Modeling the angular correlation function and its full covariance in Photometric Galaxy Surveys

0.97 MB·English

by  Martin Crocce

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed31January2011 (MNLATEXstylefilev2.2)
Modeling the angular correlation function and its full
covariance in Photometric Galaxy Surveys
1
Mart´ın Crocce1, Anna Cabr´e2 & Enrique Gaztan˜aga1
1
0
2 1 Institut de Ci`enciesde l’Espai, IEEC-CSIC, Campus UAB, Facultat de Ci`encies, Torre C5 par-2, Barcelona 08193, Spain
2 Centerfor Particle Cosmology, Universityof Pennsylvania, 209, South 33rd Street, Philadelphia, PA, 19104, USA
n
a
J

8 31January2011
2

] ABSTRACT
O
C Nearfuturecosmologywillseetheadventofwideareaphotometricgalaxysurveys,
. likethe DarkEnergySurvey(DES), thatextenttohighredshifts(z ∼1−2)butwith
h
poor radial distance resolution. In such cases splitting the data into redshift bins and
-p using the angular correlation function w(θ), or the Cℓ power spectrum, will become
o the standard approach to extract cosmological information or to study the nature
r of dark energy through the Baryon Acoustic Oscillations (BAO) probe. In this work
st we present a detailed model for w(θ) at large scales as a function of redshift and
a bin width, including all relevant effects, namely nonlinear gravitational clustering,
[
bias,redshiftspace distortionsandphoto-z uncertainties.We alsopresenta modelfor

2 the full covariance matrix characterizing the angular correlation measurements, that
v takes into accountthe same effects as for w(θ) and also the possibility of a shot-noise
0 componentandpartialskycoverage.ProvidedwithalargevolumeN-bodysimulation
4 from the MICE collaboration we built severalensembles of mock redshift bins with a
6 skycoverageanddepthtypicalofforthcomingphotometricsurveys.Themodelforthe
4
angular correlationand the one for the covariance matrix agree remarkably well with
.
4 the mock measurements in all configurations. The prospects for a full shape analysis
0 of w(θ) at BAO scales in forthcoming photometric surveyssuch as DES are thus very
0 encouraging.
1
: 1 INTRODUCTION pling of baryon and photons prior to recombination known
v
astheBaryon AcousticOscillations (BAO).TheBAOhave
i
X The statistical analysis of the distribution of structure at alreadybeendetectedinthespectroscopicsamplesofLumi-
largeastronomical scaleshasplayedakeyroleinadvancing nousRedGalaxies(LRGs)inbothSDSSand2dFGRS(Cole
r
a the field of Cosmology over the last 20 years. From shap- etal. 2005;Eisensteinetal. 2005),andstudiedintheearly
ing our understanding of complex processes driving galaxy imagining data of SDSS(Padmanabhan et al. 2007).
formation and evolution to constraining the energy density Buttheobservationalquesthasonlystarted.Severalof
content of the Universe. the next-generation surveys will gain in area and depth, in
The completion of large extra-galactic surveys such at exchange for a poorer determination of radial positions. In
the Sloan Digital Sky Survey (SDSS, York et al. 2000) turnthisimposestheneedforangularclusteringanalysisin
and the 2dF Galaxy Redshift Survey (2dFGRS, Colless redshiftbinsofwidthfewtimeslargerthatofthephotomet-
et al. 2003) have bolstered our general knowledge in the ricerroruncertaintyatthegivenredshifts.Thedifficultylies
field. Particularly more so when combined with the precise in that the projection in redshift bins lowers the clustering
measurements of theCosmic Microwave Background or the amplitude,erasinganyparticularfeatureandincreasingthe
increasingly reach data from Supernova data (S´anchez et noise-to-signal ratio. The achievable precision of our pho-
al. 2009; Percival et al. 2010; Reid et al. 2010; Komatsu tometrically estimated redshift will play a crucial role. We
et al. 2010) . One of the most promising, and eventually thusneedtounderstandwhataffectstheangularclustering
rewarding,challengesforthefieldoflargescalestructureto- pattern more severely.
dayistheprospectfordeterminingwhatdrivesthelatetime The aim of our work is to tackle this problem, pro-
acceleration of the Universe (Riess et al. 1998; Perlmutter viding a well calibrated model for the clustering signal at
et al. 1999). This isprobed bythepresence, in thecluster- large-scales as a function of angle, radial distance and bin
ing pattern of galaxies, of remanent features from the cou- width,deepeningtheavailableliteratureinthesubject(e.g.
(cid:13)c 0000RAS
2 Crocce et al.
Padmanabhanetal. 2007,Blakeetal. 2007andreferences whereθ istheanglebetweendirectionsnˆ andnˆ+θˆ,related
therein). We put particular effort in stressing the most rel- to thepair-separation through
evanteffects,redshift distortions andphoto-zuncertainties, r (θ)= r(z )2+r(z )2 2r(z )r(z )cos(θ) 1/2 (3)
and how they interplay. 12 1 2 − 1 2
(cid:8) (cid:9)
Anequallyimportantproblemistohavethecapability and r(z) is the co-moving distance toredshift z given by
of estimating the full errors of the measurements. We thus z c
provideawelltesteddescriptionofthecomplexerrormatrix r(z)= du, (4)
Z H(u)
characterizingthemeasurementsofthecorrelation function 0
in real situations, i.e. including effects of partial sky cover- where H(z)/H0 = Ωm(1+z)3+ΩDE(1+z)3(1+w) is the
age, photo-z,redshift distortions, bias and shot-noise. Hubble parameter,pΩm and ΩDE = 1 Ωm are the matter
−
Both, themodelforthecorrelation andtheonefor the and dark energy densities respectively, and w is the dark
error matrix, will be extensively tested against a very rich energy equation of state 6. Since we are interested in red-
set of mocks redshift bins. This work should therefore be shift binsandnot in extendedselections wecan neglect the
relevant for ongoing projects that use photometric redshift growthevolutionwithinthebinandsimplyevaluatethe3-d
estimates liketheDarkEnergy Survey1 (DES),thePhysics correlation in some fiducial redshift z¯ (e.g. the mean red-
of the Accelerating Universe collaboration2 (PAU) and the shift of the bin,weighted by φ).In addition we will assume
the Panoramic Survey Telescope and Rapid Response Sys- a local and linear bias relation between fluctuations in the
tem3 (PanStarrs).Butalsoforupcomingimagingproposals tracer (e.g. galaxies) and matter density field, δg = b(z)δ
such as the Large Synoptic Survey Telescope4 (LSST) and (see Sec. 5.2 for a justification). Under these assumptions
theESA/Euclid5 survey. Eq.(2) is converted to
This paperis organized as follows. In Sec.2 and Sec. 3
w(θ)= dz f(z ) dz f(z )ξ(r(z ),r(z ),θ,z¯) (5)
wediscussthemodelsproposedinthisworkfortheangular Z 1 1 Z 2 2 1 2
correlation function and its full error matrix respectively.
where f(z) b(z)φ(z) and ξ is the matter 3-d correlation
In Sec. 4 we describe the set of ensembles of mock redshift ≡
function.
bins implemented using a large volume N-body simulation.
