How well do thirdorder aperture mass statistics separate E and Bmodes?
^{1} MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStraße 1, 85740 Garching bei München, Germany
email: xun@mpagarching.mpg.de
^{2} Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK
^{3} ArgelanderInstitut für Astronomie (AIfA), Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
Received: 4 September 2013
Accepted: 25 November 2013
With thirdorder statistics of gravitational shear it will be possible to extract valuable cosmological information from ongoing and future weak lensing surveys that is not contained in standard secondorder statistics because of the nonGaussianity of the shear field. Aperture mass statistics are an appropriate choice for thirdorder statistics because of their simple form and their ability to separate E and Bmodes of the shear. However, it has been demonstrated that secondorder aperture mass statistics suffer from E/Bmode mixing because it is impossible to reliably estimate the shapes of close pairs of galaxies. This finding has triggered developments of several new secondorder statistical measures for cosmic shear. Whether the same developments are needed for thirdorder shear statistics is largely determined by how severe this E/Bmixing is for thirdorder statistics. We tested thirdorder aperture mass statistics against E/Bmode mixing and found that the level of contamination is well described by a function of θ / θ_{min}, where θ_{min} is the cutoff scale. At angular scales of θ > 10 θ_{min}, the decrease in the Emode signal due to E/Bmode mixing is lower than 1 percent, and the leakage into Bmodes is even less. For typical smallscale cutoffs this E/Bmixing is negligible on scales larger than a few arcminutes. Therefore, thirdorder aperture mass statistics can safely be used to separate E and Bmodes and infer cosmological information, for groundbased surveys as well as forthcoming spacebased surveys such as Euclid.
Key words: gravitational lensing: weak / methods: statistical / largescale structure of Universe / cosmological parameters
© ESO, 2014
1. Introduction
Forthcoming largefield multicolor imaging surveys, such as KiDS^{1}, DES^{2}, HSC^{3}, LSST^{4}, and Euclid^{5}, will obtain galaxy shape and photometric redshift information for a huge number of galaxies. This will boost weak lensing statistical power, especially in constraining the properties of dark matter, dark energy, and the laws of gravity.
The cosmological information obtained from cosmological weak lensing can be enhanced by going beyond the standard analysis of secondorder (twopoint) statistics. The most straightforward path to the exploitation of higherorder statistical information is the use of threepoint functions of gravitational shear. These can probe nonGaussian signatures in the underlying matter density field, and thus are key tools for better exploiting the wealth of information on small, nonlinear scales. Moreover, adding thirdorder statistics to the weak lensing analysis may substantially improve the selfcalibration of systematic effects (Huterer et al. 2006).
Previous studies (e.g. Takada & Jain 2004) suggested that the strength of the constraints on cosmological parameters from thirdorder weak lensing statistics alone are comparable to those from twopoint statistics. Recently, Kayo et al. (2013) and Kayo & Takada (2013) estimated the cosmological information from combined two and threepoint statistics taking into account nonGaussian error covariances as well as the crosscovariance between the power spectrum and the bispectrum. They found that adding the thirdorder information improves the dark energy figureofmerit of weak lensing twopoint statistics alone by about 60%. This potential benefit comes without the need for additional observations, so that an efficient extraction of weak lensing threepoint information is desirable.
The increasingly precise measurements of future weak lensing observations call for synchronously improving measurement accuracy. The major sources of weak lensing systematics lie in the measurement process, specifically, in the galaxy shape measurement (e.g. Kitching et al. 2012) and the determination of photometric redshifts (e.g. Abdalla et al. 2008). Additionally, there are systematics originating from astrophysical processes, the most worrisome being the intrinsic alignment of galaxy shapes (see Semboloni et al. 2008; Shi et al. 2010 for work at the threepoint level). How much weak lensing threepoint statistics are affected by these systematic effects is still uncertain to a large degree.
In this situation, sensitive and reliable systematics tests are of utmost importance. One such test is the decomposition of statistics of the gravitational shear into electric fieldlike Emode components and magnetic fieldlike Bmode components (Crittenden et al. 2002; Schneider et al. 2002). The cosmological weak lensing signal only generates Emodes to first order, with the Bmode signal making less than a permil level contribution for twopoint statistics (Hilbert et al. 2009). Hence, a significant Bmode signal serves as a smoking gun for the presence of systematics in the data, also at the threepoint level.
