A direct probe of cosmological power spectra of the peculiar velocity field and the gravitational lensing magnification from photometric redshift surveys

0.2 MB·English

by  Adi Nusser

A direct probe ofcosmologicalpower spectra ofthe peculiar velocityfield and
thegravitationallensing magnificationfrom photometric redshift surveys
Adi Nusser,1,2,∗ Enzo Branchini,3,4,5,† and Martin Feix1,‡
1Department of Physics, Israel Institute of Technology - Technion, Haifa 32000, Israel
2Asher Space Science Institute, Israel InstituteofTechnology - Technion, Haifa32000, Israel
3Department of Physics, Universita` RomaTre, ViadellaVascaNavale84, 00146, Rome, Italy
4INFN Sezione di Roma 3, Via della Vasca Navale 84, 00146, Rome, Italy
5INAF, Osservatorio Astronomico di Brera, Via Brera 28, 20121 Milano, Italy
(Dated:July26,2012)
Thecosmologicalpeculiarvelocityfield(deviationsfromthepureHubbleflow)ofmattercarriessignificant
2
informationondarkenergy,darkmatterandtheunderlyingtheoryofgravityonlargescales. Peculiarmotions
1
ofgalaxiesintroducesystematicdeviationsbetweentheobservedgalaxyredshiftszandthecorrespondingcos-
0
mologicalredshiftsz .Anovelmethodforestimatingtheangularpowerspectrumofthepeculiarvelocityfield
2 cos
basedonobservationsofgalaxyredshiftsandapparentmagnitudesm(orequivalentlyfluxes)ispresented.This

l methodexploitsthefactthatameanrelationbetweenz andmofgalaxiescanbederivedfromallgalaxies
u cos
inaredshift-magnitude survey. Givenagalaxymagnitude, itisshownthatthez (m)relationyieldsitscos-
J cos
4 mologicalredshiftwitha1σerrorofσz ∼ 0.3forasurveylikeEuclid(∼ 109 galaxiesatz . 2),andcanbe
usedtoconstraintheangularpowerspectrumofz−z (m)withahighsignal-to-noiseratio. Atlargeangular
2 cos
separationscorresponding tol . 15, weobtain significant constraintsonthepower spectrumof thepeculiar

velocityfield.At15.l.60,magnitudeshiftsinthez (m)relationcausedbygravitationallensingmagnifica-
] cos
O tiondominate,allowingustoprobetheline-of-sightintegralofthegravitationalpotential.Effectsrelatedtothe
environmentaldependenceintheluminosityfunctioncaneasilybecomputedandtheircontaminationremoved
C
fromtheestimatedpowerspectra. Theamplitudeofthecombinedvelocityandlensingpowerspectraatz∼1
h. canbemeasuredwith.5%accuracy.
p
- PACSnumbers:98.80.-k,98.80.Es
o
r
t
s I. INTRODUCTION galaxydistanceswhichareavailableforonlyasmallfraction
a ofgalaxies. Presently,thenumberofgalaxieswithmeasured
[
Darkenergyandthetheoryofgravitationdictatetheevolu- distancesisseveralordersofmagnitudebelowthatofgalaxies

1 tionoflarge-scalestructureintheUniverse.Thephysicalcon- in redshift surveysused for clustering studies. Furthermore,
v ditionsallowingfortheformationofgalaxies,ultimatelylead although the underlyingpeculiar velocity is an honest tracer
0
to a bias between the distribution of galaxies and the under- ofthegeneralmatterflow,theinferenceofpeculiarvelocities
0
8 lyingmassdensity. However,large-scalemotionsofgalaxies fromobservationaldata is plaguedwith observationalbiases
5 are most certainly locked to the peculiar velocity field asso- [1]. Traditionalpeculiarvelocitycatalogsareexpectedtoim-
. ciatedwith the gravitationaltugof the totalunderlyingmass prove within the next few years, but it remains questionable
7
fluctuations. This assumes that gravity is the only relevant how well observational biases will be controlled, especially
0
2 large-scale force and neglects the contribution from decay- atlargedistances. Analternativeprobeofthepeculiarveloc-
1 inglinearmodes.Despitethebiasbetweenthedistributionof ityfieldmaybeastrometricmeasurementsofgalaxiesbythe
: galaxiesandtheunderlyingmatterfield,theclusteringproper- Gaia space mission [2, 3]. This probe is essentially free of
v
tiesofgalaxieshavebeenthemaintoolfortestingcosmologi- the classic biases contaminating traditional peculiar velocity
i
X calmodels.Currentlyplannedgalaxysurveyswillallowusto measurements,butitisalsolimitedtonearbygalaxieswithin
r quantifythe clustering of galaxieson hundredsof comoving ∼100h−1Mpc.
a
Mpcsandtoevenmeasurecoherentdistortionsofgalaxyim-
Herewe describeamethodforderivingstrongconstraints
ageswhicharisefromgravitationallensingbytheforeground
onthepowerspectrumofthegalaxies’peculiarvelocityfield
matter.
independent of direct distance measurements and any bias-
On the other hand, the peculiar motions of galaxies have
ing relation between galaxies and mass. The method is an
traditionallybeenlesssuccessfulasacosmologicaltool,and
extension of the approaches we have recently proposed [4–
thereareseveralreasonsforthat. Neglectingotherpotentially
6], and it relies on using the observed fluxes of galaxies as
importanteffects,peculiarvelocitiesareapproximatelyequal
a proxy for their cosmological distance [7]. Although this
to the redshifts less the correspondingHubble expansionre-
mostbasic“distanceindicator”isverynoisy,thelargenumber
cession velocities. The latter requiredirectmeasurementsof
of galaxiesavailable in futuresurveyswill allow oneto beat
down this noise to a sufficiently low level. Planned galaxy
redshiftsurveyssuchasEuclid[8,9]willprobethestructure
∗[email protected] of the Universe over thousands of (comoving) Mpcs, com-
†branchin@fis.uniroma3.it prising . 108 − 109 galaxies at z ∼ 1 and beyond. These
‡[email protected] observations will provide redshifts and fluxes, z and f, re-
2
spectively. The observed redshifts deviate from the cosmo- as[10]
logicalredshiftswhichwouldbeobservedinapurelyhomo-
geneousuniverse. Thelargenumberofgalaxiesandthelarge Θ= V(t,r) − Φ(t,r) − 2 t0dt∂Φ[rrrˆr(t),t], (1)
sky coverage of these surveys can be used to derive a mean c c2 c2 ∂t
Zt(r)
globalrelationbetweenthemeanredshiftofagalaxyandits
whererrrˆ is a unit vectoralong the line-of-sightto the object.
