How to optimally constrain galaxy assembly bias: supplement projected correlation functions with count-in-cells statistics

Most models for the statistical connection between galaxies and their haloes ignore the possibility that galaxy properties may be correlated with halo properties other than halo mass, a phenomenon known as galaxy assembly bias. And yet, it is known that such correlations can lead to systematic errors in the interpretation of survey data that are analysed using traditional halo occupation models. At present, the degree to which galaxy assembly bias may be present in the real Universe, and the best strategies for constraining it remain uncertain. We study the ability of several observables to constrain galaxy assembly bias from redshift survey data using the decorated halo occupation distribution (dHOD), an empirical model of the galaxy-halo connection that incorporates assembly bias. We cover an expansive set of observables, including the projected two-point correlation function $w$_p(r_p), the galaxy-galaxy lensing signal ΔΣ(r_p), the void probability function VPF(r), the distributions of counts-in-cylinders P(N_CIC), and counts-in-annuli P(N_CIA), and the distribution of the ratio of counts in cylinders of different sizes P(N₂/N₅). We find that despite the frequent use of the combination $w$_p(r_p) + ΔΣ(r_p) in interpreting galaxy data, the count statistics, P(N_CIC) and P(N_CIA), are generally more efficient in constraining galaxy assembly bias when combined with $w$_p(r_p). Constraints based upon $w$_p(r_p) and ΔΣ(r_p) share common degeneracy directions in the parameter space, while combinations of $w$_p(r_p) with the count statistics are more complementary. Therefore, we strongly suggest that count statistics should be used to complement the canonical observables in future studies of the galaxy-halo connection.