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arXiv:astro-ph/0511705v4 1 Mar 2006

Astronomy & Astrophysics manuscript no. 4586

February 5, 2008

c ? ESO 2008

Principal Component Analysis of Weak Lensing Surveys

Dipak Munshi1,2and Martin Kilbinger3

1Institute of Astronomy, Madingley Road, Cambridge, CB3 OHA, United Kingdom

2Astrophysics Group, Cavendish Laboratory, Madingley Road, Cambridge CB3 OHA, United Kingdom

3Argelander-Institut f¨ ur Astronomie⋆, Universit¨ at Bonn, Auf dem H¨ ugel 71, D-53121 Bonn, Germany

Received / Accepted

ABSTRACT

Aims. We study degeneracies between cosmological parameters and measurement errors from cosmic shear surveys. We simulate

realistic survey topologies with non-uniform sky coverage, and quantify the effect of survey geometry, depth and noise from

intrinsic galaxy ellipticities on the parameter errors. This analysis allows us to optimise the survey geometry.

Methods. We carry out a principal component analysis of the Fisher information matrix to assess the accuracy with which

linear combinations of parameters can be determined. Using the shear two-point correlation functions and the aperture mass

dispersion, which can directly be measured from the shear maps, we study various degeneracy directions in a multi-dimensional

parameter space spanned by Ωm, ΩΛ, σ8, the shape parameter Γ, the spectral index ns, along with parameters that specify the

distribution of source galaxies.

Results. A principal component analysis is an effective tool to probe the extent and dimensionality of the error ellipsoid. If

only three parameters are to be obtained from weak lensing data, a single principal component is dominant and contains

all information about the main parameter degeneracies and their errors. For four or more free parameters, the first two

principal components dominate the parameter errors. The degeneracy directions are insensitive against variations in the noise

or survey geometry. The variance of the dominant principal component of the Fisher matrix, however, scales with the noise.

Further, it shows a minimum for survey strategies which have small cosmic variance and measure the shear correlation up to

several degrees. This minimum is less pronounced if external priors are added, rendering the optimisation less effective. The

minimisation of the Fisher error ellipsoid can lead to slightly different results than the principal component analysis.

Key words. Cosmology: theory – gravitational lensing – large-scale structure of Universe – Methods: analytical, statistical,

numerical

1. Introduction

Recent

Anisotropy Probe

standard cosmological model with a very high degree of

accuracy (Spergel et al. 2003). In particular, these obser-

vations confirmed that the universe is spatially flat and

dominated by dark energy and dark matter. Regarding

the initial power spectrum of scalar perturbations, the

predictions of the simplest inflationary models were

strengthened, i.e. the near scale-invariance, adiabaticity

and Gaussianity of the initial density perturbations.

However, certain outstanding issues remain to be solved,

observationsbytheWilkinson

mission

Microwave

confirmed(WMAP)the

Send offprint requests to: munshi@ast.cam.ac.uk

⋆Founded by merging of the Sternwarte, Radioastro-

nomischesInstitutandInstitut

Extraterrestrische Forschung der Universit¨ at Bonn

f¨ urAstrophysikund

such as the running of spectral index αs which can be

addressed in more detail with additional data from galaxy

surveys such as SDSS (York et al. 2000), 2dF (Colless et

al. 2001) and the Lyman-α forest (see Seljak et al. 2003

and references therein).

Weak lensing surveys are expected to make impor-

tant and complementary contributions to high-precision

measurements of cosmological parameters. Contaldi et al.

(2003) used the Red Cluster Sequence (RCS) to show

that the Ωm-σ8 degeneracy direction is nearly orthogo-

nal to the one from CMB measurements, making weak

lensing particularly suitable for combined analyses (van

Waerbeke et al. 2002). Ishak et al. (2003) argued that a

joint CMB-cosmic shear survey provides an optimal data

set for constraining the amplitude and running of spec-

tral index which helps to probe various inflationary mod-

els. Tereno et al. (2004) studied cosmological forecasts

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2Dipak Munshi and Martin Kilbinger: PCA of weak lensing surveys

for joint CMB and weak lensing data. Clearly, the po-

tential of weak lensing surveys (Mellier 1999; Bartelmann

& Schneider 2001; R´ efr´ egier 2003; van Waerbeke & Mellier

2003; Schneider 2005) as a cosmological probe is now well

established (Contaldi et al. 2003; Hu & Tegmark 1999).