apparent magnitude m = −2.5logf + const. We interpret
Here the radial peculiar velocity V and the usual metric po-
this relation as yielding the cosmologicalredshift z (m) for
cos tential Φ are assumed as relative to their present-day values
agivenapparentmagnitudem. Angularpowerspectraofthe
at r = 0 (t = t ). The first and second terms on the right-
difference z − z (m) between the observed redshift z of a 0
i cos i i handsideofEq. (1)aretheDopplerandgravitationalshifts,
galaxyanditsexpectedcosmologicalredshiftz shouldcon-
cos respectively,whilethethirdtermdescribestheenergychange
tainvaluableinformation,mainlyonthepeculiarvelocityfield
of light as it passes througha time-varyinggravitationalpo-
whichisthemaincosmologicalsourceforz −z (m).Inthis
i cos i tential. Notethatthisthirdtermisequivalenttothelate-time
work,wewillshowthatthevelocitypowerspectrumonlarge
integrated Sachs-Wolfe effect experiencedby photonsof the
scales of a few 100 Mpcs could be constrained with signifi-
cosmic microwave background(CMB). In what follows, we
cant signal-to-noise ratio (S/N) at the effective depth of the
survey. Another contributionto z −z (m) results from the will denote the three terms as ΘV, ΘΦ and ΘΦ˙, respectively.
timeevolutionofthegravitationalipoteconstiali alongthephoton Also, we will consider angular power spectra (equivalent to
path, but is significantly smaller than that induced by pecu- angular correlations) of Θ on large scales where the corre-
liar velocities as we will show below. There are two addi- spondingsignalissignificantonlyrelativetotheexpecteder-
tional,indirecteffectswhichmodifytherelationz (m)along ror. Throughoutthispaper,wethereforerelyonlineartheory
a given line of sight. The first effect is related tcoos the envi- ina ΛCDM modelwhereperturbationsonallscales growat
ronmental dependence between galaxy luminosities and the thesame rate. IntroducingD(t) asthegrowthrateoftheun-
large-scale structure in which they reside. Since this depen- derlyingmassdensitycontrastδ=ρ/ρ¯−1,lineartheoryyields
dence is closely connectedto the underlyingdensity field, it thewell-knownrelations
canbeself-consistentlyremovedinouranalysis. Thesecond
δ(t,rrr)= D(t)δ (rrr), (2)
effectiscaused bygravitationallensing magnificationwhich 0
changestheapparentmagnitudesofgalaxiesinagivendirec- D(t)
Φ(rrr,t)= Φ (rrr), (3)
0
tion.Thislattercontributioncouldactuallybeveryrewarding a
since gravitationallensing providesa directprobeof the un- 2aD˙(t) ∂Φ
V(t,rrr)=− 0, (4)
derlyingmassdistribution.Consideringtheanalysispresented 3Ω H2 ∂r
below,wewillthereforetreatitaspartofthesoughtsignal. 0 0
The paper is structured as follows: We begin with a de- where D(t0) = 1. The second relation is obtained from the
taileddescriptionofthemethodanditsapplicationtogalaxy first using Poisson’s equation, i.e. ∇∇∇2rΦ = 3H02Ω0δ/2a. For
redshiftsurveysinSec. II.InSec. III,weconsiderpredictions thespecialcaseΩ0 =1,wehaveD=aandfluctuationsinthe
forthestandardΛCDMmodelanddiscussthemethod’svia- gravitationalpotentialremainconstantwithtime.
bilityaswellasitsexpectedperformance.Finally,wepresent
ourconclusionsinSec. IV.Forclarity,someofthetechnical
A. Cosmologicalredshiftversusapparentmagnitude
material is given separately in an appendix. In the follow-
ing, we adoptthe standard notation. The matter density and
the cosmologicalconstant in units of the critical density are WeaimtoderivethefieldΘsampledatthepositionsofall
denotedbyΩ andΛ, respectively. The scale factora is nor- galaxiesinafluxlimitedredshiftsurveycoveringasignificant
malizedto unity atthe presenttime (t = t ), andthe Hubble regionoftheskytogetherwithalargenumberofgalaxies.As
0
functionisdefinedasH =a˙/a.Further,r =c t0dt′/a(t′)will an example, we consider the planned Euclid redshift survey
t [8,9].InordertoobtainanestimateofΘ foreachgalaxy,we
bethecomovingdistancetoanobjectandzitscorresponding i
R shalluseapparentgalaxymagnitudes(orequivalentlyfluxes)
redshift, assuming a homogeneousand isotropic cosmologi-
asaproxytothecosmologicalredshiftordistance. Although
calbackground. Throughoutthepaper,thesubscript“0”will
refer to quantities given at t = t , and a dot symbol denotes this most trivial distance indicator is very noisy, we will see
0
partialderivativeswithrespecttotimet,i.e. A˙ ≡∂A/∂t. thatthehighnumberofavailablegalaxiesallowsonetobeat
downitsscatter.
Themeancosmologicalredshiftz (m)correspondingtoa
cos
givenapparentmagnitudemis
II. METHODOLOGY z2
z (m)=z¯ = z P(z |m)dz , (5)
cos cos cos cos cos
Zz1
Inaninhomogeneousuniverse,theobservedredshiftzofa andtherootmeansquare(rms)scatteraroundthisrelationis
givenobjectdiffersfromitscosmologicalredshiftz (defined
forthe unperturbedbackground). Therelativediffceosrencebe- z2
σ (m)= (z −z¯ )2P(z |m)dz , (6)
tweentheseredshifts,Θ≡(z−z )/(1+z )canbeexpressed z cos cos cos cos
cos cos Zz1
3
where z and z are the limiting redshifts in the survey, and forallgalaxiesinthesurvey. Notethatinthedenominatorof
1 2
P(z |m)denotestheprobabilitythatagalaxywithmeasured theabove,wehaveusedz insteadofz (m).Thissubstitution
cos i cos
apparent magnitude m has a cosmological redshift z . The is consistent at linear order and motivated by the fact that z
cos i
z (m) relation can be linked to certain characteristics of the isactuallyabetterestimateofthetruez thanz (m)which
cos cos cos
galaxysurvey.Theluminosityofagalaxyatz isL=4πfd2 additionally includes deviations from the actual value of z
cos L cos
whered (z )istheluminositydistancetothegalaxy,andits duetolargerandomerrorswithanrmsofσ .
L cos z
absolute magnitude is defined as M = −2.5logL+const = Environmentaldependencesof the luminosity distribution
m − 5logd . Let Φ(M) be the underlying luminosity func- onthelarge-scaledensityfieldmaysystematicallyshiftallΘ
L i
tion suchthatΦdM is the numberdensityof galaxieswithin for galaxies lying in the direction of a certain line of sight.
themagnitudeinterval[M,M+dM]. Generally,thefunction However, we will show below that the resulting signal con-
Φ(M)dependsoncosmictime,butinfavorofasimplifiedde- tamination is small and in any case, it can be removedfrom
scription, we will assume for the momentthat it varies little the correlations since information on the underlying density
throughoutthedepthofthesurveyconsidered. Notethatitis field attherelevantscaleswill bedirectlyavailablefromthe
trivialtoincludeatime-dependentevolutionaryterminΦ(M). observations(seeSec. IIC).
