WiggleZ parent catalogue and targeted catalogue for the four survey
regions analyzed in this paper. This ratio indicates the fraction of close pairs surviving the restrictions of fibre collisions as a function
of angular scale; pairs are lost for angular separations less than θ = 2 arcmin which is indicated by the vertical dashed line. The solid
curve indicates the best fit of the 2-parameter model {1 + erf[(log10θ − µ)/σ]}/2.
Figure 6. The dependence of the redshift blunder rate of galaxies
with Q = 3 spectra on the (true) galaxy redshift, determined by
comparing repeat observations consisting of pairs of spectra with
Q = 3 and Q ≥ 4. Poissonian error bars are shown.
dependence is displayed in Figure 6 based on the comparison
of Q = 3 and Q ≥ 4 pairs of repeat observations. The total
blunder rate for all reliable (Q ≥ 3) redshifts is below 5%
for the range z < 0.7, increasing to 20% by z = 1. The
redshift blunder rate does not depend on galaxy continuum
magnitude.
Redshift mis-identification reduces the measured value
of the galaxy correlation function because a fraction of true
close data-data pairs are lost as one or both of the red-
shifts is randomized. If fbad is the redshift blunder rate,
the correction to the correlation function is a constant fac-
tor (1 − fbad)−2assuming the blunder redshift is randomly
distributed. We applied this correction to the measured cor-
relation function to deduce the final value:
ξz(σ,π)corrected= ξz(σ,π)measured× (1 − fbad)−2
(10)
When measuring the galaxy correlation function for a
particular redshift or luminosity range, we re-calculated the
redshift blunder rate for the corresponding sample in each
region as explained below. We corrected the correlation func-
tion for that region using Equation 10, before combining
together the correlation functions for the different regions.
We determined the redshift blunder rate for each region by
weighting the blunder probabilities of the N individual ob-
jects in that region:
fbad=
1
N
?
N
?
i=1
fbad,i
?
(11)
For objects with Q = 3 we assigned the probability for each
object based on its redshift using Figure 6. For objects with
Q ≥ 4 we assumed a blunder rate of 1%.
Page 9
WiggleZ survey: small-scale clustering
9
3.5Jack-knife re-samples
In order to determine the error in the measured correlation
function we must characterize the statistical fluctuations in
the data sample. It is well-known that these fluctuations are
not well-described by Poisson statistics, for which the error
in the pair count DD in a separation bin would be equal
to
same galaxy participating in pairs in different separation
bins cause the statistical variance of the galaxy pair count
to exceed the Poisson prediction and induce covariances be-
tween the bins.
In this analysis we use jack-knife re-sampling to deter-
mine the correlation function error. In this technique the
dataset is divided into N equal-area sub-regions on the sky.
The correlation function analysis is repeated N times, in
each case omitting one of the sub-regions in turn. Labelling
the different correlation function measurements at separa-
tion s as ξi(s) from i = 1 to i = N, the covariance between
separation bins j and k was deduced as:
√DD. Sample variance, geometrical edge effects and the
Cjk
≡?ξ(sj)ξ(sk)? − ?ξ(sj)??ξ(sk)?
??N
where ξ(sj) =?N
independent, sharing a high fraction of common sources.
We defined the jack-knife samples by splitting each sur-
vey region into N = 49 sub-regions using constant bound-
aries of right ascension and declination. We tried the al-
ternative technique of using the GALEX tiles to define the
jack-knife regions; this produced a result that did not differ
significantly. Future analyses of the WiggleZ survey clus-
tering will quantify the statistical fluctuations using mock
galaxy catalogues constructed from N-body simulations.
(12)
≈(N − 1)
i=1ξi(sj)ξi(sk)
N
− ξ(sj)ξ(sk)
?
(13)
i=1ξi(sj)/N. The factor (N − 1) in equa-
tion 13 is required because the jack-knife re-samples are not
4RESULTS
4.12D correlation function
Figure 7 illustrates the dependence of the 2D redshift-space
correlation function ξz(σ,π) of equation 1 on the separa-
tions π and σ perpendicular and parallel to the line-of-sight
for the sample of WiggleZ galaxies spanning the full red-
shift range 0.1 < z < 1.3. We measured the correlation
function separately for the four independent survey regions
and combined the results using inverse-variance weighting.
