Cosmic Shear Tomography and Efficient Data Compression using COSEBIs

1.6 MB·English

by  Marika Asgari

Astronomy & Astrophysics manuscript no. COSEBIs2 c ESO 2012
(cid:13)
January 13, 2012
Cosmic Shear Tomography and Efficient Data Compression
using COSEBIs
Marika Asgari1,2 , Peter Schneider1 , Patrick Simon1
1 Argelander-Institut fürAstronomie, Bonn University
2 SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ,
U.K.e-mail: [email protected]
Preprint online version: January 13, 2012
2
1
ABSTRACT
0
2 Context. Gravitational lensing is one of the leading tools in understanding the dark side of the Universe. The need

n for accurate, efficient and effective methods which are able to extract this information along with other cosmological
a parametersfromcosmicsheardataisevergrowing.COSEBIs,CompleteOrthogonalSetsofE-/B-Integrals,isarecently
J developed statistical measurethat encompasses thecomplete E-/B-modeseparable information contained in theshear
2 correlation functions measured on a finiteangular range.
1 Aims. The aim of the present work is to test the properties of this newly developed statistics for a higher-dimensional
parameter space and togeneralize and test it for shear tomography.
] Methods.WeuseFisheranalysistostudytheeffectivenessofCOSEBIs.Weshowourresultsintermsoffigure-of-merit
O quantities, based on Fisher matrices.
Results. We find that a relatively small number of COSEBIs modes is always enough to saturate to the maximum
C
informationlevel.Thisnumberisalwayssmallerfor‘logarithmicCOSEBIs’thanfor‘linearCOSEBIs’,andalsodepends
.
h onthenumberofredshiftbins,thenumberandchoiceofcosmologicalparameters,aswellasthesurveycharacteristics.
p Conclusions. COSEBIs provide a very compact way of analyzing cosmic shear data, i.e., all the E-/B-mode separable
- second-order statistical information in the data is reduced to a small number of COSEBIs modes. Furthermore, with
o thismethod thearbitrariness in databinningis nolonger an issue since theCOSEBIs modes are discrete. Finally, the
r small number of modes also implies that covariances, and their inverse, are much more conveniently obtainable, e.g.,
t
s from numerical simulations, than for the shear correlation functions themselves.
a
[ Key words. Gravitational lensing– cosmic shear: COSEBIs– methods: statistics

1
v
9 1. Introduction relationfunctionsξ±(ϑ),whichinrealitycanbedetermined
6 Aslighttravelsthroughthe Universe,the gravitationalpo- only on a finite interval ϑmin ≤ϑ≤ϑmax. These, however,
6 cannot be used for a comparison with theoretical models,
tential inhomogeneities distort its path; these distortions
2 since the shear field is in general composed of two modes:
result in sheared galaxy images and carry invaluable in-
. B-modes cannot be due to leading-order lensing effects,
1 formation about the matter distribution between the ob-
although they provide a measure of other effects such as
0 server and the source. Cosmic shear analysis is the study
2 shape measurement errors and intrinsic alignment effects
of the effects of large-scale structures on light bundles (see
1 (see Joachimi & Schneider 2010; also Schneider et al. 1998
Bartelmann & Schneider 2001). Consequently, it is one of
: and Schneider et al. 2002b for other effects). On the other
v the mostpromisingprobesforunderstandingtheUniverse,
hand, E-modes are the only relevant modes when it comes
i especiallydarkenergy.Theupcomingcosmicshearsurveys
X to comparing the cosmic shear data with models.
(e.g. Pan-STARRS1, KIDS2, DES3, LSST4, and Euclid5)
r will have better statistical precision compared to current Almost all of the recent analysis of cosmic shear
a
surveys, which means lower noise levels, larger fields of data employ methods of E-/B-mode separation (e.g.
view, deeper images, and more accurate redshift estima- Benjamin et al.2007andFu et al.2008).Thesestudiesare
tions. Trustworthy and accurate methods are able to ex- doneineitherFourierorrealspace.ForFourierspaceanaly-
tract all the potential information in these future observa- sisonehastofindanestimateofthepowerspectrum,which
tions and make the effort put into launching them worth- is sensitive to gaps and holes in the survey and in general
while. the survey geometry, which complicates such analysis. On
The most direct second-order statistical measurement the other hand the studies in real space do not share the
from any weak lensing survey are the shear two-point cor- same complications, since estimators of the shear correla-
tion functions are unaffected by such gaps. Most of these
1 http://pan-starrs.ifa.hawaii.edu/public/ studies use the aperture mass dispersion (Schneider et al.
2 http://www.astro-wise.org/projects/KIDS/ 1998), which applies compensated circular filters to the
3 http://www.darkenergysurvey.org shear field. As was shown in Crittenden et al. (2002) and
4 http://www.lsst.org/ Schneider et al. (2002a), the aperture statistics, in prin-
5 http://sci.esa.int/euclid/, Laureijs et al. 2011 ciple, cleanly separates the shear two-point correlations
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M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
(2PCFs) into E-/B-mode contributions. Furthermore, in the full information, but also provide a highly efficient and
thetwopapersjustmentioned,adecompositionoftheshear simple method for data compression.
