Mon. Not. R. Astron. Soc. 000, 1–?? (2012) Printed 3 July 2014 (MN L A T E X style file v2.2) The Gigaparsec WiggleZ Simulations: Characterising scale dependant bias and associated systematics in growth of structure measurements Gregory B. Poole 1,2? , Chris Blake 1 , Felipe A. Mar´ ın 1 , Chris Power 3 , Simon J. Mutch 2 , Darren J. Croton 1 , Matthew Colless 4 , Warrick Couch 5 , Michael J. Drinkwater 6 , Karl Glazebrook 1 1 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia 2 School of Physics, University of Melbourne, Parksville, VIC 3010, Australia 3 ICRAR, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia 4 Research School of Astronomy & Astrophysics, Australian National University, Weston Creek, ACT 2600, Australia 5 Australian Astronomical Observatory, P.O. Box 915, North Ryde, NSW 1670, Australia 6 School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia draft version 3 July 2014 ABSTRACT We present the Gigaparsec WiggleZ (GiggleZ) simulation suite and use this re- source to characterise the effects of galaxy bias and its scale dependence on the two point correlation function of dark matter halos for a range of redshifts (z ∼ <1.2) and dark matter halo masses (100[km/s]<V max <700[km/s]) in a standard ΛCDM cos- mology. Under the ansatz that bias converges to a scale independent form at large scales, we develop an 8-parameter phenomenological model which fully expresses the mass and redshift dependence of bias and its scale dependence in real or red- shift space. Lastly, we use this fitting formula to illustrate how scale-dependent bias can systematically skew measurements of the growth-rate of cosmic structure as obtained from redshift-space distortion measurements. When data is fit only to scales less than k max =0.1[h -1 Mpc] -1 , we find that scale dependent bias effects are significant only for large biases (b ∼ >3) at large redshifts (z ∼ >1). However, when smaller scales are incorporated (k max ∼ >0.2[h -1 Mpc] -1 ) to significantly increase measurement precision, the combination of reduced statistical uncertainties and increased scale dependent bias effects can result in highly significant systematics for most large halos across all redshifts. We identify several new interesting aspects of scale dependent bias, including a significant halo bias boost for small halos at low-redshifts due to substructure effects (approximately 20% for Milky Way-like systems) and a halo mass that is nearly independent of redshift (corresponding to a redshift-space bias of approximately 1.5 at all redshifts) for which halo bias has no scale dependence on scales greater than 3 [h -1 Mpc]. This suggests an optimal strategy of targeting bias ∼1.5 systems for clustering studies which are dominated more by systematic effects than statistical precision, such as cosmological mea- surements of neutrino masses. Code for generating our fitting formula is publicly available at http://gbpoole.github.io/Poole_2014a_code/. Key words: surveys, large-scale structure, cosmological parameters, theory 1 INTRODUCTION Maps of the distribution of galaxies across enormous cos- mic volumes – as determined from galaxy redshift surveys – have become extremely rich resources for a variety of powerful examinations of cosmological models. These in- clude (but are certainly not limited to) precise standard ruler measurements of the cosmic expansion history using ? E-mail: [email protected]harmonic features induced by “Baryon Acoustic Oscilla- tions” (BAOs) in the Universe’s matter density field and measurements of the growth rate of cosmic structure as probed by the imprints of the cosmic peculiar velocity field on redshift-derived (i.e. redshift-space) distributions of galaxies. Our ability to perform these and other cosmo- logical examinations using redshift surveys is based upon our ability to connect observed galaxy distributions to our highly developed and robust models of the distribution of matter in the early Universe, its evolution with redshift c 2012 RAS arXiv:1407.0390v1 [astro-ph.CO] 1 Jul 2014
Transcript
Mon. Not. R. Astron. Soc. 000, 1–?? (2012) Printed 3 July 2014 (MN LATEX style file v2.2)
The Gigaparsec WiggleZ Simulations:Characterising scale dependant bias and associatedsystematics in growth of structure measurements
Gregory B. Poole1,2?, Chris Blake1, Felipe A. Marın1, Chris Power3,Simon J. Mutch2, Darren J. Croton1, Matthew Colless4, Warrick Couch5,Michael J. Drinkwater6, Karl Glazebrook1
1 Centre for Astrophysics & Supercomputing, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC 3122, Australia2 School of Physics, University of Melbourne, Parksville, VIC 3010, Australia3 ICRAR, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia4 Research School of Astronomy & Astrophysics, Australian National University, Weston Creek, ACT 2600, Australia5 Australian Astronomical Observatory, P.O. Box 915, North Ryde, NSW 1670, Australia6 School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia
draft version 3 July 2014
ABSTRACTWe present the Gigaparsec WiggleZ (GiggleZ) simulation suite and use this re-source to characterise the effects of galaxy bias and its scale dependence on the twopoint correlation function of dark matter halos for a range of redshifts (z∼<1.2) and
dark matter halo masses (100[km/s]<Vmax<700[km/s]) in a standard ΛCDM cos-mology. Under the ansatz that bias converges to a scale independent form at largescales, we develop an 8-parameter phenomenological model which fully expressesthe mass and redshift dependence of bias and its scale dependence in real or red-shift space. Lastly, we use this fitting formula to illustrate how scale-dependentbias can systematically skew measurements of the growth-rate of cosmic structureas obtained from redshift-space distortion measurements. When data is fit only toscales less than kmax=0.1 [h−1Mpc]−1, we find that scale dependent bias effectsare significant only for large biases (b∼>3) at large redshifts (z∼>1). However, when
smaller scales are incorporated (kmax∼>0.2 [h−1Mpc]−1) to significantly increasemeasurement precision, the combination of reduced statistical uncertainties andincreased scale dependent bias effects can result in highly significant systematicsfor most large halos across all redshifts. We identify several new interesting aspectsof scale dependent bias, including a significant halo bias boost for small halos atlow-redshifts due to substructure effects (approximately 20% for Milky Way-likesystems) and a halo mass that is nearly independent of redshift (corresponding toa redshift-space bias of approximately 1.5 at all redshifts) for which halo bias hasno scale dependence on scales greater than 3 [h−1Mpc]. This suggests an optimalstrategy of targeting bias ∼1.5 systems for clustering studies which are dominatedmore by systematic effects than statistical precision, such as cosmological mea-surements of neutrino masses. Code for generating our fitting formula is publiclyavailable at http://gbpoole.github.io/Poole_2014a_code/.
Key words: surveys, large-scale structure, cosmological parameters, theory
1 INTRODUCTION
Maps of the distribution of galaxies across enormous cos-mic volumes – as determined from galaxy redshift surveys– have become extremely rich resources for a variety ofpowerful examinations of cosmological models. These in-clude (but are certainly not limited to) precise standardruler measurements of the cosmic expansion history using
harmonic features induced by “Baryon Acoustic Oscilla-tions” (BAOs) in the Universe’s matter density field andmeasurements of the growth rate of cosmic structure asprobed by the imprints of the cosmic peculiar velocityfield on redshift-derived (i.e. redshift-space) distributionsof galaxies. Our ability to perform these and other cosmo-logical examinations using redshift surveys is based uponour ability to connect observed galaxy distributions to ourhighly developed and robust models of the distribution ofmatter in the early Universe, its evolution with redshift
and the dependence of both on background cosmology.Of course, the success of this endeavour rests completelyon our ability to relate observed galaxy distributions totheir underlying matter distributions; a relationship gen-erally referred to as “galaxy bias”. However, it has longbeen understood observationally that galaxy bias has acomplicated dependancy on galaxy luminosity, colour andmorphology (Loveday et al. 1995; Hermit et al. 1996) withmodern studies still continuing to refine this understand-ing (e.g. Norberg et al. 2001; Zehavi et al. 2005; Ross,Brunner & Myers 2007; Swanson et al. 2008; Cresswell &Percival 2009, see Baugh 2013 for a review).
