Statistical and systematic errors in redshift-space distortion measurements from large surveys

2.3 MB·English

by  Davide Bianchi

Mon.Not.R.Astron.Soc.000,1–19(2012) Printed18January2013 (MNLATEXstylefilev2.2)
Statistical and systematic errors in redshift-space
distortion measurements from large surveys
D. Bianchi1,2⋆, L. Guzzo2, E. Branchini2,3,4, E. Majerotto2,5,6, S. de la Torre7,
F. Marulli8,9,10, L. Moscardini8,9,10, and R. E. Angulo11
1Dipartimento di Fisica, Universita` degli Studi di Milano, via Celoria 16, I-20133 Milano, Italy
3
2INAF - Osservatorio Astronomico di Brera, Via Bianchi 46, I-23807 Merate (LC), Italy
1
3Dipartimento di Fisica, Universita` degli Studi “Roma Tre”, viadella Vasca Navale 84, I-00146 Roma, Italy
0
4INFN, Sezione di Roma Tre, viadella Vasca Navale 84, I-00146, Roma, Italy
2
5Instituto de F´ısica Te´orica (UAM/CSIC), Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain

n 6Departamento de F´ısica Te´orica (UAM), Universidad Auto´noma de Madrid, Cantoblanco, 28049 Madrid, Spain
a 7SUPA†, Institute forAstronomy, Universityof Edinburgh, Royal Observatory, Blackford Hill, EH9 3HJ Edinburgh, UK
J 8Dipartimento di Astronomia, Alma MaterStudiorum-Universit`a di Bologna, via Ranzani 1, I-40127 Bologna, Italy

7 9INAF - Osservatorio Astronomico di Bologna, viaRanzani 1, I-40127 Bologna, Italy
1 10INFN, Sezione di Bologna, viale Berti Pichat 6/2, I-40127 Bologna, Italy
11Max-Planck Institut fu¨rAstrophysics, D-85748, Garching b. Mu¨nchen, Germany

]
O
C
. Accepted 2012September 9.Received2012September 9;inoriginalform2012March8
h
p
-
o ABSTRACT
r We investigate the impact of statistical and systematic errors on measurements of
t
s linear redshift-space distortions (RSD) in future cosmological surveys by analysing
a large catalogues of dark-matter halos from the BASICC simulation. These allow us
[ to estimate the dependence of errors on typical survey properties, as volume, galaxy

2 densityandmass(i.e.biasfactor)ofthe adoptedtracer.We findthatmeasuresofthe
v specific growth rate β = f/b using the Hamilton/Kaiser harmonic expansion of the
−1
5 redshift-space correlation function ξ(rp,π) on scales larger than 3 h Mpc are typi-
4 cally under-estimated by up to 10% for galaxy sized halos. This is significantly larger
5 than the corresponding statistical errors, which amount to a few percent, indicating
1 the importance of non-linear improvements to the Kaiser model, to obtain accurate
3. measurementsofthegrowthrate.Thesystematicerrorshowsadiminishingtrendwith
0 increasing bias value (i.e. mass) of the halos considered. We compare the amplitude
2 and trends of statistical errors as a function of survey parameters to predictions ob-
1 tained with the Fisher information matrix technique. This is what is usually adopted
: to produce RSD forecasts,based on the FKP prescriptionfor the errorson the power
v
i spectrum. We showthat this produces parametererrorsfairly similar to the standard
X deviations from the halo catalogues, provided it is applied to strictly linear scales in
−1
r Fourierspace(k <0.2hMpc ).Finally,wecombineourmeasurementstodefineand
a
calibrate an accurate scaling formula for the relative error on β as a function of the
sameparameters,whichcloselymatchesthesimulationresultsinallexploredregimes.
This provides a handy and plausibly more realistic alternative to the Fisher matrix
approach, to quickly and accurately predict statistical errors on RSD expected from
future surveys.
Key words: cosmological parameters – dark energy – large-scale structure of the
Universe.
1 INTRODUCTION
Galaxyclusteringasmeasuredinredshift-spacecontainsthe
imprint of the linear growth rate of structure f(z), in the
⋆ E-mail:[email protected] form of a measurable large-scale anisotropy (Kaiser 1987).
† ScottishUniversitiesPhysicsAlliance This is produced by the coherent peculiar velocity flows
©2012RAS
2 D. Bianchi, et al.
towards overdensities, which add an angle-dependent con- ative error on β as a function of survey volume and mean
tribution to the measured redshift. In linear theory, these density. The range of parameters explored in this case was
redshift-space distortions (RSD) in the clustering pattern however limited, and one specific class of galaxies only (i.e.
can be quantified in terms of the ratio β(z) = f(z)/b(z) bias) was analyzed.
(where b is the linear bias of the sample of galaxies con- Thesecondcrucialaspecttobetakenintoconsideration
sidered). A value for β can be obtained by modeling the whenevaluatingFishermatrixpredictions,isthattheyonly
anisotropy of the redshift-space two-point correlation func- considerstatisticalerrorsandcannotsayanythingaboutthe
tionξ(rp,π)(whererp andπ aretheseparationsperpendic- importance of systematic effects, i.e. on theaccuracy of the
ularandparallel tothelineofsight) or,equivalently,ofthe expected estimates. This is clearly a key issue for projects
power spectrum (see Hamilton (1998) for a review). Since aiming at percent or sub-percent precisions, for which sys-
b can be defined as the ratio of the rms galaxy clustering tematic errors will bethe dominant source of uncertainty.
