Issue 
A&A
Volume 638, June 2020



Article Number  A96  
Number of page(s)  15  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201936915  
Published online  19 June 2020 
The importance of magnification effects in galaxygalaxy lensing
^{1}
ArgelanderInstitut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany
email: sandra@astro.unibonn.de
^{2}
Exzellenzcluster Universe, Boltzmannstr. 2, 85748 Garching, Germany
^{3}
LudwigMaximiliansUniversität, UniversitätsSternwarte, Scheinerstr. 1, 81679 München, Germany
^{4}
PhysikDepartment, Technische Universität München, 85748 Garching, Germany
Received:
14
October
2019
Accepted:
16
April
2020
Magnification changes the observed local number density of galaxies on the sky. This biases the observed tangential shear profiles around galaxies: the socalled galaxygalaxy lensing (GGL) signal. Inference of physical quantities, such as the mean mass profile of halos around galaxies, are correspondingly affected by magnification effects. We used simulated shear and galaxy data from the Millennium Simulation to quantify the effect on shear and mass estimates from the magnified lens and source number counts. The former is due to the largescale matter distribution in the foreground of the lenses; the latter is caused by magnification of the source population by the matter associated with the lenses. The GGL signal is calculated from the simulations by an efficient fast Fourier transform, which can also be applied to real data. The numerical treatment is complemented by a leadingorder analytical description of the magnification effects, which is shown to fit the numerical shear data well. We find the magnification effect is strongest for steep galaxy luminosity functions and high redshifts. For a KiDS+VIKING+GAMAlike survey with lens galaxies at redshift z_{d} = 0.36 and source galaxies in the last three redshift bins with a mean redshift of _{¯}z_{s} = 0.79, the magnification correction changes the shear profile up to 2%, and the mass is biased by up to 8%. We further considered an even higher redshift fiducial lens sample at z_{d} = 0.83, with a limited magnitude of 22 mag in the rband and a source redshift of z_{s} = 0.99. Through this, we find that a magnification correction changes the shear profile up to 45% and that the mass is biased by up to 55%. As expected, the sign of the bias depends on the local slope of the lens luminosity function α_{d}, where the mass is biased low for α_{d} < 1 and biased high for α_{d} > 1. While the magnification effect of sources is rarely more than 1% of the measured GGL signal, the statistical power of future weak lensing surveys warrants correction for this effect.
Key words: gravitational lensing: weak / methods: analytical / methods: numerical
© ESO 2020
1. Introduction
Gravitational lensing is a powerful tool in unveiling the true distribution of matter in the Universe and probing cosmological parameters (see, e.g. Kilbinger 2015 for a recent review). The lensing signal is sensitive to all matter, regardless of its nature, and is observed as the distortion of light bundles travelling through the Universe. In the weak lensing regime, this distortion is small and must be studied with large statistical samples. Therefore, large and deep surveys are required, for example, the Kilo Degree Survey^{1} (KiDS); the Dark Energy Survey^{2} (DES); the Hyper SuprimeCam Subaru Strategic Program^{3} (HSC SSP); or the nearfuture surveys with Euclid^{4} and the Large Synoptic Survey Telescope^{5} (LSST). To maximise the scientific output from these surveys, the scientific community is currently putting great efforts into understanding nuances in the theoretical framework.
Galaxygalaxy lensing (GGL) correlates the position of foreground galaxies to the distortion of background galaxies (see, e.g. Hoekstra 2013). The distortion is typically measured in terms of mean tangential shear with respect to the lens position. This shear signal, as a function of separation from the lens centre, can be related to the underlying mass properties of the parent halo. A major challenge is obtaining an unbiased mass estimate. Biases arise if the underlying model does not describe all contributions to the mattershear correlation function sufficiently. Also, the galaxy bias that connects the position of galaxies to its surrounding matter distribution must be carefully taken into account. For current and future surveys, we must further consider secondorder effects to the galaxymatter correlation, such as, for example, magnification effects (Ziour & Hui 2008; Hilbert et al. 2009) and intrinsic alignment of galaxies (Troxel & Ishak 2015). In this work, we focus on the former effect.
Magnification is the change of the observed solid angle of an image compared to the intrinsic solid angle, or, since the surface brightness remains constant, the ratio of observed flux to the intrinsic one. Like the shear, it is a local quantity, a direct prediction of the lensing formalism, and is caused by all matter between the observed galaxy population and us. However, direct measurements of magnification are challenging because the intrinsic flux is typically unknown. Yet, the change in size and magnitude results in a changed spatial distribution of the galaxy population. This socalled number count magnification has been measured (e.g. Chiu et al. 2016; GarciaFernandez et al. 2018). Consequently, magnification by the largescale structure (LSS) also changes the GGL signal compared to a signal that is just given by matter correlated with the lens galaxies. We stress that the magnification changes the number counts of the source as well as lens galaxies on the sky. The impact of magnification of the lens galaxies on the GGL signal for surveys like CFHTLenS is ∼5% (Simon & Hilbert 2018), but can be as large as 20% for other lens samples (Hilbert et al. 2009). Although these results suggest a fairly large impact of magnification on GGL lensing estimates, quantitative analyses widely neglect the influence of magnification. Unruh et al. (2019) studied the impact of the number count magnification of lens galaxies on the shearratio. They found that the shearratio test (Jain & Taylor 2003) is affected by lens magnification and that its effect must be mitigated, especially for highlens redshifts.
In this paper, we quantify the impact of magnification on observed tangential shear profiles and halomass estimates from GGL. We consider both the effect of magnification of the sources by the lenses, as well as the effect of magnification of the lenses by the LSS. For this, we compared the GGL signal with and without magnification using simulated data. We then derived mean halo masses in both cases employing a halo model to quantify the expected mass bias. We complemented the numerical results with analytic estimates of the effects.
This article is organised as follows. The theoretical framework is briefly described in Sect. 2. Section 3 features an analytical description of how magnification affects shear estimates, as well as a brief discussion of how numerical results were obtained in this study. The numerical procedure is then more thoroughly explained in Appendix A. In Sect. 4, we discuss the impact of magnification of source galaxies by the lenses, and in Sect. 5 we discuss magnification of the lens galaxies by the LSS. We study the magnification bias on mass estimates in Sect. 6 and conclude in Sect. 7.
2. Theory
In the following, we introduce the theoretical concepts of gravitational lensing for this work. For a more general and extensive overview, the reader is kindly referred to Bartelmann & Schneider (2001).
2.1. Cosmological distances
For a flat universe, the Hubble parameter H(z) can be written as
where z is the redshift, H_{0} denotes the Hubble constant, and Ω_{m} is the matter density in units of today’s critical density ; with the vacuum speed of light c and the gravitational constant G. The comoving distance travelled by a photon between redshift z_{1} and z_{2} reads
and the angulardiameter distance is
For a redshift z_{1} = 0, which is the observer’s position, we write D(0, z) = : D(z). In addition, the dimensionless Hubble parameter h is used to parametrise our ignorance about the true value of today’s Hubble parameter, defined as H_{0} = 100 h km s^{−1} Mpc^{−1}. In the following, all distances are angulardiameter distances.
2.2. Gravitational lensing distortions and magnification
Gravitational lensing distorts the appearance of galaxy images. In the weak lensing regime, this distortion can locally be described as a linear mapping from the background (source) plane to the foreground (lens) plane. The Jacobian 𝒜 of the local mapping can be written as
where κ is the convergence, γ_{1, 2} are the two Cartesian shear components, and g_{1, 2} = γ_{1, 2}/(1 − κ) are the two Cartesian reduced shear components, which all depend on the position in the lens plane. The convergence causes an isotropic scaling of the galaxy image, while the shear leads to an anisotropic stretching, and thus causes an initially circular object to appear elliptical.
This scaling of the galaxy image changes the apparent solid angle ω of the image, compared to one in the absence of lensing, which we denote by ω_{0}. Likewise, the flux is affected by gravitational lensing, the unlensed flux s_{0} is enhanced or reduced to the observed flux s. The ratio of these quantities defines the magnification μ and can also be calculated from the Jacobian by
Magnification changes the observed local number density of galaxies on the sky. The cumulative observed number density of galaxies on the sky n(> s), brighter than flux s, is locally
where n_{0} denotes the cumulative number density in absence of lensing. The prefactor 1/μ is due to the scaling of the solid angle. The flux in the argument of n_{0} must also be scaled by 1/μ to account for the flux enhancement or reduction.
Magnification effects in the weak lensing limit are small, specifically μ − 1 ≪ 1, and we Taylor expand Eq. (6) in (μ − 1) to obtain to first order
where the exponent α is the local slope at the flux limit s_{lim}, it is defined as
For α > 1, the galaxy counts are enhanced, and for α < 1 they are depleted. In the case of α = 1, no magnification bias is present.
2.3. Galaxygalaxy lensing
In GGL, the positions of foreground galaxies (lenses) are correlated with the shear of background galaxies (sources). For a position θ, the complex shear is written as γ(θ) = γ_{1}(θ)+iγ_{2}(θ). The tangential shear γ_{t} and the cross shear γ_{×} at source position θ_{s} for a given lens at position θ_{d} are
where an asterisk denotes complex conjugation. The GGL signal ⟨γ_{t}⟩(θ) is defined as the correlator between the positions of foreground galaxies and the tangential shear,
where κ_{g}(θ) is the fractional numberdensity contrast of foreground lens galaxies on the sky. The corresponding correlator for the crosscomponent of the shear is expected to vanish, due to parity invariance.
A practical estimator for the GGL signal averages the tangential and cross shear over many lenssource pairs in bins of separation θ:
Here, denotes the position of the ith source, denotes the position of the jth lens, and the binning function Δ(θ, θ′) is unity if θ′ falls into the corresponding θ bin, and zero if it does not.
3. Magnification effects in GGL
In this section, we consider the effect of magnification on the GGL signal. As we show, magnification of sources and lenses leads to a bias of the estimator (11), which is a function of limiting magnitudes for the lens and source population, as well as their redshifts, since these determine the local slope (8) at the limiting magnitude. Magnification is typically assumed to be a minor effect in GGL measurements, and most theoretical predictions do not account for it. While the impact of the magnification of lenses has already received some attention (e.g. Ziour & Hui 2008; Hartlap 2009), the source magnification is less well known.
3.1. Magnification of lenses by largescale structure
Magnification, caused by the LSS between us and the lenses, changes the number density of the lens galaxy sample, while simultaneously inducing a shear on background galaxies. Thus, the observed shear signal differs from what is typically considered as the GGL signal, which is a correlation of lens galaxy positions and the shear on background galaxies (Eq. (11)). In this correlation, the lens galaxies are connected by the galaxy bias to their surrounding matter, which induces a shear in the background galaxies. Magnification by intervening matter structures alters this rather simple picture. Since for larger lens redshifts more intervening matter is present, the impact of magnification effects grows with increasing redshift. On the other hand, the impact is reduced with increasing lineofsight separations of lenses and sources. In the following, we consider a lowestorder correction for the magnification of the lenses by the LSS for the GGL signal of a flux or volumelimited lens sample (see, e.g. Ziour & Hui 2008; Hartlap 2009; Thiele et al. 2020). We stress that this correction ignores the magnification of sources, which is treated in the next subsection.
In the weak lensing regime, we can approximate the magnification by μ ≈ 1 + 2κ, valid if κ ≪ 1, γ ≪ 1. Then, the number count magnification (7) of the observed number density of lenses n_{d}(θ) at redshift z_{d} on the sky is, for a fluxlimited sample,
where n_{d, 0} denotes the lens number density without magnification, denotes the mean lens number density, α_{d} denotes the local slope of the lenses at the limiting magnitude, and κ^{LSS} denotes the convergence due to matter structures between us and the lenses. Thus, in the presence of magnification, the expected signal is modified to
where denotes the tangential shear signal without magnification, and the LSS shear signal is
which is shown explicitly in Hartlap (2009) and Simon & Hilbert (2018). We set D(z_{s}) = D_{s} and D(z_{d}) = D_{d}, and by J_{n}(x) we denote the nthorder Bessel function of the first kind. In this work, we use the revised Halofit model (Takahashi et al. 2012) for the spatial matter powerspectrum P_{m}(k; z) at wavenumber k and redshift z. The argument in the matter power spectrum arises through the application of the wideangle corrected Limber projection, which was recently put forward by Kilbinger et al. (2017); the denominator is the comoving angulardiameter distance f_{k}(z) = (1 + z)D(z) at redshift z. The corresponding expression of Eq. (13) in the absence of a flux limit, meaning for a volumelimited lens sample, can be obtained by setting α_{d} = 0.
For α_{d} < 1, the magnification by the LSS suppresses the GGL signal. For α_{d} = 1, the magnification effect vanishes. For α_{d} > 1, the LSS contribution enhances the GGL signal.
3.2. Magnification of sources by lenses
Galaxies are correlated with the mass distribution, and thus the location of the galaxies correlates with the magnification induced on the background sources. This implies that the number density of sources is correlated with the positions of the lens galaxies. Assuming for a moment that the numbercount slope of sources α_{s} is larger than unity, one expects that the number density of sources is more enhanced close to lens galaxies living in a dense environment. The estimator (11) therefore contains a disproportionately high number of lenssource pairs for those lenses living in a dense environment compared to those located in less dense regions.
The expected number density of sources is
where in the second step we used the firstorder Taylor expansion leading to Eq. (7), and in the last step we again made the weak lensing approximation μ ≈ 1 + 2κ.
The expectation value of the estimator (11) of the GGL signal is therefore affected by the local change of the source number density and becomes
Thus, in the case of small magnifications, the bias is given by a thirdorder crosscorrelation between the number density of foreground (lens) galaxies and the shear and convergence experienced by the background galaxies. This correlation is caused by the lensing effect of matter associated with the lens galaxies; hence, the bias is caused by magnification of sources by the matter at z_{d}. Equation (16) ignores the effect of intervening matter since this is subdominant for the source galaxy sample. Given that the bias term differs from the GGL signal by one order in the convergence, and that the characteristic convergence dispersion is of order 10^{−2}, we expect that magnification of sources biases the GGL signal at the level of ∼1%.
Interestingly, the thirdorder correlator in the final expression of Eq. (16) is related to the galaxyshearshear correlator that was introduced by Schneider & Watts (2005) as one of the G_{±} quantities of galaxygalaxygalaxy lensing, since κ and γ are linearly related. Thus, from measurements of the galaxyshearshear correlations in a survey, this bias term can be directly estimated. We note that such measurements have already been successfully conducted (e.g. Simon et al. 2008, 2013). A more quantitative description of this correction, which will be relevant for precision GGL studies in forthcoming surveys like Euclid and LSST, is beyond the scope of this paper and will be done at a later stage.
An approximate, more intuitive way of describing the magnification of sources by lenses is provided by assuming that each lens is located at the centre of a halo of mass m. In the case of no magnification, the expected tangential shear signal can be expressed as^{6}
for a population of sources with redshift distribution p_{zs}(z_{s}), a population of lenses with redshift distribution p_{zd}(z_{d}), and a conditional distribution p_{mzd}(mz_{d}) of the masses m of the halos in which the lens galaxies reside. The mean tangential shear profile γ_{t}(θm, z_{d}, z_{s}) for lenses with halo mass m at redshift z_{d} and sources at redshift z_{s} can be factorised,
where D_{ds} = D(z_{d}, z_{s}), and γ_{∞} is the mean shear profile for (hypothetical) sources at infinite distance.
Equation (17) assumes that the source number density is statistically independent of the lens positions. However, the observed number density of sources may change behind lenses due to magnification by the lenses. The expected magnification is a function of angular separation, the source and lens redshift, and the lens halo mass. For a fluxlimited sample, the expected shear signal (17) then changes to:
where μ(θm, z_{d}, z_{s}) denotes the mean magnification of sources at redshift z_{s} and separation θ by lenses at redshift z_{d} with halo mass m, and α_{s}(z_{s}) denotes the slope of the source counts at redshift z_{s} at the source flux limit as in Eq. (8). We obtained the corresponding expression for a volumelimited source sample by replacing α_{s} by zero in Eq. (19).
We may also assume that γ_{t}(θm, z_{d}, z_{s}) and μ(θm, z_{d}, z_{s}) are larger when the halo mass m is larger (in the weaklensing regime). Then, lens galaxies in more massive halos appear under represented in the estimator (11), and the expected shear signal (19) is lower than the prediction (17) ignoring magnification when α_{s} < 1. For α_{s} = 1, the effect vanishes. For α_{s} > 1, the number of sourcelens pairs with more massive lenses in the GGL estimator (11) is enhanced more by magnification, and thus one expects a shear signal (19) that is larger than the prediction (17). When neglecting magnification, this may cause biases in the estimation of the mean halo mass.
As Eq. (19) indicates, magnification may also affect the observed redshift distributions of the lenses and sources. Furthermore, the magnification profile μ(θm, z_{d}, z_{s}) usually has a strong radial dependence, being large for radii close to the Einstein radius, but rapidly dropping to values close to unity for larger radii. Thus, the magnification effects on the GGL signal are stronger for smaller radii. When neglecting magnification, this may cause additional biases when estimating parameters such as the halo concentration, and also when estimating the width of the halomass distribution.
3.3. Mock data production
We made use of raytracing results through the Millennium Simulation (MS, Springel 2005; Hilbert et al. 2009), from which we obtained tangential shear profiles using a fastFouriertransform (FFT) method. The method follows and improves the one presented in Unruh et al. (2019). Specific details are given in Appendix A.
For the radial binning of the tangential shear profile, we chose N_{bin} = 16 logarithmically spaced bins between θ_{in} = 0.6′ and θ_{out} = 17.5′. We estimated the error on the shear signal with a jackknife method that measures the fieldtofield variance of the 64 fields. Then, we repeated the whole process while replacing the lens galaxies with random positions to obtain the shear estimate that is caused by the long modes in the matter density field, as well as boundary effects as recommended by Singh et al. (2017). The random signal was subtracted from original signal . For convenience, we dropped the hat to distinguish the estimator from theoretical expectations in the following.
For our analyses, we wanted to obtain samples of galaxies with different local slopes α of the source counts, which depend on redshift and limiting magnitude. For a given lens galaxy sample with a magnitude cut corresponding to a flux limit s_{lim}, we estimated the local slope (8) by finite differencing around s_{lim}. Figure 1 illustrates the cumulative number of galaxies at redshift z_{d} = 0.41 for the whole simulated field of 64 × 16 deg^{2}. Several magnitude cuts are indicated, as well as one example of a tangential curve at s_{lim} = 21 mag with slope α = 1.06. For all GGL measurements in this work, we fixed the source redshift to z_{s} = 0.99. To find the local slopes α_{s} of the source galaxy counts, we varied the limiting magnitude in the rband filter (see Table 1a). For the local slope α_{d} of the lens galaxy counts, we further varied the lens redshift z_{d}, as well as the limiting magnitude, as shown in Tables 1b and c.
Fig. 1. Cumulative number of galaxies N_{0} as a function of rband magnitude for a field of 64 × 16 deg^{2} at redshift z = 0.41. The dotted vertical lines indicate the magnitude cuts listed in Table 1b. Finally, the tangential curve shows the local slope α at r = 21 mag. 

