Error analysis in cross-correlation of sky maps: application to the ISW detection

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by  Anna Cabre

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed5February2008 (MNLATEXstylefilev2.2)
Error analysis in cross-correlation of sky maps:
application to the ISW detection
Anna Cabr´e, Pablo Fosalba, Enrique Gaztan˜aga & Marc Manera
Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, F. de Ci`encies, Torre C5 par-2, Barcelona 08193, Spain
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5February2008
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n
a ABSTRACT
J

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ConstrainingcosmologicalparametersfrommeasurementsoftheIntegratedSachs-
1
Wolfeeffectrequiresdevelopingrobustandaccuratemethodsforcomputingstatistical

1 errors in the cross-correlationbetween maps. This paper presents a detailed compar-
v ison of such error estimation applied to the case of cross-correlation of Cosmic Mi-
3 crowave Background (CMB) and large-scale structure data. We compare theoretical
9
models for error estimation with montecarlo simulations where both the galaxy and
3
the CMB maps vary around a fiducial auto-correlation and cross-correlation model
1
0 which agreeswell with the current concordanceΛCDM cosmology.Our analysis com-
7 pares estimators both in harmonic and configuration (or real) space, quantifies the
0 accuracy of the error analysis and discuss the impact of partial sky survey area and
/
h the choice of input fiducial model on dark-energy constraints. We show that purely
p analytic approaches yield accurate errors even in surveys that cover only 10% of the
- sky and that parameter constraints strongly depend on the fiducial model employed.
o
r Alternatively, we discuss the advantages and limitations of error estimators that can
st be directly applied to data. In particular, we show that errors and covariances from
a the Jack-Knife method agree well with the theoretical approaches and simulations.
:
v We also introduce a novel method in real space that is computationally efficient and
i canbeappliedtorealdataandrealisticsurveygeometries.Finally,wepresentanum-
X
ber of new findings and prescriptions that can be useful for analysis of real data and
r
a forecasts, and present a critical summary of the analyses done to date.
1 INTRODUCTION the redshift evolution of the measured signal already pro-
vides first constraints on alternative cosmological scenarios
The ISW effect (Sachs & Wolfe, 1967) has emerged as a ( Corasaniti et al. 2005; Gaztan˜aga et al. 2006).
new and powerful tool to probe our universe on the largest
physical scales, testing deviations from General Relativity However,samplevariancefromtheprimaryCMBaniso-
and the existenceof dark-energy independentof other clas- tropies limits the ability with which one can detect CMB-
sical probes (e.g, Crittenden and Turok 1996; Bean and LSScorrelations.FortheobservationallyfavoredflatΛCDMmodel,
Dore 2004; Lue et al 2004; Cooray et al. 2004; Garriga et evenanoptimalmeasurement ofthecross-correlation could
al. 2004; Song et al. 2006). Recently, a number of groups onlyachieveasignal-to-noise ratio of 10(Crittendenand
∼
haveobtainedthefirstdetectionsoftheISWeffectbycross- Turok1996;PeirisandSpergel2000;Afshordi2004,seealso
correlating low redshift tracers of the large scale structure Fig.17 below). Given the low significance level of ISW de-
(LSS) with the cosmic variance limited cosmic microwave tections,agoodunderstandingofthesystematicandstatis-
background(CMB)mapsobtainedbyWMAP(e.g,Boughn tical errors is crucial to optimally exploit CMB-LSS corre-
andCrittenden2004;Noltaetal.2004;FosalbaandGaztan˜aga lation data that will be collected in future surveys such as
2004; Fosalba et al. 2003; Scranton et al. 2003; Afshordi et PLANCK,DES,SPT,LSST,etc.,forcosmologicalpurposes
al2004,Rassatetal2006,Cabr´eetal.2006).Althoughcur- (seee.g., Pogosian et al.2005). Recentwork hasfocused on
rent detectionsare only claimed at the2-4 σ level, all anal- the impact of known systematics on cross-correlation mea-
yses coherently favor a flat ΛCDM model that is consistent surements (Boughn and Crittenden 2005; Afshordi 2004),
with WMAP observations (Spergel et al. 2006). Moreover, however no detailed analysis has been carried out to assess
(cid:13)c 0000RAS
2 Cabr´e et al.
how different error estimates compare or what is the accu- Armed with this machinery, we address some of the open
racydeliveredbyeachofthem.Sofar,mostofthepublished questions that have been raised in previous work on the
analyseshaveimplementedonespecificerrorestimator(pri- ISW effect detection: how accurate are JK errors? are er-
marilyinrealorconfigurationspace) withoutjustifyingthe ror estimators different dueto theinput theoretical models
choiceof that particularestimator orquantifyingits degree or the data themselves? how many Montecarlo simulations
of accuracy. shouldoneusetogetaccurateresults?canwesafelyneglect
Inparticular,mostofthegroupsthatfirstclaimedISW thecross-correlation signalinthesimulations?doharmonic
detections(Boughn and Crittenden2004; Nolta etal. 2004; and real space methods yield compatible results? What is
Fosalba and Gaztan˜aga 2004; Fosalba et al. 2003; Scranton theuncertaintyassociated to thedifferent error estimates?
