Issue 
A&A
Volume 542, June 2012



Article Number  A122  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201218828  
Published online  19 June 2012 
Cosmic shear tomography and efficient data compression using COSEBIs
^{1} ArgelanderInstitut für Astronomie, Bonn University, 53121 Bonn, Germany
^{2} SUPA, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK
email: ma@roe.ac.uk
Received: 16 January 2012
Accepted: 20 April 2012
Context. Gravitational lensing is one of the leading tools in understanding the dark side of the Universe. The need for accurate, efficient and effective methods, which are able to extract this information along with other cosmological parameters from cosmic shear data is ever growing. Complete Orthogonal Sets of E/BIntegrals (COSEBIs) is a recently developed statistical measure that encompasses the complete E/Bmode separable information contained in the shear correlation functions measured on a finite angular range.
Aims. The aim of the present work is to test the properties of this newly developed statistics for a higherdimensional parameter space and to generalize and test it for shear tomography.
Methods. We use Fisher analysis to study the effectiveness of COSEBIs. We show our results in terms of figureofmerit quantities, based on Fisher matrices.
Results. We find that a relatively small number of COSEBIs modes is always enough to saturate to the maximum information level. This number is always smaller for “logarithmic COSEBIs” than for “linear COSEBIs”, and also depends on the number of redshift bins, the number and choice of cosmological parameters, as well as the survey characteristics.
Conclusions. COSEBIs provide a very compact way of analyzing cosmic shear data, i.e., all the E/Bmode separable secondorder statistical information in the data is reduced to a small number of COSEBIs modes. Furthermore, with this method the arbitrariness in data binning is no longer an issue since the COSEBIs modes are discrete. Finally, the small number of modes also implies that covariances, and their inverse, are much more conveniently obtainable, e.g., from numerical simulations, than for the shear correlation functions themselves.
Key words: gravitational lensing: weak / methods: data analysis / methods: statistical
© ESO, 2012
1. Introduction
As light travels through the Universe, the gravitational potential inhomogeneities distort its path; these distortions result in sheared galaxy images and carry invaluable information about the matter distribution between the observer and the source. Cosmic shear analysis is the study of the effects of largescale structures on light bundles (see Bartelmann & Schneider 2001). Consequently, it is one of the most promising probes for understanding the Universe, especially dark energy. The upcoming cosmic shear surveys (e.g. PanSTARRS^{1}, KIDS^{2}, DES^{3}, LSST^{4}, and Euclid^{5}) will have better statistical precision compared to current surveys, which means lower noise levels, larger fields of view, deeper images, and more accurate redshift estimations. Trustworthy and accurate methods are able to extract all the potential information in these future observations and make the effort put into launching them worthwhile.
The most direct secondorder statistical measurement from any weak lensing survey are the shear twopoint correlation functions ξ_{ ± }(ϑ), which in reality can be determined only on a finite interval ϑ_{min} ≤ ϑ ≤ ϑ_{max}. These, however, cannot be used for a comparison with theoretical models, since the shear field is in general composed of two modes: Bmodes cannot be due to leadingorder lensing effects, although they provide a measure of other effects such as shape measurement errors and intrinsic alignment effects (see Joachimi & Schneider 2010; also Schneider et al. 1998; and Schneider et al. 2002b, for other effects). On the other hand, Emodes are the only relevant modes when it comes to comparing the cosmic shear data with models.
Almost all of the recent analysis of cosmic shear data employ methods of E/Bmode separation (e.g. Benjamin et al. 2007; Fu et al. 2008). These studies are done in either Fourier or real space. For Fourier space analysis one has to find an estimate of the power spectrum, which is sensitive to gaps and holes in the survey and in general the survey geometry, which complicates such analysis. On the other hand the studies in real space do not share the same complications, since estimators of the shear correlation functions are unaffected by such gaps. Most of these studies use the aperture mass dispersion (Schneider et al. 1998), which applies compensated circular filters to the shear field. As was shown in Crittenden et al. (2002) and Schneider et al. (2002a), the aperture statistics, in principle, cleanly separates the shear twopoint correlations (2PCFs) into E/Bmode contributions. Furthermore, in the two papers just mentioned, a decomposition of the shear 2PCFs into E and Bmode correlation function ξ_{E,B}(ϑ) has been derived, which also has been employed in cosmic shear analyses of survey data (Lin et al. 2011).
However, both the aperture statistics and the E/Bmode correlation functions are unobservable in practice. The aperture mass dispersion requires shape measurements of galaxy pairs down to arbitrarily small angular scales. Since this is not feasible in real data, usually raytracing simulations fill in the gap, resulting in biases and E/Bmode mixing (see Kilbinger et al. 2006). On the other hand, the determination of ξ_{E,B}(ϑ) requires the knowledge of ξ_{ − }(ϑ′) out to infinite ϑ′. Hence, in both cases, determining E/Bmode separated statistics requires some sort of data invention.
To overcome these problems, Schneider & Kilbinger (2007) derived general conditions and relations for E/Bstatistics based upon twopoint statistical quantities, namely 2PCFs and convergence power spectra. They defined the quantities provided that the filter functions satisfy (3)E depends only on the Emode shear, and B depends only on the Bmode shear (with the aperture dispersion being one particular example). Moreover, they have shown that in order to obtain these statistics from the shear 2PCFs on a finite angular interval, 0 < ϑ_{min} < ϑ < ϑ_{max} < ∞, the filter function T_{ + } should have finite support on the same angular interval and satisfy (4)Whereas all solutions to the above relations provide statistics which cleanly separate E/Bmodes on a finite interval, different solutions may vary in their information contents. For example, the ring statistics introduced in Schneider & Kilbinger (2007) has a lower signaltonoise for a fixed angular range than the aperture dispersion, which, however, is compensated by its more diagonal noisecovariance matrix resulting in comparable Fisher matrices with aperture mass dispersion (Fu & Kilbinger 2010).