The non-circularity of the contours of constant ξz trace the
imprint of galaxy peculiar velocities; we use linear scales
of σ and π in this plot to focus on the large-scale distor-
tions. In particular, for scales > 10h−1Mpc the increase
in the value of ξz with increasing angle to the line-of-sight
θ = arctan(σ/π) for fixed total separation
signature of coherent galaxy infall and can be quantified to
measure the redshift-space distortion parameter β (see Sec-
tion 4.3). We also detect some evidence for “fingers of god”,
in the form of elongation of the contours of ξz along the
π-axis, due to the virialized motions of galaxies in clusters.
There is some similarity here with the results of Coil et al.
(2008) Figure 7 for luminous blue galaxies, except that our
sample size is significantly larger.
√σ2+ π2is a
Figure 7. The 2D redshift-space correlation function ξz(σ, π)
as a function of separation σ perpendicular to the line-of-sight
and π parallel to the line-of-sight. The function is represented
using both greyscale and contours. Results for the four survey
regions analyzed in this paper have been combined for the galaxy
redshift range 0.1 < z < 1.3. The non-circularity of the contours
encodes the imprint of galaxy peculiar velocities, as discussed in
the text. The red line (3rd contour from the bottom left) is the
ξz = 1 contour which lies at approximately
Mpc; the blue line (8th contour from the bottom left) is the ξz =
0.1 contour.
√σ2+ π2≈ 5h−1
4.2Clustering length of the sample
Galaxy peculiar velocities change values of π but not σ. The
real-space clustering properties of the galaxies may there-
fore be deduced by integrating ξz(σ,π) along the π-axis,
as discussed in Section 3.1. We summed the 2D correlation
function for the 0.1 < z < 1.3 sample in 5 logarithmic bins
of π between πmin = 0.5h−1Mpc and πmax = 20h−1Mpc.
The result is plotted in Figure 8 for the projected separa-
tion range 1 < σ < 100h−1Mpc, with errors obtained from
the jack-knife re-sampling. The full covariance matrix C de-
duced from the jack-knife re-samples is displayed in Figure
9 by plotting in greyscale the correlation coefficient between
two separation bins i and j:
r(i,j) =
Cij
?CiiCjj
(14)
We employed the methodology of Section 3.1 to fit a
power-law real-space correlation function ξr = (r0/r)γto
the redshift-space data over the range 1.5 < σ < 15h−1
Mpc, by minimizing the χ2statistic using the covariance
matrix:
χ2=
?
i,j
δyi(C−1)ijδyj
(15)
Page 10
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Blake et al.
Figure 8. The projected correlation function Ξ(σ)/σCγ as a
function of projected separation σ for galaxies in the redshift
range 0.1 < z < 1.3, combining the results for the four survey
regions analyzed in this paper. The solid line is the best-fitting
power-law for the separation range 1.5 < σ < 15h−1Mpc. The
y-axis is normalized by a factor which produces numerical results
approximating (r0/σ)γ.
Figure 9. Greyscale plot of the correlation coefficient r of equa-
tion 14, indicating the degree of covariance between different sep-
aration bins for each redshift slice.
where δyi is the vector of offsets between the data and the
model, and C−1is the inverse of the covariance matrix. The
fitting range was motivated by our wish to estimate the clus-
tering length r0 for which ξ(r0) = 1. A power-law provides
a good fit to the data with a best-fitting χ2= 7.1 (for 8
degrees of freedom). The marginalized measurements of the
power-law parameters are r0 = 4.40 ± 0.12h−1Mpc and
γ = 1.92 ± 0.08 for the 0.1 < z < 1.3 sample. We compare
these measurements to previous studies of UV-selected and
optically-selected galaxies in Section 4.5.
In Figure 10 we plot the separate projected correlation
function measurements for each of the four survey regions
analyzed in this paper. The four regions give consistent re-
sults.
