Issue 
A&A
Volume 624, April 2019



Article Number  A134  
Number of page(s)  33  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201834379  
Published online  25 April 2019 
Consistent cosmic shear in the face of systematics: a Bmode analysis of KiDS450, DESSV and CFHTLenS
^{1}
Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
email: ma@roe.ac.uk
^{2}
ArgelanderInstitut für Astronomie, Auf dem Hügel 71, 53121 Bonn, Germany
^{3}
Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK
^{4}
Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA
^{5}
Center for Cosmology and AstroParticle Physics, The Ohio State University, 191 West Woodruff Avenue, Columbus, OH 43210, USA
^{6}
Leiden Observatory, Leiden University, Niels Bohrweg 2, 2333 CA Leiden, The Netherlands
Received:
4
October
2018
Accepted:
26
March
2019
We analyse three public cosmic shear surveys; the KiloDegree Survey (KiDS450), the Dark Energy Survey (DESSV) and the Canada France Hawaii Telescope Lensing Survey (CFHTLenS). Adopting the “COSEBIs” statistic to cleanly and completely separate the lensing Emodes from the nonlensing Bmodes, we detect Bmodes in KiDS450 and CFHTLenS at the level of ∼2.7σ. For DESSV we detect Bmodes at the level of 2.8σ in a nontomographic analysis, increasing to a 5.5σBmode detection in a tomographic analysis. In order to understand the origin of these detected Bmodes we measure the Bmode signature of a range of different simulated systematics including PSF leakage, random but correlated PSF modelling errors, camerabased additive shear bias and photometric redshift selection bias. We show that any correlation between photometricnoise and the relative orientation of the galaxy to the pointspreadfunction leads to an ellipticity selection bias in tomographic analyses. This work therefore introduces a new systematic for future lensing surveys to consider. We find that the Bmodes in DESSV appear similar to a superposition of the Bmode signatures from all of the systematics simulated. The KiDS450 and CFHTLenS Bmode measurements show features that are consistent with a repeating additive shear bias.
Key words: gravitational lensing: weak / methods: data analysis / methods: statistical / surveys / cosmology: observations
© ESO 2019
1. Introduction
Weak gravitational lensing is recognised as a powerful probe of the largescale structure of the Universe. Its reach, however, will always be limited by the accuracy to which terrestrial and astrophysical contaminating signals can be controlled. Known sources of astrophysical systematics include the intrinsic alignment of neighbouring galaxies (see Joachimi et al. 2015, and references therein) and the impact of baryon feedback when modelling the nonlinear matter power spectrum (Semboloni et al. 2011) as well as the more subtle effect of the clustering of background “source” galaxies (Schneider et al. 2002a). Known sources of terrestrial systematics arise from residual distortions resulting from uncertainty in the pointspread function (PSF) model (Hoekstra 2004), biases in the adopted source redshift distributions (Hildebrandt et al. 2012), object selection bias (Hirata & Seljak 2003), shear calibration bias (Heymans et al. 2006) and detectorlevel effects (Massey et al. 2014; Antilogus et al. 2014). As weak lensing surveys have grown in size, the list of known sources of error has also grown, with accompanying mitigation strategies (see Mandelbaum 2018). This progress is impressive, but there will always be the possibility that hitherto unknown sources are contaminating the cosmic shear signals that we observe. In this paper we therefore explore the sensitivity of the “COSEBIs” weak lensing statistic to blindly uncover a range of different contaminating signals.
Complete Orthogonal Sets of E/BIntegrals, “COSEBIs”, were defined by Schneider et al. (2010). They provide a complete set of filter functions which cleanly separate a measured cosmic shear signal into its curlfree (Emode) and divergencefree (Bmode) distortion patterns over a finite angular range. Weak lensing can only produce Emodes^{1}, and as such any detected Bmodes in the measured cosmic shear signal will have a nonlensing origin. The most popular statistic used in current cosmic shear analyses are the shear twopoint correlation functions, ξ_{±}, (2PCFs; Jee et al. 2016; Joudaki et al. 2017a; Hildebrandt et al. 2017; Troxel et al. 2018a). As these direct measurements of the cosmic shear signal mix E and B modes, other methods are required in order to extract and identify any contaminating nonlensing signal through its Bmode distortion pattern.
A range of different statistics exist to filter E/Bmodes in 2PCFs, for example, aperture mass statistics (Schneider et al. 2002b), ξ_{E/B} (Crittenden et al. 2002) and ring statistics (Schneider & Kilbinger 2007). The aperture mass statistics and ξ_{E/B} rely on knowing the 2PCFs at either very small angular separations (where the galaxy images blend) or very large angular scales, (beyond the surveyed area). Using these statistics therefore results in biased estimates of E/Bmodes. For the aperture mass statistic, the E/Bmode leakage is ∼10% for a typical case (Kilbinger et al. 2013). Both ring and aperture mass statistics suffer from a loss of information due to their filtering method.
Alternatives to realspace estimators decompose the cosmic shear signal into its E and Bmode convergence power spectrum. Quadratic estimators can be used (Köhlinger et al. 2016), but this method is sensitive to the modelling of the noise and is also challenging to use to estimate the power at large Fourier modes due to its computational speed (Köhlinger et al. 2017). Faster methods estimate “pseudo” power spectra where in an ideal case the E/B power spectra can be easily separated. Unfortunately the presence of masks mixes Fourier modes, and hence E/Bmodes, making this method sensitive to the modelling of the mask (Asgari et al. 2018; Hikage et al. 2019). Power spectra can also be estimated from 2PCFs, if the 2PCFs are known over all scales. In practice this is not feasible, hence the integrals over 2PCFs are truncated, which can produce biases in the estimates (van Uitert et al. 2018). Alternatives to bandpower spectrum estimation from 2PCFs have also been suggested (Becker & Rozo 2016), which attempt to minimize the information leakage from the outofrange angular scales. Another approach to power spectrum inference uses hierarchical Bayesian modelling (Alsing et al. 2016, 2017). Although this method is not sensitive to masking effects, it is highly computationally expensive as it relays on estimating posterior density distributions for all combinations of power spectra, which in turn can produce inaccuracies in the analysis.
In this paper we adopt the COSEBIs statistic as it is the only method that can cleanly, without loss of information, separate E and Bmodes over a finite angular range from realistic lensing survey data. They are also efficient as a small number of COSEBIs modes (∼5 per tomographic redshift bin) can essentially capture the full cosmological information (Asgari et al. 2012). With data compression, using linear combinations of the tomographic COSEBIs modes that are most sensitive to the parameters to be estimated, the total number of data points can also be significantly decreased (Asgari & Schneider 2015). This compression then makes the method less sensitive to the accuracy to which the covariance matrix of the data can be estimated from numerical simulations.
COSEBIs have been used to analyse the CanadaFranceHawaii Telescope Lensing Survey (CFHTLenS; Kilbinger et al. 2013; Asgari et al. 2017), finding significant Bmode signals in the tomographic analysis that were not detected by a range of other systematic analyses (Heymans et al. 2012). The COSEBIs statistic is therefore more sensitive and stringent in detecting Bmode distortions. It is not immediately apparent, however, how a COSEBIs Bmode detection can be used in order to uncover the origin of the observed nonlensing distortions. In contrast, the ξ_{B} and aperture mass statistics are rather intuitive. For example. a peak in the measured Bmode at the angular scale of the CCD chip can be readily associated with an issue on the chiplevel. It is also unclear how detected COSEBIs Bmodes impact the cosmological parameters from the measured Emodes. For example, is a significant highorder COSEBIs Bmode detection an issue, when all the cosmological information is contained in the first five COSEBIs Emodes?
By using a range of different simulated systematic errors and analysing three public weak lensing surveys, this paper explores how Bmode statistics can be used to diagnose datarelated systematic errors as follows. We describe the COSEBIs, ξ_{E/B} and compressed COSEBIs (CCOSEBIs) statistics as well as their covariance matrices in Sect. 2. In Sect. 3, we introduce the three public weak lensing surveys that we analyse; the KiloDegree Survey (KiDS450, Hildebrandt et al. 2017), the science verification data from the Dark Energy Survey (DESSV, Dark Energy Survey Collaboration 2016) and CFHTLenS (Heymans et al. 2013), presenting a full Bmode analysis of these surveys in Sect. 4. We then use mock weak lensing surveys to explore how the COSEBIs and ξ_{E/B} statistics respond to a range of different observationally motivated systematics, introduced in Sect. 5, with results presented in Sect. 6. We compare the results for the mocks and real data in Sect. 7 and conclude in Sect. 8. In Appendix A we discuss the biases that exist in published 2PCF analyses that arise from the angular binning of the 2PCFs. We also show how these biases can be mitigated. Appendix B determines the σ_{8} − Ω_{m} degeneracy direction for a CCOSEBIs analysis of KiDS data. We discuss how to optimise Bmode nulltests using differing selections of the data vector in Appendix C and present supplementary material for the tomographic data analysis in Appendix D.
2. Methods
The most familiar twopoint statistics used in cosmic shear analysis are the shear twopoint correlation functions, ξ_{±}, which correlate γ_{t/×}, the tangential and cross components of shear, of two galaxies separated by an angle θ in the sky. They are defined as
In practice, galaxy ellipticities, ϵ, are measured with differing accuracies, accounted for using weights, w. In this case, an unbiased estimator for ξ_{±} is given by
where the sum goes over all galaxy pairs in an angular bin labelled as θ (see Appendix A for binning choices). w_{a} is the weight associated with the measured ellipticity at x_{a} and ϵ_{t/×} are the tangential and cross components of the measured ellipticity (Schneider et al. 2002b). Here the ellipticity is defined such that its expectation value is equal to the reduced shear, in absence of systematics (Schramm & Kayser 1995; Seitz & Schneider 1997). If the ellipticity measurements require a multiplicative correction, m (see for example Miller et al. 2013), then the correlation functions may be calibrated by dividing them with the following correction,
Theoretically the 2PCFs can be calculated through their relation to the shear power spectrum, P_{γ},
where ℓ is the Fourier conjugate of θ and J_{0} and J_{4} are the ordinary Bessel functions of zeroth and fourth order (see Kaiser 1992, for example). The shear power spectrum is in turn related to the threedimensional matter power spectrum. This relation can be simplified assuming a flatsky and Limber approximation (Kaiser 1998), although, these approximations start to fail for small ℓmodes (corresponding to large scales). Various approximations and corrections are investigated in Kilbinger et al. (2017). Their “hybrid” case which is used in most of the recent cosmic shear data analysis (see also Loverde & Afshordi 2008), can be written for redshift bins i and j as follows,
where H_{0} is the Hubble constant, Ω_{m} is the matter density parameter, c is the speed of light in vacuum, a is the scale factor normalized to one at the present, P_{δ} is the 3D matter power spectrum and χ is the comoving radial coordinate. The geometric factor for redshift bin i, g^{i}(χ), is given by
where χ_{h} is the comoving horizon scale, is the probability density of sources in comoving distance for redshift bin i and f_{K}(χ) is the comoving angular diameter distance, which is equal to χ for a Universe with flat spatial geometry.
The correlation functions calculated using Eq. (4) need to be binned in θ before they can be compared to the measurement. As we usually compress the data by binning ξ_{±} into broad θbins, we should apply the same binning to the theory, to take the functional form of the 2PCFs over the angular bin into account. Additionally, the number of galaxy pairs is roughly proportional to their angular separation. Therefore in the binned data, 2PCFs values for larger θ contribute a larger weight to the mean signal in the bin. In Appendix A we calculate the biases introduced by binning 2PCFs data, showing that using a point estimate for the expected values of ξ_{±} can produce biases of up to ±10% for the angular range and binning adopted in Hildebrandt et al. (2017) and Troxel et al. (2018b).
In practice we need to modify the relation between the 2PCFs and shear power spectrum in Eq. (4), to accommodate any Bmode power spectra that may exist in the data,
where^{2}P_{γ}(ℓ)=P_{E}(ℓ) and P_{B}(ℓ) is the Bmode power spectrum (Schneider et al. 2002a). In the following subsections we introduce three methods that separate E/Bmodes in cosmic shear data: ξ_{E/B}, COSEBIs and compressed COSEBIs. COSEBIs, being the most robust twopoint statistic method, will be used as our primary E/Bmode separation method.
2.1. E/Bmode 2PCFs
The correlation functions, ξ_{±}, can be separated into E/Bmodes following Crittenden et al. (2002) and Schneider et al. (2002a), where
with
The above definition makes ξ_{E/B} pure E/Bmodes and hence we can write
From Eqs. (7)–(10) we can immediately see that for a Bmode free case ξ_{E} = ξ_{+}(θ).
In Schneider et al. (2002a), ξ_{E/B}(θ) is denoted ξ_{E + /B+}(θ) as they also provide an alternative definition for E/B two point correlation functions, ξ_{E − /B−}(θ), in terms of integrals over ξ_{+}(ϑ). In that case the integrals are taken from ϑ = 0 up to ϑ = θ, instead. Although in both cases the integral is taken over a range of angular separations that are not observable, it is preferable to use Eq. (8) since, at least for a Bmode free case, ξ_{−}(θ)/θ is very small for large θ (ξ_{−}(θ)∝θ^{−3} at large scales). In this case we can truncate the integrals in Eq. (9) without needing to extrapolate to infinitely large ϑ. However, we may lose some Bmode information by this truncation, as there is no guarantee that the Bmode signal is negligible for large angular scales. One way to extend the integral to large angular scales that are not available in the data is to use the theoretical value of ξ_{−}(θ) for these angular ranges. In this paper we use measurements over an angular range of [0.5′,300′] and a theoretical ξ_{−}(θ) from θ = 300′ out to θ = 1000′. We find that the inclusion of the theoretical extension of the integral has less than 5% effect on the largest angular bin (used in KiDS450) centred at 50′ and drops to subpercent level for θ ≲ 20′.
2.2. COSEBIs
COSEBIs (Complete Orthogonal Sets of E/BIntegrals) modes live neither in Fourier nor real space. The filter functions for COSEBIs form sets of basis functions which transform 2PCFs and shear power spectra to the COSEBIs modes. The two sets of COSEBIs basis functions are the Lin and LogCOSEBIs filters, which are written in terms of polynomials in ϑ and ln(ϑ) in real space, respectively. In this analysis we use the LogCOSEBIs, as they require fewer modes compared to the LinCOSEBIs to capture essentially all the cosmological information (see Schneider et al. 2010 for a single redshift bin and Asgari et al. 2012 for the tomographic case).
The COSEBIs can be written in terms of the 2PCFs as
where and are the E and Bmode COSEBIs for redshift bins i and j, and n, a natural number, is the order of the COSEBIs modes. T_{±n}(ϑ) are the COSEBIs filter functions, (given in Eqs. (28)–(37) in Schneider et al. 2010). These are oscillatory functions with n + 1 roots in their range of support, as shown in Fig. 1. Therefore, the COSEBIs modes with larger n values have more oscillations in their range of support and can pick up features in the 2PCFs that appear as smallerscale variations, compared to the modes with small n and few oscillations. These small nmodes are more sensitive to the largerscale variations in the input ξ_{±} or the overall behaviour of these functions.
Fig. 1.
LogCOSEBIs filter functions, T_{±n}(θ). These filter functions convert ξ_{±} to COSEBIs E and B modes through Eqs. (11) and (12). We show four example nmodes for each filter for the angular separation range of [0.5′,100′]. By definition T_{±n}(θ) are equal to zero outside of the range of their support. 

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The E/BCOSEBIs can also be expressed as a function of the convergence power spectra,
where are the E(B)mode convergence power spectra and the W_{n}(ℓ) are the Hankel transforms of T_{±n}(ϑ),
Figure 2 shows the W_{n}(ℓ) functions corresponding to the T_{±n}(θ) filters shown in Fig. 1. The first peak in W_{n}(ℓ) is set by the value of ϑ_{max} and n. As can be seen, the higher order W_{n} pick up more power from larger ℓ. We use Eq. (13) to calculate the theoretical value of the Emode COSEBIs as theories, in general, give their predictions in terms of the power spectrum. However, in practice the shear 2PCFs are more straightforward to measure from data, hence Eqs. (11) and (12) are used to calculate the E/Bmode COSEBIs from data and simulations. To evaluate these integrals in the angular range of [0.5′,100′] we use 4 × 10^{5} linear angular bins (see Asgari et al. 2017, for a discussion on optimising the number of bins for this type of analysis).
Fig. 2.
LogCOSEBIs weight functions, W_{n}(ℓ), normalized to their maximum value. These weight functions convert E and B shear power spectra to COSEBIs modes through Eq. (13). Four example nmodes are shown for the angular range of [0.5′,100′]. 

