© ESO 2020

1. Introduction

A central goal of modern cosmology is to discover whether the dark energy that appears to fill the Universe is associated with its matter-energy content, laws of gravity, or some alternative physics. A compelling means of distinguishing between these scenarios is to analyse the different observational signatures that are present in the clumpy, inhomogeneous Universe, which powerfully complements measurements of the expansion history of the smooth, homogeneous Universe (e.g. Linder 2005; Wang 2008; Guzzo et al. 2008; Weinberg et al. 2013; Huterer et al. 2015).

Two important observational probes of the inhomogeneous Universe are the peculiar velocities induced in galaxies by the gravitational collapse of large-scale structure, which are statistically imprinted in galaxy redshift surveys as redshift-space distortions (e.g. Hamilton 1998; Scoccimarro 2004; Song & Percival 2009), and the gravitational lensing of light by the cosmic web, which may be measured using cosmic shear surveys (e.g. Bartelmann & Schneider 2001; Kilbinger 2015; Mandelbaum 2018). These probes are complementary because they allow for the differentiation between the two space-time metric potentials which govern the motion of non-relativistic particles, such as galaxy tracers, and the gravitational deflection of light. The difference or “gravitational slip” between these potentials is predicted to be zero in general relativity, but it may be significant in modified gravity scenarios (e.g. Uzan & Bernardeau 2001; Zhang et al. 2007; Jain & Khoury 2010; Bertschinger 2011; Clifton et al. 2012).

Recent advances in weak gravitational lensing datasets, including the Kilo-Degree Survey (KiDS, Hildebrandt et al. 2020), the Dark Energy Survey (DES, Abbott et al. 2018) and the Subaru Hyper Suprime-Cam Survey (HSC, Hikage et al. 2019), have led to dramatic improvements in the quality of these observational tests. Gravitational lensing now permits the accurate determination of, and combinations of, important cosmological parameters such as the matter density of the Universe and normalisation of the matter power spectrum, and thereby detailed comparisons with other cosmological probes such as galaxy clustering (Alam et al. 2017a) and the cosmic microwave background radiation (Planck Collaboration VI 2020). Some of these comparisons have yielded intriguing evidence of “tension” on both small and large scales (e.g. Joudaki et al. 2017; Leauthaud et al. 2017; Lange et al. 2019; Hildebrandt et al. 2020; Asgari et al. 2020a), which are currently unresolved.

In this paper we perform a new study regarding this question using the latest weak gravitational lensing dataset from the Kilo-Degree Survey, KiDS-1000 (Kuijken et al. 2019), in conjunction with overlapping galaxy spectroscopic redshift survey data from the Baryon Oscillation Spectroscopic Survey (BOSS, Reid et al. 2016) and the 2-degree Field Lensing Survey (2dFLenS, Blake et al. 2016a). In particular, we focus on a simple implementation of the lensing-clustering test which compares the amplitude of gravitational lensing around foreground galaxies (commonly known as galaxy-galaxy lensing), tracing low-redshift overdensities, with the amplitude of galaxy velocities induced by these overdensities and measured by redshift-space distortions, which constitutes an amplitude-ratio test. This diagnostic was first proposed by Zhang et al. (2007) as the E_G statistic and implemented in its current form by Reyes et al. (2010) using data from the Sloan Digital Sky Survey. These measurements have subsequently been refined by a series of studies (Blake et al. 2016b; Pullen et al. 2016; Alam et al. 2017b; de la Torre et al. 2017; Amon et al. 2018; Singh et al. 2019; Jullo et al. 2019) which have used new datasets to increase the accuracy of the amplitude-ratio determination, albeit showing some evidence of internal disagreement.

The availability of the KiDS-1000 dataset and associated calibration samples has allowed us to perform the most accurate existing amplitude-ratio test, on projected scales up to 100 h⁻¹ Mpc, including rigorous systematic-error control. As part of this analysis, we use these datasets and representative simulations to study the efficacy of different corrections for the effects of source photometric redshift errors, comparing different galaxy-galaxy lensing estimators and the relative performance of angular and projected statistics. Our analysis sets the stage for future per-cent level implementations of these tests using new datasets from the Dark Energy Spectroscopic Instrument (DESI, DESI Collaboration 2016), the 4-metre Multi-Object Spectrograph Telescope (4MOST, de Jong et al. 2019), the Rubin Observatory Legacy Survey of Space and Time (LSST, Ivezić et al. 2019), and the Euclid satellite (Laureijs et al. 2011).

This paper is structured as follows: in Sect. 2 we review the theoretical correlations between weak lensing and overdensity observables, on which galaxy-galaxy lensing studies are based. In Sect. 3 we summarise the angular and projected galaxy-galaxy lensing estimators derived from these correlations, with particular attention to the effect of source photometric redshift errors. In Sect. 4 we introduce the amplitude-ratio test between galaxy-galaxy lensing and clustering observables, constructed from annular differential surface density statistics, and in Sect. 5 we derive the analytical covariances of these estimators in the Gaussian approximation, including the effects of the survey window function. We introduce the KiDS-1000 weak lensing and overlapping Luminous Red Galaxy (LRG) spectroscopic datasets in Sect. 6. We create representative survey mock catalogues in Sect. 7, which we use to verify our cosmological analysis in Sect. 8. Finally, we describe the results of our cosmological tests applied to the KiDS-LRG datasets in Sect. 9. We summarise our investigation in Sect. 10.

2. Theory

In this section we briefly review the theoretical expressions for the auto- and cross-correlations between weak gravitational lensing and galaxy overdensity observables, which form the basis of galaxy-galaxy lensing studies.

2.1. Lensing convergence and tangential shear

The observable effects of weak gravitational lensing, on a source located at co-moving co-ordinate χ_s in sky direction $\widehat{\mathrm{\Omega }}$, can be expressed in terms of the lensing convergence κ (for reviews, see Bartelmann & Schneider 2001; Kilbinger 2015; Mandelbaum 2018). The convergence is a weighted integral over co-moving distance χ of the matter overdensity δ_m along the line-of-sight, which we can write as

$$\begin{array}{c}\hfill \kappa ({\chi }_{\mathrm{s}},\widehat{\mathrm{\Omega }})=\frac{3{\mathrm{\Omega }}_{\mathrm{m}}{H}_0^2}{2c^2}{\displaystyle {\int }_0^{{\chi }_{\mathrm{s}}}\mathrm{d}\chi \frac{\chi ({\chi }_{\mathrm{s}}-\chi )}{{\chi }_{\mathrm{s}}}\frac{{\delta }_{\mathrm{m}}(\chi ,\widehat{\mathrm{\Omega }})}{a(\chi )},}\end{array}$$(1)

assuming (throughout this paper) a spatially-flat Universe, where Ω_m is the matter density as a fraction of the critical density, H₀ is the Hubble parameter, c is the speed of light, and a = 1/(1 + z) is the cosmic scale factor at redshift z. We can conveniently write Eq. (1) in terms of the critical surface mass density at a lens plane at co-moving distance χ_l,

$$\begin{array}{c}\hfill {\mathrm{\Sigma }}_{\mathrm{c}}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}})=\frac{c^2}{4\pi G}\frac{{\chi }_{\mathrm{s}}}{(1+z_{\mathrm{l}}){\chi }_{\mathrm{l}}({\chi }_{\mathrm{s}}-{\chi }_{\mathrm{l}})},\end{array}$$(2)

where G is the gravitational constant, and χ_s >  χ_l. Hence,

$$\begin{array}{c}\hfill \kappa ({\chi }_{\mathrm{s}},\widehat{\mathrm{\Omega }})=\overline{{\rho }_{\mathrm{m}}}{\displaystyle {\int }_0^{{\chi }_{\mathrm{s}}}\mathrm{d}\chi {\mathrm{\Sigma }}_{\mathrm{c}}^{-1}(\chi ,{\chi }_{\mathrm{s}}){\delta }_{\mathrm{m}}(\chi ,\widehat{\mathrm{\Omega }}),}\end{array}$$(3)

where $\overline{{\rho }_{\mathrm{m}}}$ is the mean matter density¹.

Suppose that the overdensity is associated with an isolated lens galaxy at distance χ_l in an otherwise homogeneous Universe. In this case, Eq. (3) may be written in the form

$$\begin{array}{c}\hfill \kappa ({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}},\widehat{\mathrm{\Omega }})\approx \overline{{\rho }_{\mathrm{m}}}{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}}){\displaystyle {\int }_0^{{\chi }_{\mathrm{s}}}\mathrm{d}\chi {\delta }_{\mathrm{m}}(\chi ,\widehat{\mathrm{\Omega }}).}\end{array}$$(4)

Equation (4) motivates that the weak lensing observable can be related to the projected mass density around the lens, Σ = ∫ρ_m dχ, where ${\delta }_{\mathrm{m}}={\rho }_{\mathrm{m}}/\overline{{\rho }_{\mathrm{m}}}-1$. The convergence may be written in terms of this quantity as

$$\begin{array}{c}\hfill \kappa ({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}},\widehat{\mathrm{\Omega }})\approx {\mathrm{\Sigma }}_{\mathrm{c}}^{-1}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}})(\mathrm{\Sigma }-\overline{\mathrm{\Sigma }}),\end{array}$$(5)

where $\overline{\mathrm{\Sigma }}=\int \overline{\rho }\mathrm{d}\chi$ represents the average background, emphasising that gravitational lensing traces the increment between the mass density and the background.

