KiDS+VIKING+GAMA: Testing semi-analytic models of galaxy evolution with galaxy-galaxy-galaxy lensing

The main observable for the lens-lens-shear correlation is the three-point correlation function $\tilde{\mathcal{G}}$. This function correlates the projected lens galaxy number density and the tangential gravitational lensing shear $\gamma_{\textrm{t}}$, measured with respect to the bisector of the angle $\phi$ between the lens-source separations $\overrightarrow{{\bf\it\vartheta}}_{1}$ and $\overrightarrow{{\bf\it\vartheta}}_{2}$ (see Fig. 1). For unmixed lens pairs, whose galaxies have the projected number density $N(\overrightarrow{{\bf\it\vartheta}})$, $\tilde{\mathcal{G}}$ is

where $\overline{N}$ is the mean number density of lenses. For mixed lens pairs, where the lenses are from samples with number densities $N_{1}$ and $N_{2}$, the correlation function is

Instead of measuring $\tilde{\mathcal{G}}$, we estimated a redshift-weighted correlation function $\tilde{\mathcal{G}}_{Z}$, which includes a redshift weighting function $Z$, which depends on the redshift difference $\Delta z_{12}=z_{1}-z_{2}$ of the lenses in a pair. We chose the redshift weighting such that it is large for small $\Delta z_{12}$ and vanishes for large $\Delta z_{12}$. It, therefore, weights true pairs with small redshift differences higher than chance pairs with large redshift differences and increases the S/N (40). To define the redshift-weighted correlation function $\tilde{\mathcal{G}}_{Z}$, we used that the projected lens number densities $N_{1,2}$ are related to the three-dimensional number densities $n_{1,2}(\overrightarrow{{\bf\it\theta}},z)$ at redshift $z$ by the selection functions $\nu_{1,2}(z)$,

The selection functions give the fraction of galaxies at redshift $z$ included in the lens sample. For a flux-limited galaxy sample, this corresponds to the fraction of galaxies brighter than the magnitude limit. The selection functions $\nu_{1,2}(z)$ are related to the galaxies’ redshift distributions $p_{1,2}(z)$ by

With Eq. (3), the redshift-weighted correlation function is

where $\Delta z_{12}:=z_{1}-z_{2}$ is the redshift difference between lenses in a pair and

Additionally, we also measured the physical correlation function $\tilde{\mathcal{G}}_{\textrm{phys}}(\overrightarrow{{\bf\it r}}_{1},\overrightarrow{{\bf\it r}}_{2})$, which gives the projected excess mass around lens pairs with physical lens-source separations $\overrightarrow{{\bf\it r}}_{1}$ and $\overrightarrow{{\bf\it r}}_{2}$ projected on a plane midway between the lenses. To find $\tilde{\mathcal{G}}_{\textrm{phys}}$, instead of averaging the tangential shear $\gamma_{\textrm{t}}$, we averaged the projected excess mass density $\Delta\Sigma$ around lens pairs with

where

and

with the angular diameter distance $D_{\mathrm{A}}(z_{1},z_{2})$ between redshifts $z_{1}$ and $z_{2}$. The projected excess mass density $\Delta\Sigma$ is

with the source-averaged inverse critical surface mass density $\bar{\Sigma}^{-1}_{\mathrm{crit}}$. Using the source redshift distribution $p(z_{\textrm{s}})$, $\bar{\Sigma}^{-1}_{\mathrm{crit}}$ is

where $D_{\textrm{A}}(z):=D_{\textrm{A}}(0,z)$. This critical surface mass density is not the comoving critical surface mass density $\bar{\Sigma}_{\textrm{crit, com}}$, defined by

used in some weak lensing studies. Appendix C in Dvornik et al. (2018) discusses the implications of different definitions of the critical surface mass density.

The correlation functions $\tilde{\mathcal{G}}_{Z}$ and $\tilde{\mathcal{G}}_{\textrm{phys}}$ depend only on the lens-source distances $\vartheta_{1}$, $\vartheta_{2}$ and $r_{1}$, $r_{2}$, and the opening angle $\phi$ between the lens-source separations, because of the statistical isotropy of the matter and galaxy density fields. Therefore, we define

and

The three-point correlation function $\tilde{\mathcal{G}}$ contains second- and third-order statistics. This can be seen by writing $\tilde{\mathcal{G}}$ as

where $\kappa_{\textrm{g}}(\overrightarrow{{\bf\it\vartheta}})=N(\overrightarrow{{\bf\it\vartheta}})/\bar{N}-1$ is the two-dimensional galaxy number density contrast. The second and third term in Eq. (14) are GGL statistics which correspond to the shear around individual lens galaxies.