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Local slopes for the source galaxy counts α_{s} and the lens galaxy counts α_{d} given as a function of redshift and as a function of the limiting flux.
4. Magnification effects on background galaxies
Lens and source galaxies are affected by magnification. To understand this impact in more detail, we first discuss how the shear profile changes when only source galaxies are magnified. In the following, we describe how we generated an arbitrary number of mock source catalogues, with and without magnification bias included. Results from the Millennium Simulations are then presented, and the magnificationinduced bias in the GGL signal is compared to the prediction of the analytical model presented in Sect. 3.2.
4.1. Magnification switched off
To switch magnification off, we simply chose random source positions. We set the number of galaxies to N_{s, 0} = 10^{7} per 16 deg^{2} field to keep the impact of noise low.
4.2. Magnification switched on
Magnification changes the number counts of observed galaxies on the sky. Using the raytracing data, we obtained the cumulative number counts of the galaxies as a function of magnificationcorrected flux, n_{s, 0}(> s_{0}). We obtained the local expected number counts of galaxies by adjusting the flux limit to s_{lim, s}/μ(θ) at each position θ and using the first equality in Eq. (15). We further scaled the number counts so that for μ = 1 the expected number of source galaxies is N_{s} = 10^{7} per field of solid angle A = 4° ×4°. The threshold of finding a source at a grid position θ is then T(θ) = n_{s}(> s_{lim, s}; θ) A/N_{pix}, where we restricted T(θ) to be smaller than unity. Finally, we drew a uniform random number P(θ) between zero and one for each position. A source galaxy was placed at a position θ if T(θ) > P(θ). The “magnification off” method can be recovered if we insert μ = 1 for all θ in Eq. (15).
4.3. Results
The relative impact of magnification of sources on a tangential shear profile is shown by the orange “upward” triangles in Fig. 2. As expected, the net effect depends on the local slope α_{s}; the effect is typically of the order of 1 to 2% per bin. In the two panels on the lefthand side in Fig. 2, the local slope α_{s} is larger than unity, and the shear signal is enhanced; while in the two panels on the right, α_{s} < 1, which reverses the effect. Also, the magnification effect is stronger for smaller separations θ. This is seen more clearly in Fig. 3, where the absolute difference of sourcemagnified to magnificationcorrected shear profiles is compared. The shear profiles vary with α_{s} for constant redshifts z_{d, s} according to Table 1a. The difference between the expected and “measured” shear profiles in Fig. 3 is the bias that we estimated in Sect. 3.2 and gave in Eq. (16). We calculated the first and the third line of (16) using the numerical data and show them as solid and dotted lines, respectively. Both models are in good agreement with the numerical data for moderate α_{s}. However, for very steep α_{s} ≳ 2, the weak lensing approximation μ − 1 ≪ 1 is not sufficient anymore; although large magnifications are rare, they affect the number counts significantly.
Fig. 2. Relative difference between shear profiles with and without magnification. The redshifts for the lenses are z_{d} = 0.41 in the upper panels and z_{d} = 0.62 in the lower panels; the source redshift is kept fixed at z_{s} = 0.99. The limiting magnitude of the lenses is 22 mag in the rband, and for the sources it is 23 and 25 mag for α_{s} = 1.75 and 0.58, respectively. The red “downward” triangles indicate shear profiles that only have magnification in the lens galaxy population, while the orange “upward” triangles show the influence of magnification for source galaxies only. It can be seen that a local slope > 1 of lens or source population leads to an enhanced signal, whereas α_{d, s} < 1 causes a reduced signal. The green crosses display a measurement closest to real observations, i.e. where magnification affects both source and lens galaxy populations. A reduction or enhancement depends on both slopes α_{d, s}, as well as the redshifts of lenses and sources z_{d, s}. In all cases, the shape of the shear profile changes. 

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Fig. 3. Absolute difference between shear profiles with and without magnification of sources, for different local slopes α_{s} shown in the legend. The solid lines correspond to the first expression in Eq. (16) and dotted lines correspond to its approximation in the third line. The upper scale shows comoving transverse separation, and the redshifts are z_{s} = 0.99 for the sources and z_{d} = 0.41 for the lenses. The brown triangles with α_{s} = 1.75 and the orange ones α_{s} = 0.58 are directly comparable to the orange triangles in the upper panels of Fig. 2. We also show the goodnessoffit parameter with 16 degrees of freedom for the solid line. 