The average tangential shear γ_t at angular separation θ from an axisymmetric lens is related to the convergence as

$$\begin{array}{c}\hfill ⟨{\gamma }_{\mathrm{t}}(\theta )⟩=⟨\overline{\kappa }(<\theta )⟩-⟨\kappa (\theta )⟩,\end{array}$$(6)

where $\overline{\kappa }(<\theta )$ is the mean convergence within separation θ. At the location of the lens, angular separations are related to projected separations as R = χ(z_l) θ. Defining the differential projected surface mass density around the lens as a function of projected separation,

$$\begin{array}{c}\hfill \mathrm{\Delta }\mathrm{\Sigma }(R)=\overline{\mathrm{\Sigma }}(<R)-\mathrm{\Sigma }(R),\end{array}$$(7)

where,

$$\begin{array}{c}\hfill \overline{\mathrm{\Sigma }}(<R)=\frac{2}{R^2}{\displaystyle {\int }_0^R{R}^{\prime }\mathrm{\Sigma }(R^{\prime })\mathrm{d}{R}^{\prime },}\end{array}$$(8)

we find that for a single source-lens pair at distances χ_l and χ_s (omitting the angled brackets),

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{t}}(\theta )={\mathrm{\Sigma }}_{\mathrm{c}}^{-1}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}})\mathrm{\Delta }\mathrm{\Sigma }(R).\end{array}$$(9)

2.2. Galaxy-convergence cross-correlation

For an ensemble of sources with distance probability distribution p_s(χ) (normalised such that ∫p_s(χ) dχ = 1), the total convergence in a given sky direction is

$$\begin{array}{cc}\hfill \kappa (\widehat{\mathrm{\Omega }})& ={\displaystyle \int }\mathrm{d}{\chi }_{\mathrm{s}}{p}_{\mathrm{s}}({\chi }_{\mathrm{s}})\kappa ({\chi }_{\mathrm{s}},\widehat{\mathrm{\Omega }})\hfill \\ \hfill & =\overline{{\rho }_{\mathrm{m}}}{\displaystyle {\int }_0^{\infty }\mathrm{d}\chi \overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}(\chi ){\delta }_{\mathrm{m}}(\chi ,\widehat{\mathrm{\Omega }}),}\hfill \end{array}$$(10)

where,

$$\begin{array}{c}\hfill \overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}(\chi )={\displaystyle {\int }_{\chi }^{\infty }\mathrm{d}{\chi }_{\mathrm{s}}{p}_{\mathrm{s}}({\chi }_{\mathrm{s}}){\mathrm{\Sigma }}_{\mathrm{c}}^{-1}(\chi ,{\chi }_{\mathrm{s}}),}\end{array}$$(11)

with the lower limit of the integral applying because ${\mathrm{\Sigma }}_{\mathrm{c}}^{-1}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}})=0$ for χ_s <  χ_l. We consider forming the angular cross-correlation function of this convergence field with the projected number overdensity of an ensemble of lenses with distance probability distribution p_l(χ),

$$\begin{array}{c}\hfill {\delta }_{\mathrm{g},2\mathrm{D}}(\widehat{\mathrm{\Omega }})={\displaystyle \int }\mathrm{d}\chi p_{\mathrm{l}}(\chi ){\delta }_{\mathrm{g}}(\chi ,\widehat{\mathrm{\Omega }}).\end{array}$$(12)

The galaxy-convergence cross-correlation function at angular separation θ is,

$$\begin{array}{c}\hfill {\omega }_{\mathrm{g}\kappa }(\mathit{\theta })=⟨\kappa (\widehat{\mathrm{\Omega }}){\delta }_{\mathrm{g},2\mathrm{D}}(\widehat{\mathrm{\Omega }}+\mathit{\theta })⟩.\end{array}$$(13)

Expressing the overdensity fields in terms of their Fourier components we find, after some algebra,

$$\begin{array}{c}\hfill {\omega }_{\mathrm{g}\kappa }(\mathit{\theta })={\displaystyle \int }\frac{{\mathrm{d}}^2\mathit{\ell }}{{(2\pi )}^2}{C}_{\mathrm{g}\kappa }(\mathit{\ell }){\mathrm{e}}^{-\mathrm{i}\mathit{\ell }·\mathit{\theta }},\end{array}$$(14)

where ℓ is a 2D Fourier wavevector, and the corresponding angular cross-power spectrum C_gκ(ℓ) is given by (Guzik & Seljak 2001; Hu & Jain 2004; Joachimi & Bridle 2010),

$$\begin{array}{c}\hfill C_{\mathrm{g}\kappa }(\ell )=\overline{{\rho }_{\mathrm{m}}}{\displaystyle \int }\mathrm{d}\chi p_{\mathrm{l}}(\chi )\frac{\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}(\chi )}{{\chi }^2}{P}_{\mathrm{gm}}(\frac{\ell }{\chi },\chi ),\end{array}$$(15)

where P_gm(k, χ) is the 3D galaxy-matter cross-power spectrum at wavenumber k and distance χ. Taking the azimuthal average of Eq. (14) over all directions θ, the complex exponential integrates to a Bessel function of the first kind, J₀(x), such that,

$$\begin{array}{c}\hfill {\omega }_{\mathrm{g}\kappa }(\theta )={\displaystyle \int }\frac{{\mathrm{d}}^2\mathit{\ell }}{{(2\pi )}^2}{C}_{\mathrm{g}\kappa }(\mathit{\ell })J_0(\ell \theta )=\int \frac{\mathrm{d}\ell \ell }{2\pi }{C}_{\mathrm{g}\kappa }(\ell )J_0(\ell \theta ).\end{array}$$(16)

Using Eq. (6) and Bessel function identities, we can then obtain an expression for the statistical average tangential shear around an ensemble of lenses,

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{t}}(\theta )=\overline{{\omega }_{\mathrm{g}\kappa }}(<\theta )-{\omega }_{\mathrm{g}\kappa }(\theta )={\displaystyle \int }\frac{\mathrm{d}\ell \ell }{2\pi }{C}_{\mathrm{g}\kappa }(\ell )J_2(\ell \theta ).\end{array}$$(17)

Likewise, we can generalise Eq. (9) to apply to broad source and lens distributions:

$$\begin{array}{c}\hfill {\gamma }_{\mathrm{t}}(\theta )={\displaystyle \int }\mathrm{d}\chi p_{\mathrm{l}}(\chi )\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}(\chi )\mathrm{\Delta }\mathrm{\Sigma }(R,\chi ).\end{array}$$(18)

Comparing the formulations of Eqs. (17) and (18) allows us to demonstrate that,

$$\begin{array}{c}\hfill \mathrm{\Sigma }(R)=\overline{{\rho }_{\mathrm{m}}}{\displaystyle {\int }_{-\infty }^{\infty }\mathrm{d}\mathrm{\Pi }[1+{\xi }_{\mathrm{gm}}(R,\mathrm{\Pi })],}\end{array}$$(19)

in terms of the 3D galaxy-matter cross-correlation function ξ_gm(R, Π) at projected separation R and line-of-sight separation Π, where the constant term “1+” cancels out in the evaluation of the observable ΔΣ. After some algebra we find,

$$\begin{array}{c}\hfill \mathrm{\Delta }\mathrm{\Sigma }(R)=\overline{{\rho }_{\mathrm{m}}}{\displaystyle {\int }_0^{\infty }\mathrm{d}rW(r,R){\xi }_{\mathrm{gm}}(r),}\end{array}$$(20)

where,

$$\begin{array}{c}\hfill W(r,R)=\frac{4r^2}{R^2}-[\frac{4r\sqrt{r^2-R^2}}{R^2}+\frac{2r}{\sqrt{r^2-R^2}}]H(r-R),\end{array}$$(21)

where H(x) = 0 if x <  0 and H(x) = 1 if x >  0 is the Heaviside step function. The relations in this section make the approximations of using the Limber equation (Limber 1953) and neglecting additional effects such as cosmic magnification (Unruh et al. 2020) and intrinsic alignments (Joachimi et al. 2015).

2.3. Auto-correlation functions

In order to determine the analytical covariance in Sect. 5, we also need expressions for the auto-correlation functions of the convergence, ${\omega }_{\kappa \kappa }(\mathit{\theta })=\int \frac{{\mathrm{d}}^2\mathit{\ell }}{{(2\pi )}^2}{C}_{\kappa \kappa }(\mathit{\ell }){\mathrm{e}}^{-\mathrm{i}\mathit{\ell }·\mathit{\theta }}$, and the galaxy overdensity, ${\omega }_{\mathrm{gg}}(\mathit{\theta })=\int \frac{{\mathrm{d}}^2\mathit{\ell }}{{(2\pi )}^2}{C}_{\mathrm{gg}}(\mathit{\ell }){\mathrm{e}}^{-\mathrm{i}\mathit{\ell }·\mathit{\theta }}$. Given two source populations with distance probability distributions p_s, 1(χ) and p_s, 2(χ), and associated integrated critical density functions $\overline{{\mathrm{\Sigma }}_{\mathrm{c},1}^{-1}}$ and $\overline{{\mathrm{\Sigma }}_{\mathrm{c},2}^{-1}}$, the angular power spectrum of the convergence is given by,

$$\begin{array}{c}\hfill C_{\kappa \kappa }(\ell )={\overline{{\rho }_{\mathrm{m}}}}^2{\displaystyle \int }\mathrm{d}\chi \frac{\overline{{\mathrm{\Sigma }}_{\mathrm{c},1}^{-1}}(\chi )\overline{{\mathrm{\Sigma }}_{\mathrm{c},2}^{-1}}(\chi )}{{\chi }^2}{P}_{\mathrm{mm}}(\frac{\ell }{\chi },\chi ),\end{array}$$(22)

where P_mm(k, χ) is the 3D (non-linear) matter power spectrum at wavenumber k and distance χ. Likewise, for two projected galaxy overdensity fields with distance probability distributions p_l, 1(χ) and p_l, 2(χ), the angular power spectrum is,

$$\begin{array}{c}\hfill C_{\mathrm{gg}}(\ell )={\displaystyle \int }\mathrm{d}\chi \frac{p_{\mathrm{l},1}(\chi )p_{\mathrm{l},2}(\chi )}{{\chi }^2}{P}_{\mathrm{gg}}(\frac{\ell }{\chi },\chi ),\end{array}$$(23)

where P_gg(k, χ) is the 3D galaxy power spectrum.