In the last few years there have been many studies which

have detected cosmic shear in random patches of the sky

(Brown et al. 2003; Bacon et al. 2003; Bacon, R´ efr´ egier

& Ellis 2000; Hamana et al. 2003; H¨ ammerle et al. 2002;

Hoekstra et al. 2002a; Hoekstra, Yee & Gladders 2002a;

Jarvis et al. 2002; Kaiser, Wilson & Luppino 2000; Maoli

et al. 2001; R´ efr´ egier, Rhodes, & Groth 2002; Rhodes,

R´ efr´ egier & Groth 2001; van Waerbeke et al. 2000, 2001,

2002; Wittman et al. 2000). While early studies were pri-

marily concerned with the detection of the weak lensing

signal, present generations of weak lensing observations

are putting constraints on cosmological parameters, in

particular the matter density parameter Ωmand the power

spectrum normalisation σ8.

Inspired by the success of these surveys, there are many

other ongoing, planned and proposed weak lensing sur-

veys which are currently in progress, including the Deep

Lens Survey (Wittman et al. 2002), the Canada-France-

Hawaii Telescope Legacy Survey (Hoekstra et al. 2005;

Semboloni et al. 2005), the Panoramic Survey Telescope

and Rapid Response System, the Supernova Acceleration

Probe (Massey et al. 2003), the NOAO Deep Wide-Field

Survey (Groch et al. 2002) and the Large Synoptic Survey

Telescope (Tyson et al. 2002). Future cosmic shear sur-

veys will be able to probe much larger scales in the linear

regime which will provide more stringent bounds on cos-

mological parameters such as the equation of sate of dark

energy and its time variations.

In a recent work (Kilbinger & Schneider 2004, hereafter

KS04), the impact of the survey design on cosmological

parameter constraints was analysed using a likelihood and

Fisher matrix analysis, extending previous studies based

on the assumption of uniform sky coverage (Schneider et

al. 2002a). Earlier work in this direction by Kaiser (1998)

considered a singe 3◦× 3◦-field and studied the effect of

sparse sampling and intrinsic ellipticity dispersion. The

motivation for the present work remains the same, al-

though we concentrate on the eigenvalues of the Fisher

matrix. Therefore, in contrast to KS04 where the 1σ errors

on individual parameters have been used to optimise the

survey geometry, we consider all parameter combinations

corresponding to the eigenvectors of the Fisher matrix.

We study how noise due to the intrinsic ellipticity disper-

sion of galaxies σǫ, the number density of galaxies ngal,

the survey depth and marginalisation affects the determi-

nation of the parameter combinations for various survey

strategies.

Cosmic shear is sensitive to a large number of cosmo-

logical parameters. However, the dependency on these pa-

rameters is partially degenerate (although these degenera-

cies can be broken by the use of external data sets such as

CMB, galaxy surveys and Lyman-α surveys). A principal

component analysis (PCA) can be used as an efficient tool

to identify the degeneracy directions and linear combi-

nations of cosmological parameters, rank-ordered accord-

ing to the accuracy with which they can be determined

from a given survey set-up. Indeed, in recent years there

has been a renewed interest in applying principal com-

ponent analysis techniques to various cosmological data

sets, a technique pioneered by Efstathiou & Bond (1999).