When studying G3L we are interested in the excess shear around lens pairs, which is given only by the first term in Eq. (14). Therefore, we converted $\tilde{\mathcal{G}}_{Z}$ and $\tilde{\mathcal{G}}_{\textrm{phys}}$ to the third-order aperture statistics $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}$ and $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{\textrm{phys}}$, which only include the third-order statistics, as shown in 40. For this, we use a compensated filter function with an aperture scale $\theta$,

which fulfills

With this filter function, the third-order aperture statistics are defined as

and

with the lensing convergence $\kappa(\overrightarrow{{\bf\it\theta}})$ and the surface mass density $\Sigma$, given by

We use the exponential filter function

for which the aperture statistics can be calculated from $\tilde{\mathcal{G}}_{Z}$ and $\tilde{\mathcal{G}}_{\textrm{phys}}$ with

and

The kernel function $\mathcal{A}_{\mathcal{N}\mathcal{N}\mathcal{M}}$ is defined in the appendix of Schneider & Watts (2005).

The aperture statistics can be used to constrain the galaxy bias, which is the relation between the galaxy number density contrast and the matter density contrast (Schneider & Watts, 2005, e.g). The simplest assumption for this bias is a linear deterministic relation,

where $b$ is a scale-independent bias factor (Kaiser, 1984). A larger bias factor $b$ implies a higher galaxy number density for a given matter overdensity. For two galaxy populations with bias factors $b_{1}$ and $b_{2}$, this simple model predicts for the aperture statistics

From this follows, that

We measure $R$ in the observation and simulation to assess the assumption of linear deterministic bias.

To measure $\tilde{\mathcal{G}}_{Z}$ and $\tilde{\mathcal{G}}_{\textrm{phys}}$, we used the estimators from 40 for $N_{\mathrm{s}}$ source and $N_{\mathrm{d}}$ lens galaxies. These estimators measure the correlation functions by averaging the ellipticities of the source galaxies over all lens-lens-source galaxy triplets. For $\tilde{\mathcal{G}}_{Z}$, in the bin $B$ of $(\vartheta_{1},\vartheta_{2},\phi)$, the estimator is the real part of

where $w_{k}$ is the weight of the source ellipticity $\epsilon_{k}$, $\omega_{Z}$ is the redshift-weighted angular two-point correlation function of the lens galaxies, and

The angles $\varphi_{ik}$ and $\varphi_{jk}$ are the polar angles of the lens-source separation vectors $\overrightarrow{{\bf\it\theta}}_{i}-\overrightarrow{{\bf\it\theta}}_{k}$ and $\overrightarrow{{\bf\it\theta}}_{j}-\overrightarrow{{\bf\it\theta}}_{k}$ (corresponding to $\varphi_{1}$ and $\varphi_{2}$ in Fig. 1) and $\phi_{ijk}=\varphi_{ik}-\varphi_{jk}$ is the opening angle between the lens-source separation vectors (corresponding to $\phi$ in Fig. 1).

Source galaxies with more precise shape measurements receive a higher ellipticity weight $w_{k}$. The weight, therefore, increases the contribution of source galaxies with more exact shapes to the estimator. For the simulated shear data, we set the weights to $w_{k}=1$ for all sources.

We obtained the lens two-point correlation function $\omega_{Z}$, with the estimator by Szapudi & Szalay (1998) which is

for two different observed lens samples with $N_{\textrm{d}_{1}}$ and $N_{\textrm{d}_{2}}$ galaxies and two "random samples". These random samples contain $N_{\textrm{r}_{1}}$ and $N_{\textrm{r}_{2}}$ unclustered galaxies following the same selection function as the observed galaxies.

The ${D_{1}D_{2}}_{Z}$, ${D_{1}R_{2}}_{Z}$, ${D_{2}R_{1}}_{Z}$, and ${R_{1}R_{2}}_{Z}$ are the pair counts of observed and random galaxies. For two equal lens samples and $DD=D_{1}D_{2}$, $DR=D_{1}R_{2}=D_{2}R_{1}$, and $RR=R_{1}R_{2}$, the estimator in Eq. (29) reduces to the usual Landy-Szalay estimator (Landy & Szalay, 1993)

Usually, pair counts are defined as the number of pairs within a certain angular separation. However, we used redshift-weighted pair counts to account for the stronger clustering of true lens pairs due to the redshift weighting function $Z$. They are defined for a bin centred at $\theta$ with bin size $\Delta\theta$ with the Heaviside step function $\Theta_{\textrm{H}}$ as

and

with $a,b\in{1,2}$.

