Issue 
A&A
Volume 520, SeptemberOctober 2010



Article Number  A116  
Number of page(s)  16  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201014235  
Published online  13 October 2010 
COSEBIs: Extracting the full E/Bmode information from cosmic shear correlation functions
P. Schneider^{1}  T. Eifler^{2,1}  E. Krause^{3}
1  ArgelanderInstitut für Astronomie,
Universität Bonn, Auf dem Hügel 71,
53121 Bonn, Germany
2 
Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 W. Woodruff Ave., Columbus, OH 43210, USA
3 
California Institute of Technology, Dept. of Astronomy,
MC 10524, Pasadena CA 91125, USA
Received 10 February 2010 / Accepted 29 June 2010
Abstract
Context. Cosmic shear is considered one of the most powerful
methods for studying the properties of dark energy in the Universe. As
a standard method, the twopoint correlation functions
of the cosmic shear field are used as statistical measures for the shear field.
Aims. In order to separate the observed shear into E and
Bmodes, the latter being most likely produced by remaining systematics
in the data set and/or intrinsic alignment effects, several statistics
have been defined before. Here we aim at a complete E/Bmode
decomposition of the cosmic shear information contained in the
on a finite angular interval.
Methods. We construct two sets of such E/Bmode measures,
namely Complete Orthogonal Sets of E/Bmode Integrals (COSEBIs),
characterized by weight functions between the
and the COSEBIs which are polynomials in
or polynomials in
,
respectively. Considering the likelihood in cosmological parameter
space, constructed from the COSEBIs, we study their information
content.
Results. We show that the information grows with the number of
COSEBI modes taken into account, and that an asymptotic limit is
reached which defines the maximum available information in the Emode
component of the .
We show that this limit is reached the earlier (i.e., for a smaller
number of modes considered) the narrower the angular range is over
which
are measured, and it is reached much earlier for logarithmic weight functions. For example, for
on the interval
,
the asymptotic limit for the parameter pair
is reached for 25 modes
in the linear case, but already for 5 modes in the logarithmic case.
The COSEBIs form a natural discrete set of quantities, which we suggest
as method of choice in future cosmic shear likelihood analyses.
Key words: largescale structure of Universe  gravitational lensing: weak  cosmological parameters  methods: statistical
1 Introduction
The shear field in weak lensing is caused by the tidal component of the gravitational field of the mass distribution between us and a distant population of sources (see Munshi et al. 2008; Mellier 1999; Bartelmann & Schneider 2001; Refregier 2003; Schneider et al. 2006, for recent reviews). If the shear, estimated from the image shapes of distant galaxies, is solely due to gravitational lensing, then it should consist only of a ``gradient component'', the socalled Emode shear (see Schneider et al. 2002; Crittenden et al. 2002). Bmodes (or curl components) cannot be generated by gravitational light deflection in leading order, and higherorder effects from lensing are expected to be small, as seen in raytracing simulations through the cosmological density field (e.g., Jain et al. 2000; Hilbert et al. 2009).
Therefore, the splitting of the observered shear field into its E and Bmodes is of great importance to isolate the gravitational shear from the shear components most likely not due to lensing, in order to (i) have a measure for the impact of other effects besides lensing (such as insufficient PSF correction for the shape measurements, or intrinsic alignment effects) on the observed shear field; and to (ii) isolate the lensing shear and to compare it with the expectation from cosmological models. Indeed, almost all more recent cosmic shear surveys perform such an E/Bmode decomposition of secondorder shear measures (e.g., Hoekstra et al. 2002; Fu et al. 2008; Jarvis et al. 2003; Hetterscheidt et al. 2007).
The standard technique for this separation is the aperture dispersion and (Schneider et al. 1998), which can be calculated in terms of the shear twopoint correlation functions (2PCFs) on a finite interval . Alternatively, one can construct E and Bmode shear correlation functions (Crittenden et al. 2002), which, however, can be calculated only if the shear correlation function is known for arbitrarily large separations. As was pointed out by Kilbinger et al. (2006), the fact that the calculation of the aperture dispersion requires the knowledge of the shear correlation functions down to zero separation, together with the inability to measure the shape of image pairs with very small angular separation, leads to biases in the estimated values for the aperture dispersions, in particular to an effective E/Bmode mixing.
For that reason, Schneider & Kilbinger (2007)  hereafter SK07  developed a new secondorder shear statistics, that can be calculated from the shear correlation functions on a finite interval and which provides a clean separation of E and Bmodes. In particular, SK07 derived general expressions for the relation between E/Bmode secondorder shear quantities and the shear 2PCFs. They considered one particular example of such a relation, leading to the socalled the ring statistics, based solely on geometric considerations. Eifler et al. (2010) and Fu & Kilbinger (2010)  hereafter FK10  have shown that, although the signaltonoise at fixed angular scale is smaller for the ring statistics than for the aperture dispersion, the correlation matrix between measurements at different angular scales is considerably narrower in the case of the ring statistics, yielding that the information contents of the two measures are quite comparable. Applying the ring statistics to the same cosmic shear correlation functions as used by Fu et al. (2008) in their measurement from the CanadaFranceHawaii Telescope Legacy Survey, Eifler et al. (2010) obtained a clear signal, as well as a better localization of the remaining Bmodes.
In FK10, more general E/Bmode measures have been considered, based on the general transformation derived in SK07. Specifically, FK10 have constructed Emode quantities which maximize the signaltonoise for a given interval , or which maximize the figure of merit in parameter space, as obtained from considering the Fisher matrix. Both of the resulting Emode statistics are by construction superior to the ring statistics, and also yield higher signaltonoise, or a higher figureofmerit, than the aperture dispersion.
In this paper, we construct sets of E/Bmode measures, E_{n} and B_{n}, based on shear correlation functions on a finite interval. In a welldefined sense, for a given angular interval , these secondorder E/Bmode measures form a complete set each, so that all EBseparable information contained in the is also contained in this complete set. With these complete sets of secondorder shear measures, we propose a new approach to compare observed shear correlations with model predictions. Whereas all such comparisons done hitherto define a secondorder shear measure as a function of angular scale [such as or ], the choice of the grid points in the angular scale being arbitrary, the complete set of the E_{n} are a ``natural'' discrete set of quantities that can be used in a likelihood analysis. One can hope that a finite and possibly rather small number of the E_{n} contains most of the cosmological information, depending on the choice of the set.
In Sect.2 we summarize the general equations for E/Bmode measures obtained from the twopoint correlation functions of the shear field over a finite interval, and derive the covariance matrix for a set of such EBmode measures. We then construct in Sect.3 two examples of Complete Orthogonal Sets of E/Bmode Integrals (COSEBIs), one of them using weight functions which are polynomials in , the others being polynomials in . In the former case, explicit relations for the corresponding weight functions are obtained for any polynomial order, whereas in the logarithmic case the coefficients have to be obtained through a matrix inversion. In Sect.4, we then investigate the information content of these COSEBIs, by calculating the likelihood of cosmological parameter combinations and the corresponding Fisher matrix for a fiducial cosmic shear survey, using the two COSEBIs constructed, as well as the original shear correlation functions. We conclude by discussing the advantages of the COSEBIs over the other secondorder shear measures that have been suggested in the literature. In AppendixB, we show how COSEBIs can be used to maximize the signaltonoise of a cosmic shear Emode measure. In addition we show how to construct pure E/Bmode correlation functions from the COSEBIs and relate them to the 2PCF.
2 E/Bmode decomposition
In SK07 we have shown than an E/Bmode separation of secondorder shear statistics is obtained from the 2PCFs byprovided the two weight functions are related through
or, equivalently,
In this case, E contains only Emodes, whereas B depends only on the Bmode shear. Furthermore, it was shown in SK07 that an Emode secondorder statistics is obtained from the shear correlation functions on a finite interval if the function T_{+} vanishes outside the same interval, and in addition, the two conditions
are satisfied; in this case, the function as calculated from Eq.(3) also has finite support on the interval . In SK07, a particular set of functions was introduced, originating from the geometrical construction of crosscorrelating the shear in two nonoverlapping annuli, and the corresponding estimators were termed ``ring statistics''.
The origin of the conditions expressed in Eq.(4) can be understood as follows: a uniform shear field cannot be assigned an E or Bmode origin. Such a shear field gives rise to shear correlation functions of the form and . According to the first of Eq.(4), this component is filtered out in Eq.(1). Furthermore, one possibility to distinguish between E and Bmodes is the consideration of the vector field constructed from partial derivatives of the shear field (Kaiser 1995). A pure Emode shear yields a vanishing curl of , whereas a pure Bmode shear leads to ; a shear field which yields cannot be uniquely classified as E or Bmode.
If we now consider a shear field which depends linearly on , then the vector field is constant, and thus it cannot be uniquely split into E and Bmodes. On the other hand, such a shear field gives rise to correlation functions of the form , , where A and B are constants. Again, the correlation function of such a shear field is filtered out due to the conditions in Eq.(4).
2.1 E/Bmodes from a set of functions
Of course, there are many functions
which satisfy the
constraints in Eq. (4). Assume we construct a set of
functions
which all satisfy Eq. (4)
and which are, in a way specified later, orthogonal. Then one can
construct the corresponding
from
Eq. (3), and thus one obtains the set E_{n} and
B_{n} of secondorder shear measures with a clean E/Bmode
separation. Each of the E_{n} and B_{n} measures an integral over the
power spectrum of E and Bmodes, respectively,
where the filter functions are
and where we made use of the relation between the shear correlation functions and the power spectra (see, e.g., Schneider et al. 2002)
We next calculate the covariance of the E and Bmode measures making use of Eq. (5),
where in the final step we have assumed a Gaussian shear field and used the corresponding expression for the covariance of the power spectrum from Joachimi et al. (2008). Here, A is the survey area, is the amplitude of the white noise power spectrum resulting from the intrinsic ellipticity distribution of sources, is the dispersion of the intrinsic ellipticity, and is the mean number density of sources. The covariance of the B_{n}, , has exactly the same form, with replaced by , and the covariance between the E_{n}and B_{m} vanishes.
As a consistency check, we calculate the covariance in a different
form, starting from the relation between the E_{n} and the shear
correlation functions. We then obtain
where is the covariance of the shear correlation function . Using the relations of Joachimi et al. (2008) for the covariance of the , assuming a Gaussian shear field, and making use of Eq.(2), the result (8) is reobtained.
The comparison of the
obtained from observations
with those of a model
,
where
denotes a set of
M model parameters, can then be done via
(10) 
where N is the maximum number of Emodes considered, or with a likelihood function
(11) 
2.2 Calculation of Emode secondorder statistics from raytracing simulations
Due to the limited range of validity of analytic approximations for the calculation of cosmic shear statistics, ray tracing through Nbody simulated threedimensional density distributions are carried out (see, e.g., Jain et al. 2000; Hilbert et al. 2009, and references therein). As shown in these papers, the resulting Bmode shear is several orders of magnitude smaller than the Emode shear, so that the resulting shear field can be described very accurately in terms of an equivalent surface mass density . It is often faster to derive statistical properties of the resulting shear field from the corresponding properties of the field. For example, the aperture mass (Schneider 1996) can be obtained from the shear field through a radial filter function Q, but also from the field through a related radial filter function U. Hence, one can calculate the field of from the equivalent surface mass density, convolved with the filter U, and the aperture mass dispersion is then given as the dispersion of this field. In this way, no correlation functions of the shear need to be obtained for making predictions, saving computation time.
Here we will show that, similar to the case of the aperture mass
dispersion, the Emode secondorder shear statistics defined in
Eq.(1) can be obtained from a simulated
field, without the need to calculate the shear correlation
functions. For that we note that,
in the absence of Bmodes, one has
and that the correlation functions of and agree,
If we smooth the convergence field with a radial filter function F, obtaining
(12) 
the correlator of the smoothed field with the unsmoothed field at zero lag becomes
(13) 
Setting , we see that
(14) 
if we choose . Hence, the calculation of E from simulations can proceed by convolving the field with the function , and correlating the resulting field with the original field, dropping a band of width along the boundaries of the field where the convolution via FFT causes artifacts.
3 Complete sets of weight functions
Here, we construct complete sets of functions which satisfy the constraints (4) for the weight function on the interval . It should be noted that, once a complete set of such functions is known, the maximization of the signaltonoise of the secondorder Emode shear  a problem considered in FK10  reduces to a linear algebra problem, as shown in AppendixB.
Readers less interested in the explicit construction of these COSEBIs can go directly to Sect.4.
3.1 Polynomial weight functions
Figure 1: The linear filter functions for , . Note that the shape of the curves depends only on the ratio . 

