The covariance of cosmic shear correlation functions and cosmological parameter estimates using redshift information

0.43 MB·English

by  Patrick Simon

Astronomy & Astrophysics manuscript no.
(will be inserted by hand later)
The covariance of cosmic shear correlation functions and
cosmological parameter estimates using redshift information
4 Patrick Simon, Lindsay King & Peter Schneider
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Institut fu¨r Astrophysik undExtraterrestrische Forschung,Universit¨at Bonn, Aufdem Hu¨gel 71, D-53121 Bonn,
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Germany
n
a
J

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Abstract. Cosmological weak lensing by thelarge scale structureof theUniverse,cosmic shear, iscoming of age
1
asapowerful probeof theparameters describingthecosmological modeland matterpowerspectrum.Itcomple-

2 mentsCosmicMicrowaveBackgroundstudies,bybreakingdegeneraciesandprovidingacross-check.Furthermore,
v upcomingcosmicshearsurveyswithphotometricredshiftinformation willenabletheevolutionofdarkmatterto
2 bestudied,andevenacrudeseparationofsourcesintoredshiftbinsleadstoimprovedconstraintsonparameters.
3 Animportant measure of thecosmic shearsignal are theshear correlation functions;thesecan bedirectly calcu-
0 lated from data,and compared with theoretical expectations fordifferent cosmological models andmatterpower
9
spectra. We present a Monte Carlo method to quickly simulate mock cosmic shear surveys. One application of
0
this method is in the determination of the full covariance matrix for the correlation functions; this includes red-
3
shift binningandis applicabletoarbitrarysurveygeometries. Termsarising from shotnoiseand cosmicvariance
0
(dominantonsmallandlargescalesrespectively)areaccountedfornaturally.Asanillustrationoftheuseofsuch
/
h covariance matrices, we consider to what degree confidence regions on parameters are tightened when redshift
p binningisemployed.Theparametersconsideredarethosecommonlydiscussedincosmicshearanalyses-themat-
-
o ter density parameter Ωm, dark energy density parameter (classical cosmological constant) ΩΛ, power spectrum
r normalisation σ8 and shape parameter Γ. We incorporate our covariance matrices into a likelihood treatment,
t and also use the Fisher formalism to explore a larger region of parameter space. Parameter uncertainties can be
s
a decreased by a factor of 4 8 ( 5 10) with 2 (4) redshift bins.
: ∼ − ∼ −
v
i Key words.large scale structure- cosmology:theory - weak gravitational lensing - methods:numerical
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a
1. Introduction obtained. For example, the CFHT Legacy survey
(http://www.cfht.Hawaii.edu/Science/CFHLS) will cover
The statistics of the distorted images of distant galaxies,
172 deg2 in 5 optical bands, with a smaller area to be
gravitationally lensed by the tidal gravitational field of
observed in J and K bands.
intervening matter inhomogeneities, contain a wealth of
In order to compare these observations with predic-
information about the power spectrum of the dark and
tions for various cosmological models and matter power
luminous matter in the Universe,and the underlying cos-
spectra,differenttwo-pointstatisticsofgalaxyellipticities
mological parameters. The importance of “cosmic shear”
can be employed, all of which are filtered versions of the
as a cosmologicaltool was proposed in the early 1990sby
convergencepowerspectrum.Here,wefocusonthegravi-
Blandfordetal.(1991),Miralda-Escud´e(1991)andKaiser
tationalshearcorrelationfunctions,whichcanbe directly
(1992). Further analytic and numerical work (e.g.. Kaiser
obtained from the data as described in Sect.2.2.
1998; Schneider et al. 1998; White & Hu 2000) took into
accounttheincreasedpoweronsmallscales,resultingfrom This quest for the parameters describing the matter
thenon-linearevolutionofthepowerspectrum(Hamilton content and geometry of the Universe is limited by sev-
et al. 1991; Peacock & Dodds 1996). eral sources of error, dominated by the dispersion in the
The feasibility of cosmic shear studies was demon- intrinsic ellipticities of galaxies and by cosmic (sampling)
strated in 2000, when four teams announced the first variance. The covariance (error) matrix is thus an essen-
observational detections (Bacon et al. 2000; Kaiser tial ingredient in the extraction of parameters from data,
et al. 2000; van Waerbeke et al. 2000; Wittman et or in parameter error estimate predictions. Schneider et
al. 2000). Upcoming surveys will cover much larger al.(2002a)provideanalyticalapproximationsforthe con-
areas, and multicolour observations will enable pho- tributions to the covariance matrix. They consider the
tometric redshift estimates for the galaxies to be case when the mean redshift of the population is known,
2 Simon, King& Schneider:3D correlations of cosmic shear
and data taken in a single contiguous area. Kilbinger & p(w′)dw′ is the comoving distance probability distribu-
Schneider (2004) use a numerical approach to investigate tionforthesources.a(w)isthescalefactornormalisedto
the impact of survey geometry on parameter constraint. a(w =0)=1 and H is the Hubble constant.
0
Using a Fisher matrix approach, which provides a lower- Splitting up the weak lensing survey in redshift, as
bound estimate of covariance, Hu (1999) has shown that in Fig.1, defines a set of effective convergence and shear
evencruderedshiftinformationenablesmuchtightercon- maps instead of a single one, giving more information on
straints to be placed on cosmological parameters, com- theevolutionofthedarkmatterfluctuationsandtherefore
pared with the case when only the mean redshift of the enabling tighter constraints to be placed on cosmological
populationisknown.Thisstudyconcentratedonthecon- parameters.Auto- andcross-correlationofthese maps in-
vergence power spectrum as the vehicle of cosmological troduceawholesetofpowerspectra,generalisingEq.(1):
information.