This method can reveal the detailed statistical structure

of cosmological parameter space which is lacking in an

one-dimensional confidence level presentation. Efstathiou

(2002) studied PCA in the context of the tensor degener-

acy in CMB. For a recent work see Rocha et al. (2004),

where the possibility of measurement of the fine-structure

constant α has been explored in the context of CMB data

with analysis based on Fisher matrix and PCA. Hu &

Keeton (2002) applied this technique to map the density

distribution along the radial direction from weak lensing

surveys. Jarvis & Jain (2004) used PCA to correct for the

point spread function (PSF) variation in weak lensing sur-

veys. In the context of SN Ia observations to constrain the

dark energy equation of state, Huterer & Starkman (2003)

and Huterer & Cooray (2004) employed PCA and its vari-

ants (see Wang & Tegmark 2005; Crittenden & Pogosian

2005 for more recent results). Tegmark et al. (1998) used

PCA for decorrelating the power spectrum of galaxies.

This idea was initially proposed by Hamilton (1997) and

further discussed in the context of galaxy surveys by

Hamilton & Tegmark (2000).

This paper is organised as follows: in Sect. 2, a very

brief overview of our notations is provided; in particular,

we introduce how the covariance matrix and the Fisher

matrix is constructed for a given estimator and a given sur-

vey strategy. This section also outlines the basics of prin-

cipal components analysis. In the next section (Sect. 3) we

provide the details of survey geometries and the numerical

results of the PCA considering three survey set-ups. For a

small number of cosmological parameters, (≤ 4) we con-

sider various survey strategies and try to optimise those.

For a larger set of parameters, we consider a ten times

larger shear survey. Effects of various noise sources on the

principal components are investigated. Section 4 is left for

discussions and future prospects.

2. Notation and formalism

2.1. Second-order shear statistics

In our numerical studies presented here, we use the two-

point correlation functions of shear ξ± and the aperture

mass dispersion ?M2

ap? to predict constraints on cosmolog-

ical parameters. Both these statistics depend linearly on

the convergence power spectrum Pκ(Kaiser 1992; Kaiser

1995; Schneider 1996; Schneider et al. 1998)

ξ±(θ) =

1

2π

∞

?

0

dℓℓPκ(ℓ)J0,4(ℓθ);

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Dipak Munshi and Martin Kilbinger: PCA of weak lensing surveys3

?M2

where Jνis the first-kind Bessel function of order ν.

Estimators of these statistics and their covariances are

defined in Schneider et al. (2002a). We use the Monte-

Carlo-like method from KS04 to integrate the analytical

expressions of the covariances, which are exact in case of

a Gaussian shear field. Our result is expected to underes-

timate the covariance due to non-Gaussian contributions

on scales between ∼ 1 and 10 arc minutes.

ap(θ)?

=

1

2π

∞

?

0

dℓℓPκ(ℓ)

?24J4(ℓθ)

(ℓθ)2

?2

, (1)

2.2. Fiducial cosmological model

We calculate the convergence power spectrum and the

shear estimators using the non-linear fitting formulae of

Peacock & Dodds (1996). Our cosmological model has

seven free parameters: These are the five cosmological pa-

rameters Ωm, ΩΛ, the power spectrum normalisation σ8,

the spectral index of the initial scalar fluctuations nsand

Γ, which determines the shape of the power spectrum. The

fiducial model is assumed to be a flat ΛCDM cosmology

with Ωm = 0.3, σ8 = 1, Γ = 0.21 and ns = 1. The two

parameters z0and β characterise the redshift distribution

of background galaxies (Brainerd et al. 1996),

p(z)dz =

β

z0Γ(3/β)

?z

z0

?2

e−(z/z0)βdz, (2)

with fiducial values z0= 1 and β = 1.5.

2.3. Principal components analysis of the Fisher matrix

We use the expression for the Fisher matrix (see Tegmark,

Taylor & Heavens 1997 for a review) in the case of

Gaussian errors and parameter-independent covariance,

Fij=

?

ij

?∂xk

∂Θi

?

(C−1)kl

?∂xl

∂Θj

?