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First, we construct a complete set of weight functions which are
polynomials in .
To do so, we transform the interval
onto the unit interval
,
by defining
with , . In addition, we define the relative interval width . Thus, as varies from to , x goes from 1 to +1. Then we set , and . The t_{+n} are chosen to be polynomials in x; as Eq. (15) is a linear transformation, the polynomial order is preserved. Furthermore, we require that the set of functions are orthonormal, i.e.,
The first two functions of the set are constructed ``by hand'': the lowestorder polynomial which can satisfy the constraints (4) and the normalization constraint (16) is of second order. Hence, we choose t_{+1}(x) to be a secondorder polynomial, and determine its three coefficients from the three constraints. The lowestorder polynomial which can satisfy the two constraints (4) and the orthonormality relation (16) for m=1,2 is of third order, and its four coefficients are determined accordingly; this yields
t_{+1}(x)  =  
t_{+2}(x)  =  
(17) 
with
X_{1}  =  
X_{2}  =  (18) 
To obtain the higherorder functions of this set, we note that the Legendre polynomials P_{n}(x) are orthogonal, and that
This shows that the constraints (4), written in terms of x, are satisfied if we choose for all . Furthermore, the P_{n}(x) for are orthogonal to t_{+1}(x) and t_{+2}(x), since the latter are polynomials of order . Thus, choosing the normalization such as to satisfy Eq.(16), we find for ,
In the upper panel of Fig.1, we have plotted the filter function for three values of n. For , they are simply proportional to the Legendre polynomials. Note that has (n+1) roots in the interval , and the normalization is chosen such that . The corresponding filter functions which relate the COSEBIs to the power spectrum are displayed in Fig.2, for several values of n and for two different values of the relative width parameter B (corresponding to two different values of ).
Figure 2: The functions W_{n} as defined in Eq.(6) which relate the COSEBIs to the underlying power spectrum, calculated from the . The upper panel corresponds to , whereas the lower panel is calculated using , both for . 

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For this set of functions t_{+n}(x), we can obtain the corresponding
t_{n}(x) using Eq.(3),
where
and the coefficients A_{k} are given explicitly as
A_{0}  =  
A_{2}  =  (22) 
For the first two functions, the integral is carried out explicitly, yielding
t_{1}(x)  =  
t_{2}(x)  =  (23) 
with the coefficients U
U_{10}  =  5+19 B^{2}15 B^{4}+3 B^{6} ,  
U_{11}  =  
U_{12}  =  15+7 B^{2}+B^{4}3 B^{6} ,  
U_{13}  =  
U_{20}  =  
U_{21}  =  525 + 215 B^{2}30 B^{4}+38 B^{6}+18B^{8} ,  
U_{22}  =  
U_{23}  =  
U_{24}  =  
U_{25}  =  
U_{26}  =  
U_{27}  = 
For , we first define
(24) 
For k=0, one obtains
(25) 
whereas for , we make use of the recurrence relation for Legendre polynomials, (2n+1) y P_{n}(y)=(n+1)P_{n+1}(y)+n P_{n1}(y), to find
(26) 
Making use of Eqs.(20) and (21), we then find, for ,
(27) 
For three different values of n and , , the functions are displayed in the lower panel of Fig.1.
3.2 Logarithmic weight functions
Choosing the T_{+n} to be polynomials in implies that the structure of these weight functions is similar on all angular scales from to . For example, the roots of the T_{+n} are fairly evenly spread on the interval . On the other hand, we expect the correlation function to show more structure on small scales than on large scales. Hence, for a given maximum number N of modes, the large angular scales will be sampled on finer scales than needed, whereas small angular scales may not be sufficiently well resolved to extract all information contained in the correlation function.Figure 3: Mathematica (Wolfram 1991) program to calculate the roots in Eq.(36)  they are stored with 8 significant digits in the lower left halve of the table ROOTS. Furthermore, the array norm[n] contains the normalization coefficients N_{n}. 

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In order obtain a finer sampling of the smallscale correlation
function for a given N, we now construct a set of weight functions
which are polynomials in
.
Hence the roots of these weight
functions are approximately evenly spaced in
,
thus the weight
functions sample small angular scales with higher resolution than
large angular scales. As in Sect.3.1, this set
of functions must fulfill the constraints (4), and
we require the functions to be orthonormal. Hence, the lowestorder
weight function again is of secondorder. We parametrize this set of
weight functions as
where we choose
(29) 
which varies from 0 to as goes from to . Furthermore, we defined with , so that . In this way, the relative amplitude of the c's is decoupled from the overall normalization N_{n}. As before, we set and . With this transformation of variables the constraints (4) become
and an orthonormality condition analogous to Eq.(16) can be written as
To write these constraints in a more compact form we
define the set of coefficients
(32) 
where is the incomplete Gamma function.
With the representation (28), the constraints
(30) become
These two equations determine the two coefficients , needed to obtain . We then obtain the corresponding coefficients by iterating in n. Thus, for a given n, we assume that the have been determined for all m<n. Then, the are obtained from the two Eqs.(33), and the (n1) orthogonality conditions (31) for , which read in the representation (28)
or
where we used that . Thus, together we have n+1linear equations for the n+1 unknown coefficients , , which in principle can be readily solved (but see below). Finally, to obtain the normalization of the functions, we use Eq.(31) for m=n, which together with Eq. (28) yields
(35) 
which determines N_{n} (and thus the ) up to an (arbitrary) sign. For definiteness, we choose the sign such that , implying that N_{n}=c_{n(n+1)}>0.
It turns out that the solution of the system of linear equations for
the c's requires very high numerical accuracy for even moderately
large n, in particular for large values of
.
We used
Mathematica (Wolfram 1991) with large setting of WorkingPrecision for calculating the incomplete Gamma function and
for carrying out the sums in Eq. (34). Once the c's have
been determined, the integrals in Eqs. (30) and
(31)  the latter for m<n  have been calculated to
check the accuracy of the solution. We found that, for
and for
,
one needs to determine the c's to 40 significant digits, in order for all these integrals, which
should be zero, to attain values less than 0.1. We then calculated
the n+1 roots r_{n,i} of the
,
and represented
the functions as
For the same parameters as before, using only five significant digits for the r's renders all the integrals zero to better than 10^{6}, and with eight significant digits, the integrals are zero to better than 10^{17} even for . Thus, the representation (36) is the adequate one for practical work. A short Mathematica program for calculating the r_{n} is displayed in Fig.3.
Figure 4: The logarithmic filter functions for and . The left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small . 