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2.3. CCOSEBIs
We use the data compression method of Asgari & Schneider (2015) to explore the effect of systematics on cosmological parameter estimation, as this method is informed by the sensitivity of the data to the parameters. This method, which can be applied to any statistic, reduces the number of data points, which is important to minimise errors when estimating covariance matrices from simulations (see Hartlap et al. 2007, for example).
To compress COSEBIs we need to have an estimate for their inverse covariance matrix (see Sect. 2.4), as well as their first and secondorder derivatives with respect to the parameters to be measured. We then linearly combine the COSEBIs modes using the sensitivity of each mode to the given parameter(s) as their coefficient. For the firstorder compressed ECOSEBIs we have,
where μ is a cosmological parameter, ℂ^{−1} is the inverse covariance matrix of E_{n} and n_{max} is the number of COSEBIs modes considered in the compression. This first order compression is equivalent to a Karhunen–Loeve compression where the covariance matrix is known (see Tegmark et al. 1997, for example), but using the first order compression alone can result in a loss of information when the covariance matrix estimate is inaccurate. We therefore follow Asgari & Schneider (2015) by adding the following secondorder compressed quantities to the data,
where ν is a second cosmological parameter and second order derivatives of E_{n} are taken. In short, we can write both first and secondorder CCOSEBIs as the following matrix equation,
where the elements of the compression matrix, Γ, are formed from combinations of the derivatives of E_{n} with respect to the parameters and their inverse covariance matrix.
2.4. Covariance matrix
To quantify the significance of the Bmodes measured from the data or in the simulations, we need to know the covariance matrix of the data vector. Aside from currently negligible physical effects that can produce Bmodes (discussed in Sect. 1) and in the absence of systematics, we expect any observed Bmodes to be consistent with random noise arising from galaxy shapenoise. Therefore, to calculate the Bmode covariance, we assume that they are only due to noise and find the covariance matrix for each of the above statistics.
Assuming a field of galaxies with ellipticities randomly picked from a Gaussian with zero mean and variance, we can write the covariance of the 2PCF as
where is the number of galaxy pairs, in redshift bin pair ij, within an angular separation bin with the label θ. The Kronecker symbols, δ_{ij} and δ_{θϑ}, are equal to unity if their arguments are equal and are otherwise zero (for example see Eq. (34) in Joachimi et al. 2008).
An approximation for N_{pair}(θ) can be determined by calculating the number of pairs in an infinite field, scaled by the true finite field area, A, where
and is the mean number density of galaxies in redshift bin i. This approximation fails, however, as it does not account for intricate smallscale survey geometry, source clustering or any variable depth effects. Furthermore, as we get closer to the field size, it does not account for the pairs of galaxies which are lost due to the discontinuities in the observed field (see for example Joachimi et al. 2008). As the significance of any measured Bmodes is determined entirely by the shotnoise, we therefore choose to use a direct measurement of N_{pair}(θ) from the data. We follow the method of Schneider et al. (2002b), who determine the full covariance matrix for 2PCFs for a weighted ellipticity field, to find the shapenoiseonly term of the covariance matrix, with the number of galaxy pairs given as
Here the sums are over galaxies in the given angular separation bin. Determining N_{pair} from Eq. (20) instead of the approximation in Eq. (19), enlarges the covariance at large scales where there are fewer pairs of galaxies due to geometry effects. On small scales where variable depth and source clustering become important, the covariance is decreased.
Inserting Eq. (18) into the following expression for the COSEBIs covariance (Schneider et al. 2010)
where C_{±±}(θ, θ′) is the covariance of ξ_{±}, we find the Bmode covariance for COSEBIs,
where , δ_{D} is the Dirac delta function and we have used δ_{θθ′} = δ_{D}(θ − θ′) Δθ to remove the Kronecker symbol. Taking the inner integral in Eq. (22) results in,
We calculate the COSEBIs Bmode covariance using trapezoidal integration with fine θbins and verified that these equations accurately predict the noiseonly covariance, by analysing 1000 shapenoiseonly mock simulations. We find that for a 100 deg^{2} field the noise term for the COSEBIs covariance is underestimated by 30% if we use from Eq. (19), while using Eq. (20) recovers the measured covariance from the simulation.
The corresponding covariance for CCOSEBIs is simply equal to the COSEBIs covariance sandwiched between two compression matrices,
where ^{t} denotes a transposed matrix^{3}.
The covariance matrix of ξ_{B} can also be calculated from Eq. (18),
Note that the only difference between Eqs. (25) and (18) is a factor of 2, which arises from the fact that ξ_{±} depends on both E/Bmodes and their associated noise, while ξ_{E/B} only depends on a single component, as can be seen in Eqs. (10) and (7). As a result, ξ_{E/B} is only sensitive to the noise components that resemble E/Bmodes. The power spectrum of the noise can be equally divided into an Emode and a Bmode component, and as such the noise covariance for ξ_{E/B} is half the amplitude of the corresponding covariance for 2PCFs.
In addition to Bmodes, we show Emode measurements for the data with error bars calculated assuming Gaussian covariances. We choose not to include the nonGaussian and super sample terms in the error calculation which primarily affect the offdiagonal terms of the covariance matrix^{4}. As we do not analyse the Emodes in a quantitative way in this study, and use the Emode covariances solely for plotting purposes, our chosen Gaussian treatment of the covariance is sufficient. We can write the Gaussian covariance for the Emodes in terms of three contributors,
where the Mixed term depends on both cosmology and noise. The Noise term here is estimated in the same manner as the Bmodes covariance, (Eqs. (23)–(25)), taking all the survey effects into account. For the other two contributions, however, we assume a simple survey geometry and follow Eqs. (53) and (54) in Joachimi et al. (2008) for the covariance of power spectra and correlation functions^{5}, respectively. The Gaussian mixed and cosmic variance terms for COSEBIs covariance are given in Eq. (11) in Asgari et al. (2012) for the tomographic case.
3. Data
We use three sets of cosmic shear catalogues that are in the public domain, KiDS450, DESSV and CFHTLenS. Our focus in this paper is the analysis of their Bmode signal, but we also compare the corresponding measured Emode signals to theoretical predictions, based on the published best fitting cosmological parameters from each survey, as given in Table 1. This allows for the level of Bmodes to be assessed, relative to the Emodes, but we leave a full Emode cosmological parameter analysis to a future paper.
Published bestfitting cosmological parameters for the surveys (KiDS450, CFHTLenS, and DESSV: Hildebrandt et al. 2017; Heymans et al. 2013; Abbott et al. 2016), and the simulation (SLICS, HarnoisDéraps et al. 2018), that we use in this paper.
The theoretical predictions are calculated using COSMOSIS (Zuntz et al. 2015)^{6} with linear matter power spectra calculated with CAMB (Lewis et al. 2000; Howlett et al. 2012)^{7}. Takahashi et al. (2012) is used to model the nonlinear evolution of the matter power spectrum. A Limber approximation is employed to estimate the lensing power spectrum as described in Sect. 2. For the intrinsic alignment of galaxies we adopt the nonlinear model from Bridle & King (2007)^{8}, which is equivalent to the models used in the analysis of all three surveys. The 2PCFs are measured from the data and the simulations using ATHENA^{9} (Kilbinger et al. 2014).
3.1. CFHTLenS
Heymans et al. (2012) present the CFHTLenS, a completed survey with 154 square degrees of observed data in 5 photometric bands. The public data products that we analyse here are processed by THELI (Erben et al. 2013), with galaxy ellipticities measured using lensfit (Miller et al. 2013) and photometric redshifts determined using the Bayesian photometric redshift code BPZ (Benítez 2000; Hildebrandt et al. 2012).
The 2PCFs cosmic shear analysis for CFHTLenS is presented in Kilbinger et al. (2013) and Heymans et al. (2013). As summarised in Kilbinger et al. (2017), however, several improvements have been recognised since these publications, in particular with respect to the calibration of the photometric redshifts (see for example Choi et al. 2016; Joudaki et al. 2017b) and the shear measurements (see Kuijken et al. 2015; Fenech Conti et al. 2017). The resulting uncertainty in these calibrations will impact the Emode cosmological parameter constraints from this survey. As our focus is on a Bmode analysis however, which is independent of these calibration corrections, we choose to use the redshift distributions and calibration corrections adopted by Heymans et al. (2013) for this study.
We follow Heymans et al. (2013) by dividing the data into six photometric redshift bins: z_{1} ∈ (0.2, 0.39], z_{2} ∈ (0.39, 0.58], z_{3} ∈ (0.58, 0.72], z_{4} ∈ (0.72, 0.86], z_{5} ∈ (0.86, 1.02] and z_{6} ∈ (1.02, 1.3], also including a single bin case that uses the full range of z ∈ (0.2, 1.3]. In Asgari et al. (2017), we analysed CFHTLenS using COSEBIs to find a significant level of Bmodes. We extend this analysis to explore higher modes in COSEBIs, in addition to ξ_{E/B}, and we use an exact noise covariance (Eq. (20)) in contrast to our earlier work which used Eq. (19).
3.2. KiDS450
The KiloDegree Survey (KiDS) will collect 1350 square degrees and in combination with VIKING (VISTA Kilodegree Infrared Galaxy survey) will present data in nine photometric bands (see Kuijken et al. 2015 and de Jong et al. 2017). We analyse the data products released for the first 450 square degrees (KiDS450), that has been processed by THELI (Erben et al. 2013) and AstroWISE (Begeman et al. 2013). Galaxy ellipticities are measured with lensfit (Miller et al. 2013) and calibrated using the image simulations described in Fenech Conti et al. (2017). The 4band photometric redshifts are calibrated using external overlapping spectroscopic surveys (Hildebrandt et al. 2017) and galaxies are binned into tomographic bins using BPZ.
The KiDS450 2PCFs cosmic shear analysis is shown in Hildebrandt et al. (2017) and Joudaki et al. (2017a); Joudaki et al. (2018), with complementary cosmic shear power spectrum analyses calculated using quadratic estimators in Köhlinger et al. (2017), and integrals over 2PCFs in van Uitert et al. (2018). All these analyses reported significant but lowlevel traces of Bmodes in the data.
As in the KiDS450 cosmic shear analyses we divide the data into four photometric redshift bins: z_{1} ∈ (0.1, 0.3], z_{2} ∈ (0.3, 0.5], z_{3} ∈ (0.5, 0.7] and z_{4} ∈ (0.7, 0.9], including a single bin case that uses the full range of z ∈ (0.1, 0.9].
3.3. DESSV
The Dark Energy Survey Collaboration (2005) introduce the Dark Energy Survey (DES) project which will produce 5000 square degrees of gravitational lensing data in five bands. The science verification data also known as DESSV^{10} is the public dataset that we analyse here. The galaxy ellipticities in DESSV are measured using NGMIX (Jarvis et al. 2016) and photometric redshifts are determined using a machine learningbased pipeline, SKYNET (Bonnett et al. 2016).
Becker et al. (2016) present the primary cosmic shear analysis of the DESSV data using 2PCFs along with cosmic shear power spectrum measurements (also see Troxel et al. 2018a, for the analysis of the first 1300 square degrees of DES data). Fourier space Bmode measurements detected no significant Bmodes on scales ℓ < 2500.
We divide the data into three photometric redshift bins following Becker et al. (2016): z_{1} ∈ (0.3, 0.55), z_{2} ∈ (0.55, 0.83) and z_{3} ∈ (0.83, 1.3) and also consider a single bin case that uses the full range of z ∈ (0.3, 1.3). In order to compare our measured Emode signal to the published bestfitting cosmological parameters, listed in Table 1, we also take into consideration the bestfitting DESSV shear calibration and photometric redshift biases in our predictions, which Abbott et al. (2016) include as nuisance parameters in their fit. For our single bin analysis of DESSV data we adopt zero bias for the photometric redshift and the same value as the first tomographic bin for the shear calibration bias, which is similar to the average of the biases measured for the three bins (see Table D.1).
4. Results: survey E/Bmodes
In this section we present the measured COSEBIs (Eq. (11) and Eq. (12)), CCOSEBIs (Eq. (17)) and ξ_{E/B} (Eq. (8)) for KiDS450, DESSV and CFHTLenS. In Fig. 3 we show the COSEBIs measurement for a single redshift bin encompassing the full range of redshifts adopted by each survey. For the COSEBIs statistics we need to choose an angular range and throughout this paper we show results for three sets of angular ranges: the full angular range: [0.5′,100′], large scales: [40′,100′] and small scales: [0.5′,40′]. These were chosen to span both the surveyadopted ξ_{+}(θ) angular ranges: KiDS450 (0.5′< θ_{+} < 72′), DESSV (2′≲θ_{+} ≲ 60′), CFHTLenS (1.5′< θ_{+} < 35′), whilst also probing some of the larger angular scales used in the corresponding ξ_{−}(θ) analysis: KiDS450 (8.6′< θ_{−} < 300′), DESSV (24.5′≲θ_{−} ≲ 245.5′) and CFHTLenS (1.5′< θ_{−} < 35′). The large scale cut for ξ_{+}(θ) is generally employed to avoid biasing the results, when a constant additive bias term (cterm) is present in the shear catalogues. The same large scale angularcut is not applied to ξ_{−}(θ), since this statistic is not sensitive to a constant cterm. COSEBIs share this insensitivity with ξ_{−}(θ) and hence any measured COSEBIs Bmodes that use scales beyond the maximum θ_{+} range are not a result of a constant cterm.
Fig. 3.
COSEBIs Emodes (left) and Bmodes (right) for a single broad redshift bin. Results for DESSV are shown with blue squares, KiDS450 with black stars and CFHTLenS with magenta triangles. The angular ranges are shown for each row in the upper right corner. In addition, the significance of the Bmodes is shown as pvalues for each survey and angular range. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). Note that COSEBIs modes are discrete and the theory values are connected to each other only as a visual aid. A zeroline is also shown for reference. 