As with most aspects of large scale structure, a greatdeal of theoretical insight can be obtained through ex-cursion set analyses. The earliest successful theory of thistype was that of Kaiser (1984, subsequently extended byBardeen et al. 1986) who illustrated how the two-pointclustering statistics of collapsed cosmological objects be-comes enhanced if associated with early overdensities inthe cosmological matter field. The first model to build ex-plicitly upon the popular framework of Press & Schechter(1974) and its extensions (EPS) was that of Mo & White(1996, MW) which was subsequently confronted by thenumerical investigation of Jing (1998) who identified sig-nificant discrepancies in this model’s treatment of lower-mass systems. These discrepancies were traced to incor-rect assumptions about the form of the halo mass func-tion in MW by Sheth & Tormen (1999) who were able tobuild a successful analytic model constructed from massfunctions calibrated by numerical simulations, thus es-tablishing an intimate link between the mass-dependentclustering bias of a halo population and its associatedmass function. This was soon followed by Sheth, Mo &Tormen (2001, SMT) who added an account of the dy-namics of ellipsoidal collapse to the traditional EPS ap-proach through the adoption of a mass-dependence forthe spherical collapse overdensity, leading to significantimprovements in the excursion set results for both massfunctions and the mass dependence of large-scale bias (al-though, see Borzyszkowski, Ludlow & Porciani 2014, fora recent challenge to this interpretation).
Generally, two approaches to the analysis of halobias exist: Eulerian approaches (which dominate the lit-erature) focus on the contemporaneous relationship ofhalo and matter clustering and Lagrangian approacheswhich relate the evolving clustering of halos to their ini-tial linear-regime matter field. Interesting challenges tothe conclusions of Eulerian studies have emerged from La-grangian studies. For example, Porciani, Catelan & Lacey(1999) utilised simulations to show that the low-mass biasmodifications of Jing (1998, mentioned above) to the an-alytic model of MW reflects conditions embedded in theinitial state of the simulations, and not exclusively subse-quent non-linear processes. Such findings motivate a care-ful examination of traditional excursion set descriptionsof halo formation; a conclusion echoed by Jing (1999)and subsequently built upon by several studies includingLudlow & Porciani (2011) and Elia, Ludlow & Porciani(2012).
While analytic progress continues to be made (e.g.Ma et al. 2011, who employ a Non-Markovian extensionand a stochastic collapse barrier within the frameworkof traditional EPS approaches to obtain improved massfunction and bias models), the work of SMT makes itclear that treatment of the detailed structure of collaps-ing cosmological fields are important to obtaining accu-
rate estimates of volume-averaged clustering statistics.As a result, most significant progress has been driven oflate by improved calibrations of analytic models usingN-body simulations (e.g. Seljak & Warren 2004; Tinkeret al. 2005). This effort has culminated in Tinker et al.(2010, TRK) who examine a more generalised form of theSMT model and perform a careful numerical calibrationof its parameters. Recent studies have validated the TRKmodel (Papageorgiou et al. (2012); see Basilakos & Plio-nis 2001; Basilakos, Plionis & Ragone-Figueroa 2008, for asimilarly successful model) which we will use as our maincomparison for the large-scale bias calculations which an-chor the scale-dependent bias analysis in this work.
While large-scale galaxy bias has received a greatdeal of study, relatively few inquiries have been madeinto its scale dependence. Early examinations (e.g. Sheth& Lemson 1999; Casas-Miranda et al. 2002; Zehavi et al.2004; Seo & Eisenstein 2005) have discussed some generalexpectations and presented evidence of scale-dependantbias in observed datasets but the work of Tinker et al.(2005) is the first to present a general model. Subse-quently, in their clustering analysis of the 2dF GalaxyRedshift Survey, Cole et al. (2005) introduced the “Q-model”; a phenomenological Fourier-space model whichhas subsequently found applications in the analysis ofSLOAN LRGs (e.g. Padmanabhan et al. 2007). Employ-ing arguments based on the halo model, other Fourier-space accounts of scale-dependent bias include the modelof Schulz & White (2006, subsequently extended by Huffet al. 2007) and Smith, Scoccimarro & Sheth (2007) whoclearly illustrate the existence of scale dependent bias anda dependence on halo mass and galaxy type. Lastly, Pol-lack, Smith & Porciani (2013) have recently explored scaledependent bias within standard perturbation theory find-ing that the non-linear processes giving rise to such effectsare not sufficiently described in popular second-order lo-cal Eulerian schemes.
Observationally, an important additional complica-tion arises. Positions for very large ensembles of galax-ies are generally not determined through direct distancemeasurements but are rather inferred from redshifts. Dis-tributions measured in this way are said to be constructedin “redshift-space” and the presence of peculiar velocitiesimprinted upon the background Hubble-flow by accelera-tions from local density gradients is known to induce sig-nificant bias effects in this space. The classic treatment byKaiser (1987, K87 henceforth) predicts that coherent bulkflows on large scales induce a “Kaiser-boost”; a significantincrease in clustering bias over that which would be in-ferred in real-space due to halo assembly effects alone.On large scales, this model has been validated by numeri-cal simulations (e.g. Montesano, Sanchez & Phleps 2010)but on small scales – where incoherent motions such asthose giving rise to the “Fingers of God” effect can leadto a suppression of bias – significant scale-dependence tothese redshift-space effects have been identified (e.g. Sel-jak 2001).
The primary consequence of scale dependent bias isthat it introduces a source of systematic uncertainty tocosmology constraints at a level which is now importantfor ongoing and future surveys. While several investiga-tions have found that the consequences for constraintsbased on the scale of the BAO peak should not be sig-nificant (e.g. Eisenstein, Seo & White 2007; Crocce &Scoccimarro 2008; Angulo et al. 2008; Smith, Scoccimarro& Sheth 2008, a conclusion supported by our investiga-
tions) there is concern that constraints sensitive to thefull shape of scale-dependent clustering statistics (suchas power spectra or correlation functions) will be moresusceptible, particularly when pushed to smaller scales.Two notable such cases include measurements of neu-trino masses from the cosmological power spectrum (e.g.Riemer-Sørensen et al. 2012) and measurements of thegrowth rate of cosmic structure (e.g. Blake et al. 2011a)which we focus on in this study.
Beyond the issue of systematic bias, several inter-esting physical processes can lead to scale dependentbias providing new opportunities for the study of otherphysics. These include the induction of scale-dependentbias from departures from non-Gaussianity in the earlyuniverse (Dalal et al. 2008; Slosar et al. 2008; Taruya,Koyama & Matsubara 2008) or from subtle environmentaleffects induced by the physics of galaxy formation (Coles& Erdogdu 2007; Barkana & Loeb 2011). Firmly estab-lishing an accurate and robust theory in the absence ofthese effects will be essential for their search in observa-tional datasets.
In this work, we take a distinctly different approachfrom past studies, performing a straight-forward phe-nomenological characterisation of Eulerian bias in con-figuration space. Surprisingly little theoretical investiga-tion of scale dependent bias within this framework hasbeen performed in the recent literature despite the factthat it’s the space in which most observational analysisis performed. As noted by Huff et al. (2007) (also seeGuzik, Bernstein & Smith 2007), configuration space of-fers an important advantage over Fourier space: a loweramplitude of scale dependent bias. Interpreted within theframework of the halo model, they note that this is due tothe fact that most scale dependent bias is a product of thedifferent scales on which matter and galactic halos transi-tion from the 1-halo regime to the 2-halo regime. This oc-curs on relatively small scales as far as most cosmologicalstudies are concerned, thus isolating its effects in config-uration space. In Fourier space, such broad-spectrum fea-tures become spread across a wider range of scales trans-ferring signal from the small scales on which the phenom-ena occurs, to larger scales where most of the clusteringsignal resides.