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We define the mean fractional difference between a shear profile with and without magnification for all bins as
where we stress that a difference of δ_{γ} = 0 is not necessarily equivalent to an unaltered shear profile. However, we only applied this estimator to the orange “upward” and red “downward” triangles seen in Fig. 2, which display either a positive or a negative sign for all angular scales investigated.
Results for δ_{γ} as a function of α_{s} for constant z_{s} can be seen in Fig. 4. We selected the source galaxies according to Table 1a, for which δ_{γ} is almost linear in α_{s}, as expected from Eq. (16) in the weak lensing approximation, although the slope of the linear relation depends on different z_{d} (Table 1c). The maximum shear difference of 4% is found for the largest α_{s}.
Fig. 4. Fractional change of the shear profile by magnification, where the magnification is only turned on for the source galaxies at z_{s} = 0.99 (Table 1a). At a local slope of α_{s} ≈ 1 the impact of magnification flips its sign. The relation between α_{s} and δ_{γ} is almost linear. Different redshifts of the lenses z_{d} = 0.41 (plus) and 0.62 (cross) affect the impact of the sources’ magnification; the rband limiting magnitude of lenses is fixed to 22 mag. 

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5. Magnification effects on foreground galaxies
In this section, we investigate the influence of magnification on lens galaxies. We follow the structure from the previous section, meaning we obtain and analyse results from the raytracing data and then compare those to the analytic estimate presented in Sect. 3.1.
5.1. Magnification switched on
The galaxies in the Henriques catalogue are affected by magnification by design. Hence, to create a catalogue including magnification, we simply extracted lens positions from galaxies brighter than a magnitude limit s_{lim, d} and assigned them to their nearest grid point.
5.2. Magnification switched off
To switch off magnification in the mock data, we undid the magnification as follows. As was done for the sources, the magnificationcorrected flux s_{0} can be easily recovered from the magnification given in the raytracing catalogue and the apparent flux of lens galaxies. The galaxy’s apparent position, however, is shifted on the sky compared to its unlensed position. Unfortunately, the unlensed position cannot be recovered from the simulated data, and the absolute amount of shifting is of the order of an arcminute and depends on redshift (Chang & Jain 2014). Hence, we must aim to create a sample of galaxies that is not affected by magnification in a different way.
As was outlined in Sect. 2.2, for both local slopes α_{d, s} = 1, the magnification leaves the observed number counts unaffected, and thus the shear profile γ_{t} unbiased. Therefore, we transformed the magnificationcorrected flux distribution such that the galaxy counts obey . As a reference point, we chose the number of galaxies at limiting magnitude . This results in the following mapping from observed magnificationcorrected flux, s_{0}, to the transformed flux:
In other words, this new flux scale distorts the number counts of lenses in such a way that in the transformed flux system, α′ = 1, the lens galaxy counts are unaffected by magnification bias (although the individual lens positions are not). We can now calculate the observed transformed number density n′( > s′), where . To create a lens galaxy sample free from magnification, we again chose only those galaxies that are brighter than the given flux limit s′> s_{lim, d}. This leads to a different selection of galaxies for the original and the transformed number density.
We tested this approach with two consistency checks. The first is based on the fact that in the method described above, the total number of galaxies has to be conserved. This is true for all the lens redshifts used. For a magnitude cut of 22 mag in the rband, a lens redshift of z_{d} = 0.41, and the full field of view of 64 × 16 deg^{2}, 595 348 lenses are found with a cut in observed magnitude, and 595 355 lenses are found with observed transformed magnitude. Compared to the original fluxes, a detailed analysis showed that in the transformed flux system, 917 galaxies became brighter than 22 mag, while 924 galaxies became dimmer, leaving the overall number count almost unchanged. The tiny difference in seven galaxies is due to the fact that the number count function is discretely sampled.
The second consistency check uses a null test: the socalled shearratio test (SRT, Jain & Taylor 2003), which is based on Eq. (18),
for which we expect T(z_{d}, z_{s1}, z_{s2}) = 0 for two source populations at redshifts z_{s1} and z_{s2} in the absence of magnification effects. For the test, the location of the same lens galaxies and the shear from two source galaxy populations at different distances D_{s1, 2} were used. A ratio of the tangential shear estimates is equal to the ratio of the corresponding angulardiameter distances, while the lens properties drop out. As shown in Unruh et al. (2019), the SRT is strongly affected by lens magnification. The impact is stronger for higher lens redshifts and smaller lineofsight separation of lenses and sources. Therefore, we performed the SRT for lenses selected with and without magnification for two different lens redshifts. We performed the SRT by taking a weighted integral of T(θ; z_{d}, z_{s1}, z_{s2}) over θ from θ_{in} to θ_{out} as in Unruh et al. (2019). In the case that includes magnification, we recovered the results from Unruh et al. (2019). For lenses that are selected with corrected magnification, the SRT performs better by a factor of ≳100. We give results for two example redshift combinations in Table 2. The corrected SRT still shows a slight scatter due to the statistical noise in the data, coming from the lensing by the largescale structure in each of the 64 fields, and a bias that arises from shifting lens galaxies to their nearest grid point.
5.3. Results
The red “downward” triangles in Fig. 2 show the relative impact of magnification on γ_{t}. Similar to the results given in Sect. 4.3, where the magnification of the source galaxies is discussed, the local slope α_{d} determines whether the shear signal is enhanced or reduced. The upper panels of Fig. 2 show that the shear profiles are reduced for a local slope α_{d} that is smaller than unity at redshift z_{d} = 0.42, while the lower panels display shear profiles with α_{d} > 1 at a higher redshift z_{d} = 0.62, where the reverse effect is observed. The panels with higher z_{d} show larger magnification effects; relative deviations by up to 7% in a single bin can be seen. In general, the shear signal is more strongly affected at larger separations θ from the lens centre until it reaches a maximum at ≈8′ for z_{d} = 0.62 and ≈10′ for z_{d} = 0.41; for even larger separations, the magnification effect becomes relatively weaker.
A comparison of numerical results to the analytic estimate (13) can be seen in Fig. 5. The absolute difference between shear profiles affected by a magnification of lens galaxies and those unaffected by magnification is plotted. The triangles show numerical results, while lines indicate our analytical model for . We employed the reduced χ^{2}test as an estimator for the goodness of our model and find that all models are in good agreement with the data for the considered angular scales. However, the local slope is not necessarily a sufficiently good quantity for the analytic correction if the local slope α_{d} becomes very steep, meaning when the luminosity function is not well approximated by a power law anymore (cf. Fig. 1). However, the analytic correction still reduces the impact of magnification significantly.
Fig. 5. Absolute difference between shear profiles with and without magnification of lenses. The upper scale shows comoving transverse separation and the shear difference is shown for several limiting magnitudes of the lens galaxies, which yield the local slopes α_{d} shown in the legend. Also, the goodnessoffit parameter with 16 degrees of freedom is indicated in the legend. The redshifts are z_{s} = 0.99 for the sources and z_{d} = 0.41 for the lenses. 

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To explore the dependencies of the mean fractional shear difference δ_{γ} (Eq. (20)) on α_{d}, we altered the lens properties according to Table 1b. Results can be seen in Fig. 6 and show the impact of magnification for constant lens redshift z_{d} and varying limiting magnitude, for two different local slopes α_{s} of the sources. The impact on the shear profile is almost exactly the same in both local slopes, with a small residual noise that is present in the data. The dependence of δ_{γ} on α_{d} is similar to the one in Fig. 4, which shows the magnification for source galaxies only. The shear profile changes by up to 3% in the mean.
Fig. 6. Behaviour of mean fractional shear difference δ_{y} as a function of local slope for lens galaxies α_{d} and rbandlimiting magnitude, to change the lens properties according to Table 1b. The source and lens redshift is the same for both cases, i.e., z_{d} = 0.41 and z_{s} = 0.99. The effect is independent of the sources local slope α_{s}. 