Hence,inordertopredictw(θ)weneedamodelforthe
In Sec. 5 and 6 we test the models against the mocks, un-
spatial clustering accurate in a sufficiently large range of
der different regimes and assumptions. Section 7 contains
scales to allow theprojection in Eq. (5),in particular when
ourconclusionsandfuturelinesofresearch.Wealsoinclude
photo-zerrorsbroadenstheextentoftheradialdistribution
several appendixes. In AppendixA we give a description of
(see (S´anchez et al. 2010) for a empirical parametric fit
ourmodelforthe3-dnonlinearmattercorrelationfunction.
suited for BAO scales).
In Appendix B we study the limitations of the widely used
Inwhatfollowswediscusshowtoincludephoto-zeffects
Limber approximation. Finally, Appendix C gives a brief
andthemodelforspatialclusteringinrealandredshiftspace
note on the covariance of the angular power spectrum in-
that we will use throughout this paper.
duced bypartial sky coverage.
2.1 Photo-z
Weincorporatethewayuncertaintiesinthetrueredshiftpo-
2 AN ANALYTIC MODEL FOR THE
sitions obtained from photometric estimates affect angular
ANGULAR CORRELATION FUNCTION
clustering by means of the radial selection function, follow-
Let us start by considering the projection of the spatial ing Budavariet al. 2003 (see also Ma et al. 2006).
galaxy fluctuations δ (x,z) along a given direction in the The radial selection φ is the probability to include a
g
sky nˆ galaxyinourredshiftbin.Iftheselection ofobjects isdone
accordingtotheirtrueredshifts, thenφisequaltothetrue
δ(nˆ)= dzφ(z)δ (nˆ,z), (1) numberofgalaxiesperunitredshifttimesawindowfunction
g
Z W encoding selection characteristics (e.g. redshift cuts),
where φ is the radial selection function. The angular corre- dN
φ(z)= g W(z). (6)
lationfunctionisthenobtainedasasimpleprojectionofthe dz
3-d correlation function ξ (Peebles 1973), Instead, if the selection is done according to photomet-
w(θ) δ (nˆ)δ (nˆ+θˆ) = ric redshift estimates, one must incorporate the probability
g g
≡ h i P(zz ) for the true redshift to be z when the photomet-
p
|
= dz1φ(z1) dz2φ(z2)ξgg(r(z1),r(z2),θ) (2) ric one is zp. The ending result is theproduct (Budavari et
Z Z
al. 2003),
dN
φ(z)= g dz P(zz )W(z ), (7)
dz Z p | p p
1 www.darkenergysurvey.org
2 www.pausurvey.org
3 pan-stars.ifa.hawaii.edu 6 Theseexpressions explicitlyassume aflat cosmology and con-
4 www.lsst.org stantw,formoregeneralcasessee(Matsubara 2004)andrefer-
5 www.euclid-imaging.net encestherein
(cid:13)c 0000RAS,MNRAS000,000–000
Modelingthe angularcorrelationfunction and its full covariancein Photometric GalaxySurveys 3
where W(z ) is the photometric redshift window function. thegrowth factor squared D2(z).The second term arises
p
∼
Throughout this paper we will only consider top-hat win- from leading order mode-mode coupling and thus scales as
dow functions both in true and photometric redshifts (i.e. D4(z). In turn the damping of BAO is proportional to
∼
W = 1 within a given redshift range, and 0 otherwise). In theamplitudeoflarge-scale velocityflows,Eq.(A1),andso
addition we will only consider the idealized case where the s D(z).Puttingtheseconsiderations togetherwescale
bao
∼
photometric estimate is Gaussianly distributed around the ourparametric model with redshift as,
true redshift (e.g. Ma et al. 2006). Although this might
befar from reality,it servesas an interesting starting point ξ(r,z) = D2(z)[ξ (r) e−(r/D(z)sbao)2](r)
formorerealisticscenarios(Hearinetal. 2010;Bernstein& Lin,0 ⊗
Huterer 2010).Lastly,werecallthatφshouldbenormalized + AmcD4(z)ξL(1in),0(r)ξL′in,0(r) (10)
to unitywithin the redshift range of interest.
where sub-script 0 means (linear) quantities evaluated at
z = 0. The values for s and A can be taken from a
bao mc
2.2 Spatial clustering and redshift evolution
best-fit analysis to ξ(r) at any given redshift (or to w(θ)
We now turn into the discussion of the 3-d matter corre- in any given redshift bin, after the projection in Eq. (5)).
lation model accounting for nonlinear gravitational effects, In our case will be those from the best-fit at z = 0.3,
redshiftspacedistortions,andthewayweevolveitwithred- sbao = 6.37h−1Mpc and Amc = 1.55. This is detailed in
shift.WepostponetoAppendixAthetestingofthismodel AppendixA,wherewepresentadetailedcomparisonofour
against measurements of 3-d matter clustering in N-body model against numerical simulations, with particular em-
simulations. phasis on thescaling introduced in Eq. (10).