We use the Gigaparsec WiggleZ (GiggleZ) Simula-tion Suite for this study. GiggleZ was constructed to sup-port the science program of the WiggleZ Dark EnergySurvey (Drinkwater et al. 2010) – a large redshift sur-vey of UV-selected galaxies conducted with the multi-object AAOmega fibre spectrograph at the 3.9-m Aus-tralian Astronomical Telescope – and has been used inseveral WiggleZ-related publications to date (e.g. Blakeet al. 2011b; Riemer-Sørensen et al. 2012; Contreras et al.2013; Marın et al. 2013; Blake, James & Poole 2013). Wetake this opportunity to present details related to the con-struction of the GiggleZ simulation program and subse-quently present a simple and direct model of the massand redshift dependence of both large-scale and scale-dependent bias of dark matter halos. We examine for thefirst time the effects of substructure on models of galaxybias of this form, finding significant (∼20%) effects on low-bias systems at low redshift. We then use this model tobuild upon previous studies of systematic biases in growthof structure measurements (Okumura & Jing 2011; Jen-nings, Baugh & Pascoli 2011; Contreras et al. 2013), cal-culating the potential magnitude of systematic errors in-
duced in the absence of corrections for scale-dependentbias effects.
In Section 2 we present the GiggleZ simulation suite;the simulations involved, our approach to initialising, run-ning and analysing them, and the results of a convergencestudy run to determine the optimal integration propertiesof our adopted simulation code. In Section 3 we presentour scale dependent bias model, stepping through the jus-tifications for each of our chosen parameterisations. InSection 4 we present the consequences of scale depen-dent bias for growth of structure measurements. Lastly,we summarise and discuss our conclusions in Section 5.
Our choice of fiducial cosmology throughout will be astandard spatially-flat WMAP-5 ΛCDM cosmology (Ko-matsu et al. 2009): (ΩΛ, ΩM , Ωb, h, σ8, n)=(0.727, 0.273,0.0456, 0.705, 0.812, 0.960).
2 SIMULATIONS
The GiggleZ simulation suite consists of 5 simulations: alarge GiggleZ-main run consisting of 21603 particles dis-tributed in a periodic box 1 [h−1Gpc] on-a-side, and 4simulations of an identical 125 [h−1Mpc] on-a-side con-trol volume spanning a factor of 512 in mass resolutionwith snapshot temporal resolutions as fine as 15 Myrs.The basic specifications for these 5 runs are listed in Ta-ble 1. The large scale of the GiggleZ-main simulation wasmotivated by the unprecedented combination of large vol-ume and low halo mass of the low-bias UV-selected galax-ies targeted by WiggleZ. Such observational programspresent a demanding challenge for theoretical supportof clustering studies, leading us to create (at the time)one of the highest-resolution gigaparsec-scale cosmolog-ical simulations available, comparable to modern simu-lation programs such as the Multi-dark BigBolshoi Sim-ulation (Prada et al. 2012). The control-volume simula-tions were designed to conduct systematic studies of theresolution requirements for semi-analytic galaxy forma-tion studies. In this paper we focus on the GiggleZ-mainsimulation only. A companion paper will present the con-trol volume simulations in detail where they are used topresent our method of merger tree construction and theirconvergence properties.
We have run our simulations with GADGET-2(Springel 2005), a Tree-Particle Mesh (TreePM) code wellsuited to large distributed memory systems. We havemodified the publicly available version to conserve RAMin dark matter only simulations by removing all supportfor hydrodynamics, ’FLEXSTEP’ time stepping and vari-able particle masses (along with all associated memoryallocations). All simulations were run on the Green Ma-chine at Swinburne University, with the largest run con-suming all the resources of 124 nodes, each housing dualquad core Intel Clovertown 64-bit processors (for a totalof 992) with 16GB of RAM.
2.1 Initial Conditions
To initialise our simulations we use the Parallel N-bodyInitial Conditions (PaNICs) code developed at Swin-burne for this project. PaNICs follows the approach ofBertschinger (2001) to construct a displacement fieldwhich, when applied to a uniform distribution of parti-cles, yields a distribution with our desired power spec-trum. This power spectrum was generated using CAMB
Table 1. Box sizes (L), particle counts (Np), particle mass (mp), number of snapshots (nsnap), approximate snapshot temporalresolution (∆t) and gravitational softening length (ε) for the GiggleZ simulations.
0.8
0.9
1.0
1.1
1.2 (i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.005d, 0.010)
(i, ε, η)=(2, 0.010d, 0.010)
(i, ε, η)=(3, 0.020d, 0.010)
(i, ε, η)=(4, 0.040d, 0.010)
(i, ε, η)=(5, 0.080d, 0.010)
0.8
0.9
1.0
1.1
1.2 (i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.005d, 0.010)
(i, ε, η)=(2, 0.010d, 0.010)
(i, ε, η)=(3, 0.020d, 0.010)
(i, ε, η)=(4, 0.040d, 0.010)
(i, ε, η)=(5, 0.080d, 0.010)
0.8
0.9
1.0
1.1
1.2
Φi(>M
)/Φ
0(>M
)
(i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.005d, 0.020)
(i, ε, η)=(2, 0.010d, 0.020)
(i, ε, η)=(3, 0.020d, 0.020)
(i, ε, η)=(4, 0.040d, 0.020)
(i, ε, η)=(5, 0.080d, 0.020)
0.8
0.9
1.0
1.1
1.2
Pi(k
)/P
0(k
)
(i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.005d, 0.020)
(i, ε, η)=(2, 0.010d, 0.020)
(i, ε, η)=(3, 0.020d, 0.020)
(i, ε, η)=(4, 0.040d, 0.020)
(i, ε, η)=(5, 0.080d, 0.020)
1011 1012 1013 1014 1015
M[M]
0.8
0.9
1.0
1.1
1.2 (i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.020d, 0.040)
(i, ε, η)=(2, 0.040d, 0.040)
(i, ε, η)=(3, 0.080d, 0.040)
0.0 0.2 0.4 0.6 0.8 1.0
k[h−1Mpc−1]
0.8
0.9
1.0
1.1
1.2 (i, ε, η)=(0, 0.005d, 0.005)
(i, ε, η)=(1, 0.020d, 0.040)
(i, ε, η)=(2, 0.040d, 0.040)
(i, ε, η)=(3, 0.080d, 0.040)
Figure 1. A plot comparing the effects of variations in gravitational softening parameter (ε) and time step integration accuracy (η)
on the mass function (Φi; left) and power spectrum of halos more massive than 1012 M (Pi; right) of a (L,N)=(250 [h−1Mpc], 5403)
simulation (i.e. the same mass resolution as the GiggleZ-main and GiggleZ-NR runs). In each case, we normalise the mass functionand power spectrum to the case (ε,η)=(0.005d,0.005). Based on these results, we selected (ε,η)=(0.02d,0.01) for all GiggleZ simulations(where d is the mean interparticle spacing of the simulation). Runs with ε=0.02d are labeled in red with the η=0.01 case additionally
highlighted with a thick line (top panels). Grey shaded regions indicate the magnitude of the Poisson statistical uncertainty of the(ε,η)=(0.005d,0.005) case used as reference in all cases.
(Lewis, Challinor & Lasenby 2000) with our standardspatially-flat WMAP-5 ΛCDM cosmology given above.This power spectrum was normalised for a starting red-shift zinit=49 for the GiggleZ-main run and zinit=499 forthe control volume simulations. These starting redshiftsensure that initial particle displacements are smaller thanthe grid cell size of the displacement field for all simula-tions, a condition advocated by Lukic et al. (2007). Thishigh starting redshift may introduce some numerical noisefor the lower resolution control volume runs affecting de-tailed halo structure, but should have a negligible effecton the mass accretion histories which will be the mainfocus of their use. This was verified for the GiggleZ-NRmass resolution during our convergence testing in whichwe performed a run with zinit=49 and found no signif-icant effect on the simulation’s halo power spectrum ormass function.