9H4Ω2
Motivated by these studies, in this paper we demon- P(ij)(ℓ) = 0 m
strate how numericalsimulations can be used to estimate κ 4c4
the full covariance matrix for the shear correlation func- wH W¯(i)(w)W¯(j)(w) ℓ
dw P ,w
tions in the presence of redshift information, and for ar- × a2(w) δ f(w)
Z0 (cid:18) (cid:19)
bitrary survey geometries. We consider auto- and cross- wi f(w w′)
correlationsfor redshift bins (as in Hu 1999)and in addi- W¯(i)(w) ≡ dw′ p(i)(w′) f(−w′) , (4)
tion allowfor cross-correlationsbetweenmeasurements of Zwi−1
the shear signal at different angular scales. With covari- with p(i)(w) being the normalised distribution in comov-
ance matrices in hand, we then investigate the improve- ing distance inside the ith bin, where i runs between
ment in parameter estimates due to redshift binning. 1 and the number of redshift bins N . P(ii) are auto-
z κ
Further details and derivations of the equations rele- correlation power spectra, whereas P(ij) with i = j are
κ
vant to cosmic shear and weak lensing can be found in 6
cross-correlationpower spectra.
Bartelmann & Schneider (2001). For a recent review of
cosmic shear and future prospects see van Waerbeke &
2.2. Shear correlation functions
Mellier (2003).
Constraints can be placed on cosmological parameters
using the directly observable shear correlation functions,
2. Power spectrum and correlation functions
which we now turn to.
Accesstocosmologicalparametersisprovidedthroughthe Thebasiswhichunderpinstheuseofthedistortedim-
observabletwo-pointstatisticsoftheellipticitiesofdistant ages of distant galaxiesin weak lensing studies is a trans-
galaxies.Inthissection,wedescribehowthesearerelated formationrelatingthesource,ǫ(s),andimage,ǫ,(complex)
tothematterpowerspectrum,andtotheunderlyingden- ellipticities to the tidalgravitationalfield of density inho-
sity field. mogeneities (for definitions see Bartelmann & Schneider
2001). We focus on the non-critical regime where
2.1. The convergence power spectrum ǫ(s)+g
ǫ= ǫ(s)+γ , (5)
The power spectrum P (ℓ) of the effective convergence, 1+g∗ǫ(s) ≈
κ
or equivalently of the shear P (ℓ) (e.g. Bartelmann &
γ where g γ/(1 κ) is the reduced shear.
Schneider 2001), is related to that of the density fluctu- ≡ −
Empiricallytheprobabilitydistributionfunction(pdf)
ations, P (ℓ), through a variant of Limber’s equation in
δ of the galaxies’ intrinsic ellipticities is a truncated
Fourier space (Kaiser 1998)
Gaussian for both the real and imaginary parts of ǫ(s):
P (ℓ) = 9H04Ω2m wH dw W¯2(w)P ℓ ,w (1) exp ǫ(s) 2/σ2
κ 4c4 Z0 a2(w) δ(cid:18)f(w) (cid:19) pǫ(s) = πσ2 1−|exp| 1ǫ/(sσ)2 , (6)
W¯(w) wHdw′p(w′)f(w′−w) (2) ǫ(s) (cid:0)− − ǫ(cid:1)(s)
≡ Zw f(w′) where σǫ(s) is(cid:2)the intri(cid:0)nsic ellipt(cid:1)ic(cid:3)ity dispersion.
K−1/2sin(K1/2w) (K >0) As in Schneider et al. (2002b), the shear correlation
functions are defined as
f(w) = w (K =0) , (3)
( K)−1/2sinh[( K)1/2w] (K <0) ∞ dℓℓ
− − ξ±(θ)= γtγt γ×γ× = J0,4(ℓθ)Pκ(ℓ), (7)
h i±h i 2π
where ℓ isthe angular wave-vector, Fourier space con- Z0
jugate to θ. w is the comoving radial distance, K the where J are n-th order Bessel functions of the first kind;
n
curvature parameter. A value K = 0 corresponds to γt and γ× are the tangential and cross shear components
Ω + Ω =1,whereΩ isthecosmologicalconstantand respectively. From now on we focus on ξ , since this con-
m Λ Λ +
Ω thematterdensityparameter.ThefunctionW¯(w)ac- tains most of the cosmological information on the scales
m
countsforthe sourcesbeing distributedinredshift,where of interest.
Simon, King& Schneider:3D correlations of cosmic shear 3
2.3. Choice of cosmology and matter power spectrum Gaussian field is in our case simply the area on the sky
covered by the field.
Unless otherwise stated, our cosmology throughout is a
In ourwork,however,the situation is a bit more com-
ΛCDM model with Ω = 0.3, Ω = 0.7 and H =
70kms−1Mpc−1. A scmale-invariantΛ(n = 1, Harri0son- plicatedthanthat:theellipticitiesofgalaxiesbelongingto
differentredshiftbinsarecorrelatedaswellastheelliptic-
Zel’dovich) spectrum of primordial fluctuations is as-
ities of galaxies at different angular positions. Therefore,
sumed.Predictingtheshearcorrelationfunctionsrequires
when more than one redshift bin is considered, several
a model for the redshift evolution of the 3-D power spec-
Gaussian fields - cosmic shear maps - with prescribed
trum. We use the fitting formula of Bardeen et al. (1986;
cross-correlationshave to be simulated simultaneously. A
BBKS) for the transfer function, and the Peacock and
waytodothisingeneralonaregulargridforrealGaussian
Dodds (1996) prescription for evolution in the nonlin-
fieldsisshowninthefollowingsubsection.Thesubsection
earregime.Thepowerspectrumnormalisationisparame-
thereafter explains how we used this approach for simu-
terisedwithσ =0.9,andΓ=0.21.Quantitiescalculated
8 lating mock cosmic shear surveys.
for this fiducial cosmology/powerspectrum will be super-
scripted with a “t”.