For secondorder statistics, several E/Bmode separating statistical measures in configuration space have been developed, all of which can be obtained via a linear transformation of the twopoint shear correlation functions. The aperture mass statistics (Schneider et al. 1998) are conceptually, and in practice, the easiest to apply, but require a measurement of the shear correlation function down to lag zero for perfect separation into E and Bmodes. However, the correlation functions are not measurable at small separation, for instance because of the overlap of galaxy images (Van Waerbeke et al. 2000). Note that it is impractical to extract aperture mass statistics directly from data because of gaps and masked areas in the images.
Kilbinger et al. (2006) demonstrated that the lower limit on the angular galaxy separation available for correlation function measurements leads to a significant leakage of Emodes into the Bmode aperture mass statistic on small scales. This E/Bmode mixing reduces the effectiveness of the Bmode signal as a channel for detecting systematics and, if unaccounted for, causes biases in any cosmological analyses performed with the Emode signal.
To eliminate this undesirable E/Bmode mixing, more sophisticated twopoint statistics have been developed, including the ring statistics and the complete orthogonal sets of EB mode integrals (Schneider & Kilbinger 2007; Eifler et al. 2010; Fu & Kilbinger 2010; Schneider et al. 2010; Asgari et al. 2012; Kilbinger et al. 2013). The corresponding weight functions for these statistics used in the transformation from correlation functions have support only on finite intervals [θ_{min};θ_{max}] where θ_{min} > 0, which implies that the shear twopoint correlation functions (2PCFs) are not required for θ < θ_{min} to calculate these statistics, for which therefore no E/Bmode mixing results from a cutoff in the 2PCFs.
The statistics of choice at the threepoint level for the direct application to data are again the shear correlation functions, since they are most straightforward to measure from the data (since they are unaffected by holes and gaps in the data field). However, matters are complicated by the existence of 2^{3} = 8 components of the shear threepoint correlation functions (3PCFs), and three arguments instead of one, as well as some freedom in the choice of reference frame for the triplet of angular positions (Schneider & Lombardi 2003). Unsurprisingly, it is much more difficult to construct threepoint statistics that can be accurately decomposed into E and Bmodes (see Shi et al. 2011 for a derivation of the general conditions).
Nonetheless, thirdorder ring statistics have been developed that allow for a clean E/Bmode separation (Krause et al. 2012), albeit at the price of processing data with a complicated filter function whose practicability still needs to be demonstrated. In contrast, thirdorder aperture statistics, which were the first E/Bmode separating statistics generalized to the threepoint level (Jarvis et al. 2004; Schneider et al. 2005), are relatively easy to compute theoretically and to derive from data. Consequently, they have been predominantly employed in observational analyses, from early detections (e.g. Pen et al. 2003) to recent results from the COSMOS^{6} (Semboloni et al. 2011) and CFHTLenS^{7} (Kilbinger et al., in prep.) surveys.
Threepoint aperture statistics are susceptible to E/Bmode mixing caused by the unavailability of shear 3PCF measurements at small angular separation, just like their twopoint counterparts. To test whether they remain viable as simple and wellestablished cosmological probes for forthcoming weak lensing analyses, we determine analytically the amount of E/Bmode leakage expected for typical lensing surveys.
2. Separating E and Bmodes in the cosmic shear signal
2.1. General E/Bmode separation
Mathematically speaking, E and Bmodes are general decompositions of a spin2 polarization field based on parity symmetry, just as curlfree and divergencefree components are those of a spin1 vector field. Whereas the Emode can be derived from a scalar potential, the Bmode can be derived from a pseudoscalar potential (Stebbins 1996; Kamionkowski et al. 1997, 1998; Zaldarriaga & Seljak 1997; Hu & White 1997; Crittenden et al. 2002; Schneider et al. 2002).
We follow the notation of Schneider et al. (2002) and describe the E and Bmode components of the cosmic shear field by defining the complex lensing potential (1)Then the Cartesian components of the shear can be defined as (2)Shear components defined in this way are not invariant under coordinate rotation. One common way to amend this is to define the tangential (t) and cross (×) components of the shear relative to a reference point on the twodimensional plane (Kamionkowski et al. 1998; Crittenden et al. 2002; Schneider et al. 2002), (3)where φ_{r} is the polar angle of the position vector connecting the reference point to the point where the shear is measured. For second and thirdorder statistics, we choose the reference point to be the center of mass throughout this paper.
The goal of cosmic shear E/Bmode separation is to find statistics that respond only to the Emode shear component of the field, and Bmode statistics that are affected solely by the Bmode signal. Taking correlation functions of the shear field as the “observables”, the general method to construct E/Bmode statistics is to weight the correlation functions, and find the conditions that these weight functions need to satisfy to separate E and Bmodes.
