Abstract:
We investigate the effect of weak gravitational lensing in the limit of small angular scales where projected galaxy clustering is strongly nonlinear. This is the regime likely to be probed by future weak lensing surveys. We use well-motivated hierarchical scaling arguments and the plane-parallel approximation to study multi-point statistical properties of the convergence field. These statistics can be used to compute the vertex amplitudes in tree models of hierarchical clustering; these can be compared with similar measurements from galaxy surveys, leading to a powerful probe of galaxy bias.

Abstract:
A joint analysis of the clustering of galaxies and their weak gravitational lensing signal is well-suited to simultaneously constrain the galaxy-halo connection as well as the cosmological parameters by breaking the degeneracy between galaxy bias and the amplitude of clustering signal. In a series of two papers, we perform such an analysis at the highest redshift ($z\sim0.53$) in the literature using CMASS galaxies in the Sloan Digital Sky Survey-III Baryon Oscillation Spectroscopic Survey Eleventh Data Release (SDSS-III/BOSS DR11) catalog spanning 8300~deg$^2$. In this paper, we present details of the clustering and weak lensing measurements of these galaxies. We define a subsample of 400,916 CMASS galaxies based on their redshifts and stellar mass estimates so that the galaxies constitute an approximately volume-limited and similar population over the redshift range $0.47\le z\le 0.59$. We obtain a signal-to-noise ratio $S/N\simeq 56$ for the galaxy clustering measurement. We also explore the redshift and stellar mass dependence of the clustering signal. For the weak lensing measurement, we use existing deeper imaging data from the CFHTLS with publicly available shape and photometric redshift catalogs from CFHTLenS, but only in a 105~deg$^2$ area which overlaps with BOSS. This restricts the lensing measurement to only 5,084 CMASS galaxies. After careful systematic tests, we find a highly significant detection of the CMASS weak lensing signal, with total $S/N\simeq 26$. These measurements form the basis of the halo occupation distribution and cosmology analysis presented in More et al. (Paper II).

Abstract:
We discuss the second order contributions to lensing statistics resulting from the clustering of background sources from which galaxy shape measurements are made in weak lensing experiments. In addition to a previously discussed contribution to the lensing skewness, background source clustering also contributes to the two-point correlation function, such as the angular power spectrum of convergence or shear. At arcminute scales or above, the second order contribution to the angular power spectrum of convergence due to source clustering is below the level of a few percent. The background clustering of sources also results in a non-Gaussian contribution to the power spectrum covariance of weak lensing convergence through a four-point correlation function or a trispectrum in Fourier space. The increase in variance is, at most, a few percent relative to the Gaussian contribution while the band powers are also correlated at the few percent level. The non-Gaussian contributions due to background source clustering is at least an order of magnitude smaller than those resulting from non-Gaussian aspects of the large scale structure due to the non-linear evolution of gravitational perturbations. We suggest that the background source clustering is unlikely to affect the precision measurements of cosmology from upcoming weak lensing surveys.

Abstract:
I investigate the effects of source clustering on the weak lensing statistics, more particularly on the statistical properties of the local convergence, kappa, at large angular scales. The Perturbation Theory approach shows that the variance is not affected by source clustering at leading order but higher order moments such as the third and fourth moments can be. I compute the magnitude of these effects in case of an Einstein-de Sitter Universe for the angular top-hat filtered convergence. In these calculations the so-called Broadhurst and multiple lens coupling effects are neglected. The source clustering effect is found to be particularly important when the redshift distribution is broad enough so that remote background sources can be significantly lensed by closer concentrations of galaxy sources. The source clustering effects are shown to remain negligible, for both the skewness and the kurtosis, when the dispersion of the redshift of the sources is less than about 0.15.

Abstract:
We study how 21 cm intensity mapping can be used to measure gravitational lensing over a wide range of redshift. This can extend weak lensing measurements to higher redshifts than are accessible with conventional galaxy surveys. We construct a convergence estimator taking into account the discreteness of galaxies and calculate the expected noise level as a function of redshift and telescope parameters. At $z \sim 2-3$ we find that a telescope array with a collecting area $\sim 0.2 \, {\rm km}^2$ spread over a region with diameter $\sim 2 \, {\rm km}$ would be sufficient to measure the convergence power spectrum to high accuracy for multipoles between 10 and 1,000. We show that these measurements can be used to constrain interacting dark energy models.

Abstract:
The available probes of the large scale structure in the Universe have distinct properties: galaxies are a high resolution but biased tracer of mass, while weak lensing avoids such biases but, due to low signal-to-noise ratio, has poor resolution. We investigate reconstructing the projected density field using the complementarity of weak lensing and galaxy positions. We propose a maximum-probability reconstruction of the 2D lensing convergence with a likelihood term for shear data and a prior on the Fourier phases constructed from the galaxy positions. By considering only the phases of the galaxy field, we evade the unknown value of the bias and allow it to be calibrated by lensing on a mode-by-mode basis. By applying this method to a realistic simulated galaxy shear catalogue, we find that a weak prior on phases provides a good quality reconstruction down to scales beyond l=1000, far into the noise domain of the lensing signal alone.