For the GiggleZ-main simulation, the displacementfield was computed on a 43203 grid while the control vol-ume simulations used a common displacement field com-puted on a 21603 grid. Uniform distributions in all caseswere computed from integral periodic tilings of a 1353
glass configuration (see White 1994, for more details) gen-erated using GADGET.
Particle velocities were computed from the PaN-ICs displacement field using the Zeldovich approximation(Zel’Dovich 1970; Buchert 1992). Higher-order correc-tions to this calculation (e.g. Scoccimarro 1998; Crocce,Pueblas & Scoccimarro 2006) could not be implementedin a timely fashion for this project, but will certainly beincorporated in future projects.
2.2 Halo finding
The majority of the analysis in this study will utilise thebound dark matter halos which emerge from our simula-tions. To extract these structures we use the well testedcode SUBFIND of Springel et al. (2001). This code firststarts by finding friends-of-friends (FoF) structures forwhich we use the standard linking length criterion of 0.2d(where d = L/ 3
√Np denotes the mean interparticle spac-
ing of the simulation). It subsequently identifies boundsubstructures within these FoF groups as locally over-dense collections of particles, removing unbound particlesthrough an unbinding procedure.
This procedure leads to two classes of halo: FoFgroups and substructure halos. In the work which follows,we perform our analyses on both classes of halo sepa-rately. Since FoF groups are more closely related to theoverdensity peaks forming the basis of Extended Press-Schechter analyses, results derived from study of theseobjects should form a better comparison to models devel-oped within that framework. However, observed galaxypopulations are more closely related to our substructurehalos and results derived from analyses of this class ofhalo should be more straight-forwardly related to ob-served galaxy distributions. Later in Section 3.4 we willfind that there are interesting differences between the biasproperties of the two.
2.3 Convergence tests
Being principally responsible for the accuracy and run-time of our simulations, we carefully considered the set-tings of two GADGET parameters in particular whensetting-up our calculations: the gravitational softening (ε;
we will express this in units of d henceforth) and thedimensionless parameter controlling the accuracy of thetimestep criterion (η; referred to as ErrTolIntAccuracy inthe GADGET manual).
We ran a grid of (L,N)=(250 [h−1Mpc], 5403) simu-lations (i.e. the same mass resolution as the GiggleZ-mainand GiggleZ-NR run), varying combinations of these pa-rameters over the ranges ε=0.005d to 0.08d and η=0.005to 0.04. Since our primary science interests in WiggleZ in-volve studies of L* galaxy formation and clustering on 100[h−1Mpc] scales, we seek convergence based on the sub-structure halo mass function and substructure halo powerspectrum of halos in the range M>1012 [h−1 M].
The results are presented in Fig. 1. Expected trendsare realised: larger softenings in particular have a strongimpact on small scales (i.e. low-mass and high-k). Fur-thermore, we find that the power spectrum is a morestringent condition in these tests than the mass function.When P (k) is converged, the mass function is converged.Using the power spectrum at z=0 as our metric of fit-ness, we can immediately rule out softenings ε>0.04d bydemanding that deviations from our fiducial P (k) remainless than 5% over the range k=0.1 to 1 [h−1Mpc]−1.
There is a degeneracy in these tests between ε andη: moderate increases in ε can be compensated for bydecreasing η. Reducing η has a significant impact on therun-time of the simulation however, placing practical con-straints on how far it can be lowered. Taken in combina-tion, we use these constraints to settle upon the com-bination (ε,η)=(0.02d,0.01) for all runs in this project.From these experiments, we expect the mass function tobe accurate to ∼2% on M* scales. We expect the powerspectrum to be accurate to ∼2% over the range k= [0.1, 1][h−1Mpc]−1.
2.4 Halo groupings
For this study, we are interested in the mass and red-shift dependence of halo clustering properties. To facili-tate our analysis, we have assembled a number of ’group-ings’ of both our FoF and substructure halos for a set ofseven redshifts from z=0 to z∼1.2 in steps of dz∼0.2. Ineach case we have rank-ordered the structures by theirmaximum circular velocities (denoted Vmax) and selectedcontiguous groupings of ni (zi, Vmax,i) systems (yielding
grouping number densities of ni per[h−1Gpc
]3) for each
’i’th grouping. This is done such that Vmax,i are medianvalues for their respective groupings, starting at 150 km/sfor i=0 and extending upwards in steps of 10 km/s untilwe run out of massive halos (at a value of Vmax,i whichdeclines with redshift). We use Vmax as our metric of halomass to render our results less sensitive to peculiaritiesof our chosen halo finder and to increase reproducibility.Furthermore, subhalo abundance matching has suggestedthat Vmax may more directly parameterise the stellar massof galaxies (Reddick et al. 2013), potentially improvingthe degree to which our Vmax-selected subhalo groupingsrepresent the clustering characteristics of stellar-mass se-lected galaxy samples. See Figure 2 for an illustration ofthe relationship between Vmax and Mvir.
We set ni for each grouping to yield correlation func-tions of roughly equivalent signal-to-noise despite thegrowth of structure moderated by the linear growth fac-tor (denoted D and given by D=δ(z)/δ(0), where δ(z)is the evolving matter density contrast) and the mass-dependent bias which we estimate using the Tinker et al.
Figure 2. A plot presenting the number densities (ni) adoptedfor the halo ’groupings’ and the relationship between Vmax
and Mvir used for all analysis in this work. Three redshifts
evenly spanning the range of this study (z∼<1.2) are depicted.Number densities are chosen such that they scale inversely with
large scale bias (as estimated from the model of Tinker et al.
2010, TRK) and the linear growth factor (i.e. ni∝1/ (bTRKD))normalised such that ni (zi=0.6,bTRK=1) =105 [(h/Gpc)3].
(2010, TRK) model† (denoted bTRK). This approach hasthe added benefit of naturally reducing our bin size asmass and redshift increase, adapting to regimes where ha-los densities are low and clustering properties are rapidlyevolving. More specifically, we choose groupings for whichbTRK (Vmax,i,zi) =1 to have ni=105 halos at z=0.6 andscale ni for other cases by 1/ (bTRKD). The resulting val-ues of ni used for this study are illustrated in Figure 2.
To add redshift-space distortion effects to our cata-logs we assume a flat-sky approximation, taking the posi-tions of each halo grouping and adding a 1D displacementin the x-direction (δx) given by:
δx=vxh
a(z)H(z)(1)
where vx is the x-component of the physical centre-of-mass velocity of the halo, a(z) is the cosmological expan-sion factor and H(z) is the redshift-dependant Hubbleparameter.
† Throughout this paper we will convert the overdensity pa-rameterising this model to an effective Vmax assuming stan-dard Navarro, Frenk & White (1997) scaling properties and the
mass-concentration relation of Munoz-Cuartas et al. (2011).
3 ANALYSIS
For the analysis presented in this paper, we will use the2-point correlation function as our measure of clusteringstrength and its scale dependence. The method of Landy& Szalay (1993) is used throughout and is applied to allof the halo groupings described in Section 2.4 as well as torandomly sampled subsets of 106 particles from each rel-evant snapshot of our simulations. This method requiresa large number of randomly distributed points and weuse 250000 points for halo analysis and 5×106 for mat-ter field analysis, ensuring that there are at least 5 timesmore random points than data points in all cases.
Examples of our computed correlation functions arepresented in Figure 3 where we show results at three red-shifts evenly spanning the range of our study (z=0,0.593and 1.224) for distributions of matter and for FoFhalo groupings of three masses (Vmax=150,300, and 450[km/s]) in the GiggleZ-main simulation. Expected trendsof increasing clustering amplitude with halo mass and in-creased redshift-space clustering (particularly on scalesless than ∼2 [h−1 Mpc] where ”halo exclusion” effectsbecome significant) are apparent.