Recently, a complete solution of this issue was obtained (Schneider et al. 2010, hereafter SEK) by defining Complete Orthogonal Sets of E/BIntegrals (COSEBIs). COSEBIs capture the full information of the shear 2PCFs on a finite interval which is E/Bmode separable. In fact, SEK have shown that a small number of COSEBIs contain all the information about the cosmological dependence in their twoparameter model. Furthermore, they showed that COSEBIs in fact put tighter constraints on these parameters compared to the aperture mass dispersion. Eifler (2011) obtained a similar conclusion for a fiveparameter cosmological model. Therefore, the set of COSEBIs not only capture the full information, but also provide a highly efficient and simple method for data compression.
In this paper we further generalize the analysis in SEK to seven cosmological parameters, σ_{8}, Ω_{m}, Ω_{Λ}, w_{0}, n_{s}, h, and Ω_{b}, and investigate the effect of tomography on the results. Tomography, the joint analysis of shear auto and cross2PCFs of galaxy populations with different redshift distributions, is a powerful tool for cosmological analysis (Albrecht et al. 2006; Peacock et al. 2006), in particular in multidimensional parameter space (see Schrabback et al. 2010 for a recent paper on constraints on dark energy from cosmic shear analysis with tomography). We use Fisher analysis throughout our paper to represent the constraining power of COSEBIs, and compare the results from a mediumsized with that of a large cosmic shear survey.
In Sect.2 we summarize the method used in SEK and write the corresponding relations for shear tomography. In Sect.3 we briefly explain our choice of cosmology, and in Sect.4 the covariance of COSEBIs is shown. We present our figureofmerit based on Fisher analysis and show the results for the seven cosmological parameters and up to eight redshift bins in Sect.5. Finally we conclude by summarizing the most important results of the previous sections and emphasizing the advantages of COSEBIs over other methods of cosmic shear analysis. We have also derived an analytic solution to the linear COSEBIs weight functions presented in Appendix A.
2. COSEBIs
There is an infinite number of filter functions T_{ + }(ϑ) satisfying Eq. (4). Such filters can be expanded in sets of orthogonal functions, labeled T_{ + n}(ϑ); the corresponding T_{ − n}(ϑ) are obtained from solving Eq.(3)which can be inverted explicitly (Schneider et al. 2002a). Accordingly, the corresponding E/Bstatistics are denoted by E_{n} and B_{n}, respectively. Here we will also consider the case that different galaxy populations can be distinguished (mainly by their redshifts); therefore, one can measure auto and crosscorrelations functions of the shear, . We denote the corresponding COSEBIs by and . They are related to the auto and crosspower spectra of the convergence, by where are the E/Bcross convergence power spectra of galaxy populations i and j (see Schneider et al. 2002a), and are related to the 2PCFs by Inserting the above relations into Eq.(1), one can find relations connecting W_{n} to T_{ ± n}Any type of cosmic shear analysis needs some sort of error assessment. In particular Fisher analysis, used in the present work, depends on the noisecovariance of the statistics employed. The noisecovariance of COSEBIs for several galaxy populations assuming Gaussian shear fields (see Joachimi et al. 2008) is (11)where (12)and X stands for either E or B. The survey parameters are also included in Eq.(11)with the survey area, A, the galaxy intrinsic rms ellipticity, σ_{ϵ}, and the mean number density of galaxies in each redshift bin, .
In a recent paper, Sato et al. (2011) have shown that the Gaussian covariance model in Joachimi et al. (2008) overestimates the true Gaussian covariance for surveys with small area (A ≲ 1000 deg^{2}), and they have developed a fitting formula to correct for this discrepancy; in spite of their findings we will stick to the estimation of Joachimi et al. (2008), since the fitting formula in the latter paper depends on source redshift and is developed for a single source galaxy redshift, making it nonapplicable for this work.
Alternatively, one can write the covariance (Eq.(11)) in terms of T_{ ± n} and the twopoint correlation functions’ covariance (see SEK). However, in this approach double integrals over the covariance of 2PCFs slow down the calculations.
2.1. The COSEBIs filter and weight functions
SEK constructed two complete orthogonal sets of functions, linear and logarithmic COSEBIs (hereafter Lin and LogCOSEBIs respectively), by considering Eq.(4), and imposing orthogonality conditions on the T_{ + n} filters. Once the T_{ + n} filters are known, the T_{ − n} filters can be calculated via Eq.(3). The LinCOSEBIs filters are polynomials in ϑ, the angular separation of galaxies, while the LogCOSEBIs filters are polynomials in ln(ϑ).
The output of theoretical cosmological models which is of relevence here is the power spectrum. Hence, the quickest way to treat COSEBIs in theory is to work in ℓspace and to use Eq.(11)for the covariance, without taking the detour of calculating the shear 2PCFs and their covariance. As a result we need to calculate the W_{n}(ℓ) functions which are the Hankel transform (Eq.(9)) of their realspace counterparts, T_{ ± n}. For convenience, we choose to evaluate W_{n}(ℓ) from their integral relation with J_{0} and T_{ + n}. Since both J_{0} and T_{ + n} are oscillating functions, evaluating these integrals is rather challenging, in particular for large ℓ. A piecewise integration, from one extremum to the next, is used in the present work to evaluate W_{n}(ℓ). AppendixA contains more details about the numerical integrations and also a (semi)analytic formula for the linear W_{n} functions.
As is explained in SEK, the LogCOSEBIs are more efficient for a cosmic shear analysis. The reason is that unlike the linear filter functions which oscillate fairly uniformly in linear scale, the logarithmic have their roots fairly uniformly distributed in log (ϑ), i.e., they are more sensitive to variations of ξ_{ ± } on smaller scales. Combining this property with the fact that most of the cosmic shear information is contained in these smaller scales shows that it is more reasonable to employ LogCOSEBIs. In the next section we will show the difference of the Log and LinCOSEBIs using our figureofmerit.
In Figs. 1 and 2 the behavior of linear and logarithmic COSEBIs weight functions, and , for three angular ranges can be seen. The and have different yet similar oscillatory properties. They both die out rapidly with increasing ℓ but the lower frequency oscillations of are more prominent. They show approximately the same inverse relation to ϑ_{max} and ϑ_{min} for their lower and upper limits.
Fig. 1 Weight functions . They are the Hankel transforms of as in Eq.(9). In the blowups, the two modes of oscillation for each can be seen, the lower frequency mode and the higher frequency mode which are inversely proportional to ϑ_{min} and ϑ_{max}, respectively. The overall amplitude of the oscillations strongly depends on n and ϑ_{max}. 