At the twopoint level, the “observables” of the shear field are the shear 2PCFs (4)and (5)for an angular separation r, where φ_{r} is the polar angle of the vector r connecting the two points. The imaginary component of ξ_{−} is expected to vanish because of parity invariance.
The general secondorder E and Bmode statistics can be defined as (Schneider & Kilbinger 2007) (6)Under the condition that (7)EE only contains Emodes, BB only Bmodes. One can also define a mixedterm EB which is derivable from a mixture of E and Bmodes ⟨ψ_{E}ψ_{B} ⟩, and observable as the imaginary part of ξ_{−}. However, this EB term violates parity. It generally vanishes since the shear field is expected to be parity symmetric.
E/Bmode separation needs to be performed separately at each statistical order. The general approach to constructing E/Bseparating statistics remains the same for higherorder statistics, only the conditions for the weight functions that connect the shear correlation functions to the E/Bseparating statistics need to be individually derived at each order. For thirdorder statistics, the conditions for general E/Bmode separation are given in Shi et al. (2011).
2.2. Aperture mass statistics
The aperture mass M_{ap} was first introduced by Kaiser (1995) and Schneider (1996) to estimate masses of galaxy clusters from gravitational lensing signals. It is a filtered version of both the shear γ and the real part of the convergence, κ_{E} ≡ ∇^{2}ψ_{E} / 2 with axisymmetric filter functions, (8)with tangential shear γ_{t} being specified relative to the center of the aperture. The function U_{θ} is an compensated filter function, and the filter functions U_{θ} and Q_{θ} are interrelated by the relation (9)The two most often used sets of filter function forms, the polynomial one proposed by Schneider et al. (1998), and the exponential one by Crittenden et al. (2002), have (nearly) finite support in both real and Fourier space. This feature makes the variance of aperture mass useful also in cosmic shear studies. This statistic provides a welllocalized probe of the power spectrum and is easy to determine from observational data.
Tangential shear averaged over a circle is only sensitive to Emodes, whereas cross shear over a circle is only sensitive to Bmodes. Therefore the aperture mass M_{ap} is a measure of the Emode shear. A corresponding quantity, (10)accordingly is a measure of the Bmode shear.
At the threepoint level, it is convenient to combine the eight correlation functions into the natural components (Schneider & Lombardi 2003) (11)where γ_{cen} is defined relative to the center of mass of the triangle characterized with two side lengths y_{1},y_{2} and the angle between them φ_{y} (see Fig. 1). Because they are measured relative to a center of the triangle formed by the three positions of shear measurements, these natural components are independent of the orientation of the triangle.
Fig. 1 Sketch of the halo model (1halo term) for shear threepoint functions and the notations used. In the text, the three shears in threepoint shear correlator e.g. ⟨γγγ ⟩ correspond to γ_{1}, γ_{2}, and γ_{3}, accordingly. Note that the polar angle of any vector k in Cartesian coordinates is denoted as φ_{k}, but φ_{y} is defined to be the angle between y_{1} and y_{2}. 

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Jarvis et al. (2004) and Schneider et al. (2005) have derived the relations between the natural components of the shear threepoint functions (shear 3PCFs) and the aperture mass statistics using the Crittenden et al. (2002) filter functions Here we present the relations in the specific case of three equal filter sizes, and refer to Eqs. (62), (68) and (73) in Schneider et al. (2005) for the general form: (14)is the pure Emode (EEE) statistics, with (15)and (16)where q_{i} indicates the vectors pointing from the center of mass of the triangle (y_{1},y_{2},φ_{y}) to its vertices (see Fig. 1), (17)Here we used a complex notation so that a vector (a,b) corresponds to the complex number q = a + ib. The asterisk denotes the complex conjugation, so q^{∗} = a − ib corresponds to the vector (a, − b). For the forms of the weight functions T_{0,3}, a Gaussian filter was assumed. Without loss of generality, we choose y_{1} to be parallel to the xaxis, so that φ_{y1} = 0 and φ_{y} is the polar angle of y_{2}.
Similarly, the Bmode threepoint aperture mass statistics include the EEB term (18)the EBB term (19)and the BBB term (20)Among them, and violate parity, whereas conserves parity symmetry and therefore is harder to distinguish from pure Emode statistics.
3. E/Bmode mixing with threepoint aperture mass statistics
Here we investigate the degree of EBmixing in threepoint aperture mass statistics due to the aforementioned smallscale information loss. Since threepoint statistics probe more into the nonlinear, smallscale regime than their twopoint counterparts, one might naively expect threepoint statistics to be more severely affected by the smallscale information loss.
