Galaxy-galaxy Lensing Research Articles

Emulating galaxy clustering and galaxy–galaxy lensing into the deeply non-linear regime: methodology, information, and forecasts

The combination of galaxy-galaxy lensing (GGL) with galaxy clustering is one of the most promising routes to determining the amplitude of matter clustering at low redshifts. We show that extending clustering+GGL analyses from the linear regime down to $\sim 0.5 \, h^{-1}$ Mpc scales increases their constraining power considerably, even after marginalizing over a flexible model of non-linear galaxy bias. Using a grid of cosmological N-body simulations, we construct a Taylor-expansion emulator that predicts the galaxy autocorrelation $\xi_{\text{gg}}(r)$ and galaxy-matter cross-correlation $\xi_{\text{gm}}(r)$ as a function of $\sigma_8$, $\Omega_m$, and halo occupation distribution (HOD) parameters, which are allowed to vary with large scale environment to represent possible effects of galaxy assembly bias. We present forecasts for a fiducial case that corresponds to BOSS LOWZ galaxy clustering and SDSS-depth weak lensing (effective source density $\sim 0.3$ arcmin$^{-2}$). Using tangential shear and projected correlation function measurements over $0.5 \leq r_p \leq 30 \, h^{-1}$ Mpc yields a 1.8% constraint on the parameter combination $\sigma_8\Omega_m^{0.58}$, a factor of two better than a constraint that excludes non-linear scales ($r_p > 2 \, h^{-1}$ Mpc, $4 \, h^{-1}$ Mpc for $\gamma_t,w_p$). Much of this improvement comes from the non-linear clustering information, which breaks degeneracies among HOD parameters that would otherwise degrade the inference of matter clustering from GGL. Increasing the effective source density to $3$ arcmin$^{-2}$ sharpens the constraint on $\sigma_8\Omega_m^{0.58}$ by a further factor of two. With robust modeling into the non-linear regime, low-redshift measurements of matter clustering at the 1-percent level with clustering+GGL alone are well within reach of current data sets such as those provided by the Dark Energy Survey.

Cosmological simulations for combined-probe analyses: covariance and neighbour-exclusion bias

We present a public suite of weak lensing mock data, extending the Scinet Light Cone Simulations (SLICS) to simulate cross-correlation analyses with different cosmological probes. These mocks include KiDS-450- and LSST-like lensing data, cosmic microwave background lensing maps and simulated spectroscopic surveys that emulate the GAMA, BOSS and 2dFLenS galaxy surveys. With 844 independent realisations, our mocks are optimised for combined-probe covariance estimation, which we illustrate for the case of a joint measurement involving cosmic shear, galaxy-galaxy lensing and galaxy clustering from KiDS-450 and BOSS data. With their high spatial resolution, the SLICS are also optimal for predicting the signal for novel lensing estimators, for the validation of analysis pipelines, and for testing a range of systematic effects such as the impact of neighbour-exclusion bias on the measured tomographic cosmic shear signal. For surveys like KiDS and DES, where the rejection of neighbouring galaxies occurs within ~2 arcseconds, we show that the measured cosmic shear signal will be biased low, but by less than a percent on the angular scales that are typically used in cosmic shear analyses. The amplitude of the neighbour-exclusion bias doubles in deeper, LSST-like data. The simulation products described in this paper are made available at http://slics.roe.ac.uk/.