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Fig. 2 Weight functions . They are the Hankel transformation of as in Eq.(9). Similar to the , the position of the first peak depends mainly on ϑ_{max} and is rather insensitive to ϑ_{min}. The difference between the two sets of linear and logarithmic function can be seen most prominently in the blowups; the lower frequency oscillations are more pronounced in this case. 

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3. Cosmological model
The cosmological model assumed in the present work is a wCDM model (Peebles & Ratra 2003, and references therein), i.e., a cold dark matter model including a dynamical dark energy with an equationofstate parameter w_{0}. The fiducial value of the parameters involved are listed in Table1.
Fiducial cosmological parameters consistent with WMAP 7years results.
The starting point in the analysis is to derive the matter power spectrum. For the linear power spectrum we used the Bond & Efstathiou (1984) transfer function, and the halo fit formula of Smith et al. (2003) for a fit of the nonlinear regime.
To calculate the convergence power spectrum we need the redshift distribution of galaxies. The overall redshift probability distribution is parametrized by (13)which represents the galaxy distribution fairly well (it is a generalization of Brainerd et al. 1996). The parameters, α, β, and z_{0} depend on the survey. We consider a medium and a large survey (hereafter MS and LS respectively). The MS has the same area as the CFHTLS (Fu et al. 2008), a current survey, while the LS covers the whole extragalactic sky and represents future surveys. The parameters of our two model surveys are given in Table2, and the corresponding redshift distributions are plotted in Fig.3.
Constructing the Fisher matrix requires the derivatives of the Emode COSEBIs and of their covariances with respect to the parameters. For example, to take the derivative with respect to Ω_{m}, its relation to the shape parameter, Γ, should be notified. In the present work we use the Sugiyama (1995) relation, (14)In their derivatives with respect to Ω_{m}, SEK assumed a constant Γ, equivalent to allowing h or Ω_{b} to vary accordingly (the only dependence of the convergence power spectrum on h or Ω_{b} comes through Γ). In the present work h and Ω_{b} are independent parameters and Γ depends explicitly on Ω_{m}. The difference between the two approaches is not negligible, as shown in Fig.4 which displays the derivative of the power spectrum with respect to Ω_{m} in both cases. This difference is due to the nonlinear relation between h and Γ. To justify our choice of parametrization, we just mention that the constraints from cosmological probes on h is tighter compared to Γ, and that makes it a more natural choice especially when priors are used.
Fig. 3 Overall source redshift probability distribution of source galaxies assumed for the two surveys. LS has a deeper source distribution compared to MS. 

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Fig. 4 Absolute value of the derivative of the convergence power spectrum with respect to Ω_{m}. Both of the curves rely on a five point stencil method where 4 nearby points have to be evaluated. The solid curve is drawn assuming all parameters are fixed except Ω_{m} and Γ, in contrast to the dotted curve where instead of Γ, h or Ω_{b} are variable. 

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Redshift distribution parameters and the survey parameters for our medium and large surveys.
4. COSEBIs covariance
Figure 5 shows the noisecovariance of linear and logarithmic Emode COSEBIs for the model parameters of the MS (the covariance has a similar behavior in the case of the LS but with a different amplitude). This covariance is calculated from Eq.(11)assuming a single source redshift distribution (Eq.(13)). Moreover, by defining the correlation coefficients of COSEBIs, (15)the behavior of the offdiagonal terms becomes clearer. (The capital subscripts N and M can be different from the COSEBIs subscripts, if several source populations are considered; see below for more details.) Figure 6 compares the correlation coefficients for three different choices of the angular range, [1′,400′] , [20′,400′] , and [1′,20′] , at a fixed M = 9.
Fig. 5 3D representation of the nontomographic covariance of 15 Emode COSEBIs for an angular range of [1′,400′] , for MS parameters. The x and yaxes correspond to the elements of the covariance matrix, and the value of the vertical axis shows the value of the covariance of the corresponding element. A contour representation of the covariance is shown for each plot at its base. 

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Fig. 6 Correlation coefficients of nontomographic COSEBIs for different angular ranges [ϑ_{min},ϑ_{max}] at m = 9, for the MS parameters. Here M, the capital subscripts, are equal to the COSEBIs mode, m. 

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Fig. 7 Representation of a tomographic covariance. In this diagram 3 redshiftbins (1, 2, 3) and 5 COSEBIs modes are assumed to be present. The blowup shows one of the covariance building blocks; the numbers 1−5 show the COSEBIs mode considered, e.g. 15 means the covariance of E_{1} and E_{5}. The numbers on the sides of the matrix show which combination of redshift bins is considered, e.g., 12 means the covariance of redshiftbins 1 and 2 is relevant. Due to symmetry, only a part of the covariance elements have to be calculated, here shown in pink. 

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Fig. 8 Correlation coefficients of COSEBIs for an angular range of [1′,400′] and 4 redshift bins. In total, 15 COSEBIs modes are considered for each graph. The r_{MN} is shown for M = 7 corresponding to , and for M = 82 corresponding to . 

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Cosmic shear analysis, as we will see in Sect.5, provides more information when redshift information is available. In practice the redshifts of galaxies are estimated using several photometric filters (see e.g. Hildebrandt et al. 2010), from which an overall distribution for the source galaxies is obtained. The distribution is then divided into a number of photometric redshift bins. The photometric redshifts are not exact, so the true redshift distributions will overlap. Therefore, instead of redshift bins, in general one has to consider redshift distributions. However, for simplicity, in the present work we have assumed redshift bins with sharp cuts and no overlap. In addition, the bins are selected such that the number of galaxies in each bin is the same.
In general, a tomographic covariance for r redshift bins consists of [r(r + 1)/2] ^{2} building blocks, each of which is a covariance matrix of where i,j,k,l are fixed and n, m = 1,2,...,n_{max}. This means in total the covariance matrix has [r(r + 1)n_{max}] ^{2}/4 elements, where n_{max} is the maximum number of COSEBIs modes considered.
Nevertheless, a covariance matrix is by definition symmetric and a tomographic covariance is made up of smaller covariances, i.e., only x(x + 1)/2 × n_{max}(n_{max} + 1)/2 elements, with x = r(r + 1)/2, have to be calculated, the rest are equal to these (see Fig.7).
The covariance of the depends on six indices; in order to apply normal matrix operations, the three indices of are combined into one “superindex” N, given by (16)where r is the total number of redshift bins and n_{max} is the total number of COSEBIs modes.
Using the new labeling, the correlation coefficients of and (corresponding to N = 7 and N = 82, respectively) with the other is shown in Fig.8, where 15 COSEBIs modes and 4 redshift bins are considered. Each of the peaks in the figure correspond to the correlation coefficient of and . The highest peak with r = 1 occurs for M = N, while the rest of the peaks are correlations between different redshift bins. The LogCOSEBIs show larger noisecorrelations between different modes compared to LinCOSEBIs, which may persuade one to choose the LinCOSEBIs for cosmic shear analysis. However, the LogCOSEBIs compensate this apparent disadvantage by requiring fewer modes to saturate the Fisher information level for relevant cosmological parameters compared to the linear ones, i.e., the number of covariance elements that have to be calculated for LinCOSEBIs is higher and hence determining their covariance matrix is more time consuming, especially when redshift binning is considered. Consequently, in Sect.5 we mainly employ LogCOSEBIs to analyze tomographic Fisher information.
5. Results of Fisher analysis
5.1. Figureofmerit
Figure 10 shows the sensitivity of the first five LogCOSEBIs to three cosmological parameters. Since the value of the COSEBIs alone do not have an obvious meaning, we show their values, normalized with respect to those for the fiducial parameters. For each COSEBIs mode, its normalized variance is shown at the fiducial point as well. The signaltonoise of the COSEBIs can thus be inferred from these plots. As one infers for the plot, the relative variance increases with increasing n, due to the stronger oscillations of the weight function T_{ + n} or, equivalently, the W_{n}. Although it provides a more intuitive way for understanding COSEBIs, this figure by itself cannot be used for constraining parameters, due to the nondiagonal covariance between the E_{n}’s. Therefore, a likelihood or Fisher analysis is required.
Hence, in this section we carry out a figureofmerit analysis to demonstrate the capability of COSEBIs to constrain cosmological parameters from cosmic shear data. Our figureofmerit, f, based on the Fisher matrix, quantifies the credibility of the estimated parameters. In general, for any unbiased estimator, the Fisher matrix gives the lower limit of the errors on parameter estimations (see e.g. Kenney & Keeping 1951; and Kendall & Stuart 1960, for details).
The Fisher matrix is related to the COSEBIs by (17)where C is the COSEBIs covariance, , E is the vector of the Emode COSEBIs, and the commas followed by subscripts indicate partial derivatives with respect to the cosmological parameters (see Tegmark et al. 1997, for example). We define our figureofmerit, f, in a very similar manner to SEK (18)where n_{p} is the number of free parameters considered. In the following analysis, we will assume for simplicity that the first term in Eq.(17)is much smaller than the second and can thus be neglected. Note that this approximation becomes more realistic in the case of a large survey area, since the first term on the r.h.s. of Eq.(17)does not depend on the survey area, while the second term is proportional to it (recall that C ∝ 1/A or in other words C^{1} ∝ A). We checked that our medium survey is already big enough for this approximation to hold (see Fig.9).
Fig. 9 Comparison between a simplified and complete Fisher analysis, using LogCOSEBIs. The asterisks show the case where the derivatives of the covariance is taken into account (the first part of Eq.(17)) while the squares show the simplified case where we assume these derivatives are zero, in calculating f. Here σ_{8} is the only free parameter, whereas the rest of the parameters is fixed to their fiducial values. 