amplitudetothatoftheunderlyingmatter,b σgal/σmass, In fact, a number of works in recent years sug-
themeasuredproductβ σgal isequivalentto≈the8pred8icted gest that the standard linear Kaiser description
combination f(z) σma×ss(8z) (Song& Percival 2009). The of RSD is not sufficiently accurate on quasi-linear
× 8
latter is a prediction dependingon the gravity theory, once scales ( 5 50 h−1Mpc) where it is routinely ap-
≈ −
normalized to the amplitude of matter fluctuations at the plied (Scoccimarro 2004; Tinker, Weinberg, & Zheng
given epoch, e.g. using CMB measurements. 2006; Taruya, Nishimichi, & Saito 2010;
Measurements of the growth rate f(z) are crucial to Jennings, Baugh, & Pascoli 2011). Various non-linear
pinpoint the origin of cosmic acceleration, distinguishing corrections are proposed in these papers, the difficulty
whether it requires the addition of “dark energy” in often being their practical implementation in the anal-
the cosmic budget, or rather a modification of General ysis of real data, in particular in configuration space
Relativity. These two radically alternative scenarios are (dela Torre & Guzzo 2012). One may hope that in the
degenerate when considering the expansion rate H(z) future, with surveys covering much larger volumes, it will
alone, as yielded, e.g., by the Hubble diagram of Type Ia be possible to limit the analysis to very large scales, where
supernova (e.g. Riess et al. 1998; Perlmutteret al. 1999) the simple linear description should be adequate. Still,
or Baryonic Acoustic Oscillations (BAO, e.g Percival et al. ongoing surveys like Wigglez (Blake et al. 2011), BOSS
2010). Although the RSD effect is well known since (Eisenstein et al. 2011) and VIPERS (Guzzo et al., in
long, its important potential in the context of dark en- preparation), will still need to rely on the clustering signal
ergy studies has been fully appreciated only recently at intermediate scales to model RSD.
(Zhanget al. 2007; Guzzo et al. 2008). This led to a true Here, we shall address in a more systematic and ex-
renaissanceofinterestinthistechnique(Wang2008;Linder tended way the impact of random and systematic errors on
2008; Nesseris & Perivolaropoulos 2008; Acquavivaet al. growthratemeasurementsusingRSDinfuturesurveys.We
2008; Song & Percival 2009; White, Song, & Percival shall compare the results directly to Fisher matrix predic-
2009; Percival & White 2009; Cabr´e & Gaztan˜aga 2009; tions,thoroughlyexploringthedependenceofstatisticaler-
Blake et al. 2011),such that RSDhavequicklybecome one rors on the survey parameters, including also, in addition
of the most promising probes for future large dark energy to volume and density, the bias parameter of the galaxies
surveys. This is the case of the recently approved ESA used. This is also relevant, as one could wonder which kind
Euclid mission (Laureijs et al. 2011), which is expected ofobjects would bebestsuited tomeasureRSDinafuture
to reach statistical errors of a few percent on measure- project.Thesewillincludeusinghalosofdifferentmass(i.e.
ments of f(z) in several redshift bins out to z = 2 using bias),uptothosetraced bygroupsand clustersofgalaxies.
this technique (coupled to similar precisions with the Potentially,usinggroupsandclusterstomeasureRSDcould
complementary weak-lensing experiment). beparticularlyinterestinginviewofmassivegalaxyredshift
Ingeneral,forecastsofthestatisticalprecisionreachable surveysasthat expectedfrom Euclid (Laureijs et al.2011),
by future projects on the measurements of different cosmo- which can be used to build large catalogues of optically-
logical parameters havebeen produced through widespread selected clusters with measured redshifts. A similar oppor-
application of theso-called Fisher information matrix tech- tunitywillbeofferedbyfutureX-raysurveys,suchasthose
nique (Tegmark 1997). This has also been done specifically expectedfromtheE-Rositamission(Cappelluti et al.2011),
forRSDestimatesofthegrowth rateandrelatedquantities althoughinthatcase,meanclusterredshiftswillhavetobe
(Wang 2008; Linder 2008; White, Song, & Percival 2009; measured first.
Percival & White2009;McDonald & Seljak2009).Onelim- This paper is complementary to the parallel work of
itationoftheseforecastsisthattheynecessarilyimplysome Marulli et al. (2012), where we investigate the impact on
idealizedassumptions(e.g.ontheGaussiannatureoferrors) RDS of redshift errors and explore how to disentangle ge-
andhavenotbeenverified,ingeneral,againstsystematicnu- ometrical distortions introduced by the uncertainty of the
merical tests. This is not easily doable in general, given the underlying geometry of the Universe – i.e. the Alcock-
largesizeofplannedsurveys.Afirstattempttoproducegen- Paczynski effect (Alcock & Paczynski 1979) – on measure-
eralforecastsbasedonnumericalexperimentswaspresented ments of RSD. Also, while we were completing our work,
by Guzzo et al. (2008), who used mock surveys built from independent important contributions in the same direction
the Millennium simulation (Springel et al. 2005) to numer- appeared in the literature by Okumura& Jing (2011) and
ically estimate the random and systematic errors affecting Kwan, Lewis, & Linder (2012).
theirmeasurementofthegrowthratefromtheVIMOSVLT The paper is organized as follows. In § 2 we describe
DeepSurvey.Usingagridofreferencesurveyconfigurations, the simulations used and the mass-selected subsamples we
theycalibratedanapproximatedscalingrelationfortherel- defined; in § 3 we discuss the technical tools used to esti-
©2012RAS,MNRAS000,1–19
Errors in redshift-space distortion measurements 3
mate and model the two-point correlation function in red- Ncut Mcut [h−1 M⊙] Ntot n[h3 Mpc−3]
shift space, ξ(rp,π), and to estimate the intrinsic values of 20 1.10×1012 7483318 3.11×10−3
biasanddistortiontobeusedasreference;in§4wepresent 30 1.65×1012 4897539 2.04×10−3
themeasuredξ(rp,π)andshowtheresultingstatistical and 45 2.47×1012 3158088 1.31×10−3
systematicerrorsonβ,asafunctionofthehalobias;herewe 63 3.46×1012 2164960 9.00×10−4
discussindetailhowwellobjectsrelatedtohigh-biashalos, 91 5.00×1012 1411957 5.87×10−4
as groupsand clusters, can beused tomeasure RSD;in §5 136 7.47×1012 866034 3.60×10−4
we organise all our results into a compact analytic formula 182 9.99×1012 597371 2.48×10−4
as a function of galaxy density,bias and survey volume; we 236 1.30×1013 423511 1.76×10−4
310 1.70×1013 290155 1.21×10−4
then directly compare these results to the predictions of a
364 2.00×1013 230401 9.58×10−5
Fishermatrix code;finally wesummarizeourresultsin §6.