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Each row in Fig. 3 corresponds to one angular range, as denoted in the right panels, with Emodes on the left and Bmodes on the right. The different symbols show the results for DESSV (squares), KiDS450 (stars) and CFHTLenS (triangles). Overlaid are the theoretical predictions, given the published bestfitting survey cosmological parameters from Table 1. We show these Emode predictions as curves for ease of comparison, even though the COSEBIs modes are discrete. As the COSEBIs modes are correlated to their neighbouring modes (see Asgari et al. 2012, 2017, for plots of the covariance matrices), we caution that the goodnessoffit to the model should not be deduced by simply looking at the graphs, a practice commonly known as “χbyeye”. Any goodnessoffit exercise must take into account the significant correlations between the points.
Throughout this paper we truncate the COSEBIs measurements at n = 20. In principle the COSEBIs can be estimated for an infinite number of modes. However as can be seen in the left hand panels of Fig. 3 the Emode predictions are equivalent to zero for n ≳ 7. Therefore, we do not expect to gain any cosmological information from these modes. On the other hand the signal from systematic effects does not necessarily follow the same behaviour. We expect a significant signal at larger nmodes for certain systematics (see Sect. 6). As a result we choose n = 20 as our maximum nmode, which encompasses modes that are important for both cosmological and systematic analyses. Future analyses may however want to extend their diagnostic Bmode analysis to even higher nmodes, depending on the signature of the systematic that they are searching for.
Focusing first on the Emode measurements (left panels of Fig. 3) we expect to measure signal in the lower nmodes and none for the modes n ≳ 8, as seen in the theoretical predictions. This arises from the fact that both the 2PCFs and shear power spectrum are relatively smooth functions with a few features that are captured, almost entirely, by the first few COSEBIs modes. Any significant detection of highorder COSEBIs modes indicates highfrequency variations in the 2PCFs, which are unexpected in a ΛCDM cosmology and therefore indicative of systematics. We remind the reader that our Emode errors, which include both sampling variance assuming a Gaussian shear field and shot noise, will be slightly underestimated as we have not included the subdominant super sample and nonGaussian contributions to the sampling variance terms (see Sect. 2.4).
Turning to the Bmode measurements (right panels of Fig. 3), we determine the significance of the measured Bmodes using “pvalues”, for each dataset and angular range, listed in Table 2. The pvalue is equal to the probability of randomly producing a Bmode that is equally or more significant than the measured Bmode signal, given the model that Bmodes are equal to zero and their distribution is Gaussian (see Appendix C for the mathematical definition of pvalue). This model is appropriate for Bmodes generated from random noise. The degreesoffreedom here is equal to the number of COSEBIs modes (20 modes in the single redshift bin case), as the model has no free parameters to be fitted. The pvalues take into account the correlations between the COSEBIs modes. Our error analysis for the Bmodes is accurate, taking into account the weighted number of galaxy pairs in each dataset. We consider the Bmodes to be significant when the measured pvalues are p < 0.01 (highlighted in bold), corresponding to greater than 2.3σ detection of Bmodes. We find that the Bmodes of KiDS450 and CFHTLenS are consistent with zero, finding p > 0.1 in all cases. DESSV, however, shows significant 2.8σBmodes with p = 0.0026, when the full angular range is considered. We discuss the complexity of linking Bmode features with Emode features in Sect. 6.
Probability of zero Bmode contamination for each survey, given the measured COSEBIs Bmodes.
In Table 2 we also list the significance of the measured COSEBIs Bmodes for a tomographic analysis of the three angular ranges, using the surveydefined photometric redshift bins (see Sect. 3). The COSEBIs tomographic measurements for each survey, adopting the full angular range, are shown in Appendix D. For all angular ranges, we find no significant COSEBIs Bmodes for KiDS450. In contrast, for DESSV data we find a 4.0σ detection of Bmodes for the largescale angular range that includes angular scales used in the DESSV cosmic shear analysis. For the full angular range, including smallscale information that was excluded from the DESSV cosmic shear analysis, the significance of the detection increases to 5.5σ. For CFHTLenS we find a significant Bmode detection for small scales, but not at large scales. This result is in contrast to Asgari et al. (2017) who found significant CFHTLenS Bmodes for large, but not small scales. We do however recover this result if we limit our pvalue analysis to the first 7 COSEBIs modes adopted by Asgari et al. (2017). This demonstrates that the pvalues are sensitive to the choice of modes considered in the analysis, motivating the study of how different systematics impact different COSEBIs E/Bmodes in Sect. 6.
In Fig. 4 we show the measured compressed COSEBIs, where the COSEBIs modes are combined to produce a set of Emode CCOSEBIs that, in a systematicfree dataset, are only sensitive to cosmological parameters (Eq. (17)). We compress the Bmode COSEBIs using the same compression matrix. Cosmic shear is mainly sensitive to a combination of σ_{8} and Ω_{m} (see for example Jain & Seljak 1997), hence we choose only these two parameters to form the CCOSEBIs modes^{11}. The CCOSEBIs modes are highly correlated as σ_{8} and Ω_{m} are degenerate in cosmic shear data, and we hence caution the reader, again, against a “χbyeye” analysis.
Fig. 4.
CCOSEBIs E and Bmodes for nontomographic (left) and tomographic (right) analyses. The Emodes are shown as empty symbols, with the Bmodes shown as filled symbols, for DESSV (blue squares), KiDS450 (black stars) and CFHTLenS (magenta triangles). The analysis is conducted over three different angular ranges, denoted in the upper right corner of each panel. The CCOSEBIs mode is indicated on the horizontalaxis. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). A zeroline is also shown for reference. 

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Figure 4 shows the results for a singlebin analysis (left) and a tomographic analysis (right) for the three sets of angular ranges indicated in the right panels. The Emodes are shown as open symbols and the Bmodes as filled symbols for DESSV, KiDS450 and CFHTLenS. Overlaid is the theoretical expectation for the Emode signal, shown as curves for visual aid even though the CCOSEBIs modes are discrete. The horizontal axis shows which parameter (for the firstorder modes) or two parameters (for the secondorder modes) the CCOSEBIs mode is sensitive to. We highlight that CCOSEBIs represent a significant data compression, particularly in the tomographic case where, for example, we compress the 3bin 120 datapoint DESSV analysis, and the 6bin 420 data point CFHTLenS analysis, down to the same 5 CCOSEBIs modes.
Comparing the measured Emodes with the level of Bmodes in Fig. 4 we find that, aside from the largest angular range that also has the lowest signaltonoise ratio, the Emodes are about an order of magnitude larger than the Bmodes. In all panels we see that the KiDS450 Emode signal is lower than DESSV and CFHTLenS, resulting from a smaller upper photometric redshift cut of z_{phot} < 0.9 in this dataset.
Table 3 shows the pvalues for CCOSEBIs Bmodes. We do not show pvalues for the Emodes since for this analysis we have not included the super sample covariance term and our Emode errors are therefore underestimated. Readers concerned by the apparent offset between the highly correlated [40′,100′] Emode measurements and expectation values, should note that the similarly highly correlated Bmodes, for the nontomographic case, are all consistent with zero, even for the cases where they look inconsistent. The significance of the Bmodes is different from the values shown in Table 2, where we have used the first 20 COSEBIs modes to measure the pvalues. This apparent inconsistency is not unexpected, as the bulk of the CCOSEBIs signal comes from the first few COSEBIs modes, which contain the cosmological signal and different levels of systematics in comparison to the full set of 20 COSEBIs modes. A good example of this difference comes in the tomographic analysis of DESSV where we find a significant ∼5.5σ nonzero Bmode signal for COSEBIs, but the CCOSEBI Bmode is not significant at 2.2σ. This shows that the first few COSEBIs modes have a smaller contribution to the total DESSV Bmode signal compared to the higher order modes, which can also be seen in Fig. D.1. KiDS450, however, shows an insignificant Bmode signal when we consider both high and low COSEBIs modes, in contrast to a 2.7σBmode detection when only the low nmodes are used to construct the CCOSEBIs. As can be seen in Table 3, and the upper right panel of Fig. 3, the lownBmodes for KiDS450 data only become significant when the small angular scales are included.
If the origin of the Bmodes detected in the COSEBIs analysis was known to impact the E and B modes equally, then the CCOSEBIs result would be the most relevant for cosmic shear studies. If the systematics impact the E and B modes differently, however, then the compressed CCOSEBIs result, focused on only lown modes, could lead to a false nulltest for the survey. It is therefore important to study how different systematics impact the full range of E and B COSEBIs, which we carry out in Sect. 6, and discuss this matter further in Sect. 7. In Appendix C we also discuss how analysis choices, for example in this case tomographic or nontomographic, COSEBIs or CCOSEBIs, can optimise or dilute the power of a Bmode nulltest.
Finally we turn to Fig. 5 which shows the measured ξ_{E/B} statistic across the full redshift range for each survey. Overlaid are the best fitting theory curves for each dataset derived from the published cosmological parameters in Table 1. The pvalues corresponding to the zero Bmode model are low in all cases, as given in the legend of the figure, with all surveys showing a tendency for increasing Bmode power and decreasing Emode power at large scales, which, we discuss further in Sect. 7. Bmodes are detected at greater than ∼2.6σ for all surveys. For DESSV the significance of the Bmodes is particularly high at ∼9σ, but this reduces to 2.3σ, or p = 0.012, when we select the angular scales [4.2′,72′] which roughly correspond to the angular cuts applied to ξ_{+} in the DESSV cosmic shear analysis^{12}. In Appendix D we present the ξ_{E/B} tomographic analysis for each survey where we find a significant Bmode detection for DESSV (p ∼ 4 × 10^{−19}), but no detection of Bmodes for KiDS or CFHTLenS (p ∼ 0.7).
Fig. 5.
ξ_{E} and ξ_{B}E/Bmodes for a single broad redshift bin. The Emodes are shown as empty symbols, with the Bmodes shown as filled symbols, for DESSV (blue squares), KiDS450 (black stars) and CFHTLenS (magenta triangles). The DESSV and CFHTLenS results are horizontally offset relative to KiDS450 to aid visualisation. Emode predictions for ξ_{E} are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). A zeroline is also shown for reference. We detect significant Bmodes in all cases as shown by the pvalues, in the legend, which determine the probability of the data Bmodes given a null Bmode model. 

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Given the required extrapolation of the data in order to calculate the ξ_{E/B} statistic (see Eq. (9)) we emphasize that these results are, by nature, a biased measurement of ξ_{E/B}, which may not represent the data accurately. For this statistic, the errors on ξ_{B} are uncorrelated (see Eq. (25)) but also biased as the integral truncation when estimating ξ_{E/B} also affects its noise properties, which we have not taken into account. We therefore do not place too strong an emphasis on the high significance of the measured Bmodes, or the lack of Emode power on largescales for all surveys, particularly as these are the scales that are most impacted by the choices made when extrapolating the data. That said, if surveys continue to use 2PCFs as a standard cosmic shear statistic, then it is still relevant to measure ξ_{E/B} as it is the Bmode measurement that is most closely related to the 2PCFs.
5. Modelling systematics
In Sect. 4 we detected significant Bmodes in the DESSV data as well as in certain tomographic combinations of CFHTLenS and KiDS450 data. In this section we introduce models for a range of datarelated systematic effects that are appropriate for the datasets described in Sect. 3. We consider three models of systematics that affect the shear measurement of all galaxies independently of their redshift. In addition we model one photometric redshiftdependent systematic, demonstrating how catastrophic errors in photometric redshifts can lead to shape selection bias. We add these systematic models to mock data to explore their effect on the 2 point statistics introduced in Sect. 2. We are particularly interested in measuring the Bmodes associated with each systematic (see Sect. 6). By comparing simulated results with and without these systematic effects, the Bmode signatures can then be used as a tool for diagnosing the source of the Bmodes in the surveys analysed in Sect. 4.
In this analysis, we do not model masking effects, since all of the methods we use rely on measuring 2PCFs, which are insensitive to masking, provided the mask is uncorrelated with the shear field. If such correlations exist, all statistics will be affected by them. This is in contrast to methods that rely on Fourier transforms of the shear field, where masks can cause significant systematic effects (see for example Asgari et al. 2018).
5.1. Shear measurement errors
For the case of weak shear with γ≪1, we can model the observed ellipticity as
where ϵ^{int} is the intrinsic galaxy ellipticity, γ is the shear, η is random noise on the ellipticity measurement, ϵ^{*} is the PSF model ellipticity, is the residual ellipticity between the model and true PSF, and c is an additive shear that is uncorrelated with the PSF. For all these quantities we use complex notation where, for example, γ = γ_{1} + i γ_{2}. For the two PSFdependent terms, αϵ^{*} quantifies the fraction of the model PSF ellipticity that leaks into the shape measurement, and β δϵ^{*} quantifies the fraction of the residual PSF arising from PSF modelling errors, that leaks into the shape measurement. The term m is a multiplicative shear bias that is traditionally calibrated using image simulations.
We simulate each of the systematic terms in Eq. (27) in isolation, in order to characterise their Bmode signature. One exception is the shear calibration correction, m, which we set to zero, as an isotropic shear bias cannot introduce a Bmode signal, only scale an Emode signal.
5.1.1. Point spread function (PSF) leakage: αϵ^{*}
In order to mimic the effect of the PSF leakage on cosmic shear measurements we use PSF models from KiDS to make a realistic spatially varying PSF model spanning 100 square degrees. We construct this largescale PSF pattern on a 1 arcmin^{2} resolution grid, mapping the KiDS PSF measurements onto a 10 ° ×10° field by stitching together two 5 ° ×10° sections from the G12 and G15 regions in KiDS450 data (see Hildebrandt et al. 2017, for details). This provides us with a model for , where is shown in the left panel of Fig. 6. In KiDS the PSF is modelled with polynomials of third order within each pointing, where the lowest order is allowed to vary between CCDs to allow for discontinuities between CCD chips (see Kuijken et al. 2015, for more details). Similar modelling approaches are taken by CFHTLenS and DESSV. The mean of the PSF ellipticity and its one sigma deviation is and its full range is covered by for both components. Figure 6 shows how the PSF pattern changes within and across each ∼square degree pointing. In areas where the KiDS data are masked and the PSF model unconstrained, we linearly interpolate the value of the PSF ellipticity to accommodate all galaxy positions in our unmasked mock data analysis.
Fig. 6.
First ellipticity component of the spatially varying systematic effects, simulated over a 10 ° ×10° field. Here the effects are normalized to their maximum value for a better visual comparison. From the left, the first panel shows the point spread function pattern used to model PSF leakage (). The second panel shows a regular pattern using the detector chip bias model from OmegaCam multiplied by a factor of 5 (0.001 < c_{1} < 0.025). The third panel shows the random correlated noise PSF residual model with a smoothing length similar to the chip size (), while the last panel shows the same systematic for a roughly pointing size smoothing length (). 