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Figure 5: The logarithmic filter functions for and . As in Fig.4, the left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small . 

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Figure 6: The W_{n}functions calculated from . The upper panel corresponds to , whereas the lower panel is calculated using , and in both cases. 

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The corresponding
are constructed from
Eq.(3), by defining
,
which yields
=  
=  
=  (37)  
Given the remarks above, the first of these expressions (i.e., numerical integration) is the method of choice if the are given in the form (36). Alternatively, making use of the representation
one can write the as
(38) 
where the coefficients are given as
a_{n2}  =  
d_{nm}  =  (39) 
In Figs.4 and 5, we have plotted the filter functions for and . The left panels show these filter functions over the whole angular range, the right panels show an enlargement for small values of . As expected, the roots of the weight functions are clustered towards lower values of . Thus, for a fixed maximum number of n, these functions resolve those scales better than the linear filter functions. Figure 6 shows the filter functions which, according to Eq.(5), relates the COSEBIs to the underlying power spectrum . With increasing n, the COSEBIs are sensitive to power at increasingly larger values of .
3.3 E/Bmode correlation functions
Crittenden et al. (2002) and Schneider et al. (2002) constructed E/Bmode correlation functions, which consist of the original correlation function plus a correction term which is again an integral over correlation functions. However, these correction terms are unobservable, since the integral extends over an infinite angular range. Thus, these E/Bmode correlation functions cannot be obtained in practice and are of little use.With the full E/Bmode decomposition provided by the COSEBIs, we can
define new pure E/Bmode correlation functions,
obviously, the only depend on the Emode shear, whereas the contains information only from Bmodes. Owing to the constraints (4) which the functions T_{+n} have to obey, one finds that
In fact, as shown in SK07, the function T_{} also obeys analogous constraints, namely
so that
In Fig.7, we have plotted the pure Emode correlation functions , together with the orinial 2PCFs , for a fiducial CDM cosmological model that will be described in the next section; the overall shape of these functions, however, does not depend on the details of the choice of cosmological parameters. Although not easily visible, both have two roots, as required by the constraints (41) and (42). The function is rather similar in shape to the original 2PCF , modified in a way as to obey Eq.(41). However, has a very different shape than . In fact, it is easy to see from Eqs.(3) and (4) that , . In Appendix C, we show how these new puremode correlation functions are related to the original 2PCFs. As is obvious from their definition, these puremode correlation functions can be obtained from the 2PCFs over a finite interval, hence their estimation does not require extrapolations or ``inventing data''.
Figure 7: The 2PCFs and the corresponding pure Emode correlation functions , for , , and the fiducial cosmological model described in Sect.4. 

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4 Likelihood analysis
We calculate the posterior likelihood in the  parameter space for four cases of COSEBIs ( , , each for and ). Note that, unless stated otherwise, we choose as the minimum separation in the 2PCF. For each of the four cases we are interested in two main questions: first, how does the information content evolve when including more modes n in the likelihood analysis? Second, once it saturates, how large is the difference to the information content of the 2PCFs?4.1 Model choice
In the likelihood analysis we assume a flat universe, and vary the
matter density
(and simultaneously
to
preserve flatness) and the normalization
of the density
fluctuations; all other parameters are held fixed, i.e. the
dimensionaless Hubble constant h=0.73, the density parameter in
baryons
,
and the slope of the primordial
fluctuation power spectrum
.
We choose
and
as our fiducial model which enters the likelihood
analysis in this section and represents the cosmological model used in
Fig.7. The Bmode power spectrum is set to zero,
,
whereas the shear power spectra are obtained from the threedimensional density power spectra
using Limber's equation (see, e.g., Kaiser 1998). The
power spectrum
is calculated with the transfer function
from Efstathiou et al. (1992). For the nonlinear evolution we use the fitting
formula of Smith et al. (2003). In the calculation of
we choose a
redshift distribution of source galaxies similar to that of
Benjamin et al. (2007),
with , , z_{0}=1.171. The corresponding 2PCFs are calculated from Eq.(7), and from these, the COSEBIs are calculated according to Eq. (1) for various modes n using linear and logarithmic filter functions. The covariances used in our likelihood analysis are calculated from the power spectrum as described in Joachimi et al. (2008), assuming our fiducial cosmology. This method does not account for the nonGaussianity of the shear field or the cosmologydependence of the covariance (Eifler et al. 2009), however these issues are not crucial for our purpose as we are only interested in the relative performance of COSEBIs and the 2PCFs. More important is that we can choose an arbitrary binning in the 2PCF covariance. The latter aspect in combination with the speed of the calculation is decisive to resolve the numerical issues in the calculation of the COSEBIs' covariance. The survey parameters read A=170 , , and , and correspond to those of the upcoming cosmic shear analysis of the full CFHTLS survey area.
The exact method to calculate the posterior likelihood from the data vectors and covariances is described in Eifler et al. (2010). Similar to their analysis we assume flat priors inside the intervals and , and zero prior otherwise.
Figure 8: The correlation coefficients (44) for linear (top) and logarithmic (bottom) weight functions , calculated for , , and the fiducial cosmological model described in the text. 