455 2.50×1013 165267 6.87×10−5
546 3.00×1013 124497 5.17×10−5
Table 1.Properties of the halo catalogues used inthe analysis.
2 SIMULATED DATA AND ERROR
NcutisthethresholdvalueofNpart,e.g.thecatalogueNcut=20
ESTIMATION isthesetofgroups(i.e.halos)withatleast20DMparticles;Mcut
isthecorrespondingthresholdmass;Ntot isthetotal numberof
2.1 Halo catalogues from the BASICC
halos (i.e. the number of halos with Mhalo > Mcut); n is the
simulations number density (i.e.n=Ntot/V,where V =13403 h−3Mpc3 is
thesimulationvolume).
The core of this study is based on the high-resolution
Baryonic Acoustic-oscillation Simulations at the Institute
for Computational Cosmology (BASICC) of Angulo et al.
at z = 1, from which we select sub-samples with different
(2008), which used 14483 particles of mass 5.49
1010h−1M⊙ to follow thegrowth of structurein dark ma×t- massthresholds(i.e.numberofparticles).Thiscorresponds
ter in a periodic box of side 1340h−1Mpc. The simulation to samples with different bias values. Table 1 reports the
main features of these catalogues. In the following we shall
volumewas chosentoallow forgrowth of fluctuationstobe
modelledaccuratelyonawiderangeofscalesincludingthose refer to a given catalogue by its threshold mass Mcut (i.e.
the mass of the least massive halo belonging to that cata-
of BAO.Theverylarge volumeof thebox also allows usto
logue).Wealsousethecompletedarkmattersample(here-
extract accurate measurements of the clustering of massive
afterDM),includingmorethan3 109 particles1.Foreach
halos. Themass resolution of thesimulation is high enough ×
catalogue,wesplitthewhole(cubical)boxofthesimulation
to resolve halos that should host the galaxies expected to
into N3 sub-cubes (N = 3 unless otherwise stated).
be seen in forthcoming high-redhift galaxy surveys (as e.g. split split
Eachsub-cubeideallyrepresentsadifferentrealizationofthe
Luminous Red Galaxies in the case of SDSS-III BOSS).
sameportionoftheUniverse,sothatweareabletoestimate
Thecosmologicalparametersadoptedarebroadlyconsistent
theexpectedprecisiononaquantityofcosmologicalinterest
with recent data from the cosmic microwave background
through its scatter among the sub-cubes. Using N = 3
andthepowerspectrumofgalaxyclustering(S´anchezet al. split
is a compromise between having a better statistics from a
2006): the matter density parameter is ΩM = 0.25, the larger number of sub-samples (at the price of not sampling
cosmological constant density parameter Ω = 0.75, the
Λ someverylargescales),andcoveringevenlargerscales(with
normalization of density fluctuations, expressed in terms of
their linear amplitude in spheres of radius 8h−1Mpc at the Nsplit = 2), but with fewer statistics. In general, there are
large-scalemodessharedbetweenthesub-cubes.Asaconse-
present day σ8 =0.9, the primordial spectral index ns =1, quence,ourassumptionthateachsub-samplecanbetreated
thedark energy equation of state w= 1, and thereduced
Hubble constant h = H /(100kms−1M−pc−1) = 0.73. We as an independent realization breaks down on such scales.
0 To overcome this problem, we limit our analysis to scales
note the high value of normalization of the power spec-
much smaller than thesize of thesub-cubes.
trum σ , with respect to more recent WMAP estimates
8 Thisanalysisconcentratesatz=1,becausethisiscen-
(σ = 0.801 0.030, Larson et al. 2011). This has no ef-
8 ± tral to the range of redshifts that will become more and
fect on the results discussed here (but see Angulo& White
more explored by surveys of the next generation. This in-
(2010)foramethodtoscaleself-consistentlytheoutputofa
cludes galaxies, but also surveys of clusters of galaxies, as
simulationtoadifferentbackgroundcosmology).Outputsof
thosethatshouldbepossiblewiththeeRositasatellite,pos-
theparticlepositionsandvelocitiesarestoredfromthesim-
siblyduetolaunchin2013.Exploringtheexpectationsfrom
ulations at selected redshifts. Dark matter halos are identi-
RSD studies using high-bias objects, corresponding e.g. to
fiedusingaFriends-of-Friends(FOF)percolation algorithm
groups of galaxies, is one of the main themes of this paper.
(Daviset al. 1985) with a linking length of 0.2 times the
meanparticleseparation.Positionandvelocityaregivenby
the values of the center of mass. In this paper, only groups
1 Suchanumberofpointsinvolvesverylongcomputationaltimes
withalleastNpart =20particlesareconsidered(i.eonlyha-
loswithmassMhalo >1.10×1012 h−1 M⊙).Thislimitpro- wcohmene ctahlicsuplartoibnlge,me,.gw.,eaotfwteon-puosinetacosrpraerlsaetliyonsafumnpclteidons.uTbo-soevteor-f
vides reliable samples in term of their abundance and clus-
theDMcatalogue. Inordertolimittheimpactofshot-noise,we
tering, which we checkedby comparing thehalo mass func-
nevertheless always keep the DM samples denser than the least
tion and correlation function against Jenkins et al. (2001) densehalocatalogue(i.e.Mcut=1.10×1012 h−1 M⊙).Weveri-
and Tinker et al. (2010) respectively. fieddirectlyonasubsetthatourresultsdonoteffectivelydepend
Weusethecompletecatalogueofhalosofthesimulation onthelevelofDMdilution.
