  Liquids near the glass transition exhibit dynamical heterogeneity, i.e.
correlated regions in the liquid relax at either a much faster rate or a much
slower rate than the average. This collective phenomenon has been characterized
by measurements of a dynamic susceptibility $\chi_4(t)$, which are sometimes
interpreted in terms of the size of those relaxing regions and the intensity of
the fluctuations. We show that the results of those measurements can be
affected not only by the collective fluctuations in the relaxation rate, but
also by density fluctuations in the initial state and by single-particle
fluctuations. We also show that at very long times the average overlap $C(t)$
probing the similarity between an initial and a final state separated by a time
interval $t$ decays as a power law $C(t) \sim t^{-d/2}$. This is much slower
than the stretched exponential behavior $C(t) \sim {\rm e}^{-(t/\tau)^{\beta}}$
previously observed at times within one or two orders of magnitude of the
$\alpha$-relaxation time $\tau_{\alpha}$. We find that for times longer than
$10-100 \tau_{\alpha}$, the dynamic susceptibility $\chi_4(t)$ is dominated by
single particle fluctuations, and that $\chi_4(t) \approx C(t) \sim t^{-d/2}$.
Finally, we introduce a method to extract the collective relaxation
contribution to the dynamic susceptibility $\chi_4(t)$ by subtracting the
effects of single-particle fluctuations and initial state density fluctuations.
We apply this method to numerical simulations of two glass forming models: a
binary hard sphere system and a Kob-Andersen Lennard-Jones system. This allows
us to extend the analysis of numerical data to timescales much longer than
previously possible, and opens the door for further future progress in the
study of dynamic heterogeneities, including the determination of the exchange
time.