2PCFsintoE-andB-modecorrelationfunctionξ (ϑ)has In this paper we further generalize the analysis in SEK
E,B
beenderived,whichalsohasbeenemployedincosmicshear to seven cosmological parameters, σ , Ω , Ω , w , n , h,
8 m Λ 0 s
analyses of survey data (Lin et al. 2011). and Ω , and investigate the effect of tomography on the
b
However, both the aperture statistics and the E-/B- results. Tomography, the joint analysis of shear auto- and
mode correlation functions are unobservable in practice. cross-2PCFs of galaxy populations with different redshift
Theaperturemassdispersionrequiresshapemeasurements distributions, is a powerful tool for cosmological analysis
of galaxy pairs down to arbitrarily small angular scales. (Albrecht et al. 2006; Peacock et al. 2006), in particular in
Since this is not feasible in real data, usually ray-tracing multi-dimensional parameter space (see Schrabback et al.
simulations fill in the gap, resulting in biases and E-/B- 2010for arecentpaperonconstraintsondarkenergyfrom
modemixing(seeKilbinger et al.2006).Ontheotherhand, cosmic shear analysis with tomography). We use Fisher
the determination of ξ (ϑ) requires the knowledge of analysis throughout our paper to represent the constrain-
E,B
ξ−(ϑ′) out to infinite ϑ′. Hence, in both cases, determin- ing power of COSEBIs, and compare the results from a
ing E-/B-mode separated statistics requires some sort of medium-sized with that of a large cosmic shear survey.
data invention. In Sect.2 we summarize the method used in SEK and
To overcome these problems, Schneider & Kilbinger write the corresponding relations for shear tomography.In
(2007) derived general conditions and relations for E- Sect.3 we briefly explain our choice of cosmology, and in
/B-statistics based upon two-point statistical quantities, Sect.4thecovarianceofCOSEBIsisshown.Wepresentour
namely 2PCFs and convergence power spectra. They de- figure-of-merit based on Fisher analysis and show the re-
fined the quantities sultsforthe sevencosmologicalparametersanduptoeight
redshiftbinsinSect.5.Finallyweconcludebysummarizing
∞
1 themostimportantresultsoftheprevioussectionsandem-
E = 2 dϑϑ[T+(ϑ)ξ+(ϑ)+T−(ϑ)ξ−(ϑ)], (1) phasizing the advantages of COSEBIs over other methods
Z0
∞ of cosmic shear analysis. We have also derived an analytic
1
B = dϑϑ[T+(ϑ)ξ+(ϑ) T−(ϑ)ξ−(ϑ)]; (2) solution to the linear COSEBIs weight functions presented
2 −
Z0 in App.A.
provided that the filter functions satisfy
2. COSEBIs
∞ ∞
dϑϑT+(ϑ)J0(ℓϑ)= dϑϑT−(ϑ)J4(ℓϑ), (3) ThereisaninfinitenumberoffilterfunctionsT (ϑ)satisfy-
+
Z0 Z0 ing Eq.(4). Such filters can be expanded in sets of orthog-
E depends only on the E-mode shear,and B depends only onal functions, labeled T+n(ϑ); the corresponding T−n(ϑ)
on the B-mode shear (with the aperture dispersion being are obtained from solving Eq.(3) which can be inverted
one particular example). Moreover, they have shown that explicitly (Schneider et al. 2002a). Accordingly, the corre-
in orderto obtain these statistics fromthe shear2PCFs on sponding E/B-statistics are denoted by En and Bn, re-
a finite angular interval, 0 < ϑ < ϑ < ϑ < , the spectively. Here we will also consider the case that dif-
min max
filter function T should have finite support on th∞e same ferent galaxy populations can be distinguished (mainly by
+
angular interval and satisfy theirredshifts);therefore,onecanmeasureauto-andcross-
correlations functions of the shear, ξij(ϑ). We denote the
±
ϑmaxdϑϑT+(ϑ)=0= ϑmaxdϑϑ3T+(ϑ). (4) corresponding COSEBIs by Enij and Bnij. They are related
totheauto-andcross-powerspectraoftheconvergence,by
Zϑmin Zϑmin
∞
dℓℓ
Whereas all solutions to the above relations provide Eij = Pij(ℓ)W (ℓ), (5)
n 2π E n
statistics which cleanly separate E-/B-modes on a finite Z0
∞
interval, different solutions may vary in their information dℓℓ
Bij = Pij(ℓ)W (ℓ), (6)
contents. For example, the ring statistics introduced in n 2π B n
Schneider & Kilbinger (2007) has a lower signal-to-noise Z0
where Pij are the E-/B-cross convergence power spectra
for a fixed angular range than the aperture dispersion, E/B
which,however,iscompensatedbyitsmorediagonalnoise- of galaxy populations i and j (see Schneider et al. 2002a),
covariance matrix resulting in comparable Fisher matrices and are related to the 2PCFs by
with aperture mass dispersion (Fu & Kilbinger 2010). ∞ dℓℓ