Thelinearevolution oftheclusteringpatternpreserves Lastly, we move to the inclusion of redshift space dis-
its shape but increases the overall amplitude. The main ef- tortions. The true distance to a galaxy differs from the one
fectsduetononlineargravitationalclusteringatlargescales derivedfromitsredshiftthroughtheHubblelawbecauseof
areasmoothingoftheBAOwigglesandariseinclustering theradialpeculiarvelocityofthegalaxyontopoftheHub-
amplitudeabovelinearvaluestowardssmaller scales dueto bleflow(Kaiser 1987).Atlargescales,thecoherentinfallof
mode-couplingeffects(Seo&Eisenstein 2005;Eisensteinet galaxiesintolargeoverdensities,suchasclusters,maketheir
al. 2007; Crocce and Scoccimarro 2008). Although these observed radial separation smaller, squashing the structure
processes can be modeled from first principles (Crocce and along theline-of-sight and boosting theamplitudeof the3-
Scoccimarro 2006a; Matarrese & Pietroni 2008; Matsub- d two point correlation. In this way for separations along
ara 2008; Taruya et al. 2009), it is also possible and de- theline ofsight π < 40h−1Mpcthecorrelation (ornumber
sirable to find simpler parametric approximations. In the ofpairs) increases∼dramatically, whilefor larger separations
correlation function these two effects can be parameterized thecorrelation becomes negativein suchaway that theto-
as (Crocce and Scoccimarro 2008), talnumberofpairsalong thel.o.sispreserved(e.g.Fig.A1
in Gaztan˜aga et al. 2009). This implies that, by dividing
ξ(r)=[ξLin(r)⊗e−(r/sbao)2](r)+AmcξL(1in)(r)ξL′in(r) (8) the data in redshift bins, one is discarding the leverage of
large radial separations effectively increasing the (angular)
wheres andA arefittingparameters,ξ isthelinear
bao mc Lin correlationwithinthebin(seeNocketal. 2010forarecent
correlation function at thegiven redshift,ξ′ itsderivative
Lin and detailed discussion of this effect).
and
Thelinear redshift distortions discussed above,namely
ξ(1) 4π P (k,z)j (kr)kdk. (9) the Kaiser effect, can be easily described assuming the
Lin≡ Z Lin 1 plane-parallel approximation. We will incorporate it into
our modeling of the angular correlation function by writ-
InEq.(8),thesymbol denotesaconvolution.Thismodel
have been already used⊗in the analysis of matter, halo and ing ξ(r1,r2)=ξ(σ,π) in Eq. (2), with (Hamilton 1992),
galaxyclustering(S´anchezetal. 2008;S´anchezetal. 2009).
However the standard approach for analyzing cluster-
ingdatainaphotometricredshiftsurveycoveringfrom low ξ(σ,π)=ξ0(s)P0(µ)+ξ2(s)P2(µ)+ξ4(s)P4(µ), (11)
(z 0.2)tohighredshift(z 1.4)istodividethedatainto
∼ ∼
several redshift bins (whose minimum width are ultimately
determined by the photo-z accuracy, e.g. Padmanabhan et whereπ=r r andσ2=2r r (1 cosθ)(toyields=r )
2 1 1 2 12
− −
al. 2007). If one then performs a joint analysis of all these are the pair-separation along and transverse to the line-of-
binsit isdesirable tohavetheleast numberofnuisancepa- sight, µ = π/s and P are the Legendre polynomials. The
ℓ
rameterspossibleinordertooptimizeconstraintsonderived double integrals in Eq. (2) are still performed in the r , r ,
1 2
cosmological parameters.Fromthispointofviewitisinter- variablesleaving theevaluation of theradial selection func-
esting to investigate to what extent a single set of best-fit tions unchanged.The monopoles of theanisotropic correla-
parameters can be used to describe the 3-d clustering from tion ξ (s) are given in Appendix B2 (Eqs. B10,B11,B12) in
i
low to high redshift, and hence the angular clustering after termsofthe3-dmonopolecorrelationξ(s),thatwewilltake
theprojection in Eq. (5). astheoneincludingnonlineargravitational effectsgivenby
Weimplementthisasfollows.ThefirstterminEq.(8)is Eq.(10).In AppendixB2 we discuss thiseffect in more de-
proportionaltothelinearcorrelation,thereforeitscaleswith tail, also in thecontext of theLimber approximation.
(cid:13)c 0000RAS,MNRAS000,000–000
4 Crocce et al.
3 ANALYTICAL MODELING OF THE to those in C as,
ℓ
COVARIANCE MATRIX OF W(θ)
2l+1 2
Anequallyimportantaspecttotheunderstandingofthesig- Covθθ′ = (cid:18) 4π (cid:19) Pℓ(cosθ)Pℓ′(cosθ′)Covℓℓ′ (15)
nalin clustering analysis of galaxy surveysisthecapability ℓ,Xℓ′≥0
to estimate the corresponding errors in the measurements. where
This is of particular importance for analysis that use cor-
relation functions in Configuration space because the mea- Covθθ′ ≡hw˜(θ)w˜(θ′)i, Covℓℓ′ ≡hC˜ℓC˜ℓ′i, (16)
surements are highly correlated. andw˜(θ)andC˜ denotetheestimatorsusedforw(θ)andC
ℓ ℓ
Notablythereisscarceworkintheliteratureaimingat
respectively. In a full sky situation, and assuming the a
ℓm
developing analytical estimates of the covariance matrix of spectra are Gaussianly distributed, the C˜ measurements
ℓ
angularcorrelationfunctionsbesidestheearlyworkofBern-
areuncorrelated,Covℓℓ′ =Var(Cℓ)δℓℓ′.Inaddition,onecan
stein 1994, who developed an error estimate for theLandy
estimate each ℓ power using the2ℓ+1 available modes,
& Szalay estimator in terms of higher order correlations.
Most “data analysis” papers have relied on sub- 1 ℓ
C˜ a2 (17)
sampling techniques of the data itself , such as jack-knife, ℓ ≡ 2ℓ+1 ℓm
bootstrapandfield-to-fieldvariations(e.g.Rossetal. 2007; mX=−ℓ
Meneux et al. 2009; Sawangwit et at. 2009). However as thus, Var(C ) = 2C2/(2ℓ + 1). Replacing these relations
ℓ ℓ
noted in the comprehensive work of Norberg et al. 2009 backintoEq.(15)leadstothefinalexpressionforCovθθ′ in
all theseapproaches havefailings, at least in 3-dclustering, a full sky survey.
