Measuring cosmic shear with the ring statistics
aa r X i v : . [ a s t r o - ph . C O ] J u l Astronomy&Astrophysicsmanuscript no. paper c (cid:13)
ESO 2018October 31, 2018
Measuring cosmic shear with the ring statistics
T. Eifler , P. Schneider and E. Krause ,
1) Argelander-Institut f¨ur Astronomie, Universit¨at Bonn, Auf dem H¨ugel 71, D-53121 Bonn, Germany2) California Insitute of Technology, M / C 350-17, Pasadena, California 91125, USA
ABSTRACT
Context.
Commonly used methods to decompose E- and B-modes in cosmic shear, namely the aperture mass dispersion and the E / B-modeshear correlation function, su ff er from incomplete knowledge of the two-point correlation function (2PCF) on very small and / or very largescales. The ring statistics, the most recently developed cosmic shear measure, improves on this issue and is able to decompose E- and B-modesusing a 2PCF measured on a finite interval. Aims.
First, we improve on the ring statistics’ filter function with respect to the signal-to-noise ratio. Second, we examine the ability of thering statistics to constrain cosmology and compare the results to cosmological constraints obtained with the aperture mass dispersion. Third,we use the ring statistics to measure a cosmic shear signal from CFHTLS (Canada-France-Hawaii Telescope Legacy Survey) data.
Methods.
We consider a scale-dependent filter function for the ring statistics which improves its signal-to-noise ratio. To examine theinformation content of the ring statistics we employ ray-tracing simulations and develop an expression of the ring statistics’ covariance interms of a 2PCF covariance. We perform a likelihood analysis with simulated data for the ring statistics in the Ω m - σ parameter space andcompare the information content of ring statistics and aperture mass dispersion. Regarding our third aim, we use the 2PCF of the latestCFHTLS analysis to calculate the ring statistics and its error bars. Results.
Although the scale-dependent filter function improves the S / N ratio of the ring statistics, the S / N ratio of the aperture mass dispersionis higher. In addition, we show that there exist filter functions which decompose E- and B-modes using a finite range of 2PCFs ( EB -statistics)and have higher S / N ratio than the ring statistics. However, we find that data points of the latter are significantly less correlated thandata points of the aperture mass dispersion and the EB -statistics. As a consequence the ring statistics is an ideal tool to identify remain-ing systematics accurately as a function of angular scale. We use the ring statistics to measure a E- and B-mode shear signal from CFHTLS data. Key words. cosmology: theory - gravitational lensing - large-scale structure of the Universe - methods: statistical
1. Introduction
Cosmic shear was first detected in 2000 (Bacon et al. 2000;Kaiser et al. 2000; van Waerbeke et al. 2000; Wittman et al.2000) and has progressed to a valuable source of cosmologi-cal information. Latest results (e.g., van Waerbeke et al. 2005;Semboloni et al. 2006; Hoekstra et al. 2006; Schrabback et al.2007; Hetterscheidt et al. 2007; Massey et al. 2007b; Fu et al.2008) already indicate its great potential to constrain cosmo-logical parameters, which will be enhanced by large upcomingsurveys like Pan-STARRS, KIDS, DES or Euclid.An important step in a cosmic shear analysis is the decom-position into E- and B-modes, where, to leading order, grav-itational lensing only creates E-modes. In principle, B-modescan arise from the limited validity of the Born approximation(Jain et al. 2000; Hilbert et al. 2008) or redshift source clus-tering (Schneider et al. 2002b). Another possible source areastrophysical contaminations such as intrinsic alignment ofsource galaxies; King & Schneider (2003) show how to sep-
Send o ff print requests to : tim.eifl[email protected] arate the cosmic shear signal from intrinsic alignment contam-inations if redshift information is available. The strength of B-modes coming from these e ff ects are examined through numer-ical simulations; although the results di ff er (e.g. Heavens et al.2000; Crittenden et al. 2001; Jing 2002), the observed B-modeamplitude is higher than expected from the foregoing expla-nations. Shape-shear correlation (Hirata & Seljak 2004) is an-other astrophysical contamination which can cause B-modes.Joachimi & Schneider (2008, 2009) show how to exclude thecontaminated scales, again using redshift information.Most likely, B-modes indicate remaining systematics in the ob-servations and data analysis, in particular they can result froman insu ffi cient PSF-correction. The Shear TEsting Program(STEP) has significantly improved on this issue (for latest re-sults see Heymans et al. 2006; Massey et al. 2007a); still theaccuracy of the ellipticity measurements must be improved fur-ther to meet the requirements of precision cosmology.The identification of remaining systematics (B-modes) will beimportant especially for future surveys, where the statistical er-rors will be significantly smaller. Therefore, decomposing the T. Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics shear field into E- and B-modes must not be a ff ected from in-herent deficits. The most commonly used methods for an E-and B-mode decomposition, the aperture mass dispersion andthe E / B-mode shear correlation function, require the shear two-point correlation (2PCF from now on) to be known down toarbitrary small or up to arbitrary large angular separations, re-spectively. This is not possible in practice; as a consequence thecorresponding methods do not separate E- and B-modes prop-erly on all angular scales. A detailed analysis of this issue canbe found in Kilbinger et al. (2006) (hereafter KSE06).Most cosmic shear analyses, e.g. Massey et al. (2007b) andFu et al. (2008) (hereafter FSH08), simulate 2PCFs from a the-oretical model of P κ to account for the scales on which the2PCF cannot be obtained from the data. This ansatz is problem-atic, since one explicitly assumes that the corresponding scalesare free of B-modes. In addition, the assumed cosmology in thetheoretical power spectrum can bias the results.The ring statistics (Schneider & Kilbinger 2007, hereafterSK07) provides a new method to perform an E- / B-mode de-composition using a 2PCF measured over a finite angular range[ ϑ min ; ϑ max ]. In this paper we examine the ring statistics in de-tail; more precisely we improve the ring statistics’ filter func-tion with respect to its S / N ratio and examine its ability to con-strain cosmological parameters. Furthermore, we construct afilter functions which has higher S / N ratio than the ring statis-tics but still decomposes E / B-modes with a 2PCF measuredover a finite range. We will refer to this as EB -statistics.Due to the fact that the ring statistics’ data points show sig-nificantly lower correlation than data points of the aperturemass dispersion and the EB -statistics, it provides an ideal toolto identify remaining systematics in cosmic shear surveys de-pending on the angular scale. We employ the ring statistics toidentify B-modes in the CFHTLS survey.The paper is structured as follows: In Sect. 2 we start with thebasics of second-order cosmic shear measures, followed by themain concepts of the ring statistics in Sect. 3. We derive a for-mula to calculate the ring statistics’ covariance from a 2PCFcovariance in Sect. 5 and also compare the correlation coef-ficients of ring statistics, aperture mass dispersion, and EB -statistics in this section. In the same section we examine theS / N ratio of the ring statistics and compare it to the other mea-sures. More interesting than the S / N ratio however, is the abil-ity of a measure to constrain cosmology. This, in addition tothe S / N ratio, depends on the correlation of the individual datapoints. In order to quantify this accurately, we perform a like-lihood analysis in Sect. 6 for the ring statistics, aperture massdispersion and EB -statistics using data from ray-tracing simu-lations. The results of our analysis of CFHTLS data using thering statistics are presented in Sect. 7 followed by our conclu-sions in Sect. 8.
2. Two-point statistics of cosmic shear
In this section we briefly review the basics of second-order cosmic shear measures. For more details on this topicthe reader is referred to Bartelmann & Schneider (2001);Schneider et al. (2002a,b); van Waerbeke & Mellier (2003);Munshi et al. (2008). To measure the shear signal we define ϑ as the connectingvector of two galaxy centers and specify tangential and cross-component of the shear γ as γ t = − Re (cid:16) γ e − ϕ (cid:17) and γ × = − Im (cid:16) γ e − ϕ (cid:17) , (1)where ϕ is the polar angle of ϑ . The 2PCFs depend only on theabsolute value of ϑ . They are defined in terms of the shear andcan be related to the power spectra P E and P B (Schneider et al.2002b) ξ ± ( ϑ ) ≡ h γ t γ t i ( ϑ ) ± h γ × γ × i ( ϑ ) (2) = Z ∞ d ℓ ℓ π J / ( ℓϑ ) [ P E ( ℓ ) ± P B ( ℓ )] , (3)with J n denoting the n -th order Bessel-function.Starting from the 2PCF as the basic observable quantity, thereexist several methods to decompose E-modes and B-modes,such as the E / B-mode shear correlation function or the aperturemass dispersion (e.g. Crittenden et al. 2002; Schneider et al.2002b). The latter can be calculated as h M / ⊥ i ( θ ) = Z θ d ϑ ϑθ " ξ + ( ϑ ) T + ϑθ ! ± ξ − ( ϑ ) T − ϑθ ! . (4)The filter functions read T + ( x ) = − x )5 " − π arcsin (cid:18) x (cid:19) + x √ − x π (120 + x − x + x − x (cid:17)o H(2 − x ) , (5) T − ( x ) = π x − x ! / H(2 − x ) , (6)with H being the Heaviside step function. Decomposing E- andB-modes with the either the aperture mass dispersion or theE / B-mode shear correlation function requires that the 2PCF iseither measured down to arbitrary small or large angular sep-aration, respectively. For further details on this problem thereader is referred to KSE06.