Photometric redshifts for the Kilo-Degree Survey

We present a machine-learning photometric redshift (ML photo-z) analysis of the Kilo-Degree Survey Data Release 3 (KiDS DR3), using two neural-network based techniques: ANNz2 and MLPQNA. Despite limited coverage of spectroscopic training sets, these ML codes provide photo-zs of quality comparable to, if not better than, those from the Bayesian Photometric Redshift (BPZ) code, at least up to zphot ≲ 0.9 and r ≲ 23.5. At the bright end of r ≲ 20, where very complete spectroscopic data overlapping with KiDS are available, the performance of the ML photo-zs clearly surpasses that of BPZ, currently the primary photo-z method for KiDS. Using the Galaxy And Mass Assembly (GAMA) spectroscopic survey as calibration, we furthermore study how photo-zs improve for bright sources when photometric parameters additional to magnitudes are included in the photo-z derivation, as well as when VIKING and WISE infrared (IR) bands are added. While the fiducial four-band ugri setup gives a photo-z bias 〈δz/(1 + z)〉 = −2 × 10−4 and scatter σδz/(1+z) < 0.022 at mean 〈z〉 = 0.23, combining magnitudes, colours, and galaxy sizes reduces the scatter by ~7% and the bias by an order of magnitude. Once the ugri and IR magnitudes are joined into 12-band photometry spanning up to 12 μm, the scatter decreases by more than 10% over the fiducial case. Finally, using the 12 bands together with optical colours and linear sizes gives 〈δz/(1 + z)〉 < 4 × 10−5 and σδz/(1+z) < 0.019. This paper also serves as a reference for two public photo-z catalogues accompanying KiDS DR3, both obtained using the ANNz2 code. The first one, of general purpose, includes all the 39 million KiDS sources with four-band ugri measurements in DR3. The second dataset, optimised for low-redshift studies such as galaxy-galaxy lensing, is limited to r ≲ 20, and provides photo-zs of much better quality than in the full-depth case thanks to incorporating optical magnitudes, colours, and sizes in the GAMA-calibrated photo-z derivation.

Testing redMaPPer centring probabilities using galaxy clustering and galaxy–galaxy lensing

Galaxy cluster centring is a key issue for precision cosmology studies using galaxy surveys. Mis-identification of central galaxies causes systematics in various studies such as cluster lensing, satellite kinematics, and galaxy clustering. The red-sequence Matched-filter Probabilistic Percolation (redMaPPer) estimates the probability that each member galaxy is central from photometric information rather than specifying one central galaxy. The redMaPPer estimates can be used for calibrating the off-centring effect, however, the centring algorithm has not previously been well-tested. We test the centring probabilities of redMaPPer cluster catalog using the projected cross correlation between redMaPPer clusters with photometric red galaxies and galaxy-galaxy lensing. We focus on the subsample of redMaPPer clusters in which the redMaPPer central galaxies (RMCGs) are not the brightest member galaxies (BMEM) and both of them have spectroscopic redshift. This subsample represents nearly 10% of the whole cluster sample. We find a clear difference in the cross-correlation measurements between RMCGs and BMEMs, and the estimated centring probability is 74$\pm$10% for RMCGs and 13$\pm$4% for BMEMs in the Gaussian offset model and 78$\pm$9% for RMCGs and 5$\pm$5% for BMEMs in the NFW offset model. These values are in agreement with the centring probability values reported by redMaPPer (75% for RMCG and 10% for BMEMs) within 1$\sigma$. Our analysis provides a strong consistency test of the redMaPPer centring probabilities. Our results suggest that redMaPPer centring probabilities are reliably estimated. We confirm that the brightest galaxy in the cluster is not always the central galaxy as has been shown in previous works.

Galaxy–galaxy lensing in the outskirts of CLASH clusters: constraints on local shear and testing mass–luminosity scaling relation

We present a selection of 24 candidate galaxy-galaxy lensing (GGLs) identified from Hubble images in the outskirts of the massive galaxy clusters from the CLASH survey. These GGLs provide insights into the mass distributions at larger scales than the strong lensing region in the cluster cores. We built parametric mass models for three of these GGLs showing simple lensing configurations, in order to assess the properties of their lens and its environment. We show that the local shear estimated from the GGLs traces the gravitational potential of the clusters at 1-2 arcmin radial distance, allowing us to derive their velocity dispersion. We also find a good agreement between the strength of the shear measured at the GGL positions through strong-lensing modelling and the value derived independently from a weak-lensing analysis of the background sources. Overall, we show the advantages of using single GGL events in the outskirts of clusters to robustly constrain the local shear, even when only photometric redshift estimates are known for the source. We argue that the mass-luminosity scaling relation of cluster members can be tested by modelling the GGLs found around them, and show that the mass parameters can vary up to $\sim$30% between the cluster and GGL models assuming this scaling relation.