3.1 Computing scale dependent bias andmotivating its general form
Throughout the analysis which follows, we will focus onthree correlation function ratios which capture separatecontributions to halo bias and its scale dependence. Theseratios will be between the redshift-space halo correlationfunction and the real-space halo correlation function (Rz;said to express the redshift-space ’boost’ effects on totalbias), the ratio of the real-space halo correlation functionto the real-space dark matter correlation function (Rh;said to express halo-assembly effects on the total bias) andthe ratio of the redshift-space halo correlation function tothe real-space dark matter correlation function (Rt; saidto express the total redshift-space bias). Throughout thiswork we will refer to these ratios in a general form asRx where x=‘z’,‘h’ or ‘t’ denoting the redshift-space ef-fect, halo-assembly effect or total bias ratios respectively.Conceptually, Rt=Rh×Rz, although we fit to each ratioindividually and do not enforce this relation.
In all cases, Rx(s) profiles and uncertainties are com-puted from the median and (potentially asymmetric) dis-tribution of 216 jack-knife subsamples evaluated usinga regular 63 grid. Furthermore, we concentrate only onscales larger than 3 [h−1 Mpc] for two reasons: we findthat the behaviour of Rx(s) on scales less than this iscomplicated (with a character similar to that presentedin figure 4 of Zehavi et al. 2004) and difficult to param-eterise and because it is on scales less than this wherethe morphology-density relation of observed galaxy pop-ulations becomes significant (Hansen et al. 2009; Haineset al. 2009; von der Linden et al. 2010; Lu et al. 2012;Wetzel, Tinker & Conroy 2012; Rasmussen et al. 2012;Bahe et al. 2013), greatly complicating the use of thesescales for realistic galaxy populations.
Examples of each ratio for cases spanning the range ofredshift and halo mass addressed by this study are shownin Figure 4 (for all plots henceforth, the same colourscheme is used: green to represent real-space halo bias,red to represent redshift-space ’boost’ effects and black torepresent total redshift-space bias). Several general trendsare immediately obvious from this plot. At large scales,the limited volume of our simulation results in a rapid
Figure 3. Two-point correlation functions (ξ(s)) for the total matter and for populations of dark matter halos at three halo
masses and three redshifts. Green and blue lines denote ξ(s) in real and redshift-space for the dark matter particles at each redshift
respectively. Black and red lines with error bars denote ξ(s) in real and redshift-space for the dark matter halos respectively. In allcases, the number halos involved in the represented FoF halo groupings (ni) is given. Uncertainties are computed from jack-knife
subsamples using a regular 63 grid. Values across the top denote the redshift represented by each column while values along the
right indicate the halo mass (expressed in terms of maximum halo circular velocity, Vmax) represented by each row.
0.8
1.0
1.2
1.4
1.6
1.8
z=0.0 z=0.593 z=1.224
Vm
ax =150km
/s
0.8
1.0
1.2
1.4
1.6
1.8
Rx(s
)/b2 x
Vm
ax =300km
/s
101
0.8
1.0
1.2
1.4
1.6
1.8
101
s [h−1 Mpc]101
Vm
ax =450km
/s
Figure 4. The scale dependence of three ratios taken between total matter and halo correlation functions at the same three halo
masses and three redshifts depicted in Figure 3. Red denotes the ratio of the redshift-space halo correlation function to the real-space
halo correlation function (Rz; expressing the redshift-space ’boost’ effects on the total bias), green the ratio of the real-space halocorrelation function to the real-space total matter correlation function (Rh; expressing the halo-assembly effects on the total bias)
and grey the ratio of the redshift-space halo correlation function to the real-space total matter correlation function (Rt; expressingthe total redshift-space bias). All ratios have been computed using their jack-knife subsamples to minimise cosmic variance, with
shaded regions indicating 68% confidence intervals. Thick solid lines indicate the best fit of Eqn. 2 to each dataset assuming γ=1.
In all cases, large-scale bias effects have been normalised-out such that all curves converge to a value of 1 at large values of s. Valuesacross the top denote the redshift represented by each column while values along the right indicate the halo mass (expressed in
terms of maximum halo circular velocity, Vmax) represented by each row.
increase in the variance of each Rx profile as scales beginto exceed 20-30 [h−1 Mpc]. Within these admittedly largeuncertainties, there is little evidence of scale-dependentbias effects beyond these scales, as we expect from the re-sults of previous studies. At smaller scales where our sim-ulation is adequate for quantifying Rx(s), we see clear ev-idence of scale dependence increasing in magnitude withhalo mass and redshift for Rh and Rt while trends aremore mild and less discernible for Rz. Furthermore, insome regimes we find that Rx can be enhanced on smallscales relative to large scales (generally the case for Rh
and Rt) or suppressed on small scales.Therefore, taking as an ansatz that Rx converges to
a constant value at large scales, this figure motivates usto assume the following form for Rx:
Rx= b2x(1 + S (s/sx)−γ
)
where S = ±1 (2)
This is a four parameter model (applicable on scaless>3[h−1Mpc]) where bx quantifies the amplitude of biaseffects at large scales, γ sets the slope of Rx on smallscales, sx is effectively a measurement of the amplitude ofscale dependent effects (particularly for a fixed value of γ,as we will ultimately adopt below) and S sets whether biasis suppressed by scale dependent effects on small scales(i.e. the case S=−1) or enhanced on small scales (i.e. thecase S=+1).
3.2 The mass dependence of scale dependentbias
In Figure 5 we show the results of fitting the model intro-duced in Equation 2 to each scale-dependence ratio, forall of our halo groupings at three redshifts spanning therange of our study. These preliminary illustrative fits areconstructed using a simple χ2-minimisation approach.
When allowing γ to vary freely between values of 0and 3, we find very little discernible trend for γ with Vmax
and very noisy trends for sx with Vmax. This suggeststhat the four parameter model of Equation 2 is under-constrained by these datasets. However, when we fix γ toa value of 1, clear trends in sx(Vmax) emerge as illustratedin Figure 5. Fixing γ in this way results in a minimalreduction in the quality of fit, as shown in the bottompanels of this figure where we compare the χ2 obtainedallowing γ to vary (nDoF=5) to those obtained when wefix γ to a value of 1 (nDoF=6). This value of γ was chosenas a compromise in the range of best fit values obtainedwhen allowing it to vary with mass, redshift and ratiotype. While the results of fits change in detail when otherfixed values of γ are chosen, little change results to thequality of fit or to the conclusions of our study.
For the large scale bias parameters (bx; illustrated inthe top panels of Figure 5), expected trends are apparentwith halo bias increasing with both mass and redshift.The total redshift-space bias consistently follows the real-space bias with an offset which decreases with mass but isrelatively constant with redshift. Redshift-space contribu-tions decline with mass, converging towards a value of 1(i.e. no contribution to redshift-space bias from peculiarvelocities) as masses increase. This trend is remarkablyconstant with redshift as well.
As mentioned above, when we fix γ to a value of 1,sx effectively quantifies the amplitude of scale dependentbias effects. For this choice of γ, a value of sx=1 [h−1
Mpc] results in a 10% difference in bias between scaless=100 [h−1 Mpc] and s=∞, a value of sx=2 [h−1 Mpc] a20% difference, etc.
There is a clear pattern illustrated in Figure 5 of sx
decreasing and then increasing roughly linearly with massabout a pivot point which varies with redshift and ratiotype. This is a result of bias effects being suppressed atsmall scales for small halo masses (i.e. S=−1), passing apoint at which there is no scale dependence (sx=0), andthen increasing with enhanced small-scale bias at largevalues of halo mass (i.e. S=+1). As such, the point ofminimum sx for each case indicates a halo mass at whichscale dependence of bias disappears. This behaviour isdiscernible in Figure 4.