dependingonthewaytheyareimplementedandtheregime Howeveramorerealisticandinterestingscenarioisone
of scales of interest. On the other hand, projection along inwhich theskycoverage ispartial.InCabr´eetal. 2007 it
the line-of-sight alleviates this tension leading to a good was shown, using Gaussian realizations of the a spectra,
ℓm
agreementwiththeoreticalestimates, asshownbyCabr´eet thaterrorsinconfigurationsspacescaleas1/ f (which,
sky
al. 2007inthecontextofcross-correlations betweengalaxy in turn, is the scaling of the available numbepr of harmonic
and CMB maps. modes). In what follows we will assume this scaling, and
In what follows we try to revert the lack of analytical compute the covariance matrix as (Dodelson 2003; Cabr´e
workprovidedthatweareinterestedonlargeangularscales, et al. 2007)
where nonlinear (i.e. non-Gaussian) effects are weaker and
2 2ℓ+1
tthraattewinekdnisocwushsoinwgthoomwotdoelmthodeesligenxapleicttseedlf.eWrreortshuinscaonngcuelnar- Covθθ′ = fsky Xℓ≥0 (4π)2Pℓ(cosθ)Pℓ(cosθ′)(Cℓ+1/n¯)2
clustering, including the effects of sampling variance, shot- (18)
noise,partialskycoverage,photo-zandredshiftdistortions. where we have also included the standard shot-noise con-
We put particular emphasis on the description of the full tribution arising in the variance of the C estimates (Pee-
ℓ
error matrix, and not only the diagonal components, and bles 1973) (n¯ is the number of objects per steradian). We
leaveforfurtherworktheassessment ofpossible systematic remark that the assumption leading to Eq. (18) is not that
effects. the C covariance remains diagonal in a partial sky survey
ℓ
Let us start by decomposing the fluctuations in the butinsteadthatCov(θ,θ′)canbeobtainedfromitsfullsky
number of objects “per pixel” in the sky into spherical co- expression by the scaling 1/f . We discuss this further in
sky
ordinates (Peebles 1973), AppendixC.
To proceed further wethusneed amodel for theangu-
ℓ
lar spectra. In real space the C spectra are given by (see
δ(nˆ)= a Y (nˆ), (12) ℓ
ℓm ℓm
AppendixA)
Xℓ≥0mX=−ℓ
1
wcahlehrearnˆmoisnitchs.eTlihnee-coof-esffiigchitendtisreicntitohnisaenxdpaYnℓmsiontheforsmphetrhie- Cℓ,Exact = 2π2 Z 4πk2dkP(k)Ψ2ℓ(k) (19)
angular power spectrum, where P(k) is the real space matter power spectrum and,
haℓmaℓ′m′i≡δℓℓ′δmm′Cℓ (13) Ψℓ(k)= dzφ(z)D(z)jℓ(kr(z)). (20)
Z
thatcanberelatedtotheangularcorrelationfunctionusing
theAddition theorem 7 yielding, Throughout this paper we will use the linear theory power
spectrum in Eq. (19). We have tested that the inclusion of
2ℓ+1
w(θ)= P (cosθ)C (14) small scale nonlinear effects (or the damping of the baryon
Xℓ≥0(cid:18) 4π (cid:19) ℓ ℓ acoustic features) havenoimpact in ourpredictions for the
errors at the(large) angular scales we are interested in.
where Pℓ are the Legendre polynomials of degree ℓ. The Redshift space distortions are accounted for by follow-
covarianceinthemeasurementsofw(θ)canthenberelated
ing the same procedure that leads to the C expression in
ℓ
Eq. (19) but starting from a power spectrum that includes
theKaisereffectdiscussedinSec.2.2andEq.B8.Thefinal
7 Pℓ(nˆ·nˆ′)= 2ℓ4+π1Pℓm=−ℓ=Yℓm(nˆ)Yℓ⋆m(nˆ) result is simply the following additive contribution to the
(cid:13)c 0000RAS,MNRAS000,000–000
Modelingthe angularcorrelationfunction and its full covariancein Photometric GalaxySurveys 5
kernelin Eq. (20) (Padmanabhan et al. 2007), spacingalongtwocartesianaxisequaltor¯+∆r/2,whilethe
third to √2r¯∆r. Thus we defined N as the integer part of
(2ℓ2+2ℓ 1) z
Ψrℓ(k)=βZ dzφ(z)D(z)(cid:20)(2ℓ+3)(2ℓ−−1)jℓ(kr) Ltoboixts/√ne2ar¯r∆es,twihnitleegNerx9a.nNdotNicyeatshathterorouunnddin-ogff-offofirn¯s+te∆adr/o2f
ℓ(ℓ 1) (ℓ+1)(ℓ+2) takingtheintegerpartcouldleadtoasligthvolumeoverlap
−(2ℓ 1)−(2ℓ+1)jℓ−2(kr)− (2ℓ+1)(2ℓ+3)jℓ+2(kr)(cid:21) (21) inthosedirections.Howeverwehaveexplicitlycheckedthat
−
volumeoverlap neverexceeds 1% in any of themocks.
wherer=r(z).Inturn,photo-zeffectsareincludedthrough
Note that the very large size of MICE7680 is a critical
theradial selection function φ(z),as discussed in Sec. 2.1.
pointinordertohavearobuststatisticsinthewholeredshift
Notice however that the expressions in Eqs. (19,20,21)
range of the assumed survey.
arenumerically expensivetoevaluateduetotheoscillatory
Inordertohaveaclearerunderstandingofthedifferent
behavior of j (x) for x 1. In Appendix B1 we discuss
ℓ
≫ components of the model we built ensembles of mocks in
our own approach to perform these integrals, valid at large
increasing “layers of reality”. We first selected dark-matter
scales and involving thenatural cut-off s in Eq. (8).
bao
particles directly from real space assuming a radial distri-
bution as expected in DES. We also tested, and dismissed,
effectsduetobiasedtracersbyrepeatingthisexercisestart-
4 N-BODY SIMULATION AND MOCK
ingfromhalocatalogues.Nextwemovedparticlestoredshift
SURVEY CATALOGUES
space before doing the selection. Alternatively, we imposed
As discussed previously, we aim at developing and test- a random radial uncertainty in the position of each parti-
ing analytical expressions for the signal, variance and co- cle before selection to mimic photometric error. Finally we
variance of the angular correlation function w(θ) against imposed the radial distribution in addition to redshift dis-
measurements in simulated upcoming photometric surveys. tortions and photometric error to build mocks which are
Hence, in order to have a robust statistics but at the closest to a real survey.