et al. 2003, Rassat et a 2006) estimated errors from CMB Themethodologyandresultspresentedhereshouldlargely
Gaussian montecarlo (MC) simulations alone. In this ap- apply to other cross-correlation analyses of different sky
proach statistical errors areobtained from thedispersion of mapssuch as galaxy-galaxy or lensing-galaxy studies.
thecross-correlationbetweentheCMBskyrealizationswith Thispaperisorganizedasfollows:Section2presentsthe
a(single) fixedobservedmap tracingthenearbylarge-scale Montecarloandthetheoreticalmethodsusedtocomputethe
structure. This estimator is expected to be reasonably ac- errors for the galaxy-temperature cross-correlation signal.
curate as long as the cross-correlation signal is weak and Sections3& 4shows acomparison between thenormalized
the CMB autocorrelation dominates the total variance of covariances and diagonal errors from different estimators.
the estimator. We shall call this error estimator MC1 (see Theimpact ofthechoiceoferrorestimatoron cosmological
below). parametersisdiscussedinSection5.Finally,inSection6we
Fosalba,Gaztan˜aga&Castander(2003),Fosalba&Gaztan˜agasummarize our main results and conclusions.
(2004), also used Jack-knife (JK) errors. They found that
the JK errors perform well as compared to the MC1 esti-
mator, but the JK error from the real data seems up to a
factorof twosmaller(on sub-degreescales) thantheJKer- 2 METHODS
rorestimatedfrom simulations.Thisdiscrepancyarisefrom
Weconsiderfourmethodstoestimateerrors.Thefirstoneis
thefactthatthefiducialtheoreticalmodelusedintheMC1
based on Montecarlo (MC) simulations of the pair of maps
simulationsdoesnotmatchthebestfittothedata(seecon-
wewanttocorrelate.Weconsidertwovariants:MC2,where
clusions).
pairs are correlated with a given fiducial model and MC1,
Afshordiet al(2004) criticize theMC1and JK estima-
whereonemapineachpairisfixedandnocross-correlation
torsandimplementapurelytheoreticalGaussian estimator
signalisincluded.Thenexttwomethodsrelyontheoretical
in harmonic space (which we shall call TH bellow). How-
estimation. We will use a popular harmonic space predic-
ever,theydidnotshowwhytheirchoiceofestimatorshould
tion,thatweshallcallTH(TheoryinHarmonicspace).We
be more optimal or validate it with simulations. This criti-
will also introduce a novel error estimator that is an ana-
cism to the JK approach has been spread in the literature
lytic function of the auto and cross-correlation of the fields
without any critical assessment. Vielva, Martinez-Gonzalez
in real space that we shall call TC (Theory in Configura-
& Tucci (2006) also point out the apparent limitations of
tionspace).Finally,wewillestimateJack-Knife(JK)errors
the JK method and adopted the MC1 simulations instead.
which uses sub-regions of the actual data map to calculate
However they seem to find that the signal-to-noise of their
thedispersion in our estimator.
measurement dependson the statistical method used.
Once we have errors estimated in one space, it is also
Padmanabham etal (2005) use Fisher matrix approach
possible to translate them, through Legendre transforma-
and MC1-type simulations to validate and calibrate their
tion, into the complementary space. We shall make a clear
errors. They also claim that JK errors tend to underesti-
distinction between the method for the error calculation
mateerrorsbecauseofthesmallnumberofuncorrelatedJK
(i.e.,MC,TH,TCorJK),andtheestimatorontowhichthe
patcheson thesky,but they provideno proof of that.
errors are propagated: i.e. either w(θ) or C . For example,
ℓ
Giannantonio et al (2006), use errors from MC simula- TH w means errors in w(θ) propagated from theoretical
−
tions that follow the method put forward by Boughn et al errorsoriginallycomputedinharmonicspace.Thisnotation
(1997). Intheir work theerror estimator is built from pairs is summarized in Table 1.
of simulation maps (of the CMB and large-scale structure In all cases (except for theJK) we are assuming Gaus-
fields) including the predicted auto and cross-correlation. sian statistics. In principle, it is also possible to do all this
ThisistheestimatorweshallnameMC2below.Theypoint withnon-Gaussianstatisticsbutthisrequiresparticularnon-
out that their results are consistent with what is obtained Gaussianmodels,whicharecurrentlynotwellmotivatedby
from the simpler MC1 estimator. observations. Ultimately, our focus here is on the compari-
Inthispaperwedevelopasystematicapproachtocom- son of different methodsfor a well definedset of reasonable
paredifferent error estimators in cross-correlation analyses. assumptions.