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We choose to apply a 10% PSF leakage by setting α = 0.1. This level of leakage corresponds to the α measured in the poorerseeing KiDS iband data (see Amon et al. 2018). For the highquality KiDS rband data that are used for the main cosmic shear analysis, α was found to be consistent with zero (Hildebrandt et al. 2017).
5.1.2. Regular repeating additive pattern: c(x,y)
In the absence of PSFrelated errors, the amplitude of any remaining additive bias that is uncorrelated with the PSF, c, can be estimated directly from the data. Since we expect ⟨ϵ^{int} + γ_{i} + η⟩=0 over a large area, when α = β = 0. Surveys typically correct for any significant measurement of c before any analysis, but this empirical correction usually takes an average over all galaxies and is therefore insensitive to small scale spatial variations c(x, y) (van Uitert & Schneider 2016). Systematic effects that are stable and associated with the camera or telescope would result in a repeating pattern in the survey which is built from a series of different pointings. To determine the impact of such a systematic, we model a spatially varying, but repeating additive term, which remains constant between pointings.
We use the data from Hoekstra et al. (in prep.), who present a detailed analysis of imaging from the KiDS OmegaCam camera. Hoekstra et al. (in prep.) report lowlevel but significant detector and electronic defects that introduce an additive shear contribution per CCD that is uncorrelated with the PSF and spans the range and , shown in the second panel in Fig. 6. The white “chip 15” of OmegaCam shows the largest bias at the level of ϵ_{1} = 0.005. For the purposes of this analysis we multiply the Hoekstra et al. (in prep.) detectorbias model by a factor of five to amplify its effect, as we find that the original level of this effect is too small to show any significant E/Bmodes for the current datasets.
5.1.3. Random but correlated noise: β δϵ^{*}
Errors in PSF modelling, δϵ^{*}, can be systematic if the stars used in the modelling are unrepresentative of the PSF experienced by the galaxies (Antilogus et al. 2014; Guyonnet et al. 2015; Gruen et al. 2015). In this case the resulting systematic behaves similarly to the PSF leakage model outlined in Sect. 5.1.1, and we therefore do not consider this type of PSF modelling error.
Instead we consider the random errors in the PSF modelling that arise from noise in the PSF measurement. The impact of measurement noise on the PSF model increases as the number of stars available to characterise the model at each position in the field decreases. The PSF modelling strategy (see Sect. 5.1.1) means that any local random errors from the sparse PSF measurement will propagate as random but correlated errors across the PSF model for the full field of view.
We mimic the impact of random but correlated PSF residual errors by assigning a randomly generated number from a Gaussian distribution with zero mean and unit dispersion to each 5 × 5 arcsec pixel within a 10 ° ×10° field. We first verify that the uncorrelated version of this systematic does not produce any coherent signal, as expected from a random error. We then correlate the random PSF measurement noise over each pointing using a Gaussian filter convolution defined within the boundaries of the pointing. These convolved fields are then renormalized to produce an overall dispersion equal to 10% of the shear dispersion in the mock data, σ_{RCN} = 0.1σ_{γ}. We investigate two kernel sizes with a correlation length of roughly the CCD chip level (∼1.6 arcmin) and the pointing scale level (∼43 arcmin). The resulting systematic patterns are shown in the two right panels of Fig. 6, where the systematic ranges are (chiplevel correlation) and (pointinglevel correlation). For this systematic we chose both components of the contaminating ellipticity to be equal.
5.2. Photometric redshift selection bias
Cosmic shear surveys rely on photometric measurements to estimate the redshifts of galaxies. The photometric redshift (photoz) of a galaxy can be estimated by comparing the magnitude of the galaxy in several colourbands to template catalogues of galaxy spectral energy distributions (SEDs) or to spectroscopic training samples (see Salvato et al. 2019, and references therein). The most probable value for the redshift of each galaxy, given the measured photometric colours, z_{phot}, is then used to divide the galaxy sample into tomographic redshift bins. The true redshifts of these galaxies may not all lie within the boundaries of their appointed photometric redshift bins but provided the true underlying redshift distribution of the galaxies is known, this can be accounted for in the theoretical predictions of the cosmic shear signal (Eq. (6)). The dispersion in true redshift within these tomographic bins will however depend on the precision of each galaxy’s photoz estimate, which in turn depends on the error on the measured flux of the galaxy in each photometric band. As a galaxy imaged with different noise realisations can therefore appear in different photometric redshift bins, in cases where the flux error is correlated with the shape or orientation of the galaxy, selecting a galaxy sample based on photometric redshifts could therefore lead to an ellipticityredshift selection bias and hence a systematic error in a cosmic shear analysis. In this section we explore and introduce this new concept of photometric redshift selection bias as a systematic for lensing surveys.
Consider two identical elliptical galaxies of fixed size and flux. The first galaxy is convolved with an elliptical PSF aligned with its major axis. The second is convolved with an elliptical PSF aligned perpendicular to its major axis. In the resulting image our second galaxy will appear to cover a larger area than our first galaxy and with a lower surface brightness and lower significance. It will therefore have larger photometric errors compared to the first galaxy. Kaiser (2000) recognised that this effect implied that any cuts on observed significance would introduce a PSFdependent selection bias in the ellipticity of the galaxies (see also Bernstein & Jarvis 2002). The introduction of tomographic photoz selection in a cosmic shear analysis, which implicitly depends on the significance of each detection, can therefore also lead to an ellipticitydependent selection bias.
In addition to this core effect, flux errors that are correlated with the relative orientation of the galaxy and PSF can also arise simply from the methodology used to measure the photometry in each band. DESSV use SEXTRACTOR automated aperture magnitudes where the aperture is fixed by the galaxy shape in the detection image (Bonnett et al. 2016; Rykoff et al. 2016). Whilst this method ensures that the physical apertures are matched between the bands, it does not take into account the differing PSFs. Hildebrandt et al. (2012) show that this approach leads to an overall degradation in the photometric redshifts. For example, if the PSF in the iband is perpendicular to the PSF in the detection rband, the resulting iband flux, assuming a fixeddetection aperture, will be underestimated. This approach therefore results in flux errors that are correlated with the relative orientation of the galaxy and PSF in each band. Hildebrandt et al. (2012) demonstrate the importance of homogenising the PSFs between bands before determining the matchedaperture photometry. Both CFHTLenS and KiDS Gaussianise the PSFs before measuring the photometry using the methodology proposed by Kuijken (2008) and Kuijken et al. (2015). These surveys should therefore be fairly immune to this additional error and we note that the DES photometry methodology has been significantly improved since the release of the DESSV data that we analyse in this paper (DrlicaWagner et al. 2018).
In this analysis we make the first step to examine photometricredshift selection bias, by simulating a mock galaxy catalogue where we introduce an anticorrelation between the signaltonoise ratio of the measured flux and the ellipticity of the galaxies relative to the local PSF ellipticity, , in four bands x = u, g, r, i. We use the following linear relation for the anticorrelation,
where the value for a_{x} and b_{x} are determined by fitting to KiDS450 multiband data (see Table D.2). Given that KiDS can only measure the noisy observed ellipticity, we recognise that the majority of the anticorrelation that we find in the KiDS450 data, derive from taking the mean of the absolute value of the observed ellipticity where the measurement noise, η in Eq. (27), increases with decreasing signaltonoise. Using Eq. (28) to apply a correlation between the signaltonoise of a galaxy and its relative ellipticity to the local PSF therefore provides an upper limit for this effect. Future work will use multiband image simulations to determine values for a_{x} and b_{x} where the true ellipticity is known. Our current approach is however sufficient for the purposes of examining the Bmode signature that is introduced by such an effect.
We produce mock ellipticity catalogues by randomly associating ellipticities to galaxies, using a fit to the observed KiDS450 galaxy ellipticity distribution, such that ⟨ϵ_{mock}⟩=0. We simulate 15 fields of 100 deg^{2} each with a total galaxy number density of 5.5 arcmin^{−2}. We choose a simple model of constant PSF per 1 deg^{2} pointing taken randomly from a uniform distribution between [ − 0.1, 0.1] for each component of the PSF ellipticity^{13}.
We associate noisefree multiband fluxes to the mock galaxies using simulations similar to the ones presented in Sect. 3.1 of Hildebrandt et al. (2009) but adapted to KiDS. These simulations were created with the HYPERZ package (Bolzonella et al. 2000) and are based on SED templates from the library of Bruzual & Charlot (1993), iband number counts from COSMOS (Capak et al. 2007), and redshift distributions from BPZ (Benítez 2000). These magnitude simulations contain half a million galaxies with magnitudes given in each of the four bands, selected to recover the KiDS redshift distributions given in Fig. 7.
Fig. 7.
True redshift distribution of the mock galaxies, separated into photometric redshift bins. The z_{B} selection is shown in the legend. The cyan histogram shows the true redshift distribution of the galaxies in the parent noisefree sample. In order to determine the photometric redshifts we introduce flux errors that mimic a KiDSlike survey and depend on the relative ellipticity of the mock galaxy to the mock PSF. 

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For each galaxy, we assign an error on the flux in each band using Eq. (28). Noise is then added to the mock galaxy flux, sampling from a Gaussian distribution. This approach correlates high values of observed galaxy ellipticity with high flux errors, as expected from the ellipticity measurement noise in the data. In addition, the flux error may also depend on the relativeorientation of the galaxy and the PSF, in each band, as expected from some photometry measurement methods in addition to the Kaiser (2000) effect. As this is the first investigation into photometric redshift selection bias, we have not tried to separate these effects in our analysis. We also note that this method of assigning noise to our mock galaxy sample ignores the additional dependence of the signaltonoise ratio on galaxy size and magnitude. Future work will need to investigate this in more detail, using multicolour image simulations.
We use the Bayesian Photometric Redshifts software BPZ to estimate photoz’s for each of our mock galaxies using a template fitting method (Benítez 2000; Benítez et al. 2004; Coe et al. 2006). The inputs are the noisy flux measurements and their associated errors. The output is the best fitting photometric redshift, z_{B}, which we use to then bin the galaxies into the four redshift bins that were used in the KiDS450 cosmic shear analysis, z_{i} ≤ z_{B} < z_{i + 1}, with z_{i} = {0.1, 0.3, 0.5, 0.7, 0.9} as well as a broad single bin encompassing the full redshift range of KiDS450, 0.1 ≤ z_{B} < 0.9. DESSV and CFHTLenS use a similar number of tomographic bins, spanning similar ranges in photometric redshifts. Note that the SED templates that we use in BPZ to estimate the redshift of the mock galaxies is independent from the ones used to make the mocks, which shows the robustness of this method to the choice of templates.
Figure 7 shows the true redshift distribution of the mock galaxies for each tomographic redshift bin in z_{B}. The distributions are broad due to the noise with extended high and low redshift tails which we label as catastrophic outliers in the distribution. The mean and median of each tomographic bin is similar to those in the KiDS450 data, demonstrating that our method to assign noise to our mock galaxy sample is sufficient for this analysis.
6. Results: mock E/Bmodes
In this section we present the twopoint statistic signatures of the systematics introduced in Sect. 5.1 using the statistics explained in Sect. 2. We show the effect of the three redshiftindependent shear systematics in Sect. 6.2 considering only a single redshift distribution for the galaxies. The effect of the remaining systematic, photometric redshift selection bias, is explored in Sect. 6.3 where redshift binning is applied. With these signatures identified, our goal is to use Bmode measurements as a diagnostic tool to uncover the origin of the systematic signals identified in DESSV, CFHTLenS and KiDS in Sect. 4. We can also determine the impact of these systematics on the measured Emode signals. Our approach is complementary to previous studies by Amara & Réfrégier (2008), Kitching et al. (2016), Taylor & Kitching (2018) who propagated cosmic shear systematics through to cosmological parameters in order to set requirements on their (in)significance.
6.1. SLICS shear simulations
The basis of our systematics analysis makes use of the ensemble of mock KiDS450 catalogues constructed from the SLICS^{14} simulations suite described in HarnoisDéraps & van Waerbeke (2015) and HarnoisDéraps et al. (2018). Each SLICS line of sight corresponds to a 10 ° ×10° field that includes galaxy positions, shear and their true and photometric redshifts. On average the redshift distribution and number density of galaxies in these mocks correspond to the KiDS450 data, which is not so dissimilar from the properties of both DESSV and CFHTLenS.
In Fig. 8 we present the SLICS cosmic shear measurements, ξ_{±}, ξ_{E/B} and COSEBIs, averaged over 10 shape noisefree lines of sight (i.e. ϵ^{int} = 0). On the left we show ξ_{±} and ξ_{E/B}, for θ ∈ [0.5′,300′] in 50 logarithmic bins. The top left panel shows ξ_{+} and ξ_{E} and the lower left panel shows ξ_{−} and ξ_{B}. The right panels belong to the Emode COSEBIs for a range of angular scales. The measurements from SLICS can be compared to the theoretical prediction (Eqs. (4), (10), (13)), shown as thick solid curves for ξ and pluses for COSEBIs. Here we adopt a flat ΛCDM model given by the input cosmology of the SLICS simulations in Table 1. We use a Bond & Efstathiou (1984) transfer function to estimate the linear matter power spectrum and the Smith et al. (2003) halofit model for the nonlinear scales. This combination, although dated, was chosen as the resulting theoretical predictions fit the mocks better than the more modern alternatives (HarnoisDéraps & van Waerbeke 2015). The thin coloured lines in Fig. 8 show the measured values for each lineofsight (LOS), which show a considerable scatter, especially for larger angular scales. Even with the inclusion of a larger number of LOS, however, we do not expect the theory to match the mean of the mocks perfectly, as the finite boxsize of the Nbody simulations, where L_{box} = 505 h^{−1} Mpc, results in a loss of power on large scales (HarnoisDéraps et al. 2018).
Fig. 8.
SLICS 2point statistics, ξ_{±} and ξ_{E/B} (left) and Emode COSEBIs (right), averaged over 10 noisefree linesofsight, which serves as our fiducial “systematicsfree” measurement. The mean result can be compared to the theoretical expectation (smooth solid curves). For ξ_{±} and COSEBIs we also show the measurements for each individual lineofsight with thin solid curves with matching colors between different panels. The upper left panel shows the measured ξ_{+} (magenta squares) and ξ_{E} (blue diamonds), with the lower left panel showing ξ_{−} (green pluses) and ξ_{B} (black crosses). The expectation value for ξ_{B} is zero, shown with the dashed black line. The COSEBIs Emodes (right panels) are shown for the three angular ranges indicated in each row. The COSEBIs Bmodes in SLICS are 4 orders of magnitude smaller than the Emodes and are therefore not shown. The measurements are shown as squares and their expected theory value as plus symbols. Note that COSEBIs modes are discrete and the points are only connected together as a visual aid. 

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For a Bmode free dataset, ξ_{+} = ξ_{E}. In the upper left panel of Fig. 8, we see that this is not the case for SLICS as at large angular scales ξ_{E} is smaller than ξ_{+}. Looking at the lower panel we see that ξ_{B} is nonzero for the same angular ranges. This leakage from E to B in the ξ_{E/B} statistic is a result of using the theoretical ξ_{−} at large scales for calculating the integrals in Eq. (9), which differs from the ξ_{−} of SLICS due to its finite boxsize. We find that COSEBIs do not suffer from either of these effects, with the COSEBIs Bmode signal in the mocks found to be ∼4 orders of magnitude smaller than the Emodes (not shown). The reason for the robustness of COSEBIs to the finite box bias comes from the weight functions that convert the shear power spectrum to these statistics. The low ℓ behaviour of the weight functions for COSEBIs have a leadingorder term proportional to ℓ^{4} such that the function reaches zero at small ℓvalues in contrast to the ξ_{+} kernel, J_{0}, which has power at small arguments (see Eq. (4)). At high ℓ the COSEBI weights also diminish rapidly in contrast to the 2PCFs which include some degree of power from all scales (see Fig. 4 in Kilbinger et al. 2017, for a comparison between the kernels corresponding to COSEBIs and 2PCFs).
6.2. The Bmode signature of shear measurement systematics
We add the shear measurement systematic effects, developed in Sect. 5.1, in turn to the SLICS simulations. We follow the standard approach of applying an empirical systematics correction to each mock by subtracting the average observed ellipticity from each line of sight before commencing our statistical analysis. In Fig. 9 we compare the resulting 2PCFs with the signal measured in the systematicfree fiducial data (see Fig. 8). The 2PCFs are ξ_{+} (squares), ξ_{−} (pluses), ξ_{E} (blue diamonds) and ξ_{B} (crosses). The left panels show the fractional difference of ξ_{±} and ξ_{E} to their fiducial values, calculated from systematic free mocks and shown with a “fid” superscript, for each systematic, while the right panels show the difference between ξ_{B} and its fiducial value as well as ξ_{E} − ξ_{B} (pluses) for each case. Here we use 50 logarithmic θbins, to show the angular dependence of each systematic in detail.
Fig. 9.
Impact of shear measurement systematics on ξ_{±} and ξ_{E/B} for four different types of shear measurement systematics; From the top down: PSF leakage, a repeating additive pattern, and random but correlated noise, correlated on chip and pointing scales (see Fig. 6). For ξ_{+} (magenta squares), ξ_{−} (green pluses) and ξ_{E} (blue diamonds) we present, in the left panels, the fractional difference between the measured signal in the systematicinduced KiDSlike SLICS mocks and the fiducial systematicfree case. As ξ_{B} (black crosses) and the E/B difference ξ_{E} − ξ_{B} (red pluses) tends to zero, we present, in the right panels, the difference between these measurements and the fiducial case, multiplied by the angular distance in arcminutes and scaled by 10^{4}. The measured Bmodes can be compared to the expected shapenoise error for KiDS450 (shaded area). 

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Each row in Fig. 9 shows the impact of the systematic which from topdown cover PSF leakage, a repeating additive pattern, and random but correlated noise (RCN), similar to PSF residuals, correlated over chipscales and then pointing scales. The grey regions in the right panels show the level of noise expected for KiDS450 data, which is similar to the noise in the DESSV and CFHTLenS analyses. As the simulations are free of shapenoise, the error associated with them is negligible compared to the expected errors from either of the three surveys, therefore we have excluded the simulation error from this figure (Fig. 9) and the next one (Fig. 10). We note that all these systematics also produce parity violating signals ξ_{×} = ⟨ϵ_{t}ϵ_{×}⟩, which is expected to be zero for a shear only field. We find that their amplitude is about an order of magnitude smaller than the Bmodes, however, and are therefore harder to detect in the data. As a result, we limit our systematics study to the effect of systematics on E/Bmodes.
Fig. 10.
Impact of shear measurement systematics on Emode (left) and Bmode (right) COSEBIs for four different types of shear measurement systematics; PSF leakage (blue squares), a repeating additive pattern (black stars), and random but correlated noise on chip (green triangles) and pointing (magenta diamonds) scales (see Fig. 6). The analysis is conducted for three different angular ranges spanning [0.5′,40′] (upper panels), [0.5′,100′] (middle panel), and [40′,100′] (lower panels). We present the difference between the measured signal in the systematicinduced KiDSlike SLICS mocks and the fiducial systematicfree case scaled by 10^{10}. The measured Bmodes and the resulting change to the Emode can be compared to the expected shapenoise error for KiDS450 (shaded area). 