same time be representative of such future surveyswe used Oncetheparticles were selected webuilt angular num-
partitions of a very large N-body simulation, provided by ber density maps in the Healpix format with Nside = 256
the MICE collaboration8, to build survey mocks. The sim- (G´orski et al. 1999)10. This Nside corresponds to 98304
ulation, named MICE7680, tracked the gravitational evo- pixels in 1/8 of sky with an angular resolution of 13.75
lution of 20483 dark-matter particles within an unprece- arc-min. In this way, and given our mass resolution mp =
dented comoving volume of Lbox = 7680h−1Mpc using 3.65×1012h−1M⊙, we obtained mocks with ∼ 0.1−0.5
the Gadget-2 code (Springel 2005). Initial conditions were galaxies per squared arc-min (depending on the bin-width)
set at z = 150 using the Zeldovich dynamics and assum- in the low redshift bins (z = 0.3) and 1 4 galaxies per
i ∼ −
ing a flat LCDM cosmology with parameters Ω = 0.25, squared arc-min at high redshift (z = 1.1). Once the den-
m
Ω = 0.75, Ω = 0.044 and h = 0.7. The spectral tilt was sityfieldwaspixelizedwemeasuredthecorrelationfunction
Λ b
setton =0.95andtheinitialamplitudeoffluctuationsset using the standard estimator (Barriga & Gaztan˜aga 2002;
s
to yield σ =0.8 at z =0. The resulting particle mass was Eriksen et al. 2002)
8
3.65×1012h−1M⊙ (see Fosalba et al. 2008 and Crocce et 1
al. 2010 for furtherdetails). wˆ(θ)= N δiδj, (22)
Without loss of generality we next assumed a survey pairs Xij
covering a continuous 5000deg2 of sky (i.e. a sky fraction
where δ =(n n )/ n is the density fluctuations in the
i i
f = 1/8), and redshift coverage in the range 0.2 < z < −h i h i
sky i-th pixeland N is the numberof pixelpairs.
pair
1.4.Inbroadterms,thismatchesthespecificationsforDES.
In what follows we give a more detailed discussion of
In turn, the fact that redshifts are estimated photo-
the different cases considered, while Table 1 contains a de-
metrically implies that much of the “small-scales” radial
scriptivesummaryofthefinalsuiteofmockensemblesused
information is lost. In such scenario the best approach is
throughout this paper.
to study angular clustering in redshift bins of width larger
or comparable to the mean photo-z error (e.g. Simpson et
al. 2009).Wethusbuiltphoto-zsurveymocksbyextracting
4.1 Real Space Mocks
spherical shells of varying width from “comoving” outputs
of MICE7680. Each shell is restricted to span only one oc- Let us start by describing our mocks in real (or configu-
tant in angular size and its radius r¯(z) is set to match the ration) space. To fairly sample the range 0.2 < z < 1.4
comoving distance to theredshift of theoutput. we select comoving outputs of MICE7680 at the redshifts
Toplacethesphericalshellswithinthesimulationbox- z¯ = 0.3, 0.5, 0.73 and 1.1. In each output we extract all
sizewedefinedobserversin aregulargrid and setthenum- the particles within spherical shells of radius equal to the
berofgrid-pointsineachdirectionN inordertohavenone
i
or minimal volume overlap between different mocks (while
placing the shells in the positive octant of the observer). 9 Exceptwhenweincluderedshiftdistortionsand/orphotomet-
Strictlynon-overlapingmockscanbeachievedbysettingthe ric error where take Nx = Ny = Nz and round off Lbox/r(z⋆)
instead, withz⋆ the redshiftat which the true redshiftdistribu-
tioninEq.(7)isnegligible(seeFig.2)
8 http://www.ice.cat/mice 10 http://healpix.jpl.nasa.gov
(cid:13)c 0000RAS,MNRAS000,000–000
6 Crocce et al.
comoving distance to thegiven redshift. In turn, to be rep-
Real Space Mocks
resentative of typical photo-z errors we set 4 different bin
widthforeachz¯,namely∆z/(1+z)= 0.03, 0.05, 0.1, 0.15. z¯ ∆z r¯ ∆r n¯ Mocks
The comparison of these 16 mock configurations with the (1+z)
approximate σ (z) expected for DES (Banerji et al. 2008)
z 0.3 0.03 845.7 102.65 0.11 1344
is shown in Fig. 1. 0.3 0.10 843.3 342.14 0.33 441
Translated to comoving distance these bins range from 0.3 0.15 839.8 513.20 0.47 392
100 to 500h−1Mpc in width. These spherical shells are re-
stricted tohaveright ascension anddeclination in 0◦ 90◦, 0.5 0.05 1345.8 178.23 0.48 324
therefore covering 1/8 of sky. Lastly, by randomly s−elect- 0.5 0.10 1343.2 356.50 0.88 175
ingparticles within each binwe imposethefollowing radial 0.5 0.15 1338.7 534.83 1.21 150
distribution
0.73 0.05 1859.7 181.50 0.92 104
0.73 0.10 1856.5 363.10 1.68 96
dN/dz (z/0.5)2exp (z/0.5)1.5 , (23)
∝ − 0.73 0.15 1851.1 544.88 2.30 80
(cid:2) (cid:3)
which is also what is expected in DES (we thank DES LSS 1.1 0.10 2558.2 360.07 2.60 36
working group for providing this). 1.1 0.15 2551.9 540.56 3.90 36
The resulting number of mocks depends on z¯ and
∆z/(1+z), but it ranges from tens to thousands, making
Real Space Halo Mocks
thesesetof ensemblesverysuitableforerrorstudiesaswell
asfor testingmodels andsystematics. Thetop panelofTa- z¯ (1∆+zz) Mhalo bias n¯ Mocks
ble 1 includes the main characteristics for 11 of these bins,
whicharetheonesthat forconcretenesswefocuson inthis 0.3 0.15 1013 2.35 4.010−4 392
paper, although our conclusions extend to thefull set. 0.5 0.10 2×1013 2.95 3.210−4 175
0.5 0.10 1014 4.40 2.910−4 175
Redshift Space Mocks
4.2 Real Space Mocks for biased tracers
z¯ ∆z f ≡ ∂lnD Mocks
(1+z) ∂lna
Inordertostudydifferentialfeaturesintheangularcluster-
ingofbiasedtracersascomparedtothatofdark-matterwe 0.5 0.05 0.705 125
also built mocks starting from halo catalogues. 0.5 0.15 0.705 125
We concentrated in two characteristic redshifts and
built mocks in exactly the same manner as described in Photo-z Space Mocks
Sec. 4.1. We note however that from the numerical point
ofviewitisveryhardtoresolvehalosof1012−13h−1M⊙ in z¯ (1∆+zz) σz Mocks
avolumeaslarge astheoneweareconsideringhere, 450
∼
cubic Gpc/h. Thus, to be able to reproduce the mass-scale 0.5 0.05 0.06 125
ofLRGhaloswechoosepoorlyresolvedhalos(orgroups)as 0.5 0.15 0.06 125
tracers. Note that, although poorly resolved, these groups
have the same clustering amplitude and abundance of real Photo-z + Redshift Space Mocks
LRGgalaxies (Cabr´e and Gaztan˜aga 2009).