2.4. Bias model

We computed the linear matter power spectrum P_L(k) in our models using the CAMB software package (Lewis et al. 2000), and evaluated the non-linear matter power spectrum P_mm(k) including the “halofit” corrections (Smith et al. 2003; Takahashi et al. 2012, we define the fiducial cosmological parameters used for the simulation and data analysis in subsequent sections). We adopted a model for the non-linear galaxy-galaxy and galaxy-matter 2-point functions, appearing in Eqs. (15) and (23), following Baldauf et al. (2010) and Mandelbaum et al. (2013). This model assumes a local, non-linear galaxy bias relation via a Taylor expansion of the galaxy density field in terms of the matter overdensity, ${\delta }_{\mathrm{g}}=b_{\mathrm{L}}{\delta }_{\mathrm{m}}+\frac{1}{2}{b}_{\mathrm{NL}}{\delta }_{\mathrm{m}}^2+\dots$, defining a linear bias parameter b_L and non-linear bias parameter b_NL. The auto- and cross-correlation statistics in this model can be written in the form (McDonald 2006; Smith et al. 2009),

$$\begin{array}{cc}& {\xi }_{\mathrm{gg}}=b_{\mathrm{L}}^2{\xi }_{\mathrm{mm}}+2b_{\mathrm{L}}{b}_{\mathrm{NL}}{\xi }_{\mathrm{A}}+\frac{1}{2}{b}_{\mathrm{NL}}^2{\xi }_{\mathrm{B}},\hfill \\ \hfill & {\xi }_{\mathrm{gm}}=b_{\mathrm{L}}{\xi }_{\mathrm{mm}}+b_{\mathrm{NL}}{\xi }_{\mathrm{A}},\hfill \end{array}$$(24)

where ξ_mm is the correlation function corresponding to P_mm(k), and ξ_A and ξ_B are obtained by computing the Fourier transforms of,

$$\begin{array}{cc}\hfill A(k)& ={\displaystyle \int }\frac{{\mathrm{d}}^3\mathit{q}}{{(2\pi )}^3}{F}_2(\mathit{q},\mathit{k}-\mathit{q})P_{\mathrm{L}}(q)P_{\mathrm{L}}(|\mathit{k}-\mathit{q}|),\hfill \\ \hfill B(k)& ={\displaystyle \int }\frac{{\mathrm{d}}^3\mathit{q}}{{(2\pi )}^3}{P}_{\mathrm{L}}(q)P_{\mathrm{L}}(|\mathit{k}-\mathit{q}|),\hfill \end{array}$$(25)

which depend on the mode-coupling kernel in standard perturbation theory,

$$\begin{array}{c}\hfill F_2({\mathit{q}}_1,{\mathit{q}}_2)=\frac{5}{7}+\frac{1}{2}\frac{{\mathit{q}}_1.{\mathit{q}}_2}{q_1{q}_2}(\frac{q_1}{q_2}+\frac{q_2}{q_1})+\frac{2}{7}{\left(\frac{{\mathit{q}}_1.{\mathit{q}}_2}{q_1{q}_2}\right)}^2.\end{array}$$(26)

We evaluated these integrals using the FAST software package (McEwen et al. 2016) and note that ${\xi }_{\text{B}}={\xi }_{\text{L}}^2$, where ξ_L is the correlation function corresponding to P_L(k). This model is only expected to be valid on scales exceeding the virial radius of dark matter haloes, since it does not address halo exclusion, the distribution of galaxies within haloes, or other forms of stochastic or non-local effects (Asgari et al. 2020b). However, this 2-parameter bias model is adequate for our large-scale analysis, which we verify using representative mock catalogues in Sect. 8.

3. Estimators

In this section we specify estimators that may be used to measure γ_t(θ) and ΔΣ(R) from ensembles of sources and lenses, and discuss how estimates of ΔΣ(R) are affected by uncertainties in source distances.

3.1. Average tangential shear γ_t(θ)

We can estimate the average tangential shear of a set of sources (s) around lenses (l) by evaluating the following expression (Mandelbaum et al. 2006), which also utilises an unclustered random lens catalogue (r) with the same selection function as the lenses:

$$\begin{array}{c}\hfill \widehat{{\gamma }_{\mathrm{t}}}(\theta )=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{ls}}-\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{rs}}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}}.\end{array}$$(27)

The sums in Eq. (27) are taken over pairs of sources and lenses with angular separations within a bin around θ, w_i are weights applied to the different samples (normalised such that ∑_lw_l = ∑_rw_r), and e_t indicates the tangential ellipticity of the source, projected onto an axis normal to the line joining the source and lens (or random lens).

Equation (27) involves the random lens catalogue in two places. First, the tangential shear of sources around random lenses is subtracted from the data signal. The subtracted term has an expectation value of zero, but significantly decreases the variance of the estimator at large separations (Singh et al. 2017). Second, the estimator is normalised by a sum over pairs of sources and random lenses, rather than data lenses. This ensures that the estimator is unbiased: the alternative estimator, $\widehat{{\gamma }_{\text{t}}}=\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{ls}}/\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}$, is biased by any source-lens clustering (if the angular cross-correlation function ω_ls(θ)≠0), which would modify the denominator of the expression but not the numerator. The magnitude of this effect is sometimes known as the “boost” factor (Sheldon et al. 2004),

$$\begin{array}{c}\hfill B(\theta )=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}},\end{array}$$(28)

where the sums are again taken over source-lens pairs with angular separations within a given bin. We note that ⟨B(θ)⟩ = 1 + ω_ls(θ) for unity weights.

3.2. Projected mass density ΔΣ(R)

Assuming the source and lens distances are known, each source-lens pair may be used to estimate the projected mass density around the lenses by inverting Eq. (9):

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}(R)=e_{\mathrm{t}}(R/{\chi }_{\mathrm{l}}){\mathrm{\Sigma }}_{\mathrm{c}}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}}).\end{array}$$(29)

For an ensemble of sources and lenses, the mean projected mass density may then be estimated by an expression analogous to Eq. (27) (Singh et al. 2017),

$$\begin{array}{cc}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}(R)& =\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}{w}_{\mathrm{ls}}{e}_{\mathrm{t},\mathrm{ls}}(R/{\chi }_{\mathrm{l}}){\mathrm{\Sigma }}_{\mathrm{c}}({\chi }_{\mathrm{l}},{\chi }_{\mathrm{s}})}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{w}_{\mathrm{rs}}}\hfill \\ \hfill & -\frac{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{w}_{\mathrm{rs}}{e}_{\mathrm{t},\mathrm{rs}}(R/{\chi }_{\mathrm{r}}){\mathrm{\Sigma }}_{\mathrm{c}}({\chi }_{\mathrm{r}},{\chi }_{\mathrm{s}})}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{w}_{\mathrm{rs}}},\hfill \end{array}$$(30)

where we have allowed for an additional pair weight between sources and lenses, w_ls, and random lenses, w_rs. Assuming a constant shape noise in e_t, the noise in the estimate of ΔΣ(R) = e_t Σ_c from each source-lens pair is proportional to Σ_c, hence the optimal inverse-variance weight is $w_{\mathrm{ls}}\propto {\mathrm{\Sigma }}_{\mathrm{c}}^{-2}$, and the weighted estimator may be written,

$$\begin{array}{cc}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}(R)& =\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{ls}}(R/{\chi }_{\mathrm{l}}){\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}^{-1}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-2}}\hfill \\ \hfill & -\frac{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{rs}}(R/{\chi }_{\mathrm{r}}){\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-1}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-2}}.\hfill \end{array}$$(31)

3.3. Photo-z dilution correction for ΔΣ(R)

The difficulty faced when determining ΔΣ is that source distances are typically only accessible through photometric redshifts and may contain significant errors, leading to a bias in the estimate through incorrect scaling factors Σ_c (we assume in this discussion that spectroscopic lens distances are available). For example, sources may apparently lie behind lenses according to their photometric redshift, whilst in fact being positioned in front of the lenses and contributing no galaxy-galaxy lensing signal, creating a downward bias in the measurement.

