The Low-Mass X-ray Binary and Globular Cluster Connection in Virgo Cluster Early-type Galaxies: Optical Properties

(Abridged) Chandra and Hubble ACS observations of 11 early-type galaxies probe the low-mass X-ray binary (LMXB) - globular cluster (GC) connection. We explore the optical properties of 270 GCs with LMXBs and 6,488 GCs without LMXBs. More massive, redder, and more compact GCs are more likely to contain LMXBs. Unlike Galactic GCs, a large number of GCs with LMXBs have half-mass relaxation times > 2.5 Gyr. We fit the dependence of the expected number of LMXBs per GC, \lambda_t, on the GC mass M, color (g-z), and half-mass radius r_{h,cor}, and find that \lambda_t \propto M^{1.24\pm0.08} 10^{0.9^{+0.2}_{-0.1} (g-z)} r_{h,cor}^{-2.2^{+0.3}_{-0.4}}. Our fit rules out that the number of LMXBs per GC is linearly proportional to GC mass and that most GCs with high X-ray luminosities contain a single LMXB. The detailed dependence of \lambda_t on GC properties appears essentially equivalent to a dependence on the encounter rate \Gamma_h and the metallicity Z, \lambda_t \propto \Gamma_h^{0.82\pm0.05} Z^{0.39\pm0.07}. Our analysis provides strong evidence that dynamical formation and metallicity play the primary roles in determining the presence of an LMXB in extragalactic GCs. The shallower than linear dependence for our sample requires an explanation by theories of dynamical binary formation. The abundance dependence is consistent with a metallicity-dependent variation in the number o neutron stars and black holes per unit mass GC, effects from irradiation induced winds or suppression of magnetic braking in metal-poor stars.