Recently, a complete solution of this issue was ob- ξ+ij(ϑ)= 2π J0(ℓϑ)[PEij(ℓ)+PBij(ℓ)], (7)
tained (Schneider et al. 2010, hereafter SEK) by defining Z0
∞
dℓℓ
Complete Orthogonal Sets of E-/B-Integrals (COSEBIs). ξij(ϑ)= J (ℓϑ)[Pij(ℓ) Pij(ℓ)]. (8)
COSEBIs capture the full information of the shear 2PCFs − 2π 4 E − B
Z0
on a finite interval which is E-/B-mode separable. In fact, Inserting the above relations into Eq.(1), one can find re-
SEK haveshownthata smallnumber ofCOSEBIs contain lations connecting Wn to T±n
all the information about the cosmological dependence in
theirtwo-parametermodel.Furthermore,theyshowedthat ϑmax
W (ℓ)= dϑϑT (ϑ)J (ℓϑ) (9)
n +n 0
COSEBIs in fact put tighter constraints on these parame-
Zϑmin
terscomparedtotheaperturemassdispersion.Eifler(2011) ϑmax
obtainedasimilarconclusionforafive-parametercosmolog- = dϑϑT−n(ϑ)J4(ℓϑ). (10)
icalmodel.Therefore,thesetofCOSEBIsnotonlycapture Zϑmin
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M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
Any type of cosmic shear analysis needs some sort of Table 1. Thefiducialcosmological parametersconsistentwith
error assessment. In particular Fisher analysis, used in the WMAP 7-years results. The normalization of the power spec-
presentwork,dependsonthenoise-covarianceofthestatis- trum,σ8,isthestandarddeviationofperturbationsinasphere
tics employed. The noise-covariance of COSEBIs for sev- of radius 8 h−1Mpc today. Ωm, ΩΛ, and Ωb are the matter,
the dark energy and the baryonic matter density parameters,
eralgalaxypopulationsassumingGaussianshearfields(see
Joachimi et al. 2008) is respectively.w0 isthedarkenergy equationofstateparameter,
which is equal to the ratio of dark energy pressure to its den-
sity.Thespectralindex,ns,istheslopeoftheprimordialpower
CX(ij,kl) XijXkl Xij Xkl spectrum. The dimensionless Hubble constant, in H0 = 100h
mn ≡h m n i−h mih n i km s−1 Mpc−1, characterizes the rate of expansion today.
∞
1
= dℓℓW (ℓ)W (ℓ)
2πAZ0 m n 0σ.88 0Ω.2m7 0Ω.7Λ3 −w10.0 0n.9s7 0.h70 0.Ω0b45
P¯ik(ℓ)P¯jl(ℓ)+P¯il(ℓ)P¯jk(ℓ) , (11)
× X X X X
(cid:16) (cid:17)
where
the linearfilterfunctionswhichoscillatefairlyuniformlyin
P¯Xik(ℓ):=PXik(ℓ)+δik2σn¯ǫ2 , (12) flionremalryscdaisletr,itbhuetleodgainriltohgm(ϑic),Ti+L.eno.g,hthaevyetahreeirmrooroetssefnasiritlyivuentio-
i
and X stands for either E or B. The surveyparametersare variationsofξ± onsmallerscales.Combiningthis property
alsoincludedinEq.(11)withthesurveyarea,A,thegalaxy with the fact that most of the cosmic shear information
intrinsic r.m.s ellipticity, σ , and the mean number density is contained in these smaller scales shows that it is more
ǫ
of galaxies in each redshift bin, n¯ . reasonable to employ Log-COSEBIs. In the next section
i
Inarecentpaper,Sato et al.(2011)haveshownthatthe we will show the difference of the Log- and Lin-COSEBIs
using our figure-of-merit.
GaussiancovariancemodelinJoachimi et al.(2008)overes-
In Fig.1 and Fig.2 the behavior of linear and logarith-
timatesthetrueGaussiancovarianceforsurveyswithsmall
area(A.1000deg2),andtheyhavedevelopedafittingfor- mic COSEBIs weight functions, WnLin(ℓ) and WnLog(ℓ), for
mula to correct for this discrepancy; in spite of their find- threeangularrangescanbeseen.TheWnLog(ℓ)andWnLin(ℓ)
have different yet similar oscillatory properties. They both
ingswewillsticktotheestimationofJoachimi et al.(2008),
since the fitting formula in the latter paper depends on die out rapidly with increasing ℓ but the lower frequency
source redshift and is developed for a single source galaxy oscillationsofWnLog(ℓ)aremoreprominent.Theyshowap-
proximatelythe sameinverserelationto ϑ andϑ for
redshift, making it non-applicable for this work. max min
their lower and upper limits.
Alternatively, one can write the covariance (Eq.11) in
terms of T±n and the two-point correlation functions’ co-
variance(see SEK).However,inthisapproachdouble inte-
3. Cosmological Model
grals over the covariance of 2PCFs slow down the calcula-
tions. The cosmological model assumed in the present work is
a wCDM model (Peebles & Ratra 2003 and references
therein),i.e.,acolddarkmattermodelincluding adynam-
2.1. The COSEBIs filter and weight functions
ical dark energy with an equation-of-state parameter w .
0
SEKconstructedtwocompleteorthogonalsetsoffunctions, The fiducial value of the parameters involved are listed in
linear and logarithmic COSEBIs (hereafter Lin- and Log- Tab.1.