Unveiling galaxy bias via the halo model, KiDS, and GAMA

We measure the projected galaxy clustering and galaxy-galaxy lensing signals using the Galaxy And Mass Assembly (GAMA) survey and Kilo-Degree Survey (KiDS) to study galaxy bias. We use the concept of non-linear and stochastic galaxy biasing in the framework of halo occupation statistics to constrain the parameters of the halo occupation statistics and to unveil the origin of galaxy biasing. The bias function $\Gamma_{\text{gm}}(r_{\text{p}})$, where $r_{\text{p}}$ is the projected comoving separation, is evaluated using the analytical halo model from which the scale dependence of $\Gamma_{\text{gm}}(r_{\text{p}})$, and the origin of the non-linearity and stochasticity in halo occupation models can be inferred. Our observations unveil the physical reason for the non-linearity and stochasticity, further explored using hydrodynamical simulations, with the stochasticity mostly originating from the non-Poissonian behaviour of satellite galaxies in the dark matter haloes and their spatial distribution, which does not follow the spatial distribution of dark matter in the halo. The observed non-linearity is mostly due to the presence of the central galaxies, as was noted from previous theoretical work on the same topic. We also see that overall, more massive galaxies reveal a stronger scale dependence, and out to a larger radius. Our results show that a wealth of information about galaxy bias is hidden in halo occupation models. These models should therefore be used to determine the influence of galaxy bias in cosmological studies.

Measuring the scale dependence of intrinsic alignments using multiple shear estimates

We present a new method for measuring the scale dependence of the intrinsic alignment (IA) contamination to the galaxy-galaxy lensing signal, which takes advantage of multiple shear estimation methods applied to the same source galaxy sample. By exploiting the resulting correlation of both shape noise and cosmic variance, our method can provide an increase in the signal-to-noise of the measured IA signal as compared to methods which rely on the difference of the lensing signal from multiple photometric redshift bins. For a galaxy-galaxy lensing measurement which uses LSST sources and DESI lenses, the signal-to-noise on the IA signal from our method is predicted to improve by a factor of $\sim 2$ relative to the method of Blazek et al. 2012, for pairs of shear estimates which yield substantially different measured IA amplitudes and highly correlated shape noise terms. We show that statistical error necessarily dominates the measurement of intrinsic alignments using our method. We also consider a physically motivated extension of the Blazek et al. 2012 method which assumes that all nearby galaxy pairs, rather than only excess pairs, are subject to IA. In this case, the signal-to-noise of the method of Blazek et al. 2012 is improved.

General relativistic corrections in density-shear correlations

We investigate the corrections which relativistic light-cone computations induce on the correlation of the tangential shear with galaxy number counts, also known as galaxy-galaxy lensing. The standard-approach to galaxy-galaxy lensing treats the number density of sources in a foreground bin as observable, whereas it is in reality unobservable due to the presence of relativistic corrections. We find that already in the redshift range covered by the DES first year data, these currently neglected relativistic terms lead to a systematic correction of up to 50% in the density-shear correlation function for the highest redshift bins. This correction is dominated by the fact that a redshift bin of number counts does not only lens sources in a background bin, but is itself again lensed by all masses between the observer and the counted source population. Relativistic corrections are currently ignored in the standard galaxy-galaxy analyses, and the additional lensing of a counted source populations is only included in the error budget (via the covariance matrix). At increasingly higher redshifts and larger scales, these relativistic and lensing corrections become however increasingly more important, and we here argue that it is then more efficient, and also cleaner, to account for these corrections in the density-shear correlations.

Halo mass and weak galaxy-galaxy lensing profiles in rescaled cosmological N-body simulations

We investigate 3D density and weak lensing profiles of dark matter haloes predicted by a cosmology-rescaling algorithm for $N$-body simulations. We extend the rescaling method of Angulo & White (2010) and Angulo & Hilbert (2015) to improve its performance on intra-halo scales by using models for the concentration-mass-redshift relation based on excursion set theory. The accuracy of the method is tested with numerical simulations carried out with different cosmological parameters. We find that predictions for median density profiles are more accurate than $\sim 5\,\%$ for haloes with masses of $10^{12.0} - 10^{14.5} h^{-1}\,M_{\odot}$ for radii $0.05 < r/r_{200\text{m}} < 0.5$, and for cosmologies with $\Omega_\text{m} \in [0.15,\,0.40]$ and $\sigma_8 \in [0.6,\,1.0]$. For larger radii, $0.5 < r/r_{200\text{m}} < 5$, the accuracy degrades to $\sim20\,\%$, due to inaccurate modelling of the cosmological and redshift dependence of the splashback radius. For changes in cosmology allowed by current data, the residuals decrease to $\lesssim2\,\%$ up to scales twice the virial radius. We illustrate the usefulness of the method by estimating the mean halo mass of a mock galaxy group sample. We find that the algorithm's accuracy is sufficient for current data. Improvements in the algorithm, particularly in the modelling of baryons, are likely required for interpreting future (dark energy task force stage IV) experiments.