3. The ring statistics
To circumvent the aforementioned di ffi culties SK07 in-troduced the ring statistics whose second-order moments( hRR E i , hRR B i ) decompose E- and B-modes properly using2PCFs measured on a finite interval [ ϑ min ; ϑ max ]. The quan-tity hRR E i can be interpreted as the correlator of the shearmeasured from galaxy pairs which are located inside two con-centric rings (see Fig. 1). Their annuli are chosen as follows: ζ ≤ θ ≤ ζ for the first ring and ζ ≤ θ ≤ ζ for the sec-ond. The rings are non-overlapping, i.e. ζ i < ζ j if i < j . Theargument of the rings statistics is named Ψ = ζ + ζ and only2PCFs with ϑ ≤ Ψ enter in the calculation of hRR E i . In addi-tion, the ring statistics depends on a parameter η quantifyingthe separation between outer and inner ring, i.e. η/ Ψ = ζ − ζ .In order to calculate the ring statistics properly from a set of2PCFs within [ ϑ min ; ϑ max ] it is required that Ψ does not exceed ϑ max and that ϑ min / Ψ ≤ η < . Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics 3 Fig. 1.
This figure illustrates the basic idea of the ring statis-tics and how it can be obtained from the 2PCF of cosmic shear.We measure the 2PCF of each galaxy in the inner ring withall galaxies in the outer ring. For a given argument of the ringstatistics Ψ , the angular separation of the required 2PCFs ex-tends over η Ψ ≤ ϑ ≤ Ψ . The meaning of η and its possible val-ues are further explained in the text. The ring statistics is thencalculated as an integral over the 2PCF with the filter functions Z ± ( ϑ, η ).position of the ring statistics can be obtained from the 2PCFas hRR E i ( Ψ ) = Z Ψ η Ψ d ϑ ϑ (cid:2) ξ + ( ϑ ) Z + ( ϑ, η ) + ξ − ( ϑ ) Z − ( ϑ, η ) (cid:3) , (7) hRR B i ( Ψ ) = Z Ψ η Ψ d ϑ ϑ (cid:2) ξ + ( ϑ ) Z + ( ϑ, η ) − ξ − ( ϑ ) Z − ( ϑ, η ) (cid:3) . (8)The functions Z ± are defined in SK07; we plot them in Fig. 2for four di ff erent η , i.e. ϑ min / Ψ = . , . , . , . hRR E i canbe related to the E-mode power spectrum. Inserting Eq. (3),into Eq. (7) gives hRR E i ( Ψ ) = Z ∞ d ℓ ℓ π P E ( ℓ ) W E ( ℓ Ψ , η ) (9)with W E ( ℓ Ψ , η ) = Z Ψ η Ψ d ϑ ϑ (cid:2) J ( ℓϑ ) Z + ( ϑ, η ) + J ( ℓϑ ) Z − ( ϑ, η ) (cid:3) . (10)When calculating hRR E i for di ff erent arguments Ψ , we distin-guish two cases for η . It can be fixed to a specific value or itcan vary according to Ψ , in particular η = ϑ min / Ψ . We willrefer to the latter case as a scale-dependent η . Here, the lowerlimit in the integrals of Eqs. (7) and (8) is equal to ϑ min whichimplies that all 2PCFs in the interval [ ϑ min ; Ψ ] are included in the calculation. The choice of η = ϑ min / Ψ should give a higherS / N ratio compared to a fixed η for the reason that more galaxypairs are included which reduces the statistical noise. In SK07the authors hold η fixed; in order to obtain a high signal thisimplies that η must be chosen as small as possible.Choosing a fixed η has a second disadvantage. The lower limitin the integrals Eqs. (7) and (8) cannot be smaller than ϑ min , i.e. η Ψ ≥ ϑ min . Vice versa, this implies that Ψ ≥ ϑ min /η . Fixing η to a small value (in order to increase the S / N ratio) implies that Ψ is restricted to larger scales. This trade-o ff between S / N ratioand small-scale sensitivity can be overcome when relaxing thecondition of a fixed η .
4. General E/B-mode decomposition on a finiteinterval
The ring statistics described in the last section is the specialcase of a general E / B-mode decomposition. According to SK07this general EB -statistics can be defined as E = Z ∞ d ϑ ϑ (cid:2) ξ + ( ϑ ) T + ( ϑ ) + ξ − ( ϑ ) T − ( ϑ ) (cid:3) , (11) B = Z ∞ d ϑ ϑ (cid:2) ξ + ( ϑ ) T + ( ϑ ) − ξ − ( ϑ ) T − ( ϑ ) (cid:3) . (12)To provide a clean separation of E- and B-modes using a 2PCFmeasured over a finite interval, the following conditions mustbe fulfilled (see SK07 for the exact derivation). Starting froman arbitrary function T + ( ϑ ), which is zero outside the interval[ ϑ min ; ϑ max ], the constraints Z ϑ max ϑ min d ϑ ϑ T + ( ϑ ) = = Z ϑ max ϑ min d ϑ ϑ T − ( ϑ ) (13)must hold. For a so constructed filter function T + ( ϑ ) a corre-sponding filter function T − ( ϑ ) can be calculated as T − ( ϑ ) = T + ( ϑ ) + Z ϑϑ min d θ θϑ T + ( θ ) " − (cid:18) θϑ (cid:19) . (14)Conversely, one can construct T + for a given T − .The expressions for T + and T − used in this paper are given inthe Appendix. We calculate the EB -statistics according to Eq.(11) and compare the results to the ring statistics. Note that this EB -statistics can be optimized, e.g., with respect to its S / N ratioor its ability to constrain cosmology. For more details on thistopic the reader is referred to Fu & Kilbinger (2009).In this paper, the EB -statistics is calculated as a function of Ψ .Similar to the ring statistics, Ψ denotes the maximum angularscale of 2PCFs which enter in the calculation of E ( Ψ ).