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One interesting result from Fig. 9 comes from the nonzero signal in the curves. If a systematic adds equal power to both observed E and Bmodes, P_{sys}, from Eq. (7) and Eq. (8) we find that the observed ξ_{B} is equal to the excess signal in the observed ξ_{E} due to this systematic,
where are the observed E/Bmode 2PCFs and is the Emode signal produced from a shear only field. Furthermore as ξ_{−} is proportional to P_{E} − P_{B}, an equal P_{sys} contribution to the E and Bmodes will cancel such that . In this case there is a clear route to correct the measured Emode by the measured Bmode or to select “clean” angular scales for the Emode analysis which are Bmode free (see for example Hildebrandt et al. 2017). For the sample of systematics that we have simulated, however, we see that this common assumption of the equal contribution of systematic power to the E and B modes is far from reality, especially for larger angular scales^{15}. This implies that if a ξ_{B} signal is detected at any angular scale, its origin should be identified and mitigated at the catalogue or image level. Without understanding the origin it is unclear how that systematic will contaminate the ξ_{±} signal.
In Fig. 10 we present the COSEBIs analysis of the mocks. We show the relative effect of systematics on the Emode (left) and Bmode (right) COSEBIs, as the difference between their fiducial values and those estimated from the systematic induced mocks including PSF leakage, αϵ^{*} (blue squares), repeating additive pattern, c(x, y) (black stars), and random but correlated noise (RCN), βδϵ^{*}, correlated on chip scales (green triangles) and pointing scales (magenta diamonds). All values are shown for the mean of the 10 SLICS linesofsight. The grey regions show the one sigma errors corresponding to a KiDS450like survey. Random but correlated noise at the chip level shows small deviations from the fiducial values in agreement with Fig. 9. Within a single pointing, this systematic has a similar form to the additive pattern, where we find similar lown behaviour between these two systematics with similar peaks at n = 2 and a dip at n = 4, 5, albeit at different amplitudes. PSF leakage and the random but correlated noise at the pointing level are more significant, exhibiting a similar signal from the lower COSEBIs modes. The repeating additive pattern has the most chaotic effect on COSEBIs, in comparison to the other systematics that we have simulated. The erratic high frequency changes that can be seen in the 2PCFs in Fig. 9 are reflected in the significant power seen in the higher COSEBIs modes. As these systematics produce varying correlations for different angular scales, COSEBIs modes are affected by them in differing amplitudes, which also depend on the angular range they probe. Comparing Fig. 9 and Fig. 10 gives insight into the sensitivity of COSEBIs modes to correlation at various angular scales.
For all four systematics, we find that their effects on the COSEBIs are more prominent when the full angular range is used (middle panels). Comparing the right and left panels we see that all systematics affect both E and Bmodes, but not equally. In general a significant Bmode signal translates to a significant contamination to the Emodes on the same scales. The repeating additive pattern forms a clear exception to this rule though. This draws us to the same conclusion as the ξ_{B} analysis, the origin of any COSEBIs Bmode signal should be traced back to its source and corrected for at the pixeldata product level where phase information is still available, since it is unclear how these systematics will impact the Emode at another angular scale.
The characteristic patterns that we have identified should be used in any future approach to diagnose and correct shearmeasurement systematics. As an example, if COSEBI Bmodes are found to be oscillatory and extend to highn, the survey should investigate additive biases that repeat on a fixed angular scale across the survey, for example detectorlevel effects. If the Bmodes are localised at lown with little highn power, the survey should investigate the PSF modelling. With COSEBIs alone we cannot distinguish between PSF leakage, αϵ^{*} or correlated noise in the PSF model, βδϵ^{*}, but these two effects can be separated by measuring the correlation between galaxy shape and PSF ellipticity, which will be significant if α is nonzero (Bacon et al. 2003). If our ellipticity model in Eq. (27) is reasonable in its approach to add systematic terms linearly, we would expect the Bmodes from each individual effect to also add linearly. When we see both significant power at n < 7 and highn oscillatory power, as we do for DESSV for example, we can conclude that there is a likely superposition of systematics from both the PSF modelling errors and repeating additive biases.
The simulation approach that we use here should be specialised to the survey in question in future work. This would allow for a more precise exploration of how surveyspecific issues flow through to cosmological biases. In this analysis we have presented results that use KiDS to motivate the angular dependence of the systematics that we have simulated. In addition, we have also tested a variety of alternative schemes in the development of this work such that we are confident that the global behaviour of the Bmode signatures, presented in Fig. 10, are broadly representative of how these systematics would feature in any weak lensing survey. These tests included modelling different possible patterns and fixed amplitudes for each of the systematics, with the pointing size fixed to 1 deg^{2} in all cases in accordance to both KiDS and CFHTLenS. DES pointings are however hexagons of width 2.2°.
6.3. The Bmode signature of photometric redshift selection bias
We simulate photometric redshift selection bias following the approach developed in Sect. 5.2. As this effect is subtle, we choose to analyse the correlations in a random intrinsic ellipticity field alone, in contrast to the analysis in Sect. 6.2. where we also include the correlated SLICS cosmic shear field. Starting with the traditional statistics in Fig. 11, we compare two cases. The top panels show the 2PCFs using the full galaxy sample which are consistent with zero by construction. The lower panels show the same 2PCF analysis, including a photometric redshift selection with 0.1 ≤ z_{phot} ≤ 0.9. Comparing these two results we immediately see that the photometric redshift selection has produced a significant signal in all but the ξ_{−} statistic. As such we find similar levels of E/Bmodes in the lower right panel where the difference between ξ_{E} − ξ_{B} ≈ 0. Comparing the bottom panel of Fig. 11 with the top panel of Fig. 9 we see that the location of the peak in ξ_{+} is similar. This is because the PSFleakage systematic (Fig. 9) is related to photometric redshift selection bias, although, here we have used a simplified PSF model without the small scale variations that affect the PSFleakage systematic (Fig. 6).
Fig. 11.
Photometric redshift selection bias: the upper panels show the correlations measured from a random intrinsic ellipticity field using the full galaxy sample. Lower panels: correlations measured from the same random intrinsic ellipticity field after a 0.1 ≤ z_{phot} ≤ 0.9 photometric redshift selection has been applied. The 2PCFs shown on the left are ξ_{+} (squares) and ξ_{−} (pluses). On the right panel, ξ_{E} (diamonds), ξ_{B} (crosses) and the difference between the two (pluses) are shown. The onesigma error bars correspond to the level of ellipticity noise in the mock data. The signals are shown multiplied by θ in arcminutes and scaled by 10^{4}. 

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To see the full effect of the photometric redshift selection bias on the measured correlation function, we use COSEBIs to analyse four photometric redshift bins, corresponding to those chosen in the KiDS450 analysis. Fig. 12 shows the results revealing significant signal in the different tomographic slices, with the strongest effect in the highest redshift bins. To quantify the significance of this effect we use a theoretical covariance to estimate a χ^{2} value for COSEBIs relative to the null hypothesis of zero signal. We then calculate the pvalue corresponding to that χ^{2} finding vanishingly small values of p ∼ 10^{−28} and p ∼ 10^{−15} for the E and B modes respectively, clearly confirming the significance of the bias.
Fig. 12.
Impact of photometric redshift selection bias for a COSEBIs 4bin tomographic analysis of a random intrinsic ellipticity field. The upper triangle shows the Emodes and the lower triangle shows the Bmodes, where the error bars in both cases correspond to the level of ellipticity noise in the mocks. Each panel shows COSEBIs for a tomographic redshift bin pair, z − ij, corresponding to the correlation between photometric redshift bins i and j. As our photometric redshift mocks are devoid of any cosmological correlations, in the absence of any selection bias, we would expect both the E and B modes to be consistent with zero. 

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As with the analysis of the shear measurement systematics in Sect. 6.2 we see similarities (generally lown) and differences (generally highn) in the measured E and Bmodes, again leading us to the conclusion that Bmodes can be used as a diagnostic but cannot blindly be used to correct the Emodes.
6.4. Cosmological parameter inference
Although we can see the signature of each systematic in Figs. 9 to 12, it is not immediately clear how they would affect cosmological parameter inference. One could carry out a likelihood analysis to find any biases introduced by these systematics (see for example Amara & Réfrégier 2008), but our preference is to use compressed COSEBIs (CCOSEBIs, see Sect. 2.3) as a faster alternative approach. CCOSEBIs are formed of linear combinations of COSEBIs that are sensitive to cosmological parameters. If the systematics that we identify in the COSEBIs analyses are null in both the E and B mode CCOSEBIs case, then we can conclude that the systematics are unlikely to be detrimental to the cosmological inference.
In this analysis we focus on the CCOSEBIs that are sensitive to Σ_{8} = σ_{8}(Ω_{m}/0.3)^{α}, as this is the combination of parameters that cosmic shear data are mostly sensitive to. We find that for a KiDS450 redshift distribution, α = 0.65 best describes the COSEBIs degeneracy direction (see Appendix B). Because we are interested in Σ_{8} we only consider the 5 first and secondorder CCOSEBIs for σ_{8} and Ω_{m}; , , , and , where E^{c} is defined in Eq. (15). Although it is possible to construct a CCOSEBIs mode that is sensitive to Σ_{8} directly, we choose to look at these 5 modes instead in order to also provide an internal consistency check. For each of the five modes we calculate a single compressed value, , that can be compared to its expectation value, given a set of cosmological parameters, noise covariance and source redshift distribution.
Figure 13 shows the measured Emode (left panel) and Bmode (right panel) CCOSEBIs for the full angular range of [0.5′,100′]. The symbols correspond to the range of shear measurement systematics^{16} simulated using the SLICS cosmic shear simulations in Sect. 5.1. The lines connecting the Emode points indicate which of the 5 CCOSEBIs modes are shown and also show their theoretical value. Each measured Emode can be compared to the value of Σ_{8} = σ_{8}(Ω_{m}/0.3)^{0.65} that would be inferred from the measurement. In the absence of systematic errors, we would expect to find the inferred parameters to be consistent with each other and the input SLICS cosmology with Σ_{8} = 0.808. We would also expect to find the Bmode signal consistent with zero, but looking at the fiducial “nosystematics” mocks (circles), we do recover a very small residual Bmode and a slightly high bestfit Σ_{8} = 0.815. This result is expected, however, given the imperfect match between the twopoint statistics measured from SLICS and the theoretical expectation shown in Fig. 8. Here no errors are associated to the Emode CCOSEBIs, because the mock data used to produce them are free of shapenoise.
Fig. 13.
Left panel: inferred values of Σ_{8} = σ_{8}(Ω_{m}/0.3)^{0.65} from an Emode CCOSEBIs analysis of four mock cosmic shear surveys that suffer from PSF leakage (blue squares), a repeating additive pattern (black stars), and random but correlated noise on chip (green triangles) and pointing (magenta diamond) scales. The curves show the theoretical values of the 5 Emode CCOSEBIs when varying Σ_{8}, calculated using the KiDS450 noise only covariance matrix and redshift distribution; E_{σ8σ8} (dashed), E_{ΩmΩm} (dotted), E_{σ8} (middle solid), E_{Ωm} (dotdashed) and E_{σ8Ωm} (lower solid). The inferred cosmology for each mock systematic survey can be compared between the 5 different modes. The higher the recovered Emode is relative to the fiducial “nosystematics case” (circles), the stronger the bias is on the inferred value of Σ_{8} and the more discrepant the inferred cosmology is between the different CCOSEBIs modes. The bias in Σ_{8} can be compared to the grey region which shows the onesigma error for Σ_{8} from the KiDS450 cosmic shear analysis, centred on the fiducial case. Right panel:Bmode CCOSEBIs from the four mock cosmic shear surveys. The measured Bmode signal, which does not depend on Σ_{8}, can be compared to the shape noise on a KiDS450like survey (shown in grey). 