3.1. Realisations of Correlated Gaussian Fields
3. Simulating cosmic shear surveys
According to condition (9) the pair ck and c−k is corre-
In this section we describe the method we used to make lated.InthissectionwerestrictourselvestorealGaussian
Monte-Carlosimulationsofcosmicshearsurveys.Theim- fields with δ(r) = δ∗(r). This introduces an additional
plementation makes the simulations as computationally condition that follows from the definition (8) of the ck:
inexpensiveaspossible,i.e.withoutinvokingN-bodysim-
∗
c =c . (10)
ulations. k −k
Calculating the lensing signal by ray tracing through
In particular, for real Gaussian fields we thereby have
N-bodysimulationshasbecomeacommontoolformaking
simulated weak lensing surveys (see e.g. Blandford et al. ∗ 1
ckck′ = Pkδkk′ , (11)
1991;Wambsganss,Cen&Ostriker1998;Jainetal.2000). h i V
We take a different path here, because only the two-
δkk′ being the Kronecker symbol. The conditions (9) and
point statistics of weak lensing is considered. This allows
(10) are easily accounted for if, say, only the ck for half
us to reduce the computational effort by expressing the
of the spatial frequencies k are worked out and the c−k
fields of the shear and convergence as random Gaussian
frequenciesaresetaccordingly.Hence,forourchoice,ifwe
fieldshavingthe samepowerspectrumasthe correspond-
talk about ck we actually mean only Fourier coefficients
ing fields from the N-body approach. In the weak lens-
in the right half-plane.
ing regime, ray-tracing is well described by the Born ap-
Furthermore, the real and imaginary parts of ck are
proximationwhichignorestheeffectsoflens-lenscoupling
uncorrelated,andbothfollowthesameGaussianpdf.This
and deviations of light rays from the fiducial path (see
pdf haszeromean1 anda varianceσk thatis expressedin
White & Hu 2000). The task of calculating the required
terms of the power spectrum Pk describing the two-point
powerspectrathenbecomesrelativelystraightforward,be-
correlations of the fluctuations in the Gaussian field (see
cause these can be shown to be linear functions of the
e.g. Peacock 2001)
three-dimensional evolving dark matter power spectrum.
The accuracy of the results depends on how accurately 1
that three-dimensional power spectrum is known. σk2 = 2V Pk . (12)
Sinceweconsiderthetwo-pointcosmicshearstatistics
The procedure for making one Gaussian field realisa-
inanareaofrelativelysmallangularsize,wecanrepresent
tion requires two steps: 1. drawing numbers for the real
thecosmicshearfieldsbyrandomGaussianfieldsinaflat
sky approximation. andimaginarypartsforeveryck withaGaussianrandom
number generator, and 2. transformation of this Fourier
Thepracticaladvantageofsimulatingasinglerandom
space representation to real space in order to obtain the
Gaussianfieldδ(r)-ahomogeneous andisotropicrandom
field realisation. For the second step we used an FFT al-
Gaussian to be exact - is the fact that from the Fourier
gorithm from Press et al. (1992)2.
coefficients
This procedure also holds when realisations of more
1
ck = dr δ(r)exp(ik r) (8) than one, but uncorrelated Gaussian fields are desired.
V ·
of such aZVrandom field only a pair of coefficients is corre- 1 In the case that δ(r) has a non zero mean, ck for k=0
h i
becomes different from zero.
lated
2 As in FFT the matrix of the Fourier coefficients contains
1
ckc−k′ = δkk′Pk , (9) ck that share the same matrix elements with c−k, one has for
h i V
these particular coefficients to set the imaginary parts to zero
where δD is the Dirac delta function and Pk is the power andtoincreaseσk bythefactor√2.Thelatterisnecessaryto
spectrum of the random field. The volume V of the guaranteethatthevarianceofthemodulusofckisstillcorrect.
4 Simon, King& Schneider:3D correlations of cosmic shear
“Uncorrelated” means that if we denote the Fourier co- where ATk denotes the transpose of Ak. Hence, together
efficients of, say, N Gaussian random fields by c(i) with with equation (14) this puts another constraint on the
k
i=1..N then we expect for those fields the relation matrix Ak, namely
c(ki) ck(j′) ∗ = 1Pk(ii)δijδkk′ , (13) V1Pk(ij) = [Ak]iq ATk qj . (17)
V q
D h i E X (cid:2) (cid:3)
where δ is also a Kronecker symbol, this time for the Forconvenienceweintroducethepower matrix definedas
ij
Gaussian field indices. Pk(ii) is the previously introduced [Pk]ij ≡ V1Pk(ij) to abbreviate this equation:
power spectrum, or auto-correlation power spectrum, of Pk =AkATk . (18)
the ith random field. Thus, here correlations between ck
of different random fields vanish. The power matrix is the covariance matrix between the
For the purposes of this work, however, we need to FouriercoefficientsofasetofGaussianfields foracertain
be able to allow for cross-correlations between different k.
random fields i=j in a defined manner, like This shorthandofN2 equationsdoes not uniquely de-
6 termine the matrix Ak, because it contains only N(N +
c(ki) ck(j′) ∗ = 1Pk(ij)δkk′ . (14) 1)/2linearlyindependentequations,sinceboththematrix
V onthelhsandthematrixproductontherhsaresymmet-
D h i E
P(ij) is for i = j the cross-correlation power spectrum. ric. As there are no further constraints on Ak, we are al-
k 6 lowedtosettheremainingN2 N(N+1)/2=N(N 1)/2
Likefortheauto-correlations,onlycertainpairsofFourier − −
constraintsofAk aswe like.We dothis by assumingthat
coefficientsofdifferentGaussianfieldsarecorrelated.This
Ak is symmetric, so that we finally obtain
followsfromtheassumptionthatthecross-correlationsare
homogeneous, too. Note that P(ij) =P(ji). Pk =A2k Ak = Pk . (19)
k k ⇒
Inordertofindarecipeformakingrealisationsofthat In general the squpare root is not unique (see e.g.