Scale dependence of galaxy biasing investigated by weak gravitational lensing: An assessment using semi-analytic galaxies and simulated lensing data

Galaxies are biased tracers of the matter density on cosmological scales. For future tests of galaxy models, we refine and assess a method to measure galaxy biasing as a function of physical scalekwith weak gravitational lensing. This method enables us to reconstruct the galaxy bias factorb(k) as well as the galaxy-matter correlationr(k) on spatial scales between 0.01hMpc−1≲k≲ 10hMpc−1for redshift-binned lens galaxies below redshiftz≲ 0.6. In the refinement, we account for an intrinsic alignment of source ellipticities, and we correct for the magnification bias of the lens galaxies, relevant for the galaxy-galaxy lensing signal, to improve the accuracy of the reconstructedr(k). For simulated data, the reconstructions achieve an accuracy of 3–7% (68% confidence level) over the abovek-range for a survey area and a typical depth of contemporary ground-based surveys. Realistically the accuracy is, however, probably reduced to about 10–15%, mainly by systematic uncertainties in the assumed intrinsic source alignment, the fiducial cosmology, and the redshift distributions of lens and source galaxies (in that order). Furthermore, our reconstruction technique employs physical templates forb(k) andr(k) that elucidate the impact of central galaxies and the halo-occupation statistics of satellite galaxies on the scale-dependence of galaxy bias, which we discuss in the paper. In a first demonstration, we apply this method to previous measurements in the Garching-Bonn Deep Survey and give a physical interpretation of the lens population.

The effects of assembly bias on the inference of matter clustering from galaxy–galaxy lensing and galaxy clustering

The combination of galaxy-galaxy lensing (GGL) and galaxy clustering is a promising route to measuring the amplitude of matter clustering and testing modified gravity theories of cosmic acceleration. Halo occupation distribution (HOD) modeling can extend the approach down to nonlinear scales, but galaxy assembly bias could introduce systematic errors by causing the HOD to vary with large scale environment at fixed halo mass. We investigate this problem using the mock galaxy catalogs created by Hearin & Watson (2013, HW13), which exhibit significant assembly bias because galaxy luminosity is tied to halo peak circular velocity and galaxy colour is tied to halo formation time. The preferential placement of galaxies (especially red galaxies) in older halos affects the cutoff of the mean occupation function $\langle N_\text{cen}(M_\text{min}) \rangle$ for central galaxies, with halos in overdense regions more likely to host galaxies. The effect of assembly bias on the satellite galaxy HOD is minimal. We introduce an extended, environment dependent HOD (EDHOD) prescription to describe these results and fit galaxy correlation measurements. Crucially, we find that the galaxy-matter cross-correlation coefficient, $r_{gm} \equiv \xi_{gm} \cdot [ \xi_{mm} \xi_{gg} ]^{-1/2}$, is insensitive to assembly bias on scales $r \gtrsim 1 \; h^{-1}\text{Mpc}$, even though $\xi_{gm} $ and $\xi_{gg} $ are both affected individually. We can therefore recover the correct $\xi_{mm} $ from the HW13 galaxy-galaxy and galaxy-matter correlations using either a standard HOD or EDHOD fitting method. For $M_r \leq -19$ or $M_r \leq -20$ samples the recovery of $\xi_{mm}$ is accurate to 2% or better. For a sample of red $M_r \leq -20$ galaxies we achieve 2% recovery at $r \gtrsim 2\;h^{-1}\text{Mpc}$ with EDHOD modeling but lower accuracy at smaller scales or with a standard HOD fit.