(cid:13)c 0000RAS,MNRAS000,000–000
Error analysis in cross-correlation of sky maps 3
notation random Gaussian fields ψ ,
Table 1.Notationusedinthispaper.
aT = CTT ψ , (2)
ℓm ℓ 1,ℓm
TC TheoryinConfigurationspace qCTG [CTG]2 1/2
TH TheoryinHarmonicspace aG = ℓ ψ + CGG ℓ ψ
MC Montecarlosimulations ℓm CℓTT 1,ℓm „ ℓ − CℓTT « 2,ℓm
MC2 MCofthe2fields,withcorrelationsignal
p
MC1 MCof1fieldalone(CMB),nocorrelationsignal for the CMB temperature (T) and galaxy over-density (G)
JK Jack-knifeerrors fields, respectively, and CXY is the (cross) angular power
ℓ
MC2-w errorsinwTG fromMC2simulations spectrum of theX and Y fields.
MC1-w errorsinwTG fromMC1simulations Oursimulations assumeamodelthatisbroadlyspeak-
MC2-Cℓ errorsinCℓ fromMC2simulations
ing in agreement with current observations, although the
TH-w errorsinwTG fromTHtheory
TC-w errorsinwTG fromTCtheory precisechoiceofparametervaluesisnotcriticalforthepur-
TH-Cℓ errorsinCℓ fromTHtheory pose of this paper. We assume adiabatic initial conditions
JK-w errorsinwTG fromJKsimulations and a spatially flat FRW model with the following fiducial
cosmological parameters: Ω = 0.7, Ω = 0.05, Ω = 0,
DE B ν
n=1,h=0.7,σ =0.9.Althoughwewillbasemostofour
8
analysesonthisfiducialmodel,wehavealsorunasetof1000
2.1 Montecarlo Simulations (MC)
MCsimulations foramorestrongly dark-energydominated
We have run 1000 Montecarlo (MC) simulation pairs of ΛCDMmodelwithΩDE =0.8(otherparametersremainas
theCMBtemperatureanisotropyandthedark-matterover- inourfiducialmodel).Thiswillallow ustotest howrobust
density field, including its cross-correlation, following the are ourmain results to changes around our fiducial model.
approach presented in Boughn et al. 97 (see Eq.2 below). Galaxies aredistributedin redshiftaccording toanan-
These simulations are produced using the synfast routine alytic selection function,
of the Healpix package1. We assume that both fields are
Gaussian: this is a good approximation for the CMB field dN = 3z2e−(z/z0)3/2 (3)
dz 2z3
on thelargest scales (i.e few degrees on the sky),which are 0
the relevant scales for the ISW effect. However, the mat- where zm = 1.412z0 is the median redshift of the source
ter density field is weakly non-linear on these scales (eg distribution, and by definition, dN/dz =1. Note that for
see Bernardeau etal 2002) and therefore non-Gaussian, i.e such selection function, one can show that its width simply
R
it has non-vanishing higher-order moments. Therefore our scaleswithitsmedianvalue,σz zm/2.Forconveniencewe
≃
simulationsarerealisticaslongasnon-Gaussianitydoesnot shall take zm = 0.33 as our fiducial model. For all the sky
significantlyalter theCMB-mattercross-correlation andits we have set the monopole (ℓ = 0) and dipole (ℓ = 1) con-
associatederrors.2 Wetakegalaxiestobefairtracersofthe tribution to zero in orderto be consistent with theWMAP
underlyingspatialdistributionofthematterdensityfieldon data.
large-scales: we assume that a simple linear bias model re- We have run simulations for surveys covering different
latesbothfields,sothatwGG =b2wMM andwTG =bwTM. areas,rangingfrom anall-skysurvey(fsky =1)toasurvey
Therefore, in what follows, we make no difference between that covers only 10% of the sky (fsky = 0.1). The latter is
◦
matter and galaxies in our analysis (other than b), without realized byintersectingaconewith anopeningangleof37
loss of generality. fromthenorthpolewiththesphere.Largersurveyareasare
Decomposing our simulated fields on the sphere, we obtainedbytakinglarger openingangles.Forfsky =0.1we
have havedonethesameanalysistakingacompactsquareinthe
equator (with galactic coordinates l = 0o to l = 66o and
f(nˆ)= a Y (nˆ) (1)
ℓm ℓm b = 33o to b = 33o) and have found similar results. We
Xℓm note−that the wide f =0.1 survey is comparable in area
where a are the amplitudes of the scalar field projected sky
ℓm anddepthtothedistributionofmainsamplegalaxiesinthe
on the spherical harmonic basis Y . In our simulations,
ℓm SDSSDR2-DR3.