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Figure 9: Likelihood contours for a fiducial cosmic shear survey, with parameters described in Sect.4.1. The upper ( lower) six panels correspond to (20'). Shown in the first and third rows are the likelihood as obtained from the COSEBIs with linear filter functions and various , in the second and fourth rows the likelihood as obtained from the logarithmic filter functions, and in comparison, we show the likelihood obtained from the shear twopoint correlation functions. 

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Figure 10: The values of q  see Eq.(46)  calculated from the COSEBIs for the case of linear (circles) or logarithmic (triangles) T_{n}functions, as a function of the maximum mode which was included in the likelihood analysis. The results in the left (right) panel correspond to ( ), and the filled symbols are calculated for ; in the left panel, we also plot corresponding results for , indicated by the open triangles. The dashed (dashdotted) line represents the optimal q for ( ), obtained when using the 2PCFs directly. The dotted lines shows the asymptotic value of q achieved for large . 

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4.2 The covariance of the COSEBIs
In Fig.8 we have plotted the correlation matrix of the COSEBIs, defined asfor several values of m, using both linear (upper panel) and logarithmic (lower panel) weight functions. The value of m can be identified as the point where r_{mn} = 1. For the linear weight functions, we see that the correlation matrix declines quickly for , reaches a (negative) minimum at , and essentially is zero for . Thus, the covariance matrix is in essence a band matrix. For the logarithmic COSEBIs, the nonzero correlations between the E_{n} span a larger range in mn. One therefore expects that the inversion of the covariance matrix for a given is more difficult for the logarithmic COSEBIs than for the linear ones. However, as we will show below, a smaller number of logarithmic COSEBIs are needed to extract all the cosmological information contained in the shear correlation functions, compared to the linear COSEBIs.
4.3 Figures of Merit: a short discussion
In order to illustrate the information content one usually calculates the socalled credible regions, inside of which the true set of parameters is located with a probability of e.g. 68%, 95%, 99.9%. Instead of showing likelihood contours for all cases considered, we use two different measures to quantify the size of these credible regions, where each measure characterizes the information contents through a single number.
The first measure, q, is calculated from the determinant of the
secondorder moment of the posterior likelihood
,
where are the parameters of the model, and are the parameters of the fiducial model (here, i=1,2, corresponding to and ). We quantify the size of the credible region by the square root of the determinant of ,
Smaller credible regions in parameter space correspond to smaller values of q. In this paper, all q's are given in units of 10^{4}.
Our second figure of merit is obtained from the Fisher information
matrix (Tegmark et al. 1997)
where subscripts separated by a comma denote partial derivatives with respect to , and , where is the dimensional vector of the first E_{n}'s. The dimensional covariance matrix has the elements , as given in Eq.(8). We consider a constant covariance in parameter space, so that the first term of Eq.(47) vanishes. Since the Fisher matrix is the Hessian of the (negative of the) loglikelihood function at its maximum, its elements describe the size and shape of ellipses of constant likelihood near the maximum. If the likelihood was strictly Gaussian, the Fisher matrix would completely describe its functional form. We define our second figure of merit f as
For a better comparison with q we chose to modify the more commonly used figure of merit definition (see e.g., Albrecht et al. 2006)  we consider the area of the error ellipse itself, not its inverse. Similar to q, f is given in units of 10^{4}. With the definition (48), q and f give the same result if (1) the likelihood in the parameter space considered is Gaussian and (2) if the likelihood outside the region where we set a flat prior is negligible. We note that f and q can be significantly different if these two assumptions are not satisfied. Then, the Gaussian defined by the Fisher matrix is only a useful approximation close to the fiducial model, and the resulting values of f can be rather bad approximations for q. In contrast, q is sensitive to parameter regions far from the fiducial model and we therefore consider q as the more useful measure for the information contents. In order to give an impression of the meaning of different q and f we show a sample of likelihood contours in Fig.9  it is obvious that the likelihood function in our case is far from Gaussian.
4.4 Results of the likelihood analysis
Figure 10 shows the values of q for the case of (left panel) and for (right panel). The triangles correspond to the COSEBIs from , whereas the circles correspond to the COSEBIs calculated using the linear T_{n}. For comparison we show the information content of the 2PCF (dashed line), which serves as an upper limit on the information content of any secondorder E/Bdecomposing measure  since the 2PCFs contain all information from secondorder shear measurements and the COSEBIs are derived from them (Eifler et al. 2008).For the minimum value of q obtainable from the COSEBIs  and thus the available information on the two cosmological parameters considered  is extremely close to that obtained from the 2PCFs. The logarithmic E_{n} reach this threshold already for , whereas the linear E_{n} saturate around , indicating that the logarithmic modes capture the bulk of cosmological information in significantly fewer data points compared to the linear case. This property can be particularly important in higherdimensional parameter spaces, where data compression and computing time become important.
The COSEBIs for saturate much earlier; the information content of E_{n} is hardly increased when going beyond n=4 (n=3 for the logarithmic weight functions). More important, however, is the large difference between the saturation limit of the COSEBIs and the corresponding information content of the 2PCFs (which is also seen in the likelihood contours of Fig.9). Obviously, the choice of has a significant impact on the information content, and on the relative information contained in the COSEBIs and the 2PCFs.
This latter difference is not due to a deficiency of the COSEBIs  since they form a complete set of E/Bmode measures, they contain all the information that can uniquely be split into the two modes. If, however, one assumes that the shear field has no Bmode contribution, and thus using of the full 2PCFs obviously yields tighter parameter constraints. But, this assumption will hardly be justifiable in any of the forthcoming surveys. The fact that the measured Bmodes are compatible with zero within the error bars in a data set is not a justification  since any realistic survey may contain Bmodes which cannot be identified as such, for example a uniform shear field which can either be E or Bmode. Therefore, the loss of information due to a clean mode separation is inevitable, but a small price to pay relative to a potential bias of results due to undetected Bmodes. Fortunately, for surveys which allow shear correlation measurements on large angular scales, this information loss is seen to be almost negligible.
Figure 11: The q of the COSEBIs as a function , for . The COSEBIs are calculated from , where n ranges from 1  10. 

