  We study theoretically and numerically a family of multi-point dynamic
susceptibilities that quantify the strength and characteristic lengthscales of
dynamic heterogeneities in glass-forming materials. We use general theoretical
arguments (fluctuation-dissipation relations and symmetries of relevant
dynamical field theories) to relate the sensitivity of averaged two-time
correlators to temperature and density to spontaneous fluctuations of the local
dynamics. Our theoretical results are then compared to molecular dynamics
simulations of the Newtonian, Brownian and Monte-Carlo dynamics of two
representative glass-forming liquids, a fragile binary Lennard-Jones mixture
and a model for the strong glass-former silica. We justify in detail the claim
made in [Science 310, 1797 (2005)], that the temperature dependence of
correlation functions allows one to extract useful information on dynamic
lengthscales in glassy systems. We also discuss some subtle issues associated
to the choice of microscopic dynamics and of statistical ensemble through
conserved quantities, which are found to play an important role in determining
dynamic correlations.