For a single source-lens pair, the estimated value of Σ_c for the pair based on the source photometric redshift, Σ_c, lp, may differ from its true value based on the source spectroscopic redshift, Σ_c, ls,

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}=e_{\mathrm{t}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}=\left(\frac{\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathrm{true}}}{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}}\right){\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}=\left(\frac{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}}{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}}\right)\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathrm{true}}.\end{array}$$(32)

Combining many source-lens pairs allowing for a pair weight w_ls we find,

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{ls}}\left(\frac{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}}{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}}\right)\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathrm{true}}}{\sum _{\mathrm{ls}}{w}_{\mathrm{ls}}}.\end{array}$$(33)

Using the optimal weight $w_{\mathrm{ls}}\propto {\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}^{-2}$ this expression may be written,

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}=\frac{\sum _{\mathrm{ls}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}^{-1}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}^{-1}\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathrm{true}}}{\sum _{\mathrm{ls}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}^{-2}}.\end{array}$$(34)

The estimated value of ΔΣ hence contains a multiplicative bias, $\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathrm{true}}=f_{\mathrm{bias}}⟨\widehat{\mathrm{\Delta }\mathrm{\Sigma }}⟩$ where,

$$\begin{array}{c}\hfill f_{\mathrm{bias}}=\frac{\sum _{\mathrm{ls}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}^{-2}}{\sum _{\mathrm{ls}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}^{-1}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}^{-1}}=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{ls}}}{\sum _{\mathrm{ls}}{w}_{\mathrm{ls}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{lp}}{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}^{-1}}.\end{array}$$(35)

This multiplicative correction factor may be estimated at each lens redshift from a representative subset of sources with complete spectroscopic and photometric redshift information, by evaluating the sums in the numerator and denominator of Eq. (35) (Nakajima et al. 2012).

An alternative formulation of the photo-z dilution correction may be derived from the statistical distance distribution of the sources. Provided that the lens distribution is sufficiently narrow, Eq. (18) indicates that an unbiased estimate of ΔΣ from each lens-source pair is,

$$\begin{array}{c}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}(R)=e_{\mathrm{t}}(R/{\chi }_{\mathrm{l}}){[\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}({\chi }_{\mathrm{l}})]}^{-1},\end{array}$$(36)

where $\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}$ is evaluated from Eq. (11) using the source distribution p_s(χ). This motivates an alternative estimator mirroring Eq. (31) (Sheldon et al. 2004; Miyatake et al. 2015; Blake et al. 2016b),

$$\begin{array}{cc}\hfill \widehat{\mathrm{\Delta }\mathrm{\Sigma }}(R)& =\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{ls}}(R/{\chi }_{\mathrm{l}})\overline{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}^{-1}}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{\left(\overline{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-1}}\right)}^2}\hfill \\ \hfill & -\frac{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{e}_{\mathrm{t},\mathrm{rs}}(R/{\chi }_{\mathrm{r}})\overline{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-1}}}{\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{\left(\overline{{\mathrm{\Sigma }}_{\mathrm{c},\mathrm{rs}}^{-1}}\right)}^2}.\hfill \end{array}$$(37)

The accuracy of these potential photo-z dilution corrections must be assessed via simulations, which we consider in Sect. 8. We trialled both point-based and distribution-based correction methods in our analysis.

4. Amplitude-ratio test

In this section we construct test statistics which utilise the relative amplitudes of galaxy clustering and galaxy-galaxy lensing to test cosmological models. We first define the input statistics for these tests.

4.1. Projected clustering w_p(R)

The amplitude of galaxy-galaxy lensing is sensitive to the distribution of matter around lens galaxies, projected along the line-of-sight. We can obtain an analogous projected quantity for lens galaxy clustering by integrating the 3D galaxy auto-correlation function, ξ_gg along the line-of-sight,

$$\begin{array}{c}\hfill w_{\mathrm{p}}(R)={\displaystyle {\int }_{-\infty }^{\infty }\mathrm{d}\mathrm{\Pi }{\xi }_{\mathrm{gg}}(R,\mathrm{\Pi }),}\end{array}$$(38)

where Π is the line-of-sight separation. This formulation has the additional feature of reducing sensitivity of the clustering statistics to redshift-space distortions, which modulate the apparent radial separations Π between galaxy pairs.

We can estimate w_p(R) for a galaxy sample by measuring the galaxy correlation function in (R, Π) separation bins, and summing over the Π direction in the range 0 <  Π <  Π_max:

$$\begin{array}{c}\hfill \widehat{w_{\mathrm{p}}}(R)=2{\displaystyle \sum _{\mathrm{bins}i}\mathrm{\Delta }{\mathrm{\Pi }}_i{\widehat{\xi }}_{\mathrm{gg}}(R,\mathrm{\Pi }).}\end{array}$$(39)

4.2. The Upsilon statistics, Υ_gm(R) and Υ_gg(R)

Equation (7) demonstrates that the amplitude of ΔΣ(R) around lens galaxies depends on the surface density of matter across a range of smaller scales from zero to R, and hence on the galaxy-matter cross-correlation coefficient at these scales. Given that this cross-correlation is a complex function which is difficult to model from first principles, it is beneficial to reduce this sensitivity to small-scale information using the annular differential surface density statistic (Reyes et al. 2010; Baldauf et al. 2010; Mandelbaum et al. 2013),

$$\begin{array}{cc}\hfill {\mathrm{{\rm Y}}}_{\mathrm{gm}}(R,R_0)& =\mathrm{\Delta }\mathrm{\Sigma }(R)-\frac{R_0^2}{R^2}\mathrm{\Delta }\mathrm{\Sigma }(R_0)\hfill \\ \hfill & =\frac{2}{R^2}{\displaystyle {\int }_{R_0}^R\mathrm{d}R\prime R\prime \mathrm{\Sigma }(R\prime )-\mathrm{\Sigma }(R)+\frac{R_0^2}{R^2}\mathrm{\Sigma }(R_0),}\hfill \end{array}$$(40)

which is defined such that Υ_gm = 0 at some small-scale limit R = R₀, chosen to be large enough to reduce the main systematic effects (typically, R₀ is somewhat larger than the size scale of dark matter haloes). In this sense, the cumulative effect from the cross-correlation function at scales R <  R₀ is cancelled, although it is not the case that this small-scale suppression translates to Fourier space (Baldauf et al. 2010; Asgari et al. 2020b; Park et al. 2020). In any case, the efficacy of these statistics and choice of the R₀ value must be validated using simulations, as we consider below.

The corresponding quantity suppressing the small-scale contribution to the galaxy auto-correlations is (Reyes et al. 2010),

$$\begin{array}{cc}& {\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0)={\rho }_{\mathrm{c}}\hfill \\ \hfill & {\displaystyle [\frac{2}{R^2}{\int }_{R_0}^R\mathrm{d}R\prime R\prime w_{\mathrm{p}}(R\prime )-w_{\mathrm{p}}(R)+\frac{R_0^2}{R^2}{w}_{\mathrm{p}}(R_0)],}\hfill \end{array}$$(41)

where ρ_c is the critical matter density. We note that if w_p is defined as a step-wise function in bins R_i (with bin limits R_i, min and R_i, max) then Eq. (41) may be written in the useful form,

$$\begin{array}{c}\hfill {\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0)=\frac{{\rho }_{\mathrm{c}}}{R^2}{\displaystyle \sum _{i=j}^k{C}_i{w}_{\mathrm{p}}(R_i),}\end{array}$$(42)

where (k, j) are the bins containing (R, R₀), and

$$\begin{array}{c}\hfill C_i=\{\begin{array}{cc}{R}_{i,\max }^2\hfill & i=j\hfill \\ R_{i,\max }^2-R_{i,\min }^2\hfill & j<i<k\hfill \\ -R_{i,\min }^2\hfill & i=k\hfill \end{array}\end{array}$$(43)

For convenience we chose R₀ to coincide with the centre of a separation bin, such that we could use the direct measurements of ΔΣ(R₀) and w_p(R₀) in Eqs. (40) and (41) without interpolation between bins (we will show below that our results are not sensitive to the choice of R₀).

4.3. The E_G test statistic

The relative amplitudes of weak gravitational lensing and the rate of assembly of large-scale structure depend on the “gravitational slip” or difference between the two space-time metric potentials. This signature is absent in general relativity but may be significant in modified gravity scenarios (Uzan & Bernardeau 2001; Zhang et al. 2007; Jain & Khoury 2010; Bertschinger 2011; Clifton et al. 2012).

Zhang et al. (2007) proposed that these amplitudes might be compared by connecting the velocity field and lensing signal generated by a given set of matter overdensities, probed via redshift-space distortions and galaxy-galaxy lensing, respectively. Reyes et al. (2010) implemented this consistency test by constructing the statistic,

$$\begin{array}{c}\hfill E_{\mathrm{G}}(R)=\frac{1}{\beta }\frac{{\mathrm{{\rm Y}}}_{\mathrm{gm}}(R,R_0)}{{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0)},\end{array}$$(44)

where β = f/b_L is the redshift-space distortion parameter which governs the observed dependence of the strength of galaxy clustering on the angle to the line-of-sight, in terms of the linear growth rate of a perturbation, f = dlnδ/dlna. Equation (44) is independent of the linear galaxy bias b_L and the amplitude of matter clustering σ₈, given that β ∝ 1/b_L, ${{\rm Y}}_{\text{gm}}\propto b_{\text{L}}{\sigma }_8^2$ and ${{\rm Y}}_{\text{gg}}\propto b_{\text{L}}^2{\sigma }_8^2$. The prediction of linear perturbation theory for general relativity in a Λ cold dark matter (CDM) Universe is a scale-independent value E_G(z) = Ω_m(z = 0)/f(z), although see Leonard et al. (2015) for a detailed discussion of this approxmation.

5. Covariance of estimators

In this section we present analytical formulations in the Gaussian approximation for the covariance of estimates of γ_t(θ) and ΔΣ(R), and model how this covariance is modulated by the presence of a survey mask (that is, by edge effects). Our covariance determination hence neglects non-Gaussian and super-sample variance components. This is a reasonable approximation in the context of the current analysis as these terms are subdominant (we refer the reader to Joachimi et al., in prep. for more details on the relative amplitude of the different covariance terms in the context of KiDS-1000).