TheprobabilityP(z |m)satisfies
cos
P(z |m)∝ P(m|z )n(z ), (7) 1. ExpectationsforEuclid
cos cos cos
wheren(z )istheunderlyingmeannumberdensityofgalax-
cos Asatestcaseforafuturesurvey,weconsiderEuclidwhich
iesat z . In the absenceofgalaxypopulationevolution,we
cos aimsatmeasuringphotometricandspectroscopicredshiftsof
have
galaxieswithz ∼ 1over15,000deg2. Moredetailsonthe
n(z )∝d (z )2ddc(zcos), (8) Euclid missioncocsan be found in [9]. Here we focus on the
cos c cos dz photometric redshift survey since we aim at good statistics
cos
rather than precise redshift measurements. Photometric red-
where d is the comoving distance from the observer to z .
The relaction in Eq. (7) can be easily derived using Bayecoss’ shiftswillbemeasuredwithanerrorofσphot ≤0.05(1+zphot)
for ∼ 30 galaxies per arcmin2, with z ≥ 0.7 and magni-
theoremwhichyields cos
tudesinthebroadR+I+Zband(550-920nm)RIZ ≤ 24.5.
AB
P(z |m)P(m)= P(m|z )P(z ), (9) Estimatesofthephotometricredshiftswillrelyonthreenear-
cos cos cos infrared(NIR) bands(Y, J, and H in the range 0.92-2.0µm)
where P(m)isanobservedquantity, P(z )isproportionalto forobjectswithY ≤ 24, J ≤ 24,andH ≤ 24. These
cos AB AB AB
n(z ),andP(m|z )isrelatedtoΦ(M)through willfurtherbecomplementedbyground-basedphotometryin
cos cos
thevisiblebandsderivedfrompublicdataorthroughengaged
Φ(M)
P(M|z )= . (10) collaborations.
cos Ml(zcos)Φ(M)dM TheexpectednumberdensityofEuclidgalaxieswithmea-
−∞
sured photometric redshift can be parametrized as follows
HereMl =ml−5logdL(zcosR)istheabsolutemagnitudewhich [11]:
corresponds to the limiting apparent magnitude m of the
l
galaxysurvey. n(zphot)∝z2phote−(zphot/z0)3/2, (12)
Note that the scatter of z about the mean relation is not
cos wherez =z /1.412isthepeakoftheredshiftdistribution
Gaussian. However, the associated 1σ error in the derived 0 mean
andz themedian. Herewe assume thatz = 0.9. We
angular power spectra depends only on the rms quantity σ . mean mean
z
have compared the expected distribution n(z) of Eq. (12) to
Moreover,thecentrallimittheoremimpliesthattheerrorsin
that measured from galaxies with H ≤ 24 and RIZ ≤
thepowerspectratendtobeGaussian.Consideringtheactual AB AB
24.5 in the zCOSMOS catalog [12–14]. Indeed, we have
observations,thetwoquantitiesz (m)andσ (m)canbeesti-
cos z foundthatEq. (12)doesprovideagoodfittothedatainthe
matedfromthefullsurveybydividingthedatainmagnitude
range0.7<z <2.0whichwewillconsiderinouranalysis.
binswithoutactuallycomputingP(z |m). Inthiscase,σ (m) phot
cos z The observed z − H relation of zCOSMOS galaxies
willincludeapositivecontributionfromthecosmologicalde- phot
hasbeen used to derivez¯ (H) and σ (H) in differentH-
viations,butthisisoverwhelmedbyboththeintrinsicscatter phot zphot
magnitude bins. To match Euclid constraints, we have only
in the z (m) relation and the uncertainties in the photomet-
cos considered galaxies with RIZ ≤ 24.5, H ≤ 24, and
ric redshifts. The underlying assumption is that the sought AB AB
0.7 < z < 2.0. The additionalconstraintsY ≤ 24 and
cosmologicaldeviationsbetweenobservedandcosmological phot AB
J ≤ 24 do not significantly modify our results and have
redshiftscanceloutwhenallgalaxiesoftheentiresurveyare AB
therefore not been enforced. Our results for the zCOSMOS
used. Clearly, a global constant mode should persist in this
dataareshowninFig. 1. The(blue)solidcurveandthe(red)
procedure,butweshallneglectitinthispaper,assumingthat
dashedcurverepresenttheexpectedz (H)andσ (H)forEu-
itdoesnotaffectmodesinthepowerspectraonsmallerscales. cos z
clidgalaxies,respectively. Sinceerrorsonthemeasuredred-
Onceweobtainz (m),wecompute
cos shift, i.e. σ , dominatethe scatterforH< 20, theywillbe
phot
z −z (m) added in quadrature to determine the effective scatter in the
Θ = i cos (11)
i 1+z z −H relation.
i cos
4

1.2
1
0.8
z
cos
z
0.6 σ /(1+z)
z
0.4
0.2
0
18 19 20 21 22 23 24
H
FIG.1. Themean relationz (m) andthecorresponding rmsscatter σ(m)(1+z) for Euclidgalaxies withRIZ ≤ 24.5, H ≤ 24, and
cos z AB AB
photometricredshiftsintherange0.7<z <2.0:TheshownresultsarebasedonzCOSMOSdata.
phot
B. Sphericalharmonicsdecompositionfordiscretenoisydata observedpartofthesky,andthenumberoftermsinthesumof
thesecondrelationis(2l+1)f . Asisobvious,theserelations
sky
Asusual,angularpowerspectraaredefinedintermsofthe shouldbeunderstoodtoholdintheapproximatesense[17].