5. Covariance and signal-to-noise ratio
For our further analysis we have to derive a formula to calcu-late the covariance of ring statistics and EB -statistics. A corre-sponding expression for h M i reads (see e.g. Schneider et al.2002b).C M ( θ k , θ l )) = I X i = J X j = ∆ ϑ i ∆ ϑ j θ k θ l ϑ i ϑ j T. Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics − − − J [arcmin] Z + h = h = h = h = − − − J [arcmin] Z − h = h = h = h = Fig. 2.
This plot shows the filter functions Z + ( left ) and Z − ( right ) depending on ϑ for four di ff erent choices of η : ϑ min / Ψ = . solid ), 0.1 ( dashed ), 0.4 ( dotted ), 0.7 ( dotted dashed ). × X m , n =+ , − T m ϑ i θ k ! T n ϑ j θ l ! C mn ( ϑ i , ϑ j ) , (15)with C mn ( ϑ i , ϑ j ) denoting the 2PCF covariance. Here, the upperlimits I and J are chosen such that ϑ i ≤ θ k and ϑ j ≤ θ l . Thering statistics’ covariance is defined asC R ( Ψ k , Ψ l ) = D ˆ R ( Ψ k ) ˆ R ( Ψ l ) E − hRR E i ( Ψ k ) hRR E i ( Ψ l ) , (16)where ˆ R denotes the estimator of the ring statistics. To cal-culate this estimator from a binned 2PCF data vector with binwidth ∆ ϑ i we replace the integrals in Eq. (7) by a sum over thebinsˆ R ( Ψ ) = I X i = ∆ ϑ i ϑ i h ˆ ξ + ( ϑ i ) Z + ( ϑ i , η ) + ˆ ξ − ( ϑ i ) Z − ( ϑ i , η ) i , (17)with ˆ ξ ± ( ϑ i ) denoting the estimator of the i -th 2PCF bin. Theupper limit I in Eq. (17) denotes the bin up to which ϑ i ≤ Ψ .Inserting Eq. (17) into Eq. (16) we deriveC R ( Ψ k , Ψ l ) = I X i = J X j = ∆ ϑ i ∆ ϑ j ϑ i ϑ j × X m , n =+ , − Z m ( ϑ i , Ψ k ) Z n ( ϑ j , Ψ l ) C mn ( ϑ i , ϑ j ) , (18)where I and J denote the bins up to which ϑ i ≤ Ψ k ( ϑ j ≤ Ψ l )holds.Similarly a covariance for the general EB -statistics can be cal-culated asC E ( Ψ k , Ψ l ) = I X i = J X j = ∆ ϑ i ∆ ϑ j ϑ i ϑ j × X m , n =+ , − T m ( ϑ i , θ k ) T n (cid:16) ϑ j , θ l (cid:17) C mn ( ϑ i , ϑ j ) . (19) In order to illustrate the correlation between the individual datapoints we calculate the correlation matrix R for hRR E i , E , and h M i from the corresponding covariance matrix. For C beingthe covariance of either hRR E i , E , or h M i the correlation co-e ffi cients are defined asR i j = C i j p C ii C j j . (20)The covariances are calculated from a 2PCF ray-tracing co-variance via Eqs. (15), (18), and (19), respectively; finally thecorrelation matrix is obtained via Eq. (20). The ray-tracing sim-ulations (175 realizations) have the following underlying cos-mology: Ω m = . , Ω Λ = . , σ = . , h = . , Ω b = . , n s = .
0. From now on we refer to this cosmological pa-rameter set as our fiducial cosmological model π fid . Survey pa-rameters which enter in the calculation read as follows: galaxydensity n gal = / arcmin , survey area A =
36 deg , and in-trinsic ellipticity noise σ ǫ = .