We follow these steps: (i) we model the “observable” shear 3PCFs which contain only Emode signal; (ii) we derive the aperture statistics from the shear 3PCFs; (iii) we introduce a cutoff in the 3PCFs at a small angular scale, mimicking the effect of information loss due to unreliable shear estimates caused by overlapping galaxy images (this generates a Bmode signal that results in a nonzero Bmode aperture statistic that conserves parity), and we repeat step (ii) using only correlation functions above the cutoff scale; (iv) finally, we compare the aperture mass statistics resulted from steps (ii) and (iii).
3.1. Modeling
We use the realspace halo model (Zaldarriaga & Scoccimarro 2003; Takada & Jain 2003b) to model shear threepoint correlation functions. Since our study focuses on small scales, the halo model is a natural choice because it is expected to provide the most precise predictions on small scales. For the same reason, slightly imprecise modeling on large scales will not affect our main results. We take advantage of this and neglect the two and threehalo terms, which significantly reduces the computational load. In fact, as shown by Takada & Jain (2003b), the onehalo term already captures most of the features of the shear 3PCFs measured from raytracing simulations.
In the realspace halo model the convergence profile for a halo of mass M can be expressed as (21)where s is the angular distance from the center of the halo, f_{K} indicates the comoving angular diameter distance, a is the scale factor, and χ and χ_{s} are the comoving distance to the halo and the source, respectively. The surface mass density Σ contributed by the halo is computed from an NFW profile ρ_{NFW} (Navarro et al. 1996) with cutoff at the virial radius r_{vir}(22)where r_{∥} is the proper distance along the line of sight. Σ can be expressed analytically, (23)with c_{h} being the halo concentration parameter, and θ_{vir}(M,χ) = r_{vir}(M,χ) / f_{K}(χ) / a(χ). The explicit form of the function F(x) is given by Eq. (27) in Takada & Jain (2003a).
The halo model tangential shear profile can then be expressed using the relation (24)with being the mean surface mass density inside a circle of radius s. The tangential shear profile can also be expressed analytically as (25)see Eq. (17) in Takada & Jain (2003b) for the explicit form of the function G(x). Note that comoving coordinates have been used to express the surface mass density and the distances in Takada & Jain (2003a,b).
The natural components of shear 3PCFs can then be derived by averaging shear triplets over the triangle angular position s, and summing contributions from all halos with a comoving distance χ smaller than that of the source χ_{s} (see also Takada & Jain 2003a),
where Ω is the angular coverage of the full sky, V is the comoving volume element, and n is the comoving halo mass function. We have taken the form of the comoving differential volume , which is true for a flat universe. Again we chose y_{1} to be parallel to the xaxis. This does not affect generality since the last integral does not depend on φ_{y1} because of the spherical symmetry of a halo. The mass and distance dependencies of γ_{t} are not shown to keep the notation concise. We define to be the projection operator for , which projects the three shears (or their complex conjugates) measured relative to the center of the halo to those measured relative to the center of mass of the triangle formed by y_{1} and y_{2} (see Fig. 1). To derive its form, we first consider the projection of one of the shears, for example, γ_{3}. The shear value γ(s) can be seen as γ_{3} measured relative to the center of the halo. Using Eq. (3), one can express γ_{3} in Cartesian coordinates and then project it relative to the center of mass of the triangle (27)Taking the definition of the natural components (11) into account, the explicit form of reads (28)The different polar angles can be expressed as functions of s, y_{1} and y_{2}.
We adopt a flat ΛCDM cosmology with dark matter content Ω_{m0} = 0.28, dark energy content Ω_{Λ} = 0.72, Hubble parameter h_{0} = 0.7, slope of the initial power spectrum n_{s} = 0.96, and the normalization of the matter power spectrum σ_{8} = 0.8. We use the Tinker et al. (2008) formula to model the halo mass function, and the Duffy et al. (2008) fitting formula for the concentration parameter of NFW halos. To reduce the computational load even more, we assume a single plane of source galaxies at redshift z_{s} = 1 (see the discussion section for a generalization from this case).
Fig. 2 Dependence of the modeled shear 3PCFs on angular separation. The two parityinvariant components ttt (stands for ⟨γ_{t}γ_{t}γ_{t} ⟩) and × ×t (stands for ⟨γ_{×}γ_{×}γ_{t} ⟩) are shown for equilateral triangles, see Eq. (11). The × ×t signal becomes negative beyond r ≈ 2 arcmin, and its absolute value is plotted for larger angles. 