spherical harmonics Y (rrrˆ). For an all-sky continuous field Considering a function f(rrrˆ) with limited sky coverage in
lm
f(rrrˆ),thedecompositionis thecontinuouslimit,wedefine
1
flm = dΩf(rrrˆ)Ylm(rrrˆ), flm = f1/2 dΩf(rrrˆ)Ylm(rrrˆ), (15)
Z sky Z
∞ +l (13)
wheretheintegrationisagaintakenovertheobservedregion
f(rrrˆ)= f Y∗ (rrrˆ),
lm lm only.TheangularpowerspectrumC isdefinedasthevariance
l
Xl=0 mX=−l ofthe flm’sandgivenby
Inourcase,however,wehavetodealwithpartialskycoverage
C =<|f |2 > , (16)
(around30%oftheskyforEuclid)andconsiderthatthefield l lm ens
is sampled at discrete points given by the galaxy positions. wheretheaverageistakenovermanydifferentrealizationsof
The limited sky coveragecould formallybe describedby an
a field with the same power spectrum as f. For brevity, we
appropriatemaskingofthesphere[15,16]. However,thede- willusethesymbol<·>withoutanysubscripttorefertothis
scriptionintermsofmaskswillunnecessarilycomplicatethe
kind of averaging. Up to cosmic variance, the values of C
l
notationandsomewhatobscurethephysicalinterpretationof obtainedfrom the ensemble average < |f |2 > should equal
lm
the results. We shall thereforeresortto a simplified descrip- those computedby averagingoverthe (2l+1)f modesfor
tionandassumethatweareprovidedwithasurveycovering sky
eachl. Thus,theangularpowerspectrummaybeestimatedas
4πf steradiansofthesky.Foreachdegreel,weassumethat
tfhuellrseskykayreco(2vle+rag1e).fsCkyoinnsdiedpeernindgenmtomdoedsewsi,tihnsatnegaudlaorfr2elso+lu1tifoonr Cl =<|flm|2 >m≡ (2l+11)f |flm|2. (17)
sky m
muchsmallerthantheextentofthesurvey,wethenhave X
Inthefollowing,weshallinterchangebetweenthesedifferent
Y (rrrˆ)Y∗ (rrrˆ)dΩ= f δKδK , kindsofaveragingwheneverappropriate.Thefactor1/f1/2in
lm l′m′ sky ll′ mm′ Eq. (15)andtherulesinEq. (14)guaranteethat sky
Z (14)
(2l+1)f
Y (rrrˆ)Y∗ (rrrˆ′)= skyP(rrrˆ·rrrˆ′), 2l+1
m lm lm 4π l C(cosθ)= Cl 4π Pl(cosθ),
X Xl (18)
whereP istheLegendrepolynomialofdegreel. Throughout
l C = dΩC(cosθ)P(cosθ),
thepaper,theangularintegrationiscarriedoutonlyoverthe l l
Z
5
which can be shown by decomposing P into Y according galaxy according to the σ in the sum of Eq. (20). To min-
l lm i
to the second relation in Eq. (14) and assuming thatC(θ) is imize the effects of shot noise, we weight each galaxy by a
negligibleforangularseparationslargerthantheextentofthe factorofw whichisgivenby
i
survey(seeAppendixA).
Foradiscretesamplingof f atthepositionsof N galaxies Nσ−2
w2 = i . (26)
distributedovertheobservedpartofthesky,wewrite[18] i σ−2
j j
N ThisparticularweightingschePmeyields
f(rrrˆ)≈ n(rrrˆ)f(rrrˆ)dΩ
i
Xi Z (19) N
σ2 = , (27)
≈n¯ f(rrrˆ)dΩ, f σ−2
j j
Z
wheren(rrrˆ)istheprojectednumberdensityofobjectsandn¯ = wherewehaveassumedthatthPattheunderlyingsignalmakes
a negligible contribution to σ2. The weighting does not af-
N/(f 4π)isthecorrespondingmeannumberdensityoverthe f
sky
observedpartofthesky. Herewehaveassumedthat f itself fect the ensemble average of |f |2, and its net effect is that
lm
dependsonthedensitycontrastδ=n/n¯−1,sothatthelaststep σ should be computed as given by Eq. (27), i.e. it merely
f
appliestolinearorderinthefluctuations. Again,theangular reducesshotnoiseerrors(foradditionaldetails,seeAppendix
integration is carried out only over the observed part of the B).Inprinciple,onemayuseanyweightingscheme.
sky.InanalogytoEq. (15),wefurtherdefine
1 N C. Environmentaldependencesintheluminosityfunctions
f = f(rrrˆ)Y (rrrˆ). (20)
lm n¯f1/2 i lm i andmagnificationbygravitationallensing
sky Xi=1
In this case, one can show that the angular power spectrum So far, we have assumed that the systematic shifts in the
takestheform(seeAppendixB) z (m)relationforgalaxiesinagivendirectionaresolelydue
cos
tothetermsappearingontheright-handsideofEq.(1).How-
σ2 ever, additional shifts may arise from changes in the mean
C =<|f |2 >− f, (21)
l lm magnitudesdue to large-scaledensityfluctuationsin a given
n¯
direction,i.e.environmentaldependencesandevolutioninthe
where the second term represents the contribution of shot luminosity function, and the magnification effect caused by
noiseduetothediscretesamplingof f andσ2f = i f(rrrˆi)2/N. gravitationallensing. In what follows, we shall denote their
ToestimatethevaluesofCl,wethereforeuse contribution as Θenv and Θlens, respectively. Both effects re-
P
sultinamagnitudeshiftwhichtranslatesintoadditionalcor-
σ2
Cl =<|flm|2 >m − n¯f. (22) rTeolamteodddeelvtihaetiocnosntorfibtuhteioenstoimfathteedsezceosfffercotms iintstmheeacnorrreellaattiioonn.
functionofΘ,weneedtotranslateamagnitudeshift∆mintoa
If f =S(rrrˆ)+ǫ whereS isanunderlyingcosmologicalsignal
i i i correspondingshift∆z. Thiscandirectlybereadoffthe(blue)
andǫ isanuncorrelatedrandomerror,thissimplyreflectsthe
i solidcurvez (m)showninFig.1inwhichashift∆m = 0.2
factthatC =< |S |2 >. The expectederrorin this estimate cos
l lm leadstomeanshift∆z ≈0.017[19].
ofC isgivenby(again,seeAppendixB) cos
l To quantifythe impactof environmentaldependences,we
2 σ2 2 assumethatthesystematicshiftingalaxymagnitudedepends
Σ2 = f +C (23) on the density contrast δ in regions where they reside, irre-
(2l+1)f n¯ l spective of the smoothingscale [20, 21]. Here we adoptthe
sky  
which includes contributions from both shot noise Σsn and flirnoemarthreelavtiisoibnle∆mban=d[02.22]δ.wThhiechacitsuaolbdseeprveantdioennaclelyisinaffeurrnecd-
cosmicvarianceΣcv, tion of the photometric band, and it is expected to be much
weakerinEuclid’sH-bandasindicatedbytheweakenviron-
Σ = 2 σ2f, Σ = 2 C (24) mental dependence of the Schechter parameter M∗ fitted to
sn (2l+1)f n¯ cv (2l+1)f l NIR luminosities [23]. The remaining Schechter parameter,
s sky s sky
usually dubbed α, exhibits a stronger dependence, but since
Thevarianceofthescatterσ2, this parameter fixes the shape at the faint end, it will have
i
onlyasmallimpactindeepsurveyslikeEuclid. Nonetheless,
σ2 (z)+σ2(m) itwillbepossibletoremovemostofthecontaminationcaused
σ2i =<ǫi2 >= phot(1i+z)2z i , (25) bythiseffectusingtheobserveddistributionofgalaxiesinthe
i
Euclidsurvey.