,(3)

where xk is either ξ+(θk), ξ−(θk), ?M2

of the combined correlation function ξtot = (ξ+,ξ−). C

denotes the covariance matrix of the estimator of the cor-

responding shear statistics, Θ = (Θ1,...Θn) is the vector

of cosmological parameters. The inverse of the Fisher ma-

trix is the covariance of the parameter vector at the point

of maximum likelihood,

ap(θk)? or an entry

F−1= ?∆Θ∆Θt? = ?ΘΘt? − ?Θ??Θt?.(4)

The standard deviation of the ithparameter obtained from

the Fisher matrix, ∆Θi= (?Θ2

is called the minimum variance bound (MVB). According

to the Cram´ er-Rao inequality, the variance of any unbi-

ased estimator is always larger or equal to the MVB.

Any real matrix W is called a decorrelation matrix if

it satisfies

i?−?Θi?2)1/2= [(F−1)ii]1/2,

F = WtΛW, (5)

where Λ is a diagonal matrix (Hamilton & Tegmark 2000).

The quantities Φ = WΘ are uncorrelated because their

covariance matrix is diagonal,

?∆Φ∆Φt? = W?∆Θ∆Θt?Wt= Λ−1.(6)

By multiplying W with the square root of the diagonal

matrix Λ, the quantities Φ can be scaled to unit variance

without loss of generality. In this case, (5) is written as

F =˜ Wt˜ W, (7)

where˜ W = Λ1/2W. Note that the choice of˜ W is not

unique. If some matrix˜ W satisfies (7), the same is true

for any orthogonal rotation O˜ W with O ∈ SO(n) and

therefore, there are infinitely many decorrelation matrices

satisfying (7).

If W is an orthogonal matrix, its rows are the eigen-

vectors piof F and Λ = diag(λi) is the diagonal matrix of

the corresponding eigenvalues. In that case, (5) is called a

principal component decomposition. We assume the eigen-

values to be in descending order.

The eigenvectors or principal components of F deter-

mine the principal axes of the n-dimensional error ellipsoid

in parameter space. The eigenvectors represent orthogonal

linear combinations of the physical (cosmological)parame-

ters that can be determined independently from the data.

The more these vectors are aligned with the parameter

axes, the less are the degeneracies between those parame-

ters. The accuracy with which these linear parameter com-

binations can be determined is quantified by the variance

σi ≡ σ(pi) = ∆Φi = Λ−1/2

ii

pal component decomposition of the Fisher matrix gives

us information about which (linear) parameter combina-

tions can be determined with what accuracy from a given

data set. Since the eigenvalues are in descending order,

the first eigenvector p1having the smallest variance cor-

responds to the best constrained parameter combination.

The last eigenvector pn is the direction with the largest

uncertainty.

From (6), one can calculate the MVB from the eigen-

vectors and eigenvalues of F,

= λ−1/2

i

. Thus, a princi-

∆Θj=

?

n

?

i=1

W2

jiλ−1

i

?1/2

. (8)

3. Numerical results

3.1. Survey strategies

We simulate shear surveys consisting of P circular, uncor-

related patches of radius R on the sky, each in which N

individual fields of view of size 13′× 13′are distributed

randomly but non-overlapping. The total number of fields

of view is n = PN = 300, corresponding to a total survey

area of A = 14.1 square degree. Different surveys with

N = 10, 20, 30, 50, and 60 are considered, correspond-

ing to geometries with P = 30, 15, 10, 6 and 5 patches,

respectively (see KS04).

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4Dipak Munshi and Martin Kilbinger: PCA of weak lensing surveys

We denote these survey strategies with (N,R), e.g.

(50,100′) corresponds to a survey with N = 50,R = 100′

and P = 6. Further, we consider a survey consisting of 300

uncorrelated lines of sight ` a 13′×13′, which are randomly

distributed on the sky. This survey, denoted by 300·13′2,

has smaller cosmic variance than any of the patch strate-

gies, but does not sample intermediate and large angular

scales.