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Fig. 3 Dependence of the modeled shear 3PCFs on triangle configuration. A subset of triangle configurations with two equal sidelengths y_{1} = y_{2} = 1 arcminute is shown, and plotted are the dependencies of the shear 3PCFs on the angle between them. The left panel “parity even” shows shear 3PCFs that involve zero or two tangential shears and are thus invariant under a parity transformation, while the right panel shows shear 3PCFs that contain an odd number of tangential shears and therefore change sign under parity transformation. The notation in the labels is analogous to Fig. 2. 

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Figures 2 and 3 present the modeled shear threepoint correlation functions. Shown are the threepoint correlators of shear components that can be seen as parts of the natural components, see (11). As can be seen from Fig. 2, the pure tangential component ⟨γ_{t}γ_{t}γ_{t} ⟩ peaks at about 30 arcsec for equilateral triangle configurations. In comparison, the virial radius for a 10^{14} M_{⊙} halo at z = 0.5 corresponds to an angular size of approximately 100 arcsec. In contrast to shear 2PCFs, all shear 3PCFs vanish when r → 0. This is expected since a statistically isotropic spin2 field has vanishing skewness. For equilateral triangles, ⟨γ_{×}γ_{×}γ_{t}⟩ = ⟨γ_{×}γ_{t}γ_{×}⟩ = ⟨γ_{t}γ_{×}γ_{×} ⟩, and all the components involving 1 or 3 γ_{×} vanish, owing to the additional symmetry of three equal sidelengths. Therefore only two components are plotted in Fig. 2.
Figure 3 shows a strong dependence of shear 3PCFs on the triangle shape. After taking account of the different ordering of the three shears and the different definitions of φ, the shapes are consistent with Fig. 6 in Zaldarriaga & Scoccimarro (2003). The detailed shapes of the curves are difficult to understand in detail, but several features do appear as expected. First, for equilateral triangles (φ_{y} = π / 3) one sees the additional symmetry ⟨γ_{×}γ_{×}γ_{t}⟩ = ⟨γ_{×}γ_{t}γ_{×}⟩ = ⟨γ_{t}γ_{×}γ_{×} ⟩ and ⟨γ_{×}γ_{t}γ_{t}⟩ = ⟨γ_{t}γ_{×}γ_{t}⟩ = ⟨γ_{t}γ_{t}γ_{×} ⟩. Second, because we plotted isosceles triangles, for all opening angles ⟨γ_{×}γ_{t}γ_{×}⟩ = ⟨γ_{t}γ_{×}γ_{×} ⟩, ⟨γ_{×}γ_{t}γ_{t}⟩ = − ⟨γ_{t}γ_{×}γ_{t} ⟩, and ⟨γ_{t}γ_{t}γ_{×}⟩ = ⟨γ_{×}γ_{×}γ_{×}⟩ = 0. Third, for collapsed triangles (θ = 0), the ⟨γ_{t}γ_{t}γ_{t} ⟩ and ⟨γ_{×}γ_{×}γ_{t} ⟩ components have the same sign. This is expected since for collapsed triangles two of the three shear components have the same value and thus their product is always positive. The sign of the shear 3PCF then depends on the third shear component, which is shared by ⟨γ_{t}γ_{t}γ_{t} ⟩ and ⟨γ_{×}γ_{×}γ_{t} ⟩. Judging from this feature, some of the shear 3PCFs in Fig. 10 in Takada & Jain (2003b) have incorrect signs. Apart from that figure shows the same dependencies of the shear 3PCFs. Finally, the triangle configuration with opening angle φ_{y} = ψ and that with φ_{y} = −ψ can be considered as mirror images of each other, so are the triangle configuration with opening angle φ_{y} = π − ψ and that with φ_{y} = π + ψ. This implies that γ3PCF(y_{1},y_{2},ψ) = γ3PCF(y_{1},y_{2}, − ψ) = γ3PCF(y_{1},y_{2},2π − ψ) when γ3PCF is a parityeven shear 3PCF, and therefore the slopes of the parityeven shear 3PCFs at φ_{y} = 0 and φ_{y} = π are zero; on the other hand, γ3PCF(y_{1},y_{2},ψ) = −γ3PCF(y_{1},y_{2}, − ψ) = −γ3PCF(y_{1},y_{2},2π − ψ) when γ3PCF is a parityodd shear 3PCF, and the shear 3PCF values at φ = 0 and φ = π for parityodd shear 3PCFs are zero, as shown in the figure.
Fig. 4 Threepoint aperture mass statistic modeled with the halo model with the onehalo term. Contributions from halos below different maximum mass limits are shown. 

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Fig. 5 Same as Fig. 4, but showing contributions from halos below different redshift limits. The line with “all redshifts” contains contributions from all halos below the source redshift z_{s} = 1. 