dependsonbothmagnitudeandredshift. Notethatthefactor Intheweak-fieldlimit,themagnificationinducedbygrav-
of(1+z)arisesfromthedefinitionofΘinEq. (1). Consid- itational lensing is proportionalto 1+2κ, where κ is the ef-
i
eringtheapplicationtorealdata,itisprudenttoweighteach fectiveconvergencefield,i.e. anintegraloverthe(weighted)
6
densitycontrastalongthelineofsight[24]. ForaflatΛCDM where j is the usual first-kind spherical Bessel function of
l
modelandafixedsourceredshiftcorrespondingtoacomov- degreel,weget
ingsourcedistancer,onefinds
il r2
Θ˜Φ = drW d3kΦ j(kr)Y (kkkˆ). (33)
κ(θ,r)= 3H02Ω0 rdr′r′(r−r′)δ(r′θ,r′), (28) lm 2π2c2 Zr1 φZ kkk l lm
2c2 a(r′)r
Z0 Therefore,wefinallyarriveat
wherethetwo-dimensionalangularvectorθ isperpendicular
CΦ =<|Θ˜Φ |2 >
to the lineof sight. Therefore,itis straightforwardto model l lm
themagnificationanditseffectinouranalysis. Themagnifi- 2 r2 2 (34)
cationfield containsvaluableinformationsince it probesthe = πc4 dkk2PΦ(k) drWφjl(kr) ,
growth of the angular derivatives of the gravitational poten- Z (cid:12)(cid:12)Zr1 (cid:12)(cid:12)
(cid:12) (cid:12)
tial[e.g.24,25]. Itcanalsobeusedtoconstrainthegravita- wherewehaveused<ΦkkkΦkkk′ >=(cid:12)(cid:12)(cid:12)(2π)3δD(kkk−kkk′)P(cid:12)(cid:12)(cid:12)Φ(k). Sim-
tionalslipwhicharisesincertainmodificationsofthegeneral ilarly,usingthelinearrelationinEq.(4),weobtain
theory of relativity [e.g. 26, 27]. Given a theory of gravity,
much of the contribution to the power spectra from gravita- CV = 2 dkk2P (k) r2drW ljl −kj 2, (35)
tionallensingcan, in principle, be removedusing the under- l πc2 Φ V r l+1
lyinglarge-scaledensityfieldwhichisinferredfromthefore- Z (cid:12)(cid:12)Zr1 !(cid:12)(cid:12)
(cid:12) (cid:12)
groundgalaxydistribution. Consideringthefollowinganaly- where W = 2aD˙p(r)/3Ω (cid:12)(cid:12)H2. As for ΘΦ˙, the last(cid:12)(cid:12) term ap-
sis,however,wewilltreattheeffectsoflensingmagnification V 0(cid:12) 0 (cid:12)
pearing in Eq. (1), we will assume that the signal is mostly
aspartofthesignal. Aswewillseebelow,itscontributionis
causedbythelarge-scalestructurebetweentheobservedhigh
notnegligible,andcanbeconstrainedtogetherwiththepower redshiftgalaxysampleandtheobserver.Therefore,ΘΦ˙ isap-
spectrumofthevelocityfield.
proximatelythesameforallgalaxiesalongacommonline-of-
sightsuch thatΘ˜Φ˙ ≈ ΘΦ˙. The angularpowerspectrumthen
reads
III. THEORETICALANGULARPOWERSPECTRA
CΦ˙ =<|ΘΦ˙ |2 >
l lm
Inthefollowing,wewillpresentpredictionsfortheangular
8 r 2 (36)
powerspectra correspondingtothe varioustermsinEq. (1). = dkk2P (k) dr′W j(kr′) ,
Sincetheobservedgalaxiescoveraredshiftrangez1 <z<z2, πc6 Z Φ (cid:12)Z0 φ˙ l (cid:12)
wdeefidnoednaottacospnesicdifierctrheedsahnigfut,labrutpionwsteerasdpwecetruaseofaquantity f whereWφ˙(t)/a=(d/dt)[D/a].Th(cid:12)(cid:12)(cid:12)(cid:12)eintegrationove(cid:12)(cid:12)(cid:12)(cid:12)rr′istaken
fromr = 0outtoadistancebeyondwhichΦbecomesnearly
r2 constantwithtime. Forsimplicity,wewillassumethatmost
f˜(rrrˆ)= f [rrrˆr,t(r)]p(r)dr, (29) ofthecontributiontotheintegralcomesfromtheinneredge
Zr1 oftheconsideredgalaxysample,andadoptz = 1inEq. (36)
forallgalaxiesinthesurvey. ForEuclid,thisseemstobethe
where r and r are the comoving distances at the survey’s
1 2 casesincethemeanredshiftisexpectedtobez∼1.Notethat
limiting redshifts, z and z , respectively, and p(r)dr is the
1 2
the same assumptionsare used when calculating the angular
probabilityofobservingagalaxywithintheinterval[r,r+dr].
powerspectrumClens.
Note that the function f represents any of the terms on the
right-handside of Eq. (1), i.e. ΘV, ΘΦ, and ΘΦ˙. Beginning
withΘΦ,wewrite
A. PredictionsoftheΛCDMscenario
Θ˜Φ = dΩΘ˜ΦY (rrrˆ)
lm lm Havingderivedtherelevantexpressionsabove,wearenow
Z1 r2 (30) readytomakepredictionsfortheframeworkofΛCDM.Here
= dΩY (rrrˆ) W (r)Φ (rrrˆr)dr, weadoptaspatiallyflatmodelwithbest-fitparametersbased
c2 lm φ 0
Z Zr1 on the CMB anisotropies measured by the Wilkinson Mi-
crowave Anisotropy Probe (WMAP) [28]. In this case, the
wherewehaveusedthelinearrelationΦ(rrr,t)=(D/a)Φ (rrr,t )
0 0 totalmassdensityparameterisΩ =0.266,thebaryonicden-
and defined W = Dp(r)/a, with D(t) and a(t) evaluated at m
φ sity parameter Ω = 0.0449, the Hubble constant h = 0.71
t=t(r). ExpandingΦ (rrr)inFourierspace, b
0 in units of 100 kms−1Mpc−1, the scalar spectral index n =
s
Φ0(rrr)= (21π)3 d3kΦkkkeikkk·rrr, (31) 0w.i9t6h3in,asnpdheσre8s=of08.8h0−1foMrpthce.rWmeswofolriknewairthdeanpsaitryamfluectrtiucaftioornms
Z ofthepowerspectrumtakenfromRef. [29](seeEqs. 29-31
andusing intheirpaper). Forthecalculationofangularpowerspectra,
we use the expressionsfromSec. III togetherwith p(r) cor-
eikkk·rrr =4π ilj(kr)Y∗ (nnnˆ)Y (kkkˆ), (32) respondingtotheredshiftdistributionofgalaxiesappropriate
l lm lm
forEuclidwhichisgivenbyEq. (12).