Abstract:
We formulate the concept of non-linear and stochastic galaxy biasing in the framework of halo occupation statistics. Using two-point statistics in projection, we define the galaxy bias function, b_g(r_p), and the galaxy-dark matter cross-correlation function, R_{gm}(r_p), where r_p is the projected distance. We use the analytical halo model to predict how the scale dependence of b_g and R_{gm}, over the range 0.1 Mpc/h < r_p < 30 Mpc/h, depends on the non-linearity and stochasticity in halo occupation models. In particular we quantify the effect due to the presence of central galaxies, the assumption for the radial distribution of satellite galaxies, the richness of the halo, and the Poisson character of the probability to have a certain number of satellite galaxies in a halo of a certain mass. Overall, brighter galaxies reveal a stronger scale dependence, and out to a larger radius. In real-space, we find that galaxy bias becomes scale independent, with R_{gm} = 1, for radii r > 1 - 5 Mpc/h, depending on luminosity. However, galaxy bias is scale-dependent out to much larger radii when one uses the projected quantities defined in this paper. These projected bias functions have the advantage that they are more easily accessible observationally and that their scale dependence carries a wealth of information regarding the properties of galaxy biasing. To observationally constrain the parameters of the halo occupation statistics and to unveil the origin of galaxy biasing we propose the use of the bias function Gamma_{gm}(r_p)=b_g(r_p)/R_{gm}(r_p). This function is obtained via a combination of weak gravitational lensing and galaxy clustering, and it can be measured using existing and forthcoming imaging and spectroscopic galaxy surveys.

Abstract:
Weak gravitational lensing by large scale structure affects the number counts of faint galaxies through the ``magnification bias'' and thus affects the measurement of the angular two-point correlation function $\w $. At faint magnitudes the clustering amplitude will decrease differently with limiting magnitude than expected from Limber's equation. The amplitude will hit a minimum and then rise with limiting magnitude. This behavior occurs because $\w$ due to clustering decreases with distance, while the ``magnification bias'' due to weak lensing increases with distance. The apparent magnitude $m_{min}$ at which the magnification bias starts to dominate the observed clustering is model and color dependent. It is given by $\omega(m=m_{min},\theta=5^\prime) \approx (1\ -\ 2)\times 10^{-3}(5s-2)^2 \Omega_0^2 \sigma_8^2$, where $s$ is the logarithmic slope of the number counts. Already published measurements of $\w$ at $R=25$ may be strongly influenced by the ``magnification bias''. An experiment using the ratio of blue and red number counts across the sky can be designed such that the effects of the ``true'' clustering is minimized. The magnification bias is a measurement of the clustering of the mass. This weak lensing experiment does not require measuring shapes and position angles of galaxies. I derive a revised Limber's Equation including the effects of magnification bias.

Abstract:
We develop an algorithm for the reconstruction of the two-dimensional mass distribution of a gravitational lens from the observable distortion of background galaxies. From the measured reduced shear, the lens mapping is obtained, from which a mass distribution is derived. This is unlike other methods where the convergence ("kappa") is directly obtained. We show that this method works best for sub-critical lenses, but can be applied to a critical lens away from the critical lines. For finite fields the usual mass-sheet degeneracy is shown to exist in this method as well. We show that the algorithm reproduces the mass distribution within acceptable limits when applied to simulated noisy data.

Abstract:
In the theory of structure formation, galaxies are biased tracers of the underlying matter density field. The statistical relation between galaxy and matter density field is commonly referred as galaxy bias. In this paper, we test the linear bias model with weak-lensing and galaxy clustering measurements in the 2 square degrees COSMOS field (Scoville et al. 2007). We estimate the bias of galaxies between redshifts z=0.2 and z=1, and over correlation scales between R=0.2 h^-1 Mpc and R=15 h^-1 Mpc. We focus on three galaxy samples, selected in flux (simultaneous cuts I_814W < 26.5 and K_s < 24), and in stellar-mass (10^9 < M_* < 10^10 h^-2 Msun and 10^10 < M^*< 10^11 h^-2 Msun). At scales R > 2 h^-1 Mpc, our measurements support a model of bias increasing with redshift. The Tinker et al. (2010) fitting function provides a good fit to the data. We find the best fit mass of the galaxy halos to be log(M_200 h^-1 Msun) = 11.7^+0.6_-1.3 and log(M_200 h^-1 Msun) = 12.4^+0.2_-2.9 respectively for the low and high stellar-mass samples. In the halo model framework, bias is scale-dependent with a change of slope at the transition scale between the one and the two halo terms. We detect a scale-dependence of bias with a turn-down at scale R=2.3\pm1.5 h^-1 Mpc, in agreement with previous galaxy clustering studies. We find no significant amount of stochasticity, suggesting that a linear bias model is sufficient to describe our data. We use N-body simulations to quantify both the amount of cosmic variance and systematic errors in the measurement.