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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.

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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

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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-

Page 13

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

Page 14

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.