38. The survey parameters di ff erslightly from those of the covariance used in the latest CFHTLSsurvey; FSH08 use A = . , n gal = . / arcmin , and σ ǫ = . ff erent angular range corre-sponding to the data vectors of hRR E i , E , and h M i , which wedefine as hRR E i = [ hRR E i ( Ψ ) , ..., hRR E i ( Ψ n )] t , (21) E = [ E ( Ψ ) , ..., E ( Ψ n )] t , (22) h M i = h h M i ( θ ) , .., h M i ( θ m ) i t . (23)Whereas hRR E i and E extend from 1 ′ ≤ Ψ ≤ ′ , h M i extends from 6 ′ ≤ θ ≤ ′ . The di ff erent maximum angularseparation of the aperture mass dispersion result from the fact . Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics 5 Y [arcmin] Y [ a r c m i n ] < RR E > h = J min Y Y [arcmin] Y [ a r c m i n ] E q [arcmin] q [ a r c m i n ]
10 20 50 100 200 < M ap2 > Fig. 3.
This figure shows the correlation matrices of hRR E i ( left ), E ( middle ), and h M i ( right ) derived from ray-tracing 2PCFcovariance matrix. In each panel the n -th contour line (starting with n = . n .that hRR E i ( Ψ ) and E ( Ψ ) contain information on the 2PCF with ϑ ≤ Ψ , whereas h M i ( θ ) contains information on the 2PCFup to ϑ ≤ θ . The lower limit of 6 ′ was chosen to circumventthe problem of E / B-mode mixing for the h M i covariance. Therange of the original 2PCF ray-tracing covariance extends from0 . ′ ≤ ϑ ≤ ′ . Below 6 ′ it is not possible to calculate the h M i covariance properly.Figure 3 shows the correlation matrices of the ring statistics(left), the EB -statistics (middle), and of the aperture mass dis-persion (right). Starting from the diagonal, where R ii =
1, the n -th contour line corresponds to values of 0 . n . It is clearly no-ticeable that data points of the ring statistics are significantlyless correlated than those of the aperture mass dispersion andthe EB -statistics.The boxy contours in Fig. 3 result from the small number ofbins we choose in the covariances. The reason for this is thatthe ray-tracing covariance is an estimated quantity; its inverse,needed for the likelihood analysis in Sect. 6, is in general af-fected from numerical artifacts. These artifacts become moresevere in case of a high dimension matrix. In order to guar-antee a stable inversion process we choose a small number ofbins. We now use the above derived covariances to quantify the S / Nratio of the ring statistics, EB -statistics and compare both tothat of the aperture mass dispersion.We calculate a set of 2PCFs via Eq. (3) for an angular rangesimilar to that of the ray-tracing simulations (see Sect. 5.1), i.e. ϑ ∈ [0 . ′ , ′ ]. The required shear power spectra P E are ob-tained from the density power spectra P δ employing Limber’sequation. As underlying cosmology we choose our fiducialmodel (see Sect. 5). The power spectrum P δ is calculated froman initial Harrison-Zeldovich power spectrum ( P δ ( k ) ∝ k n s )with the transfer function from Efstathiou et al. (1992). For thenon-linear evolution we use the fitting formula of Smith et al. (2003). In the calculation of P E we choose a redshift distribu-tion of source galaxies similar to that of Benjamin et al. (2007) n ( z ) = β z Γ ((1 + α ) /β ) zz ! α exp − zz ! β , (24)with α = . β = . z = . hRR E i , E , and h M i according to Eqs. (7) and (11), and (4), respec-tively. The angular range of these data vectors are chosen sim-ilar to the range of the corresponding covariances (Sect. 5.1),i.e. 0 . ′ ≤ Ψ ≤ . ′ hRR E i and E , and 6 . ′ ≤ θ ≤ . ′ h M i . The S / N ratio is calculated asS / N = hRR E i ( Ψ i )[C R ( Ψ i , Ψ i )] / and S / N = h M i ( θ i )[C M ( θ i , θ i )] / . (25)The results are illustrated in Fig. 4. We compare the ringstatistics for with scale-dependent η and η = . EB -statistics and the aperture mass dispersion. The figure showsthe anticipated behavior (Sect. 3); the ring statistics with scale-dependent η gives a larger S / N ratio compared to the case where η is fixed. In addition, it can be measured down to arbitrarysmall values of Ψ (above ϑ min ), which is not possible whenchoosing a fixed η . For the case considered here, i.e. ϑ min = . ′ η = . Ψ to scales ≥ ′ ; decreasing η further in order to increase the S / N ratio willlimit hRR E i to larger Ψ .When comparing the ring statistics to the aperture mass dis-persion, we find that the ring statistics’ signal is lower. Evenwith the scale-dependent filter function the S / N ratio of thering statistics is on average by a factor of ≈ / N ratio of the aperture mass dispersion. This di ff erence canbe explained when comparing the filter functions of hRR E i and h M i , Z ± (Fig. 2) and T ± (e.g. Fig. 1 in Schneider et al. 2002b),respectively. The Z -functions have two roots at their boundarieswhereas the T + -function becomes particularly large for small x .However, we point out that the S / N ratio does not solely deter-mine the ability of a measure to constrain cosmology, but onehas to account for the fact that the data points of the hRR E i areless correlated than those of h M i . For a full comparison of the T. Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics Y [arcmin] S / N < M ap2 >< RR E > ( h = J min Y ) < RR E > ( h = Fig. 4.