Motivated by these results, we choose the followingparameterisation for the halo mass dependence of scaledependent bias:
log10 b2x(z, Vmax) = b0x(z) + bVx (z)Vmax
sx(z, Vmax) = sVx (z) |Vmax − VSF,x(z)|
S(z, Vmax) =
−1 if Vmax < VSF,x,
+1 if Vmax > VSF,x
(3)
This represents a 4 parameter model describing the massdependence of bias and its scale dependence at a fixed red-shift. Two parameters describe a linear Vmax dependencefor the logarithmic bias (b0x and bVx ), one sets the strengthof the mass dependence of scale dependent bias (sVx ) andone sets the mass at which bias becomes scale free atthe regime between the suppression (at Vmax<VSF,x) andthe enhancement of bias at small scales (at Vmax>VSF,x).The results of fitting this model to the cases illustratedin Figure 5 are illustrated with solid lines. For this andall cases which follow, these fits are applied directly tothe Rx profiles and their (possibly asymmetric) distribu-tion obtained in the manner described in Section 3.1 (andnot to the individual points depicted in Figure 5 result-ing from our χ2 fits to individual cases) using the MCMCmachinery introduced in Poole et al. (2013).
3.3 The redshift dependence of scale dependentbias and the final full model
Finally, we now seek a parameterisation of the full massand redshift dependence of scale dependent bias. This isachieved by parameterising the redshift dependence of the4 parameters in the model given by Equation 3 for eachratio type.
In Figure 6 we present a series of fits (in colouredpoints) of the model presented in Eqn. 3 at several red-shifts spanning the range of our study for both our FoF(solid points) and substructure halos (open points). Theseare equivalent to the fits shown with solid lines in Fig-ure 5 but applied to a larger number of redshifts (andto both halo types). We find that the parameters of ourmass-dependence model vary smoothly with redshift, mo-tivating the following form for the redshift dependence ofscale-dependent bias:
Figure 5. The results of fitting the scale dependent bias model of Eqn. 2 to halo groupings of various masses (expressed in terms
of maximum halo circular velocity, Vmax) at three redshifts spanning the range utilized in this study. Coloured points indicate fitsfor redshift-space ’boost’ effects (Rz; red), halo bias (Rh; green) and total redshift-space bias (Rt; black). Solid lines indicate the
fit of the halo mass dependence model expressed by Eqn. 3 to each of these three datasets (with matching colours). Dotted lines
similarly indicate the results of our full mass-and-redshift dependent bias model expressed by Eqns. 2, 3 and 4 and Table 2. Thebottom 2 panels in each column indicate the values of χ2/nDoF (where nDoF=6 in all cases) obtained from the fit assuming γ = 1
(second from bottom) and the difference in χ2 obtained when allowing γ to vary over the range 0 to 3 (bottom; nDoF=5 in this
case). Values across the top denote the redshift represented by each column.
0.0 0.3 0.6 0.9 1.2
0.5
1.0
2.0
VSF
Total z-space Bias
0.0 0.3 0.6 0.9 1.2
Halo Bias
0.0 0.3 0.6 0.9 1.2
z-space ’boost’
0.0 0.3 0.6 0.9 1.2
0.5
1.0
1.5
2.0
s V
0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2
0.0 0.3 0.6 0.9 1.2
−0.4
−0.2
0.0
0.2
0.4
b o
0.0 0.3 0.6 0.9 1.2 0.0 0.3 0.6 0.9 1.2
0.0 0.3 0.6 0.9 1.2z
−0.15
0.00
0.15
0.30
0.45
b V
0.0 0.3 0.6 0.9 1.2z
0.0 0.3 0.6 0.9 1.2z
Figure 6. The results of fitting our final redshift-and-mass dependent model to the profiles of redshift-space ’boost’ effects, halobias and total z-space bias computed for this study. Individual points denote fits of Eqn. 3 to Rx(s, Vmax) for each ratio type
at several redshifts with solid points indicating fits to FoF halos and open points indicating fits to substructure halos. Solid lines
denote our full mass and redshift dependent model expressed by Eqn. 4 when fit to FoF halos and dashed lines denote this fit tosubstructure halos. Note that the solid lines are not fits to the data points, but rather a single joint MCMC fit to all Rx(s) profiles
used in this study. The agreement validates our chosen parameterisation of the redshift dependence of the parameters in our finalmodel.
Table 2. Parameters for our full scale dependent bias model, as expressed by Eqns. 2, 3 and 4. Values for both halo types
(friends-of-friends halos and substructure) are given.
0.1 0.3 0.5 0.7 0.9 1.1
200
300
400
500
600
700
Vm
ax[k
m/s
]
Total z-space Bias
0.1 0.3 0.5 0.7 0.9 1.1z
Halo Bias
0.1 0.3 0.5 0.7 0.9 1.1
z-space ’Boost’
0.0
0.5
1.0
1.5
2.0
χ2/n
DoF
Figure 7. Values of χ2/nDoF (where nDoF=6 in all cases) obtained from fitting our scale-dependent bias model – bt(Vmax, z) on
the left, bh(Vmax, z) in the middle, bz(Vmax, z) on the right – to the friends-of-friends (FoF) halos of the GiggleZ-main simulation.These planes represent the full range in Vmax and z over which our model has been constrained, with the white region in the top
right being due to a lack of dark matter halos of sufficient density at corresponding masses and redshifts.
log10 VSF,x(z) = V 0SF,x + V zSF,xz
sVx (z) = sV,0
x + sV,zx z
b0x(z) =
b0,0x + b0,zx z if x=‘h’ or ‘t’,
b0,0z + b0,zzz (z − zb,z)2 if x=‘z’
bVx (z) = bV,0x + sV,zx z (4)
This represents a linear redshift dependence for all of theparameters in Equation 3 with the exception of the pa-rameters for the redshift-space ’boost’ which we find re-quires a quadratic dependence for b0x(z) centred on red-shift zb,z (hence introducing an extra parameter in thiscase). Although very-nearly constant with redshift, wefind this refined form of redshift dependence is necessarydue to the strong dependence of sx on bx when Rx is onlyweakly scale dependent (which is always the case for Rz).
Also presented on Figure 6 (with lines; solid for FoFhalos and dotted for substructure) is the results of a globalMCMC fit to our full dataset. This fit is applied simulta-neously to all of the Rx profiles measured for every group-ing at all redshifts employed for this study (and not to theplotted points). We find that our chosen parameterisationforms an excellent fit to the individual fits presented withcoloured points, validating our assumed form for the red-shift dependancies of each parameter. The resulting pa-rameters describing our full scale-dependent bias model,as described by Equations 2 (under the assumption thatγ=1), 3 and 4 are presented in Table 2 for both the FoF
and substructure halos of our simulation‡. The quality offit across the whole range of redshifts and masses used toconstrain this model are presented in Figure 7. Over thevast majority of the probed mass and redshift range, thequality of fit is very good. At the highest masses, the qual-ity of fit declines presumably due to overly coarse massbinning demanded by the limited volume available to usfor this study.
3.4 Qualitative trends with mass, redshift andhalo type
Several interesting general trends regarding the depen-dence of bias (and its scale dependence) on mass, redshiftand halo type emerge at this point. Commenting first onthe halo mass at which bias becomes scale free (VSF), wesee that for all bias ratios VSF declines with redshift ata similar rate in all cases and in a nearly identical wayfor both FoF groups and substructure. This mass scaleis higher for redshift-space ’boost’ effects however, lead-ing to a significant increase in this mass scale for the totalredshift-space results over that from halo assembly effectsalone. For the full redshift range of our study (z∼<1.2), VSF
is restricted to the range 150[km/s] to 350[km/s]. From
‡ A Python script with the full model and its coefficients
has been made available online at http://gbpoole.github.