l,m
X
7
−9
10
Ctot
Clens
−10
10 Cenv
Cl Σ CV
sn
CΦ
−11
10
CΦ˙
10Σ
env
−12
10
5 10 15 20 25 30
l
FIG.2. AngularpowerspectraintheΛCDMmodelasexplainedinthetext: TheshadedarearepresentsthecosmicvarianceuncertaintyΣ
cv
onCtot,andΣ isthe1σshot-noiseerrorfortheEuclidsurvey. ThedesiredsignaliscontaminatedbyCenvwhichresultsfromenvironmental
sn
dependences in the luminosity function and can be removed from the actual Euclid data with high precision represented by Σ . Only
env
contributions from ΘV (peculiar velocity), ΘΦ (gravitational shift), and Θlens (lensing magnification) are included inCtot. At l ∼ 60 (not
shown),Σ andCtotbecomecomparable.
sn
Theangularpowerspectraandassociatederrorsareplotted B. Signal-to-noiseandexpectederroronmodelnormalization
inFig. 2. Theshot-noiseerrorΣ iscomputedusingEq.(27)
sn
and assuming σ = 0.17 as inferred from the (red) dashed
f SupposethatobservationsyieldanestimateCobsfortheto-
curve in Fig. 1. For completeness, we also show the power l
spectrumCenvwhichresultsfromenvironmentaldependences talpowerspectrumCltot. LetClH0 =0andClH1 betheexpected
totalpowerspectraforthenullhypothesiswithnocorrelations
in the luminosity function. As explained in Sec. IIC, the
(H )andoftheΛCDMmodel(H ),respectively.Theratioof
contaminationarising from this effect can be removedgiven 0 1
probabilitiesforH andH isgivenby
theobserveddistributionofgalaxiesinthesurvey. Theaccu- 1 0
racy to which this can be achieved for Euclid is represented
byΣ , the1σerrorwithinwhichCenv canbeestimatedex- −2lnP(H )/P(H )=
env 0 1
pclleicairtlfyrofmromCletnhseindathtae(fisgeuereA,pmpaegnndiifixcBatifoonrcdaeutsaeilds)b.yAgrsavis- Clobs−ClH0 2 − Clobs−ClH1 2 +2ln Σsn . (37)
i(tzait−iozncaosl)l/e(n1s+inzgi)i.ntroducessignificantangularcorrelationsin Xl (cid:16) Σ2sn (cid:17) (cid:16) Σ2 (cid:17) Σ !
The quantityCtot is the power spectrum of the sum of the If H holds, it followsthatCH1 is equalto theCtot shown in
threesignalsΘlens,ΘV,andΘΦ,correspondingtothelensing Fig.12,andCobs−CH1 isaralndomvariablewithlvarianceΣ2
magnification, the Doppler shift, and the gravitational shift, l l
givenbyEq. (23). Therefore,thesignal-to-noiseratio(S/N)
respectively. Note that the calculation of Ctot does include
forrejectingH is
0
covariancebetween the three individualsignals. The contri-
butionfromΘΦ˙ isnegligibleasisindicatedbyCΦ˙,andwedo
S 2
not include it in Ctot. At l∼< 10, CV is the dominant contri- =−2lnP(H )/P(H )=
0 1
bution, but the lensing term Clens takes over at l & 10−15, N 0
rTohueghshlyaduendtilalre∼a6r0epwrehseernetisttdhreop1sσbeelrorworthoef sChtoott-dnuoeisetolecvoesl-. (cid:18) (cid:12)(cid:12)(cid:12)(cid:12) ((cid:19)2l+1)fskyCl2 + 1+ n¯Cl 2−2ln 1+ n¯Cl , (38)
mtoicprvoavriidaencme.eaFsourrepmraecnttiscaolfpCurVpoasneds,Citleniss tbhyeraenfoarpepproospsriibaltee Xl  2σ4f/n¯2  σ2f   σ2f 
fittingprocedureofthetwocorrespondingcurvestothemea- whereC = CH1 = Ctot. Substituting the relevantquantities
l l l
suredCtot. Unfortunately,atlowl,cosmicvarianceissolarge into the above, we obtain (S/N)| = 101, where the above
0
thataccurateconstraintsonCΦdonotseempossible. sumrapidlyconvergesbyl = 30. Ontheotherhand,ifH is
0
8
1.4 9.2
6.17
1.3
2.3
1.2
1.1
v
a
1
0.9
0.8
0.7
0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2
a
lens
FIG. 3. Confidence levels for measuring the amplitudes of the velocity and lensing power spectra by fitting a2CV +a2 Clens to random
v lens
realizationsofCtotwhichincludebothshotnoiseandcosmicvariance: Thefittingprocedureignoresphysicalcross-correlationsbetweenthe
usedquantities.Contoursareshownfor∆χ2 =2.3,6.17,and9.2,correspondingtoconfidencelevelsof68%,95.4%and99%,respectively.
true,thentheS/N forrejectingH isgivenby two-parametermodel
1
S 2 =−2lnP(H1)/P(H0)= Ctot,m =a2VCV+a2lensClens. (41)
N 1
(cid:18) (cid:12)(cid:12)(cid:12)(cid:12) (cid:19)(2l+1)fskyCl2 + 1+ n¯Cl −2+2ln 1+ n¯Cl , (39) aTshwisemlloadsealnfyorcCovtoat,rmiannecgelbecettswteheenctohnetrreibmuatiionninogfsΘigΦnatols,CΘobVs
l 2(σ2f/n¯ +Cl)2  σ2f   σ2f  and lensing magnification Θlens. Using the above expres-
whXichquicklyconvergesatl∼ 50andyields(S/N)|1= 14.7. sion, we have repeated the procedure described above for
The fact that (S/N)|1 ≪ (S/N)|0 is a result of cosmic vari- constraining σ8 with the 1000 random realizations of Cobs.
ancewhichiszerofortheH hypothesis,butsignificantinthe The result is presented in Fig. 3 which shows contours of
0
ΛCDMscenario,i.e. Σ2 =2C2/(2l+1)f . ∆χ2 = −2lnP + 2lnmax(P) as a function of aV and alens.