kind, we make the Ansatz that the N Fourier coefficients Higham1997).However,wearealreadysatisfiedwithone
c(ki) are a linear transformation Ak of N different uncor- particular solution to this problem.In order to determine
relatedcoefficientsd(i) withanequalGaussianpdfforthe such a solution, note that Pk is a symmetric positive
k
(semi)definite matrix, which is ensured by the properties
real and imaginary parts, zero mean and a 1/√2 disper-
of the power spectra the power matrix consists of:
sion
∗ P(ij) = P(ji) , (20)
d(ki) d(kj) =δij ; c(ki) = [Ak]iq dk(q) . (15) k 2 k
(ij) (ii) (jj)
D h i E Xq Pk ≤ Pk Pk . (21)
Ak isaN N lineartransformationmatrix.Thelinearity Thherefiore, this matrix can uniquely be decomposed into
×
ofthistransformationaccountsforthefactthattheresult-
ing set of coefficients c(i) still obeys a Gaussian statistics, Pk =RTk Dk Rk , (22)
k
becauselinearcombinationsofGaussianrandomvariables where Rk is an orthogonal matrix whose column vectors
are also Gaussian. are the eigenvectors of Pk, while their corresponding, al-
Since for realGaussianfields realandimaginaryparts ways real and positive, eigenvalues λi are on the diagonal
of the Fourier coefficients are not correlated, and by our of the diagonal matrix Dk = diag(λ1,λ2,...,λN). As one
Ansatz neither are the real and imaginary parts of d(i), particular square root we pick out
k
only realnumbers for the components [Ak]i are allowed; Ak = RTk Dk Rk
q
aimnaagdindaitriyonpaalrtismoafgidn(ai)r,ythpearretbyofpAosksibwloyuilndtrmodixucrienagl caonrd- Dk ≡ diagp λ1, λ2,..., λN (23)
k
relations between real and imaginary parts in ck. A ma- pwhich is a solu(cid:16)tpion dupe to √Dpk√D(cid:17)k =Dk.
trix Ak that is purely imaginary would be an alternative To sum up, for every k mode considered, the process
choice, though. for the realisation of correlated Gaussian random fields
Eqs.(15) can now be combined to give requires one to find the square root Ak of the power ma-
∗ ∗ trix Pk. This defines a linear transformation for a vec-
c(ki) ck(j) = [Ak]iqdk(q)[Ak]jr d(kr) tor of uncorrelated random complex numbers (real and
D h i E Xq,r D h i E imaginary part of the same coefficient are uncorrelated,
∗ too) with zero mean, real and imaginary parts obeying a
= [Ak]iq[Ak]jr dk(q) d(kr) Gaussian pdf with 1/√2 variance. Applying Ak yields a
Xq,r D h i E vector of Fourier coefficients belonging to the realisations
= [Ak]iq[Ak]jrδqr ofthecorrelatedGaussianrandomfields.Due to (10)this
q,r is performed only for one half of the spatial frequencies
X
= [Ak]iq ATk qj , (16) ccoonnsdiidtieorned.. The other half is set accordingly to fulfil this
q
X (cid:2) (cid:3)
Simon, King& Schneider:3D correlations of cosmic shear 5
3.2. Simulating the weak lensing survey γ˜ℓ andκ˜ℓ aretheFouriercoefficientsoftheshearandcon-
vergence fields for the angular frequency ℓ, respectively.
This relation stems from the fact that both shear and
convergence are linearly related to a potential function.
p(z)0.4 z z z z z
0 1 2 3 4 For every galaxy, shear and convergence are then com-
0.35 bined with the intrinsic ellipticity ǫ(s), randomly drawn
i
0.3 fromthepdfEq.(6)usingσ(s) =0.3,tocomputethefinal
ǫ
ellipticity of the galaxy via Eq.(5).
0.25
Both angular size ∆ and number of pixels N along
P
0.2 one axis - the sampling size - limits the number of fluc-
tuation modes accounted for in the simulated data. This
0.15
means, since we are lacking fluctuations on scales outside
0.1 of 5◦/(2N ) Θ 5◦, equivalent to ℓ ℓ ℓ ,
P min max
≤ ≤ ≤ ≤
that we have less correlation in the cosmic shear fields
0.05
than expected (Eq.7)
0
0 0.5 1 1.5 2 2.5 3 3.5 4 ℓmax dℓℓ
izt p(z) ξ±(θ) = 2π J0,4(ℓθ)Pκ(ℓ) (26)
Zℓmin
π
Fig.1. Galaxies are binned together according to their ℓmin = ; ℓmax =NP ℓmin . (27)
√2∆
redshift, the boundaries of the pairwise adjacent redshift
bins are w with i=0...N (here as an example N =4). The values for the limits are estimates for a square field;
i z z
Foreveryredshiftbinthereducedshearfieldiscalculated, the limits are not clearly defined, because the number of
averagingover the redshift distribution inside the bin. ℓ-modes in the FFT matrix becomes very small near the
cutoffs. One solution to this problem is to artificially set
a clearly defined range within the interval[ℓ ,ℓ ], or,
min max
as we havedone,to finda best fit cutoff. This is foundby
Each galaxy in the mock galaxy catalogue is defined by varying the cutoffs to obtain closest agreement between
an angular position, an ellipticity ǫ and a redshift bin it thetheoreticaltwo-pointcorrelationandtheensembleav-
belongs to. The ellipticity of the isophotes of a galaxy is erage of all Monte-Carlo realisations.
determined by the intrinsic shape of a galaxyǫ(s) and the Intotal,wesimulatedtwodatasets.Thefirstdataset
reduced shear g at the position of the galaxy (isee Eq.5). consistsofNf =795independentrealisationseach5◦ 5◦.