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We find that the introduction of the random, but correlated noise (RCN) increases the recovered Σ_{8} value, but within the statistical tolerance of KiDS (shown as a grey bar) in the case of the chipscale correlation. The PSF leakage and repeating additive pattern result in the largest bias in cosmological parameters with a ∼5% deviation from the true input cosmology. Applying this level of bias to either σ_{8} or Ω_{m} can produce excess correlations of only up to 13% which is significantly less than the up to 40% biases seen in the twopoint correlation function analysis in Fig. 9, from which we can conclude that the impact of these systematics on the data can be only weakly correlated with the impact of varying cosmological parameters.
We find that the stronger the bias in the recovered cosmology, the larger the inconsistency between the 5 CCOSEBIs modes, providing another important diagnostic tool. We also note that all the shear measurement systematics tested in this analysis serve to increase the inferred value of Σ_{8}. If these types of systematics were present in the weak lensing data, correcting for them would decrease the recovered Σ_{8}, exacerbating the current hints of cosmological parameter tension between weak lensing surveys and Planck (see for example Troxel et al. 2018b).
Comparing the power in the measured Emodes (left panel) and Bmodes (right panel) reveals a close connection, where a larger bias in the Emodes corresponds to larger Bmodes. We note that although the magnitude of the E/Bmodes are connected, they can take opposite signs. For example, in Fig. 10 we see that for the large angular scale analysis with the repeating additive pattern systematic, the sign of the first four Emodes differs from the sign of the first four Bmodes. Although we do not show the largescale CCOSEBIs result, we can confirm that, this difference in sign is also reflected there, as the CCOSEBIs are sensitive to the first few modes that contain a large proportion of the cosmological information. This example demonstrates that although the large angular scales have the lowest signal to noise, they can and should be used as an investigative tool for hunting systematics that could also impact small angular scales.
7. Discussion
In this section we discuss how we can use the measured COSEBIs Bmode signatures from our systematics mocks in Sect. 6 to diagnose the origin of the Bmodes recovered in the CFHTLenS, DES and KiDS surveys in Sect. 4. Firstly, we focus on the nontomographic COSEBIs Bmode measurements in Fig. 3. One feature that stands out for all three surveys is the high nmode oscillatory pattern in the full angular range, shown in the middle right panel. This oscillatory pattern is the signature of a repeating additive systematic, shown in Fig. 10, which we find to be the systematic that was most detrimental to cosmological parameter estimation in Fig. 13.
We find the level of Bmodes for KiDS450 and CFHTLenS for higher nmodes to be small and hence the repeating additive signature is not highly significant in these cases. The similarity between the Bmodes in the data and this systematic signature does however warrant further exploration, particularly as we also see similarities in the Emodes for KiDS and the repeating pattern Emode signature. Here the unexpected Emode “secondary peak” seen at n ∼ 6 in the small angular scales of KiDS450 Emodes (upper left panel of Fig. 3) is replicated at n ∼ 6 in the Emode analysis of the same angular scales of the repeating additive bias mocks (upper left panel of Fig. 10). If a repeating additive systematic persists it would likely become significant in future releases of KiDS. It could also be responsible for the power seen in the lown modes that lead to the significant detection of the KiDS CCOSEBIs Bmodes.
For DESSV we find a significant detection of Bmodes, noting that in addition to the highn oscillatory pattern, DESSV presents significant additional signal for modes around n = 8 and n = 4. For an instrumentbased repeating additive pattern, the resulting Bmode signature will depend on the dithering strategy and camera fieldofview. Both KiDS450 and CFHTLenS have a fieldofview of ∼1 deg^{2} with small dithers. DESSV, however, has a hexagonal fieldofview, 2.2° across, and uses halffield dithers. This means the frequency of any repeating additive pattern will differ for DESSV in comparison to the KiDSlike imaging strategy that we have simulated in our mocks. Looking at only the first few modes for DESSV data, however, we find that the signal resembles the signature of both PSF leakage, and random but correlated noise on the pointing level. This result is consistent with the findings of Zuntz et al. (2018) who report and correct for a small but significant PSF residual in their analysis of the first year of DES observations. We therefore conclude that the Bmode signature recovered for DESSV is likely a superposition of different shear measurement systematics.
By comparing the pvalues in Table 2 we can see that for DESSV the significance of the COSEBIs Bmodes substantially increases when the data are separated into tomographic bins. This could be understood by considering the photometric redshift selection bias explained in Sect. 5.2. This systematic correlates the PSF ellipticity with the redshift estimation for a galaxy, and can produce significant Bmodes when the data are binned into smaller photometric redshift bins. It is likely that all surveys will suffer from this systematic to some degree, but the level will depend on how the multiband photometry is measured in each survey and how the PSF ellipticity varies in each optical band. We cannot directly compare our mock analysis with the Bmodes in the DESSV tomographic analysis, but our firstlook at this effect certainly motivates further exploration with more detailed simulations that fully mimic the photometric redshift measurement in each survey.
Interestingly, comparing the DESSV tomographic and nontomographic pvalues in Table 3, we find that for the analyses that include smallscale information, the significance of the Bmodes, measured using the cosmologicalparametersensitive CCOSEBIs, decreases when the data are separated into tomographic bins. This promising result means that if the systematic that was introduced when the DESSV tomographic selection is applied adds equal power to the E and B modes, that systematic would not introduce modifications to the Emode signal that would bias the inferred cosmological parameters. Unfortunately, however, the photometric redshift selection bias systematic was found to exhibit different E and B mode signals in Fig. 12. Passing the CCOSEBIs Bmode nulltest therefore cannot validate the CCOSEBIs Emode measurement. In addition, this CCOSEBIs Bmode result does not hold for the large angular scales, [40′,100′], where again we see a substantial increase in the measured Bmode when the data are separated into tomographic bins.
Our findings for DESSV contradict Becker et al. (2016) who conclude that the Bmodes in DESSV are insignificant using two Fourier space methods. We argue that as power spectra pick up features of the data at different scales compared to ξ_{±}, they are not suitable statistics for verifying the absence of Bmodes in a ξ_{±} cosmic shear analysis. In addition, power spectra measurements are binned in a range of Fourier modes, such that any highfrequency variations in Fourier space will average out. COSEBIs are sensitive to these variations and can therefore be used to diagnose the origin of the Bmodes in the data.
At first sight our findings for KiDS also contradict Hildebrandt et al. (2017) who report a lowlevel but significant detection of ξ_{B}. This is in contrast to our tomographic ξ_{B} analysis which concludes that ξ_{B} is consistent with zero (see Fig. D.5). We find that the ξ_{B} statistic is sensitive to the choice of the maximum θscale measured from the data and the maximum θscale used for completing the integral to infinity using a theoretical prediction (in this analysis we use 1000′ instead of 3000′ used in Hildebrandt et al. 2017). We also find that ξ_{B} is sensitive to the method used to bin ξ_{±} as explained in Appendix A (see Eq. (A.10)). This sensitivity to data analysis choices provides another reason to archive the traditional ξ_{E/B} approach. In this paper we promote COSEBIs as the optimal statistic for both E and B mode measurements as it can be estimated accurately and free of any biases connected to binning and extrapolating the data. Analysing all 20 COSEBIs modes, we find no significant evidence for Bmodes in KiDS. In our compressed CCOSEBIs analysis, however, we arrive at the same conclusion of both Hildebrandt et al. (2017) and van Uitert et al. (2018), that lowlevel but significant Bmodes are present in KiDS450. In our [0.5′,100′] CCOSEBIs analysis, we find a ∼2.7σ detection of a Bmode signal that is less than 10% of the amplitude of the Emode. This difference between the significance of the COSEBIs and CCOSEBIs Bmode analysis might seem confusing or even contradictory. We therefore refer the reader to Appendix C where we explore how choices over the number of modes used in a nulltest can dilute or optimise the detection of systematics.
8. Conclusions
Twopoint shear correlation functions (2PCFs) have been the primary observables in cosmic shear analysis to date, but they are not immune to systematics. These statistics mix E and Bmodes in the data, giving rise to a mixed lensing and nonlensing signal in the presence of systematic errors. In order to test for systematics most surveys turn to alternative statistics to separate E/Bmodes, using ξ_{E, B} or power spectrum measurements. We argue that these alternative statistics are biased as they depend on infinite integrals over 2PCFs and are sensitive to binning choices. In addition, treating the E/Bmode decomposition with a statistic that has a different scaledependence to the statistic used in the cosmological parameter inference, causes a disparity in the analysis. For future cosmic shear analyses, we therefore advocate the use of COSEBIs for both parameter inference and systematic analyses (see Sect. 2.2). COSEBIs cleanly and completely separate E/Bmodes over a finite angular range, without loss of information. They have discrete modes and therefore are insensitive to binning choices. The first few modes of COSEBIs contain almost all of the cosmological information and as such a COSEBIs analysis is also an efficient approach to data compression. For a Bmode analysis, however, a larger number of modes need to be considered, as systematics can affect the E and Bmode at different scales.
In this paper we analysed the E and Bmode signals in three public cosmic shear surveys, CFHTLenS, DESSV and KiDS450. We compared the ξ_{E, B} statistic with COSEBIs and CCOSEBIs, using pvalues to quantify the level of Bmodes in the data. To determine COSEBIs filter functions we need to first define an angular range of interest. For this study we chose three sets of angular separation ranges: small separations, [0.5′,40′], large separations, [40′,100′], and the overall separation range, [0.5′,100′]. We measure COSEBIs up to mode n = 20. We considered two cases for each survey; one using the same redshift bins as used in each survey’s primary cosmic shear analysis, and another combining those bins into a single redshift bin. We see that for DESSV data the tomographic cases show significant Bmodes at a level between 4σ and 5.5σ. For the nontomographic DESSV analyses, Bmodes are detected at the level of 2.8σ. For KiDS450 and CFHTLenS, we find no significant detection of Bmodes for the majority of our analyses. There is however some exceptions in each case. The CCOSEBIs analysis of the small separations (nontomographic case only) and the analysis of the full angular range show Bmodes at up to 2.7σ for KiDS450. The tomographic COSEBIs analysis over the small angular range [0.5′,40′] detects a Bmode signal at 2.8σ for CFHTLenS.
In order to diagnose the origin of the Bmodes detected in each survey, we modelled several nonastrophysical systematic effects relevant to current data in order to determine their E/B mode signature and assess their impact on cosmological parameter inference. We modelled four shear measurement systematics. PSFleakage, was modelled using the mosaic PSF pattern from KiDS450 assuming a 10% leakage with α = 0.1. An instrumentbased additive bias term resulting in a repeating pattern from pointing to pointing. Here we used the lowlevel CCD bias of OmegaCam (Hoekstra et al., in prep.), multiplied by a factor of five to amplify and model this effect. To model biases arising from random PSF modelling errors, we correlated low levels of random noise using two kernel sizes, corresponding to roughly KiDS CCD and pointing scales. In addition to these shear measurement systematics, we have introduced a new effect by modelling the impact of photometric redshift selection bias that arises from the correlation between the relative orientation of PSF ellipticity and galaxy ellipticity, and the measured signaltonoise of the galaxy.
All of the systematics simulated were detected in our Bmode analysis. The PSFleakage and random but correlated noise systematics introduced low nmode COSEBIs signal. This was in contrast to the repeating additive bias which introduced high frequency variations in the shear field which are picked up as oscillatory behaviour in the high nmode COSEBIs measurements. Photometric redshift selection bias also resulted in high nmode power in the high photometric redshift bins. Comparing the Bmode signatures recovered by our mocks to the Bmodes measured in each survey we conclude that DESSV is likely to suffer from a combination of all the systematics that we have simulated. The significant increase in DESSV Bmodes when the tomographic redshift selection is applied is particularly striking, motivating future work to enhance the realism of the firstlook photometric redshift simulations that we have analysed in this paper. KiDS450 and CFHTLenS show oscillatory behaviour in the high nmode indicating a repeating additive bias in the data, although this result is not significant.
The simulated systematics produce Emodes that would bias cosmological parameter inference. For the analysed shear measurement systematics we found that Σ_{8} = σ_{8}(Ω_{m}/0.3)^{0.65} is biased high in all cases. As a result, we conclude that these types of systematics, if present, cannot explain the mild tension between some current cosmic shear and Planck results and could in principle exacerbate the tension as they bias Σ_{8} to even higher values (see Fig. 13). It is interesting to note that the DESSV cosmological parameter constraints on S_{8} = σ_{8}(Ω_{m}/0.3)^{0.5} are higher than those from KiDS450, CFHTLenS and the first year DES results, which include a number of improvements over the DESSV analysis. Given the significant DESSV Bmodes detected in our analysis, the direction of the difference in S_{8} between the surveys is expected. The published cosmological parameter constraints from all three surveys are, however, in good agreement.
For the analysis of KiDS450, we find an interesting case where the survey formally passes the COSEBIs Bmode analysis, but a flag is raised with a 2.7σBmode detection in the compressed CCOSEBIs analysis. Here the COSEBIs Bmodes that are insignificant overall are weighted in such a way that the resulting CCOSEBIs signal becomes significant. We therefore recommend measuring both CCOSEBIs and COSEBIs Bmodes in future analyses, ensuring that both are consistent with zero. The CCOSEBIs Bmodes will robustly identify systematics that will lead to a bias in the cosmological parameter inference, if the systematic impacts the E/Bmodes in the same way. In contrast the COSEBIs Bmodes detect systematics that can affect E and Bmodes differently. For this reason we have shown that it is not sufficient to correct data by simply subtracting the Bmodes from the Emodes. As we have seen from our repeating additive bias systematic, where the E and B mode behaviour is very different, it will be crucial to look at both a COSEBIs and CCOSEBIs analysis and return to correct the input catalogues if a significant Bmode is detected in either case.
Contributions beyond the firstorder Born approximation (Schneider et al. 1998) and source clustering (Schneider et al. 2002a) can produce insignificant levels of Bmodes for the current generation of shear surveys.
Semboloni et al. (2007) find the transition between the Gaussian and nonGaussian terms occurs at θ ∼ 20′. At this scale the cosmic variance and mixed term roughly double the size of the error bars. At θ ∼ 1′ the nonGaussian term is an order of magnitude larger than the Gaussian cosmic variance term, but as the noise term is dominant here the effect of the nonGaussian term on the error bars is only ∼10%.
COSMOSIS: bitbucket.org/joezuntz/cosmosis
CAMB: http://camb.info
We chose this strong PSF over the very low ellipticity KiDS450 PSF model, ϵ^{*} = −0.006 ± 0.018, shown in the lefthand panel of Fig. 6, as initial KiDSlike studies did not result in a significant photometric redshift selection bias.
Available here: http://slics.roe.ac.uk/
Note that as we do not include a cosmic shear signal in the photometric redshift selection bias mocks developed in Sect. 5.2, we do not present a CCOSEBIs analysis of this systematic.
Acknowledgments
We thank Joe Zuntz for his help with DESSV data and COSMOSIS. We also thank Benjamin Joachimi, Alex Hall, Tilman Troester, Michael Troxel and Angus Wright for useful discussions. CH, MA, AA and JHD acknowledge support from the European Research Council under grant number 647112. This work was carried out in part at the Aspen Center for Physics, which is supported by the National Science Foundation grant PHY1607611, where CH, HH and KK were also supported by a grant from the Simons Foundation. PS is supported by the Deutsche Forschungsgemeinschaft in the framework of the TR33 “The Dark Universe”. KK acknowledges support by the Alexander von Humboldt Foundation. HH is supported by Emmy Noether (Hi 1495/21) and Heisenberg grants (Hi 1495/51) of the Deutsche Forschungsgemeinschaft as well as an ERC Consolidator Grant (No. 770935). LM acknowledges support from STFC grant ST/N000919/1. AC acknowledges support from NASA grant 15WFIRST150008. Computations for the Nbody simulations were performed in part on the Orcinus supercomputer at the WestGrid HPC consortium (www.westgrid.ca), in part on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund – Research Excellence; and the University of Toronto. The KiDS450 results in this paper are based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A3016, 177.A3017 and 177.A3018, and on data products produced by Target/OmegaCEN, INAFOACN, INAFOAPD and the KiDS production team, on behalf of the KiDS consortium. The CFHTLenS results in this paper are based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/IRFU, at the CanadaFranceHawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l’Univers of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency. CFHTLenS data processing was made possible thanks to significant computing support from the NSERC Research Tools and Instruments grant program. This DESSV results in this paper are based on public archival data from the Dark Energy Survey (DES). Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at UrbanaChampaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Científico e Tecnológico and the Ministério da Ciência, Tecnologia e Inovação, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones Energéticas, Medioambientales y TecnológicasMadrid, the University of Chicago, University College London, the DESBrazil Consortium, the University of Edinburgh, the Eidgenössische Technische Hochschule (ETH) Zürich, Fermi National Accelerator Laboratory, the University of Illinois at UrbanaChampaign, the Institut de Ciències de l’Espai (IEEC/CSIC), the Institut de Física d’Altes Energies, Lawrence Berkeley National Laboratory, the LudwigMaximilians Universität München and the associated Excellence Cluster Universe, the University of Michigan, the National Optical Astronomy Observatory, the University of Nottingham, The Ohio State University, the OzDES Membership Consortium, the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. Based in part on observations at Cerro Tololo InterAmerican Observatory, National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. Author contributions: All authors contributed to the development and writing of this paper. The authorship list is given in three groups: the lead authors (MA, CH) followed by two alphabetical groups. The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products. The second group covers those who have either made a significant contribution to the data products, or to the scientific analysis.
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Appendix A: Binning bias with ξ_{±}
When using the traditional ξ_{±} statistic an issue arises from the choices that can be made when binning the data. ξ_{±}(θ) is usually binned in broad θbins as a form of data compression. As we expect the number of galaxy pairs to roughly scale with θ, see Eq. (19), the sampling of ξ_{±}(θ) within the bin is nonuniform. If we bin these functions into broad angular bins, their value will therefore be biased towards larger θ scales in each bin. This is not an issue provided the theory is treated in the same way, but this is not the standard approach that is taken, as it is computationally more expensive. Troxel et al. (2018b) compare the differences in KiDS450 cosmological parameter inference if one takes the logarithmic midpoint of the bin or the weighted mean value of θ in each bin and evaluate the theoretical 2PCFs at each θ value. They argue that the latter approach is correct, supported by Krause et al. (2017) who conclude that this approach is sufficiently accurate for the first year DES analysis. Here we provide more detail on the question of binning bias, quantifying how inexact each treatment of the theory is, where we find up to 10% biases in both approaches. A full integration of the theory within the bin is the correct approach to this problem. If future surveys wish to use an approximation, however, we demonstrate that using the linear midpoint of the bin provides the closest match to the binned data, with less than 2.5% bias.
A.1. Binning theory
Consider making measurements of a function f(x) from noisy data, with samples drawn nonuniformly in x. We denote the sampled data points by f_{data}(x) and the distribution of measured x by 𝒟(x). Given that the sampled data points are noisy, we want to combine them to find an estimate for the function for a given binning in x. One way to bin the data is to write it as
where N_{bin} is the number of data points in the given bin and Δ(x − x_{b}) is the binning function defined as
This estimate for the binned function corresponds to a weighted binning, with more weight given to the values where there are more sampled points. The expectation value of is in general not equal to f(x_{b}),
where x_{min}(x_{b}) and x_{max}(x_{b}) are the edges of bin x_{b}. Even if we define x_{b} as the weighted mean of the xvalues in the bin, as advocated by Troxel et al. (2018b),
we would only recover the true value of the binned data, if f(x) is either a constant or has a linear relation to x.
Let’s now take very fine xbins, such that the variation in the sampling of f(x) is negligible, i.e. 𝒟(x) ≈ constant within each bin. The expectation value of the finely binned function , where the subscript f represents “fine” and x_{f} is the midpoint of the fine bin, is given by
where 2 δx is the width of the fine bin. If we first measure finely binned from the data, then we have the flexibility to rebin the measurements as desired,
where w(x_{f}) is a weight function assigned to each fine bin. If we chose to set w(x_{f}) to 𝒟(x_{f}) we would recover the weighted binning defined in Eq. (A.1).
If we choose w(x_{f}) such that it does not vary between different realisations of the data, the expectation value of is given by
and the covariance of the binned data for bins x_{b} and y_{b} is given by
where ℂ_{f}(x_{f}, y_{f}) is the covariance of the finely binned measurements, f_{f}(x_{f}) and f_{f}(y_{f}). If we assume no crosscorrelation between the bins, which is the case for a shapenoise only covariance, then Eq. (A.8) simplifies and the variance of f_{w}(x_{b}) can be written as,
From this equation we can see that the variance of the binned data is also not equal to the variance of the function at x_{b}, , which complicates the calculation of covariance matrices for binned data.
To simplify covariance calculation we can set the weights in Eq. (A.6) equal to unity and obtain an unweighted rebinned estimate,
In this case the expectation value of the estimator is,
where 2 Δx is the width of the bin. If the relative variation of the sampled points within a broad bin is large, then this estimator may not be optimal and can produce larger errors compared to the estimator in Eq. (A.6).
Equation (A.7) is useful for predicting the theoretical value of the binned function, especially when the sampling frequency of the data points, 𝒟(x), is derived from the data itself. In the case of cosmic shear ξ_{±}(θ) the sampling of the data points roughly scales with θ, however, survey geometry and masking effects together with variations in the depth of the images complicates the analytical estimation for the distribution of data in angular scale. Hence we suggest measuring 𝒟(θ) from the data and use Eq. (A.7) to predict the binned ξ_{±} values.
A.2. Application to cosmic shear
To demonstrate the level of bias introduced by partial treatment of the theory in a ξ_{±} cosmic shear analysis we use a theoretical prediction for ξ_{±} as our function, f(x), assuming a single KiDS450 redshift bin. To sample ξ_{±}(θ) in a nonuniform way, we randomly pick θ values from a 𝒟(θ) = θ/arcmin × 2000 distribution in the angular range of [0.5′,300′]. We then add a constant Gaussian random noise with σ = 0.01 to each sampled point to produce the noisy sampled data points, ξ_{±data}(θ) and then bin ξ_{±data}(θ) into 1000 fine logarithmic bins to produce and 9 broad logarithmic bins to get (see Eq. (A.1)).
Fig. A.1.
2PCF binning bias introduced for a range of analysis choices, shown as the ratio between the measured and their proposed theoretical value as a function of angular scale, θ. The legends in this figure are shared between the two panels. For the weighted broad bin estimator, , the bias is calculated assuming θ_{b} is given by the logarithmic midpoint of the bin, θ_{mid, log} (green squares), the weighted mean of the bin, θ_{w} (black pluses, under the blue diamonds), the geometric mean or linear midpoint of the bin, θ_{mid, lin} (dark grey triangles), or an areaweighted bin centre, θ_{area} (blue diamonds). These estimators can be compared to the fine binning case, , where the theory is estimated at the weighted mean of each bin (light grey circles), and the exact case (red crosses) where the theoretical value is calculated as a weighted integral over the signal within the bin (Eq. (A.7)). All points are plotted with errorbars, but in the case of broad binning the errors are too small to be visible. 