5.1. Covariance of average tangential shear

In Appendix A we derive the covariance of γ_t averaged within angular bins θ_m and θ_n:

$$\begin{array}{c}\hfill \mathrm{Cov}[{\gamma }_{\mathrm{t}}^{\mathit{ij}}({\theta }_m),{\gamma }_{\mathrm{t}}^{\mathit{kl}}({\theta }_n)]=\frac{1}{\mathrm{\Omega }}\int \frac{\mathrm{d}\ell \ell }{2\pi }{\sigma }^2(\ell )\overline{J_{2,m}}(\ell )\overline{J_{2,n}}(\ell ),\end{array}$$(45)

where ${\gamma }_{\mathrm{t}}^{\mathit{ij}}$ denotes the average tangential shear of source sample j around lens sample i, Ω is the total survey angular area in steradians, and $\overline{J_{2,n}}(\ell )={\int }_{{\theta }_{1,n}}^{{\theta }_{2,n}}\frac{2\pi \theta \mathrm{d}\theta }{{\mathrm{\Omega }}_n}{J}_2(\ell \theta )$, where the integral is between the bin limits θ₁ and θ₂ and Ω_n is angular area of bin n (i.e. the area of the annulus between the bin limits). The variance σ²(ℓ) is given by the expression for Gaussian random fields (e.g. Hu & Jain 2004; Bernstein 2009; Joachimi & Bridle 2010; Krause & Eifler 2017; Euclid Collaboration 2020),

$$\begin{array}{c}\hfill {\sigma }^2(\ell )=C_{\mathrm{g}\kappa }^{\mathit{il}}(\ell )C_{\mathrm{g}\kappa }^{\mathit{kj}}(\ell )+[C_{\kappa \kappa }^{\mathit{jl}}(\ell )+N_{\kappa \kappa }^j{\delta }_{\mathit{jl}}^{\mathrm{K}}][C_{\mathrm{gg}}^{\mathit{ik}}(\ell )+N_{\mathrm{gg}}^i{\delta }_{\mathit{ik}}^{\mathrm{K}}],\end{array}$$(46)

where ${\delta }_{\mathit{ij}}^{\mathrm{K}}$ is the Kronecker delta. The angular auto- and cross-power spectra appearing in Eq. (46) may be evaluated using the expressions in Sect. 2, and the noise terms are $N_{\kappa \kappa }^i={\sigma }_{\mathrm{e}}^2/{\overline{n}}_{\mathrm{s}}^i$ and $N_{\mathrm{gg}}^i=1/{\overline{n}}_{\mathrm{l}}^i$, where σ_e is the shape noise and ${\overline{n}}_{\mathrm{l}}^i$ and ${\overline{n}}_{\mathrm{s}}^i$ are the angular lens and source densities of sample i in units of per steradian.

5.2. Covariance of projected mass density

The covariance of ΔΣ may be deduced from the covariance of γ_t using $\mathrm{\Delta }\mathrm{\Sigma }(R)={\gamma }_{\mathrm{t}}(\theta )/\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}$, and by scaling angular separations to projected separations at an effective lens distance χ_l using θ = R/χ_l (Singh et al. 2017; Dvornik et al. 2018; Shirasaki & Takada 2018). We can map multipoles ℓ to the projected wavevector k = ℓ/χ_l such that,

$$\begin{array}{c}\hfill \mathrm{Cov}[\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathit{ij}}(R),\mathrm{\Delta }{\mathrm{\Sigma }}^{\mathit{kl}}(R\prime )]=\frac{1}{\mathrm{\Omega }}\int \frac{\mathrm{d}kk}{2\pi }{\sigma }^2(k)\overline{J_2}(kR)\overline{J_2}(kR\prime ),\end{array}$$(47)

where we now express the variance in terms of projected power spectra,

$$\begin{array}{c}\hfill {\sigma }^2(k)=P_{\mathrm{g}\kappa }^{\mathit{il}}(k)P_{\mathrm{g}\kappa }^{\mathit{kj}}(k)+[P_{\kappa \kappa }^{\mathit{jl}}(k)+N_{\kappa \kappa }^j{\delta }_{\mathit{jl}}^{\mathrm{K}}][P_{\mathrm{gg}}^{\mathit{ik}}(k)+N_{\mathrm{gg}}^i{\delta }_{\mathit{ik}}^{\mathrm{K}}].\end{array}$$(48)

The power spectra are given by the following relations:

$$\begin{array}{c}\hfill P_{\mathrm{g}\kappa }(k)=\frac{{\chi }_{\mathrm{l}}^2{C}_{\mathrm{g}\kappa }(k{\chi }_{\mathrm{l}})}{{[\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}({\chi }_{\mathrm{l}})]}^2}\approx \overline{{\rho }_{\mathrm{m}}}\int \mathrm{d}\chi p_{\mathrm{l}}(\chi )P_{\mathrm{gm}}(k,\chi ),\end{array}$$(49)

$$\begin{array}{cc}& P_{\kappa \kappa }(k)={\chi }_{\mathrm{l}}^2{C}_{\kappa \kappa }(k{\chi }_{\mathrm{l}})\hfill \\ \hfill & ={\chi }_{\mathrm{l}}^2{\overline{{\rho }_{\mathrm{m}}}}^2\int \mathrm{d}\chi \left[\frac{\overline{{\mathrm{\Sigma }}_{\mathrm{c},1}^{-1}}(\chi )\overline{{\mathrm{\Sigma }}_{\mathrm{c},2}^{-1}}(\chi )}{\overline{{\mathrm{\Sigma }}_{\mathrm{c},1}^{-1}}({\chi }_{\mathrm{l}})\overline{{\mathrm{\Sigma }}_{\mathrm{c},2}^{-1}}({\chi }_{\mathrm{l}})}\right]\left(\frac{{\chi }_{\mathrm{l}}^2}{{\chi }^2}\right)P_{\mathrm{mm}}(\frac{k{\chi }_{\mathrm{l}}}{\chi },\chi ),\hfill \end{array}$$(50)

$$\begin{array}{c}\hfill P_{\mathrm{gg}}(k)={\chi }_{\mathrm{l}}^2{C}_{\mathrm{gg}}(k{\chi }_{\mathrm{l}})\approx \int \mathrm{d}\chi p_1(\chi )p_2(\chi )P_{\mathrm{gg}}(k,\chi ),\end{array}$$(51)

and the noise terms are,

$$\begin{array}{c}\hfill N_{\kappa \kappa }=\frac{{\sigma }_{\mathrm{e}}^2{\chi }_{\mathrm{l}}^2}{{\overline{n}}_{\mathrm{s}}{[\overline{{\mathrm{\Sigma }}_{\mathrm{c}}^{-1}}({\chi }_{\mathrm{l}})]}^2},N_{\mathrm{gg}}=\frac{{\chi }_{\mathrm{l}}^2}{{\overline{n}}_{\mathrm{l}}}.\end{array}$$(52)

5.3. Covariance of remaining statistics

The expression for the analytical covariance of w_p(R) may be derived as (see also, Singh et al. 2017),

$$\begin{array}{cc}\hfill \mathrm{Cov}[w_{\mathrm{p}}& (R),w_{\mathrm{p}}(R\prime )]=\hfill \\ \hfill & \frac{2L_{\Vert }{\mathrm{\Pi }}_{\max }}{\mathrm{\Omega }}\int \frac{\mathrm{d}kk}{2\pi }{\sigma }^2(k)J_0(kR)J_0(kR\prime ),\hfill \end{array}$$(53)

where L_∥ is the total co-moving depth of the lens redshift slice and the expression for the variance is,

$$\begin{array}{c}\hfill {\sigma }^2(k)={[P_{\mathrm{gg}}(k)+N_{\mathrm{gg}}]}^2,\end{array}$$(54)

where P_gg(k) and N_gg are the 2D projected power spectra and noise as defined in Sect. 5.2.

We determined the analytical covariance of Υ_gm(R, R₀) from the covariance of ΔΣ(R):

$$\begin{array}{cc}\hfill \mathrm{Cov}[{\mathrm{{\rm Y}}}_{\mathrm{gm}}& (R,R_0),{\mathrm{{\rm Y}}}_{\mathrm{gm}}(R\prime ,R_0)]=\mathrm{Cov}[\mathrm{\Delta }\mathrm{\Sigma }(R),\mathrm{\Delta }\mathrm{\Sigma }(R\prime )]\hfill \\ \hfill & -\frac{R_0^2}{R^{\prime 2}}\mathrm{Cov}[\mathrm{\Delta }\mathrm{\Sigma }(R),\mathrm{\Delta }\mathrm{\Sigma }(R_0)]-\frac{R_0^2}{R^2}\mathrm{Cov}[\mathrm{\Delta }\mathrm{\Sigma }(R\prime ),\mathrm{\Delta }\mathrm{\Sigma }(R_0)]\hfill \\ \hfill & +\frac{R_0^4}{R^2{R}^{\prime 2}}\mathrm{Var}[\mathrm{\Delta }\mathrm{\Sigma }(R_0)].\hfill \end{array}$$(55)

For the case of Υ_gg(R, R₀), we propagated the covariance using Eq. (42):

$$\begin{array}{cc}\hfill \mathrm{Cov}[{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0),{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R\prime ,R_0)]& =\frac{{\rho }_{\mathrm{c}}^2}{R^2{R}^{\prime 2}}{\displaystyle \sum _i\sum _j}\hfill \\ \hfill & \times C_i{C}_j\mathrm{Cov}[w_{\mathrm{p}}(R_i),w_{\mathrm{p}}(R_j)].\hfill \end{array}$$(56)

