How to Optimally Constrain Galaxy Assembly Bias: Supplement Projected
  Correlation Functions with Count-in-cells Statistics
  Most models for the connection between galaxies and their haloes ignore the
possibility that galaxy properties may be correlated with halo properties other
than mass, a phenomenon known as galaxy assembly bias. Yet, it is known that
such correlations can lead to systematic errors in the interpretation of survey
data. 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 $w_{\mathrm{p}}(r_{\mathrm{p}})$,
the galaxy--galaxy lensing signal $\Delta \Sigma(r_{\mathrm{p}})$, the void
probability function $\mathrm{VPF}(r)$, the distributions of
counts-in-cylinders $P(N_{\mathrm{CIC}})$, and counts-in-annuli
$P(N_{\mathrm{CIA}})$, and the distribution of the ratio of counts in cylinders
of different sizes $P(N_2/N_5)$. We find that despite the frequent use of the
combination $w_{\mathrm{p}}(r_{\mathrm{p}})+\Delta \Sigma(r_{\mathrm{p}})$ in
interpreting galaxy data, the count statistics, $P(N_{\mathrm{CIC}})$ and
$P(N_{\mathrm{CIA}})$, are generally more efficient in constraining galaxy
assembly bias when combined with $w_{\mathrm{p}}(r_{\mathrm{p}})$. Constraints
based upon $w_{\mathrm{p}}(r_{\mathrm{p}})$ and $\Delta \Sigma(r_{\mathrm{p}})$
share common degeneracy directions in the parameter space, while combinations
of $w_{\mathrm{p}}(r_{\mathrm{p}})$ 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.