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We analyse this more closely in Fig.11, where we show q as a function of ; here we use logarithmic weight functions with 10 modes, i.e., where the asymptotic limit is well achieved. The amount of information increases significantly when going from 20' to 100' and becomes almost constant when going to larger . This behavior, of course, depends on the parameter space considered; for  it can be understood from the functional behavior of the power spectrum. For small , it is almost fully degenerate in these two parameters, hence going to larger angular scales does not yield significantly more information  this will be different for other parameter combinations. One also sees that the difference in information content between the COSEBIs and the 2PCFs decreases for larger  the larger , the smaller is the contribution of modes to the 2PCFs which can not be uniquely decomposed into E/Bmodes. Furthermore, we have plotted the corresponding values of q for the aperture dispersion , where is the aperture radius which is calculated from the shear 2PCFs for . Values for are calculated and plotted only for , to limit the bias caused by the lack of measured correlation functions for (see Kilbinger et al. 2006) to <. We see that its information content is smaller than that of the COSEBIs, as it must be the case, owing to the completeness of the latter.
Figure 12: The value of f  see Eq.(48)  for the case of linear (circles) or logarithmic (triangles) T_{n}functions as a function of the maximum mode which was included in the likelihood analysis. The results in the left (right) panel correspond to ( ), and the filled symbols are calculated for ; in the left panel, we also plot corresponding results for , indicated by the open triangles. The dashed (dashdotted) line represents the optimal f for ( ), obtained when using the 2PCFs directly. The dotted lines shows the asymptotic value of f achieved for large . 