We evaluated the covariance of the E_G statistic, where required, by assuming small fluctuations in the variables in Eq. (44) with respect to their mean, neglecting any correlations between the measurements:

$$\begin{array}{cc}\hfill \frac{\mathrm{Cov}[E_{\mathrm{G}}(R)E_{\mathrm{G}}(R\prime )]}{E_{\mathrm{G}}(R)E_{\mathrm{G}}(R\prime )}& =\frac{\mathrm{Cov}[{\mathrm{{\rm Y}}}_{\mathrm{gm}}(R,R_0),{\mathrm{{\rm Y}}}_{\mathrm{gm}}(R\prime ,R_0)]}{{\mathrm{{\rm Y}}}_{\mathrm{gm}}(R,R_0){\mathrm{{\rm Y}}}_{\mathrm{gm}}(R\prime ,R_0)}\hfill \\ \hfill & +\frac{\mathrm{Cov}[{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0),{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R\prime ,R_0)]}{{\mathrm{{\rm Y}}}_{\mathrm{gg}}(R,R_0){\mathrm{{\rm Y}}}_{\mathrm{gg}}(R\prime ,R_0)}+\frac{{\sigma }_{\beta }^2}{{\beta }^2},\hfill \end{array}$$(57)

where σ_β is the error in the measurement of β. This neglect of correlations is an approximation, justified in the case of our dataset by the fact that the sky area used for the galaxy clustering measurement is substantially different to the sub-sample used for galaxy-galaxy lensing (see Joachimi et al., in prep. for a detailed justification of this approximation), and that the projected lens clustering measurement (Υ_gg) is largely insensitive to redshift-space distortions (β) owing to the projection over the line-of-sight separations. We note that in our fiducial fitting approach, we determined the scale-independent statistic ⟨E_G⟩ through direct fits to Υ_gm and Υ_gg as discussed in Sect. 9, without requiring the covariance of E_G(R).

5.4. Modification of noise term

We can replace the noise terms in Sects. 5.1 and 5.2 with a more accurate computation using the survey source and lens distributions. Neglecting the random lens term (which is not important on the small scales for which the noise term is significant), we find that the variance associated with the γ_t estimator in Eq. (27) is (e.g. Miyatake et al. 2019),

$$\begin{array}{c}\hfill \mathrm{Var}[{\gamma }_{\mathrm{t}}(\theta )]=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}^2{w}_{\mathrm{s}}^2{\sigma }_{\mathrm{e}}^2}{{\left(\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}\right)}^2}.\end{array}$$(58)

Likewise, the variance associated with the ΔΣ estimator in Eq. (30) is,

$$\begin{array}{c}\hfill \mathrm{Var}[\mathrm{\Delta }\mathrm{\Sigma }(R)]=\frac{\sum _{\mathrm{ls}}{w}_{\mathrm{l}}^2{w}_{\mathrm{s}}^2{w}_{\mathrm{ls}}^2{\sigma }_{\mathrm{e}}^2{\left({\mathrm{\Sigma }}_{\mathrm{c},\mathrm{ls}}\right)}^2}{{\left(\sum _{\mathrm{rs}}{w}_{\mathrm{r}}{w}_{\mathrm{s}}{w}_{\mathrm{rs}}\right)}^2}.\end{array}$$(59)

We adopted these noise terms in our covariance model.

5.5. Modification for survey window

Equations (45) and (47) for the analytical covariance are modified by the survey window function. We can intuitively understand the need for this modification by considering that, whilst Fourier transforms assume periodic boundary conditions, the boundaries of the survey restrict the number of source-lens pairs on scales that are a significant fraction of the survey dimensions.

In Appendix B we derive how the covariance of a cross-correlation function ξ(r) between two Gaussian fields is modified by the window function of the fields, W₁(x) and W₂(x) (see also, Beutler et al. 2017). We find,

$$\begin{array}{cc}\hfill \mathrm{Cov}[\xi (r),\xi (s)]& \approx \frac{A_3(r,s)}{A_2(r)A_2(s)}\frac{1}{2\pi }\int \mathrm{d}kk\hfill \\ \hfill & \times [P_{11}(k)P_{22}(k)+P_{12}^2(k)]J_0(kr)J_0(ks),\hfill \end{array}$$(60)

where P₁₁, P₂₂ and P₁₂ are the auto- and cross-power spectra of the fields and the pre-factors are given by,

$$\begin{array}{cc}& A_2(r)={\int }_{\mathrm{bin}r}{\mathrm{d}}^3\mathit{r}\int {\mathrm{d}}^2\mathit{x}{W}_1(\mathit{x})W_2(\mathit{x}+\mathit{r})\hfill \\ \hfill & A_3(r,s)={\int }_{\mathrm{bin}r}{\mathrm{d}}^3\mathit{r}{\int }_{\mathrm{bin}s}{\mathrm{d}}^3\mathit{s}\int {\mathrm{d}}^2\mathit{x}{A}_{12}(\mathit{x},\mathit{r})A_{12}(\mathit{x},\mathit{s}),\hfill \end{array}$$(61)

where the integrals over r and s are performed within the separation bin, and we have written A₁₂(x, r) = W₁(x) W₂(x + r). We hence approximated the dependence of the covariance on the survey window by replacing the survey area in Eqs. (45) and (47) by the expression A₂(r) A₂(s)/A₃(r, s).

We calculated the terms A₂ and A₃ using the mean and covariance of the pair count R_sR_l(r) between random source and lens realisations (Landy & Szalay 1993), which have respective densities ${\overline{n}}_{\mathrm{s}}$ and ${\overline{n}}_{\mathrm{l}}$. The mean pair count in a separation bin at scale r (between r₁ and r₂), containing bin area $A_{\text{bin}}(r)=\pi (r_2^2-r_1^2)$, is

$$\begin{array}{c}\hfill ⟨R_{\mathrm{s}}{R}_{\mathrm{l}}(r)⟩={\overline{n}}_{\mathrm{s}}{\overline{n}}_{\mathrm{l}}{A}_{\mathrm{bin}}(r)A_2(r),\end{array}$$(62)

which allows us to find A₂(r), given that the other variables are known. The covariance of the pair count between separation bins r and s is,

$$\begin{array}{cc}\hfill \mathrm{Cov}[R_{\mathrm{s}}{R}_{\mathrm{l}}(r),R_{\mathrm{s}}{R}_{\mathrm{l}}(s)]& ={\overline{n}}_{\mathrm{s}}{\overline{n}}_{\mathrm{l}}{A}_{\mathrm{bin}}(r)\hfill \\ \hfill & \times [A_2(r){\delta }_{\mathit{rs}}^K+({\overline{n}}_{\mathrm{s}}+{\overline{n}}_{\mathrm{l}})A_{\mathrm{bin}}(s)A_3(r,s)],\hfill \end{array}$$(63)

which allows us to determine A₃(r, s). For all the source-lens configurations and separation bins considered in this study, we find that the area correction factor differs from 1.0 by less than 10%.

5.6. Propagation of errors in multiplicative corrections

Galaxy-galaxy lensing measurements are subject to multiplicative correction factors arising from shape measurement calibration (see Sect. 6.1) and, in the case of ΔΣ, owing to photometric redshift dilution (see Sect. 3.3). We propagated the uncertainties in these correction factors, which are correlated between different source and lens samples, into the analytical covariance of the measurements. Taking ΔΣ as an example and writing a general amplitude correction factor as α, the relation between the corrected and analytical statistics (denoted by the superscripts ‘corr’ and ‘ana’, respectively) is,

$$\begin{array}{c}\hfill \mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{corr}}=\frac{{\alpha }_{\mathit{ij}}\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{ana}}}{⟨{\alpha }_{\mathit{ij}}⟩},\end{array}$$(64)

which is normalised such that $⟨\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{corr}}⟩=\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{ana}}$, where i denotes the lens sample, j the source sample and k the separation bin. We hence find,

$$\begin{array}{cc}\hfill \mathrm{Cov}[\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{corr}},\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{lmn}}^{\mathrm{corr}}]& =\mathrm{Cov}[\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{ana}},\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{lmn}}^{\mathrm{ana}}](1+C_{ij,lm})\hfill \\ \hfill & +⟨\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{ijk}}^{\mathrm{ana}}⟩⟨\mathrm{\Delta }{\mathrm{\Sigma }}_{\mathit{lmn}}^{\mathrm{ana}}⟩C_{ij,lm},\hfill \end{array}$$(65)

where,

$$\begin{array}{c}\hfill C_{ij,lm}=\frac{⟨{\alpha }_{\mathit{ij}}{\alpha }_{\mathit{lm}}⟩-⟨{\alpha }_{\mathit{ij}}⟩⟨{\alpha }_{\mathit{lm}}⟩}{⟨{\alpha }_{\mathit{ij}}⟩⟨{\alpha }_{\mathit{lm}}⟩}=\frac{\mathrm{Cov}[{\alpha }_{\mathit{ij}},{\alpha }_{\mathit{lm}}]}{⟨{\alpha }_{\mathit{ij}}⟩⟨{\alpha }_{\mathit{lm}}⟩}.\end{array}$$(66)

We describe our specific implementation of these equations in the case of the KiDS dataset in Sect. 9.