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We further investigated the dependence of the magnification on the values of α_{d} and z_{d} for fixed z_{s} in Fig. 7. δ_{γ} is calculated for various lens redshifts with constant limiting magnitude for the lenses (see Table 1c) and the same two values of α_{s} as before. The figure shows that the sources’ local slope does not affect the magnification induced by lens galaxies. The signal is again moderately reduced by 1% for α_{d} < 1 but is not monotonic anymore. For α_{d} > 1, the relation deviates significantly from a linear one. Magnification is stronger for larger α_{d} and larger z_{d}, leading to deviations of up to 45% in the most extreme case.
Fig. 7. Impact on the shear profile as a function of α_{d} and z_{d} (Table 1c) for fixed flux limit of r = 22 mag, for a magnification of lens galaxies only. The redshift z_{d} is given on the (nonlinear) top axis. Again, the effect switches its sign at α_{d} ≈ 1. It then rises up to 45% for larger α_{d}, while less impact is seen for α_{d} < 1. Different local slopes of the sources α_{s} at z_{s} = 0.99 leave the lens’ magnification unaffected. 

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Lastly, we combined the numerical methods and results that were addressed individually in Sects. 4 and 5. The green crosses in Fig. 2 show numerical results from source and lens galaxy populations that are both affected by magnification, meaning the case of relevance for observational studies of GGL. The fractional change is approximately the sum of the bias of lens galaxies only plus the bias of source galaxies only. Hence, to first order, it is determined by α_{d} and α_{s}. The sign of δ_{γ} cannot easily be predicted if α_{d} < 1 and α_{s} > 1, and vice versa. For δ_{γ} the sign also depends on θ since the change of shear signal per bin behaves differently for magnified lenses and magnified sources.
6. Magnification bias in halomass estimates
6.1. Estimating the mean halo mass of lenses
The GGL signal is sensitive to the surfacemass density around lenses that differs from the average projected cosmological matter density. To infer from this the mean mass of a parent halo that hosts a typical lens galaxy, we used a halomodel prescription to describe the relation between galaxies and matter (Cooray & Sheth 2002). In this prescription, we expand the galaxymatter power spectrum at redshift z_{d} of lenses and at comoving wave number k,
in terms of a onehalo term,
and a twohalo term,
In this model, denotes the Fourier transform of a NavarroFrenkWhite (NFW; Navarro et al. 1996) density profile for a virial halo mass m, truncated at the virial radius and normalised to for k = 0 (Scoccimarro et al. 2001) for the massconcentration relation in Bullock et al. (2001). We further denote the mean comoving number density of halos in the mass interval m_{1} ≤ m < m_{2} by (Sheth & Tormen 1999), the bias factor of halos of mass m by b_{h}(m) (Scoccimarro et al. 2001), and the linear matter power spectrum by P_{lin}(k) (Eisenstein & Hu 1998). Finally, the mean number density of lenses is
This version of the halo model assumes a central galaxy at the centre of a halo whenever there are lens galaxies inside the halo, and satellite galaxies with a number density profile equal to the NFW matter density. For the mean number of central and satellite galaxies for a halo mass m, we follow Clampitt et al. (2017) but with centralgalaxy fraction f_{cen} ≡ 1,
where Θ = (m_{1}, m_{th}, σ_{ log m}, β) are four model parameters that determine the halooccupation distribution (HOD) of our lenses, and is the error function. The model parameters have the following meaning: m_{th} determines at which mass scale ⟨N_{cen}m⟩ = 1/2; at halo mass m_{1} the mean number of satellites equals that of central galaxies; σ_{ log m} is the width of the HOD of centrals; and β is the slope of the satellite HOD.
The mattergalaxy crosspower spectrum is related to the mean tangential shear by a Limber projection, which for lenses at z_{d} and sources at z_{s} is
We employed a maximumlikelihood estimator (MLE) to infer the mean halo mass of lenses from GGL. For this, we set {γ_{t, i}(Θ)i = 1…N_{bin}} as a set of N_{bin} measurements of the mean tangential shear in our mock data, obtained for different lenssource angular separation bins i; the error covariance estimated from the measurements for bin θ_{i} and θ_{j} is C_{ij}, and its inverse [C^{−1}]_{ij}. For the MLE of Θ, we then minimised
with respect to Θ, where γ_{t, i}(Θ) is the halo model prediction of γ_{t}(θΘ) averaged over the size of the ith separation bin that corresponds to γ_{t, i}. We refer to Θ_{mle} as the parameter set that minimises χ^{2}(Θ). Finally, given the MLE Θ_{mle}, we obtain the MLE of the mean halo mass by the integral
which has to be evaluated for the HOD of galaxies that is determined by Θ_{mle}.
In Fig. 8, we show four examples of tangential shear estimates and their best shear profile fit from the halo model. For the shear profiles, we switched off magnification effects of lens and source galaxies. It can be seen that the fitting procedure works reasonably well. The halo model itself is only an approximation of the inhomogeneous matter distribution in the Universe, and in the presence of our simulated data with almost vanishing Poisson noise, we do not expect the halo model to work perfectly. We further used approximations such as a truncated NFW model and a specific massconcentration relation, that certainly further limits the accuracy we can obtain with the halo model. The mean relative difference between the data and the model for the 16 bins is 2.6% for z_{d} = 0.41, and 1.3% for z_{d} = 0.62. As expected, the shear profiles are almost independent of the flux limit of sources if their redshift is fixed, as can be seen for the red and blue crosses, as well as for the magenta and green crosses. On the other hand, there are three main reasons for the difference between the red/blue and magenta/green shear profiles. The lensing efficiencies D_{ds}/D_{s} are different for different z_{d}. Thus, the red/blue shear profile with z_{d} = 0.41 has a larger lensing efficiency than the magenta/green one with z_{d} = 0.62. Moreover, we observed fixed angular scales, which corresponds to different physical scales. Lastly, the lens galaxy population might evolve between the two redshifts.
Fig. 8. Upper panel: magnificationcorrected shear profiles for two lens redshifts z_{d} with a rband magnitude cut of 22 mag and two differently selected source galaxies at redshift z_{s} = 0.99 (crosses) as well as their best fit according to the halo model (solid line). Lower panel: absolute difference Δγ_{t} between data and halo model fit (open circles with the same colour). For each lens redshift, source galaxies were chosen with two limiting magnitudes that have the local slopes α_{s} = 0.58 and 1.75, respectively. As expected, the shear profile does not depend on the choice of source galaxies. The offset between the red/blue points as well as the magenta/green points is for better visibility only. 