KiDS-i-800: comparing weak gravitational lensing measurements from same-sky surveys

We present a weak gravitational lensing analysis of 815 square degree of $i$-band imaging from the Kilo-Degree Survey (KiDS-$i$-800). In contrast to the deep $r$-band observations, which take priority during excellent seeing conditions and form the primary KiDS dataset (KiDS-$r$-450), the complementary yet shallower KiDS-$i$-800 spans a wide range of observing conditions. The overlapping KiDS-$i$-800 and KiDS-$r$-450 imaging therefore provides a unique opportunity to assess the robustness of weak lensing measurements. In our analysis, we introduce two new `null' tests. The `nulled' two-point shear correlation function uses a matched catalogue to show that the calibrated KiDS-$i$-800 and KiDS-$r$-450 shear measurements agree at the level of $1 \pm 4$\%. We use five galaxy lens samples to determine a `nulled' galaxy-galaxy lensing signal from the full KiDS-$i$-800 and KiDS-$r$-450 surveys and find that the measurements agree to $7 \pm 5$\% when the KiDS-$i$-800 source redshift distribution is calibrated using either spectroscopic redshifts, or the 30-band photometric redshifts from the COSMOS survey.

Mapping stellar content to dark matter haloes – III. Environmental dependence and conformity of galaxy colours

Recent studies suggest that the quenching properties of galaxies are correlated over several mega-parsecs. The large-scale "galactic conformity" phenomenon around central galaxies has been regarded as a potential signature of "galaxy assembly bias" or "pre-heating", both of which interpret conformity as a result of direct environmental effects acting on galaxy formation. Building on the iHOD halo quenching framework developed in Zu & Mandelbaum (2015, 2016), we discover that our fiducial halo mass quenching model, without any galaxy assembly bias, can successfully explain the overall environmental dependence and the conformity of galaxy colours in SDSS, as measured by the mark correlation functions of galaxy colours and the red galaxy fractions around isolated primaries, respectively. Our fiducial iHOD halo quenching mock also correctly predicts the differences in the spatial clustering and galaxy-galaxy lensing signals between the more vs. less red galaxy subsamples, split by the red-sequence ridge-line at fixed stellar mass. Meanwhile, models that tie galaxy colours fully or partially to halo assembly bias have difficulties in matching all these observables simultaneously. Therefore, we demonstrate that the observed environmental dependence of galaxy colours can be naturally explained by the combination of 1) halo quenching and 2) the variation of halo mass function with environment --- an indirect environmental effect mediated by two separate physical processes.

Auto-detection of strong gravitational lenses using convolutional neural networks

We propose a method for the automated detection of strong galaxy-galaxy gravitational lenses in images, utilising a convolutional neural network (CNN) trained on 210 000 simulated galaxy-galaxy lens and non-lens images. The CNN, named LensFinder, was tested on a separate 210 000 simulated image catalogue, with 95% of images classied with at least 98.6% certainty. An accuracy of over 98% was achieved and an area under curve of 0.9975 was determined from the resulting receiver operating characteristic curve. A regional CNN, R-LensFinder, was trained to label lens positions in images, perfectly labelling 80% while partially labelling another 10% correctly.

The VIMOS Public Extragalactic Redshift Survey (VIPERS)

We carry out a joint analysis of redshift-space distortions and galaxy-galaxy lensing, with the aim of measuring the growth rate of structure; this is a key quantity for understanding the nature of gravity on cosmological scales and late-time cosmic acceleration. We make use of the final VIPERS redshift survey dataset, which maps a portion of the Universe at a redshift of $z \simeq 0.8$, and the lensing data from the CFHTLenS survey over the same area of the sky. We build a consistent theoretical model that combines non-linear galaxy biasing and redshift-space distortion models, and confront it with observations. The two probes are combined in a Bayesian maximum likelihood analysis to determine the growth rate of structure at two redshifts $z=0.6$ and $z=0.86$. We obtain measurements of $f\sigma_8(0.6) = 0.48 \pm 0.12$ and $f\sigma_8(0.86) = 0.48 \pm 0.10$. The additional galaxy-galaxylensing constraint alleviates galaxy bias and $\sigma_8$ degeneracies, providing direct measurements of $[f(0.6),\sigma_8(0.6)] = [0.93 \pm 0.22, 0.52 \pm 0.06]$ and $f(0.86),\sigma_8(0.86)] = [0.99 \pm 0.19, 0.48 \pm 0.04]$. These measurements are statistically consistent with a Universe where the gravitational interactions can be described by General Relativity, although they are not yet accurate enough to rule out some commonly considered alternatives. Finally, as a complementary test we measure the gravitational slip parameter, $E_G$ , for the first time at $z>0.6$. We find values of $\smash{\overline{E}_G}(0.6) = 0.16 \pm 0.09$ and $\smash{\overline{E}_G}(0.86) = 0.09 \pm 0.07$, when $E_G$ is averaged over scales above $3 h^{-1} \rm{Mpc}$. We find that our $E_G$ measurements exhibit slightly lower values than expected for standard relativistic gravity in a {\Lambda}CDM background, although the results are consistent within $1-2\sigma$.