the a ’s are given by linear combinations of unit variance
ℓm
1 http://healpix.jpl.nasa.gov/
2 As a check, we will compare below, in Fig.10, the results of
2.1.1 Clustering in the simulations
theMC1simulations,whichhaveafixgalaxymapwithGaussian
statistics,withtheresultsusingtheobservedSDSSDR5,whichis
Wehavecomputedtheangular2-point correlation function
not Gaussian. We find no significant differences, indicating that
the level of non-Gaussianity in observations does not influence for the galaxy over-density wGG, thetemperature wTT and
muchtheerrorestimation. theircross-correlationwTG,aswellastheir(inverse)Legen-
(cid:13)c 0000RAS,MNRAS000,000–000
4 Cabr´e et al.
Figure 1.2-pointcorrelation function(top panels)and angular Figure 2.SameasFig.1butforawidefieldsurveyfsky =0.1
power spectra (bottom panels) for all-sky surveys with median
depthzm=0.3.DifferentpanelscorrespondtoTT(Temperature-
Temperature), GG (Galaxy-Galaxy) and TG (Temperature- as the expectation value of galaxy number density δ and
G
Galaxy)cross-correlation.Errorsshowncorrespondtodispersion
temperature∆ fluctuations:
overMontecarloMC2-typesimulationsbinnedwith∆ℓ=18(see T
textfordetails) δ = NG 1 (6)
G <NG > −
∆ = T T (in µK) (7)
T 0
−
dretransforms, i.e, the angular power spectra,
at two positions nˆ and nˆ in the sky:
1 2
2ℓ+1
w(θ) = 4π CℓPℓ(cosθ) (4) wTG(θ)≡h∆T(nˆ1)δG(nˆ2)i, (8)
Xℓ 1 where θ =|nˆ2−nˆ1|, assuming that the distribution is sta-
C = 2π dcosθw(θ)P (cosθ) (5) tisticallyisotropic.Toestimatew (θ)fromthepixelmaps
ℓ ℓ TG
Z−1 we use:
wherewedenotebyPℓ theLegendrepolynomialoforderℓ. w (θ)= i,j∆T(nˆi) δG(nˆj), (9)
In real space, we define the cross-correlation function TG Npairs
P
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Error analysis in cross-correlation of sky maps 5
Figure 3. Convergence in normalized covariance matrix for
Monte-Carlosimulations(MC2).Wecomparecovariancewhenwe
increase the number of simulations. Herewe show the difference
between a)thefirst100withrespecttothefirst200simulations Figure4.ConvergenceindiagonalerrorforMonte-Carlosimula-
(100-200), b) 200-400, c)400-700, d)700-1000. Results show that tions.Itisshowntheerrorforthefirst100simulations(dotted),
oneneeds at least700simulations forthe normalizedcovariance 200 (dashed), 400 (dash-dot), 700 (dash-dot-dot) and 999 (solid
toconverge(weuse1000inouranalysis). line).With200simulationstheaccuracyinerrorsisabout20%.
wherethesumextendstoallpairsi,j separated byθ ∆θ.
± 1 M
Survey mask and pixel window function effects have been C = ∆wk (θ )∆wk (θ ) (10)
ij M TG i TG j
appropriately taken into account using SpICE (Szapudi et
Xk=1
al. 2001a,b). This code has been probed to yield correct re-
∆wk (θ )=wk (θ ) w (θ ) (11)
sults not only on simulations but also on real data from TG i TG i − TG i
surveys with partial sky coverage and complex survey ge- wherewk (θ )isthemeasureinthek-thsimulation(k=1,...M)
TG i
b
ometries (Fosalba and Szapudi 2004). In Fig.1 we show re- and w (θ ) is the mean over M realizations. The case i=j
TG i
sults from the all-sky MC2 simulations, whereas Fig.2 dis- gives thediagonal error (i.e, variance).
plays the same for a survey covering 10% of the sky alone Ibnordertocheckthenumericalconvergenceinthecom-
(fsky =0.1). Errorbars are computed as the rms dispersion putation of the covariance matrix, we compare the results
over the MC2 simulations. For the Cℓ’s, we use linear bins using all 1000 (MC2) simulations with the ones using the
with ∆ℓ = 18, to get approximately uncorrelated errobars first 100, 200, 400 or 700 simulations. For clarity we sepa-
forfsky =0.1(seeFig.7and 3).Asshownintheplots,our rate our converge analysis into the diagonal elements (the
§
all-sky simulations are unbiased with respect to the input variance) and normalized covariance, where we divide the
fiducialmodel(continuouslines): themean over1000 simu- covariance by the diagonal elements (see Eq.22). As shown
lationsliesontopofthetheoretical(inputmodel)curve.For inFig.3,wefindthatthereisnonoticeabledifference( 5%
≃
finiteareasurveys,samplevariancemakesmeasurementson acuracy) in the normalized covariance from 700 and 1000
the largest scales (i.e, lower ℓ’s) fluctuate around the input simulations. This suggests that 700 simulations are enough
theoretical model 3. forourpurposes.Tobesafe,weshalluseall1000simulations
toderive ourmain results.