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The modeled thirdorder aperture mass statistic under different mass and redshift cuts are presented in Figs.4 and 5. Since only the onehalo term is considered in the model, the contributions from different halos to the signal are additive. The dominant contribution to the signal on and above arcminute scales comes from z = 0.3 to z = 0.7, and from massive halos corresponding to galaxy groups and clusters. In particular, when the filter scale is larger than 80 arcsec, more than half of the signal is contributed by halos with masses larger than 10^{14} M_{⊙}. The peak of the signal is located at θ ≈ 20 arcsec. The signal at larger aperture radii is dominated by larger halos at lower redshifts. Therefore the signal shifts to lower θ as an upper mass cutoff is introduced, and to higher θ as an upper redshift cutoff is introduced.
Fig. 6 E/Bmode mixing with threepoint aperture mass statistics due to smallscale cutoff at θ_{min} = 1, 2 and 5 arcsec. In the upper panel, the black lines represent the Emode aperture mass statistic , and the magenta lines represent the paritypreserving Bmode . The error bars present the shape noise contribution to the uncertainty in at θ = 0.1,0.2,0.4,0.7, and 1 arcmin. To compute the shape noise we have adopted an ellipticity dispersion σ_{ϵ} = 0.35, a galaxy number density n = 30 arcmin^{2}, and a survey area of 15 000 deg^{2}, which are typical values for the Euclid survey. The lower panel presents the relative decrease in signal when a smallscale information loss is present. All ratios follow the same curve when plotted against θ / θ_{min}. The thin blue line presents a fitting formula of this curve r_{M} = 1 − 1 / (0.8 + 0.2x + 2.3x^{2} − 0.04x^{3}) with x = θ / θ_{min}. The parityviolating Bmode signals, which are not explicitly shown in the figure, are zero over all angular scales, both with and without cutoff. 

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3.2. Effect of smallscale cutoff
As described in Kilbinger et al. (2006), the smallscale information loss is attributed to the fact that the shapes of close galaxy pairs cannot be estimated reliably (see also the detailed discussion in Miller et al. 2013). The angular scale below which this information loss occurs depends on the true angular sizes of galaxies, on the point spread function of the observation, and the ability of the shear measurement algorithm to separate overlapping isophotes. The typical size of this cutoff used in current shear measurement methods is a few arcseconds for spacebased observations (e.g. 3 arcsec in the COSMOS analysis of Schrabback et al. 2010), and slightly larger scales for groundbased observations (e.g. 9 arcsec in CFHTLenS as discussed in Kilbinger et al. 2013).
We introduce a small angular scale cutoff in the threepoint correlation function at 1, 2, 5, and 10 arcsec, and show the E/Bmode mixing introduced into the aperture mass statistics and in Fig. 6.
As expected, smallscale information loss leads to a decrease in the Emode aperture mass signal , and introduces a spurious parityconserving Bmode signal . The parityviolating Bmode signals are consistent with zero over all angular scales, both with and without cutoff. This is expected since aperture mass statistics are computed over shear correlation functions with triangle configurations of both parities.
Somewhat surprisingly, the decrease in the Emode aperture mass signal due to a smallscale cutoff is less significant for thirdorder statistics than at second order (see Fig. 1 of Kilbinger et al. 2006). The angular scales that are significantly affected are below 10θ_{min}. This means that even for a conservative smallscale cutoff at 10 arcsec, the effect is limited to ≲arcmin scales. Since these scales are strongly influenced by complicated baryonic physics which prevents precise theoretical predictions, signals on these small scales are usually not used to infer cosmology. At secondorder, however, the decrease at 1 arcmin angular separation can be as high as 10% for θ_{min} = 5 arcsec, compared to <0.5% at the thirdorder with the same cutoff scale. This difference can be understood by comparing the relations between aperture mass statistics and shear correlations function for the second and thirdorder case. In the former case, one of the two correlation functions ξ_{+} and its corresponding weight function T_{+} both peak at zero angular separation. In contrast, none of the shear 3PCFs peak at small scales, and both weight functions T_{0} and T_{3} vanish when θ approaches 0. Accordingly, a smallscale cutoff in the threepoint shear correlation functions causes a weaker effect on the aperture mass statistics.
Leakage to Bmodes is even weaker than the decrease in Emode signal. This is also very different from the twopoint case, where the growth in the Bmode signal on small scales due to a cutoff approximately equals the decrease in Emode signal in size. This small leakage enables the parity conserving Bmode to remain a good test for systematics in thirdorder statistics even in the presence of a smallscale cutoff in the 3PCFs.