The S / N ratio of the ringstatistics (for η = . ′ / Ψ and η = . EB -statistics, and theaperture mass dispersion calculatedfrom a set of theoretical 2PCFs with ϑ ∈ [0 . ′
5; 460 ′ ]. The di ff erent an-gular range of the measures is ex-plained in the text. information content we examine both measures in a likelihoodanalysis.Compared to the ring statistics the S / N ratio of the EB -statisticsis significantly larger on all scales, which again can be ex-plained by the fact that the filter function of the EB -statisticsdoes not have roots at their boundaries. Compared to the aper-ture mass dispersion, the EB -statistics’ S / N ratio is slightlylower. However, we point out that the EB -filter function, wechose here, is a simple second-order polynomial. We willpresent an extended analysis of this general EB -filter functionsin a future paper.
6. Comparison of the information content of hRR E i and h M i We now perform a likelihood analysis in the Ω m vs. σ pa-rameter space in order to compare the ability of hRR E i , E ,and h M i to constrain cosmological parameters. We calculate2PCF data vectors for various combinations of σ ∈ [0 .
4; 1 . Ω m ∈ [0 .
01; 1 . hRR E i , E , and h M i and test these against the correspondingdata vectors obtained from our fiducial model (Sect. 5.1). Weassume that all data vectors are normally distributed in param-eter space and calculate the posterior likelihood according toBayes theorem. Our likelihood function p ( d | π ) then reads p ( d | π ) = exp h − (cid:16) ( d ( π ) − d ( π fid )) t C − ( d ( π ) − d ( π fid )) (cid:17)i (2 π ) n / | C | , (26)where d must be replaced by the considered data vector, either hRR E i (Eq. 21) , h M i (Eq. 22), or E (Eq. 23).To illustrate the information content we calculate the so-calledcredible regions, where the true parameter is located with a probability of 68%, 95%, 99,9%, respectively. In addition, wequantify the size of these credible regions through the deter-minant of the second-order moment of the posterior likelihood(see e.g. Eifler et al. 2008a) Q i j ≡ Z d π p ( π | ξ ) ( π i − π f i )( π j − π f j ) , (27)where π i are the varied parameters, π f i are the parameter of thefiducial model ( i = ,
2, corresponding to Ω m and σ ). Thesquare root of the determinant is given by q = q |Q i j | = q Q Q − Q , (28)and it can be considered as our figure-of-merit quantity.Smaller credible regions in parameter space correspond to asmaller value of q . In this paper all q ’s are given in units of10 − .For the likelihood analysis in this section we employ the ray-tracing covariances and choose the angular range of the datavectors correspondingly (Sect. 5.1), i.e. Ψ ∈ [1 ′ ; 460 ′ ] and θ ∈ [6 ′ ; 230 ′ ]. We further assume a flat prior probability withcuto ff s, which means p ( π ) is constant for all parameters insidea fixed interval ( Ω m ∈ [0 .
01; 1 . σ ∈ [0 .
4; 1 . p ( π ) = hRR E i . Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics 7 . . . . . . s W m < RR E > h = J min YY min = . . . . . . s W m < RR E > h = J min YY min = . . . . . . s W m < RR E > h = . . . . . . s W m < M ap2 > q min = . . . . . . s W m E Y min = Fig. 5.
The 68%-, 95%-, 99 . EB -statistics, and the aperture massdispersion. We compare 5 di ff erent cases,namely in the upper row: η = Ψ /ϑ min for Ψ min = ′ ( left ), and for Ψ min = ′ ( mid-dle ), and hRR E i with η = . right ). Inthe lower row we see h M i ( left ), and the EB -statistics for Ψ min = ′ ( right ). The datavectors are calculated analytically from apower spectrum; the covariance is obtainedfrom ray-tracing simulations. The filled cir-cle marks our fiducial cosmology. with η = ϑ min / Ψ and Ψ ∈ [1 ′ ; 460 ′ ] (left). Second, hRR E i with η = ϑ min / Ψ and Ψ ∈ [6 ′ ; 460 ′ ] (middle). Third, hRR E i with η = . Ψ ∈ [1 ′ ; 460 ′ ] (right). The lower row shows asimilar analysis for h M i with an angular range θ ∈ [6 ′ ; 230 ′ ](left) and the EB -statistics for Ψ ∈ [1 ′ ; 460 ′ ] (right). The black,filled circle indicates the fiducial cosmology; the contours cor-respond to the aforementioned credible regions. In addition wequantify the information content by the values of q , defined inEq. (28), which are summarized in Table 1.The ring statistics with η = ϑ min / Ψ is a clear improvement over hRR E i with η = . / Nratio of the first compared to the second. Considering the ringstatistics with scale-dependent η , we find that adding informa-tion below 6 ′ increases the information content of hRR E i , suchthat it gives tighter constraints than the h M i data vector. Thestrength of this gain in information can be explained by thesmall correlation of ring statistics’ data points.In our analysis it was not possible to calculate h M i for θ ≤ ′ due to the aforementioned E / B-mode mixing, however this canchange if the 2PCF is measured on smaller angular scales. Forthis case we expect the improvement of ring statistics over theaperture mass dispersion to be even more significant. Due tothe lower correlation of the ring statistics’ data points an in-clusion of smaller scales will enhance constraints from hRR E i stronger than those from h M i .The EB -statistics gives tighter constraints on cosmology thanthe optimized ring statistics, which can be explained by its Table 1.