Ontheotherhand,Fig.4showstheconvergenceonthe
2.1.2 Convergence in simulations varianceestimation(diagonalelementsofthecovariancema-
trix) for an increasing number of simulations. One needs
The MC covariance is definedas:
about 200 simulations to converge within 20% accuracy.
This is similar to the dispersion in the errors for a given
realization due to sampling variance, see Fig.12 below. We
3 When we calculate the cross-correlation in a fraction of the will use1000 simulations which will give us betterthan 5%
sky, there is aresidual monopole inthe galaxy and temperature
accuracy in theerror estimation from these simulations.
maps,whichchangesthenormalizationofbothfluctuations.Ina
realsurveywearelimitedbythesurveyareacoveredbygalaxies
and weneed to normalizethe fluctuations using the local mean,
whichisingeneraldifferentfromthemeaninallsky(becauseof 2.1.3 Simulations with a fixed galaxy map (MC1)
samplingvariance).Wefindthatthecross-correlationcalculated
Wecanalsocalculatemontecarloerrorsbycross-correlating
withthelocalnormalizationwithfsky =0.1isabout10%lower
forourfiducialΛCDM model,butthevaluecanvaryforothers 1000simulationsofCMBwithafixedskyforgalaxies(MC1).
modelsanddifferentfsky. This is a common practice because it is quite easy to sim-
(cid:13)c 0000RAS,MNRAS000,000–000
6 Cabr´e et al.
ulate CMB maps and not so easy to simulate galaxy maps.
In this case the common (and easiest) thing to do is not to
include any cross-correlation signal in the simulated CMB
maps.Thus,thisapproachrepresentstwolevelsofadditional
approximations: no variance coming from the galaxy maps
andnocross-correlation within themaps.Despitetheseap-
proximationsoneexpectsMC1errorstobereasonablyaccu-
ratebecausemostofthevarianceshouldcomefromthelarge
scale primary CMB anisotropies, and the cross-correlation
signal is small in comparison.
Here we want to test in detail what is the accuracy of
thisapproach.Wehavetakenthemeanof20differentcases.
Figure 5.Compactsquarewith36zones,coveringa10/usedto
Eachcasehasadifferentfixgalaxymapwhichispairedwith calculateJK erroringalacticcoordinates (l=0o tol=66o and
999 CMB maps, which are not correlated. For each fixed b = −33o to b = 33o). We see that the shapes and sizes of the
galaxy case we obtain a MC1 error, so wecan calculate the zonesaresimilar.
dispersion of this error with the 20 different galaxy maps.
This will be discussed in more detail in 4.3. allthedataremovingoneof thisJK zones, thismeansthat
§
we remove all the pairs that fall completely or partially in
theJKzonethatisremoved.Tocompensateforthecorrela-
2.2 Jack-knife errors (JK) tionbetweentheJKsub-samples,wemultiplytheresulting
covariance by M 1. The covariance for this case is thus:
The JK method is closely related to the Boostrap method −
(Pressetal1992)whichundercertaincircumstancescanpro- M 1 M
videaccurateerrors.Theideaisthatthedataisgroupedin Cij = M− ∆wkTG(θi)∆wkTG(θj) (12)
sub-regions or zones which are more or less independent.4 Xk=1
We then use the fair sample hypothesis (ie ergodicity) to ∆wTkG(θi)=wTkG(θi)−wTG(θi) (13)
estimate the error (variance between zones) for the quan- where wk (θ ) is the measurement in the k-th sub-sample
TG i
tity under study.In the Boostrap methods one defines new b
(k = 1,...M) and w (θ ) is the mean for the M sub-
TG i
sub-samples(whichapproachstatistically independentreal-
samples. For each of the MC2 pair of simulated maps we
izations) by a random selection of sub-regions. In theJack-
have a JK estimationb of Cij. We can therefore calculate a
knife method each new sub-sample contains all sub-regions
JK mean and its dispersion (and distribution) to compare
butone.ApotentialdisadvantageoftheJKerroristhatone
tothe trueMC2 covariance in themaps.