©2012RAS,MNRAS000,1–19
4 D. Bianchi, et al.
2.2 Simulating redshift-space observations where rp and π are the separations perpendicular and par-
alleltothelineofsight,µisthecosineoftheanglebetween
For our measurements we need to simulate redshift-space
the separation vector and the line of sight µ=cosθ =π/r,
observations.Inotherwords,wehaveto“observe”thesim-
i are Legendre Polynomials and ξi are the multipole mo-
ulations as if the only information about the distance of an P
mentsof ξ(rp,π), which can be expressed as
object was given by its redshift. For this purpose we center
thesample(i.e.oneofthesub-cubes)atadistancegivenby 2 1
ξ (r) = 1+ β+ β2 ξ(r) (6)
0 (cid:18) 3 5 (cid:19)
z=1 c
D1 = D(z=1)=Z0 H(z′)dz′ ξ2(r) = (cid:18)43β+ 74β2(cid:19)[ξ(r)−ξ¯(r)] (7)
z=1 c
= Z0 H0 ΩM +ΩΛ(1+z′)3dz′, (1) ξ4(r) = 385β2(cid:20)ξ(r)+ 52ξ¯(r)− 72ξ¯¯(r)(cid:21), (8)
q
where the last equality holds for the flat ΛCDM cosmology where
ofthesimulation.Moreexplicitly,wetransformthepositions 3 r
(Xi,Yi,Zi) of an object in a sub-cube of side L, into new ξ¯ = r3 Z ξ(t)t2dt (9)
0
comoving coordinates
ξ¯¯ = 5 rξ(t)t4dt. (10)
L L r5 Z
6 Xi 6 , 0
− 2 2
The superscript L reminds us that Eq. (5) holds only in
L L
D1− 2 6 Yi 6D1+ 2 , (2) linear regime. A full model, accounting for both linear and
non-linear motions, is obtained empirically, through a con-
L L
6 Zi 6 , volution with the distribution function of random pairwise
−2 2
velocities along theline of sight ϕ(v):
where we arbitrarily choose the direction of the Y axis
fboer ctohnefusteradnswlaitthionthe(Zredrsehpirfetsezn)t.sTahiscoporrodciendauter,e nasostigntos ξS(rp,π)=Z +∞ξS(L)(cid:20)rp,π− v(H1(+z)z)(cid:21)ϕ(v)dv, (11)
−∞
to each object a comoving distance in real space Di = where z is the redshift and H(z) is the Hubble function
X2+Y2+Z2, hence, inverting Eq. (1), a cosmological
i i i (Davis& Peebles 1983; Fisher et al. 1994; Peacock 1999).
(pundistorted) redshift zi. We then add the Doppler contri- We represent ϕ(v) by an exponential form, consistent
bution to obtain the“observed” redshift, as
withobservationsandN-bodysimulations(e.g.Zurek et al.
zˆi=zi+ vr(1+zi), (3) 1994),
c
√2v
where vr is the line-of-sight peculiar velocity. Using zˆi in- 1 − | |
stead of zi to compute the comoving distance of an object ϕ(v)= σ √2e σ12 , (12)
12
givesitsredshift-spacecoordinate. Finally,inordertoelim-
inate the blurring effect introduced at the borders of the where σ12 is a pairwise velocity dipersion. Wenotein pass-
cube, we trim a slice of 10 h−1Mpc from all sides, a value ingthattheuseofaGaussianformforϕ(v)isinsomecases
about three times larger than typical pairwise velocity dis- tobepreferred,ase.g.whenlargeredshiftmeasurementer-
persion. rors affects thecatalogues to be analyzed. This is discussed
in detail in Marulli et al. (2012) Hereafter we shall refer to
Eq. (5) and Eq. (11) as the linear and linear-exponential
model,respectively.Moreover,inordertosimplifythenota-
3 MEASURING REDSHIFT-SPACE tions, we shall refer to the real- and redshift-space correla-
DISTORTIONS tionfunctionsjustasξ(r)andξ(rp,π)respectively,removing
thesubscripts R and S.
3.1 Modelling linear and non-linear distortions
In a fundamental paper, Kaiser (1987) showed that, in the
3.2 Fitting the redshift-space correlation function
linearregime,theredshift-spacemodificationoftheobserved
clustering pattern due to coherent infall velocities takes a We can estimate β (and σ , for the linear-exponential
12
simple form in Fourier space: model) through this modelling, by minimizing the follow-
PS(k,µk)=(1+βµ2k)2PR(k), (4) ing χ2 function overa spatial grid:
where P is the power spectrum (subscripts R and S de- χ2 = 2ln = (yi(jm)−yij)2 , (13)
note respectively quantities in real and redshift space), µk − L Xi,j δi2j
is the cosine of the angle between the line of sight and the
wave vector ~k and β = f/b is the distortion factor, where where L is thelikelihood and we havedefined thequantity
f = ddllooggGa and G is the linear growth factor of density yij =log[1+ξ(rpi,πj)]. (14)
perturbations. Hamilton (1992) translated this result into
configurationspace(i.e.intermsofcorrelation function,ξ): Here the superscript m indicates the model and δi2j repre-
sents the variance of yij. The use of log(1+ξ) in Eq. (14)
ξS(L)(rp,π)=ξ0(r)P0(µ)+ξ2(r)P2(µ)+ξ4(r)P4(µ), (5) has the advantage of placing more weight on large (linear)
©2012RAS,MNRAS000,1–19
Errors in redshift-space distortion measurements 5
101
b=3.18
t
100 10
10-1 bt=3.01 9 ξDM
ξ 10-2 MMcut == 32..0500××11001133MMsun//hh bt=2.81 8 ξ / halo
cut sun
10-3 MMcut == 21..0700××11001133MMsun//hh bt=2.69
cut sun 7
DM
10-4 7
101
b=2.49
t
100 6
b=2.32
t M
ξ 1100--21 MMcut == 19..3909××11001132MMsun//hh bt=2.15 5 ξξ / haloD
cut sun
M = 7.47×1012M /h
10-3 Mcut = 5.00×1012Msun/h bt=1.95 4
cut sun
DM
10-4
101
b=1.80
t
100 3
bt=1.67 M
10-1 ξD
ξ 10-2 MMcut == 23..4476××11001122MMsun//hh bt=1.54 ξ / halo
cut sun
10-3 MMccuutt == 11..6150××11001122MMssuunn//hh bt=1.44 2
DM
10-4
1 10 10 20 30 40 50
r [Mpc/h] r [Mpc/h]
Figure1.Left:thereal-spacecorrelationfunctionsofthehalocatalogues,comparedtothatofthedark-matterparticlesintheBASICC
simulation. Right: the ratio of ξhalo(r) and ξDM(r) for each catalogue, with the resulting best-fit linear bias b2t =ξhalo(r)/ξDM(r)=
const,fittedovertherange10<r<50h−1Mpc.Errorbarscorrespondtothestandarddeviation(ofthemean)over27sub-cubes.