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Fig. 10 Dependence of three cosmological parameters, σ_{8}, Ω_{m}, n_{s}, on the first five Emode LogCOSEBIs for a single galaxy redshift distribution. Both the parameters and the E_{n} values are normalized to their fiducial values. The errorbars show the normalized noise, . The parameters of the LS are assumed for this figure with an angular range of [1′,400′] . 

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Fig. 11 Comparison between the Log and LinCOSEBIs Fisher analysis results for two sets of assumptions. σ_{8} and Ω_{m} are the free parameters and the rest is fixed to their fiducial values. In one case the shape parameter Γ is held fixed, while in the other it is left as a variable depending on Ω_{m} and the fiducial values of h and Ω_{b} (according to Eq.(14)). The same analysis is also carried out for the FullCOSEBIs and the shear 2PCFs. 

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With the definition (18)we compress the Fisher matrix into a onedimensional quantity, which provides a measure of the geometric mean of the standard deviations of the parameters; e.g. in the case of one free parameter φ, f(φ) is equal to the standard deviation σ(φ) of that parameter^{6}.
5.2. Assumptions and parameter settings
For our cosmic shear analysis we considered a medium (MS) and a large survey (LS) as explained in Sect.3. We have also studied the effect of a Gaussian prior, in the form of a Fisher matrix. This prior is the inverse of the WMAP7 parameter covariance matrix from the final iteration (5000 sample points) of a Population Monte Carlo (PMC) run (see Kilbinger et al. 2010), called the CMB prior from here on. We implement the prior by adding the Fisher matrices of our COSEBIs analysis and the CMB prior. The value of the CMB prior is shown in terms of f(φ) in the first column of Table3 for each of the parameters. Since the LS is a future survey we also compared its results with a prediction of the Planck Fisher matrix (see Heavens et al. 2007). The predicted Fisher matrix in Heavens et al. (2007) assumes zero curvature (Ω_{m} + Ω_{Λ} = 1), therefore, we had to impose this constraint on our COSEBIs Fisher Matrices to add them. We conclude that the combination of Planck and LS has about twice tighter constraints on marginalized parameters, than WMAP7 and LS. Of course the exact value depends on the setup considered. However, here we only show the results for WMAP7 prior.
We consider three different angular ranges, [1′,20′] , [1′,400′] , and [20′,400′] . The motivation for this choice is as follows: we consider a total interval of [1′,400′] where the flat sky approximation is still valid up to the maximum separation and galaxy shapes are easily distinguishable for the minimum separation; also, ϑ_{min} = 1′ avoids the scales where baryonic effects are expected to have the strongest influence. We further divide this interval into two nonoverlapping parts with ϑ_{max}/ϑ_{min} = 20, to compare cosmic shear information on small and large scales. The smallscale range, [1′,20′] may apply for a cosmic shear survey of individual one square degree fields. The large scale interval, [20′,400′] could be used for very conservative analyses where nonlinear and baryonic effects are to be avoided.
In Sect.5.3 we show the value of f for two parameters while the rest are fixed to their fiducial values for the MS, and also for all seven parameters for the LS. In principle we could show all of the possible combinations for parameters, nevertheless finding the error on each of the parameters seems a more relevant task. Therefore, the rest of our analysis, carried out in Sect.5.4, is done for a single parameter, φ, where f(φ) = σ_{φ}.
Fig. 12 Comparison between the Lin and LogCOSEBIs results. These plots show one of our consistency checks. We consider the LS parameter with a single (top panel) and two galaxy redshift distributions (bottom panel), including all of the 7 parameters. Apart from very small numerical inaccuracies, both sets of COSEBIs saturate to the same value, as expected. There are two solid lines in each plot. The line with the higher value shows the value of LogCOSEBIs at n_{max} = 20, and the other line shows the value of f as obtained from the shear 2PCFs. The slightly smaller value of f in the latter case (this difference is not visible in the plot) is related to the fact that the analysis from the shear 2PCFs implicitly assume the absence of Bmodes, and thus contains information from very largescale modes which, however, cannot be uniquely assigned to either E or Bmodes. The comparison of the two plots shows that dividing the galaxies into two redshift bins not only increases the information content of the Fisher analysis but also decreases the number of COSEBIs modes needed. 