Figure 8. A plot comparing the large scale bias (left) and amplitude of the scale-dependence of bias (right) of friends-of-friends
(FoF) halos (solid points) and substructure halos (open points) in the GiggleZ-main simulation. Solid lines show our bias model and
dashed lines show the simulation-calibrated excursion set model of Tinker et al. (2010, TRK; dashed green), the redshift-space biasboost model of Kaiser (1987, K87; dashed red) and the total z-space bias resulting from both (dashed black). Horizontal dotted
lines in the right panel denote the values of sx which yield differences in bias between scales s=100 [h−1 Mpc] and s=∞ of 10%
(sx=1 [h−1 Mpc]) and 20% (sx=2 [h−1 Mpc]).
Equation 3 we can see that this trend in VSF(z) acts todrive an increase in the amplitude of scale dependent bias(sx) with redshift at masses above this range (where scaledependent bias always results in enhanced bias at smallscales) and a suppression of its amplitude on mass scalesbelow it (where scale dependent bias always results insuppressed bias at small scales).
Augmenting these trends in sx driven by the evolu-tion of VSF(z), the mass dependence of sx (given by sVx )also increases with redshift. Interestingly, this is the onlyparameter for which redshift-space contributions to to-tal bias differ between FoF and substructure halos; beingsignificantly higher for substructure, driving an enhancedmass dependence in the total bias as well.
We now focus on our results for large-scale bias(bx). In Figure 8 we illustrate this quantity for all ofour groupings at three redshifts spanning the range ofour study. In this case, we directly compare results forFoF (solid points) and substructure halos (open points).In this figure we also compare our results to the suc-cessful simulation-calibrated excursion set model of TRK(dashed green lines), the redshift-space distortion modelof K87 (dashed red) and the redshift-space model thatemerges from combining the two (dashed black). TheK87 model predicts a redshift-space ’boost’ given by (hisEquation 3.8, cast here in terms of our notation):
b2z=1 +2
3β +
1
5β2 (5)
where β=f/bh with f being the logarithmic derivative ofthe linear growth factor with respect to expansion factorgiven by:
f=d lnD
d ln a(6)
Lastly, we also combine the TRK and K87 models to pro-duce a reference total bias model (dashed black lines).
Over most of the range of masses and redshiftsprobed by our study we find very good agreement be-tween these reference models and our FoF large-scale biasresults. Since the FoF catalogs most straightforwardly re-late to the density structures described by excursion setmodels, this is as expected. At the highest masses and
redshifts, there is a tendency for the TRK model to pre-dict higher real-space biases than our model predicts. It ispossible that the calibration of the TRK model has beenbiased high from the very strong scale dependant bias ofhalos in this regime, but this is difficult to discern sincetheir study is conducted in Fourier space and since it isunclear from the presentation of their analysis what exactscale they have fit to.
Additionally, looking at substructure we find signifi-cant enhancements in our large scale real-space (and byextension, total redshift-space) biases at low redshift. Thisdifference is approximately 20% for Milky Way sized sys-tems (∼220 km/s) at redshift zero and increases with de-clining mass.
Interestingly (but perhaps not unexpected), there isabsolutely no difference between the two halo populationsin terms of their redshift-space bias boosts. We interpretthis similarity as a reflection of the fact that non-linearpairwise velocities are unimportant on the large scalesof our study. Furthermore, there is extremely little red-shift dependence and only a slight mass dependence forbz. We see excellent agreement with the K87 model andinterpret the lack of evolution in the Kaiser Boost as aremarkable cancelling of the effects on β from evolutionin the growth of structure (via evolution in f) and in real-space halo bias (via evolution in bh). We note that thislevel of agreement with the K87 model was also found byMontesano, Sanchez & Phleps (2010, see their table 4) intheir Fourier-space study of bias.
4 SYSTEMATIC BIASES IN GROWTH OFSTRUCTURE MEASUREMENTS
Having developed our full parameterisation of scale de-pendant bias, we seek now to quantify the systematic biasthat results in growth of structure measurements whenthe scale-dependence of bias is not taken into account.This is done by applying an extension of the Fisher ma-trix formalism to our bias model in Fourier space wherecovariance is minimised and measurement uncertaintiesare more straightforwardly modelled. We intend for this
Figure 9. The systematic bias induced in measurements of the growth rate of cosmic structure (∆f) due to an incorrect assumption
of constant galaxy bias (∆fb; solid lines) for several cutoff measurement scales (kmax; for values 0.1,0.2,0.3 [h−1Mpc]−1 in red, blueand magenta respectively) compared to the statistical uncertainty in this measurement (∆fs) for a fiducial survey with volume
1 [h−1Gpc]3 and number density 3×105 [h−1Gpc]−3 (shaded regions; pink, blue and magenta for kmax=0.1,0.2,0.3 [h−1Mpc]−1
respectively).
150 250 350 450 550 6500.0
0.2
0.4
0.6
0.8
1.0
1.2
z
-0.5∆fs
0.0
kmax=0.1
150 250 350 450 550 650
Vmax [km/s]
-4.5∆fs
-4.0∆fs-3.5
∆fs
-3.0∆fs
-2.5∆fs
-2.0∆fs
-1.5∆fs
-1.0∆fs
-0.5∆fs
0.5∆fs
0.0
kmax=0.2
150 250 350 450 550 650
-5∆fs
-4∆fs
-3∆fs
-2∆fs
-1∆fs
1∆fs
0
kmax=0.3
-0.150
-0.120
-0.090
-0.060
-0.030
0.000
0.030
∆fb
1.0 1.5 2.0 2.5 3.0 3.50.0
0.2
0.4
0.6
0.8
1.0
1.2
z -0.5
∆f s
0.0
kmax=0.1
1.0 1.5 2.0 2.5 3.0 3.5
bt
-4∆f s
-3∆f s
-2∆f s
-1∆f s
0
kmax=0.2
1.0 1.5 2.0 2.5 3.0 3.5
-5∆f s
-3∆f s
-1∆f s
1∆f s
0
kmax=0.3
-0.150
-0.120
-0.090
-0.060
-0.030
0.000
0.030
∆fb
Figure 10. The systematic bias induced in measurements of the growth rate of cosmic structure (∆fb) due to an incorrectassumption of constant galaxy bias for several measurement scale cutoffs (kmax; values given in text above each pannel) as a function
of redshift and halo mass (quantified by maximum circular velocity, Vmax; top) or total redshift-space bias (bt; bottom). Blackcontours express this in units of the statistical uncertainty in this measurement (∆fs; dashed contours for negative systematic biases,solid lines for positive systematic biases) for a fiducial survey with volume 1 [h−1Gpc]3 and number density 3×105 [h−1Gpc]−3.Thick solid contours indicate the cases where scale dependent bias vanishes on scales larger than 3 [h−1Mpc] resulting in nosystematic bias in f .
to be an illustration of the effects of scale dependant biason measurements of this sort and caution that our esti-mates here may be somewhat pessimistic. This is becausewe will assume a specific and fixed redshift-space distor-tion model for this calculation whereas fits to data usu-ally marginalise over a velocity-dispersion parameter (σv)which can absorb some of the systematic we present here.Nevertheless, we expect the general trends and effects pre-sented here to be an informative illustration of the cir-cumstances in which systematic bias should be taken intoaccount in growth of structure studies.
4.1 Estimation of systematic bias
To express our bias model in Fourier space, we first com-pute an unbiased 2D power spectrum (P (k, µ), whereµ= cos(θ) with θ being the angle between the line ofsight and a halo’s peculiar velocity vector) by applyingthe K87 redshift-space distortion model to a 1D CAMBpower spectrum:
Pmodel(k, µ) = (bh + fµ2)2PCAMB(k) (7)
We then convert this 2D Fourier space model to configu-ration space using Equation 11 of Reid et al. (2012) whichrelates correlation function multipoles (indexed by `) tothose of its associated power spectrum:
ξ`(s) =i`
2π2
∫P`(k)j`(ks)k
2 dk (8)
and apply our bias model to the result. This is done forboth our scale dependent bias model and a constant biasmodel, yielding (once we convert back to Fourier space)the biased power spectra Pmodel(k, µ) and Psys(k, µ) re-spectively.
