From Galaxy-Galaxy Lensing to Cosmological Parameters

Galaxy-galaxy lensing uses the weak distortion of background sources to measure the mean excess surface density profile, ΔΣ(r), around a sample of foreground lensing galaxies. We develop a method for combining ΔΣ(r) with the galaxy-galaxy correlation function ξ_gg(r) to constrain the matter density parameter Ω_m and the matter fluctuation amplitude σ₈, going beyond the linear biasing model to reach the level of accuracy demanded by current and future measurements. We adopt the halo occupation distribution (HOD) framework and test its applicability to this problem by examining the effects of replacing satellite galaxies in the halos of a smoothed particle hydrodynamics (SPH) simulation with randomly selected dark matter particles from the same halos. After accounting for the slight differences between the predicted radial profile of dark matter and satellite galaxies, the residual effects of individual subhalos around satellite galaxies and environmental dependence of the HOD at fixed halo mass are ≲5% in ΔΣ(r) for 0.1 h^-1 Mpc < r < 15 h^-1 Mpc. We develop an analytic approximation for calculating ΔΣ(r), improving on previous work with more accurate treatments of halo bias and halo exclusion. We demonstrate its accuracy at the few percent level with tests against a suite of populated N-body simulations. We use the analytic model to investigate the dependence of ΔΣ(r) and the galaxy-matter correlation function ξ_gm(r) on Ω_m and σ₈, once HOD parameters for a given cosmological model are pinned down by matching ξ_gg(r). The linear bias prediction that ξ_gm(r)/ξ_gg(r) = constant is accurate for r ≳ 2 h^-1 Mpc but fails at the 30%-50% level on smaller scales. The linear bias prediction that ΔΣ(r) ∝ Ω_mσ₈ breaks down at r < 10 h^-1 Mpc. We present predictions of ΔΣ(r) for SDSS galaxy samples with M_r ≤ - 20 and -21. These can be combined with future lensing measurements for these samples to constrain Ω_m and σ₈.