Following 40, we chose a Gaussian weighting function,

with width $\sigma_{z}=0.01$. This width is larger than typical galaxy correlation lengths and redshifts induced by the peculiar motion of galaxies. Accordingly, true lens pairs are not affected by the redshift weighting, while chance pairs are down-weighted. Choosing a different $\sigma_{z}$ influences the magnitude of the measured aperture statistics as well as the S/N of the measurement. Nonetheless, as long as the same width is chosen for the observation and the simulation, their G3L signals can be compared.

The estimator for $\tilde{\mathcal{G}}_{\textrm{phys}}$ for the bin $B$ of $(r_{1},r_{2},\phi)$ is

with

We measured $\tilde{\mathcal{G}}_{Z}$ and $\tilde{\mathcal{G}}_{\textrm{phys}}$ initially for $128\times 128\times 128$ bins, which were linearly spaced along $\phi$ and logarithmically spaced along $\vartheta_{1,2}$ and $r_{1,2}$. For $\tilde{\mathcal{G}}_{Z}$, the $\vartheta_{1,2}$ are between $0.15\prime$ and $200\prime$ for the observed and between $0.15\prime$ and $320\prime$ for the simulated data. For $\tilde{\mathcal{G}}_{\textrm{phys}}$, we chose $r_{1,2}$ between $0.02\,\textrm{Mpc}$ and $40\,\textrm{Mpc}$. We then applied the adaptive binning scheme of 40, by which the parameter space was tessellated to remove bins for which no galaxy triplet is in the data.

The correlation function was measured individually for 24 tiles of the observational data of size $$$\times$$$ and 64 fields-of-view of the MR of size $$$\times$$$, leading to estimates $\tilde{\mathcal{G}}_{\textrm{est}}^{i}$ and $\tilde{\mathcal{G}}_{\textrm{est, ph}}^{i}$ for each tile and field-of-view, respectively. The division into small patches allowed us to project the observational measurements to Cartesian coordinates and to estimate the uncertainty of the measurement with jackknife resampling. For each data set, the individual estimates were combined to form the total correlation functions with

where

and

where

To compute $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}$ and $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{\textrm{phys}}$, we integrated over $\tilde{\mathcal{G}}_{Z}$ using Eq. (21) and (22). We numerically approximated the integrals by summing over all $N_{\textrm{bins}}$ of $\tilde{\mathcal{G}}_{Z}$ after tessellation with

and

Here, $b_{i}$ is the $i$th bin, which has size $V(b_{i})$, and $\mathcal{A}_{\mathcal{NNM}}$ is the kernel function evaluated at the tessellation seed of bin $b_{i}$.

The aperture statistics were measured only for equal scale radii $\theta_{1}=\theta_{2}=\theta_{3}$, so we abbreviate

We estimated the statistical uncertainty of the aperture statistics in the observational data with jackknife resampling. For this we assumed that the 24 tiles are statistically independent. Although this assumption is not correct for noise due to sample variance, we expect our noise to be dominated by shape noise which is independent for each tile. In the jackknife resampling, we combined the $\tilde{\mathcal{G}}^{i}_{Z}$ of all $N$ tiles to the total $\tilde{\mathcal{G}}_{Z}$, and also create $N$ jackknife samples, for which all but one tile are combined. The aperture statistics $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}(\theta)$ are calculated for the total $\tilde{\mathcal{G}}_{Z}$, as well as for each of the $N$ jackknife samples to get $N$ $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{k}(\theta)$. The covariance matrix of $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}(\theta)$ was then estimated with

where $\overline{\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{k}}(\theta_{i})$ is the average of all $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{k}(\theta_{i})$.

As discussed by Hartlap et al. (2007) and Anderson (2003), the inverse of this estimate of the covariance matrix is not an unbiased estimate of the inverse covariance matrix. Following their suggestion, we instead estimated the inverse covariance matrix with

where $p$ is the number of data points. This gives an unbiased estimate of the inverse covariance matrix if the realisations are statistically independent and have Gaussian errors.