6. Data

6.1. KiDS-1000

The Kilo-Degree Survey is a large optical wide-field imaging survey optimised for weak gravitational lensing analysis, performed with the OmegaCAM camera on the VLT Survey Telescope at the European Southern Observatory’s Paranal Observatory. The survey covers two regions of sky each containing several hundred square degrees, KiDS-North and KiDS-South, in four filters (u, g, r, i). The companion VISTA-VIKING survey has provided complementary imaging in near-infrared bands (Z, Y, J, H, K_s), resulting in a deep, wide, nine-band imaging dataset.

Our study is based on the fourth public data release of the project, KiDS-1000 (Kuijken et al. 2019), which comprises 1006 deg² of multi-band data, more than doubling the previously-available coverage. We used an early-science release of the KiDS-1000 shear catalogues, which was created using the exact pipeline version and PSF modelling strategy implemented in Hildebrandt et al. (2017) for the KiDS-450 release. We note that these catalogues have not undergone any rigorous assessment for the presence of cosmic shear systematics, but they are sufficient for the galaxy-galaxy lensing science presented in this paper, as this is less susceptible to systematic errors in the lensing catalogues. The raw pixel data was processed by the THELI and ASTRO_WISE pipelines (Erben et al. 2013; de Jong et al. 2015), and source ellipticities were measured using lensfit (Miller et al. 2013), assigning an optimal weight for each source, and calibrated by a large suite of image simulations (Kannawadi et al. 2019). Photometric redshifts z_B were determined from the nine-band imaging for each source using the Bayesian code BPZ (Benítez 2000), calibrated using spectroscopic sub-samples (Hildebrandt et al. 2020), and used to divide the sources into tomographic bins according to the value of z_B.

6.2. BOSS

The Baryon Oscillation Spectroscopic Survey (BOSS, Dawson et al. 2013) is the largest existing galaxy redshift survey, which was performed using the Sloan Telescope between 2009 and 2014. BOSS mapped the distribution of 1.5 million Luminous Red Galaxies (LRGs) and quasars across ∼10 000 deg², inspiring a series of cosmological analyses including the most accurate existing measurements of baryon acoustic oscillations and redshift-space distortions in the galaxy clustering pattern (Alam et al. 2017a). The final (Data Release 12) large-scale structure catalogues are described by Reid et al. (2016); we used the combined LOWZ and CMASS LRG samples in our study².

6.3. 2dFLenS

The 2-degree Field Lensing Survey (2dFLenS, Blake et al. 2016a) is a galaxy redshift survey performed at the Australian Astronomical Observatory in 2014–2015 using the 2-degree Field spectroscopic instrument, with the goal of extending spectroscopic-redshift coverage of gravitational lensing surveys in the southern sky, particularly the KiDS-South region. The 2dFLenS sample covers an area of 731 deg² and includes redshifts for 40 531 LRGs in the redshift range z <  0.9, selected by applying BOSS-inspired colour-magnitude cuts to the VST-ATLAS imaging data³. The 2dFLenS dataset has already been utilised in conjunction with the KiDS-450 lensing catalogues to perform a previous implementation of the amplitude-ratio test (Amon et al. 2018), a combined cosmological analysis of cosmic shear tomography, galaxy-galaxy lensing and galaxy multipole power spectra (Joudaki et al. 2018) and to determine photometric redshift calibration by cross-correlation (Johnson et al. 2017; Hildebrandt et al. 2020). In our study we utilised the 2dFLenS LRG sample which overlapped with the KiDS-1000 pointings in the southern region. Figure 1 illustrates the overlaps of the KiDS-1000 source catalogues in the north and south survey regions with the BOSS and 2dFLenS LRG catalogues.

Fig. 1. KiDS source and LRG lens catalogues within the KiDS-N region (top panel) and KiDS-S region (bottom panel). The grey squares represent the KiDS-1000 pointings, and the fluctuating background is the gridded number density of 2dFLenS (blue) and BOSS (red) galaxies. The open rectangles outline the footprint of the full KiDS survey, indicating that the LRG overlap will continue to increase as the survey is completed.

7. Mocks

We used the MICECATv2.0 simulation (Fosalba et al. 2015a,b; Crocce et al. 2015) to produce representative KiDS lensing source catalogues and LRG lens catalogues for testing the estimators, models and covariances described above. The Marenostrum Institut de Ciencias de l’Espai (MICE) catalogues cover an octant of the sky (0 <  RA <  90°, 0 <  Dec <  90°) for redshift range z <  1.4. We used boundaries at constant RA and Dec to divide this area into 10 sub-samples, each of area 516 deg². The fiducial set of cosmological parameters for the mock is Ω_m = 0.25, h = 0.7, Ω_b = 0.044, σ₈ = 0.8 and n_s = 0.95.

7.1. Mock source catalogue

We constructed the representative mock source catalogue by applying the following steps (see van den Busch et al., in prep. for a full description of the MICE KiDS source mocks). The MICE catalogue is non-uniform across the octant: the region Dec <  30° AND [(RA <  30°) OR (RA >  60°)] has a shallower redshift distribution than the remainder. Firstly, we homogenised the catalogue with the cut des_asahi_full_i_true < 24, such that we could construct mocks using the complete octant. The MICE catalogue shears (γ₁, γ₂) are defined by the position angle relative to the declination axis. Given the MICE system for mapping 3D positions to (RA, Dec) co-ordinates, the KiDS conventions can be recovered by the following transformations: RA → 90° −RA, γ₁ → −γ₁ (γ₂ is effectively negated twice and therefore unchanged).

We constructed a KiDS-like photometric realisation based on the galaxy sizes and shapes, median KiDS seeing and limiting magnitudes, including photometric noise (see van den Busch et al., in prep.). We ran BPZ photometric redshift estimation (Benítez 2000) on the mock source magnitudes and sizes, assigning z_B values for each object. We used a KDTree algorithm to assign weights to the mock sources on the basis of a nearest-neighbour match to the data catalogue in magnitude space, and randomly sub-sampled the catalogue to match the KV450 effective source density. We produced noisy shear components (e₁, e₂) as e = (γ + n)/(1 + nγ^*) (Seitz & Schneider 1997) where γ = γ₁ + γ₂ i, e = e₁ + e₂ i and n = n₁ + n₂ i, where n₁ and n₂ are drawn from Gaussian distributions with standard deviation σ_e = 0.288 (Hildebrandt et al. 2020). The redshift distribution estimates of the KiDS data and MICE mock source tomographic samples are displayed in the left panel of Fig. 2, illustrating the reasonable match between the two catalogues.

Fig. 2. Left panel: redshift distribution estimates of the KiDS source catalogue (solid lines) and MICE mock source catalogue (dashed lines) in the five source tomographic bins split by photometric redshift. Right panel: number density as a function of redshift of the BOSS LRG dataset (black solid line), the MICE mock lens catalogue with the original BOSS colour selection (red dashed line), and the MICE mock lens catalogue with the adjusted BOSS colour selection (blue dotted line).

7.2. Mock lens catalogue

We constructed the representative mock LRG lens catalogue from the MICE simulation as follows. We used the galaxy magnitudes sdss_g_true, sdss_r_true, sdss_i_true and first applied the MICE evolution correction to these magnitudes as a function of redshift, m → m − 0.8 * [arctan(1.5 * z) − 0.1489] (Crocce et al. 2015). We then constructed the LRG lens catalogues using the BOSS LOWZ and CMASS colour cuts in terms of the variables,

$$\begin{array}{cc}& c_{\Vert }=0.7(g-r)+1.2(r-i-0.18),\hfill \\ \hfill & c_{\perp }=(r-i)-(g-r)/4-0.18,\hfill \\ \hfill & d_{\perp }=(r-i)-(g-r)/8.\hfill \end{array}$$(67)

Applying the original BOSS colour-magnitude selection cuts (Eisenstein et al. 2011) to the MICE mock did not reproduce the BOSS redshift distribution (which is unsurprising, since this mock has not been tuned to do so; the BOSS data is also selected from noisy observed magnitudes). Our approach to resolve this issue, following Crocce et al. (2015), was to vary the colour and magnitude selection cuts to minimise the deviation between the mock and data redshift distributions. We applied the following LOWZ selection cuts (where we indicate our changed values in bold font, and the previous values immediately following in square brackets):

$$\begin{array}{cc}& 16.0<r<\mathbf{20.0}[19.6],\hfill \\ \hfill & r<\mathbf{13.35}[13.5]+c_{\Vert }/0.3,\hfill \\ \hfill & |c_{\perp }|<0.2.\hfill \end{array}$$(68)

We applied the following CMASS selection cuts:

$$\begin{array}{cc}& 17.5<i<\mathbf{20.06}[19.9],\hfill \\ \hfill & r-i<2,\hfill \\ \hfill & d_{\perp }>0.55,\hfill \\ \hfill & i<\mathbf{19.98}[19.86]+1.6(d_{\perp }-0.8).\hfill \end{array}$$(69)

The resulting redshift distributions of the MICE lens mock (original, and after adjustment of the colour selection cuts) and the BOSS data are shown in the right panel of Fig. 2, illustrating that our modified selection produced a much-improved representation of the BOSS dataset. The clustering amplitude of the MICE LRG mock catalogues was consistent with a galaxy bias factor b ≈ 2, although did not precisely match the clustering of the BOSS dataset, since it was not tuned to do so. However, these representative catalogues nonetheless allowed us to test our analysis procedures.

8. Simulation tests

In this section we analyse the representative source and lens catalogues constructed from the MICE mocks described in Sect. 7. Our specific goals are to:

– Test that the non-linear galaxy bias model specified in Sect. 2.4 is adequate for modelling the galaxy-galaxy lensing and clustering statistics across the relevant scales.