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Figure 12 shows a similar analysis based on f. The results confirm our foregoing findings. Similar to the case of q, the Fisher matrix analysis shows that the logarithmic E_{n}reach the saturation limit much earlier than the linear E_{n} and again, the saturation limit for is closer to the optimal information content than for .
Table 1: Values of q and f as obtained by considering the full 2PCFs, and by using the COSEBIs , and
Note that in Figs.10 and 12 we choose a similar scale for the vertical axis in the right and the left panels to enable for an easier comparison between the different cases of . We point out the good agreement between the saturation limits of E_{n} and in all cases, which shows that our results are numerically robust. In Table1, we have listed the values of q and f as shown in Figs.10 and 12 for the maximum number of modes. The small difference between these values as obtained from the linear and logarithmic weight functions for the COSEBIs is due to the fact that for these values of , the linear ones have not yet reached their full asymptotic value.
The underlying reason why the formal loss of information of the COSEBIs, relative to the full 2PCFs, is larger for smaller is due to the filter functions that relates the 2PCFs to the power spectrum. This filter function is J_{0}(x) for the case of , i.e. a function that tends towards +1 for small arguments. This implies that the correlation function is sensitive to longrange modes, i.e., modes of small . In particular, this means that is also sensitive to the power spectrum for modes satisfying , corresponding to scales which are in fact not probed by the 2PCFs directly  and for which no E/Bmode separation is possible from the data. The relative cosmological information content of the power spectrum in the ranges and decreases with increasing , which explains the difference in ``relative information loss'' in Figs.10 through 12.
Up to now we have always chosen . However, one may ask whether cosmic shear measurements down to this angular scale can be compared to sufficient accuracy with cosmological predictions, since at the corresponding length scales, baryonic physics can have a significant influence on the projected power spectrum. Of course, modeling the behavior of baryons in a cosmological simulation is much more difficult, and burdened with higher uncertainty, than dark matteronly simulations. Jing et al. (2006) compared pure dark matter simulations with hydrodynamic simulations to conclude that for , corresponding to , the predicted power spectra differ by about 10%  much more than the predicted statistical uncertainty of future cosmic shear surveys.
Fortunately, the largest effect of baryons on the total mass distribution seems to be a change of the halo concentration parameter as a function of halo mass (Rudd et al. 2008), in that baryons render halos more concentrated. If this is the case, then this effect can be calibrated from the weak lensing data themselves. Zentner et al. (2008) studied such a selfcalibration method for future surveys and concluded that the concentrationmass relation can be determined from the weak lensing data. In the framework of the halo model for the largescale structure, the power spectrum can then be calculated using this modified concentrationmass relation, and fairly accurate model predictions can be made.
Dropping the small angular scales from future surveys implies considerably weaker cosmological constraints. In the left panels of Figs.10 and 12, we have plotted the values of q and f, respectively, for surveys with . Independent of whether the ``optimal'' constraints from the 2PCFs or the COSEBIs are employed, the resulting constraints are weaker than for . Therefore, it is of considerable interest to improve the accuracy of predictions for the matter power spectrum to small scales, to make full use of the information contained in cosmic shear surveys on small angular scales.
5 Summary and discussion
We have defined pure E and Bmode cosmic shear measures from correlation functions over a finite interval . These are complete orthonormal sets of such measures, implying that they contain all cosmic shear information in the twopoint correlation functions which can be uniquely split into E and Bmodes. For these COSEBIs, we have calculated their relation to the underlying power spectrum and their covariance matrix. Two different sets of COSEBIs have been explicitly constructed, those with weight functions which are polynomials in the angular scale, and those with polynomial weight functions in the logarithm of the angular scale. For the former case, analytic expressions were obtained for all orders, whereas in the logarithmic case, a linear system of equations needs to be solved numerically.
5.1 Advantages of the COSEBIs
Comparing the COSEBIs with earlier cosmic shear measures, we point out a number of advantages. First, using the correlation functions themselves does not provide an E/Bmode separation. The construction of E/Bmode correlation functions as described in (Crittenden et al. 2002) requires knowledge of the correlation functions over an infinite angular range, and is therefore not applicable in practice (extrapolating to infinite separation using fiducial cosmological models corresponds to ``inventing data'', and implicitly assumes that there are no longrange Bmodes). In fact, the generalization of pure E/Bmode correlation functions based on data over a finite angular range has been derived here (see Sect.2 and AppendixC); however, we expect these to be of limited use in practice.
Whereas the aperture mass dispersion (Schneider et al. 1998) provides a clean separation into E and Bmodes (Schneider et al. 2002; Crittenden et al. 2002), it requires the knowledge of the correlation function to arbitrarily small angular separation. There are at least two aspects which render this impractical: first, galaxy images need a minimum separation for their shapes to be measurable. Second, on very small scales baryonic effects will affect the power spectrum and render model predictions very uncertain. The inevitable bias of the aperture mass dispersion (Kilbinger et al. 2006) motivated the ring statistics (Schneider & Kilbinger 2007). The latter removes the bias, depends only on the correlation function over a finite interval, and has potentially higher sensitivity to cosmological parameters (Eifler et al. 2010, FK10). However, the weight function of the ring statistics is largely arbitrary.
The COSEBIs contain all available modeseparable information from the correlation functions on a finite interval, and are therefore guaranteed to provide highest sensitivity to cosmological parameters. Furthermore, they form a discrete set of measures, whereas the other cosmic shear statistics include a somewhat arbitrary grid of variables, like the outer scale of the ring statistics: if the grid is too coarse, information gets lost, whereas a finer grid renders the measures largely redundant, implying large and significantly nondiagonal covariances. In contrast, the discreteness of COSEBIs leaves no freedom, and for the linear weight functions, the covariances have a narrow band structure. The information clearly saturates after a number of modes, and this number is surprisingly small for the logarithmic weight function. Therefore, determining covariance matrices from numerical simulations (as was done for the COSMOS analysis of Schrabback et al. 2009) appears considerably simpler than for other cosmic shear measurements, which is particularly true for an unbiased estimate of their inverse (see Hartlap et al. 2007, for a discussion of this point). Based on these properties of the COSEBIs, we would like to advertise them as the method of choice for future cosmic shear analyses.
5.2 Generalizations
In case photometric redshift information of the lensed galaxies is available and several source populations can be defined based on their redshift estimates, the COSEBIs can be generalized to a tomographic version. Furthermore, under the same assumption, intrinsic alignment effects between the tidal gravitational field and the intrinsic galaxy orientation (e.g., Catelan et al. 2001; Jing 2002; Crittenden et al. 2001; Hirata & Seljak 2004) can be filtered out by properly choosing redshiftdependent weight functions, such as to avoid physically close pairs of galaxies (King & Schneider 2002; Heymans & Heavens 2003; King & Schneider 2003) or make use of the specific redshift dependence of the shearintrinsic alignments (Joachimi & Schneider 2009; Bridle & King 2007; Joachimi & Schneider 2008), possibly in combination with other data (Joachimi & Bridle 2009). We expect that these generalizations of the COSEBIs provide no real difficulties.