For our estimation of systematic bias in f (which wedenote ∆fb) we follow the method of Amara & Refregier(2008). We rewrite their Equation 8 in the following form:
∆fb =∑
(k,µ)
∆Psys(k, µ, f)
σ2P (k, µ)
dPmodel
df(k, µ, f) (9)
with σP (k, µ) being the error in each bin’s measurementgiven by:
σP (k, µ) =P (k, µ) + 1/n√
N(10)
with n being the number density of galaxies and Nthe number of Fourier modes in each bin. The quantity∆Psys=Pmodel−Psys represents the residual systematicmodelling error in the power spectrum. Lastly, dPmodel/dfgives the partial derivative of our model power spectrumwith respect to f . Throughout, we use bin widths of∆k=0.01 [h−1Mpc]−1 and ∆µ=0.1 for sums over k and µrespectively.
4.2 Effects of systematic bias
To evaluate the magnitude of this systematic bias, we ex-press it here for a fiducial survey of volume 1 [h−1Gpc]3
and number density n=3×105 [h−1Gpc]−3. This numberdensity is chosen to be similar to that of both the Wig-gleZ and BOSS Surveys and the volume is representa-tive of current large spectroscopic surveys. In Figure 9 weshow the results of this calculation at three redshifts span-ning the range z∼<1.2 for three small-scale cutoffs (denotedkmax). These are compared in each case to the statisti-cal uncertainty expected for this measurement (denoted
∆fs; shown with shaded regions) which we calculate us-ing a standard Fisher matrix forecast (see White, Song& Percival 2009; Abramo et al. 2012; Blake et al. 2013)using the same binning and range as for the systematicsforecast.
Noting first some generic trends in this figure, we seethat ∆fb is positive for low masses/biases and (more gen-erally) negative for larger masses/biases. This is due tothe transition from S=−1 (suppression of bias on smallscales) to S=+1 (enhancement of bias on small scales)with suppressed small-scale bias leading to a positive biasin f and enhanced small-scale bias (the more commoncase) leading to a negative bias in f . Additionally, we seethat increasing kmax has two distinct effects: it increasesthe precision of the measurement (particularly betweenkmax=0.1 and kmax=0.2 [h−1Mpc]−1) due to the addi-tional data involved and it increases ∆fb due to the useof scales where scale-dependent bias has an increased ef-fect on the shape of the power spectrum.
Commenting more specifically, we can see from thisfigure that when kmax=0.1 [h−1Mpc]−1, ∆fb remains sig-nificantly smaller than ∆fs for all cases with bt∼<2. In-deed, only when bt∼>3 at z∼>1 does the systematic biasbecome significant compared to the precision of the mea-surement. However, this situation dramatically changesfor larger values of kmax. When it increases to 0.2, ∆fbbecomes significant compared to ∆fs for all cases exceptthose very narrowly similar in mass to VSF, where scaledependent bias disappears.
The presentation of these results is expanded in Fig-ure 10 where we show ∆fb for the full range of cases towhich our bias model has been constrained. Across allredshifts and for all cases, we see that scale dependantbias effects are minimised when the halo population hasa bias similar to bt∼1.5.
5 SUMMARY AND CONCLUSIONS
We have used the GiggleZ-main simulation to pro-duce an 8-parameter phenomenological model quan-tifying halo bias (in both real and redshift-spaces)and its scale dependence over the range of masses100[km/s]<Vmax<700[km/s], redshifts z∼<1.2 and scales3[Mpc/h]<s<100[Mpc/h] under the ansatz that bias con-verges to a scale independent form at large scales. We findthat scale dependent bias can either enhance or suppressbias at small scales. For any given halo mass at any givenredshift, large-scale bias is given by a single constant andthe scale dependence of bias is given by two others: a bi-nary parameter determining whether bias is enhanced orsupressed on small scales (S) and a parameter setting itsamplitude (s).
While a relatively small but growing body of liter-ature has looked at scale dependent bias effects in theFourier domain, few recent studies have addressed it inconfiguration space. The results presented in this workshould not only be more directly applicable to obser-vational studies conducted in configuration space, butshould also help provide a basis upon which to build someintuition regarding the scale-dependent bias effects ob-served in Fourier-space studies.
We find several interesting trends (noted and dis-cussed in Section 3.4) which require further study to un-derstand. Most prominent among these is the fact thatscale dependence of bias transitions from the suppressionof bias at small scales for small masses to enhancement
for large masses. It does so in a narrow bias range centredon bt∼1.5 across all redshifts z∼<1.2.
It should be noted that we restrict our study to con-figuration space on scales larger than 3[Mpc/h]. On scaleslower than this, a wide variety of non-monotonic varia-tions in Rx occur (of a character similar to that presentedin figure 4 of Zehavi et al. 2004). In the Fourier domain,these features are likely to have broad spectral contentand more detailed study is required to understand theirinfluence in Fourier space.
Lastly, we compute the systematic biases induced ingrowth of structure measurements in the absence of cor-rections for scale-dependent bias effects. We find that for afiducial survey with volume 1 [h−1Gpc]3 and number den-sity n=3×105 [h−1Gpc]−3 that systematic bias is modestwhen scales only as small as kmax=0.1 are used, exceptfor highly biased halos at high redshift. Once scales asshort as kmax∼>0.2 are utilised, the situation dramaticallychanges with significant systematic biases resulting at allredshifts for biases even just slightly different from bt∼1.5.In realistic analysis where fits are generally marginalisedover a pair-wise velocity dispersion parameter, much ofthis effect is likely to be absorbed into this parameter, re-ducing the problem at the expense of compromising anymeaning given to this quantity. Further study under re-alistic conditions is clearly needed to precisely quantifythese effects on real survey results.
These results suggest that the optimal strategy at allredshifts z∼<1.2 for clustering studies which are dominatedmore by systematic effects than statistical precision (suchas the case of cosmological neutrino mass measurements)is to target bt∼1.5 systems. Fortuitously, the UV-selectedgalaxies targeted by the WiggleZ survey have a large-scalebias similar to this (Blake et al. 2009) for example.
These results reenforce the notion that scale depen-dent bias is particularly significant for studies involvingmeasurements of the shape of two-point clustering statis-tics. We have focused here on growth of structure mea-surements only, but similar analysis (following-on fromthe work of Swanson, Percival & Lahav 2010, for example)for neutrino mass measurements are clearly warranted aswell.
Of course, this study has focused on the bias proper-ties of halo tracers with complete selection properties anduniform masses. The larger bias we find for substructurecatalogs shows the importance of realistically consider-ing the sites of galaxy formation. We now need to care-fully consider the effects that can be induced by the sortsof colour selections employed during observational cam-paigns. Due to phenomena like the morphology-densityrelation, a large variety of differing results can occur ifgalaxies are selected by more observationally motivatedcriteria (e.g. luminosity or colour) which are more diffi-cult to robustly model. A great deal more study on theseissues is required to make robust statements under suchcircumstances.
ACKNOWLEDGEMENTS
We would like to thank Volker Springel for makingGADGET-2 publicly available and for permitting us touse his halo finding code (Subfind) and Aaron Ludlowfor useful comments. We would also like to thank Swin-burne University for its generous allocation of comput-ing time for this project, Jarrod Hurley and the GreenMachine Help Desk for their computing support, and
the Green Machine user community for patiently wait-ing while the GiggleZ-main simulation kept them fromtheir work. We acknowledge financial support from theAustralian Research Council through Discovery Projectgrants DP0772084 and DP1093738. GP and SM acknowl-edge support from the ARC Laureate program of StuartWyithe. CB and CP acknowledges the support of the Aus-tralian Research Council through the award of a FutureFellowship. DC acknowledge the support of an AustralianResearch Council QEII Fellowship.
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