Values of q resulting from the likelihood analyses ofthe 5 data vectors. Data vector q hRR E i ( η = ϑ min / Ψ , Ψ min = ′ ) 153.8 hRR E i ( η = ϑ min / Ψ , Ψ min = ′ ) 177.3 hRR E i ( η = .
1) 207.9 h M i ( θ min = ′ ) 169.8 E ( Ψ min = ′ ) 122.5 larger S / N ratio. However, we do not use the EB -statistics toanalyze the CFHTLS data in the next section for the reasonthat the EB -statistics’ data points are strongly correlated (seeFig. 3). In order to identify B-modes as a function of angularscale accurately, the lower correlation of the ring statistics ismore useful.
7. Ring statistics with the CFHTLS
In Sect. 5.1 we have shown that the ring statistics’ data pointsare significantly less correlated compared to data points of theaperture mass dispersion. Therefore, despite its lower S / N ra-tio, the ring statistics provides an ideal tool to analyse B-modecontaminations depending on the angular scale. In this sectionwe use the 2PCFs of the FSH08 analysis and therefrom calcu-late the ring statistics for a scale-dependent η = ϑ min / Ψ andfor η = .
1. We performed a similar analysis for other cases offixed η , which resulted in a significantly weaker signal. T. Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics − . − − . − . − . − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h = J min Y − − + − − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h = J min Y
50 100 200 500 − . − . − . − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h = J min Y − . − − . − . − . − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h = − − + − − − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h =
50 100 200 500 − . − . + . − . − . − Y [arcmin] < RR E > , < RR B > < RR E >< RR B > h = Fig. 6.
The ring statistics signal measured from the CFHTLS for the case of η = ϑ min / Ψ ( upper row). The red data points (circles)correspond to the E-mode signal, the black data points (triangles) to the B-mode signal. The three panels correspond to small( left ), intermediate ( middle ), and large ( right ) scales. The lower row shows a similar analysis but for η = . . ′ ≤ ϑ ≤ ′ ; we calculate hRR E i (Eq.7) and hRR B i (Eq. 8) in 60 logarithmic bins over a range0 . ′ ≤ Ψ ≤ . ′
0. The error for the i -th E / B-mode data point iscalculated as p C R E / B ( Ψ i , Ψ i ), where C R E / B ( Ψ i , Ψ i ) is calculatedfrom a Gaussian 2PCF covariance. This Gaussian covariancewas calculated from a theoretical model using the same cos-mology and survey parameters as in the FSH08 analysis. Wedo not employ the non-Gaussian correction of Semboloni et al.(2007) as this corrects the C ++ -term in the 2PCF covariance,but not the C −− - and C + − -terms. Here, we use the full 2PCFcovariance in the analysis. Similar to FSH08 we do not con-sider systematic errors in our analysis which might lead to anunderestimation of the error bars.The results of our analysis are illustrated in Fig. 6. The threepanels in the upper row show the ring statistics’ E- and B-modes on (from left to right) small, intermediate and largescales of Ψ for the case of η = ϑ min / Ψ . The three panels inthe lower row show the same analysis but for η = .
1. Thecircled (red) data points correspond to the E-mode signal, thetriangled (black) data points correspond to the B-mode signal.We measure a robust E-mode shear signal, however we alsofind a significant B-mode contribution on small (around 2 ′ ), in-termediate (16 ′ -22 ′ ), and large scales (right panel). On small scales E-and B-mode are of similar order. It should be stressedthat such an analysis of small-scale contaminations is not feasi-ble with the aperture mass dispersion, which, to avoid the E / B-mode mixing on small scales, involves a theoretical (thereforeB-mode free) 2PCF in its calculation. This theoretical data ex-tension, combined with the fact that the aperture mass disper-sion data points are stronger correlated (Sect. 5) can hide pos-sible small-scale contaminations in the data.The B-mode contamination on large scales is also observed inthe FSH08 analysis. In addition, we find a small B-mode onintermediate scales (between 16 ′ and 22 ′ ), otherwise these in-termediate scales are mostly free of B-modes and give a ro-bust E-mode signal. The small correlation of the individual datapoints leads to the oscillations in the amplitude of the shear sig-nal. A similar analysis with the aperture mass dispersion showsa much smoother behavior.