scales(Hawkins et al.2003).However,unlikeHawkins et al. asanextraparameterto(potentially)accountfordeviations
(2003), we simply use the sample variance of yij to esti- from linear theory2.
mate δij (as in Guzzo et al. 2008). We show in Appendix Finally, in performing the fit we have neglected an im-
A that this definition provides more stable estimates of β portantaspect,butforgoodreasons.Inprinciple,weshould
also in the low-density regime. The correlation functions consider that the bins of the correlation function are not
are measured using the minimum variance estimator of independent. As such, Eq. (13) should be modified as to
Landy & Szalay(1993).Wetesteddifferentestimators,such include also the contribution of non-diagonal terms in the
as Davis & Peebles (1983), Hewett (1982) and Hamilton covariance matrix, i.e. (in matrix form)
(1993),findingthatourmeasurementsarevirtuallyinsensi-
tivetotheestimatorchoice,atleastforr.50h−1Mpc.For 2ln = Y(m) Y TC−1 Y(m) Y , (15)
the linear-exponential model, we perform a two-parameter − L (cid:16) − (cid:17) (cid:16) − (cid:17)
fit, including the velocity dispersion, σ , as a free param-
12
eter. However, being our interest here focused on measure-
mentsofthegrowth rate(throughβ),σ12 istreatedmerely 2 See, for instance, Scoccimarro (2004) for a detailed discussion
aboutthephysicalmeaningofσ12.
©2012RAS,MNRAS000,1–19
6 D. Bianchi, et al.
Wang& Steinhardt 1998)
M = 3.00×1013M / h
cut sun
MMccuutt == 91..9190××11001122MMssuunn // hh β(z)= Ω0Mb.(5z5()z) , (16)
where, f(z)=Ω0.55(z) is the growth rate of fluctuations at
10 M
DM thegivenredshift3.Fortheflatcosmology ofthesimulation
ξ
/ o ΩM(z) is
al
h
ξ ΩM(z)= (1+z)(31Ω+Mz0)+3Ω(M10 ΩM0) . (17)
−
The linear bias can be estimated as
1 b2 = ξhalo(r) . (18)
10 ξDM(r)
r [Mpc / h]
Here ξhalo and ξDM have to be evaluated at large sepa-
Figure 2. The expected the bias factor, expressed as b2 = rations, r & 10h−1Mpc, where the linear approximation
ξhalo(r)/ξDM(r),plotted overawiderrangeofseparationsthan holds. In the following we shall adopt the notation bt and
in the previous figure. Dashed lines are obtained by fitting a
constant bias model over the range denoted by the grey area, βt for the values thus obtained. To recover the bias and
10<r <50h−1Mpc. Error bars give the standard deviation of its error for each Mcut listed in Table 1 we split each cu-
bic catalogue of halos into 27 sub-cubes. Figure 1 shows
themeanoverthe27sub-cubes.
the measured two-point correlation functions and the cor-
responding bias values for the various sub-samples. These
where Y and Y(m) are two (column) vectors containing all arecomputedatdifferentseparations r,astheaverageover
data and model values respectively (with dimension N2, 27sub-cubes,witherrorbarscorrespondingtothestandard
b deviation of the mean. Dashed lines give the corresponding
where N is the number of bins in one dimension used to
b value of b2, obtained by fitting a constant over the range
edsitmimenastieonξ(Nrp2,π))N,w2.hereasCisthecovariancematrix,with 10 < r <t 50h−1Mpc. In most cases, the bias functions
b × b show a similar scale dependence, but the fluctuations are
Thisisroutinelyusedwhenfitting1Dcorrelation func-
compatible with scale-independence within the error bars
tions(e.g.Fisher et al.1994),butitbecomesarduousinthe
case of thefull ξ(rp,π), for which Nb 100 and thecovari- (inparticularforhalo massesMcut 61.70×1013 h−1 M⊙).
ancematrixhas 108 elements.Wha≈thappensinpractice, For completeness, in Figure 2 we show that this remains
≈ valid on larger scales (r & 50h−1Mpc, whereas on small
is that the estimated functions are over- sampled, so that
scales (r . 10h−1Mpc), a significant scale-dependence is
the effective number of degrees of freedom in the data is
present. The linear bias assumption is therefore acceptable
smaller than the number of components in the covariance
for r&10h−1Mpc.
matrix, which is then singular. Still, a test with as many
In a realistic scenario, β is measured from a redshift
as 100 blockwise boostrap realizations yields a very un-
survey. Then the growth rate is recovered as f = bβ.
satisfactory covariance matrix. We tested on a smaller-size
Unfortunately in a real survey it is not possible to esti-
ξ(rp,π) theactual effect of assuming negligible off-diagonal
mate b through Eq. (18) as we described above (and as
elements in the covariance matrix, obtaining a difference of
it is done for dark matter simulations) since the real ob-
a few percent in the measured value of β, as also found
servable is the two-point correlation function of galaxies,
in de la Torre & Guzzo (2012). Part of this insensitivity is
certainlyrelatedtotheverylargevolumesofthemocksam- whereas ξDM cannot be directly observed. A possible so-
lution is to assume a model for the dependence of the
ples, with respect to the scales involved in the parameter
bias on the mass. Using groups/clusters in this context
estimations. This corroborates ourforced choice ofignoring
may be convenient as their total (DM) mass can be esti-
covariancesinthepresentwork,alsobecauseofthecompu-
mated from the X-ray emission temperature or luminosity.
tational time involved in inverting such large matrices, size
We compare our directly measured b with those calculated
multiplied by the huge number of estimates needed for the
fromtwopopularmodels:Sheth,Mo, & Tormen(2001)and
present work.
