  We examine a length scale that characterizes the spatial extent of
heterogeneous dynamics in a glass-forming binary hard-sphere mixture up to the
mode-coupling volume fraction phi_c. First, we characterize the system's
dynamics. Then, we utilize a new method [Phys. Rev. Lett. 105, 217801 (2010)]
to extract and analyze the ensemble independent dynamic susceptibility chi_4(t)
and the dynamic correlation length xi(t) for a range of times between the beta
and alpha relaxation times. We find that in this time range the dynamic
correlation length follows a volume fraction independent curve xi(t) ~ ln(t).
For longer times, xi(t) departs from this curve and remains constant up to the
largest time at which we can determine the length accurately. In addition to
the previously established correlation tau_alpha ~ exp[xi(tau_alpha)] between
the alpha relaxation time, tau_alpha, and the dynamic correlation length at
this time, xi(tau_alpha), we also find a similar correlation for the diffusion
coefficient D ~ exp[xi(tau_alpha)^theta] with theta approximately 0.6. We
discuss the relevance of these findings for different theories of the glass
transition.