COSEBIs respectively),by considering Eq.(4), and impos- Thestartingpointintheanalysisistoderivethematter
ing orthogonality conditions on the T filters. Once the powerspectrum.Forthelinearpowerspectrumweusedthe
+n
T+n filters are known,the T−n filters can be calculatedvia Bond & Efstathiou (1984) transfer function, and the halo
Eq.(3). The Lin-COSEBIs filters are polynomials in ϑ, the fit formula of Smith et al. (2003) for a fit of the non-linear
angular separation of galaxies,while the Log-COSEBIs fil- regime.
ters are polynomials in ln(ϑ). To calculate the convergence power spectrum we need
The output of theoretical cosmologicalmodels which is the redshift distribution of galaxies. The overall redshift
of relevence here is the power spectrum. Hence, the quick- probability distribution is parametrized by
est way to treat COSEBIs in theory is to work in ℓ-space
α β
and to use Eq.(11) for the covariance, without taking the β z z
p(z)= exp , (13)
detourofcalculatingtheshear2PCFsandtheircovariance. z0Γ[(1+α)/β] (cid:18)z0(cid:19) "−(cid:18)z0(cid:19) #
As a resultwe needto calculatethe W (ℓ)functions which
n
aretheHankeltransform(Eq.9)oftheirreal-spacecounter- which represents the galaxy distribution fairly well (it is a
parts, T±n. For convenience, we choose to evaluate Wn(ℓ) generalizationof Brainerd et al. 1996). The parameters,α,
from their integral relation with J and T . Since both β,andz dependonthesurvey.Weconsideramediumand
0 +n 0
J and T are oscillating functions, evaluating these inte- alargesurvey(hereafterMSandLSrespectively).TheMS
0 +n
gralsisratherchallenging,inparticularforlargeℓ.Apiece- has the same area as the CFHTLS (Fu et al. 2008), a cur-
wiseintegration,fromoneextremumtothenext,isusedin rentsurvey,whiletheLScoversthewholeextragalacticsky
the present work to evaluate W (ℓ). App.A contains more and represents future surveys. The parameters of our two
n
details about the numerical integrations and also a (semi- model surveys are given in Tab.2, and the corresponding
)analytic formula for the linear W functions. redshift distributions are plotted in Fig.3.
n
As is explained in SEK, the Log-COSEBIs are more ef- Constructing the Fisher matrix requires the derivatives
ficientforacosmicshearanalysis.Thereasonisthatunlike of the E-mode COSEBIs and of their covariances with re-
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M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
1e-03
n=10 1e-07
n=2
n=1 0e+00
5e-04
-1e-07
) 40000 60000 80000
(l
n
Li 0e+00
n
W
6e-06
-5e-04 0e+00
ϑ =1’,ϑ =400’ -6e-06
min max
3000 3100 3250
-1e-03
10 100 1000 10000
l
1e-03 1e-05
n=10
n=2
0e+00
n=1
5e-04
-1e-05
) 2000 3000 4000
(l
n
Li 0e+00
n
W
8e-06
-5e-04
0e+00
-1e-03 ϑ =20’,ϑ =400’ -8e-06
min max
3000 3100 3250
10 100 1000 10000
l
4e-06
n=10 3e-08
3e-06
n=2
n=1 0e+00
2e-06
-3e-08
1e-06
) 40000 60000 80000
(l
n
Li 0e+00
n
W
-1e-06 2e-08
-2e-06 0e+00
-3e-06 ϑ =1’,ϑ =20’
min max -2e-08
60000 62500 65000
-4e-06
100 1000 10000 100000
l
Fig.1. The weight functions WnLin(ℓ) are the Hankel transforms of T±Lin(ϑ) as in Eq.(9). In the blow-ups, the two modes of
oscillation for each WnLin can be seen, the lower frequency mode and the higher frequency mode which are inversely proportional
to ϑmin and ϑmax, respectively. The overall amplitude of theoscillations strongly dependson n and ϑmax.
4
M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
1e-03
n=10 1e-07
n=2
n=1 0e+00
5e-04
-1e-07
) 40000 60000 80000
(l
g
o 0e+00
L
n
W 6e-06
0e+00
-5e-04
-6e-06
ϑ =1’,ϑ =400’
min max
3000 3100 3250
-1e-03
10 100 1000 10000
l
1e-03 1e-05
n=10
n=2
0e+00
n=1
5e-04
-1e-05
) 2000 3000 4000
(l
g
o 0e+00
L
n
W
8e-06
-5e-04
0e+00
-1e-03 ϑ =20’,ϑ =400’ -8e-06
min max
3000 3100 3250
10 100 1000 10000
l
4e-06
n=10 3e-08
3e-06
n=2
n=1 0e+00
2e-06
-3e-08
1e-06
(l) 40000 60000 80000
g
o 0e+00
L
n
W
-1e-06 2e-08
-2e-06 0e+00
-3e-06 ϑ =1’,ϑ =20’
min max -2e-08
60000 62500 65000
-4e-06
100 1000 10000 100000
l
Fig.2. The weight functions WnLog(ℓ) are the Hankel transformation of T±Log(ϑ) as in Eq.(9). Similar to the WnLin, the position
of the first peak depends mainly on ϑmax and is rather insensitive to ϑmin. The difference between the two sets of linear and
logarithmic function can be seen most prominently in the blow-ups; the lower frequency oscillations are more pronounced in this
case.