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To find the value of f for a single parameter we use two approaches. In one approach we fix the six other parameters to their fiducial values in Table1, while in the other case we marginalized over the remaining six parameters.
For each setup we investigate the amount of information with respect to the number of COSEBIs modes considered. In addition we analyze the behavior of f with the number of redshift bins considered.
5.3. Properties of COSEBIs
For a fixed number of modes, the LogCOSEBIs are more sensitive than the LinCOSEBIs to structures of the shear 2PCFs on small scales. Here we show its effect on the Fisher analysis. We investigate the dependence of f on the number n_{max} of COSEBIs modes incorporated in the analysis.
SEK have shown the difference between the behavior of the Lin and LogCOSEBIs for two parameters, σ_{8} and Ω_{m}, with Γ fixed (their definition of f and fiducial values of parameters are slightly different from ours). Similar to their work, we here compare the values of f for the same two parameters with Lin and LogCOSEBIs. In addition, we inspect the difference between a fixed shape parameter, or its dependence as given in Eq.(14).
Figure 11 is a representation of our inspection for the MS in the angular range of [1′,400′] . Two general conclusions come out of this comparison: (1) f for fixed and dependent Γ converges to the same value for the Lin and LogCOSEBIs; (2) the values of f for a fixed or dependent Γ are different, and also the convergence rate is different. E.g., the LinCOSEBIs reach the saturated f value for n_{max} ≈ 40 for a variable Γ, while in the other case, only 25 modes are needed. This effect is less dramatic in the case of LogCOSEBIs (they need 7 modes for a variable Γ and 5 modes for a constant one), since they generally converge faster. Similarly in Fig.12, we visualize our consistency check by showing that the values of f for Log and LinCOSEBIs converge to the same value for seven parameters^{7}.
In Figs. 11 and 12, we also show the value of f as derived directly from the shear 2PCFs, i.e., without E/Bmode separation. As expected, in this case f becomes slightly smaller since it is now implicitly assumed that all the signal is due to Emodes. However, this is not justified in general; for example, very largescale modes (i.e., small ℓ) enter ξ_{ + }(ϑ) even for small ϑ, and such modes cannot be uniquely assigned to either E or Bmodes. Thus, the decrease of f, and accordingly, the information gain is just an apparent one, bought by making a strong assumption. The relative difference between the 2PCFs and the converged Lin/LogCOSEBIs values for f is larger for the variable Γ case, since here smallℓ modes, which are filtered out in the COSEBIs, contain information about the power spectrum shape.
We also considered as further possibility that the requirement of finite support for the ξ_{ − }(ϑ) is dropped, and call this “FullCOSEBIs”. They form a complete set of functions on [ϑ_{min},ϑ_{max}] , without the constraints given in Eq.(4)^{8}. Though not physically reasonable, the FullCOSEBIs are equivalent to measuring ξ_{ + } only, on the same interval. As can be seen from Fig.11, the full COSEBIs yield a slightly lower value of f than the true COSEBIs, showing that ξ_{ − } on scales larger than ϑ_{max} adds apparent information, which, however, is not observable. We stress here that the E/Bmode correlation functions ξ_{E/B}, introduced by Crittenden et al. (2002) and Schneider et al. (2002a), are essentially equivalent to the FullCOSEBIs, since they are also based on the assumption that ξ_{ − } can be measured to arbitrarily large separations – which, however, is not possible. Therefore, a cosmic shear analysis based on ξ_{E} (e.g., Fu et al. 2008; Lin et al. 2011) underestimates the uncertainties of cosmological parameters.
Fig. 13 Dependence of the COSEBIs saturated information content in form of f for each of the seven parameters while the rest is marginalized over. The above plots are relevant to MS and LS parameters for an angular ranges of [1′,400′] . The redshift dependence of f is very outstanding here and hence we needed to use logarithmic scales for the yaxes, especially in the cases where cosmic shear analysis is done without any priors. 

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Fig. 14 Plots of figureofmerit values with respect to the number of LogCOSEBIs modes considered for two surveys with the three angular ranges [1′,20′] , [1′,400′] , [20′,400′] . Here all the parameters except one is marginalized over. The first column correspond to MS (top plots) and MS+CMB (bottom plots) and the next two to LS and LS+CMB correspondingly. Eight redshift bins are used here. The CMB prior flattens the curves, especially in the case of MS, where WMAP7 puts tighter constrains on the parameters. 

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Furthermore, we compare the Lin and LogCOSEBIs for LS parameters in Fig.12, for one and two redshift bins. To constrain n_{p} parameters at least n_{p} equations are needed, i.e., if one redshift bin is considered, n_{p} COSEBIs modes should be accounted for to produce a covariance matrix with at least n_{p} × n_{p} elements. For more than one redshift bin, a smaller number of COSEBIs modes are sufficient, subsequently the saturation rate of f is faster, as is visible in the right plot in the figure. Recall that 2 redshift bins means 3 different redshift combinations, i.e., for 7 parameters, the smallest integer not less than ⌈ 7/3 ⌉ = 3 COSEBIs modes are needed.
5.4. Forecast for parameter constraints
This section is dedicated to our final results according to the assumptions and parameters explained in Sect.5.2.
Figure 13 shows the dependence of f for 20 LogCOSEBIs modes and for the [1′,400′] angular range on the number of galaxy distributions (i.e., redshift bins), where all but one parameter are marginalized over. Dividing the galaxy distribution into more than 4 redshift bins does not change the value of f considerably. Nevertheless, a much larger number of redshift bins is required to control and correct for systematic effects, e.g., coming from intrinsic alignments (see for example Joachimi & Schneider 2010, and references therein).
We also show the dependence of f on n_{max}, for 8 redshift bins and marginalized parameters, in Fig.14. Comparing the cosmic shear analysis with and without CMB prior, we see from the figure that the prior in general flattens the curves. However, the curves are flatter for MS+CMB than LS+CMB as a result of the larger difference between the LS and the CMB prior.
The constraints on each of the cosmological parameters behave differently with respect to the number of COSEBIs modes or redshift bins considered. For marginalized parameters where the behavior of parameters is entangled, their curves show a similar decline.
By comparing the different angular ranges we conclude that a wider angular range needs more modes to extract all information. We also note that the behavior of the seven parameters are not similar and each of them should be followed separately.
Based on the results from these two figures, we will report additional results for n_{max} = 20, where the value of f is converged, and for either one or eight redshift bins. These results are shown in Table3 in the form of f(φ) for different cases. We have compared these values with Debono et al. (2010), and found them fully consistent.
In the following we explain our conclusions from the two mentioned figures and Table3 in more detail:

MS vs. LS vs. CMB prior: in general, because of itsmuch larger survey area and larger galaxy number density,LS puts tighter constraints on all of the parametersthan the MS. Furthermore, sincethe LS is deeper than the MS, itallows more sensitive constraints on parameters which aresensitive to the growth of structure, inparticular w_{0}. As can be seen from Fig.14, the requested number of COSEBIs for saturation is slightly higher for the LS since this survey contains more information, but smaller than 20 in all cases. The MS constraints on parameters are weaker than the CMB alone for almost all cases. Ω_{m} and σ_{8}, the two parameters for which present cosmic shear studies provide the most relevant constraints, are the only two for which the MS constraints are comparable with CMB. As for the rest of the parameters, except for f(w_{0}, [1′,400′] ) for the case of fixed parameters, the parameter uncertainty of the CMB prior is about one order of magnitude or more smaller than that of the MS. We conclude that the MS is not large enough to be competitive with the CMB for constraining a seven parameter cosmological model. However, the combination of the two slightly tightens the constraints. In contrast to the MS, the LS yields parameter constraints which can be considerably stronger than the CMB alone, in particular when tomography is employed. To wit, for the total angular range of [1′,400′] and 8 zbins, the LS errors are smaller than those from the CMB, except for n_{s} with marginalized parameters. For fixed parameters, the f value resulting from the combination of the LS and the CMB prior is very close to the LS value, whereas it can be much smaller than that of the CMB alone. I.e., we conclude that the resulting constraints from the LS are not dominated by the assumed prior. In contrast to the MS, the LS is able to put useful constraints on the parameters, even without tomography and for marginalized parameters. This is seen in the difference between the error values obtained from CMB alone and LS+CMB. The relative value of errors on parameters is different between the two surveys. The reason is that the redshift distributions of the two surveys are different (see Table2 and Fig.3) as seen in the figures. Using redshift information in general is equivalent to using structure evolution information. The largescale evolution is more visible in the case of a wider redshift distribution which starts from z = 0, where these structures are more evolved.

Fixed vs. Marginalized: the difference between the value of f for fixed and marginalized parameters is immense, especially when prior information is not available. However, we can state that, for all cases without priors, Ω_{m} is the best constrained parameter. The relative value of the parameters is also different between fixed and marginalized cases. Also for the fixed parameter case, the convergence rate of LS is slower compared to the MS and its relative information content is higher, as expected. In contrast to the fixed parameters case, tomography substantially lowers the errors for marginalized parameters.

Angular ranges: from the table, we find the following common trends for fixed parameters: The first relation shows that there is more information at smaller scales than in the largescale angular interval, although there is some independent information at larger scales. More interesting are the second and third relations which show that there is more information at smaller scales regarding LS compared to MS, which is a consequence of their different redshift distributions and cuts (see Fig.3 and Table2). The relations between the f(ϑ_{min},ϑ_{max}) for different angular ranges change when parameters are marginalized over. In this case the above inequalities are no longer valid for all of the parameters.
Figure 15 shows constraints for two pairs of parameters. In the bottom plot the direction of the contours for the MS are determined by the CMB prior. In this case tomography slightly improves the constraints on both parameters. The top plot, on the other hand, shows the constraints on dark energy parameters. In this case tomographic improvements are more visible. Also here the direction of the contours are different for the two cases.
Values of f for 20 LogCOSEBIs modes.
Fig. 15 Contour plots of 1σ constraints for pairs of parameters. The plot legends are abbreviated to save space. M stands for Marginalized, C for the CMB prior, and #z for the number of redshift bins. Top plot: the 1σ contours of w_{0} and Ω_{Λ} for the LS. The two smaller contours are for the case where the five other parameters are held fixed to their fiducial values. Bottom plot: the 1σ contours of σ_{8} and w_{0} for the MS. 