InadditiontoS/Ncocnvsideraltions,wecaskny assesshowwella The contourshave beencomputedforone of the randomre-
measurementofCtotwouldconstrainthenormalizationofthe alizations,butthevaluesofthebest-fitparameters(givingthe
ΛCDMpowerspectrumintermsofσ . Tothisend,wewrite lowest∆χ2)obtainedforthisparticularrealizationhavebeen
8
the model’s total power spectrum as Ctot,m = (σ /0.8)2Ctot, shifted to their underlyingvalue, i.e. unity for both. This is
8
whereCtotasillustratedinFig.2isobtainedforσ =0.8.The reasonablesincetheaveragebest-fitparametersof1000real-
8
expected1σerroron σ8 isthen (−∂2lnP(H1)/∂σ28)−1, where izationsareessentiallyunbiased.Asseenfromthefigure,alens
P(H )isnowexpressedas is constrained with better accuracy than the velocity ampli-
1
tudea . ThisisconsistentwithFig. 1whichshowsthatClens
V
Cobs−Ctot,m 2 dominatesthetotalsignaloveralargerangeoflwhileCV is
lnP(H1)=− (cid:16) l 2Σ2l (cid:17) − lnΣl, (40) significantonlyforlowlwherecosmicvariancebecomesin-
l l l creasingly important. Still, both parametersare constrainted
X X
andΣ isgivenbyEq. (23)withC = Ctot,m. Usinganormal withgoodaccuracy.
l l l
probability distribution of the form given in Eq. (40) with
fixedσ = 0.8,wehavegenerated1000randomrealizations
8
of Cobs. For each of these realizations, we have maximized IV. DISCUSSION
P(H ) givenby Eq. (40) with respectto σ . Asa result, we
1 8
findthatthetruevalueσ =0.8isrecoveredwithinarelative In this paper, we have presented a novel method for de-
8
1σ error of less that4%, withoutany statistically significant rivingdirectconstraintsonthe peculiarvelocityand gravita-
bias. tionalpotentialpowerspectrafromcurrentlyplannedgalaxy
We have also inspected the possibility of constraining the redshift surveys. The large number of galaxies with photo-
velocity and lensing signal amplitudes with the help of the metricredshiftsinthesesurveysallowsonetoexploitappar-
9
entgalaxymagnitudesasaproxyfortheircosmologicalred- wide-fastsurveyisexpectedtoobserve∼ 20,000deg2inthe
shiftssinceitbeatsdownthelargescatterinthez (m)relation ugrizybands. Afterabout10yearsofoperation,itwillreach
cos
andtheuncertaintyinthephotometricredshifts. Themethod muchdeeperdepths(downtoaco-addedmagnituder = 27),
aimsatdirectlyconstrainingpowerspectraoftheunderlying detectingabout3×109galaxies[40].
fluctuationfieldsindependentofthewaygalaxiestracemass.
Othermethodsforextractingcosmologicalinformationfrom
redshiftsurveysrelyonaccuratemeasurementsofthegalaxy V. ACKNOWLEDGMENTS
power spectrum in redshiftspace (since galaxy distances re-
main unknown). The power spectrum and other statistical We are grateful to Micol Bolzonella for computing all
measuresbasedonthedistributionofgalaxieshavebeensuc- redshift-magnitude relations used in this papers from the
cessful at probing the nature of dark matter and placing im- zCOSMOS data. We also thank Henry McCracken for pro-
portantconstraintsonneutrinomasses [30–32]. Having said viding us with the H-band magnitudesof zCOSMOS galax-
that, however, they depend on a very accurate knowledgeof ies. E.B. thanks Gianni Zamorani and Massimo Meneghetti
the relation between galaxies and the full underlying matter for useful discussions and suggestions. This work was sup-
distribution. Themethodwehaveconsideredhereislesspre- ported by THE ISRAEL SCIENCE FOUNDATION (grant
cise,butitiscompletelyindependentofthegalaxyformation No.203/09),theGerman-IsraeliFoundationforResearchand
process and offers a much more sensitive assessment of the Development, the Asher Space Research Institute and the
underlying physical mechanism driving cosmic acceleration WINNIPEG RESEARCH FUND. E.B. acknowledges the
andstructureformation. Thisapproachisparticularlyworth- supportprovidedbyMIURPRIN2008“Darkenergyandcos-
whileifsuchconstraintsonthevelocityfieldandthegravita- mologywith largegalaxysurveys”andby Agenzia Spaziale
tionalpotentialarecontrastedwithlocalconstraintsobtained Italiana (ASI-Uni Bologna-Astronomy Dept. ’Euclid-NIS’
fromdataatlowredshifts. Forexample,peculiarmotionsof I/039/10/0). M.F.is supportedin partatthe Technionbythe
galaxieswithinadistanceof∼ 100h−1Mpccanbemeasured LadyDavisFoundation.
usingtightrelationsbetweenintrinsicobservablesofgalaxies
[e.g. 33, 34], and also using astrometric observations of the
Gaia space mission which is currently scheduled for launch AppendixA:ProofoftherelationsinEq.(18)
in2013. Thesepeculiarmotionsofgalaxieshavebeenuseful
for constrainingcosmologicalparameters[35] as well as the Foranyfunction f(rrrˆ),wehavedefined
amplitudeofthevelocityfieldinthenearbyUniverse[5,36].
1
AlthoughwehavepresentedpredictionsfortheEuclidsur- flm = f1/2 dΩf(rrrˆ)Ylm(rrrˆ). (A1)
vey, the science proposedin this paper will nothave to wait sky Z
for this space mission. In fact, several ground-based photo-
UsingtherelationsgiveninEq. (14),weget
metric surveysin the optical and near-infraredbands, which
willconstitutethebackboneofEuclid’sphotometry,willpro- 1 1
C = <|f |2 >=
vide photometric redshift catalogs that can be used for our l (2l+1)f lm (2l+1)f2
purposeswellbeforethelaunchofthesatellite. Onashorter sky Xm sky
timescale, the VLT Survey Telescope (VST) will be used to × dΩdΩ′ < f(rrrˆ)f(rrrˆ′)> Y (rrrˆ)Y∗ (rrrˆ′)
lm lm
carry out the Kilo Degree Survey (KiDS), one of the ESO
Z m
publicsurveys. Itwillcover1,500deg2 tou = 24,g = 24.6, 1 X
r=24.4,andi=23.1,andwillprobablycontain∼108galax- = 4πf dΩdΩ′C(rrrˆ·rrrˆ′)Pl(rrrˆ·rrrˆ′) (A2)
ieswithmeasuredphotometricredshifts[37]. Also,theDark sky Z
1
EnergySurvey(DES)willstartitsoperationssoon,anditwill = dΩ′ dΩC(rrrˆ·rrrˆ′)P(rrrˆ·rrrˆ′)
cover5000deg2oftheSouthernskywithin5years,reaching 4πf l
sky Z Z
magnitudesupto∼24inSDSSgrizfilters,comparabletothe
= dΩC(cosθ)P(cosθ),
limitingmagnitudesofEuclidandwitharedshiftdistribution l
dN/dz similar to that of Euclid galaxies. DES will measure Z
photometricredshiftsof∼3×108galaxieswithσ ∼0.12 where for the last step, we have used that the integral over
photo
at z ∼ 1 [38]. Furthermore, the first of the four planned dΩisindependentofrrrˆ′,assumingthatthecoherenceangular
Pan-STARRStelescopes[39]hasbeenoperationalsinceMay length is smaller than the spatial extent of the survey. Sim-
2010. The planned 3π area of the sky will be considerably ilarly, the first relation in Eq. (18) follows from substituting
shallower, detecting galaxies below a limiting magnitude of it into the second and exploiting the (near) orthogonality of
∼24inthegrizbands.AdeepersurveyinvolvingthePS1and Pl(cosθ).