– Test that the approaches to the photo-z dilution correction of ΔΣ described in Sect. 3.3 recovered results consistent with those obtained using source spectroscopic redshifts.

– Use the multiple mock realisations and jack-knife techniques to test that the covariance of the estimated statistics is consistent with the analytical Gaussian covariance specified in Sect. 5.

– Test that the E_G test statistics constructed from the mock as described in Sect. 4.3 are consistent with the theoretical expectation, and determine the degree to which this result depends on the choice of the small-scale cut-off parameter R₀ (see Eq. (40)).

– Test that, given our galaxy bias model, the galaxy-galaxy lensing and clustering statistics may be jointly described by a normalisation parameter σ₈ that is consistent with the mock fiducial cosmology, and use this test to assess the relative precision of angular and projected estimators.

Consistent with our subsequent data analysis, we divided the source catalogues into five different tomographic samples by the value of the BPZ photometric redshift, with divisions z_B = [0.1, 0.3, 0.5, 0.7, 0.9, 1.2] (following Hildebrandt et al. 2020). We divided the lens catalogue into five slices of spectroscopic redshift z_l of width Δz_l = 0.1 in the range 0.2 <  z_l <  0.7. This narrow spectroscopic slicing minimises systematic effects due to redshift evolution across the lens slice (Leauthaud et al. 2017; Singh et al. 2019).

8.1. Measurements

We measured the following three statistics, which form the fundamental set of measurements from which the associated statistics are derived. Firstly, we measured the average tangential shear γ_t(θ) between all tomographic pairs of source and lens samples, in 15 logarithmically-spaced angular bins in the range 0.005° < θ <  5°, using the estimator of Eq. (27). This measurement is displayed in Fig. 3 as the mean of the 10 individual mock realisations (which each have area 516 deg²), for each of the five source samples against the five lens redshift slices.

Fig. 3. Measurements of the average tangential shear, γt(θ), for the KiDS-LRG dataset (black points) and representative MICE mocks (red points), between all pairs of lens spectroscopic redshift slices (rows) and source tomographic samples split by photometric redshift (columns). The errors are derived from the diagonal elements of the full analytical covariance matrix (where we note that measurements are correlated across scales and samples). The overplotted model is not a fit to this dataset, but rather a prediction based on the galaxy bias parameter fits to the Υgg statistic of each lens redshift slice, which is inaccurate on small scales. The y-values are scaled by a factor of θ (in degrees) for clarity.

Secondly, we measured the projected mass density ΔΣ(R) between all tomographic pairs of source and lens samples, in 15 logarithmically-spaced projected separation bins in the range 0.1 <  R <  100 h⁻¹ Mpc. The mock mean measurement is displayed in Fig. 4 in units of h M_⊙ pc⁻², for each of the five source samples against the five lens redshift slices. When performing a ΔΣ measurement between source and lens samples we only included individual source-lens galaxy pairs with z_B >  z_l, for which the source photometric redshift lies behind the lens spectroscopic redshift (adopting an alternative cut z_B >  z_l + 0.1 did not change the results significantly). We applied the photo-z dilution correction f_bias computed using Eq. (35) based on the point photo-z values, and we study the efficacy of this correction in Sect. 8.2 below.

Fig. 4. Measurements of the projected mass density, ΔΣ(R), for the KiDS-LRG dataset (black points) and representative MICE mocks (red points), displayed in the same style as Fig. 3. No measurements are possible for the lower left-hand set of panels, owing to the adopted cut in source-lens pairs, zB >  zl. The units of ΔΣ are h M⊙ pc−2.

Thirdly, we measured the projected clustering w_p(R) of the lens samples in the same projected separation bins as above, using the estimator of Eq. (39) with Π_max = 100 h⁻¹ Mpc. The mock mean measurement of w_p(R) is displayed as the third row of Fig. 5, for each of the five redshift slices. Each of these measurements will be compared with cosmological model predictions as described in the subsections below.

Fig. 5. Measurements of a series of galaxy-galaxy lensing and clustering statistics (rows) for each different lens redshift slice (columns) of the MICE mocks. The first row displays the projected mass density ΔΣ(R), where measurements corresponding to the different source tomographic samples have been optimally combined following the procedure described in Appendix C. The second row shows the corresponding measurements of Υgm(R), derived from ΔΣ(R) using Eq. (40) and assuming R0 = 2 h−1 Mpc. The third and fourth rows display the galaxy clustering statistics wp(R) and Υgg(R) for each lens redshift slice, where wp(R) is the projected correlation function measurement, and Υgg(R) is the associated quantity suppressing contributions from small scales, derived using Eq. (41) and again assuming R0 = 2 h−1 Mpc. The fifth row shows the combined-probe statistic EG(R), derived from the Υgm(R) and Υgg(R) measurements using Eq. (44). Errors are displayed using the diagonal elements of the analytical covariance matrix, propagating errors where appropriate. The overplotted models are determined using the galaxy bias factors fitted to the Υgg measurements for each lens redshift slice, and χ2 statistics between the mock mean data and model are displayed in each panel.

We computed the covariance matrix for each statistic using the analytical Gaussian covariance specified in Sect. 5, where we initially used a fiducial lens linear bias factor b_L = 1.8, and iterated this value following a preliminary fit to the projected lens clustering. Figure 6 compares three different determinations of the error in ΔΣ(R) for each individual 516 deg² realisation of the MICE mocks: using the analytical covariance, using a jack-knife analysis, and derived from the standard deviation of the 10 realisations. For the jack-knife analysis, we divided the sample into 7 × 7 angular regions using constant boundaries in RA and Dec, such that each region contained the same angular area 10.5 deg². In Fig. 6 we display the comparison as a ratio between the jack-knife or realisations error, and the analytical error.

Fig. 6. Comparison of different estimates of the error in ΔΣ(R) for individual 516 deg2 realisations of the MICE mocks, between all pairs of lens spectroscopic redshift slices (rows) and source tomographic samples (columns). The black solid line shows the ratio between the error determined by a jack-knife analysis of the data and the error derived from the diagonal elements of the analytical covariance matrix, and the red dashed line is the ratio between the standard deviation across 10 realisations and the analytical error. No measurements are possible for the lower left-hand set of panels, owing to the adopted cut in source-lens pairs, zB >  zl.

We find that in the range R >  1 h⁻¹ Mpc, where the model provides a reasonable description of the measurements, the average (fractional) absolute difference between the analytical and jack-knife errors is 15%, and between the analytical and realisation scatter is 21% (which is the expected level of difference given the error in the variance for 10 realisations). Small differences between these error estimates may arise due to the Gaussian approximation in the analytical covariance, the exact details of the survey modelling, or the scale of the jack-knife regions.

Figure 7 displays the full analytical covariance matrix of ΔΣ(R) – spanning five lens redshift slices, five source tomographic samples and 15 bins of scale – as a correlation matrix with 375 × 375 entries. We note that there are significant off-diagonal correlations between measurements utilising the same lens or source sample, and between different scales. The covariance matrix is reduced in dimension if source-lens sample pairs with z_B <  z_l are excluded, as illustrated by the missing panels in Figs. 4.

Fig. 7. Full analytical covariance matrix Cij of the projected mass density ΔΣ(R) for the MICE mocks, spanning five lens redshift slices, five source tomographic samples and 15 bins of scale – i.e. 375  ×  375 entries – ordered where each subsequent bin changes separation, then source sample, then lens sample, such that the vertical and horizontal solid lines demarcate different lens redshift slices. We display the results as a correlation matrix $r=C_{\mathit{ij}}/\sqrt{C_{\mathit{ii}}{C}_{\mathit{jj}}}$, and note that there are significant off-diagonal correlations between measurements utilising the same lens or source sample, and between different scales. We note that the colour bar is saturated at r = 0.1 to reveal low-amplitude cross-correlation more clearly.

We combined the correlated ΔΣ measurements for each individual lens redshift slice, averaging over the five different source tomographic samples, using the procedure described in Appendix C. The resulting combined ΔΣ measurement for each lens redshift sample (again corresponding to a mock mean) is shown as the first row in Fig. 5.

We then used the ΔΣ(R) and w_p(R) measurements to infer the Upsilon statistics, Υ_gm(R, R₀) and Υ_gg(R, R₀), using Eqs. (40) and (41) respectively, adopting a fiducial value R₀ = 2 h⁻¹ Mpc (we consider the effect of varying this choice below). These measurements are shown in the second and fourth rows of Fig. 5. We determined the covariance of Υ_gm(R, R₀) and Υ_gg(R, R₀) using error propagation following Eqs. (55) and (56), respectively.

Finally, we determined the E_G(R) statistic for each lens redshift slice using Eq. (44) where, for the purposes of these tests focussed on galaxy-galaxy lensing, we assumed a fixed input value for the redshift-space distortion parameter β = f(z)/b_L(z), where we evaluated f(z) = Ω_m(z)^0.55 using the fiducial cosmology of the MICE simulation – we note that the exponent 0.55 is an excellent approximation to the solution of the differential growth equation in ΛCDM cosmologies (Linder 2005) – and b_L(z) is the best-fitting linear bias parameter to the Υ_gg measurements for each lens redshift slice z. Hence, systematic errors associated with redshift-space distortions lie beyond the scope of this study, and in our subsequent data analysis we will infer the required β values from existing literature. We propagated errors in E_G using Eq. (57) (and assuming no error in β in the case of the mocks). Our E_G measurements are shown as the fifth row in Fig. 5.