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6. Summary and conclusion
We have generalized the cosmic shear analysis with COSEBIs to include seven cosmological parameters, and investigated the effect of tomography on parameter constraints. For our analysis we mainly used the LogCOSEBIs, although for consistency checks we have shown that LinCOSEBIs results agree with their logarithmic counterparts. In AppendixA we show that there are analytic solutions for linear W_{n} functions, which are in principle less computationally demanding.
Besides the number of parameters and the use of redshift information, the main difference to SEK is a technical one: we calculated the COSEBIs and their covariance directly from the power spectrum, without using the 2PCFs as intermediate step. This choice is more convenient for theoretical considerations and highly speeds up the calculations of the COSEBIs covariances, an advantage in particular for the case of tomography. We also confirmed that our method can reproduce the results of SEK.
We investigated the effect of a Gaussian prior on two mock surveys, using three angular ranges on which the shear 2PCFs are asumed to be measured, by Fisher analysis methods. We considered the case that all but one parameter are fixed, as well as that where we marginalize over the other six parameters, in order to find the constraints on a single parameter. The prior was the Fisher matrix resulting from a Population Monte Carlo (PMC) analysis of WMAP7 results. We considered a medium and a large mock survey resembling the stateoftheart in cosmic shear and that of future all(extragalactic) sky surveys.
Most importantly, we found that a relatively small number of COSEBIs captures essentially all cosmological information from a cosmic shear survey. Whereas this number is larger than in SEK, due to the higherdimensional parameter space and the tomographic analysis, COSEBIs not only act as a clean E/Bmode separating shear statistics, but also as a highly efficient data compression method. We stress that this feature is extremely useful also for evaluating covariances from numerical simulations.
The required number of COSEBIs to saturate the cosmological information is considerably smaller for LogCOSEBIs than for LinCOSEBIs, which implies a clear preference for the former. It also increases with the number of free parameters, and depends on the parameters considered, the survey, and the angular range on which the 2PCFs are measured. In all cases we considered, fewer than 20 LogCOSEBIs modes were sufficient to reach information saturation. Aside from the tighter constraints of the large survey (LS) on all of the parameters compared to the mediumsized survey (MS), the order of the parameters with respect to their Fisher information is different between the two surveys. Moreover, in general LS requires more COSEBIs modes due to its higher information level. The comparison of the three angular ranges shows that most of the information in cosmic shear is contained at smaller scales which is why the LogCOSEBIs with their finer oscillations towards smaller scales are more sensitive and reach the saturated level of information with fewer modes. However, there is interesting independent information at larger scales, resulting in tighter constraints when using both angular ranges.
We have investigated the dependence of our figure of merit, f, on the number of redshift bins considered. In agreement with earlier work, we found that tomography greatly tightens parameter constraints, but the cosmological information saturates at around three or four redshift bins. This, however, does not imply that coarse redshift information is sufficient for future lensing surveys, since good redshift information is required to eliminate systematics from the data, such as intrinsic alignment effects (e.g., King & Schneider 2003; Joachimi & Schneider 2008; Joachimi & Bridle 2010).
For future work it will be interesting to investigate the effects of nulling with COSEBIs. Nulling techniques (Joachimi & Schneider 2008) eliminate the intrinsicintrinsic and intrinsicshear correlations from observed ellipticity correlations. The intrinsicintrinsic correlation can be handled by accurate redshift information to eliminate pairs with physical connections. Consequently, one can investigate how the cosmic shear information evolves by doing so, and how many redshift bins are needed in this case. Furthermore, our assumption of a Gaussian covariance becomes unrealistic at small angular scales; hence, it will be interesting to carry out a similar analysis based on more realistic covariances, either obtained from raytracing through cosmological density fields or using (semi)analytic models, such as based on lognormal fields (see Hilbert et al. 2011). As the number of tomographic bins increases, a larger number of COSEBIs modes need to be taken into account to explore the full information content. Therefore, it is worth to consider ways for a further compression of the information. This issue will be addressed in a future paper.
Another quantity, q, was also defined in SEK to measure the area of the likelihood regions. It is calculated from the secondorder moments of the posterior likelihood. q and f are equal if the posterior is a multivariate Gaussian. Eifler (2011) has shown that the difference between f and q is small, especially for a large survey area.
In general there are slight differences between the final value of f due to numerical inaccuracies, but these differences never exceed a few percent and are typically much smaller. An exception happens when the saturation is too slow, and the Fisher matrix elements are too small, which is the case for MS with one redshift bin and 7 parameters, observable especially after marginalizing over 6 parameters when the remaining parameter is w_{0}, Ω_{m}, Ω_{Λ} or σ_{8}. However, for these cases, f is much larger than unity, i.e., cases in which no meaningful constraints can be obtained anyway.
Acknowledgments
We thank Benjamin Joachimi, Andy Taylor and Tim Eifler for interesting discussions, Martin Kilbinger and Tom Kitching for sending us their CMB parameter covariances, Thierry Forveille and an anonymous referee for constructive comments. This work was supported by the Deutsche Forschungsgemeinschaft within the Transregional Research Center TR33 “The Dark Universe” and the Priority Programme 1177 “Galaxy Evolution” under the project SCHN 342/9.
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Appendix A: Analytic solutions to linear COSEBIs weight functions
Calculating the W_{n} functions and evaluating integrals involving them requires careful methods, as a result of their very oscillating nature. In this section we first show our semianalytical solutions to and at the end discuss our method of integration in more detail.
The filters, , were calculated in SEK. By a simple variable change of y = ℓϑ, Eq.(9)becomes (A.1)One can write T_{ + n} in the form (A.2)and then rewrite as: (A.3)We define the functions S_{n} as (A.4)Inserting the above equation into Eq. gives (A.5)The S_{n} functions can be obtained using standard Bessel functions relations, which in our case specialize to where the last equation results from the two equations before it.
Using Eqs. (A.6)and (A.8), the following calculations for S_{n} are carried out, (A.9)where we have carried out integration by parts twice. Note that the last term in the above equations is equal to S_{n − 2}. Consequently, the recursive formula for S_{n} is (A.10)The first two of these functions are where is the Hypergeometric functions (see Arfken & Weber 1995). Although there is an analytic formula for S_{1}, it is more convenient to solve it numerically using a similar stepwise integration method as was explained in Sect.2.1. The only difference here is that the steps are taken between zeros of the Bessel function. A plot of S_{1} can be seen in Fig.A.1.
Fig. A.1 Shape of S_{1}. The weight functions W_{n}(ℓ) depend on the integrals S_{n}, which are recursively related to each other. As is expected, S_{1} approaches zero as the argument gets small, and has a highly oscillating behavior similar to its integrand, J_{0}. 

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The recursive formula in Eq.(A.10)can be rewritten as a closed form formula of the form: (A.13)where N is (A.14)To calculate the SEK rescaled the angular separation intervals from [ϑ_{min},ϑ_{max}] to [−1,1] with a variable transformation of the form (A.15)with , Δϑ = ϑ_{min} − ϑ_{max} and calculated the results for . The explicit mathematical form of and are shown in SEK. One can find the a_{ni} coefficients in Eq.(A.2)by a change of variable from x to ϑ.
The rest of the filters are (A.16)where P_{n + 1}(x) is the Legendre polynomials of order n + 1. We rewrite the recursive relation for Legendre polynomials using Eq.(A.15)(A.17)where is the relative interval width. Note that the first term in the recursive relation has the highest polynomial order, and the rest have lower order, respectively. We write the Legendre polynomials as polynomials in ϑ(A.18)with C_{ni} coefficients (A.19)where C_{ni} = 0 if i > n or i < 0. The a_{ni} coefficients are (A.20)The weight functions computed the semianalytic way are considerably less computationally demanding, although, formulae (A.10)and (A.13)are not stable as far as we investigated. The S_{n} functions blow up for small arguments were they should reach zero and show a noisy behavior. In addition, the argument for which S_{n} starts to behave as it should grows with n, hence the resulting functions become less and less reliable for larger subscripts. However, that does not render them useless since calculating the functions from their original integral form is very time consuming for larger arguments were the S_{n} functions become reliable. Since apart from S_{1} which needs to be stored once and can be loaded for further use, the time taken to evaluate the rest of the S_{n} functions does not depend on their argument, which means one can in principle go to arbitrarily high ℓmodes to calculate .
Nevertheless, in practice one does not need to go to very high ℓmodes to evaluate the integral in Eq.(5), and find the Emode COSEBIs; since as it is evident in Fig.1, the W_{n} functions die out rapidly at large ℓ. By inspection of the plots, we deduced that the ratio of the largest peak (global maximum) and a peak at ℓ ≈ 100πn/ϑ_{max} is of around 3 − 4 orders of magnitude. This property of the weight functions makes the infinite upper limit of Eq.(5)in practice manageable, i.e., the effective limits of the integral become finite, although they also depend on the shape of the power spectrum. In the present work the integrals involving W_{n} functions are evaluated for a finite range of ℓ_{min} = 1 to ℓ_{max} ≈ 100πn_{max}/ϑ_{max}, where n_{max} is the maximum number of modes considered in the analysis of the interest.
In the piecewise method for calculating W_{n}, a Gaussian integration method (gauleg) of numerical recipes, Press et al. (2002), is used for each interval considered. The results of these integrals are summed up as the final result. There is a routine in the
code which finds the consecutive minima and maxima of the zeroth order Bessel function, and puts them as the integration limits of the pieces. However, the lowℓ values of the functions are not calculated in the same way, since in those regimes the oscillations of T_{n} is more important compared to J_{0}, and they cause numerical artifact. Instead one Gaussian integration method with higher accuracy parameter is used to evaluate them. The limit to change from one routine to the other is set by ℓ_{thresh} ≈ πn/ϑ_{max} parameter.
All Tables
Redshift distribution parameters and the survey parameters for our medium and large surveys.
All Figures
Fig. 1 Weight functions . They are the Hankel transforms of as in Eq.(9). In the blowups, the two modes of oscillation for each can be seen, the lower frequency mode and the higher frequency mode which are inversely proportional to ϑ_{min} and ϑ_{max}, respectively. The overall amplitude of the oscillations strongly depends on n and ϑ_{max}. 

