  We consider a time-dependent deformation of anti-de-Sitter (AdS) space-time
which contains a cosmological "singularity" - a space-like region of high
curvature. Making use of the AdS/CFT correspondence we can map the bulk
dynamics onto the boundary. The boundary theory has a time dependent coupling
constant which becomes small at times when the bulk space-time is highly
curved. We investigate the propagation of small fluctuations of a test scalar
field from early times before the bulk singularity to late times after the
singularity. Under the assumption that the AdS/CFT correspondence extends to
deformed AdS space-times, we can map the bulk evolution of the scalar field
onto the evolution of the boundary gauge field. The time evolution of
linearized fluctuations is well defined in the boundary theory as long as the
coupling remains finite, so that we can extend the boundary perturbations to
late times after the singularity. Assuming that the spacetime in the future of
the singularity has a weakly coupled region near the boundary, we reconstruct
the bulk fluctuations after the singularity crossing making use of generic
properties of boundary-to-bulk propagators. Finally, we extract the spectrum of
the fluctuations at late times given some initial spectrum. We find that the
spectral index is unchanged, but the amplitude increases due to the squeezing
of the fluctuations during the course of the evolution. This investigation can
teach us important lessons on how the spectrum of cosmological perturbations
passes through a bounce which is singular from the bulk point of view but which
is resolved using an ultraviolet complete theory of quantum gravity.