maythinkthatitcannotbeusedonscalesthatarecompa-
rabletothesub-regionssize.Thisisnotnecessarilyso.Rare
events(suchassuperclusters)candominatesamplingerrors 2.3 Errors in harmonic space (TH)
onallscaleseveniftheyonlyextendoversmallregions(see
Theoretical expectations for the errors are the simplest in
Baugh et al . 2001). If JK sub-regions are large enough to
Harmonic space where the covariance matrix is diagonal in
encompasstheserareevents,theycanreproducewellerrors
the all-sky limit. In particular, for Gaussian fields, one can
on all scales. Nevertheless it is clear that a danger with JK
easily see that thevariance (or diagonal error) is,
errorsisthattheresultcouldinprincipledependonthesize
1
and shape of the sub-regions. So this needs to be tested in ∆2CℓTG = f (2ℓ+1) (CℓTG)2+CℓTTCℓGG . (14)
sky
each situation. h i
Thisindicatesthatthevarianceofthepowerspectrumesti-
We can therefore calculate the error from each single
mator results from quadratic combinations of the auto and
mapusingtheJKmethod.TostudytheJK errorin afrac-
cross power, with an amplitude that depends on the num-
tionoftheskyof10%,wedivideacompactsquareareainM
berofindependentm-modesavailabletoestimatethepower
zones or sub-regions. Fig.5 shows the case M =36, but we
at a scale ℓ, which is approximately given by f (2ℓ+1).
sky
havetrieddifferentvaluesforM =20 80,andfindsimilar
− We shall emphasize that this is only approximate and rig-
results. TheJK regions haveroughly equal area and shape.
orously it is only expected to yield accurate predictions for
Thisisimportant;wehavefoundthattheJKmethodcould
azimuthal sky cuts. However, as we shall see later, this re-
giveunrealisticerrorswhentheareasorshapesarenoteven.
sult isof more general applicability. Wenotethat thedom-
Tocalculate thecovariance, we takeaJK sub-sampletobe
inant contribution to the error and covariance comes from
theauto-power of thefieldsCTTCGG involved in thecross-
ℓ ℓ
correlation,whereasthecross-correlationsignal(CTG)2only
4 It is not adequate here to consider individual points or pixels ℓ
as the units (sub-regions) to boostrap because they are highly gives a few percent contribution, depending on cosmology
correlated. and survey selection function.
(cid:13)c 0000RAS,MNRAS000,000–000
Error analysis in cross-correlation of sky maps 7
Partial sky coverage introduces a boundary which re- thesurveyareaiftheyarethrownrandomlyonthefullsky.
sults in the coupling (or correlation) of different ℓ modes: Forpartialskysurveystheseprobabilitiesdependmainlyon
the spherical harmonic basis in no longer orthonormal on the survey area and can be well approximated by the for-
animcompletesky.Thusthecovariancematrixbetweendif- mula provided in Appendix A. Particularly simple analytic
ferent modes, expressions can be obtained for a “polar cap” survey (area
obtained by intersecting a cone with the sphere) and are
Cov(Cℓ,Cℓ′)=<(Cℓ−<Cℓ >)(Cℓ′−<Cℓ′ >)> (15) given in theAppendixB.
is no longer diagonal (ie see Fig.7). Because of the partial
Thisnewmethodofcomputingerrorsinreal spacehas
sky coverage there is less power on the smaller multipoles.
several advantages. Since it takes into account the survey
This results in a systematic bias on the low multipoles of
geometry, it can provide more accurate errors at large an-
C that can sometimes be modeled with the appropriate
ℓ gleswhereboththejackknifeerrorsandtheharmonic-space
window correction of the surveymask.
errorsbecomemoreinaccurate.Comparedtomontecarloer-
Using the Legendre transform one can propagate the
rorsthismethodisfasterbecauseonedoesnotneedtogen-
error ∆C in Eq.14 aboveto configuration space,
ℓ eratealarge numberofskyrealizations. Whatismore,this
∆2w(θ)= 2ℓ+1 2 P2(µ) ∆2C , (16) estimator does not need to rely on any theoretical/fiducial
4π ℓ ℓ model and one can readily apply it to correlation functions
Xl „ « measured on thereal data to estimate the errors. 5
where µ cosθ. Forthe covariance matrix, we find:
≡
C Cov(w(θ ),w(θ )) (17)
ij i j
≡
2ℓ+1 2
= P (µ ) P (µ ) ∆2C ,
4π ℓ i ℓ j ℓ
Xl „ «
whereµ cosθ .Eq(17)andEq(16)assumesthatdifferent
i i
≡
ℓ multipoles are uncorrelated which is only strictly true for
all-sky surveys.We shall see below that this approximation
is quite accurate anyway even for surveys that cover only
10%ofthesky,i.e,cosmologicalparametercontoursderived
from this expression do not significantly differ from those
computedwithsimulationsthattakeintoaccounttheexact
covariance matrix.