×
The reduced shear is a function of the convergenceκ and The redshift distribution of the galaxies was split into 2
shearγ whichhavetobesimulatedforeachredshiftbinas bins at a redshift cut zcut = 1.25, and the distribution is
a map covering the simulated area. Here we assume that truncatedat z =3.For ourfiducial surveys,we randomly
◦ ◦
the galaxiesarebinned into N pairwiseadjacentredshift selected 10 sub-fields, each of 1.25 1.25 , from differ-
z ×
bins, chopping off the redshift distribution entlargerealisations.Thiswasdonefortworeasons:1.to
reduce the computation time, since for 10 shear maps we
β
1 1 z require only one realisation, and 2. sub-fields are less af-
p(z)= z2exp (24)
z03Γ(3/β)β "−(cid:18)z0(cid:19) # fected by ℓmin that necessarily enters the simulations due
to the finite realisation area.
as in Fig.1. This empirical distribution with β =1.5 and
The second data set has 4 redshift bins, with z =
cut
z = 1.0 is based on deep field surveys (see e.g. Smail et
0 0.75,1.5,2.25,3.0 where the last value is the truncation
al. 1995). The total number of galaxies inside the field,
redshift. It has N = 266 independent realisations. The
with chosen size of 5◦ 5◦, is set to be 2.7 106, to f
× ≈ × fiducial surveys from this data set consist of single sub-
get an average of 30 galaxies per arcmin2. Moreover, the fields of size 1.25◦ 1.25◦.
galaxies are assumed to be randomly distributed overthe ×
Forbothdatasets,ξ wasestimated(seenextsection)
+
field of view.
forN =65angularseparationbins,rangingfromabout
∆θ
The method of the last subsection is used to work 2′.0 to 40′.0. For the first (second) data set, the correlation
out the convergence maps in Fourier space on a grid of
functions weresubsequently averagedfor10 (1)sub-fields
2048 2048pixelsforallredshiftbins.Asinputthepower
× in order to simulate cosmic shear surveys consisting of 10
matrix Pk, consisting of the auto- and cross-correlation (1) independent data fields.
powerspectraoftheseconvergencemaps,specifiedbythe
In a further step, the cross- and auto-correlation of
Eqs.(4), is needed.
the cosmic shear between the shear maps were, according
Inthenextstep,theshearmapsareobtainedfromthe
to appendix A, combined to yield the cosmic shear cor-
convergence maps using the relation
relations for a coarser redshift binning; in each step the
γ˜ℓ = ℓ21−ℓℓ2122++ℓ222iℓ1ℓ2 κ˜ℓ ℓ≡(cid:18)ℓℓ12(cid:19) . (25) ntwuomnbeeirghofboreudrsinhgiftbibnisn.sTwhaissrperdoucecsesdgbayveonfoerbtyhecofimrsbtidnaintag
6 Simon, King& Schneider:3D correlations of cosmic shear
Table 1. The final column denotes the name given to a ǫ¯ . In particular, all galaxies inside this cell are assumed
ij
particular binning of data. The entries in the columns z tobeplacedatthesameposition.Theestimatorofξ for
i +
show the corresponding cuts in redshift, z . this rearrangeddata set can be shown to be
cut
1
z0 z1 z2 z3 z4 Name ξˆ+(θ) =
N (θ)
p
0 0.75 1.5 2.25 3.0 4bins NijNkl(ǫ¯ijtǫ¯klt+ǫ¯ij×ǫ¯kl×)∆θ(Θij Θkl )
0 1.5 2.25 3.0 3binsI × | − |
ij,kl
0 0.75 2.25 3.0 3binsII X
0 0.75 1.5 3.0 3binsIII Np(θ) = NijNkl∆θ(Θij Θkl ) , (29)
| − |
0 2.25 3.0 2binsI ij,kl
X
0 1.5 3.0 2binsII where Θ represents the angular position of cell ij.
ij
0 0.75 3.0 2binsIII
The advantage of this approach is obvious: instead of
0 1.25 3.0 2binsIV
considering N2 pairs (N is the number of galaxies) we
0 3.0 onebin
have to consider only N2 pairs, where N is the number
c c
of grid cells. Thus, the number of pairs depends only on
thecellsizeandnotonthenumberofgalaxies.Hence,this
set, apart from the original data, the shear correlation of method pays off once the cell size becomes large enough,
one redshift bin with boundaries z = 0 and z = 3. The making the number of cells smaller than the number of
seconddatasetallowsmorefreedomofchoiceforcombin- galaxies. Moreover, in order to find all galaxies at some
ing redshift bins, so that we are able to construct several distance from a certain cell we no longer have to check
datasetswiththreeandtworedshiftbins.Table1liststhe all galaxies,but only neighbouring cells which areeasy to
different redshift bins and reference names, all extracted find by the grid index.
from the two original data sets. The approach becomes inaccurate, however, for small
Below, this data is used to study the improvement in angular bins, because for these the assumption that cell-
thestatisticaluncertaintiesofthecosmologicalparameter galaxies are essentially concentrated into one single point
estimates, if one has more information on the redshifts of isparticularlyinaccurate.Bycomparingthe ensembleav-
the galaxies. erage of ξˆ with the theoretical ξ we find that after the
+ +
third angular bin this approximation becomes accurate
enough.Forourpurposes,thisapproachiscompletelysuf-
4. Estimating ξ+ ficient. A better and more sophisticated approach can be
To estimate the two-point correlator ξ between the found in Pen & Zhang (2003).
+
galaxyellipticities ǫi - depending on positionΘi andred- For the case with Nz = 2, with the division at zcut =
shift bin - inside of the sub-fields we use the estimator 1.25 (2binsIV) Fig.4 shows the close agreement between
the correlation and cross-correlation functions, averaged
1
ξˆ (θ) = over7950sub-fields,withtheanalyticalpredictionforthe
+
N (θ)
p fiducialΛCDMcosmologicalmodel,obtainedfromEq.(7).
wiwj(ǫitǫjt+ǫi×ǫj×)∆θ(Θi Θj ) Shown are comparisons for the lower (L) and upper (U)
× | − |
redshift bins, and cross-correlation (LU). To account for
ij
X finite field size in our numerical work, ℓ = 2π/14.9◦
N (θ) = w w ∆ (Θ Θ ) min
p i j θ | i− j| in the integration. As noted above, since there is no well-
ij
X definedcut-off,thisvalueofℓ wasdeterminedbyallow-
min
∆θ(φ) ≡ 01 for θ−∆θ/2<φ≤oθt+he∆rwθi/s2e (28) ingittovarywhileperformingaleast-squaresfitofξLt,U,LU
(cid:26) to ξˆ , so obtaining the inverse variance weighted
L,U,LU
as mentioned in Schneider et al. (2002a), with ∆θ being meaDn ℓmin.EA cut-off at high ℓ is not critical since in this
the width of the angular bins. Since we are dealing with regime the power-spectrum amplitude is much lower.
simulated data here, there is no need to weight galaxies
with respect to their ellipticity. Therefore, we set wi = 1 5. Estimating the covariance of ξˆ±
for every galaxy.