At z = 0.3 we consider groups of M > 1013h−1M⊙ z¯ (1∆+zz) f σz Mocks
(i.e. 5 or more particles) and bin width ∆z/(1 + z) =
0.15. The spatial abundance of these tracers is n¯ = 1.7 0.5 0.05 0.705 0.06 125
10−4h3Mpc−3. × 0.5 0.15 0.705 0.06 125
Atz =0.5weconsiderhaloswithM >2 1013h−1M⊙
(8 or more particles) with n¯ = 0.49 1×0−4h3Mpc−3.
×
With these selections we try to mimic LRG halos of M
1013h−1M⊙.Atz=0.5 wealso considerclustermass-sca≥le sThaifbtlesh1el.lSsuuibteteonfdiMngoc1k/c8atoaflosgkuyesa.tEaacrhadmiaolcckomcoonvsiinstgodfisatarnecde-
halos of M > 1014h−1M⊙ (35 particles or more), their and width as listed in the top panel. All mocks corresponds to
abundancegiven by n¯ =0.64 10−5h3Mpc−3. dark-matterparticles(exceptrealspacehalomocks)witharadial
×
These mocks are summarized in the second panel of distributiongiveninEq.(23).Themeandistancer¯andwidth∆r
Table 1. Wedonot consider binsat higherredshifts, as the arein h−1Mpc.Thesurfacedensityn¯inparticlespersquarearc-
galaxy bias is expected to be closer to linear and local. In min(and for agiven binis similarforreal, redshiftand photo-z
addition,sincethelinearlocalbiasmodelholdsvalidtothe space).Halo-massesarein h−1M⊙ andDisthe“linear”growth
extent we are able to investigate, see Sec. 5.2, we will next factor.Thephoto-zerrorσz equals 140h−1Mpcforour cosmol-
ogy. The bias reported was obtained from the ratio of angular
concentrateontheeffectsofphoto-zandredshiftdistortions
correlationfunctions(seeSec.5.2,alsoforassociatederrorbars).
on thematter field itself.
(cid:13)c 0000RAS,MNRAS000,000–000
Modelingthe angularcorrelationfunction and its full covariancein Photometric GalaxySurveys 7
4.3 Redshift Space Mocks
To understand the importance of redshift distortions, and
theaccuracy of themodeling, we built mockswhere we im-
posetheradialdistributioninEq.(23)butdisplacethepar-
ticles to redshift space prior to thetop-hat selection.
Weconcentrated inthecomovingoutputof MICE7680
at z = 0.5 since this redshift is the typical mean z for up-
coming photometric surveyssuch as DES or PanStarrs. We
thenidentifiedredshiftbinsofwidth∆/(1+z)=0.05(thin)
and∆/(1+z)=0.15(thick).Ineachredshiftshellthemap-
pingfromrealrtoRedshiftSpacepositionssisgivenbythe
transformation
s=r+v (1+z)/H(z)ˆr (24)
r
whereH istheHubbleparameterandv thepeculiarveloc-
r
ity of the object along the line of sight from the observer.
Thereforegiventheobserveratpositionr (thecenterofthe
0
sphericallredshiftshell)wefirstfindtheparticle’sprojected
velocity along the l.o.s. to the observer, Figure 1. Mocks configurations We built several ensembles of
mock redshift bins covering 1/8 of sky (5000 sq deg) and with
v (r r )
vr = ·r −r 0 , (25) mean redshifts z¯ = 0.3,0.5,0.73,1.1 (shown by the inset ver-
| − 0| tical arrows). For each z¯ we set four different redshift width
then displace it by δr=v (1+z)/H(z) along thel.o.s., ∆z/(1+z)= 0.03,0.05,0.1,0.15(dashedlines,bottomtotopre-
r
spectively).Amoredetaileddescriptionoftheensembleofmocks
δr=δr(r−r0) (26) for each configuration is given in Table I. The aim is to resem-
r r blewithhighstatisticalaccuracythegeometryoflargeareaand
0
| − | deep photometry galaxy surveys, such as the Dark Energy Sur-
and finally do the top-hat selection. It total, we built 125 vey(DES).Theexpectedphoto-zerrorinDES(usinggrizbands)
mockssubtendingoneoctantofangularsizeforeachofthese is shown by the long dashed line, while the solid line shows the
two bin widths (see Table 1). resulting photo-z after adding the JHKs filters from the Vista
HemisphereSurvey(fromBanerjietal.2008).
4.4 Photo-z Space Mocks
Setsofmocksincluding(Gaussian) Photo-zerrorswerealso previousmocks.Dashedlineshowsinsteadthetruedistribu-
produced in almost the same manner, except that the dis- tionofobjects(i.e.asafunctionoftheirtrueredshifts)that
placement along thel.o.s was random with a probability in our photo-z mocks entered the (top-hat) photo-z bins of
∆z/(1+z)=0.15 (left panel)and∆z/(1+z)=0.05(right
1 δr2 panel).
f(δr)= exp (27)
√2πσr (cid:20)−2σr2(cid:21) These radial selection functions were obtained from
Eq. (7) using the underlying distribution from Eq. (23)
where σ = σ c/H(z), and σ is the survey photometric
r z z
(shown by the solid line) and a Gaussian P(zz ) of width
uncertainty at the given z. We only considered σz = 0.06, | p
σ = 0.06, Eq. (27), and have been normalized to unity
which is the nominal value for DES at z = 0.5 using the z
when integrated over redshift. Their importance lies in the
grizphotometricbands(Banerji etal. 2008) as reproduced
factthatthisisalloneneedsinordertocomputethemodel
ourinFig.1.Thisisalsotheexpectedphoto-zaccuracyfor
correlation function, as discussed in Sec. 2. In addition, no-
the3πall-skysurveyofthePan-Starrscollaboration(Caiet.
ticethatwehavechosenmockbinwidthssuchthat∆z σ
al 2009)atthisredshiftusinggrizybandsalone.Inaddition, ∼ z
(i.e.a“narrow bin”of widthcomparable tothephoto-z)or
it is the approximately photo-z precision obtained for the
∆z 4σ (a“broad bin”).
opticalsampleofLRGsselectedfromtheSDSSimagingdata ∼ z
(Padmanabhanetal. 2005;Padmanabhanetal. 2007).This
isthusaveryrepresentativevalueforσ .Forourcosmology
z 4.5 “Survey” Mocks : dNdz, RSD and Photo-z
it translates to an uncertainty in the radial distance of
140h−1Mpc(seeTable1fordetails).Oncethephotometr∼ic Finally we built mocks that include all the aforementioned
errorwasaddedweselectedtheparticlesusingtop-hatcuts effects:arealistic radialdistributionEq.23,redshift distor-
inphotometricredshiftofwidth∆z/(1+z)=0.05and0.15. tions and photometric redshift with uncertainty σ =0.06.
z
The net effect of selecting a sample according to their We again concentrated in the mean redshift z = 0.5 and
photometricredshift,e.g.withatop-hatcriteria,istohavea two bins of width ∆z/(1+z) = 0.05 σ (“narrow”) and
z
∼
broaderdistributionoftrueredshiftsoftheselectedgalaxies. ∆z/(1 + z) = 0.15 4 σ (“wide”). Therefore, these
z
∼ ×
ThisisdepictedinFig. 2wherethesolidlinecorrespondsto are the closest to an actual photo-z survey such as DES
thestandardtop-hatselectionintrueredshiftasdoneinthe or PanStarrs.