KiDS-450 + 2dFLenS: Cosmological parameter constraints from weak gravitational lensing tomography and overlapping redshift-space galaxy clustering

We perform a combined analysis of cosmic shear tomography, galaxy-galaxy lensing tomography, and redshift-space multipole power spectra (monopole and quadrupole) using 450 deg$^2$ of imaging data by the Kilo Degree Survey (KiDS) overlapping with two spectroscopic surveys: the 2-degree Field Lensing Survey (2dFLenS) and the Baryon Oscillation Spectroscopic Survey (BOSS). We restrict the galaxy-galaxy lensing and multipole power spectrum measurements to the overlapping regions with KiDS, and self-consistently compute the full covariance between the different observables using a large suite of $N$-body simulations. We methodically analyze different combinations of the observables, finding that galaxy-galaxy lensing measurements are particularly useful in improving the constraint on the intrinsic alignment amplitude (by 30%, positive at $3.5\sigma$ in the fiducial data analysis), while the multipole power spectra are useful in tightening the constraints along the lensing degeneracy direction (e.g. factor of two stronger matter density constraint in the fiducial analysis). The fully combined constraint on $S_8 \equiv \sigma_8 \sqrt{\Omega_{\rm m}/0.3} = 0.742 \pm 0.035$, which is an improvement by 20% compared to KiDS alone, corresponds to a $2.6\sigma$ discordance with Planck, and is not significantly affected by fitting to a more conservative set of scales. Given the tightening of the parameter space, we are unable to resolve the discordance with an extended cosmology that is simultaneously favored in a model selection sense, including the sum of neutrino masses, curvature, evolving dark energy, and modified gravity. The complementarity of our observables allows for constraints on modified gravity degrees of freedom that are not simultaneously bounded with either probe alone, and up to a factor of three improvement in the $S_8$ constraint in the extended cosmology compared to KiDS alone.

Galaxy bias from galaxy–galaxy lensing in the DES science verification data

We present a measurement of galaxy-galaxy lensing around a magnitude-limited ($i_{AB} < 22.5$) sample of galaxies from the Dark Energy Survey Science Verification (DES-SV) data. We split these lenses into three photometric-redshift bins from 0.2 to 0.8, and determine the product of the galaxy bias $b$ and cross-correlation coefficient between the galaxy and dark matter overdensity fields $r$ in each bin, using scales above 4 Mpc/$h$ comoving, where we find the linear bias model to be valid given our current uncertainties. We compare our galaxy bias results from galaxy-galaxy lensing with those obtained from galaxy clustering (Crocce et al. 2016) and CMB lensing (Giannantonio et al. 2016) for the same sample of galaxies, and find our measurements to be in good agreement with those in Crocce et al. (2016), while, in the lowest redshift bin ($z\sim0.3$), they show some tension with the findings in Giannantonio et al. (2016). We measure $b\cdot r$ to be $0.87\pm 0.11$, $1.12 \pm 0.16$ and $1.24\pm 0.23$, respectively for the three redshift bins of width $\Delta z = 0.2$ in the range $0.2<z <0.8$, defined with the photometric-redshift algorithm BPZ. Using a different code to split the lens sample, TPZ, leads to changes in the measured biases at the 10-20\% level, but it does not alter the main conclusion of this work: when comparing with Crocce et al. (2016) we do not find strong evidence for a cross-correlation parameter significantly below one in this galaxy sample, except possibly at the lowest redshift bin ($z\sim 0.3$), where we find $r = 0.71 \pm 0.11$ when using TPZ, and $0.83 \pm 0.12$ with BPZ.