Tinkeret al. (2010) (hereafter SMT01 and T+10), in Fig-
ure3.DetailsonhowwecomputebSMT01 andbT+10 arere-
3.3 Reference distortion parameters and bias portedintheparallelpaperbyMarulli et al.(2012).Wesee
values of the simulated samples that for small/intermediate masses our measurements are
in good agreement with T+10, whereas for larger masses,
Beforemeasuringtheamplitudeofredshiftdistortionsinthe Mcut & 2 1013 h−1 M⊙, SMT01 yields a more reliable
various samples described above, we need to establish the ×
prediction of thebias.
reference values to which our measurements will be com-
pared, in order to identify systematic effects. Specifically,
we need to determine with the highest possible accuracy
the intrinsic “true” value of β for all mass-selected sam- 3 InthissectionweadoptthenotationΩM =ΩM(z)andΩM0=
ples in the simulation. This can be obtained from the rela- ΩM(z=0),nottobeconfusedwiththenotationΩM =ΩM(z=
tion (Peebles 1980; Fry 1985; Lightman & Schechter 1990; 0)adoptedelsewhereinthiswork.
©2012RAS,MNRAS000,1–19
Errors in redshift-space distortion measurements 7
N
cut
4.5 20 30 45 63 91 136 182 236 310364 455546 729 910 Ncut = 20; Mcut = 1.10 × 1012 h-1 Msun; bt = 1.44
b 10
4 T+10
30
b
SMT01
3.5 b = (ξ /ξ )1/2
t halo DM 20
3
1
2.5 h] 10
b pc / 0
M
2
π [-10 0.1
1.5 -20
-30 0.01
1e+12 1e+13
M [M / h]
cut sun -3 0 -2 0 -1 0 0 1 0 2 0 3 0
r [Mpc / h]
p
Figure3.Comparisonofthebiasvaluesmeasuredfromthesimu-
latedcataloguesasafunctionoftheirthresholdmass,Mcut,with Ncut = 182; Mcut = 9.99 × 1012 h-1 Msun; bt = 2.32
thepredictionsoftheSMT01andT+10models.Thetopaxisalso
reportsthenumber ofparticlesperhalo, Ncut,correspondingto 30
thecatalogue thresholdmass. 10
20
4 SMYESATSEUMRAETMICENETRSROOFRSTHINE GROWTH RATE pc / h] 1 00 1
M
4.1 Fitting the linear-exponential model π [-10
0.1
As in the previous section, we split each of the 12 mass- -20
selected halo catalogues of Table 1 into27 sub-cubes.Then
-30
we compute the redshift-space correlation function ξ(rp,π) 0.01
for each of them. Figure 4 gives an example of three cases
-3 0 -2 0 -1 0 0 1 0 2 0 3 0
ofdifferent mass. Following theproceduredescribed in Sec-
r [Mpc / h]
tion3.2,weobtainanestimateofthedistortionparameterβ. p
The27valuesofβ arethenusedtoestimatethemeanvalue N = 546; M = 3.00 × 1013 h-1 M ; b = 3.18
andstandarddeviationofβasafunctionofthemassthresh- cut cut sun t
old(i.e.bias).Withtheadoptedsetup(binningandrange),
30
thefitbecomesunstableforMcut >3 1013 h−1 M⊙,inthe
× 10
senseofyieldinghighlyfluctuatingvaluesforβ anditsscat- 20
ter. Very probably, this is due to the increasing sparseness
of the samples and the reduced amplitude of the distortion h] 10
(wshinecnetβhe∝m1a/sbs).grFoiwgusr(eto4petxopbliocitttloymshpoawnseltsh)etsheetswhoote-ffneocistes,: Mpc / 0 1
which depends on the number density, increases, whereas π [-10
the compression along the line of sight decreases, since it
-20
dependsontheamplitudeofβ.Forthisreason,inthiswork
0.1
we consider only catalogues below this mass threshold, as -30
listed in Table 1.
Figure 5 summarizes our results. The plot shows the -3 0 -2 0 -1 0 0 1 0 2 0 3 0
mean values of β for each mass sample, together with their rp [Mpc / h]
confidence intervals (obtained from the scatter of the sub-
cubes), compared to the expected values of the simulation
βt (also plotted with their uncertainties, due to the error Figure 4. ξ(rp,π) for the catalogues with Mcut = 1.10 ×
otanintehdeumseinagsutrhede lbiniaesarb-te,xSpeocntieonntia3l.3m).oTdhele,seEqh.av(1e1b),eewnhoicbh- t1r0a1l2pha−ne1l)Ma⊙nd(uMppcuetr=pa3n.e0l0),×M1c0u1t3=h−9.199M×⊙10(l1o2wher−1paMne⊙l).(cIesno--
correlation contours of the data are shown in cyan, whereas the
represents the standard approach in previous works, fitting
over the range 3 < rp < 35h−1Mpc, 0 < π < 35h−1Mpc bsceasltefiatnmdocdoenltcoourrrelespveolnsddsifftoerthinebthlaeckthcruerevepsa.nNelost.eTthhaetlathtteecroalorer
with linear bins of 0.5h−1Mpc. We also remark that here
arbitrarily set to {0.07, 0.13, 0.35, 1}, {0.15, 0.3, 0.7, 2.8} and
the model is built using the “true” ξ(r) measured directly {0.25, 0.5, 1.3, 5} respectively from top to bottom. When the
in real-space, which is not directly observablein thecase of mass grows, the distortion parameter β (i.e. the compression of
realdata.Thisisdoneastoclearlyseparatethelimitations thepattern alongthe lineofsight)decreases, whereasthe corre-
dependingon the linear assumption, from those introduced lationandtheshot-noiseincrease.
by a limited recontruction of the underlying real-space cor-
©2012RAS,MNRAS000,1–19
8 D. Bianchi, et al.
relation function. In Appendix B we shall therefore discuss that obtained with a galaxy-mass sample using the more
separately the effects of deriving ξ(r) directly from the ob- phenomenological linear-exponential model. This may be a
servations. reasonforpreferringtheuseofthismassrangeformeasuring
Despitetheapparentlyverygoodfits(Fig.4),wefinda β.
systematic discrepancy between the measured and the true (iii) Large masses (Mcut &2 1013 h−1M⊙)
×
value of β. The systematic error is maximum ( 10%) for This range corresponds to halos hosting what we may de-
≈
low-bias(i.e.lowmass)halosandtendstodecreaseforlarger scribe as large groups or small clusters. The random error
values (note that here with “low bias” we indicate galaxy- increases rapidlywith mass (Figure6,lower panel),regard-
sized halos with M 1012 h−1 M⊙).Inparticularfor Mcut less of the model, due to the reduction of the distortion
between7 1012 an≈d 1013 h−1 M⊙ theexpectationvalue signal (β 1/b) and to thedecreasing numberdensity.