In order to derive the bias factor of the galaxies with
respect to dark matter we generated a model non-linear mat-
ter power spectrum at z = 0 assuming a flat cosmological
model with fiducial parameters Ωm = 0.3, Ωb/Ωm = 0.15,
h = 0.7 (where H0 = 100h km s−1Mpc−1) and σ8 = 0.9,
using the “CAMB” software package (Lewis, Challinor &
Lasenby 2000) including corrections for non-linear growth of
structure using the fitting formula of Smith et al. (2003). We
used this model power spectrum to determine the non-linear
matter correlation function ξDM at z = 0. The resulting cor-
relation function satisfied ξDM(r) = 1 for r = 4.7h−1Mpc,
which we assumed as our estimate of r0,DM(0), the clustering
length of dark matter at z = 0. Given that the overall ampli-
tude of the power spectrum scales with redshift in the linear
regime as D(z)2, where D(z) is the linear growth factor, we
can approximate:
r0,DM(z) ≈ (4.7h−1Mpc) × D(z)2/γ
where γ ≈ 1.8. Hence the linear bias factor b of a population
of galaxies with clustering length r0can be approximated as:
(16)
b ≈
?
r0
r0,DM
?γ/2
=
?
r0
4.7h−1Mpc
?γ/2
× D(z)−1
(17)
Our measured clustering length r0 = 4.4h−1Mpc for a sam-
ple at median redshift z ≈ 0.6 is hence equivalent to a linear
bias factor b ≈ 1.3.
4.3Redshift-space distortions
The peculiar velocities generated by large-scale coherent in-
fall can be parameterized by β ≈ Ωm(z)0.55/b where b is the
linear bias parameter (Kaiser 1987). For a flat cosmological
constant model with Ωm(0) = 0.3, Ωm(z = 0.6) = 0.64, and
our real-space clustering measurement b = 1.3 hence pre-
dicts β = 0.6 at the median redshift of the sample. The pur-
pose of this Section is to demonstrate that our data contains
this self-consistent signal of peculiar velocities (we leave de-
tailed fits for β to a further study).
We may quantify the imprint of peculiar velocities by
measuring the quadrupole moment, Q(s), of the 2D correla-
tion function (Hamilton 1992). This statistic quantifies the
anisotropy evident in Figure 7. If we define the correlation
function moment ξℓ for multipole ℓ as:
ξℓ(s) =2ℓ + 1
2
?+1
−1
ξz(s,µ)Pℓ(µ)dµ(18)
we can then show that
Q(s) =
ξ2(s)
?3
s3
?s
0ξ0(x)x2dx?− ξ0(s)
=
4
3β +4
1 +2
7β
3β +1
5β2
(19)
which is valid for large scales s > 10h−1Mpc. Figure 11
plots the measured quantity Q(s) as a function of separa-
tion s, together with the prediction of equation 19 for vari-
ous values of β. In order to construct the quantity Q(s) we
measured the 2D redshift-space correlation function in bins
Page 11
WiggleZ survey: small-scale clustering
11
Figure 10. The projected correlation function Ξ(σ)/σCγ as a function of projected separation σ for galaxies in the redshift range
0.1 < z < 1.3, measured for the four survey regions analyzed in this paper. The solid line indicates the best-fitting power law for the
whole sample, and the number of redshifts N used for each region is displayed. The y-axis is normalized by a factor which produces
numerical results approximating (r0/σ)γ.
Figure 11. The statistic Q(s), which encodes the anisotropy
in the 2D correlation function ξ(σ,π) induced by redshift-space
distortions. The prediction of linear-theory on large-scales s >
10h−1Mpc is indicated as a function of the parameter β.
of s and µ, and summed over µ, weighting in accordance
with equation 18. The result is consistent with our estimate
β ≈ 0.6 and constitutes a statistically-significant detection
of redshift-space distortions in our sample.
4.4Redshift and luminosity dependence
Our sample of WiggleZ galaxies is large enough for us to an-
alyze the dependence of the clustering length r0 on redshift,
galaxy luminosity and colour. The situation is complicated
by our joint UV-optical selection and strong luminosity-
redshift correlation, but we can make some comparisons
with previous studies. We fix the correlation function slope
γ = 1.8 in this section of the analysis.
The variation of the clustering length with redshift is
plotted in Figure 12, dividing all WiggleZ galaxies in the
range 0.1 < z < 1.0 into redshift bins of width ∆z = 0.1.
The clustering length is roughly constant with redshift for
the range z > 0.3, with a trend to a reduced clustering
strength at low redshifts. Our interpretation of the over-
all constancy of r0(z) is that it is a product of two can-
celling effects. Galaxy luminosity increases with redshift,
which would tend to increase clustering length, but at red-
shifts z > 0.5 optically red galaxies, which possess enhanced
clustering strengths, are removed from the sample by the
optical colour cuts described in Section 2.2.