2.4 Errors in configuration space (TC)
The cross-correlation function in configuration space is es-
timated by averaging over all pairs of points separated an
angle θ in thesurvey,
′
wTG(θ)=<∆T(q)δg(q) |qˆq′=θ >survey . (18)
We have derived a formula for the covariance of the
estimator in an ensemble of sky realizations. Details of this
derivation can be found in AppendixA,
1 π K[θ ,θ ,ψ]
C = i j sinψdψ (19)
ij 8π2P(θ )P(θ ) P(ψ)
i j Z0
where thekernelK is given by:
K[θ,θ′,ψ]= 1[W (θ,ψ)W (θ′,ψ)+ (20)
2 TT GG
′ ′
W (θ ,ψ)W (θ,ψ)]+W (θ,ψ)W (θ,ψ)
TT GG TG TG
and W is a mean over the corresponding correlation w ,
X X
with X =TT,GG or TG:
π
W (θ,ψ)=2 dϕP(ψ,θ,φ) w (φ) (21)
X X 5 A FORTRAN code (named TC-ERROR) which takes as in-
Z0
where cosφ = cosθ cosψ+sinθ sinψ cosϕ. Survey geom- putwTG,wTT,andwGG andcomputethecovariancematrixand
errorsforthecross-correlationfunction,canbeobtaineduponre-
etry is encoded in P(θ) and P(ψ,θ,φ) probabilities. These
questfromtheauthors(pleasecontactMarcManera).Ofcourse,
are the probabilities for two points separated by an angle this code can also be used to estimate the autocorrelation error
θ or for a triangle of sides ψ, θ,φ to fall completely into inasinglemapbyjustplacing wTG=wTT =0.
(cid:13)c 0000RAS,MNRAS000,000–000
8 Cabr´e et al.
Figure 6. Normalized covariances in real space for different methods as labeled in the figure. No significant changes are found for
differentmethods andskyfractionsused.
Figure7.NormalizedcovariancesinCℓ spacefrommontecarlo(MC2)method.Thecovariancebecomesprogressivelydominatedbyits
diagonalelements asweincreasetheskyfractionfsky.
3 NORMALIZED COVARIANCE MATRIX logicalparamterestimation.Wewilldiscussthisindetailin
5.3.
For each one of the methods presented in the previous sec- §
tion, we next compare the normalized covariance:
C
C = ij (22)
ij
(CiiCjj) 3.2 Harmonic space
Thediagonalvalues(tbhevarpiance)andassociateddispersion InC space,thereisnocorrelationbetweendifferentl-modes
ℓ
will beinvestigated in 4.
§ (bins∆ℓ=1) for thecase of all sky (MC2) maps. Thenor-
malizedcovariancematrixisdiagonal, ascanbeseeninthe
rightpanelofFig.7.Alsoshown,intheleftandcentralpan-
3.1 Configuration space
els, are the results for 10% and 40% of the sky, where the
AsshowninFig.6allthenormalizedcovariances C inreal covariance between modes gives rise to large amplitude off-
ij
spaceareverysimilar.Theappearanceoftheplotsdoesnot diagonal elements. This is in sharp contrast to the results
seem to depend strongly on themethod we use tobestimate in configuration space (in Fig.6) where there is no signifi-
them,orthesurveyareaf .Hereweonlyshowresultsfor cantdifferencebetweennormalizedcovariancesinrealspace
sky
10% and all the sky, but intermediate values yield similar when we decrease the area. This is because the main effect
results.However,wewanttoquestionifslightdifferencesin ofincreasingtheareainconfigurationspaceisthereduction
thecovariancecouldhaveanon-negligibleimpactoncosmo- of diagonal errors (which are shown in next section), while
(cid:13)c 0000RAS,MNRAS000,000–000
Error analysis in cross-correlation of sky maps 9
Figure 8.Eigenvalues inrealspacefor10%sky(leftpanel)andallsky(right).
Figure9.Eigenvectorsinrealspacefor10%oftheskyandforallsky,aslabeledineachpanel.Firsteigenvectorisshownassolidlines,
secondasdotted, thirddashedandfourthdot-dash.ResultsforMC1−w andTC−w,notshownhere,areverysimilar.
inharmonicspacethereisatransferofpowerfromdiagonal This SVD is effectively a decomposition in different modes
to off-diagonal elements. ordered in decreasing amplitude.
We obtain very similar singular values for each mode
andforeach method,asshowin Fig.8 forsomeof thecases
3.3 Eigenvalues and Eigenvectors from SVD
(othercases give very similar results).
To calculate the distribution χ2 and the signal to noise we
need to invert the covariance matrix. We use the Singular We can understand the effect of modal decomposition
Value Decomposition method to decompose the covariance looking at the eigenvectors shown in Fig.9, where we have
in two orthogonal matrices U and V and a diagonal matrix plotted the four dominant eigenvectors as a function of an-
W which contains thesingular values λ squared on the di- gle: first mode (solid) affects only the amplitude, second
i
agonal (egseePressetal1992). Thismethod isadequateto mode(dotted)showsabimodalpattern.Thefollowingmodes,
separate the signal from thenoise: third (dashed) and fourth (dot-dash), correspond to mod-
ulations on smaller angular scales. As can be seen in the
Figure, we obtain nearly the same eigenvectors in all the
C =(UT)W V (23)
ij ik kl lj cases,inagreementtowhatwasfoundbydirectcomparison
where W = λ2δ and C is the normalized covariance of the covariance matrices in Fig.6. Again, we can ask: are
ij i ij b ij
in Eq.22. By doing this decomposition, we can choose the thesmalldifferencessignificant?Wewillstudythisindetail
number of modes that webwish to include in the analysis. in 5.