× ≈ ∝
of the measurement is veryclose to the truevalueβt.
It is interesting, and somewhat surprising, that, al-
though massive halos are intrinsically sparser (and hence 4.3 Origin of the systematic errors
disfavoured from a statistical point of view), the scatter of
The results of the previous two sections are not fully
β(i.e.thewidthofthegreenerrorcorridorinFigure5)does
unexpected. It has been evidenced in a number of recent
notincreaseinabsoluteterms,showinglittledependenceon
papers that the standard linear Kaiser description of
the halo mass. Since the value of β is decreasing, however,
RSD, Eq. (4), is not sufficiently accurate on the quasi-
therelativeerrordoeshaveadependenceonthebias,aswe
linear scales ( 5 50 h−1 Mpc) where it is normally
shall betterdiscuss in § 5. ≈ ÷
applied (Scoccimarro 2004; Tinker, Weinberg, & Zheng
2006; Taruya, Nishimichi, & Saito 2010;
Jennings, Baugh, & Pascoli 2011; Okumura& Jing 2011;
4.2 Is a pure Kaiser model preferable for
Kwan, Lewis, & Linder 2012). This involves not only the
cluster-sized halos?
linearmodel,butalsowhatwecalledthelinear-exponential
Groupsandclusterswouldseemtobenaturalcandidatesto model.SincethepioneeringworkofDavis & Peebles(1983)
trace large-scale motions based on a purely linear descrip- the exponential factor is meant to include the small-scale
tion, since they essentially trace very large scales and most non-linear motions, but this is in fact empirical and
non-linearvelocitiesareconfinedwithintheirstructure.Us- only partially compensates for the inaccurate non-linear
ing clusters as test particles (i.e. ignoring their internal de- description. The systematic error we quantified with our
greesoffreedom)weareprobingmostlylinear,coherentmo- simulations isthusmost plausibly interpretedasduetothe
tions. It makes sense therefore to repeat our measurements inadequacy of this model on such scales. Various improved
using the linear model alone, without exponential damping non-linear corrections are proposed in the quoted papers,
correction. The results are shown in Figure 6. The relative although their performance in the case of real galaxies
error(lowerpanel)obtainedinthiscaseisingeneralsmaller still requires further refinement (e.g. de la Torre & Guzzo
than when the exponential damping is included. This is a 2012). On the other hand, considering larger and larger
consequence of the fact that the linear model depends only (i.e. more linear) scales, one would expect to converge
on one free parameter, β, whereas the linear-exponential to the Kaiser limit. In this regime, however, other diffi-
model depends on two free parameters, β and σ . Both culties emerge, as specifically the low clustering signal,
12
modelsyieldsimilarsystematicerror(centralpanel),except the need to model the BAO peak and the wide-angle
forthelowermasscutoffrangewheretheexponentialcorrec- effects (Samushia, Percival, & Raccanelli 2012). We have
tionclearlyhasabeneficialeffect.Inthefollowingwebriefly exploredthis,although notinasystematicway.Wefindno
summarize how relative and systematic errors combine. To indication for a positive trend in the sense of a reduction
do this we consider three different mass ranges arbitrarily of the systematic error when increasing the minimum scale
choosen. rmin included in the fit, at least for rmin = 20 h−1 Mpc.
Systematicerrors remain present,while thestatistical error
Th(iis)rSamngaellcmorarsessepson(Mdscutot .ha5lo×s 1h0o1s2tinhg−1sMing⊙l)e L∗ galaxies. increases dramatically. The situation improves only in a
relative sense, because statistical error bars become larger
Here the linear exponential model, which gives a smaller
thanthesystematicerror.Thisisseeninmoredetailinthe
systematic error, is still not able to recover the expected
parallel work by de la Torre & Guzzo (2012). Finally, it is
valueofβ.However,anyconsiderationaboutthese“galactic
interesting to remark the indication that systematic errors
halos” may not be fully realistic since our halo catalogues
can be reduced by using the Kaiser model on objects that
are lacking in sub-structure(see Section 4.4).
are intrinsically more suitable for a fully linear description.
(ii) Intermediate masses
(5 1012 .Mcut .2 1013 h−1 M⊙)
× ×
Thisrangecorrespondstohaloshostingverymassivegalax-
4.4 Role of sub-structure: analysis of the
ies and groups. The systematic error is small compared to
Millennium mocks
that oftheothermass ranges, for bothmodels. This means
thatwearefreetousethelinearmodel, whichalways gives In the simulated catalogues we use here, sub-structures in-
a smaller statistical error (lower panel), without having to sidehalos, i.e. sub-halos,are notresolved, duetotheuseof
worry too much about its systematic error, which in any a single linking length when runningthe Friends-of-Friends
case is not larger than that of the more complex model. In algorithm (Section 2.1). As such, the catalogues do not in
particular, we notice that using the simple linear model in fact reproduce correctly the small-scale dynamics observed
this mass range, the statistical error on β is comparable to in real surveys. Although we expect that our fit (limited to
©2012RAS,MNRAS000,1–19
Errors in redshift-space distortion measurements 9
N
cut
20 30 45 63 91 136 182 236 310 364 455 546
0.6
0.5
0.4
β
0.3
lin-exp model; β ± 3σ(β)
lin-exp model; β ± 1σ(β)
β ± 3σ(β)
t t
β ± 1σ(β)
t t
0.2
1e+12 1e+13
M [M / h]
cut sun
Figure 5. The mean values of β averaged over 27 sub-cubes, as measured in each mass sample (open circles) estimated using the
“standard” linear-exponential model of Eq. (11). The dark- and light-green bands give respectively the 1σ and 3σ confidence intervals
aroundthemean.Themeasuredvalues arecomparedtotheexpected values βt,computed usingEqs.(16-18).Wealsogivethe 1σ and
3σ theoretical uncertainty aroundβt,duetotheuncertaintyinthebiasestimate(brownandredbands,respectively).
