Galaxy-galaxy lensing uses the weak distortion of background sources to measure the mean excess surface density profile , \Delta \Sigma ( r ) , around a sample of foreground lensing galaxies .
We develop a method for combining \Delta \Sigma ( r ) with the galaxy-galaxy correlation function \xi _ { gg } ( r ) to constrain the matter density parameter \Omega _ { m } and the matter fluctuation amplitude \sigma _ { 8 } , going beyond the widely used linear biasing model to reach the level of accuracy demanded by current and future measurements .
We adopt the halo occupation distribution ( HOD ) framework , and we 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 .
The difference between dark matter and satellite galaxy radial profiles has a \sim 10 \% effect on \Delta \Sigma ( r ) at r < 1 ~ { } { h ^ { -1 } { Mpc } } .
However , if radial profiles are matched , then the remaining impact of individual sub-halos around satellite galaxies and environmental dependence of the HOD at fixed halo mass is \lesssim 5 \% in \Delta \Sigma ( r ) for 0.1 < r < 15 ~ { } { h ^ { -1 } { Mpc } } .
We develop an analytic approximation to \Delta \Sigma ( r ) for a specified cosmological model and galaxy HOD , improving on previous work with more accurate treatments of halo bias and halo exclusion .
Tests against a suite of populated N -body simulations show that the analytic approximation is accurate to a few percent or better over the range 0.1 < r < 20 ~ { } { h ^ { -1 } { Mpc } } .
We use the analytic model to investigate the dependence of \Delta \Sigma ( r ) and the galaxy-matter correlation function \xi _ { gm } ( r ) on \Omega _ { m } and \sigma _ { 8 } , once HOD parameters for a given cosmological model are pinned down by matching \xi _ { gg } ( r ) .
The linear bias prediction that \xi _ { gm } ( r ) / \xi _ { gg } ( r ) = { constant } is accurate for r \gtrsim 2 ~ { } { h ^ { -1 } { Mpc } } , but it fails at the 30 - 50 \% level on smaller scales .
The scaling of \Delta \Sigma ( r ) with cosmological parameters , which we model as \Delta \Sigma ( r ) \propto \Omega _ { m } ^ { \alpha ( r ) } \sigma _ { 8 } ^ { \beta ( r ) } , approaches the linear bias expectation \alpha = \beta = 1 at r \gtrsim 10 ~ { } { h ^ { -1 } { Mpc } } , but \alpha and \beta vary from 0.8 to 1.6 at smaller r .
We calculate a fiducial \Delta \Sigma ( r ) and scaling indices \alpha ( r ) and \beta ( r ) for galaxy samples that match the observed number density and projected correlation function of Sloan Digital Sky Survey galaxies with M _ { r } \leq - 20 and M _ { r } \leq - 21 .
Galaxy-galaxy lensing measurements for these samples can be combined with our predictions to constrain \Omega _ { m } and \sigma _ { 8 } , taking full advantage of the high measurement precision on small and intermediate scales .