  We develop a systematic framework for the study of the initial collision
geometry fluctuations in relativistic heavy-ion collisions and investigate how
they evolve through different stages of the fireball history and translate into
final particle momentum anisotropies. We find in our event-by-event analysis
that only the few lowest momentum anisotropy parameters survive after the
hydrodynamical evolution of the system. The geometry of the produced medium is
found to be affected by the pre-equilibrium evolution of the medium and the
thermal smearing of the discretized event-by-event initial conditions, both of
which tend to smear out the spatial anisotropies. We find such effects to be
more prominent for higher moments than for lower moments. The correlations
between odd and even spatial anisotropy parameters during the pre-equilibrium
expansion are quantitatively studied and found to be small. Our study provides
a theoretical foundation for the understanding of initial state fluctuations
and the collective expansion dynamics in relativistic heavy-ion collisions.