We also analyzed the clustering in absolute magnitude
and rest-frame colour bins. We considered the clustering as
a function of rest-frame FUV -band and B-band absolute
magnitudes, which are well-matched in wavelength (for red-
shift z ≈ 0.5) to the observed-frame NUV -band and r-band
magnitudes which are used to define our target samples. For
Page 12
12
Blake et al.
Figure 12. Dependence of the best-fitting clustering length r0
on redshift for a fit of the power-law (r0/r)1.8to the real-space
projected correlation function.
this initial analysis we assumed redshift-dependent average
K-corrections which we applied to all galaxies regardless of
colour. These K-corrections were derived using the spectral
energy distribution of a Lyman Break Galaxy including an
intrinsic dust contribution AV = 0.14, which produces a
very good match to the redshift-dependence of the average
observed NUV − r colour of the WiggleZ targets.
We note that the FUV -band and B-band absolute mag-
nitudes of our target sample correlate strongly with redshift.
This is depicted by Figure 13 which plots the target selection
box in (MFUV,MB) for 4 different redshifts, also indicating
the characteristic absolute magnitudes (M∗
redshift obtained from Arnouts et al. (2005) and Willmer et
al. (2006). Between z = 0.25 and z = 1 the average value of
MFUV −M∗
tive at z ≈ 0.5) and the average value of MB−M∗
by 4 magnitudes (becoming positive at z ≈ 0.7).
The dependence of the clustering length r0 of the 0.1 <
z < 1.3 WiggleZ sample on MB, MFUV and MFUV − MB
is displayed in the panels of Figure 14. These measurements
show that the clustering strength of the sample increases
steadily with B-band absolute magnitude, FUV -band ab-
solute magnitude and reddening MFUV − MB colour. Sub-
samples have values of r0 ranging from 2h−1Mpc to 5h−1
Mpc.
Figure 15 plots the variation of r0 with MB for the low-
redshift and high-redshift halves of the dataset, divided at
z = 0.6. This measurement confirms that at fixed MB, the
clustering length of the sample drops slightly with redshift
as the redder galaxies are removed by the colour cuts.
FUV,M∗
B) at each
FUVbrightens by 2 magnitudes (becoming posi-
Bbrightens
4.5Comparison to previous studies
Coil et al. (2008) present clustering measurements as a func-
tion of galaxy colour and luminosity for the DEEP2 Galaxy
Redshift Survey, which has measured redshifts for ≈ 30,000
galaxies in the range 0.7 < z < 1.5 over an area of 3
deg2. The DEEP2 subset of luminous blue galaxies (Coil
et al. Table 2, line 5) has best-fitting clustering parameters
r0 = (4.27 ± 0.43)h−1Mpc and γ = 1.75 ± 0.13 at z = 1
(for a galaxy density n = 6 × 10−4h3Mpc−3and median
Figure 13. The WiggleZ UV-optical target selection box in the
space of B-band absolute magnitude MB and FUV -band abso-
lute magnitude MFUV for 4 different redshifts between z = 0.25
and z = 1 (moving from left to right in the Figure). These
absolute magnitude limits are implied by our apparent magni-
tude and colour selections NUV < 22.8, 20 < r < 22.5 and
−0.5 < NUV −r < 2. The values of M∗
shift are shown for comparison (taken from Willmer et al. 2005
and Arnouts et al. 2005). Absolute magnitudes are calculated as-
suming h = 0.7.
Band M∗
FUVat each red-
absolute magnitude MB = −22.1 assuming h = 0.7). These
results lie in good agreement with ours.
Milliard et al. (2007) and Heinis et al. (2007) present
clustering analyses of GALEX-selected samples. At low red-
shift (z < 0.3) the clustering strength of the UV-selected
sample is r0 ≈ 3.5h−1Mpc, corresponding to low-density
environments, and shows no dependence on UV luminosity
(indeed, there is tentative evidence for an anti-correlation
between r0 and luminosity). These results may naturally be
compared to clustering measurements of z ≈ 3 LBGs also
selected at rest-frame UV wavelengths (e.g. Giavalisco &
Dickinson 2001; Ouchi et al. 2001; Arnouts et al. 2002; Fou-
caud et al. 2003; Adelberger et al. 2005; Allen et al. 2005;
Ouchi et al. 2005; Lee et al. 2006; Yoshida et al. 2008). These
results show a qualitatively different conclusion: LBGs are
highly clustered and concentrated in overdense regions. Fur-
thermore, the clustering strength for galaxies brighter than
M∗
Mpc for the most luminous sub-samples. Yoshida et al.
