The effects of assembly bias on cosmological inference from
  galaxy-galaxy lensing and galaxy clusters
  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.