If not indicated otherwise, the number density of

source galaxies is ngal = 30arcmin−2. This number

density of high-redshift galaxies which are usable for

weak lensing shape measurements can be achieved with

high-quality ground-based imaging data on a 4 m-class

telescope. The source galaxy ellipticity dispersion is

σε = 0.3, if not stated otherwise. For comparison, these

quantities are varied to ngal = 20 and 40arcmin−2, and

σε = 0.2,0.4, respectively, to study the effect of noise

sources on the principal components.

3.2. Eigenvalues of the Fisher matrix

We consider the Fisher matrix F corresponding to all

seven cosmological and redshift parameters. In this sec-

tion, we demonstrate the influence of survey characteris-

tics (other than the geometry) on the eigenvalues λiof F.

The variance of the linear combination of parameters given

by the itheigenvector piof F is σ(pi) = λ−1/2

in Sect. 2.3. In Fig. 1 we show the effect of the number

of background galaxies, ngal, and the intrinsic ellipticity

dispersion, σε. In both cases, the noise variation causes a

scaling of the variance. For the intrinsic ellipticity disper-

sion, the scaling factor increases with i. The i = 1 vari-

ance scales linearly with σε, whereas the mean dependence

(averaged over all 7 eigenvalues) is quadratic in σε. In the

case of ngal, however, the variance σiof all eigenvectors is

scaled by a constant factor which is inversely proportional

to ngal. The variance of the eigenvectors are steeper func-

tions of the noise characteristics than the MVB (Kilbinger

& Munshi 2005). Note however, that in this previous study

the MVB was calculated for each parameter individually,

without taking parameter correlations into account.

i

, as defined

If boundary effects due to the finite field of the survey

are neglected, the covariance is anti-proportional to the

observed survey area. Consequently, the variance σiscales

as f−1/2

sky, where fskyis the fractional sky coverage of the

survey. As an example, we compare the 300 · 13′2survey

with a survey consisting of 50 patches with N = 60 and

R = 140′(corresponding to ten (60,140′) surveys). We

found a good agreement on the expected scaling of σias a

function of the survey area, although for the combinations

which are worst constrained by the data, the dependence

on fskyseems to be steeper (see Fig. 2).

Fig.1. The variance σ(pi) = λ−1/2

principal components of the 7×7 Fisher matrix. The

straight lines are fits to the data points. Lower left panel:

σ(pi) as a function of the intrinsic ellipticity dispersion of

galaxies σε = 0.2 and 0.4. Lower right: the variation of

σ(pi) with the number density of galaxies ngal= 20 and

40. Upper left: the variation of σ(pn) for three different

estimators, ξ+, ξ−and ξtot. Upper right: the variation of

σ(pi) for three different priors, see Sect. 3.11. The survey

strategy is (30,100′) in all cases.

i

associated with the

Fig.2. The variation of σ(pi) as a function of the eigen-

vector number i, for ξtot (left panel) and ?M2

panel). The redshift parameter β is fixed, and a Gaussian

prior with variance s(σ8) = 0.1 is added to regularise the

Fisher matrix. Two surveys with size 14.1 (solid lines) and

141 square degree (dashed), respectively, are displayed.

ap? (right

3.3. Eigenvectors of the Fisher matrix

In Tables 1–6, the eigenvectors of the Fisher matrix cor-

responding to various combinations of parameters are

shown, for the 2PCF and the aperture mass dispersion.

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Dipak Munshi and Martin Kilbinger: PCA of weak lensing surveys5

Table 5. Eigenvectors of the Fisher matrix corresponding to (Ωm, Γ, σ8, ns) and the prior ΩΛ= 0.7, for ξtot(ξ+,ξ−

in brackets) using the survey strategy (50,100′).