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In Fig. A.1 we show the binning bias introduced for a range of cases as a ratio between the measured and the proposed theoretical value of ξ_{+} (top panel) and ξ_{−} (bottom panel), as a function of angular scale, θ. The noisy finely binned data, (light grey circles), shows no significant bias relative to its expectation value (see Eq. (A.5)). As was shown in Eqs. (A.3) and (A.7) the expectation value of the broad binned, , should be calculated using an integral over ξ_{±} with the appropriate weights. The red crosses in the figure correspond to this theoretical prediction which is unbiased as expected. The remaining curves show the biases introduced when broad binning is applied to the measurements and the theory is evaluated at a single point in the bin denoted as θ_{b}. The green squares assume that θ_{b} is given by the logarithmic midpoint of the bin (as used in Hildebrandt et al. 2017), for the black pluses θ_{b} is the weighted mean of the bin (as used in Heymans et al. 2013; Troxel et al. 2018a,b), the grey triangles use the geometric mean or linear midpoint of the bin (not used to date) and finally blue diamonds assume that θ_{b} is the areaweighted bin centre (as advocated by Krause et al. 2017), where
Here θ_{min} and θ_{max} are the minimum and maximum values of the bin.
We find that the weighted midpoint and the area weighted values are similarly biased, boosting the signal at the ∼3% level at 10 arcmin, rising to ∼10% bias at large scales for ξ_{+}. Taking the logarithmic midpoint of the bin has the opposite effect, decreasing the signal at ∼7% level at 10′ and ∼10% bias at large scales for ξ_{+}. That the biases work in the opposite sense here increases the inferred impact of binning bias when comparing the two KiDS analyses in Troxel et al. (2018b).
In all cases we see that the choice of binning affects ξ_{+} more than ξ_{−}, since ξ_{+} has more curvature than ξ_{−}. We note that these biases will be smaller for narrower angular bins and as such their effect will not be as significant for the first year DES analysis (Troxel et al. 2018a) which uses the weighted mean for θ_{b} with roughly twice as many bins in the same angular range as shown here.
If future surveys conclude that it is too computationally expensive to calculate the impact of binning theoretically, especially in the case of the covariance matrix, our proposed solution is to use the linear mid point of the θbin in the binned ξ_{±} analysis. We find that this approximation presents the weakest bias with at most 2.5% bias at large and small scales and below percent level bias between 0.5′< θ < 300′. Another alternative is to move to a COSEBIs analysis. As COSEBIs are discrete they are not subject to any of the binning biases presented in this Appendix.
Appendix B: σ_{8} − Ω_{m} degeneracy
Fig. B.1.
Degeneracy direction of σ_{8} and Ω_{m} for a CCOSEBIs analysis of the KiDSlike data. The colours in the image show the value of the CCOSEBIs E_{σ8} mode, in comparison to dashed lines of constant Σ_{8} = σ_{8}(Ω_{m}/0.3)^{α} with α = 0.65. The repeating color scheme was chosen to capture the variations in the values of E_{σ8}. The lower left corner has the smallest value of E_{σ8} which gradually increases, perpendicular to the dashed curves, towards the upper right corner. 

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Cosmic shear is most sensitive to a combination of σ_{8} and Ω_{m} (Jain & Seljak 1997), where the degeneracy can be written as
Here is arbitrary but is usually taken to be 0.3. In the majority of cosmic shear analyses α has been taken to be α = 0.5, even though the optimal value of α will depend on the statistic used, the redshift distributions and the angular ranges used in the analysis. As an example, Hildebrandt et al. (2017) present joint Σ_{8} − Ω_{m} constraints with α = 0.5. The tilt seen in their Fig. 6 of these constraints demonstrates that α = 0.5 does not best represent the degeneracy direction of Ω_{m} and σ_{8} for the KiDS450 2PCF tomographic analysis.
In Fig. B.1 we show the value of the CCOSEBIs mode E_{σ8} (see colour bar) for a range of σ_{8} and Ω_{m} values assuming a KiDSlike survey. The degeneracy shown in E_{σ8} can be compared to the dashed lines of constant Σ_{8} = σ_{8}(Ω_{m}/0.3)^{α} where α = 0.65. We have carried out this test for all the CCOSEBIs modes in our analysis; E_{Ωm},E_{σ8, σ8}, E_{σ8, Ωm} and E_{Ωm, Ωm} to confirm that α = 0.65 is an optimal choice for our CCOSEBIs analysis.
Appendix C: Optimising the COSEBIs Bmode nulltest
All null tests are subject to the choices we make in our data analysis. As an example, if we limit our Bmode analysis of CFHTLenS to the first 7 COSEBIs modes, following Asgari et al. (2017), we conclude there are no significant small scale Bmodes in CFHTLenS. In contrast, our analysis of the first 20 COSEBIs modes, in Sect. 4, finds a significant Bmode detection for CFHTLenS on the same scales. In this Appendix we explore the question of how many COSEBIs modes should be used to determine the overall significance of the Bmodes in a dataset.
Fig. C.1.
Model comparison using 10 000 random samples of the PSFleakage systematic model, M_{PSF} (solid) defined in Sect. 5.1.1. The data points show the mean of the random samples, with the errors reflecting the noise on a single realisation. The data samples are analysed to determine how often the input model M_{PSF} can be distinguished from the nosystematics zero Bmode model, M_{0} (dashed). 

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As an illustrative example, we take two parameterfree models for COSEBIs Bmodes, shown in Fig. C.1: M_{0} where B_{n} = 0 for all n, and M_{PSF} where B_{n} corresponds to the measured PSFleakage systematic defined in Sect. 5.1.1. The difference between these two models is captured by the first few modes, with almost zero power for n ≳ 10. We create 10 000 random samples of B_{n} for the full angular range of [0.5′,100′] given the model M_{PSF} and the KiDS noiseonly covariance for the nontomographic case. Figure C.1 shows the mean of these samples (red squares) with errors corresponding to a single sample as well as the input model (blue curve).
Fig. C.2.
χ^{2} distribution of the mock data given the true PSF leakage model, M_{PSF}, (blue histogram) or given the null model, M_{0}, (orange histogram). Left panel: analysis of the n ≤ 20 COSEBIs modes. In the right panel only the n ≤ 5 modes are considered. 

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Fig. C.3.
Distribution of pvalues for the 10 000 data samples, showing the probability that the M_{PSF} model (blue histogram) or the M_{0} no systematics model (orange histogram) is the true underlying model, given each data sample. Left panel: pvalues from an analysis of the n ≤ 20 COSEBIs modes. In the right panel only the n ≤ 5 modes are considered. 

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We can determine which of the two models best represents the data using a Bayesian evidence analysis. If we give the same weight to both models then the ratio of the Bayesian evidences for these models is given by the Bayes factor,
where D is the data, M_{i} is model i and Φ_{i} represents the set of parameters for model i. For the simplified case of parameter free models that we consider here, Eq. (C.1) simplifies to,
The resulting Bayes factor will however depend on the number of nmodes that are included in the analysis. The Bayesian evidence can only be used when an alternative model exists, but in the case of null tests, such as a Bmode test, the only available model is the null hypothesis and therefore we need to use classical methods to identify the significance of the Bmodes. Here we use χ^{2} and pvalues to test the null hypothesis. The pvalue for the χ^{2} is defined as the probability of calculating a χ^{2} value larger than the measured one, , given the model M,
Figure C.2 shows the distribution of the measured χ^{2} across our 10 000 random samples when the data are fit using the input M_{PSF} model (blue histogram) and the M_{0} no systematics model (orange histogram). In the left panel we take the nulltest case where all modes up to n = 20 are included in the analysis (Allmodes). In the right panel, only the first 5 modes (n < 6) are analysed. As M_{PSF} is the correct model, we naturally find better fits to the data, i.e. lower χ^{2} values, for this model. The difference between the two distributions for the χ^{2} values is however enhanced when the modes analysed are limited to the range where the two models differ significantly. This means that the power of the null test is optimised over this reduced, n ≤ 5, range.
Figure C.3 shows the distribution of pvalues for the χ^{2} values shown in Fig. C.2. If the model used to fit the data is the true underlying model, any particular pvalue is as likely to be measured as the other. If the model is not representative of the data, however, then one is more likely to obtain smaller pvalues from the sample. As expected with M_{PSF} as the correct model, we find a uniform distribution of pvalues and a skewed distribution for the M_{0} model. When all 20 COSEBIs modes are included this pvalue distribution is less skewed compared to when we only include the n ≤ 5 modes. By adding more data points to the analysis, we have diluted the systematic signal of the PSF leakage, making this nulltest less effective.
Based on this analysis, we must recognise that finding that the Bmodes pass a nulltest using a large data vector does not ensure that analysing a smaller dataset will give the same result. A good example of this is KiDS450 passing the 20mode COSEBIs nulltest, but failing the CCOSEBIs nulltest which is most sensitive to the n ≤ 5 modes. In contrast DESSV and CFHTLenS fail the 20mode COSEBIs nulltest, even though they pass the CCOSEBIs nulltest. Their Bmodes therefore appear when adding in more data points to the analysis. As our example shows how increasing the size of your data set serves to reduce the stringency of the nulltest, we can therefore conclude that the significant DESSV and CFHTLenS Bmode, seen with COSEBIs and not with CCOSEBIs, is present in the highn data that is not included in the CCOSEBIs analysis. If we had only performed a COSEBIs nulltest, we would have missed the presence of a systematic signal in KiDS. If we had only performed a lown CCOSEBIs nulltest, we would have missed the presence of a systematic signal in DESSV and CFHTLenS.
To illustrate our discussion of nulltests we have used COSEBIs, but the concept holds for any statistic or nulltest. If a systematic produces a feature at a particular scale, but is otherwise identical to the standard model, by adding data from other scales we will dilute the power of the statistical test to distinguish between the two cases. As null Bmode tests are generally performed independently of alternative models, it is not clear which data points should be added to the nulltest analysis. We therefore propose that future nulltests are performed with the Bmode signatures shown in Sect. 6 in mind. In this way one can optimise the modes over which to carry out a model comparison.
Appendix D: Supplementary data and figures
Figures D.1–D.3 show the tomographic COSEBIs measurements, using the angular range of [0.5′,100′], for DESSV, KiDS450 and CFHTLenS respectively. In each figure, the upper panels present the Emodes, the lower panels present the Bmodes, and the significance of the Bmodes are indicated with a pvalue shown in the upper left corner. The pvalues for the other two angular ranges analysed are given in Table 2. The predicted Emodes, given the bestfitting cosmology parameters listed in Table 1, are shown as curves.
Figures D.4–D.6 show ξ_{E/B} for the tomographic cases for DESSV, KiDS450 and CFHTLenS respectively. We show pvalues for the significance of the Bmodes in each figure, but caution the reader that due to binning and the truncated integrals discussed in Sect. 4, this method is not robust. However, as ξ_{B} data points are uncorrelated, they can help with identifying the source of the systematic even though it was seen in Sect. 6 that systematics do not always affect the same angular ranges for E and Bmodes. The prediction for ξ_{E}, given the bestfitting cosmology parameters listed in Table 1, is shown as curves.
For DESSV, we note that the significance of the tomographic ξ_{B} signal significantly decreases when we restrict the analysis to an angular range of [4.2′,72′], as adopted by Dark Energy Survey Collaboration (2016), with the pvalue increasing from p = 4 × 10^{−19} to p = 0.012. If the systematics that source the Bmodes detected in the standard [0.5′,100′] analysis add equally to the E and B modes, then the chosen DESSV angular selection would serve to mitigate the impact of these systematics. As shown in Sect. 6, however, we find that the range of tested systematics exhibit different E and B mode responses. We would therefore caution against concluding that a choice selection of angular scales, based on the Bmode response, is sufficient to remove the systematic contamination to the Emodes within those chosen scales.
In Table D.1 we list the bestfitting values of the calibration parameters for DESSV used to calculate the Emode predictions for DESSV shown in Sect. 4 and this Appendix. The first row shows the value of the multiplicative shear calibration bias and the second row the additive photometric redshift bias for redshift bins one to three. The last column shows the values we adopted for the single bin case, which was not analysed in Abbott et al. (2016). For this case we use a multiplicative shear calibration equal to the first redshift bin value and a vanishing photometric redshift bias.
Table D.2 lists the fitted values for a_{x} and b_{x} to KiDS450 multiband data using Eq. (28). This values are used to produce a correlation between the measured ellipticity of galaxies that are binned in photometric redshift bins with their local PSF ellipticity (see Sects. 5.2 and 6.3).
In Fig. D.7 we replot the left hand side of Fig. 3 to show the difference between the measured and the fiducial Emodes, , for a single redshift bin, while keeping the right hand side as it was since Bmodes are expected to be consistent with zero. This figure has a similar format to Fig. 10, where we showed the effect of systematics on simulations. When comparing these figures note that in Fig. D.7 is not the input theory, but instead it is the result of a fit to the data which is inevitably affected by any systematics that may exist in the data. In contrast in Fig. 10 we know the correct values for the , as they are measured directly from the simulations before the systematic effect is applied to them.
Fig. D.1.
Tomographic E/B mode COSEBIs analysis of DESSV, spanning an angular range of [0.5′,100′]. Each panel shows the COSEBIs modes for the tomographic redshift bin pair z − ij, corresponding to the correlation between photometric redshift bins i and j (see Sect. 3 for the definition of the redshift bins). The Emodes (upper right panel) can be compared to the theoretical expectation given by the cosmological parameters listed in Table 1. Note that COSEBIs modes are discrete and we only connect the theoretical model in a curve as a visual aid. The Bmodes (lower left panel) can be compared to the nullcase (dashed) where the reduced χ^{2} value for the Bmodes being equal to zero is given, for the autocorrelation cases, in their corresponding panels. The reduced χ^{2} and pvalue of the full data vector is listed in the upper left. 