(2008) demonstrate that the behaviour of the clustering
length r0 depends on a combination of UV and optical lumi-
nosities: galaxies bright in optical magnitudes are strongly
clustered irrespective of UV magnitude, whereas galaxies
faint in optical magnitude have correlation lengths increas-
ing with UV luminosity (see Yoshida et al. Fig.15).
In Figure 16 we overplot the clustering measurements
of the 0.1 < z < 1.3 WiggleZ sample as a function of FUV
absolute magnitude on the compilation of low-redshift and
high-redshift clustering measurements presented by Heinis
et al. (2007). At low FUV absolute magnitudes MFUV −
M∗
selected samples agree well. This absolute magnitude range
corresponds to low redshifts z < 0.3 in the WiggleZ sam-
ple (Figure 13) for which we recover a clustering length
r0 ≈ 3h−1Mpc, similar to samples of low-redshift quies-
cent star-forming galaxies. At higher FUV luminosities and
FUVincreases with FUV luminosity, reaching r0 ≈ 15h−1
FUV > 0.5 the clustering strengths of the different UV-
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WiggleZ survey: small-scale clustering
13
Figure 14. Dependence of the best-fitting clustering length r0
on B-band absolute magnitude MB, FUV -band absolute mag-
nitude MFUV and rest-frame colour MFUV − MB, for a fit of
the power-law (r0/r)1.8to the real-space projected correlation
function. Absolute magnitudes are calculated assuming h = 0.7.
Figure 15. Dependence of the best-fitting clustering length r0on
B-band absolute magnitude MBfor the upper and lower redshift
ranges of our sample, for a fit of the power-law (r0/r)1.8to the
real-space projected correlation function. Absolute magnitudes
are calculated assuming h = 0.7.
Figure 16. Comparison of the clustering segregation with FUV
absolute magnitude observed in the WiggleZ sample with the
compilation of low-redshift and high-redshift results presented by
Heinis et al. (2007). The WiggleZ targets are more comparable
to z = 3 LBGs rather than z = 0 UV-selected galaxies. The
displayed data points are obtained from Giavalisco & Dickinson
(2001), Arnouts et al. (2002), Foucaud et al. (2003), Heinis et al.
(2004), Adelberger et al. (2005) and Heinis et al. (2007). Absolute
magnitudes are calculated assuming h = 0.7.
redshifts, the WiggleZ clustering strength is more compara-
ble to z = 3 LBGs rather than z = 0 UV-selected galaxies.
This is expected as the FUV −NUV WiggleZ selection cut
becomes effective for z > 0.3 and the nature of the result-
ing WiggleZ galaxy population changes to merger-induced
starbursts. The WiggleZ sample does not recover the very
high values of r0 present in very luminous LBGs at z = 3;
we suggest that this may be a result of the WiggleZ colour
cuts selecting against redder galaxies.
5FORECASTS FOR WIGGLEZ SURVEY
The clustering amplitude of the WiggleZ target sample is a
required input for forecasting the accuracy with which the
full 1000 deg2survey will measure the large-scale galaxy
power spectrum. The error in the power spectrum measure-
ment δPgalat a given redshift z and Fourier wavenumber k is
determined by the quantity n×Pgal, where n(z) is the galaxy
number density and Pgal(k,z) is the galaxy power spectrum
amplitude. This quantity fixes the balance between sample
variance and shot noise in the measurement error such that
?
where m is the total number of independent Fourier modes
contributing towards the measurement (e.g. Feldman, Kaiser
& Peacock 1994; Tegmark 1997). The contributions of sam-
ple variance and shot noise are equal when n×Pgal= 1. We
model the angle-averaged redshift-space linear galaxy power
spectrum as a function of k and z as:
δPgal
Pgal
=
1
√m
1 +
1
nPgal
?
(20)
Pgal(k,z) = PDM(k,0)
?
r0,gal(z)
r0,DM(0)
?γ?
1 +2β
3
+β2
5
?
(21)
where we assume r0,DM(0) = 4.7h−1Mpc, r0,gal(z) =
4.4h−1Mpc, γ = 1.9 and β = 0.6. The second term on
the right-hand-side of equation 21 describes the boost from
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14
Blake et al.