(cid:13)c 0000RAS,MNRAS000,000–000
8 Crocce et al.
tionfunction.TheresultingcorrelationsareshowninFig.3
compared with measurements in the mocks at the 4 differ-
ent mean redshifts, z = 0.3, 0.5, 0.73, 1.1 (top to bottom).
Ineachcaseforabinwidthofthesizeofthetypicalphoto-
metricerrorachievableatthegivenredshiftinasurveylike
DES (Banerji et al. 2008).
The agreement between our theoretical model and the
meanofthemeasurementsisexcellentforallconfigurations
tested,seeTable1,inparticularthoseshowninFig.3.And
we recall that we are using only two parameters obtained
from a best-fit to ξ at z =0.3 11. Hence the evolution with
redshiftisacomponentofthemodel.Ineachcasedisplayed
error bars correspond to the error on the mean of the en-
semble (i.e. σ/√N ), that given the large number of
mocks
Figure2.Radial distributionof galaxies vs.trueredshift inour ensemble memberwe count on is remarkably small.
redshift distortions and photo-z mocks. Solid lines arethe radial
The importance of nonlinear effects in front of projec-
distribution of objects for a top-hat selection in true redshift of
tiononesareminorifweconsiderthelargeerrorbarsachiev-
width ∆z/(1+z) = 0.15 (left panel) and ∆z/(1+z) = 0.05
ablein onesinglemock measurement.It ishoweverencour-
(right panel), both centered at z =0.5and assumingaconstant
aging that one is able to model accurately a large range of
spatial density. Dashed lines shows instead the distribution of
objectsasafunctionoftrueredshiftifthesametop-hatselection angular scales, given the mixing of all the distance scales
isnowdoneinphotometricredshifts.InthislatercaseaGaussian involved in the redshift bin projection. We note that the
photometricredshifterrorofσz =0.06isassumed. second term in Eq. (8) does not impact the shape and po-
sition of the BAO, but it does bring theory and mocks in
betteragreement for smaller angular separations.
One interesting result would be to have an estimate of
Before moving on we want to stress that our mocks
are obtained from comoving outputs of an N-body simula- theminimalanglethemodelisabletoreproduce.Aconser-
vative approach to investigate this is to match the smaller
tion and thuscontain all correlations inducedbynon-linear
comovingscaler involvedintheprojectionofagalaxypair
gravitational evolution as well as projection effects, partial s
subtending an angle θ within a redshift bin (where r (z)
skycoverageand realistic radial selection function.Inaddi- min
is thelower limit of theredshift range),
tionsomeaccountforphoto-zand/orRSDeffects.However,
since we do not use light-cone outputs they do not include r =r (z)(2 2cosθ)1/2 (28)
s min
evolutionaryeffectsofthesamplewithintheredshiftbin.We −
argue that, for the narrow bins under consideration, light- with the minimum scale one is capable of modeling in the
cone effects are negligible in front of photo-z and RSD (in 3-d clustering.
thesameway nonlineargravitational effectsare,seeSec5). Interestinglywehavefoundthat,ifwedefiner asthe
nl
To leading order, light-cone effects will introduce a linear scale where themodel departs from the data by some fixed
evolution,whichcanbeestimatedfromEq.(5)byweighting percentage, then our minimum scale satisfies,
theselectionbythecorrespondinglineargrowth(i.e.defined
r (z) r (z=0) D(z) (29)
withrespecttothemeanredshfit),f =D(z,z¯)b(z)φ(z).By nl ∼ nl ×
doing this, we have found no difference with the same cal- and r (0) 20h−1Mpc for a 15% error in ξ. This en-
nl
culation that instead evaluates the growth a the mean red- sures the fol∼lowing minimal angles above which the model
shift of the bin.Nonetheless, thisis certainly an interesting should perform well for the4 cases shown in Fig. 3,θ =
min
subjet that needs to be addressed more properly (i.e. with 1.3◦, 0.7◦, 0.5◦, 0.4◦, in agreement with Fig. 3. But again,
light-cone vs comoving outputs). We leave that for further thisisaconservativelimitbecausetherelativecontribution
work. of scales r to thefull redshift bin projection is minor.
s
∼
5.2 Biased tracers
5 MODEL VS. MOCKS I : THE
CORRELATION SIGNAL Wenowrevisittowhatextentthepossiblepresenceofscale
dependentbias inthespatial correlation function oftracers
In Sec. 2 we discussed in detail our model for the angular
(Smith et al. 2008; Manera and Gazatana˜ga 2009; Des-
correlation function. We now proceed to show how it per-
jacquesandSheth 2010; Desjacquesetal. 2010) translates
forms against clustering measurements in the ensembles of
intotheangularcorrelation function,dependingonthebin-
mock redshift binsdescribed in Sec. 4.
width and mean redshift. Or equivalently, to what extent
theassumptionoflinearbiasholdsintheangularclustering
5.1 Nonlinear Gravity and evolution
Using the model for ξ(r,z) in Eq. (10) we now project into 11 The level of matching does not change if we use instead the
redshiftbinsaccordingtoEq.(5)tofindtheangularcorrela- theoreticalexpectationsfortheseparameterdiscussedinSec.2.2
(cid:13)c 0000RAS,MNRAS000,000–000
Modelingthe angularcorrelationfunction and its full covariancein Photometric GalaxySurveys 9
of halos. For this exercise we thus use the suite of mocks
described in Sec. 4.2.
The top panel of Fig. 4 shows the ratio of correlation
functions(i.e.thebias)measuredinthe392mocksofz=0.3
and ∆z/(1+z)=0.15 for halos M >1013h−1M⊙. Middle
and bottom panels shows the same ratio but from the 175
mocks bins at z = 0.5 and ∆z/(1+z) = 0.1 for masses
M > 2 1013h−1M⊙ and M > 1014h−1M⊙ respectively.