It would be desirable to obtain a similar measure for thirdorder cosmic shear statistics, i.e., one that provides clear E/Bmode separations from threepoint correlation functions measured over a finite interval. Up to now, the aperture statistics is the only known such measure (Schneider et al. 2005; Jarvis et al. 2004); however, similar to the case of the aperture dispersion, thirdorder aperture statistics requires the correlation functions to be measured down to arbitrarily small separations. A generalization of the COSEBIs to third order seems challening  not only because of the higher number of independent variables (the threepoint correlation functions depend on three variables) and the larger number of modes (one pure Emode, one mixed E/Bmode, and two further modes which are not invariant under parity transformation), but also because of the more complicated relation between correlation functions and the bispectra (Schneider et al. 2005). Thus, even the analogue of the starting point of the current paper  Eqs.(1) and (3)  is not yet known for the thirdorder case.
AcknowledgementsWe thank Liping Fu and Martin Kilbinger for interesting discussions on E/Bmode separations which triggered this study, and an anonymous referee for constructive suggestions. We thank Marika Asgari for checking some of the numerical results presented here. 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.
Appendix A: Calculation of the COSEBIs: numerical problems and solutions
Several numerical issues arose during the implementation of the calculations of the COSEBIs, especially in the context of their covariance. As these issues are crucial for obtaining the correct values of q and f, we outline them in greater detail. We employ the QAG adaptive integration routine from the GNU Scientific Library^{} and obtain the E_{n} using two different methods. First, we calculate them from the set of 2PCFs according to Eq.(1) and second, we check for consistency by calculating E_{n} directly from the according to Eq.(5). The first method cleanly separates E and Bmodes, giving a Bmode residual due to numerical uncertainties which is 8 to 5 orders of magnitudes lower than the Emode, depending on the scales considered and whether one uses T_{n} or . Both methods yield results in perfect agreement, hence we are confident that there are no numerical problems in either of them.When using a binned version of the 2PCF instead, we find a nonnegligible deviation when using too few angular bins. The number of bins, above which the E/Bdecomposition becomes stable, depends on the mode E_{n}, the maximum scale of the 2PCF, and whether one uses T_{n} or . This should be checked carefully before applying the method to an actual data set. As an example, we found that for linear T_{n} and , one needs 10^{5} bins to calculate E_{30}properly and to have an accurate mode separation.
The calculation of the covariance is numerically more challenging than that of the data vectors. Again, we use two approaches and calculate from using Eq.(8), and from the 2PCF covariance using Eq.(9). Both methods have their difficulties and need to be checked carefully for consistency before using the covariance in the likelihood analysis.
The power spectrum approach using Eq.(5) involves the calculation of a onedimensional integral over multiplied by two filter functions . As can be seen from Figs.2 and 6, these filter functions are strongly oscillating, which becomes worse for large and higher modes n. We use a stepwise integration to calculate the integral and truncate the integral once the ratio of ``new contribution in step i/integral calculated until (i1) drops below a certain threshold. We vary the width of the steps as well as the truncation threshold; however, we find that the integration becomes inaccurate when going to higher modes n.
For calculation of from the covariance of the 2PCFs we find that it is too timeconsuming to calculate the 2PCF covariance for every sampling point of the integration routine separately. Instead, we calculate the 2PCF covariance for a specific binning and interpolate the values during the integration. We use a linear binning in the 2PCF covariance for the linear weight function and a logarithmic binning for the case of . In addition, we check how strongly the number of bins influences the accuracy of the integral, finding that we can calculate properly if we choose at least bins in the 2PCF covariance. The final must be symmetric, positive definite, and not illconditioned, as already small deviations from these requirements can bias the information content measures q and f.
Appendix B: S/N maximization
From a complete set of functions T_{+n} obeying the constraints (4) for given and , we can find a weight function which maximizes the signaltonoise of the Emode. This problem was also considered by FK10. In this case, we can writewhich satisfies the integral constraints (4) for any choice of the a_{n}. Then the Emode signal is, in the absence of Bmodes,
The noise N of E is obtained through the covariance of the E_{n},
yielding as signaltonoise ratio
To obtain a maximum of S/N with respect to the coefficients a_{n}, we differentiate the foregoing expression with respect to a coefficient a_{k},
Setting this derivative to zero results in
From this equation we see that the overall amplitude of the a_{n}cannot be determined, i.e., if the a_{n} are a solution, then solve the equation as well. Noting that the first term on the r.h.s. of Eq. (B.6) does not depend on k, a solution is obtained as
(B.7) 
as can be also verified by inserting this into Eq. (B.6). Thus, if the function T_{+} is expanded into a set of functions which all satisfy the constraints (4), the signaltonoise maximization can be done analytically. If different sets of functions are used for constructing the T_{+} maximizing the S/N, the resulting function should be the same in the limit ; however, different sets of functions may require different N before the asymptotic limit is reached.
Appendix C: Pure E/Bmode correlation functions
We will now explore how the puremode correlation functions introduced in Eq.(40) are related to the original . For this, we use Eq.(1) in the definition (40) to obtainwhere for Emodes, for the Bmodes, and where we defined the functions in the last step. These functions are calculated next, by noting that the normalized Legendre polynomials p_{n}(x) as defined in Eq.(19) are orthonormal,
form a complete set of functions on the interval [1,1], and therefore obey
(C.2) 
Noting that we have chosen in Sect. 3.1 t_{n}(x)=p_{n+1}(x)for , we find that
s_{++}(x,y)  =  
=  
=  
=:  (C.3) 
where in the final step we have defined the function F_{++}(x,y), which is obviously symmetric in its arguments. The explicit expression for it reads
F_{++}  
+  (C.4)  
+ 
We can now calculate the other sums in Eqs. (C.1), making use of Eq. (20) written in the form
(C.5) 
with
(C.6) 
This then yields
s_{+}(x,y)  =  
=  (C.7)  
= 
where
(C.8) 
Owing to symmetry,
(C.9) 
and
s_{}(x,y)  =  
=  (C.10)  
where the symmetric function W is defined as
W(x,y)  =  
(C.11) 
Thus, we find for the in turn, using and :
=  
=  
=  (C.12)  
=  
with F_{+}(x,y)=F_{++}(x,y)+V(x,y), F_{}(x,y)=F_{++}(x,y)+V(x,y)+V(y,x)W(x,y). We finally obtain for the pure mode correlation functions
=  
=  
(C.13) 
Hence, pure mode correlation functions can be obtained from the observed correlation functions over a finite interval. However, we believe that these pure mode correlation functions are of little practical use, since for a quantitative analysis of cosmic shear surveys the COSEBIs contain all relevant information.
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Footnotes
 ... Library^{}
 http://www.gnu.org/software/gsl/
All Tables
Table 1: Values of q and f as obtained by considering the full 2PCFs, and by using the COSEBIs , and
All Figures
Figure 1: The linear filter functions for , . Note that the shape of the curves depends only on the ratio . 