Figure 17. The dependence of n × Pgal on redshift for four
scales k representative of those important for the measurement of
baryon acoustic oscillations. If n×Pgal= 1, then the contribution
of shot noise to the power spectrum error equals that of sample
variance.
the galaxy linear bias factor b (equation 17) using the re-
lation Pgal = PDMb2D2. The third term is the result of
redshift-space distortions averaged over angles. We used the
cosmological parameters as listed in Section 4.2 to pro-
duce the z = 0 dark matter power spectrum: Ωm = 0.3,
Ωb/Ωm = 0.15, h = 0.7 and σ8 = 0.9. In order to incorporate
the fraction of redshift blunders fbad we reduced the effec-
tive value of the power spectrum by a factor (1−fbad)2[i.e.
increased the value of r0,DM(0) by a factor (1 − fbad)−2/γ].
Figure 17 plots the dependence of n × Pgal on redshift
for a set of different scales 0.05 < k < 0.2h Mpc−1relevant
for the detection of baryon acoustic oscillations, assuming
a source redshift distribution combining the survey regions
plotted in Figure 2. We further assume a total target density
of 350 deg−2with a 70% redshift completeness. We note
that over a significant range of redshifts and scales our large-
scale power spectrum measurement will be limited by sample
variance rather than shot noise, i.e. n × Pgal> 1.
A useful quantity to describe the survey is the scale-
dependent “effective volume” Veff(k) which is defined by
Veff(k) =
?∞
0
?
n(z)Pgal(k,z)
1 + n(z)Pgal(k,z)
?2dV
dzdz
(22)
where dV/dz is the co-moving volume element. The effective
volume represents an optimally-weighted stacking of power
spectrum measurements at different redshifts (Tegmark
1997). For scales k = (0.05,0.1,0.15,0.2)h Mpc−1we find
Veff = (0.65,0.41,0.25,0.15)h−3Gpc3. Thus the survey de-
sign will achieve the goal of mapping ∼ 1 Gpc3= 0.34h−3
Gpc3.
We can use the effective survey volume to forecast the
error in the final survey power spectrum δPgal(k) in a Fourier
bin of width ∆k (Tegmark 1997):
δPgal
Pgal
=
2π
k?
Veff(k)∆k
(23)
This prediction is plotted for bins of width ∆k = 0.01h
Mpc−1in Figure 18, in which we divide the power spectrum
by the “no-wiggles” reference spectrum provided by Eisen-
Figure 18. Simulation of the errors in the final WiggleZ survey
galaxy power spectrum. We have divided by a smooth “reference”
power spectrum to clarify the signature of baryon acoustic oscil-
lations.
stein & Hu (1998) in order to delineate clearly the baryon
acoustic oscillations.
We also generated 100 Monte Carlo realizations of the
final 1000 deg2survey using the methods described in Blake
& Glazebrook (2003) and Glazebrook & Blake (2005). The
scatter in the power spectrum measurements across the re-
alizations was very close to that predicted by equation 23.
We used these Monte Carlo realizations to assess the ac-
curacy with which the full WiggleZ survey will measure
the tangential and radial standard ruler scale imprinted by
the baryon acoustic oscillations via the fitting formula de-
scribed in Blake et al. (2006). Restricting ourselves to the
0.3 < z < 0.9 subset, and first considering an “angle-
averaged” measured power spectrum P(k), we found that
the scatter in the fitted acoustic wavescale was 2.8%. Mea-
suring instead a 2D power spectrum P(ktan,krad), where
ktan and krad are wavevectors measured perpendicular and
parallel to the line-of-sight, the scatters in the tangential
and radial fitted wavescales were 4.6% and 7.2%, respec-
tively. This latter pair of measurements corresponds to the
accuracy of determination of the quantities DA(z)/s and
H(z)−1/s at an effective redshift z ≈ 0.6, where DA is the
angular diameter distance, H(z) is the high-redshift Hub-
ble constant, and s is the sound horizon at recombination,
i.e. the standard ruler scale. Dividing the survey into red-
shift slices we find that the angle-averaged wavescale may
be measured with accuracy (6.6%,3.7%,6.3%) in redshift
slices (0.25 − 0.5,0.5 − 0.75,0.75 − 1). The angle-averaged
wavescale measures a quantity proportional to (D2
as discussed by Eisenstein et al. (2005).