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M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
Table2.Theredshiftdistributionparametersandthesurveyparametersforourmediumandlargesurveys.α,β,andz0determine
the total redshift distribution of sources, while zmin and zmax indicate the minimum and the maximum redshifts of the sources
considered.Aisthesurveyareainunitsofdeg2,σǫ isthegalaxyintrinsicellipticitydispersion,andn¯ isthemeannumberdensity
of sources persquare arcminute in the field.
z-distribution parameters surveyparameters
α β z0 zmin zmax A σǫ n¯
MS 0.836 3.425 1.171 0.2 1.5 170 0.42 13.3
LS 2.0 1.5 0.71 0.0 2.0 20000 0.3 35
1 1e-06
LS variable shape parameter
MS 1e-08 constant shape parameter
0.8
1e-10
p(z) 00..46 P(l)|κΩ, m 11ee--1142
| 1e-16
1e-18
0.2
1e-20
0
1e-22
0 0.2 0.5 1 1.5 2 2.5 3 1e+00 1e+01 1e+02 1e+03 1e+04 1e+05 1e+06 1e+07
z l-mode
Fig.3. The overall source redshift probability distribution of Fig.4. Theabsolute valueof thederivativeof theconvergence
source galaxies assumed for the two surveys. LS has a deeper power spectrum with respect to Ωm. Both of thecurvesrely on
source distribution compared to MS. a five point stencil method where 4 nearby points have to be
evaluated.Thesolidcurveisdrawnassumingallparametersare
fixed except Ωm and Γ, in contrast to the dotted curve where
instead of Γ, h or Ω are variable.
b
specttotheparameters.Forexample,totakethederivative
with respect to Ω , its relationto the shape parameter,Γ,
m
shouldbenotified.InthepresentworkweusetheSugiyama of COSEBIs,
(1995) relation, CE
r = MN , (15)
MN
CE CE
Γ=Ω hexp[ Ω (1+√2h/Ω )]. (14) MM NN
m b m
− q
the behavior of the off-diagonal terms becomes clearer.
IntheirderivativeswithrespecttoΩ ,SEKassumedacon- (The capital subscripts N and M can be different from
m
stant Γ, equivalent to allowingh or Ω to vary accordingly the COSEBIs subscripts, if several source populations are
b
(the only dependence of the convergence power spectrum considered;seebelowformoredetails.)Fig.6comparesthe
on h or Ω comes through Γ). In the present work h and correlationcoefficients for three different choices of the an-
b
Ω are independent parameters and Γ depends explicitly gular range, [1′,400′], [20′,400′], and [1′,20′], at a fixed
b
on Ω . The difference between the two approaches is not M =9.
m
negligible, as shown in Fig.4 which displays the derivative Cosmicshearanalysis,aswewillseeinSect.5,provides
of the power spectrum with respect to Ω in both cases. moreinformationwhenredshiftinformationisavailable.In
m
This difference is due to the non-linear relation between practice the redshifts of galaxies are estimated using sev-
h and Γ. To justify our choice of parametrization, we just eral photometric filters (see e.g. Hildebrandt et al. 2010),
mentionthattheconstraintsfromcosmologicalprobesonh from which an overall distribution for the source galaxies
is tighter comparedto Γ,andthatmakesit a morenatural is obtained.The distributionis thendividedintoa number
choice especially when priors are used. ofphotometricredshiftbins.Thephotometricredshiftsare
not exact, so the true redshift distributions will overlap.
Therefore, instead of redshift bins, in general one has to
4. COSEBIs Covariance consider redshift distributions. However, for simplicity, in
the presentworkwehaveassumedredshiftbins withsharp
Fig.5 shows the noise-covariance of linear and logarithmic cuts andno overlap.Inaddition, the bins are selectedsuch
E-modeCOSEBIsforthemodelparametersoftheMS(the that the number of galaxies in each bin is the same.
covariancehas a similar behavior in the case of the LS but In general,a tomographic covariancefor r redshift bins
with a different amplitude). This covariance is calculated consists of [r(r+1)/2]2 building blocks, each of which is a
fromEq.(11)assumingasinglesourceredshiftdistribution covariancematrix of (Eij,Ekl) where i,j,k,l are fixed and
n m
(Eq.13). Moreover, by defining the correlation coefficients n,m = 1,2,...,n . This means in total the covariance
max
6
M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
1 1’,400’
Linear COSEBIs 20’,400’
Covariance Matrix 0.8 1’,20’
6e-19
8e-19 [1’,400’] 4e-19 0.6 Lin-COSEBIs’
6e-19 2e-19 0.4 Correlation Coeficients
0
4e-19
n
2e-19 r9 0.2
0
-2e-19 0
14 -0.2
12
10
2 4 6 8 10 12 14 2 4 6 8 --00..64
2 4 6 8 10 12 14 16 18 20
n
1 1’,400’
Logarithmic COSEBIs 20’,400’
Covariance Matrix 0.8 1’,20’
6e-19
[1’,400’] 4e-19 0.6 Log-COSEBIs’
8e-19 2e-19
6e-19 0 0.4 Correlation Coeficients
4e-19
n
2e-19 r9 0.2
0
-2e-19 0
14 -0.2
12
10
2 4 6 8 10 12 14 2 4 6 8 --00..64
2 4 6 8 10 12 14 16 18 20
n
Fig.5.A3Drepresentationofthenon-tomographiccovariance Fig.6. The correlation coefficients of non-tomographic
of15E-modeCOSEBIsforanangularrangeof[1′,400′],forMS COSEBIsfordifferentangularranges[ϑmin,ϑmax]atm=9,for
parameters.Thex-andy-axescorrespondtotheelementsofthe the MS parameters. Here M, the capital subscripts, are equal
covariance matrix, and the value of the vertical axis shows the to theCOSEBIs mode, m.
valueofthecovarianceofthecorrespondingelement.Acontour
representation of the covariance is shown for each plot at its
base.
largernoise-correlationsbetweendifferentmodescompared
to Lin-COSEBIs, which may persuade one to choose the
Lin-COSEBIs for cosmic shearanalysis.However,the Log-
matrix has [r(r+1)n ]2/4 elements, where n is the
max max COSEBIs compensate this apparent disadvantage by re-
maximum number of COSEBIs modes considered.