At the small angular scales where the E/Bmode mixing occurs, shot or shape noise (the noise due to randomness of the intrinsic shapes of the galaxies) is the major source of uncertainty in the measurement of the aperture mass signals. We plot in Fig. 6 the shape noise contribution (see appendix) to the uncertainty of for the Euclid survey. The decrease in the Emode aperture mass signal caused by a smallscale cutoff is subdominant to the uncertainty of measurement for θ_{min} < 5 arcsec, even for a large and relatively deep survey like Euclid. For surveys with smaller sky coverage and/or shallower depth, the shot noise contribution is expected to be even higher. Therefore, the E/Bmode mixing effect on the thirdorder statistics is generally not observable.
4. Discussion
As mentioned above, a single sourceplane at z = 1 was used for this study. This source redshift was chosen to represent deep surveys like Euclid (which has an expected median source redshift of z ≈ 0.9). The results we obtained, however, can be generalized to shallower surveys as well. According to the halo model, if the comoving distance to the source χ_{s} is changed, the shear 3PCF contributed by a certain halo at χ changes its amplitude as [f_{K}(χ_{s} − χ) / f_{K}(χ_{s})]^{3}, whereas the angular dependence is retained (25, 26). A shallower survey will give more relative weight to lowredshift halos, which will shift the signals to larger angular scales (see e.g. Fig. 5), making slightly less affected by smallscale cutoff.
We have tested EBmixing in thirdorder aperture statistics with three equalsized filters. For the more general case with three different filter sizes, that is , we argued that the degree of EBmixing is bounded from above by that for , with θ_{1} being the smallest of the three filter sizes. To see this, it is helpful to consider the aperture mass as obtained by placing apertures on the image. In this case, the estimator of involves summation over all galaxy triplets in the field, with the three galaxies in a triplet being weighted by three different filters. A smallscale cutoff will eliminate close triplets from this sum and cause the Emode signal to be underestimated. This underestimation is most severe when all three galaxies in the triplet are given high weights from their corresponding filters. Because the filter Q_{θ} is more extended for a larger filter size θ, the eliminated close triplets are on average given a lower weight relative to the retained ones when some of the filter sizes are larger in size. Therefore, if EBmixing is negligible in at θ_{1}, it is also negligible in a general aperture mass statistic with θ_{2} ≥ θ_{1} and θ_{3} ≥ θ_{1}.
The test we performed shows how much Emode signal will leak to the Bmode under a smallscale cutoff for thirdorder aperture statistics. The opposite question needs to be addressed in a quantitative way as well: if there exists a Bmode signal, for instance, from noise and bias coming from shape measurements or intrinsic alignments of galaxy shapes, how much will the Emode signal be affected? The answer to this question depends on the actual angular and shape dependencies of the particular Bmode contamination and is therefore hard to give in general. For future lensing surveys, once the amplitude of the Bmode signal and its statistical uncertainty is known, one can study the maximum bias a smallscale cutoff could contribute to the Emode signal. Ultimately, each contribution to both E and Bmode signals needs to be understood to derive precise cosmological information from cosmic shear measurements.
5. Conclusion
We tested the degree of EBmixing in thirdorder aperture statistics due to the inevitable absence of smallscale correlation measurements. Both the decrease in Emode signal and the introduction of a spurious Bmode signal were found to be smaller for the thirdorder aperture mass statistics than for the secondorder ones. Quantitatively, the change in the Emode signal is lower than 1% at an angular separation of ten times the cutoff scale and above, and therefore is negligible on angular scales of interest to ongoing and future weak lensing surveys. Some paritypreserving Bmode signal is created on small angular scales (<1 arcmin) because of the smallscale cutoff, but with an amplitude that is even smaller than the decrease in the Emode signal. The parityviolating Bmodes remain zero in the presence of a smallscale cutoff.
These findings suggest that the aperture statistics introduce only negligible E/Bmode mixing for thirdorder shear statistics on the angular scales that will be probed by ongoing and future weak lensing surveys. Therefore, thirdorder aperture statistics can safely be used as E/Bmode separating statistics to infer cosmological information from those surveys.
Acknowledgments
We thank Martin Kilbinger for helpful discussions, and Masahiro Takada for his quick email response regarding their halo model paper (Takada & Jain 2003b). This work was supported by the Deutsche Forschungsgemeinschaft under the project B5 in the TRR33 “The Dark Universe”. B.J. acknowledges support by an STFC Ernest Rutherford Fellowship, grant reference ST/J004421/1.
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Appendix A: Shape noise contribution to the uncertainty of the measurement
Here we derive a simple formula that yields an approximate estimate of , the uncertainty of measurement contributed by the shape noise.