These forecasts should be considered a pessimistic lower
limit on expected performance for two reasons. Firstly we
have neglected the cosmological information contained in
the overall shape of the galaxy power spectrum, which is
divided out in the above analysis to focus on the “standard
ruler” aspect of the acoustic oscillations. This method pro-
duces robustness against systematic errors (which are ex-
pected to affect the shape of the power spectrum but not
the oscillatory signature). The power spectrum shape car-
ries information about Ωm and H0 which further breaks the
degeneracy in cosmological distances between these two pa-
AH−1)1/3,
Page 15
WiggleZ survey: small-scale clustering
15
rameters and the dark energy. Secondly we have neglected
the improvement offered by “reconstruction” of the density
field, which sharpens the measurement of the acoustic sig-
nature by un-doing (to first order) the large-scale coherent
galaxy motions which smooth out the acoustic peaks (Eisen-
stein et al. 2007).
We investigated improved forecasts using the method-
ology of Seo & Eisenstein (2007) which properly incorpo-
rates information from the power spectrum shape, redshift-
space distortions and density-field reconstruction. The pre-
dicted tangential and radial measurement accuracies for the
0.3 < z < 0.9 sample are 2.7% and 4.3%, respectively (and
are correlated with a correlation coefficient r ≈ 0.4, further
enhancing the power to constrain the cosmological model).
We assume here that reconstruction can improve the pa-
rameters (Σ⊥, Σ?) defined by Seo & Eisenstein (2007) by
a factor equal to 0.5 − 0.3log10(n × Pgal) (Eisenstein, priv.
comm.). Dividing the survey into redshift slices we find that
the tangential and radial wavescales may be measured with
accuracies (5.5%,8.7%) for 0.25 < z < 0.5, (3.6%,5.8%) for
0.5 < z < 0.75 and (7.9%,10.9%) for 0.75 < z < 1. This
information is collected in Table 1 for ease of reference.
We used this last set of forecasts with reconstruction
in 3 redshift bins to determine the expected accuracy of
measurement of a constant equation-of-state wcons of dark
energy (assuming the measurements of DAand H−1are cor-
related with coefficient r = 0.4). Confidence ellipses are dis-
played in Figure 19 in the space of wconsand the matter den-
sity Ωm for a flat cosmology with fiducial model wcons = −1
and Ωm = 0.27. Results are shown for each redshift bin sep-
arately and for the combination of all 3 bins. In order to
generate this Figure we have used the 5-year-WMAP mea-
surement of the CMB acoustic scale ℓA = 302.1 ± 0.9 (Ko-
matsu et al. 2009) in order to cancel the dependence of the
baryon oscillation measurement on the sound horizon at re-
combination. In Figure 19 we have not included any further
CMB information or other external datasets. The marginal-
ized errors are σ(wcons) = 0.31 and σ(Ωm) = 0.03.
In Figure 20 we add in information from the 5-year-
WMAP measurement of the CMB shift parameter R =
1.71 ± 0.02 (Komatsu et al. 2009), including the correla-
tion between R and ℓA, together with the latest supernovae
data from the Essence, SNLS and HST observations (see
Wood-Vasey et al. 2007, Astier et al. 2006, Riess et al.
2007, Davis et al. 2007). The marginalized errors in the
cosmological model from the full combination of datasets
are σ(wcons) = 0.07 and σ(Ωm) = 0.02. The forecast per-
formance of the WiggleZ survey exceeds that of the cur-
rent CMB and supernovae data, but the different mea-
surements are also complementary, breaking degeneracies
in the (Ωm,wcons)-plane through independent techniques.
Disagreement between any pair of datasets would produce
the possibility of discovering non-standard physics (if it ex-
ists) or systematic measurement errors. The final accuracy
of wcons constitutes a robust and precise test of the dark
energy model.
6CONCLUSIONS
We have measured the small-scale clustering amplitude of
high-redshift bright emission-line galaxies using the first
Figure 19. The forecast 68% confidence ellipses for measure-
ments of a constant dark energy equation-of-state wcons and the
matter density Ωm using standard ruler measurements from the
final WiggleZ survey in combination with a CMB prior on the
acoustic scale ℓA. Results are shown for 3 redshift bins (the dif-
ferent contours) and for the combination of the redshift bins (the
shaded area).
Figure 20. The forecast 68% confidence ellipse for measurement
of (Ωm,wcons) from the WiggleZ survey plus CMB acoustic scale
(the yellow ellipse), compared with existing measurements from
the CMB shift parameter (the orange ellipse) and latest super-
novae (the red ellipse). The 68% and 95% confidence regions for
the combination of all the datasets is displayed as the central
contours.