Although mathematicallysimple,ittakesquite atime We now outline how the covariance matrix of ξˆ± is es-
to evaluate the estimator due to the large number of timated, for the case of N = 2 redshift bins, with the
z
galaxy pairs. To speed up the whole procedure we put divisionatz =1.25(2binsIV).As describedabove,our
cut
a grid of rectangular cells of size ∆θ ∆θ over the sub- mock survey consists of 10 uncorrelated fields. An angle
×
field in question and compute the number N of galaxies bracket denotes averaging over all 7950 sub-fields. Note
ij
andthemeanoftheirellipticitiesǫ¯ insideeverycell.The that we may dropthe L, U andLU sub-scripts for ease of
ij
index ij indicates the position of the cell inside the grid. notation.
This means we are representing galaxies inside the same When no redshift binning is assumed, it is computa-
cell by a single data point with weight N and ellipticity tionally advantageous to determine the shear correlation
ij
Simon, King& Schneider:3D correlations of cosmic shear 7
ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ
L LU U 11 12 13 14 22 23 24 33 34 44
ξ
11
ξ ξ
12
L
ξ
13
ξ
14
ξ ξ
22
LU
ξ
23
ξ
24
ξ
ξ
U 33
ξ
34
ξ
Fig.2. Schematic illustration of the symmetric covari- 44
ance matrix C for the case where there are N = 2
z
Fig.3.ThecovariancematrixCdeterminedfromoursim-
sourceredshiftbinsandN =5angularseparationbins.
∆θ
ulations, for N = 4, N = 65 and a survey consisting
Combinations of ξ identify covariance terms of the z ∆θ
L,U,LU
of 1 sub-field. Different blocks correspond to auto- and
form given in Eq.(31).
cross-correlations between the redshift bins. Inside these
blocks are auto-and cross-correlationsfor angular separa-
function by combiningthose determined forthe casewith tion bins.
redshift binning:
ξˆ=n2ξˆ +2n n ξˆ +n2ξˆ , (30) 0.001
L L L U LU U U
where n are the fraction of sources in the lower and
L,U
upper bins respectively. A more general relation between
ξˆand the cross- and auto-correlations of the shear from 0.0001
more than two redshift bins can be found in appendix A.
Theimpactofcosmicvarianceistakenintoaccountby θ()+
determiningcorrelationfunctionsforeachoftheN =795 ξ
f
independent surveys. The covariance matrix between bin 1e-05
i and j is determined using
C = ξˆ ξˆ ξˆ ξˆ , (31)
ij
(cid:28)(cid:16) −D E(cid:17)i(cid:16) −D E(cid:17)j(cid:29)Nf 1e-06 10
where the outer average is performed over the Nf = θ (arcmin)
795 surveys. If redshift binning is considered, there are
Fig.4. Comparison of the analytical (lines) and numeri-
N (N +1)/2 combinations of correlation and cross-
z z
cal(symbols)shearcorrelationandcross-correlationfunc-
correlation functions and hence C is a matrix composed
of [N (N +1)/2]2 blocks. Fig.2 illustrates this for the tions ξL,LU,U (lower, middle and upper lines/symbols).
z z
simplified case where N = 2 and N = 5; for exam-
z ∆θ
ple the block in the upper left of the matrix corresponds
A representation of the covariance matrix determined
to elements C = ξˆ ξˆ ξˆ ξˆ , and
ij L− L i L− L j fromoursimulationswithNz =4isshowninFig.3.Note
the shaded entry to(cid:28)C(cid:16) . TDheEbl(cid:17)ock(cid:16)in thDe Emi(cid:17)dd(cid:29)le row, that C has a strong diagonal, although it is not strictly
2,3
left column, corresponds to covariance elements between diagonally dominant.
the cross-correlation and the lower redshift bin, with the Our covariance matrix for the case of no redshift bin-
shaded entry being C . The bins denoted by i and j ex- ningisconsistent(<10%difference)withthetreatmentof
8,2
tend over ∆θ bins, repeated for each redshift auto- and Schneider et al. (2002a), and with Kilbinger & Schneider
cross-correlationbin. (2004) who adopted the same assumption of Gaussianity.
8 Simon, King& Schneider:3D correlations of cosmic shear
6. An application: Constraints on cosmological
parameters
Theratherfeaturelesstwo-pointshearcorrelationfunction
ξ (θ) or corresponding convergence (shear) power spec-
+
trum P (ℓ) leads to strong degeneracies amongst the pa-
κ
rameters that can be derived from cosmic shear surveys.
Anindicationofthedegreeofdegeneracyisthebehaviour
ofthepartialderivativesofξ withrespecttoeachparam-
+
eterπ (seeKing&Schneider2003forsuchacomparison),
i
or using a Fisher matrix analysis as in Sect.6.2.1.
External sources of information often provide compli-
mentary constraints: for example, confidence regions in
the Ω σ plane derived from weak lensing are almost
m 8
−
orthogonal to those from the analysis of CMB data (e.g.
