A theoretical framework for combining techniques that probe the link between galaxies and dark matter

We develop a theoretical framework that combines measurements of galaxy-galaxy lensing, galaxy clustering, and the galaxy stellar mass function in a self-consistent manner. While considerable effort has been invested in exploring each of these probes individually, attempts to combine them are still in their infancy despite the potential of such combinations to elucidate the galaxy-dark matter connection, to constrain cosmological parameters, and to test the nature of gravity. In this paper, we focus on a theoretical model that describes the galaxy-dark matter connection based on standard halo occupation distribution techniques. Several key modifications enable us to extract additional parameters that determine the stellar-to-halo mass relation and to simultaneously fit data from multiple probes while allowing for independent binning schemes for each probe. In a companion paper, we demonstrate that the model presented here provides an excellent fit to galaxy-galaxy lensing, galaxy clustering, and stellar mass functions measured in the COSMOS survey from z=0.2 to z=1.0. We construct mock catalogs from numerical simulations to investigate the effects of sample variance and covariance on each of the three probes. Finally, we analyze and discuss how trends in each of the three observables impact the derived parameters of the model. In particular, we investigate the various features of the observed galaxy stellar mass function (low-mass slope, plateau, knee, and high-mass cut-off) and show how each feature is related to the underlying relationship between stellar and halo mass. We demonstrate that the observed plateau feature in the stellar mass function at Mstellar~2x10^10 Msun is due to the transition that occurs in the stellar-to-halo mass relation at Mhalo ~ 10^12 Msun from a low-mass power-law regime to a sub-exponential function at higher stellar mass.