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In the text 
Fig. 2 Weight functions . They are the Hankel transformation of as in Eq.(9). Similar to the , the position of the first peak depends mainly on ϑ_{max} and is rather insensitive to ϑ_{min}. The difference between the two sets of linear and logarithmic function can be seen most prominently in the blowups; the lower frequency oscillations are more pronounced in this case. 

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In the text 
Fig. 3 Overall source redshift probability distribution of source galaxies assumed for the two surveys. LS has a deeper source distribution compared to MS. 

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In the text 
Fig. 4 Absolute value of the derivative of the convergence power spectrum with respect to Ω_{m}. Both of the curves rely on a five point stencil method where 4 nearby points have to be evaluated. The solid curve is drawn assuming all parameters are fixed except Ω_{m} and Γ, in contrast to the dotted curve where instead of Γ, h or Ω_{b} are variable. 

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In the text 
Fig. 5 3D representation of the nontomographic covariance of 15 Emode COSEBIs for an angular range of [1′,400′] , for MS parameters. The x and yaxes correspond to the elements of the covariance matrix, and the value of the vertical axis shows the value of the covariance of the corresponding element. A contour representation of the covariance is shown for each plot at its base. 

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In the text 
Fig. 6 Correlation coefficients of nontomographic COSEBIs for different angular ranges [ϑ_{min},ϑ_{max}] at m = 9, for the MS parameters. Here M, the capital subscripts, are equal to the COSEBIs mode, m. 

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In the text 
Fig. 7 Representation of a tomographic covariance. In this diagram 3 redshiftbins (1, 2, 3) and 5 COSEBIs modes are assumed to be present. The blowup shows one of the covariance building blocks; the numbers 1−5 show the COSEBIs mode considered, e.g. 15 means the covariance of E_{1} and E_{5}. The numbers on the sides of the matrix show which combination of redshift bins is considered, e.g., 12 means the covariance of redshiftbins 1 and 2 is relevant. Due to symmetry, only a part of the covariance elements have to be calculated, here shown in pink. 

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In the text 
Fig. 8 Correlation coefficients of COSEBIs for an angular range of [1′,400′] and 4 redshift bins. In total, 15 COSEBIs modes are considered for each graph. The r_{MN} is shown for M = 7 corresponding to , and for M = 82 corresponding to . 

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In the text 
Fig. 9 Comparison between a simplified and complete Fisher analysis, using LogCOSEBIs. The asterisks show the case where the derivatives of the covariance is taken into account (the first part of Eq.(17)) while the squares show the simplified case where we assume these derivatives are zero, in calculating f. Here σ_{8} is the only free parameter, whereas the rest of the parameters is fixed to their fiducial values. 

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In the text 
Fig. 10 Dependence of three cosmological parameters, σ_{8}, Ω_{m}, n_{s}, on the first five Emode LogCOSEBIs for a single galaxy redshift distribution. Both the parameters and the E_{n} values are normalized to their fiducial values. The errorbars show the normalized noise, . The parameters of the LS are assumed for this figure with an angular range of [1′,400′] . 

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In the text 
Fig. 11 Comparison between the Log and LinCOSEBIs Fisher analysis results for two sets of assumptions. σ_{8} and Ω_{m} are the free parameters and the rest is fixed to their fiducial values. In one case the shape parameter Γ is held fixed, while in the other it is left as a variable depending on Ω_{m} and the fiducial values of h and Ω_{b} (according to Eq.(14)). The same analysis is also carried out for the FullCOSEBIs and the shear 2PCFs. 

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In the text 
Fig. 12 Comparison between the Lin and LogCOSEBIs results. These plots show one of our consistency checks. We consider the LS parameter with a single (top panel) and two galaxy redshift distributions (bottom panel), including all of the 7 parameters. Apart from very small numerical inaccuracies, both sets of COSEBIs saturate to the same value, as expected. There are two solid lines in each plot. The line with the higher value shows the value of LogCOSEBIs at n_{max} = 20, and the other line shows the value of f as obtained from the shear 2PCFs. The slightly smaller value of f in the latter case (this difference is not visible in the plot) is related to the fact that the analysis from the shear 2PCFs implicitly assume the absence of Bmodes, and thus contains information from very largescale modes which, however, cannot be uniquely assigned to either E or Bmodes. The comparison of the two plots shows that dividing the galaxies into two redshift bins not only increases the information content of the Fisher analysis but also decreases the number of COSEBIs modes needed. 

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In the text 
Fig. 13 Dependence of the COSEBIs saturated information content in form of f for each of the seven parameters while the rest is marginalized over. The above plots are relevant to MS and LS parameters for an angular ranges of [1′,400′] . The redshift dependence of f is very outstanding here and hence we needed to use logarithmic scales for the yaxes, especially in the cases where cosmic shear analysis is done without any priors. 

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In the text 
Fig. 14 Plots of figureofmerit values with respect to the number of LogCOSEBIs modes considered for two surveys with the three angular ranges [1′,20′] , [1′,400′] , [20′,400′] . Here all the parameters except one is marginalized over. The first column correspond to MS (top plots) and MS+CMB (bottom plots) and the next two to LS and LS+CMB correspondingly. Eight redshift bins are used here. The CMB prior flattens the curves, especially in the case of MS, where WMAP7 puts tighter constrains on the parameters. 

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In the text 
Fig. 15 Contour plots of 1σ constraints for pairs of parameters. The plot legends are abbreviated to save space. M stands for Marginalized, C for the CMB prior, and #z for the number of redshift bins. Top plot: the 1σ contours of w_{0} and Ω_{Λ} for the LS. The two smaller contours are for the case where the five other parameters are held fixed to their fiducial values. Bottom plot: the 1σ contours of σ_{8} and w_{0} for the MS. 

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In the text 
Fig. A.1 Shape of S_{1}. The weight functions W_{n}(ℓ) depend on the integrals S_{n}, which are recursively related to each other. As is expected, S_{1} approaches zero as the argument gets small, and has a highly oscillating behavior similar to its integrand, J_{0}. 

Open with DEXTER  
In the text 
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