Open with DEXTER  
In the text 
Figure 2: The functions W_{n} as defined in Eq.(6) which relate the COSEBIs to the underlying power spectrum, calculated from the . The upper panel corresponds to , whereas the lower panel is calculated using , both for . 

Open with DEXTER  
In the text 
Figure 3: Mathematica (Wolfram 1991) program to calculate the roots in Eq.(36)  they are stored with 8 significant digits in the lower left halve of the table ROOTS. Furthermore, the array norm[n] contains the normalization coefficients N_{n}. 

Open with DEXTER  
In the text 
Figure 4: The logarithmic filter functions for and . The left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small . 

Open with DEXTER  
In the text 
Figure 5: The logarithmic filter functions for and . As in Fig.4, the left panel shows the function over the whole interval, whereas the right panel provides a more detailed view for small . 

Open with DEXTER  
In the text 
Figure 6: The W_{n}functions calculated from . The upper panel corresponds to , whereas the lower panel is calculated using , and in both cases. 

Open with DEXTER  
In the text 
Figure 7: The 2PCFs and the corresponding pure Emode correlation functions , for , , and the fiducial cosmological model described in Sect.4. 

Open with DEXTER  
In the text 
Figure 8: The correlation coefficients (44) for linear (top) and logarithmic (bottom) weight functions , calculated for , , and the fiducial cosmological model described in the text. 

Open with DEXTER  
In the text 
Figure 9: Likelihood contours for a fiducial cosmic shear survey, with parameters described in Sect.4.1. The upper ( lower) six panels correspond to (20'). Shown in the first and third rows are the likelihood as obtained from the COSEBIs with linear filter functions and various , in the second and fourth rows the likelihood as obtained from the logarithmic filter functions, and in comparison, we show the likelihood obtained from the shear twopoint correlation functions. 

Open with DEXTER  
In the text 
Figure 10: The values of q  see Eq.(46)  calculated from the COSEBIs for the case of linear (circles) or logarithmic (triangles) T_{n}functions, as a function of the maximum mode which was included in the likelihood analysis. The results in the left (right) panel correspond to ( ), and the filled symbols are calculated for ; in the left panel, we also plot corresponding results for , indicated by the open triangles. The dashed (dashdotted) line represents the optimal q for ( ), obtained when using the 2PCFs directly. The dotted lines shows the asymptotic value of q achieved for large . 

Open with DEXTER  
In the text 
Figure 11: The q of the COSEBIs as a function , for . The COSEBIs are calculated from , where n ranges from 1  10. 

Open with DEXTER  
In the text 
Figure 12: The value of f  see Eq.(48)  for the case of linear (circles) or logarithmic (triangles) T_{n}functions as a function of the maximum mode which was included in the likelihood analysis. The results in the left (right) panel correspond to ( ), and the filled symbols are calculated for ; in the left panel, we also plot corresponding results for , indicated by the open triangles. The dashed (dashdotted) line represents the optimal f for ( ), obtained when using the 2PCFs directly. The dotted lines shows the asymptotic value of f achieved for large . 

Open with DEXTER  
In the text 
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