  We present a theoretical analysis of the dynamic structure factor (DSF) of a
liquid at and below the mode coupling critical temperature $T_c$, by developing
a self-consistent theoretical treatment which includes the contributions both
from continuous diffusion, described using general two coupling parameter
($F_{12}$) mode coupling theory (MCT), and from the activated hopping,
described using the random first order transition (RFOT) theory, incorporating
the effect of dynamical heterogeneity. The theory is valid over the whole
temperature plane and shows correct limiting MCT like behavior above $T_{c}$
and goes over to the RFOT theory near the glass transition temperature,
$T_{g}$. Between $T_{c}$ and $T_{g}$, the theory predicts that neither the
continuous diffusion, described by pure mode coupling theory, nor the hopping
motion alone suffices but both contribute to the dynamics while interacting
with each other. We show that the interplay between the two contributions
conspires to modify the relaxation behavior of the DSF from what would be
predicted by a theory with a complete static Gaussian barrier distribution in a
manner that may be described as a facilitation effect. Close to $T_c$, coupling
between the short time part of MCT dynamics and hopping reduces the stretching
given by the F$_{12}$-MCT theory significantly and accelerates structural
relaxation. As the temperature is progressively lowered below $T_c$, the
equations yield a crossover from MCT dominated regime to the hopping dominated
regime. In the combined theory the dynamical heterogeneity is modified because
the low barrier components interact with the MCT dynamics to enhance the
relaxation rate below $T_c$ and reduces the stretching that would otherwise
arise from an input static barrier height distribution.