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Table 3 accompanies Fig. 8 and lists the fitting parameters, the goodnessoffit values, and the mass estimates for the different shear profiles. The mean halo mass, in contrast to the shear amplitude, is larger for the highredshift lenses when the same magnitude limit is applied.
Fitting results for the halo model, with the mean halo mass ⟨m⟩_{mle}, the scatter in host halo mass σ_{ log m}, the mass scale where 50% of halos host a galaxy m_{th}, the normalisation factor for the satellite galaxies m_{1} and its slope β, and the goodnessoffit value with 12 degrees of freedom.
We quantify this bias in halo mass for fixed limiting magnitudes of lenses and sources, which means fixed α_{d, s}, and fixed redshifts as follows. Using the halo model as described above, we calculated the best mass estimate from the magnificationcorrected shear profile. As could be seen in the previous sections, the relative change of the shear profile is typically of the order of a couple of percent. Therefore, we fixed the scatter in the host halo mass σ_{ log m} and the slope of the mean number of satellite galaxies β to their best fit value in the magnificationcorrected case. Then, we only fitted the remaining two parameters m_{th} and m_{1} to estimate the mass for the three remaining shear profiles, meaning a shear profile with magnification of the sources only turned on, a profile with magnification of the lenses only turned on, and a shear profile with lens and source magnification turned on. Similar to the fractional shear difference (20), we define the bias of halomass estimates, inferred from γ_{t}, by
6.2. Numerical results
Figures 9–12 show results for the mass bias δ_{M}. Figure 9 shows the bias for magnified source galaxy counts and magnificationcorrected lens counts. The lenses have constant limiting magnitude of 22 mag in the rband and their redshifts are z_{d} = 0.41 and 0.62. Source galaxies at redshift z_{s} = 0.99 are selected for several limiting magnitudes (cf. Table 1a). The bias is of the same order of magnitude as the corresponding mean fractional difference of the shear, and α_{s} determines whether mass is overestimated or underestimated. The mean halo mass is biased by up to 3.5%.
Fig. 9. Relative mass bias δ_{M} (32) for magnified source number counts. Source galaxies are at z_{s} = 0.99 and lenses are chosen according to Table 1a with redshifts z_{d} = 0.41 (plus) and 0.62 (cross), for fixed limiting magnitude. The mass bias behaves roughly like the fractional shear difference (cf. Fig. 4). For local slopes α_{s} < 1, the underestimation of mass is stronger than for the shear profile, while for α_{s} > 1 the overestimate is similar. 

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Fig. 10. Fractional mass bias δ_{M} shown as a function of local slope α_{d} and rband limiting magnitude (cf. Table 1b). The source redshift is z_{s} = 0.99 and lens redshift is z_{d} = 0.41. For α_{d} < 1, δ_{M} the mass is biased low, while for α_{d} > 1 the mass is biased high. 

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Fig. 11. The fractional mass bias δ_{M} is shown as a function of local slope α_{d} and lens redshift z_{d} (cf. Table 1c). We keep the rband limiting magnitude for the lenses and the source redshift z_{s} = 0.99 constant. The top axis indicates the respective lens redshifts in a nonlinear scaling. Following the trend seen in Fig. 7, mass is biased low for α_{d} < 1 and shows large biases for α_{d} > 1. 

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Fig. 12. Total magnification bias for halomass estimates as a function of limiting magnitudes that yield the local slopes α_{d, s} and lens redshift z_{d} (type of symbols). The source redshift is the same in all cases, i.e., z_{s} = 0.99. The mass difference δ_{M} is shown in colour code, where blue indicates underestimation, red overestimation, and white is an unbiased result. We cut off the colour bar for the largest deviations of 55% and 58% for α_{d} = 2.41 and z_{d} = 0.83 for higher contrast in the colour scale. The plot is roughly divided by the vertical line with α_{d} = 1 into mass underestimation for α_{d} < 1 and overestimation α_{d} > 1. 