Galaxy–galaxy and galaxy–cluster lensing with the SDSS and FIRST surveys

We perform a galaxy-galaxy lensing study by correlating the shapes of $\sim$2.7 $\times$ 10$^5$ galaxies selected from the VLA FIRST radio survey with the positions of $\sim$38.5 million SDSS galaxies, $\sim$132000 BCGs and $\sim$78000 SDSS galaxies that are also detected in the VLA FIRST survey. The measurements are conducted on angular scales ${\theta}$ $\lesssim$ 1200 arcsec. On scales ${\theta}$ $\lesssim$ 200 arcsec we find that the measurements are corrupted by residual systematic effects associated with the instrumental beam of the VLA data. Using simulations we show that we can successfully apply a correction for these effects. Using the three lens samples (the SDSS DR10 sample, the BCG sample and the SDSS-FIRST matched object sample) we measure a tangential shear signal that is inconsistent with zero at the 10${\sigma}$, 3.8${\sigma}$ and 9${\sigma}$ level respectively. Fitting an NFW model to the detected signals we find that the ensemble mass profile of the BCG sample agrees with the values in the literature. However, the mass profiles of the SDSS DR10 and the SDSS-FIRST matched object samples are found to be shallower and steeper than results in the literature respectively. The best-fitting Virial masses for the SDSS DR10, BCG and SDSS-FIRST matched samples, derived using an NFW model and allowing for a varying concentration factor, are M$^{SDSS-DR10}_{200}$ = (1.2 $\pm$ 0.4) $\times$ 10$^{12}$M$_{\odot}$, M$^{BCG}_{200}$ = (1.4 $\pm$ 1.3) $\times$ 10$^{13}$M$_{\odot}$ and M$^{SDSS-FIRST}_{200}$ = 8.0 $\pm$ 4.2 $\times$ 10$^{13}$M$_{\odot}$ respectively. These results are in good agreement (within $\sim$2${\sigma}$) with values in the literature. Our findings suggest that for galaxies to be both bright in the radio and in the optical they must be embedded in very dense environment on scales R $\lesssim$ 1Mpc.

The dependence of halo mass on galaxy size at fixed stellar mass using weak lensing

Stellar mass has been shown to correlate with halo mass, with non-negligible scatter. The stellar mass-size and luminosity-size relationships of galaxies also show significant scatter in galaxy size at fixed stellar mass. It is possible that, at fixed stellar mass and galaxy colour, the halo mass is correlated with galaxy size. Galaxy-galaxy lensing allows us to measure the mean masses of dark matter haloes for stacked samples of galaxies. We extend the analysis of the galaxies in the CFHTLenS catalogue by fitting single S\'{e}rsic surface brightness profiles to the lens galaxies in order to recover half-light radius values, allowing us to determine halo masses for lenses according to their size. Comparing our halo masses and sizes to baselines for that stellar mass yields a differential measurement of the halo mass-galaxy size relationship at fixed stellar mass, defined as $M_{h}(M_{*}) \propto r_{\mathrm{eff}}^{\eta}(M_{*})$. We find that on average, our lens galaxies have an $\eta = 0.42\pm0.12$, i.e. larger galaxies live in more massive dark matter haloes. The $\eta$ is strongest for high mass luminous red galaxies (LRGs). Investigation of this relationship in hydrodynamical simulations suggests that, at a fixed $M_{*}$, satellite galaxies have a larger $\eta$ and greater scatter in the $M_{\mathrm{h}}$ and $r_{\mathrm{eff}}$ relationship compared to central galaxies.

Galaxy–galaxy lensing estimators and their covariance properties

We study the covariance properties of real space correlation function estimators -- primarily galaxy-shear correlations, or galaxy-galaxy lensing -- using SDSS data for both shear catalogs and lenses (specifically the BOSS LOWZ sample). Using mock catalogs of lenses and sources, we disentangle the various contributions to the covariance matrix and compare them with a simple analytical model. We show that not subtracting the lensing measurement around random points from the measurement around the lens sample is equivalent to performing the measurement using the lens density field instead of the lens over-density field. While the measurement using the lens density field is unbiased (in the absence of systematics), its error is significantly larger due to an additional term in the covariance. Therefore, this subtraction should be performed regardless of its beneficial effects on systematics. Comparing the error estimates from data and mocks for estimators that involve the over-density, we find that the errors are dominated by the shape noise and lens clustering, that empirically estimated covariances (jackknife and standard deviation across mocks) are consistent with theoretical estimates, and that both the connected parts of the 4-point function and the super-sample covariance can be neglected for the current levels of noise. While the trade-off between different terms in the covariance depends on the survey configuration (area, source number density), the diagnostics that we use in this work should be useful for future works to test their empirically-determined covariances.