With this estimate of the inverse covariance matrix, we calculated the S/N of our observational measurement with

We also use $C^{-1}_{ij}$ to perform a $\chi^{2}$-test, evaluating the agreement of the observational measurement $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{\rm obs}$ with the SAMs prediction $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{\rm sim}$. For this, we calculate the reduced $\chi^{2}_{\mathrm{redu}}$ as

Aside from $\expectationvalue{\mathcal{NN} M_\mathrm{ap}}_{\textrm{phys}}$, we also estimated the so-called B-mode $\expectationvalue{\mathcal{NN} M_{\perp}}$, given by applying Eq. (22) not on $\tilde{\mathcal{G}}_{\textrm{phys}}$ but on

where ${\epsilon}^{*}_{k}$ is the complex conjugate of the galaxies ellipticity. If there are no dominating systematic effects inducing a parity violation, the B-mode has to vanish (Schneider, 2003). We therefore tested for such systematics by estimating $\expectationvalue{\mathcal{NN} M_{\perp}}$.

Our observational data is the overlap of KiDS, VIKING, and GAMA (KV450 $\times$ GAMA). This overlap encompasses approximately 180 $\textrm{deg}^{2}$, divided into the three patches G9, G12 and G15, each with dimensions of $12\times 5\,\textrm{deg}^{2}$.

VIKING (Edge et al., 2013; Venemans et al., 2015) is a photometric survey in five near-infrared bands, conducted at the VISTA telescope in Paranal, Chile and covering approximately $1350\,\textrm{deg}^{2}$. It covers the same area as KiDS (Kuijken et al., 2015; de Jong et al., 2015), an optical photometric survey conducted with the OmegaCAM at the VLT Survey Telescope. The data of KiDS and VIKING were combined to form the KV450 data set, described in detail in Wright et al. (2019), which we use in the following. KV450 has the same footprint as the third data release of KiDS (de Jong et al., 2017) and was processed by the same data reduction pipelines, described in detail in Hildebrandt et al. (2017). Data are processed by THELI (Erben et al., 2005; Schirmer, 2013) and Astro-WISE (de Jong et al., 2015). Shears are measured with lensfit (Miller et al., 2013; Kannawadi et al., 2019). Photometric redshifts are obtained from PSF-matched photometry (Wright et al., 2019) and calibrated using external overlapping spectroscopic surveys (Hildebrandt et al., 2020).

We use the galaxies observed by KV450 with photometric redshift between $0.5$ and $1.2$ as source galaxies. Galaxies with a photometric redshift less than $0.5$ are excluded because most of them are in front of our lens galaxies and therefore dilute and bias the lensing signal. The averaged inverse critical surface mass density $\bar{\Sigma}^{-1}_{\mathrm{crit}}$ is calculated as described in Sect. 3 by using the weighted direct calibration redshift distributions (DIR distributions) of the KV450 galaxies as the source distribution. These DIR distributions were obtained with in-depth spectroscopic surveys overlapping with KiDS and VIKING. The spectroscopic redshift distributions from these surveys were weighted according to the photometric data in KV450 to estimate the redshift distribution of KV450 galaxies. Details of this procedure are given in Hildebrandt et al. (2017); Hildebrandt et al. (2020). We neglect the uncertainties on the redshift distribution and the multiplicative bias of the shear estimate. However, as these uncertainties are small, we do not expect them to impact our conclusions.

GAMA (Driver et al., 2009; Driver et al., 2011; Liske et al., 2015) is a spectroscopic survey carried out at the Anglo Australian Telescope with the AAOmega spectrograph. We use the data management unit (DMU) distanceFramesv14, which contains positions and spectroscopic redshifts $z$ of galaxies with a Petrosian observer-frame $r$-band magnitude brighter than 19.8 mag. The spectroscopic redshifts were flow-corrected to account for the proper motion of the Milky Way using the model by Tonry et al. (2000) according to the procedure in Baldry et al. (2012). We include all galaxies with a spectroscopic redshift lower than 0.5 and redshift quality flag N_Q$\geq$3. For the calculation of the angular two-point correlation function of lenses, we use randoms from the DMU randomsv02 (Farrow et al., 2015), which incorporates the galaxy selection function of GAMA while maintaining an unclustered galaxy distribution.

From the GAMA galaxies, we select lens samples according to their colour and stellar mass. Restframe photometry and stellar masses were obtained from the DMU stellarMassesLambdarv20. An overview of our samples is given in Table 1.

We select a ‘red’ and ‘blue’ lens sample, defined according to the galaxies’ rest-frame $(g-r)_{0}$ colour. We use the colour cut by Farrow et al. (2015), according to which a galaxy is red if its rest-frame colour $(g-r)_{0}$ and its absolute Petrosian magnitude $M_{r}$ in the $r$-band fulfil