8. Conclusions
Decomposing the shear field into E- and B-modes is an impor-tant check for systematics in a cosmic shear analysis. The mostcommonly used methods for E- and B-mode decomposition,namely the aperture mass dispersion and the E / B-mode shearcorrelation function, require the 2PCF to be known down to . Eifler, P. Schneider and E. Krause: Measuring cosmic shear with the ring statistics 9 arbitrary small or up to arbitrary large angular separations.In practice, the 2PCF is only measured over a finite interval[ ϑ min ; ϑ max ]. As a result the aforementioned methods do notseparate E- and B-modes properly, e.g. the aperture massdispersion su ff ers from E / B-mode mixing on small angularscales (see KSE06 for further details).In contrast, the ring statistics (invented in SK07) separatesE- and B-modes properly using 2PCFs measured on a finiteangular scale. As outlined in SK07 the filter functions of thering statistics, i.e. Z ± , are in general complicated to calculate;the authors restrict the free parameters this filter function toone, namely η . This parameter is held fixed, independent of theangular scale Ψ at which the ring statistics is evaluated. In thispaper, we improve on the condition of a fixed η by choosing ascale-dependent η = ϑ min / Ψ which significantly improves onthe ring statistics’ S / N ratio.Furthermore, we present a formula to calculate the ringstatistics’ covariance from a 2PCF covariance. This formulais applied to a 2PCF covariance obtained from ray-tracingsimulations. We therefrom calculate the correlation matricesof ring statistics and aperture mass dispersion and find thatthe data points of the first are significantly less correlated thanthe data points of the second. We employ these covariancesto compare the information content of the two second-orderstatistics and find that the ring statistics’ data points placetighter constraints on cosmological parameters than data pointsof the aperture mass dispersion. The reason for this is that wecan include smaller scales in the ring statistics’s data vectorwhich is not possible for h M i due to the aforementionedE / B-mode mixing. In addition, we consider a polynomialfilter function which decomposes E- and B-modes on a finiteinterval and therefrom calculate an additional second-ordermeasure, the EB -statistics. We compare the correlation ofdata points and the information content of this EB -statisticsto the ring statistics and find that it shows a significantlylarger correlation of the data points, but a higher informationcontent. This can be explained by the high S / N ratio of the EB -statistics.We apply the ring statistics with η = ϑ min / Ψ and η = . η = ϑ min / Ψ which decreases whenperforming the same analysis for η = .
1. The fact, that datapoints of the ring statistics have small correlations enables usto determine the contaminated scales very accurately. We findB-modes on large scales which is comparable to the findings ofFSH08. In addition, we detect B-modes on intermediate (16 ′ - 22 ′ ) scales and a scattered B-mode contribution on scalesbelow 3 ′ . In the latter case the shear signal is of the same orderas the B-mode contribution.A similar analysis with the aperture mass dispersion is onlypossible when including a 2PCF from a theoretical modelin order to avoid the E / B-mode mixing on small angularscales. These added theoretical data can conceal remainingsystematics (B-modes) which can be identified properly usingthe ring statistics. This property is most likely the most usefulfeature of the ring statistics. It can be used to detect remainingsystemics very accurately in future surveys. The noise-level of the ring statistics on small scales can bereduced by increasing the number of galaxy pairs within thecontributing 2PCF-bins. The number of galaxy pairs inside a2PCF-bin increases quadratically with n gal , therefore it wouldbe interesting to test the ring statistics on a data set like e.g.the COSMOS survey. Similarly, an increased survey volumewill significantly enhance the constraints, for the reasonthat the cosmic variance scales with 1 / A . For example, theCFHTLS data we consider here covers an area of 34 . with n gal = .
3. Testing the ring statistics on the full CFHTLSsample (172 deg ) would be an interesting project in the future. Appendix A: T ± -functions In order to define the T ± -functions used for the calculation ofthe EB -statistics we remap ϑ ∈ [ ϑ min ; ϑ max ] to the x ∈ [ −
1; 1]and define x = ϑ − ϑ min − ϑ max ϑ max − ϑ min , (A.1) B = ϑ max − ϑ min ϑ max + ϑ min . (A.2)We choose our filter function T + ( x ) to be the lowest order poly-nomial which fulfills the two integral constraints of Eq. (13)and the normalization R − d x T + ( x ) T + ( x ) =
2. The functionreads T + ( x ) = √ Y (cid:16) B − − B x + − B ) x (cid:17) , (A.3)where Y = + B + B )5 . (A.4)Given the analytic form of T + the corresponding T − is uniquelydetermined through Eq. (14). References
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The authors want to thank Yannick Mellier andMartin Kilbinger for useful discussions and advise. TE wants tothank Liping Fu for sharing her CFHTLS data and the Insitutd’ Astrophysique de Paris for its hospitality during the analysisof the CFHTLS data. This work was supported by the DeutscheForschungsgemeinschaft under the projects SCHN 342 / //