Fig.5.ExpectedconstraintsintheΩ σ planeplotted
van Waerbeke et al. 2002), lifting this well known degen- m− 8
for ∆χ2 = 4.61 (90% confidence) with (inner contour)
eracy (e.g. Bernardeau, van Waerbeke & Mellier 1997).
and without (outer contour)redshift binning. The survey
In this section we consider the extent to which crude
consistsof10squareuncorrelatedsub-fields,each1.25◦on
redshift information for sources used in a lensing analy-
aside.Theredshiftdistributionandbinningaredescribed
sis decreases the expected errors in the Ω σ , Ω Γ
m− 8 m− in the text.
and σ Γ planes. Since we are interested in the influ-
8
−
ence of redshift binning on parameter degeneracies, hid-
den parameters are assumed to be perfectly known. As
described above, we focus on the information provided
by the shear two-point correlation function ξ (we may
+
drop the “+” subscript). We restrict this application to
the case of N = 2 (2binsIV). A larger parameter space
z
is then exploredusing the covariancematrix derivedfrom
simulations in a Fisher analysis for the cases N =2,3,4.
z
6.1. Obtaining confidence regions in the Ωm σ8,
−
Ωm Γ and σ8 Γ planes
− −
Wenowdetermineandcomparethelikelihoodcontoursin
the Ω σ , Ω Γ andσ Γ planes for the caseswith
m 8 m 8
− − −
and without redshift binning. The likelihood function is
Fig.6. Expected constraints in the Γ Ω plane plotted
given by − m
for ∆χ2 = 4.61 (90% confidence) with (inner contour)
1 and without (outer contour)redshift binning. The survey
(π) = (32)
L (2π)n/2 C1/2 is the same as in Fig.5.
| |
1
exp ξt ξ(π) C−1 ξt ξ(π) ,
× −2 − i ij − j
ij (cid:20) (cid:21) 6.2. Fisher information
Y (cid:0) (cid:1) (cid:2) (cid:3) (cid:0) (cid:1)
where n is the number ofrows(or columns) of the covari- The Fisher matrix (Fisher 1935) gives a handle on the
ance matrix C and ξ(π) are theoretical correlation func- question as to how accurately model parameters can be
tions determined on a grid in parameter space. estimated from a given data set. In this section, we will
The log-likelihood function is distributed as χ2/2 so use this method to examine quantitatively the increase of
that information on the cosmological parameters Ω , Ω , σ
m Λ 8
and Γ when the number of redshift bins, thus the knowl-
χ2(π)= ξt ξ(π) C−1 ξt ξ(π) . (33)
− i ij − j edge of the three-dimensionaldistributionof the galaxies,
ij
X(cid:0) (cid:1) (cid:2) (cid:3) (cid:0) (cid:1) is increased.Note that we no longer impose the condition
Confidence contours can be drawn in this χ2-surface, rel- Ωm+ΩΛ = 1. After a brief introduction to this topic we
ative to the minimum (zero) at ξ(π) ξt. In Figs.5-7 the apply the Fisher statistics to our simulated data.
≡
confidence contours are shown for each of the Ω σ ,
m 8
−
Ω Γ and Γ σ planes,with and without redshift bin-
m 8 6.2.1. Fisher Formalism
− −
ning.Note thatwhile Ω isvaried,wekeepΩ +Ω =1.
m m Λ
To highlight the difference and avoid confusion, we plot Ingeneral,oneusesdatapointsξ fromameasurementto
i
contours for a single value of ∆χ2. infer model parameters π based on a theoretical model.
i
Simon, King& Schneider:3D correlations of cosmic shear 9
eigenvectors of F. This corresponds to the directions of
degeneracies.
In the application of this formalism in Sect.6.2.2 be-
low, we will look at situations where some of the model
parametersareassumedtobeknownapriori.Inthiscase,
they are no longer free parameters that have to be esti-
mated from measured data points, so that the size of the
Fisher matrix reduces accordingto the number ofparam-
eters fixed. This amounts to removing rows and columns
fromthe generalFisher matrix,one for eachfixedparam-
eter,sothatthesecasescanbeconsideredbysimplylook-
ing at sub-matrices of the largest Fisher matrix. Taking
all conceivable sub-matrices enables the explorationof all
possible combinations of fixed (strong prior) and free pa-
Fig.7. Expected constraints in the Γ σ plane plotted rameters.
8
−
for ∆χ2 = 4.61 (90% confidence) with (inner contour) In practice, one uses the approximation (32) for the
and without (outer contour) redshift binning. The survey likelihood function (where ξt ξ), so that the Fisher
L ≡
is the same as in Fig.5. information matrix is approximately
∂ξ(π) ∂ξ(π)
F = C−1 , (38)
As the measurements are polluted by noise, we cannot ij ∂π kl ∂π
expecttoexactlyobtainthedatapointsξ(π)predictedby Xkl (cid:20) i (cid:21)k(cid:2) (cid:3) (cid:20) j (cid:21)l
ourmodel.Butwecantrytofindacombinationofmodel which is exact in the case of pure Gaussianstatistics, but
parametersπˆi thatpredictdatapointsascloseaspossible may be used as a good approximation for the valley in
to the actual measurement. The closeness is decided on parameter space in which the minimum of log lies.
the grounds of a statistical estimator. The covariance of Again, C is the covariance of the measured−dataLpoints
the parameter uncertainties andξ(π) the vectorofmodelleddata points inabsenceof
noise.
Q ∆π ∆π (34)
ij i j
≡h i
1/2
with ∆π πˆ2 πˆ 2 is related to the so-called 6.2.2. Application of the Fisher formalism
i ≡ i −h ii
Fisher inform(cid:16)a(cid:10)tion(cid:11)matrix(cid:17)through Now we use the Fisher formalism to estimate constraints
∂2log [π,ξ] on various combinations of parameters, with different de-
Fij ≡− ∂πL∂π = Q−1 ij . (35) greesofredshiftbinning.First,weevaluateEq.(38)using
(cid:28) i j (cid:29) D(cid:2) (cid:3) E the covariance matrix from our fiducial survey consisting
corresponds to the likelihood for obtaining the mea- of 10 independent sub-fields, N = 2 (with z = 1.25),
z cut
L
surement ξ keeping the underlying model parameter π and N = 65. The procedure is repeated for the covari-
i ∆θ
fixed. See for example Tegmark et al. (1997) and refer- ancematrixforthecoarserN =1binning.Table2shows
z
ences therein for a more detailed description. the percentage error for N =2 as opposed to N =1 for
z z
It follows from statistics that the 1σ scatter of the the same set of free and fixed parameters.
estimated parameters is (Cram´er-Raoinequality) Weextendthetreatmenttoalargernumberofredshift
bins,inthe contextofthe surveyconsistingof1sub-field.