scalesrp >3h−1Mpc)isnotdirectlysensitivetowhathap- lection function of an IAB <22.5 magnitude-limited survey
pens on the small scales where cluster dynamics dominate, like VVDS-Wide.From each of the 100 light cones, we fur-
we have decided to perform here a simple direct check of ther consider only galaxies lying at 0.7<z <1.3 to have a
whether these limitations might play a role on the results medianredshiftclosetounity.Thecombinationofthesetwo
obtained. Essentially, we want to understand if theabsence setsofsimulationsshouldhopefully provideuswithenough
of sub-structure could be responsible for the enhanced sys- information to disentangle real effects from artifacts.
tematic error we found for thelow-mass halos. Performing the same kind of analysis applied to the
BASICC halo catalogues (Figure 7), we find a comparable
To this end, we further analysed 100 Millennium mock
systematic error, corresponding to an under-estimate of β
surveys. These are obtained by combining the output of
by10%. Werecoverβ =0.577 0.018, against an expected
thepuredark-matterMillennium run (Springel et al. 2005) ±
with the Munich semi-analytic model of galaxy formation value of βt = 0.636±0.006, suggesting that our main con-
clusions aresubstantially unaffectedbythelimited descrip-
(DeLucia & Blaizot 2007). The Millennium Run is a large
tionofsub-halosintheBASICCsamples.Anotherpotential
dark matter N-body simulation which traces the hierarchi-
source of systematic errors in the larger simulations could
cal evolution of 21603 particles between z =127 and z =0
be resolution: the dynamics of the smaller halos could be
inacubicvolumeof5003 h−3 Mpc3,usingthesamecosmol-
unrealisticsimplybecausetheycontaintoofewdark-matter
ogy of the BASICC simulation (ΩM, ΩΛ, Ωb, h, n, σ8)= particles.Ourresultsfrom theMillenniummocksandthose
(0.25, 0.75, 0.045, 0.73, 1, 0.9). The mass resolution,
of Okumura& Jing (2011), which explicitly tested for such
g8a.6la×xie1s08wihth−1aMlu⊙mianllooswitsyoonfe0.1toL∗rewsiotlhveahmailnoismcuomntaoifn1in00g effects, seem however to excludethis possibility.
particles.DetailsaregiveninSpringel et al.(2005).Theone
hundredmocks reproducethegeometry of theVVDS-Wide
“F22”surveyanalysedinGuzzo et al.(2008)(exceptforthe
5 FORECASTING STATISTICAL ERRORS IN
factthatweusecompletesamples,i.e.withnoangularselec-
FUTURE SURVEYS
tionfunction),covering2 2deg2and0.7<z <1.3.Clearly,
×
these samples are significantly smaller than the halo cata- Agalaxyredshiftsurveycanbeessentially characterizedby
loguesbuiltfromtheBASICCsimulations,yettheydescribe its volume V and the number density, n, and bias factor,
galaxies in amore realistic way and allow usto studywhat b, of the galaxy population it includes (besides more spe-
happensonsmallscales.Inaddition,whiletheBASICChalo cific effects due to sample geometry or selection criteria).
catalogues are characterized by a well-defined mass thresh- The precision in determining β depends on these parame-
old, the Millennium mocks are meant to reproduce the se- ters. Using mock samples from the Millennium run similar
©2012RAS,MNRAS000,1–19
10 D. Bianchi, et al.
N
cut
20 30 45 63 91 136 182 236 310 364 455 546
lin model
0.6 lin-exp model
β
SMT01
0.5 β
T+10
β
t
β 0.4
0.3
0.15
0.1
βt
0.05
/

β)t 0
− -0.05
β
( -0.1
-0.15
0.14
lin model
0.12
lin-exp model
βt 0.1

/
) 0.08
β
(
δ 0.06
0.04
1e+12 1e+13
M [M / h]
cut sun
Figure 6. Comparison of the performances of the linear and linear-exponential models. Upper panel: measurements of β from the
different halo catalogues, obtained wth the linear model of Eq. (5) (squares) and the linear-exponential model of Eq. (11) (trianglesl).
Meanvalues anderrorsarecomputed asinFig.5fromthe 27sub-cubes ofeach catalogue. We alsoplot theexpected values of β from
the simulation, βt = f/bt (i.e. β “true”) and from the models of Fig. 3, βT+10 = f/bT+10 and βSMT01 = f/bSMT01. Central panel:
relativesystematicerror.Lowerpanel:relativestatistical error.
to those used here, Guzzo et al. (2008) calibrated a simple by Simpson & Peacock (2010). In this section we present
scalingrelationfortherelativeerroronβ,forasamplewith the results of a more systematic investigation, exploring in
b=1.3: more detail the scaling of errors when varying the survey
parameters. This will include also the dependence on
δ(β) 50
, (19) the bias factor of the galaxy population. In general, this
β ≈ n0.44V0.5
approach is expected to provide a description of the error
While a general agreement has been found com- budget which is superior to a Fisher matrix analysis, as it
paring this relation to Fisher matrix predictions doesnotmakeanyspecificassumption onthenatureof the
(White, Song, & Percival 2009), this formula was strictly errors. All model fits presented in the following sections
valid for the limited density and volume ranges origi- are performed using the real-space correlation function
nally covered in that work. For example, the power-law ξ(r) recovered from the “observed” ξ(rp,π). This is done
dependence on the density cannot realistically be ex- through the projection/de-projection procedure described
tended to arbitrarily high densities, as also pointed out in Appendix B (with πmax = 25 h−1Mpc), which as we
©2012RAS,MNRAS000,1–19