p1

p2

p3

0.282

(0.320)+

(0.329)−

−0.217

(−0.116)+

(−0.205)−

−0.474

(−0.572)+

(−0.522)−

0.805

(0.746)+

(0.760)−

p4

△Θj

0.154

(0.158)+

(0.290)−

0.182

(0.198)+

(0.389)−

0.279

(0.289)+

(0.538)−

0.209

(0.290)+

(0.414)−

Ωm

0.657

(0.708)+

(0.591)−

0.474

(0.406)+

(0.547)−

0.542

(0.548)+

(0.533)−

0.217

(0.179)+

(0.257)−

−0.588

(−0.533)+

(−0.647)−

0.719

(0.768)+

(0.654)−

−0.062

(0.005)+

(−0.129)−

0.363

(0.352)+

(0.367)−

−0.376

(−0.332)+

(−0.349)−

−0.459

(−0.480)+

(−0.480)−

0.690

(0.611)+

(0.653)−

0.415

(0.535)+

(0.470)−

Γ

σ8

ns

λ−1/2

i

(λ−1/2

i

(λ−1/2

i

0.004

(0.005)+

(0.006)−

0.013

(0.016)+

(0.020)−

0.165

(0.255)+

(0.221)−

0.389

(0.408)+

(0.804)−

)+

)−

Table 6. Eigenvectors of the Fisher matrix corresponding to (Ωm, Γ, σ8, ΩΛ), for ξtot(ξ+,ξ−in brackets) using the

survey strategy (50,100′).

p1

0.676

(0.722)+

(0.613)−

0.479

(0.408)+

(0.558)−

0.555

(0.556)+

(0.549)−

−0.060

(−0.032)+

(−0.092)−

p2

0.556

(0.501)+

(0.619)−

−0.773

(−0.808)+

(−0.718)−

0.022

(−0.039)+

(0.088)−

0.303

(0.307)+

(0.304)−

p3

0.082

(0.030)+

(0.348)−

0.385

(0.371)+

(0.400)−

−0.339

(−0.260)+

(−0.721)−

0.854

(0.891)+

(0.445)−

p4

△Θj

0.147

(0.172)+

(0.213)−

0.094

(0.112)+

(0.156)−

0.244

(0.290)+

(0.333)−

0.220

(0.232)+

(0.448)−

Ωm

−0.475

(−0.475)+

(−0.343)−

−0.156

(−0.207)+

(0.110)−

0.759

(0.788)+

(0.441)−

0.417

(0.332)+

(0.837)−

Γ

σ8

ΩΛ

λ−1/2

i

(λ−1/2

i

(λ−1/2

i

0.005

(0.005)+

(0.006)−

0.014

(0.017)+

(0.021)−

0.210

(0.223)+

(0.364)−

0.308

(0.362)+

(0.499)−

)+

)−

Table 1. Eigenvectors of the Fisher matrix corresponding

to (Ωm, Γ, σ8) and a flat Universe as prior, for ?M2

the survey strategy (50,100′). λ−1/2

the itheigenvector, Θjthe MVB for the jthcosmological

parameter.

ap? using

i

= σiis the variance of

p1

p2

p3

△Θj

0.165

0.042

0.252

Ωm

Γ

σ8

λ−1/2

i

0.649

0.563

0.511

−0.533

0.816

−0.222

−0.542

−0.128

0.830

0.0040.0200.300

The best determined eigenvector p1is always orthogo-

nal to the Ωm-σ8degeneracy direction. The variance σnof

Table 2. Eigenvectors of the Fisher matrix corresponding

to (Ωm, Γ, σ8, ns) and the prior ΩΛ= 0.7, for ?M2

the survey strategy (50,100′).

ap? using

p1

p2

p3

0.322

−0.235

0.501

0.767

p4

∆Θj

0.254

0.333

0.473

0.329

Ωm

Γ

σ8

ns

λ−1/2

i

0.563

0.575

0.522

0.281

−0.671

0.622

−0.161

0.368

−0.358

−0.474

0.670

0.443

0.0040.0150.1470.698

the worst constrained principal component dominates the

uncertainty of all eigenvectors. In the case of (Ωm,σ8,Γ),

σ2

3= 1/λ3 constitutes more than 99% of the total un-