quiringfewermodestosaturatetheFisherinformationlevel
Nevertheless, a covariance matrix is by definition sym-
forrelevantcosmologicalparameterscomparedtothelinear
metric anda tomographiccovarianceis made upofsmaller
ones, i.e., the number of covariance elements that have to
covariances, i.e., only x(x+1)/2 n (n +1)/2 ele-
× max max be calculated for Lin-COSEBIs is higher and hence deter-
ments, with x=r(r+1)/2, have to be calculated, the rest
miningtheircovariancematrixismoretimeconsuming,es-
are equal to these (see Fig.7).
pecially whenredshiftbinning isconsidered.Consequently,
The covariance of the Eij depends on six indices; in
n in Sect.5 we mainly employ Log-COSEBIs to analyze to-
order to apply normal matrix operations, the three indices
mographic Fisher information.
of Eij are combined into one ‘superindex’ N, given by
n
(i 1)(i 2)
N = (i 1) r − − +(j i) n +n, (16) 5. Results of Fisher analysis
max
− × − 2 − ×
h i 5.1. Figure-of-merit
wherer isthetotalnumberofredshiftbinsandn isthe
max
total number of COSEBIs modes. In this section we carry out a figure-of-merit analysis to
Using the new labeling, the correlation coefficients of demonstrate the capability of COSEBIs to constrain cos-
E11 and E23 (corresponding to N = 7 and N = 82, re- mological parameters from cosmic shear data. Our figure-
7 7
spectively) with the other Eij is shown in Fig.8, where 15 of-merit, f, based on the Fisher matrix, quantifies the
n
COSEBIsmodesand4redshiftbinsareconsidered.Eachof credibility of the estimated parameters. In general, for
the peaks in the figure correspond to the correlation coef- any unbiased estimator, the Fisher matrix gives the lower
ficient of E11 and Eij. The highest peak with r =1 occurs limit of the errors on parameter estimations (see e.g.
7 7
for M = N, while the rest of the peaks are correlations Kenney & Keeping1951andKendall & Stuart1960forde-
between different redshift bins. The Log-COSEBIs show tails).
7
M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
1 M=82
M=7
0.8
0.6
0.4
N
M 0.2
r
0
-0.2 Lin-COSEBIs
4 redshift bins
-0.4
Angular range: [1’,400’]
-0.6
0 20 40 60 80 100 120 140
N
1 M=82
M=7
0.8
0.6
Fig.7.Arepresentationofatomographiccovariance.Inthisdi- 0.4
agram3redshift-bins(1,2,3)and5COSEBIsmodesareassumed N
M 0.2
tobepresent.Theblow-upshowsoneofthecovariancebuilding r
blocks; the numbers 1-5 show the COSEBIs mode considered, 0
e.g.15meansthecovarianceofE1 andE5.Thenumbersonthe
-0.2 Log-COSEBIs
sides of the matrix show which combination of redshift bins is
considered, e.g., 12 means the covariance of redshift-bins 1 and -0.4 4 redshift bins
2 is relevant. Due to symmetry, only a part of the covariance Angular range: [1’,400’]
elements haveto becalculated, here shown in pink. -0.6
0 20 40 60 80 100 120 140
N
The Fisher matrix is related to the COSEBIs by Fig.8. The correlation coefficients of COSEBIs for an angular
range of [1′,400′] and 4 redshift bins. In total, 15 COSEBIs
1
F = Tr[C−1C C−1C +C−1M ], (17) modes are considered for each graph. The rMN is shown for
ij 2 ,i ,j ij M =7 corresponding to E11, and for M =82 corresponding to
7
E23.
where C is the COSEBIs covariance, M = E ET + 7
ij ,i ,j
E ET, E is the vector of the E-mode COSEBIs, and
,j ,i
the commas followed by subscripts indicate partial deriva-
tives with respect to the cosmological parameters (see 0.06 with C derivatives
without C derivatives
Tegmark et al. 1997 for example). We define our figure-of-
merit, f, in a very similar manner to SEK 0.05
1 1/np 0.04 angular range [1’,400’]
f = , (18)
(cid:18)√detF(cid:19) f CFHTLS, free σ8
0.03
where np is the number of free parameters considered. In 1 redshift bin
the following analysis, we will assume for simplicity that
0.02
the first term in Eq.(17) is much smaller than the second
and can thus be neglected. Note that this approximation
0.01
becomes more realistic in the case of a large survey area,
5 10 15 20
sincethefirsttermonther.h.s.ofEq.(17)doesnotdepend
n
on the survey area, while the second term is proportional max
to it (recallthat C ∝1/Aor in other words C−1 ∝A). We
checked that our medium survey is already big enough for Fig.9.AcomparisonbetweenasimplifiedandcompleteFisher
this approximationto hold (see Fig.9). analysis,usingLog-COSEBIs.Theasterisksshowthecasewhere
With the definition(18)we compressthe Fisher matrix thederivativesof thecovariance is taken intoaccount (thefirst
into a one-dimensional quantity, which provides a measure partofEq.17)whilethesquaresshowthesimplifiedcasewhere
of the geometric mean of the standard deviations of the we assume these derivatives are zero, in calculating f. Here σ8
parameters; e.g. in the case of one free parameter φ, f(φ) istheonlyfreeparameter,whereastherestoftheparametersis
isequaltothe standarddeviationσ(φ) ofthatparameter.6 fixed totheir fiducial values.