PS2telescopesiscurrentlybeingplanned. Thesurveyshould
cover ∼ 7,500deg2 with limiting fluxes g = 24.7, r = 24.3,
i = 24.1, and z = 23.6. Photometric redshifts will then be AppendixB:Estimatingangularpowerspectrafordiscretedata
measured for ∼ 4.5 × 108 galaxies with similar errors. Fi-
nally, on the long run, the Large Synoptic Survey Telescope Justasinthemainbodyofthepaper,wewilltakethesym-
(LSST)isexpectedtostartoperationsin2020.Itsmaindeep- bol< (·) > ≡ (1/(2l+1)f ) (·)todenoteaveragingover
m sky m
P
10
all indices m corresponding to the degree l of the spherical Notetheremarkablefactthatduetotheuncorrelatednatureof
harmonic decomposition. Again, the symbol < · > without theerrorsandbecauseα, βaswellasα′ , β′,onlysecond-
anysubscriptwillrefertothepreviouslyintroducedensemble ordermomentsofnand f contributetoΣ2. Thisisimportant
average,andweshallinterchangebetween< · > and< · > sinceitimpliesthatthenon-Gaussiannatureofthescatterin
m
whenever appropriate. For brevity of notation, we also use thez (m)relationdoesnotaffectthevarianceoferrorsinthe
cos
thedefinitionYi ≡ Y (rrrˆ),thusomittingthesubscriptlsince estimatedC. Assuming that fn is a Gaussian random field,
m lm i l
all calculationswillrefer to a givendegreel of the spherical wefurtherhave
harmonics. Startingfromthedefinitionof flm inEq. (20),we < frnr frnrfrnr frnr >
obtain α α β β α′ α′ β′ β′
= < frfrnrnr >< fr frnr nr >
1 α β α β α′ β′ α′ β′
<|f |2 >= < f f >YiY∗j (B6)
lm n¯2f i j m m + < fαrfαr′nrαnrα′ >< fβrfβr′nrβnrβ′ >
sky i,j
X + < frfrnrnr >< fr frnr nr >.
= n¯21f < fi2 > Ymi 2+ n¯21f < fifj >YmiYm∗j (B1) Considering the aboαveβ,′ aαveβr′ages wα′ithβ nαo′ eβqual indices will
sky i sky i,j
σ2 X (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) X contribute terms proportionalto Cl2 in the sum of Eq. (B5).
f Neglectingthecontributionofshotnoise,thiswouldyieldthe
= +C,
n¯ l usual cosmic variance expression 2C2/(2l + 1)f . For the
l sky
wherethevalueofσ isinferredfromthez (m)relation,i.e. case α = α′ and β = β′, the second term on the right-hand
σ2 = σ2/N. AsoufrestimateforC,wethcoesreforetake side of Eq. (B6) turns into σ4(n¯δΩ)2, where we have again
f i i l f
used that < n2 >=< n >= n¯δΩ. The other combination
P C =<|f |2 > −σ2f. (B2) which makes tαhe same αcontribution is α = β′ and α′ = β.
l lm m n¯ A bit of algebra shows that the contribution to the variance
Another way of arriving at this result is to partition the ob- in Eq. (B5) of these two combinationsis 2σ4/(2l+1)n¯2f .
f sky
servedskyintoinfinitesimallysmallcellsofangularsizeδΩ, For α = α′ and β , β′, the second term on the right-hand
witheachcellcontainingatmostonegalaxy[16].Inthiscase, side of Eq. (B6) simplifies to σ2 < frfr > (n¯δΩ)3, and the
f β β′
wewrite totalresultofsimilarpermutationswithonlytwoequalindices
f = 1 f n Yα, (B3) reads 4σ2fCl/(2l+ 1)n¯fsky. Finally, summing up all relevant
lm n¯f1/2 α α m termsleadsto
where nα, the number of gsakylaxXiαes in cell α, is either 0 or 1. Σ2 = 2 σ2f +C 2. (B7)
Usingthat< nα >= n¯δΩinthisrepresentation,weobtainthe (2l+1)fsky  n¯ l
expression 1 UsingtheweightsdefinedinEq.(26),itiseasytosaythatthe
<|flm|2 >= n¯2f < fαfβ ><nαnβ >YmαYm∗β samInetrheesumltsaianrebovdalyidowf itthheσpfapgeivr,enwbeyhEavqe. (a2s7su).med that en-
sky α,β
X
vironmental dependences of the galaxy luminosity function
1
= n¯2f < fα2 ><n2α > Ymα2+Cl ariseduetovariationsinthelarge-scaledensitycontrast. The
sky Xα (cid:12) (cid:12) (B4) signalcontaminationassociatedwiththiseffecthasanangu-
= 1 < f2 >n¯δΩ Yα(cid:12)(cid:12) 2+(cid:12)(cid:12) C larpowerspectrumwhichisproportionaltothatofthedensity
n¯2f α m l contrast. Inthefollowing,wegiveanestimateforthispower
sky α
X (cid:12) (cid:12) spectrumandthe1σerrorwithinwhichitcanbeconstrained
= σ2f +C, (cid:12)(cid:12) (cid:12)(cid:12) fromtheobservedgalaxydistribution.Tobeginwith,wecon-
n¯ l sider[16]
wherewehaveusedtherelationn2α =nα. Wefindthispartic- δ = 1 n Yα. (B8)
ular approacha little moreintuitiveto computethe expected lm n¯f1/2 α m
error of the estimate in Eq. (B2). Let fr and nr be possible sky Xα
randomrealizationsofthedatainanotheruniverse. Thevari- Forthe f ’s,wehaveassumedthatn isadiscretesampling
lm α
ance of the error in C may then be written as an ensemble ofauniformdistribution.Thatwasconsistentwithlinearthe-
l
averageovertheserealizations,i.e. orysincewehaveassumedthat f isafunctionofthedensity
contrast. However, now we are interested in the autocorre-
Σ2 = <(Cr−C)2 >
l l lation functionof n , and thus the treatmentof δ is a little
α lm
2 different. To obtain an estimate for the angular power spec-
1
= n¯2f fαrfβrnrαnrβ <YmαYm∗β >m −Cl trumofthedensitycontrast,wewrite
= *1 sky Xα,β < frnr frnrfrnr frnr > + (B5) <|δlm|2 >= n¯21f <nαnβ >YmαYmβ
n¯4fs2ky α,Xβ,α′,β′ α α β β α′ α′ β′ β′ 1 sky Xα,β (B9)
× <YmαYm∗β >m<Ymα′′Ym∗β′′ >m′ −Cl2. = n¯ +Clδδ.