∆π [F−1] , (36) AgainEq.(38)iscalculated,thistimeusingthecovariance
i ≥ ii
matrices from the simulations with N = 4, and those
q z
where commonly the lower limit is taken to be the esti- fromcoarserbinning(N =3,2,1)ofthisdataset.Table3
z
mate for ∆πi. To quantify the degeneraciesin the param- lists the errors for Nz = 4,3,2 as a percentage of the
eterestimate,weevaluatethecorrelationoftheestimate’s N =1 error.
z
uncertainty contained in F: In order to investigate the degeneracies of the param-
F−1 eter estimates, we concentrate on the case that no priors
∆π ∆π
r h i ji = ij (37) are given. For this particular situation, the gain by intro-
ij
≡ ∆π2 ∆π2 [F(cid:2)−1] [(cid:3)F−1] ducing redshift binning is largest (see Tables2 and 3). In
h ii j ii jj
Fig.8weplotthecorrelationsoftheerrorsintheparame-
q q
(cid:10) (cid:11)
as, for example, in Huterer (2002). Highly correlated terestimatesfordifferentpairsofparametersanddifferent
or anti-correlated ∆π and ∆π are called degenerate, numbersofredshiftbins.Ifmorethanoneredshiftbinning
i j
whereasnocorrelationmeansnodegeneracy(forthefidu- for the same number of redshift bins is available in our
cial model). Another piece of information that can be ex- data set, we indicate the scatter ofcorrelationcoefficients
tracted from the Fisher matrix is the orientation of the byerrorbars.Somescatterindicatesthatthecorrelations
errorellipsoidin parameterspace,whichis definedbythe can be changed slightly by varying the bin limits. The
10 Simon, King& Schneider:3D correlations of cosmic shear
strong correlation between the estimates of Ω and σ is Table 2. Uncertainties in the parameter estimates ac-
m 8
only marginally affected by redshift binning. This is also cording to the Fisher formalism, for our fiducial survey
the case for fixed Γ and/or Ω (not shown). of 10 uncorrelated sub-fields. The first data set (2binsIV)
Λ
with N = 2 and z = 1.25 is used. Columns with dots
z cut
“.” denote fixedparameters(strong priors).Uncertainties
in the top panel are absolute values for a single redshift
bin. Those in the lower panel are for N = 2, quoted
1 z
Ω /Γ as a percentage of the single redshift bin (Nz = 1) case.
m
For instance, with no fixed parameters, ∆Ω =0.26 with
Λ
0.5 N = 2 (i.e. 13% of the N = 1 error). F−1 denotes the
ΩΛ/σ8 dezterminantoftheinversezoftheFisher|matr|ix;itssquare
j rootis proportionalto the volumeofthe errorellipsoidin
i 0
r Ω /Γ parameter space. The nth root, with n being the number
Λ
of free parameters, defines a typical size of the error el-
-0.5 Ωm/ΩΛ lipsoid; this size is proportionalto the geometric mean of
σ /Γ the lengths of the principal ellipsoid axes.
8
-1 Ωm/σ8 ∆Ωm ∆ΩΛ ∆σ8 ∆Γ F−1 1/n
| |
1 2 3 4
p
redshift bins 0.9 2.0 1.2 0.4 0.16
. 0.5 0.1 0.1 0.09
Fig.8. Correlations of the errors in the parameter esti- 0.2 . 0.4 0.2 0.04
mates for different pairs of parameters and numbers of 0.09 0.7 . 0.08 0.08
redshift bins as derived from the Fisher matrix; only the 0.3 1.0 0.3 . 0.13
case with no fixed priors is considered. Error bars denote
0.06 0.5 . . 0.09
the variance in the correlations for the different redshift
0.08 . 0.1 . 0.05
binnings for the same number of bins (only for 2 and 3
0.02 . . 0.07 0.03
bins). The data points are slightly shifted to avoid over-
. 0.3 0.06 . 0.09
lapping.
. 0.1 . 0.06 0.08
. . 0.03 0.07 0.04
0.02 . . . 0.15
. 0.10 . . 0.10
. . 0.02 . 0.02
7. Discussion
. . . 0.06 0.06
The average shear correlation functions obtained from
13% 13% 17% 26% 47%
our numerical simulations are in good agreement with
. 52% 57% 64% 72%
those obtained analytically, as was illustrated in Fig.4.
53% . 52% 49% 73%
We also pointed out that their covariance is compatible
43% 38% . 68% 65%
with Schneider et al. (2002a) and Kilbinger & Schneider
32% 24% 46% . 57%
(2003).
Our treatment is only strictly valid for Gaussian den- 50% 45% . . 64%
sity fields and is a good approximation for scales greater 86% . 85% . 88%
than 10′, giving a lower limit on the covariance at 89% . . 81% 85%
∼
smaller scales (e.g. van Waerbeke et al. 2002). A more . 65% 71% . 76%
accuratecovariancematrixis possible,though.According . 80% . 81% 81%
to Schneider et al. (2002a)(section 4 therein), the covari- . . 87% 80% 84%
ance matrix of ξ may be decomposed into three terms
+
85% . . . 85%
C =σ2X +σ4Y +Z , (39) . 81% . . 81%
ij ǫ ij ǫ ij ij
. . 89% . 89%
whereX,Y andZ aresomefunctions.X andY arefunc- . . . 82% 82%
tionsofthetwo-pointcorrelationofcosmicshearandcon-
sequentlyinsensitivetonon-Gaussianfeaturesofthefield.
Z,however,dependslinearlyonthefour-pointcorrelation
ofcosmicshearwhichinSchneideretal.(2002a)isworked therandomfielddiffersfromthatvalueonlybyaconstant
out by assuming a Gaussian field; this factorises Z into a scale-independent factor Q, the so-calledhierarchicalam-
sum of products of two-point correlators only. In the hi- plitude (see e.g. Bernardeauet al. 2002).Thus, hierarchi-
erarchical clustering regime, the four-point correlation of cal clustering increases the component Z simply by the