6 Anotherquantity,q,wasalsodefinedinSEKtomeasurethe
area of the likelihood regions. It is calculated from the second-
8
M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
0.1
Log-COSEBIs, constant Γ
Log-COSEBIs, variable Γ
0.09
Lin-COSEBIs, constant Γ
Lin-COSEBIs, variable Γ
0.08 2PCFs constant Γ
2PCFs variable Γ
0.07 Full-COSEBIs, constant Γ
Full-COSEBIs, variable Γ
f 0.06 angular range [1’,400’]
σ and Ω MS
0.05 8 m
Fig.10.AcomparisonbetweentheLog-andLin-
0.04 COSEBIs Fisher analysis results for two sets of
assumptions, where σ8 and Ωm are the free pa-
0.03 rameters and the rest is fixed to their fiducial
values.InonecasetheshapeparameterΓisheld
0.02 fixed,whileintheotheritisleftasavariablede-
10 20 30 40 50 pendingonΩmandthefiducialvaluesofhandΩb
(according to Eq.14). The same analysis is also
n
max carried out for the Full-COSEBIs and the shear
2PCFs.
0.5 0.05
Lin-COSEBIs Lin-COSEBIs
Log-COSEBIs Log-COSEBIs
2PCFs 2PCFs
0.4 0.04
angular range [1’,400’] angular range [1’,400’]
0.3 7 parameters LS 0.03 7 parameters LS
f 1 redshift bin f 2 redshift bins
0.2 0.02
0.1 0.01
0 0
10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50
n n
max max
Fig.11. These plots show one of our consistency checks. It is a comparison between the Lin- and Log-COSEBIs results for LS
with a single (left panel) and two galaxy redshift distributions (right panel), including all of the 7 parameters. Apart from very
smallnumericalinaccuracies,bothsetsofCOSEBIssaturatetothesamevalue,asexpected.Therearetwosolidlinesineachplot.
ThelinewiththehighervalueshowsthevalueofLog-COSEBIsatnmax =20, andtheotherlineshowsthevalueoff asobtained
from theshear 2PCFs. The slightly smaller value of f in the latter case (this differenceis not visible in theplot) is related to the
fact that the analysis from the shear 2PCFs implicitly assume the absence of B-modes, and thus contains information from very
large-scale modes which, however, cannot be uniquely assigned to either E- or B-modes. The comparison of the two plots shows
thatdividingthegalaxiesintotworedshiftbinsnotonlyincreasestheinformationcontentoftheFisheranalysisbutalsodecreases
the number of COSEBIs modes needed. Note that the x-axis of the single redshift distribution plot starts from 7, the other one
from 3.
5.2. Assumptions and parameter settings on. We implement the prior by adding the Fisher matrices
of our COSEBIs analysis and the CMB prior.The value of
Forourcosmicshearanalysisweconsideredamedium(MS)
theCMBpriorisshownintermsoff(φ)inthefirstcolumn
and a large survey (LS) as explained in Sect.3. We have of Tab.3 for each of the parameters.
also studied the effect of a Gaussian prior, in the form of
a Fisher matrix. This prior is the inverse of the WMAP7 We consider three different angular ranges, [1′,20′] ,
parameter covariance matrix from the final iteration (5000 [1′,400′] , and [20′,400′] . The motivation for this choice
sample points) of a Population Monte Carlo (PMC) run isasfollows:We consideratotalintervalof[1′,400′] where
(see Kilbinger et al.2010),calledthe CMB priorfromhere the flat sky approximation is still valid up to the maxi-
mum separation and galaxy shapes are easily distinguish-
able for the minimum separation; also, ϑ = 1′ avoids
order moments of the posterior likelihood. q and f are equal if min
theposteriorisamultivariateGaussian.Eifler(2011)hasshown the scales where baryonic effects are expected to have the
thatthedifferencebetweenf andqissmall,especiallyforalarge strongest ifluence. We further divide this interval into two
survey area. non-overlapping parts with ϑmax/ϑmin = 20, to compare
9
M. Asgari, P. Schneider P. Simon: Tomographic cosmic shear analysis with COSEBIs
cosmic shear information on small and large scales. The aration.Asexpected,inthiscasef becomesslightlysmaller
′ ′
small-scale range, [1,20] may apply for a cosmic shear since it is now implicitly assumed that all the signal is due
survey of individual one square degree fields. The large to E-modes. However, this is not justified in general; for
scale interval, [20′,400′] could be used for very conserva- example, very large-scale modes (i.e., small ℓ) enter ξ (ϑ)
+
tive analyses where non-linear and baryonic effects are to even for small ϑ, and such modes cannot be uniquely as-
be avoided. signed to either E- or B-modes. Thus, the decrease of f,
In Sect.5.3 we show the value of f for two parameters and accordingly, the information gain is just an apparent
while the rest are fixed to their fiducial values for the MS, one, bought by making a strong assumption. The relative
andalsoforallsevenparametersfortheLS.Inprinciplewe difference betweenthe 2PCFsandthe convergedLin-/Log-
couldshowallofthepossiblecombinationsforparameters, COSEBIsvaluesforf islargerforthevariableΓcase,since
nevertheless finding the error on each of the parameters heresmall-ℓmodes,whicharefilteredoutinthe COSEBIs,
seemsamorerelevanttask.Therefore,therestofouranal- contain information about the power spectrum shape.