§
(cid:13)c 0000RAS,MNRAS000,000–000
10 Cabr´e et al.
Figure 10. Error calculated with different methods (as labeled
Figure 11. Error at a two fix angles: 5 deg (upper lines) and
i1n0%theoffitghueres)kiyn(rteoapl slipnaecseafnodr ΩsyDmEbo=ls)0,.7t.hFeoTrCa m(caropssceosv)erainndg 20deg(lowerlines)asafunctionoffsky,thefractionofthesky
TH (dashed line) theoretical errors work well compared to the covered by the map. As predicted, errors decrease as 1/pfsky
in both cases. Note how for small areas, the TC-w prediction
montecarloMC2simulations(solidline),whileMC1simulations
(continuous line)performsbetter than the TH-wmodel (dashed
(dotted line)seemstounderestimate theerrorsby10%. TheJK
line)asitbetter reproduces theMC2simulations(squares).
method(triangles)seemsslightlybiasedup/downonlarge/small
scales,although alltheerrorsarecompatiblegiventhesampling
variancedispersionweexpect(seeFig.12).Forallskymaps(bot-
(MC1)theerrorseemstobesystematically underestimated
tomlinesandsymbols),weshowhowresultsforMC2(solid),TH
(dashed)andTC-w(cross)agreeverywell. byabout10%onallscales.Thisbiasisexpectedaswehave
neglected the variance in the galaxy field and the cross-
correlation signal. A particular case of MC1 is done with
4 VARIANCE & ERRORS real data from SDSS DR5 (shown as long dashed line in
Fig.10). Wehaveused here acompact squareof 10% of the
4.1 Variance in w(θ)
sky from the SDSS r magnitude slice of 20-21, which has
Fig.10 is one of the main result of this paper. We compare a redshift selection function similar to the one in our simu-
the variance for the different methods, which is the diago- lations (zm = 0.33). This case works surprisingly well once
nal part of the covariance matrix. By construction, in the scaled with linear bias b (estimated by comparing the mea-
limit of infinitenumberof realizations, theMC2 error from sured galaxy auto-correlation function with the one in our
simulations should provide the best approximation to the fiducial ΛCDM model). It happens to closely follow the JK
errors. We have demonstrated (in section 2.1.2) that 1000 prediction, rather than the MC1 prediction, but we believe
§
simulationsareenoughforconvergencewithin5%accuracy. this is just a fluke, given the dispersion in the errors (see
Forall skymaps(lower lines in theFigure) wecan seethat 4.3) and theuncertainties in the fiducial model.
§
the three methods used: MC2-w, TH-w and TC-w, yield
identical results, as expected. For smaller survey areas we
4.2 Effect of partial sky coverage
doexpect some deviations, because of thedifferent approx-
imationsondealingwiththesurveyboundary.Forasurvey WehavetestedMC2-w,TH-wandTC-wfordifferentpartial
covering 10% of the sky these 3 methods also agree well up skysurveyareasf andobtainedsimilarresults.InFig.11
sky
to10degrees.AtlargerscalesTH-w(dashedlines)startsto we have plotted the error for a fix angle of 5 degrees (top)
deviate,because boundary effectsare in fact not taken into and20degrees(bottom)forthedifferentvaluesoff .The
sky
accountinthismethod.TheJKerror(triangles)hasaslope three methods coincide for large areas. The error scales by
asafunctionofθthatseemslesssteepthantheothermeth- a factor 1/ f , as expected.
sky
ods, but still gives a reasonably good approximation given Noticethaterrorsatanglescomparabletothewidthof
p
thatthedispersion intheerrorsisabout20% (asdiscussin thesurveyaredifficulttoestimatetheoreticallybecauseone
4.3 below). Note how on scales larger than 10 degrees the needs to take into account the survey geometry. Even for a
§
JKmethodperformsbetter(ieitisclosertoMC2)thanthe map as wide as 10% of the sky, the survey geometry starts
TH-w error. The TC method seems to account well for the to be important for errors in the cross-correlation above 10
boundary effects, as it reproduces the MC2 errors all the degrees. This is shown in simulations as a sharp inflection
way to 50 degrees, where all other methods fail. that begins at 30 degrees in Fig. 10 (solid line) . Our new
If we only use one single realization for the galaxies TC-w method predicts well this inflection, while the more
(cid:13)c 0000RAS,MNRAS000,000–000