×
For reference the vertical blue arrow shows the position of
theBAO peak in each case.
In all cases the bias is scale independent well within
error bars (corresponding to the standard deviation of the
meanof theensemble) andat the2 3% levelfor thecases
mimickinggalaxyclustering(M 10−13h−1M⊙).Forcluster
∼
mass-scale the shot-noise of the sample is much larger and
consequentlysotheerrorbars.Nonetheless,thereisnotclear
tendency with scale. We recall that our galaxy-type halos
arepoorly resolved.Howeverwehavefoundthat increasing
the resolution (i.e. the minimum number of particles per
halo) and so the typical halo-mass, does not alter the scale
independenceof thebias shown in thetop panels of Fig. 4.
The error bars displayed in the top and middle panels
of Fig. (4) were obtained from the r.m.s of the mocks and
correspond to themean of the ensemble.
For cluster mass-scale halos (bottom panel) it was not
possible to obtain thebias ratio (w (θ)/w(θ))1/2 for every
hh
single mock and angular separation due to the large shot-
noise dominated errors that led to w < 0 in some cases.
hh
Wehavethenestimated thebiasas( w / w )1/2 andthe
hh
h i h i
errors propagating the r.m.s. ensemble errors in the halo
angular auto-correlation w and the matter angular auto-
hh
correlation w as,
δb/b=(1/2) (δw /w )2+(δw/w)2 1/2 (30)
hh hh
(cid:2) (cid:3)
andconvertingtoerrorsonthemeanbyδb δb/√N .
mocks
→
We have tested using the lower mass-scale halos that this
approach of the purely ensemble average leads to the same
bias and errors.
In summary, we conclude that there is no evidence of
scale dependentbiasforthesetracerswithin theerror bars.
5.3 Redshift Distortions and Photo-z
Let us now discuss the impact of redshift distortions and
photometricerrorsin w(θ).Figure5correspondstothean-
Figure 3. Angular Correlation Function measured in 4 ensem- gular correlation measured in the mocks in configuration
bles of mocks z-bins inReal Space compared with our nonlinear
space from Sec. 4.1 (middle green symbols), redshift space
model(solidblue)andlineartheory(reddashedline).Errorbars
from Sec. 4.3 (top red symbols) and photo-z space from
correspond to the uncertainty in the mean, the actual r.m.s. is
Sec. 4.4 (low blue symbols). In solid green, red and blue
N1/2 larger(which isspecifiedinthe insetlabel).We haveex-
mock lines we show the corresponding analytical predictions ob-
ploredmanymoreconfigurationsthanthoseshown(seedetailsin
tainedfromEqs.(5,7,10,B8).Blackdashedlineandsymbols
Table1andSec.4.1)andthesamelevelofagreementwasfound.
depictthemodelandmeasurementswhenaccountingforall
effects at once, with mocks introduced in Sec. 4.5. For red-
shift distortions we used that β = f(z = 0.5) = 0.7047 for
our cosmology (since b = 1) while for photo-z we used the
selection functions shown in Fig. 2.
Error bars displayed in Fig. 5 are true ensemble errors
corresponding to the standard deviation of the mean (i.e.
(cid:13)c 0000RAS,MNRAS000,000–000
10 Crocce et al.
Figure 4. Large Scale Halo Bias (Real Space), in the angular
correlation function for bins at z = 0.3 (top panel) and z =
0.5 (middle and bottom). The bias is linear within few percent,
and well within error bars. Except perhaps in the lower panel,
thatcorrespondstocluster-scalemasshaloswithlargeerrorsfully
dominatedbytheirlowabundance. Seetextformoredetails.
σ/√N ,whereσ is ther.m.svarianceof w(θ)measure-
mocks
ments).Theyincreasefrom 1% 2% at2◦ to 3% 4% Figure 5.Redshift Distortions vs. Photo-z effects:Weshowthe
at 4◦ (theangular BAOscal∼e). He−nce,Fig. 5 cle∼arly s−hows mean correlation over 125 mock measurements, and the corre-
sponding predictions, in configuration (middle green line and
thatthemodeldescribedinthispaperperformsremarkably
symbols), redshift (top red lineand symbols) and photo-z space
well for both thin and thick bins, in configuration, redshift,
(lowbluelineandsymbols).Blacklineandsymbolscorresponds
and/or photo-z spaces. In redshift space it tends to under-
tomocksmeasurementsandmodelwhenalleffectsarecombined
estimate themeasurements for θ<3◦ at the 2% level. together. Redshift distortions (RD) depend on the parameter
∼
Redshift space distortions produce a strong enhance- β = f/b = 0.7047 in our cosmology, while photo-z errors were
ment of the clustering signal (see (Nock et al. 2010) who settoσz =0.06.Hence,toppanelcorresponds toabinofwidth
findsimilarresultsfortheprojectedcorrelation).Thiseffect ∆z ∼ σz while bottom to ∆z ∼ 4σz. RD induces a strong and
is also a strongfunction of scale, becoming moreimportant scaledependentenhancementofthecorrelation,counteractedby
for larger separations. Indeed, it increases the amplitude of thesmearingduetothephoto-z uncertainties. Errorbarsshown
correspondtothestandarddeviationofthemeanoftheensemble.
the BAO bump by as much as a factor of 2.3 for the thin
bin of ∆=0.05(1+z) 180h−1Mpc (top panel of Fig. 5)
and 1.5 for thethick on∼e of ∆=0.15(1+z) 530h−1Mpc
∼
(bottom panel of Fig. 5). amplitudeofw(θ)decreasesby 60%whengoingfromtrue
∼
In turn, photo-z errors have the opposite effect to that to photometric redshifts (green to blue lines in the figure).
of redshift distortions, decreasing the overall amplitude of Inthecaseofawidebin(bottompanelinFig.5),although
angular correlations. This is simply because the projection its width is approximately 4 times the photo-z there is still
in Eq. (5) extends over a much larger range of scales (e.g. a reduction in amplitudeof 30%.
∼
see Fig. 2). Notably, this effect is not far from being scale Thetotalangularcorrelation,includingredshiftdistor-
independent. For the thin bin the impact is a bit stronger tions,photo-z,nonlinearevolutionandbinprojectioneffects
(toppanelofFig.5).Ifweonlyconsiderphoto-zeffects,the is depicted by the dashed line in Fig. 5. In the case of bin
(cid:13)c 0000RAS,MNRAS000,000–000