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We then explored the dependencies of the fractional mass bias δ_{M} on α_{d}, while we only considered magnificationcorrected sources. Firstly, we fixed the lens redshift to z_{d} = 0.41 and altered the lens’ limiting magnitude according to Table 1b. The result is shown in Fig. 10. The mass bias is an almost linear function of α_{d} and shows a similar dependence on α_{d} as the fractional shear difference (cf. Fig. 6). The mass is biased up to 5% for the largest α_{d}. Lastly, we calculated δ_{M} for various lens redshifts with constant limiting magnitude for the lenses (see Table 1c) and the same two α_{s} as before, which is shown in Fig. 11. The mass bias shows a strong redshift dependence, where the bias increases from a couple of percent to a mass overestimate of 55%.
To explore the observationally relevant case, we compared halomass estimates with and without magnification for both lenses and sources. Figure 12 contains all different α_{d, s} − z_{d} combinations with constant z_{s} = 0.99 from Tables 1a–c, plus some additional combinations. It shows δ_{M} as defined in Eq. (32) as a function of the local slopes α_{d, s} on the x and yaxis. Red values indicate an overestimation of the mass, where we cut off the colour bar for the highest values for better visibility. The blue values show an underestimation, while white values mean no bias in the mass estimate. The upperright quadrant shows consistently red values since α_{d, s} > 1, while the lowerleft quadrant has α_{d, s} < 1 and is consistently blue, as expected from theoretical considerations. In most of the other cases, α_{d} seems to be the decisive factor for the sign of the mass bias, for α_{d} < 1 leading to an underestimation and for α_{d} > 1 leading to an overestimation. The slope α_{s} influences the total value of the mass bias. Besides, even if two points are close in the α_{d} − α_{s} parameter space, they do not exhibit the same colour, meaning mass bias, due to their different z_{d}.
7. Discussion and conclusions
In this paper, we quantified the impact of magnification on the GGL signals and halomass estimates. Magnification changes the observed galaxy number counts on the sky, which has an impact on the measured tangential shear profiles. It is important to note that magnification affects lens galaxies as well as source galaxies. Analyses of tangential shear profiles, such as estimates of the excess mass density profiles or halo mass, are therefore biased if they ignore lens or source magnification. Our estimates with raytracing simulations and synthetic galaxy populations show that the number count slopes of sources and lenses are the most important quantities that determine the relative strength of the bias. While the analytical estimate for the bias on the GGL profile caused by magnification of lenses was known before, to our knowledge the one caused by magnification of the source population has not previously been derived.
How magnification affects tangential shear profiles can be seen in Figs. 2–7, where we varied the local slope of lens and sources galaxies α_{d, s} and the lens redshift z_{d}. We studied the magnification effect from lenses and sources individually and compared them to our analytic estimates. The latter are leadingorder estimates and describe the numerical results very well in most cases. One of the surprising results, shown in Fig. 3, is that the weak lensing approximation for the impact of magnification on source counts can significantly fail if the count slope is steep. Hence, the validity of the commonly used approximation for the number density of sources
needs to be checked, depending on its application.
Besides confirming that the shear signal is reduced if the slopes α_{d} and α_{s} are < 1 and enhanced for α_{d, s} > 1, we find: For fixed redshifts z_{d, s}, the change of the tangential shear estimate depends solely on α_{d, s}. For fixed source redshifts z_{s}, the impact of magnification of sources decreases for larger z_{d} and smaller lenssource separation D_{ds} (cf. Fig. 4). Furthermore, the relative importance of magnification of lenses increases with z_{d} and rises sharply as z_{d} approaches z_{s}. The relative change from a biased to an unbiased shear profile is a function of angular separation θ, and the mean change is typically a few percent. However, if the redshift differences between sources and lenses become small, the effect can be considerably larger (cf. Fig. 7).
For practical applications, the change in the shear signal by magnification is described by the sum of the individual effects of source magnification and the one from lens magnification. The impact of magnification depends on the galaxy’s luminosity function, the limiting magnitudes, and the redshifts z_{d, s}, but also the separation θ.
The relative mass bias δ_{M} on the average mass of a lens parenthalo inherits all trends seen in the shear profiles with magnified lenses and magnified sources (see Figs. 9–12), meaning δ_{M} > 0 for α_{d, s} > 0 and vice versa. Mediumredshift galaxies show a mass bias of less than 10%, and the higher the lens redshift, the stronger the mass bias with up to 58% for z_{d} = 0.83 and z_{s} = 0.99. The bias δ_{M} is of the same order of magnitude as the relative change of the shear profile δ_{γ}. The particular relation, however, between δ_{γ} and the mass bias δ_{M} is quite complicated. First and foremost, the amplitude of the measured shear signal determines the underlying halo mass. However, magnification changes the scale dependence of the shear signal. In the halo model, this translates to a different behaviour of the one and twohalo term, which affects the mass estimate in a highly nonlinear way. Another minor effect is that the true mean halo mass of lens galaxies affected by magnification probably differs from the true mean halo mass of unmagnified lenses due to an expected correlation between mass and luminosity. For example, for the highest z_{d} considered in this work, the mean masses differ by 0.16%. Furthermore, we fixed the observed angular scales on the sky, which relate to different physical scales at different redshifts; thus, the relative contribution of the twohalo term on the shear signal grows with redshift, which makes a comparison of mass biases from different lens redshifts more complicated. In general, when we allow for magnification effects both in source and lens galaxies, meaning, the case for real observations, the sign of the mass bias is in most cases determined by the value of α_{d}. The only exceptions shown in Fig. 12 are two cases, where either the lens redshift is low or the local slope α_{s} is very high; in such cases, the sign of the mass bias is not easily predicted and must be studied case by case.
We also considered a GGL estimate using the fluxlimits and redshift distributions as given from the combined data of KiDS+VIKING (Hildebrandt et al. 2020) and GAMA (Driver et al. 2011). KiDS and VIKING are partner surveys that probe the optical and nearinfrared sky to obtain highresolution, widefield shape and redshift information from galaxies. GAMA is a spectroscopic, fluxlimited survey with a partly overlapping footprint in the KiDS+VIKING area. In Appendix A.3, we present a detailed description of the input parameters we use for this estimate. We list the results in Table A.1. A lens galaxy sample at redshift z_{d} = 0.21 shows relative changes in the shear profile due to magnification effects of less than one percent, and relative mass bias of ≈3%, while GAMA’s highest lens sample at z_{d} = 0.36 shows that a magnification correction changes the shear profile by ≈2% and the relative halo mass bias by 8%. Although our estimate used simplifying assumptions, especially for the source population, for example, no catastrophic outliers in the redshift distribution and no further selection criteria than a cut in magnitude, we conclude that magnification effects must be carefully considered in current and future surveys.
In this paper, we assume that lenses and sources form a fluxlimited sample. While this assumption may be a realistic one for lens galaxies (e.g. galaxy redshift surveys frequently start from a fluxlimited photometric sample), it is less the case for source galaxies. Sources in weak lensing studies have rather complicated selection criteria, not merely based on flux, but also on size and signaltonoise ratio, for example. Therefore, our quantitative analysis may not apply directly to observational surveys. Besides, source galaxies typically enter a weak lensing catalogue with a weight that characterises the accuracy of the corresponding shear estimate. We ignored any such weighting scheme in our processing, but it may be relevant, since the weight of an object is also expected to depend on magnitude and size, and is thus affected by magnification.
While the relative amplitude of the bias caused by magnification is modest in most cases, and probably smaller than the uncertainties from shape noise and sample variance in previous surveys, future surveys like Euclid or LSST have such improved statistical power that magnification effects must be accounted for in the quantitative analysis of GGL.
A more comprehensive halo model is discussed in Sect. 6.
Acknowledgments
The authors would like to thank the anonymous referee for his constructive and helpful comments that improved the quality of this paper. Part of this work was supported by the Deutsche Forschungsgemeinschaft, DFG project number SCHN 342/131. Sandra Unruh is a member of the International Max Planck Research School (IMPRS) for Astronomy and Astrophysics at the Universities of Bonn and Cologne.Stefan Hilbert acknowledges support by the DFG cluster of excellence “Origin and Structure of the Universe” (www.universecluster.de).
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Appendix A: Mock data
A.1 Millennium Simulation data
To study magnification effects in GGL, we make use of raytracing results through the Millennium Simulation (MS, Springel 2005), which is an Nbody simulation of 2160^{3} dark matter particles. Each particle has a mass of 8.6 × 10^{8}h^{−1} M_{⊙} that is confined to a cube with side length of 500 h^{−1} Mpc and with periodic boundary conditions. The underlying cosmology is a flat ΛCDM model with a matter density parameter of Ω_{m} = 0.25, a baryon density parameter of Ω_{b} = 0.045, a dimensionless Hubble parameter of h = 0.73, a tilt of the primordial power spectrum of n = 1, and a variance of matter fluctuations on a scale of 8 h^{−1} Mpc extrapolated from a linear power spectrum of σ_{8} = 0.9. This cosmology is based on combined results of 2dFGRS (Colless et al. 2001) and firstyear WMAP data (Spergel et al. 2003).
The raytracing results are based on a multiplelensplane algorithm in 64 light cones constructed from 37 snapshots between redshifts z = 0 to z = 3.06, each covering a 4 × 4 deg^{2} field of view. For more information about the ray tracing, the reader is kindly referred to Hilbert et al. (2009). An important aspect of the algorithm is that the galaxymatter correlation is preserved. The raytracing results contain the Jacobians 𝒜 on a N_{pix} = 4096^{2} pixel grid, which corresponds to a resolution of 3.5 arcsec per pixel. From this, we calculated shear and magnification on a pixel grid.
We also used a catalogue of galaxies based on a semianalytic galaxyformation model by Henriques et al. (2015). This catalogue matches the GGL and galaxygalaxygalaxy lensing signal from CFHTLenS (Saghiha et al. 2017). The galaxies are listed for each redshift snapshot with various properties, for example, (magnified) flux in various filters and positions, which allows for selection according to chosen magnitude limits. However, the galaxy positions are not confined to a grid as is the Jacobi information. Thus, we shifted all selected galaxies to their nearest grid point. Therefore, analyses that are close to the centre of the galaxy suffer from discretisation effects on scales comparable to the pixel size.
A.2 Obtaining a tangential shear estimate
To extract the shear signal averaged over many lenses, a fast Fourier transform (FFT) is employed. In order to do so, we first defined lens and source number density on a grid by
with δ_{K} being one if and zero otherwise. The number of lenses and sources is N_{d, s}, and are the positions of lenses and sources, respectively. Furthermore, we define the shear field of the sources on the grid by
Then, the tangential shear estimator (11) can be expressed as
where the sums over θ′ and θ″ extend over the whole grid. The equivalence of Eqs. (A.3) and (11) can be verified by inserting the definitions of n_{d, s} and γ_{s} into the former. The sum over θ″ in the denominator calculates for each θ′ the number of lenssource pairs, ensuring that (A.3) is not affected by a potential masking or inhomogeneous survey areas. Further, the sums over θ″ in the numerator and denominator of the GGL estimator (A.3) are convolutions. Thus, the convolution theorem can be applied to these sums. If ℱ{f} is the Fourier transform of a function f and ℱ^{−1} the inverse Fourier transform, then we can rewrite the estimator (A.3) as
which can be readily solved by an FFT method. For this, we employed routines from the FFTW library by Frigo & Johnson (2005) in our code.
A FFT implicitly assumes periodic boundary conditions, which introduces a bias to the averaged shear data and, thus, must be mitigated for. We can restrict the selection of lenses to the inner (4° −2θ_{out})^{2} of the field, where θ_{out} is the maximum separation from the lens that we considered. However, for a θ_{out} ≈ 20′, we already lose approximately 30% of the lens galaxies. Alternatively, we employed a zeropadding method in which we increased the FFTarea to (4° +θ_{out})^{2} and filled the added space with zeros. In this case, we used all available lenses with the cost of slightly increased computational time and the gain of a lessnoisy shear profile. For the whole 64 × 16 deg^{2}, our FFTbased code needs a CPU time of 823 s, independent of the number of sources and lenses. We compared the performance of our method to the publicly available athena treecode (Kilbinger et al. 2014). In contrast to the FFT method, the computation time of athena is enhanced with the number of lenssource pairs. We adjusted the settings to our survey parameters while leaving the parameter that sets the accuracy of the treecode, meaning the openangle threshold, to its preset value. For 10^{7} sources and lenses at z_{d} = 0.41 with limiting magnitudes 19.5, 22, and 29 mag, athena performs with a CPU time of 889 s, 1167 s, and 2043 s, respectively.
A.3 Estimating the impact of the magnification bias on a KiDS+VIKING+GAMAlike survey
We considered GAMAlike lens galaxies with narrow redshift distributions at z_{d} = 0.21 and z_{d} = 0.36 with a flux limit of 19.8 mag. Using simulated data from the semianalytic galaxyformation model by Henriques et al. (2015), we obtain local slopes of α_{d} = 0.85 and α_{d} = 2.11, respectively. To keep the statistical error low, we still consider the data from the whole simulated area of 64 × 16 deg^{2}. To mimic the KiDS+VIKINGlike source population, we matched the last three bins of the bestestimated redshift distribution to Millennium data from 0.51 ≤ z_{s, Mil} < 1.28. We then calculated a weighted shear map from the simulated data, the final source galaxy distribution has a local slope of α_{s} = 0.51 for a limiting magnitude of 25 mag. We repeated our analysis from Sects. 4–6, and obtain the results listed in Table A.1. The results are quantitatively comparable to the results from Figs. 4 and 6.
Impact of the magnification bias on a KiDS+VIKING+ GAMAlike survey.
A KiDS+VIKING+GAMAlike survey is moderately affected. We stress that this estimate has been made with simplified assumptions, for example, there are no catastrophic outliers in the redshift distribution and no selection criteria for source galaxies other than a magnitude cut.
All Tables
Local slopes for the source galaxy counts α_{s} and the lens galaxy counts α_{d} given as a function of redshift and as a function of the limiting flux.
Fitting results for the halo model, with the mean halo mass ⟨m⟩_{mle}, the scatter in host halo mass σ_{ log m}, the mass scale where 50% of halos host a galaxy m_{th}, the normalisation factor for the satellite galaxies m_{1} and its slope β, and the goodnessoffit value with 12 degrees of freedom.
All Figures
Fig. 1. Cumulative number of galaxies N_{0} as a function of rband magnitude for a field of 64 × 16 deg^{2} at redshift z = 0.41. The dotted vertical lines indicate the magnitude cuts listed in Table 1b. Finally, the tangential curve shows the local slope α at r = 21 mag. 