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Fig. D.2.
Tomographic E/B mode COSEBIs analysis of KiDS450. See the caption of Fig. D.1 for details. 

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Fig. D.3.
Tomographic E/B mode COSEBIs analysis of CFHTLenS. See the caption of Fig. D.1 for details. 

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Best fitting shear calibration and photoz biases for different redshift bins in the DESSV data (from private communication with Joe Zuntz).
Best fitting values for a_{x} and b_{x} as defined in Eq. (28), fitted to KiDS450 data.
Fig. D.4.
Tomographic ξ_{E/B} analysis of DESSV. Each panel shows ξ_{E} (diamonds) and ξ_{B} (pluses) for the tomographic redshift bin pair z − ij, corresponding to the correlation between photometric redshift bins i and j (see Sect. 3 for the definition of the redshift bins). The Emodes, ξ_{E}, can be compared to the theoretical expectation given by the cosmological parameters listed in Table 1. The Bmodes, ξ_{B}, can be compared to the nullcase (dashed) where, for the autocorrelation cases, the reduced χ^{2} value for the Bmodes being equal to zero is listed. The reduced χ^{2} and pvalue of the full data vector is listed in the lower left. 

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Fig. D.5.
Tomographic ξ_{E/B} analysis of KiDS450. See the caption of Fig. D.4 for details. 

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Fig. D.6.
Tomographic ξ_{E/B} analysis of CFHTLenS. See the caption of Fig. D.4 for details. 

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Fig. D.7.
Difference between COSEBIs Emodes and their theory value (left) and COSEBIs Bmodes (right) for a single broad redshift bin. The only difference between this figure and Fig. 3, is the left column. Results for DESSV are shown with blue squares, KiDS450 with black stars and CFHTLenS with magenta triangles. The angular ranges are shown for each row in the upper right corner. In addition, the significance of the Bmodes is shown as pvalues for each survey and angular range. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). Note that COSEBIs modes are discrete and the theory values are connected to each other only as a visual aid. A zeroline is also shown for reference. 

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All Tables
Published bestfitting cosmological parameters for the surveys (KiDS450, CFHTLenS, and DESSV: Hildebrandt et al. 2017; Heymans et al. 2013; Abbott et al. 2016), and the simulation (SLICS, HarnoisDéraps et al. 2018), that we use in this paper.
Probability of zero Bmode contamination for each survey, given the measured COSEBIs Bmodes.
Best fitting shear calibration and photoz biases for different redshift bins in the DESSV data (from private communication with Joe Zuntz).
Best fitting values for a_{x} and b_{x} as defined in Eq. (28), fitted to KiDS450 data.
All Figures
Fig. 1.
LogCOSEBIs filter functions, T_{±n}(θ). These filter functions convert ξ_{±} to COSEBIs E and B modes through Eqs. (11) and (12). We show four example nmodes for each filter for the angular separation range of [0.5′,100′]. By definition T_{±n}(θ) are equal to zero outside of the range of their support. 

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In the text 
Fig. 2.
LogCOSEBIs weight functions, W_{n}(ℓ), normalized to their maximum value. These weight functions convert E and B shear power spectra to COSEBIs modes through Eq. (13). Four example nmodes are shown for the angular range of [0.5′,100′]. 

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In the text 
Fig. 3.
COSEBIs Emodes (left) and Bmodes (right) for a single broad redshift bin. Results for DESSV are shown with blue squares, KiDS450 with black stars and CFHTLenS with magenta triangles. The angular ranges are shown for each row in the upper right corner. In addition, the significance of the Bmodes is shown as pvalues for each survey and angular range. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). Note that COSEBIs modes are discrete and the theory values are connected to each other only as a visual aid. A zeroline is also shown for reference. 

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In the text 
Fig. 4.
CCOSEBIs E and Bmodes for nontomographic (left) and tomographic (right) analyses. The Emodes are shown as empty symbols, with the Bmodes shown as filled symbols, for DESSV (blue squares), KiDS450 (black stars) and CFHTLenS (magenta triangles). The analysis is conducted over three different angular ranges, denoted in the upper right corner of each panel. The CCOSEBIs mode is indicated on the horizontalaxis. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). A zeroline is also shown for reference. 

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In the text 
Fig. 5.
ξ_{E} and ξ_{B}E/Bmodes for a single broad redshift bin. The Emodes are shown as empty symbols, with the Bmodes shown as filled symbols, for DESSV (blue squares), KiDS450 (black stars) and CFHTLenS (magenta triangles). The DESSV and CFHTLenS results are horizontally offset relative to KiDS450 to aid visualisation. Emode predictions for ξ_{E} are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). A zeroline is also shown for reference. We detect significant Bmodes in all cases as shown by the pvalues, in the legend, which determine the probability of the data Bmodes given a null Bmode model. 

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In the text 
Fig. 6.
First ellipticity component of the spatially varying systematic effects, simulated over a 10 ° ×10° field. Here the effects are normalized to their maximum value for a better visual comparison. From the left, the first panel shows the point spread function pattern used to model PSF leakage (). The second panel shows a regular pattern using the detector chip bias model from OmegaCam multiplied by a factor of 5 (0.001 < c_{1} < 0.025). The third panel shows the random correlated noise PSF residual model with a smoothing length similar to the chip size (), while the last panel shows the same systematic for a roughly pointing size smoothing length (). 

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In the text 
Fig. 7.
True redshift distribution of the mock galaxies, separated into photometric redshift bins. The z_{B} selection is shown in the legend. The cyan histogram shows the true redshift distribution of the galaxies in the parent noisefree sample. In order to determine the photometric redshifts we introduce flux errors that mimic a KiDSlike survey and depend on the relative ellipticity of the mock galaxy to the mock PSF. 

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In the text 
Fig. 8.
SLICS 2point statistics, ξ_{±} and ξ_{E/B} (left) and Emode COSEBIs (right), averaged over 10 noisefree linesofsight, which serves as our fiducial “systematicsfree” measurement. The mean result can be compared to the theoretical expectation (smooth solid curves). For ξ_{±} and COSEBIs we also show the measurements for each individual lineofsight with thin solid curves with matching colors between different panels. The upper left panel shows the measured ξ_{+} (magenta squares) and ξ_{E} (blue diamonds), with the lower left panel showing ξ_{−} (green pluses) and ξ_{B} (black crosses). The expectation value for ξ_{B} is zero, shown with the dashed black line. The COSEBIs Emodes (right panels) are shown for the three angular ranges indicated in each row. The COSEBIs Bmodes in SLICS are 4 orders of magnitude smaller than the Emodes and are therefore not shown. The measurements are shown as squares and their expected theory value as plus symbols. Note that COSEBIs modes are discrete and the points are only connected together as a visual aid. 

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In the text 
Fig. 9.
Impact of shear measurement systematics on ξ_{±} and ξ_{E/B} for four different types of shear measurement systematics; From the top down: PSF leakage, a repeating additive pattern, and random but correlated noise, correlated on chip and pointing scales (see Fig. 6). For ξ_{+} (magenta squares), ξ_{−} (green pluses) and ξ_{E} (blue diamonds) we present, in the left panels, the fractional difference between the measured signal in the systematicinduced KiDSlike SLICS mocks and the fiducial systematicfree case. As ξ_{B} (black crosses) and the E/B difference ξ_{E} − ξ_{B} (red pluses) tends to zero, we present, in the right panels, the difference between these measurements and the fiducial case, multiplied by the angular distance in arcminutes and scaled by 10^{4}. The measured Bmodes can be compared to the expected shapenoise error for KiDS450 (shaded area). 

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In the text 
Fig. 10.
Impact of shear measurement systematics on Emode (left) and Bmode (right) COSEBIs for four different types of shear measurement systematics; PSF leakage (blue squares), a repeating additive pattern (black stars), and random but correlated noise on chip (green triangles) and pointing (magenta diamonds) scales (see Fig. 6). The analysis is conducted for three different angular ranges spanning [0.5′,40′] (upper panels), [0.5′,100′] (middle panel), and [40′,100′] (lower panels). We present the difference between the measured signal in the systematicinduced KiDSlike SLICS mocks and the fiducial systematicfree case scaled by 10^{10}. The measured Bmodes and the resulting change to the Emode can be compared to the expected shapenoise error for KiDS450 (shaded area). 

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In the text 
Fig. 11.
Photometric redshift selection bias: the upper panels show the correlations measured from a random intrinsic ellipticity field using the full galaxy sample. Lower panels: correlations measured from the same random intrinsic ellipticity field after a 0.1 ≤ z_{phot} ≤ 0.9 photometric redshift selection has been applied. The 2PCFs shown on the left are ξ_{+} (squares) and ξ_{−} (pluses). On the right panel, ξ_{E} (diamonds), ξ_{B} (crosses) and the difference between the two (pluses) are shown. The onesigma error bars correspond to the level of ellipticity noise in the mock data. The signals are shown multiplied by θ in arcminutes and scaled by 10^{4}. 

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In the text 
Fig. 12.
Impact of photometric redshift selection bias for a COSEBIs 4bin tomographic analysis of a random intrinsic ellipticity field. The upper triangle shows the Emodes and the lower triangle shows the Bmodes, where the error bars in both cases correspond to the level of ellipticity noise in the mocks. Each panel shows COSEBIs for a tomographic redshift bin pair, z − ij, corresponding to the correlation between photometric redshift bins i and j. As our photometric redshift mocks are devoid of any cosmological correlations, in the absence of any selection bias, we would expect both the E and B modes to be consistent with zero. 

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In the text 
Fig. 13.
Left panel: inferred values of Σ_{8} = σ_{8}(Ω_{m}/0.3)^{0.65} from an Emode CCOSEBIs analysis of four mock cosmic shear surveys that suffer from PSF leakage (blue squares), a repeating additive pattern (black stars), and random but correlated noise on chip (green triangles) and pointing (magenta diamond) scales. The curves show the theoretical values of the 5 Emode CCOSEBIs when varying Σ_{8}, calculated using the KiDS450 noise only covariance matrix and redshift distribution; E_{σ8σ8} (dashed), E_{ΩmΩm} (dotted), E_{σ8} (middle solid), E_{Ωm} (dotdashed) and E_{σ8Ωm} (lower solid). The inferred cosmology for each mock systematic survey can be compared between the 5 different modes. The higher the recovered Emode is relative to the fiducial “nosystematics case” (circles), the stronger the bias is on the inferred value of Σ_{8} and the more discrepant the inferred cosmology is between the different CCOSEBIs modes. The bias in Σ_{8} can be compared to the grey region which shows the onesigma error for Σ_{8} from the KiDS450 cosmic shear analysis, centred on the fiducial case. Right panel:Bmode CCOSEBIs from the four mock cosmic shear surveys. The measured Bmode signal, which does not depend on Σ_{8}, can be compared to the shape noise on a KiDS450like survey (shown in grey). 

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In the text 
Fig. A.1.
2PCF binning bias introduced for a range of analysis choices, shown as the ratio between the measured and their proposed theoretical value as a function of angular scale, θ. The legends in this figure are shared between the two panels. For the weighted broad bin estimator, , the bias is calculated assuming θ_{b} is given by the logarithmic midpoint of the bin, θ_{mid, log} (green squares), the weighted mean of the bin, θ_{w} (black pluses, under the blue diamonds), the geometric mean or linear midpoint of the bin, θ_{mid, lin} (dark grey triangles), or an areaweighted bin centre, θ_{area} (blue diamonds). These estimators can be compared to the fine binning case, , where the theory is estimated at the weighted mean of each bin (light grey circles), and the exact case (red crosses) where the theoretical value is calculated as a weighted integral over the signal within the bin (Eq. (A.7)). All points are plotted with errorbars, but in the case of broad binning the errors are too small to be visible. 

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In the text 
Fig. B.1.
Degeneracy direction of σ_{8} and Ω_{m} for a CCOSEBIs analysis of the KiDSlike data. The colours in the image show the value of the CCOSEBIs E_{σ8} mode, in comparison to dashed lines of constant Σ_{8} = σ_{8}(Ω_{m}/0.3)^{α} with α = 0.65. The repeating color scheme was chosen to capture the variations in the values of E_{σ8}. The lower left corner has the smallest value of E_{σ8} which gradually increases, perpendicular to the dashed curves, towards the upper right corner. 

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In the text 
Fig. C.1.
Model comparison using 10 000 random samples of the PSFleakage systematic model, M_{PSF} (solid) defined in Sect. 5.1.1. The data points show the mean of the random samples, with the errors reflecting the noise on a single realisation. The data samples are analysed to determine how often the input model M_{PSF} can be distinguished from the nosystematics zero Bmode model, M_{0} (dashed). 

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In the text 
Fig. C.2.
χ^{2} distribution of the mock data given the true PSF leakage model, M_{PSF}, (blue histogram) or given the null model, M_{0}, (orange histogram). Left panel: analysis of the n ≤ 20 COSEBIs modes. In the right panel only the n ≤ 5 modes are considered. 

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In the text 
Fig. C.3.
Distribution of pvalues for the 10 000 data samples, showing the probability that the M_{PSF} model (blue histogram) or the M_{0} no systematics model (orange histogram) is the true underlying model, given each data sample. Left panel: pvalues from an analysis of the n ≤ 20 COSEBIs modes. In the right panel only the n ≤ 5 modes are considered. 

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In the text 
Fig. D.1.
Tomographic E/B mode COSEBIs analysis of DESSV, spanning an angular range of [0.5′,100′]. Each panel shows the COSEBIs modes for the tomographic redshift bin pair z − ij, corresponding to the correlation between photometric redshift bins i and j (see Sect. 3 for the definition of the redshift bins). The Emodes (upper right panel) can be compared to the theoretical expectation given by the cosmological parameters listed in Table 1. Note that COSEBIs modes are discrete and we only connect the theoretical model in a curve as a visual aid. The Bmodes (lower left panel) can be compared to the nullcase (dashed) where the reduced χ^{2} value for the Bmodes being equal to zero is given, for the autocorrelation cases, in their corresponding panels. The reduced χ^{2} and pvalue of the full data vector is listed in the upper left. 

Open with DEXTER  
In the text 
Fig. D.2.
Tomographic E/B mode COSEBIs analysis of KiDS450. See the caption of Fig. D.1 for details. 

Open with DEXTER  
In the text 
Fig. D.3.
Tomographic E/B mode COSEBIs analysis of CFHTLenS. See the caption of Fig. D.1 for details. 

Open with DEXTER  
In the text 
Fig. D.4.
Tomographic ξ_{E/B} analysis of DESSV. Each panel shows ξ_{E} (diamonds) and ξ_{B} (pluses) for the tomographic redshift bin pair z − ij, corresponding to the correlation between photometric redshift bins i and j (see Sect. 3 for the definition of the redshift bins). The Emodes, ξ_{E}, can be compared to the theoretical expectation given by the cosmological parameters listed in Table 1. The Bmodes, ξ_{B}, can be compared to the nullcase (dashed) where, for the autocorrelation cases, the reduced χ^{2} value for the Bmodes being equal to zero is listed. The reduced χ^{2} and pvalue of the full data vector is listed in the lower left. 

Open with DEXTER  
In the text 
Fig. D.5.
Tomographic ξ_{E/B} analysis of KiDS450. See the caption of Fig. D.4 for details. 

Open with DEXTER  
In the text 
Fig. D.6.
Tomographic ξ_{E/B} analysis of CFHTLenS. See the caption of Fig. D.4 for details. 

Open with DEXTER  
In the text 
Fig. D.7.
Difference between COSEBIs Emodes and their theory value (left) and COSEBIs Bmodes (right) for a single broad redshift bin. The only difference between this figure and Fig. 3, is the left column. Results for DESSV are shown with blue squares, KiDS450 with black stars and CFHTLenS with magenta triangles. The angular ranges are shown for each row in the upper right corner. In addition, the significance of the Bmodes is shown as pvalues for each survey and angular range. Emode predictions are calculated using the best fitting cosmological parameter values given in Table 1 for DESSV (solid), KiDS450 (dashed) and CFHTLenS (dotted). Note that COSEBIs modes are discrete and the theory values are connected to each other only as a visual aid. A zeroline is also shown for reference. 

Open with DEXTER  
In the text 
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