  In loop quantum cosmology the quantum dynamics is well understood. We
approximate the full quantum dynamics in the infinite dimensional Hilbert space
by projecting it on a finite dimensional submanifold thereof, spanned by
suitably chosen semiclassical states. This submanifold is isomorphic with the
classical phase space and the projected dynamical flow provides effective
equations incorporating the leading quantum corrections to the classical
equations of motion. Numerical work has been done using quantum states which
are semiclassical at late times. These states follow the classical trajectory
until the density is on the order of 1% of the Planck density then deviate
strongly from the classical trajectory. The effective equations we obtain
reproduce this behavior to surprising accuracy.