Open with DEXTER  
In the text 
Fig. 2. Relative difference between shear profiles with and without magnification. The redshifts for the lenses are z_{d} = 0.41 in the upper panels and z_{d} = 0.62 in the lower panels; the source redshift is kept fixed at z_{s} = 0.99. The limiting magnitude of the lenses is 22 mag in the rband, and for the sources it is 23 and 25 mag for α_{s} = 1.75 and 0.58, respectively. The red “downward” triangles indicate shear profiles that only have magnification in the lens galaxy population, while the orange “upward” triangles show the influence of magnification for source galaxies only. It can be seen that a local slope > 1 of lens or source population leads to an enhanced signal, whereas α_{d, s} < 1 causes a reduced signal. The green crosses display a measurement closest to real observations, i.e. where magnification affects both source and lens galaxy populations. A reduction or enhancement depends on both slopes α_{d, s}, as well as the redshifts of lenses and sources z_{d, s}. In all cases, the shape of the shear profile changes. 

Open with DEXTER  
In the text 
Fig. 3. Absolute difference between shear profiles with and without magnification of sources, for different local slopes α_{s} shown in the legend. The solid lines correspond to the first expression in Eq. (16) and dotted lines correspond to its approximation in the third line. The upper scale shows comoving transverse separation, and the redshifts are z_{s} = 0.99 for the sources and z_{d} = 0.41 for the lenses. The brown triangles with α_{s} = 1.75 and the orange ones α_{s} = 0.58 are directly comparable to the orange triangles in the upper panels of Fig. 2. We also show the goodnessoffit parameter with 16 degrees of freedom for the solid line. 

Open with DEXTER  
In the text 
Fig. 4. Fractional change of the shear profile by magnification, where the magnification is only turned on for the source galaxies at z_{s} = 0.99 (Table 1a). At a local slope of α_{s} ≈ 1 the impact of magnification flips its sign. The relation between α_{s} and δ_{γ} is almost linear. Different redshifts of the lenses z_{d} = 0.41 (plus) and 0.62 (cross) affect the impact of the sources’ magnification; the rband limiting magnitude of lenses is fixed to 22 mag. 

Open with DEXTER  
In the text 
Fig. 5. Absolute difference between shear profiles with and without magnification of lenses. The upper scale shows comoving transverse separation and the shear difference is shown for several limiting magnitudes of the lens galaxies, which yield the local slopes α_{d} shown in the legend. Also, the goodnessoffit parameter with 16 degrees of freedom is indicated in the legend. The redshifts are z_{s} = 0.99 for the sources and z_{d} = 0.41 for the lenses. 

Open with DEXTER  
In the text 
Fig. 6. Behaviour of mean fractional shear difference δ_{y} as a function of local slope for lens galaxies α_{d} and rbandlimiting magnitude, to change the lens properties according to Table 1b. The source and lens redshift is the same for both cases, i.e., z_{d} = 0.41 and z_{s} = 0.99. The effect is independent of the sources local slope α_{s}. 

Open with DEXTER  
In the text 
Fig. 7. Impact on the shear profile as a function of α_{d} and z_{d} (Table 1c) for fixed flux limit of r = 22 mag, for a magnification of lens galaxies only. The redshift z_{d} is given on the (nonlinear) top axis. Again, the effect switches its sign at α_{d} ≈ 1. It then rises up to 45% for larger α_{d}, while less impact is seen for α_{d} < 1. Different local slopes of the sources α_{s} at z_{s} = 0.99 leave the lens’ magnification unaffected. 

Open with DEXTER  
In the text 
Fig. 8. Upper panel: magnificationcorrected shear profiles for two lens redshifts z_{d} with a rband magnitude cut of 22 mag and two differently selected source galaxies at redshift z_{s} = 0.99 (crosses) as well as their best fit according to the halo model (solid line). Lower panel: absolute difference Δγ_{t} between data and halo model fit (open circles with the same colour). For each lens redshift, source galaxies were chosen with two limiting magnitudes that have the local slopes α_{s} = 0.58 and 1.75, respectively. As expected, the shear profile does not depend on the choice of source galaxies. The offset between the red/blue points as well as the magenta/green points is for better visibility only. 

Open with DEXTER  
In the text 
Fig. 9. Relative mass bias δ_{M} (32) for magnified source number counts. Source galaxies are at z_{s} = 0.99 and lenses are chosen according to Table 1a with redshifts z_{d} = 0.41 (plus) and 0.62 (cross), for fixed limiting magnitude. The mass bias behaves roughly like the fractional shear difference (cf. Fig. 4). For local slopes α_{s} < 1, the underestimation of mass is stronger than for the shear profile, while for α_{s} > 1 the overestimate is similar. 

Open with DEXTER  
In the text 
Fig. 10. Fractional mass bias δ_{M} shown as a function of local slope α_{d} and rband limiting magnitude (cf. Table 1b). The source redshift is z_{s} = 0.99 and lens redshift is z_{d} = 0.41. For α_{d} < 1, δ_{M} the mass is biased low, while for α_{d} > 1 the mass is biased high. 

Open with DEXTER  
In the text 
Fig. 11. The fractional mass bias δ_{M} is shown as a function of local slope α_{d} and lens redshift z_{d} (cf. Table 1c). We keep the rband limiting magnitude for the lenses and the source redshift z_{s} = 0.99 constant. The top axis indicates the respective lens redshifts in a nonlinear scaling. Following the trend seen in Fig. 7, mass is biased low for α_{d} < 1 and shows large biases for α_{d} > 1. 

Open with DEXTER  
In the text 
Fig. 12. Total magnification bias for halomass estimates as a function of limiting magnitudes that yield the local slopes α_{d, s} and lens redshift z_{d} (type of symbols). The source redshift is the same in all cases, i.e., z_{s} = 0.99. The mass difference δ_{M} is shown in colour code, where blue indicates underestimation, red overestimation, and white is an unbiased result. We cut off the colour bar for the largest deviations of 55% and 58% for α_{d} = 2.41 and z_{d} = 0.83 for higher contrast in the colour scale. The plot is roughly divided by the vertical line with α_{d} = 1 into mass underestimation for α_{d} < 1 and overestimation α_{d} > 1. 

Open with DEXTER  
In the text 
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