We consider first the measurement of within a single aperture of angular radius θ. A convenient estimator is (A.1)where is the mean number density of galaxy images, and the sum runs over all triples of galaxy shapes in the aperture. This estimator resembles Eg. (5.24) of Schneider et al. (1998), but excludes the cases with two or three equal indices, for example, i = j. It is unbiased when the number of galaxies inside the aperture is very large.
The shape noise contribution to the dispersion of measurements in this aperture is, by definition, . Neglecting highorder correlations, the shape noise contribution to is
where is the average number of galaxies inside the aperture. In the second equation we substituted the ensemble average ⟨ ⟩ with an average over all possible galaxy positions (see Eq. (5.3) of Schneider et al. 1998), (A.3)With Crittenden et al. (2002) filter function Q_{θ} (13), the integral in the last line of (A.2) has a value of 1 / (πθ^{2}). This leads to (A.4)For a galaxy survey of size A, a number N_{f} of nearly independent apertures can be put on the field. This approximately reduces the uncertainty of measurement by a factor of . Using a rough estimation of N_{f} = A / (4θ^{2}), we obtain (A.5)We note again that this estimation of is aimed to be simple and approximate. Practically, is estimated not with (A.1) but is derived from the shear 3PCFs (14), or similarly, from the bispectrum of shear. This would lead to a slightly different prefactor in the expression of . For a more precise estimation of this prefactor, one can use Eq. (26) in Joachimi et al. (2009) as an estimate of the bispectrum covariance, and derive the covariance of with the help of Eq. (46) in Schneider et al. (2005).
All Figures
Fig. 1 Sketch of the halo model (1halo term) for shear threepoint functions and the notations used. In the text, the three shears in threepoint shear correlator e.g. ⟨γγγ ⟩ correspond to γ_{1}, γ_{2}, and γ_{3}, accordingly. Note that the polar angle of any vector k in Cartesian coordinates is denoted as φ_{k}, but φ_{y} is defined to be the angle between y_{1} and y_{2}. 

Open with DEXTER  
In the text 
Fig. 2 Dependence of the modeled shear 3PCFs on angular separation. The two parityinvariant components ttt (stands for ⟨γ_{t}γ_{t}γ_{t} ⟩) and × ×t (stands for ⟨γ_{×}γ_{×}γ_{t} ⟩) are shown for equilateral triangles, see Eq. (11). The × ×t signal becomes negative beyond r ≈ 2 arcmin, and its absolute value is plotted for larger angles. 

Open with DEXTER  
In the text 
Fig. 3 Dependence of the modeled shear 3PCFs on triangle configuration. A subset of triangle configurations with two equal sidelengths y_{1} = y_{2} = 1 arcminute is shown, and plotted are the dependencies of the shear 3PCFs on the angle between them. The left panel “parity even” shows shear 3PCFs that involve zero or two tangential shears and are thus invariant under a parity transformation, while the right panel shows shear 3PCFs that contain an odd number of tangential shears and therefore change sign under parity transformation. The notation in the labels is analogous to Fig. 2. 

Open with DEXTER  
In the text 
Fig. 4 Threepoint aperture mass statistic modeled with the halo model with the onehalo term. Contributions from halos below different maximum mass limits are shown. 

Open with DEXTER  
In the text 
Fig. 5 Same as Fig. 4, but showing contributions from halos below different redshift limits. The line with “all redshifts” contains contributions from all halos below the source redshift z_{s} = 1. 

Open with DEXTER  
In the text 
Fig. 6 E/Bmode mixing with threepoint aperture mass statistics due to smallscale cutoff at θ_{min} = 1, 2 and 5 arcsec. In the upper panel, the black lines represent the Emode aperture mass statistic , and the magenta lines represent the paritypreserving Bmode . The error bars present the shape noise contribution to the uncertainty in at θ = 0.1,0.2,0.4,0.7, and 1 arcmin. To compute the shape noise we have adopted an ellipticity dispersion σ_{ϵ} = 0.35, a galaxy number density n = 30 arcmin^{2}, and a survey area of 15 000 deg^{2}, which are typical values for the Euclid survey. The lower panel presents the relative decrease in signal when a smallscale information loss is present. All ratios follow the same curve when plotted against θ / θ_{min}. The thin blue line presents a fitting formula of this curve r_{M} = 1 − 1 / (0.8 + 0.2x + 2.3x^{2} − 0.04x^{3}) with x = θ / θ_{min}. The parityviolating Bmode signals, which are not explicitly shown in the figure, are zero over all angular scales, both with and without cutoff. 

Open with DEXTER  
In the text 