ysis,carriedoutinSect.5.4,is donefora singleparameter, We also considered as further possibility that the re-
φ, where f(φ)=σφ. quirement of finite support for the ξ−(ϑ) is dropped, and
To findthe value off forasingleparameterweusetwo call this ‘Full-COSEBIs’. They form a complete set of
approaches.Inoneapproachwefixthesixotherparameters functions on [ϑ ,ϑ ], without the constraints given
min max
to their fiducial values inTab.1, while in the othercase we in Eq.(4).8 Though not physically reasonable, the Full-
marginalized over the remaining six parameters. COSEBIsareequivalenttomeasuringξ only,onthesame
+
Foreachsetupweinvestigatetheamountofinformation interval.AscanbeseenfromFig.10,thefullCOSEBIsyield
withrespecttothe numberofCOSEBIsmodesconsidered. aslightlylowervalueoff thanthetrueCOSEBIs,showing
In addition we analyze the behavior of f with the number that ξ− on scales larger than ϑmax adds apparent informa-
of redshift bins considered. tion,which,however,isnotobservable.Westressherethat
the E-/B-mode correlation functions ξ , introduced by
E/B
Crittenden et al. (2002) and Schneider et al. (2002a), are
5.3. Properties of COSEBIs
essentially equivalent to the Full-COSEBIs, since they are
For a fixed number of modes, the Log-COSEBIs are more also based on the assumption that ξ− can be measured to
sensitive than the Lin-COSEBIs to structures of the shear arbitrarily large separations – which, however, is not pos-
2PCFsonsmallscales.HereweshowitseffectontheFisher sible. Therefore, a cosmic shear analysis based on ξ (e.g.,
E
analysis.Weinvestigatethedependenceoff onthenumber Fu et al. 2008; Lin et al. 2011) underestimates the uncer-
nmax of COSEBIs modes incorporated in the analysis. tainties of cosmologicalparameters.
SEK have shown the difference between the behavior Furthermore, we compare the Lin- and Log-COSEBIs
of the Lin- and Log-COSEBIs for two parameters, σ8 and for LS parameters in Fig.11, for one and two redshift bins.
Ωm, with Γ fixed (their definition of f and fiducial values The x-axis in the left plot starts from 7 in contrast to the
of parameters are slightly different from ours). Similar to right one which starts from 3. The reason is that to con-
their work, we here compare the values of f for the same strainn parametersatleastn equationsareneeded,i.e.,if
p p
two parameters with Lin- and Log-COSEBIs. In addition, one redshift bin is considered, n COSEBIs modes should
p
we inspect the difference between a fixed shape parameter, be accounted for to produce a covariance matrix with at
or its dependence as given in Eq.(14). least n n elements. For more than one redshift bin,
p p
Fig.10 is a representation of our inspection for the MS a smaller×number of COSEBIs modes are sufficient, subse-
in the angular range of [1′,400′]. Two general conclusions quentlythesaturationrateoff isfaster,asisvisibleinthe
come out of this comparison: (1) f for fixed and depen- right plot in the figure. Recall that 2 redshift bins means
dent Γ converges to the same value for the Lin- and Log- 3differentredshiftcombinations,i.e.,for7parameters,the
COSEBIs.(2)Thevaluesoff forafixedordependentΓare smallest integer not less than 7/3 = 3 COSEBIs modes
different,andalsotheconvergencerateisdifferent.E.g.,the are needed. ⌈ ⌉
Lin-COSEBIs reach the saturated f value for n 40
max
≈
for a variable Γ, while in the other case, only 25 modes
are needed. This effect is less dramatic in the case of Log- 5.4. Forecast for parameter constraints
COSEBIs(theyneed7modesforavariableΓand5modes
This section is dedicated to our final results according to
for a constant one), since they generally converge faster.
the assumptions and parameters explained in Sect.5.2.
Similarly in Fig.11, we visualize our consistency check by
showing that the values of f for Log- and Lin-COSEBIs Fig.12shows the dependence of f for 20 Log-COSEBIs
converge to the same value for seven parameters.7 modes and for the [1′,400′] angular range on the number
InFigs.10and11,wealsoshowthevalueoff asderived of galaxy distributions (i.e., redshift bins), where all but
directlyfromtheshear2PCFs,i.e.,withoutE-/B-modesep- one parameter are marginalized over. Dividing the galaxy
distributioninto morethan4redshiftbinsdoesnotchange
7 Ingeneralthereareslightdifferencesbetweenthefinalvalue the value of f considerably. Nevertheless, a much larger
of f due to numerical inaccuracies, but these differences never number of redshift bins is required to control and correct
exceedafewpercentandaretypicallymuchsmaller. Anexcep- forsystematiceffects,e.g.,comingfromintrinsicalignments
tion happens when the saturation is too slow, and the Fisher (seeforexampleJoachimi & Schneider2010andreferences
matrix elements are too small, which is the case for MS with
therein).
one redshift bin and 7 parameters, observable especially after
marginalizing over6parameterswhentheremainingparameter
isw0 ,Ωm,ΩΛ orσ8.However,forthesecases, f ismuchlarger 8 Theyareobtainedbyaddingtwoadditionalweightfunctions
thanunity,i.e.,casesinwhichnomeaningfulconstraintscanbe T+ to those used in the COSEBIs; for the linear case, we just
obtained anyway. takeall Legendre polynomials (see SEK).
10