  We study the evolution of density in an expanding Bose-Einstein condensate
that initially has a spatially varying phase, concentrating on behaviour when
these phase variations are large. In this regime large density fluctuations
develop during expansion. Maxima have a characteristic density that diverges
with the amplitude of phase variations and their formation is analogous to that
of caustics in geometrical optics. We analyse in detail caustic formation in a
quasi-one dimensional condensate, which before expansion is subject to a
periodic or random optical potential, and we discuss the equivalent problem for
a quasi-two dimensional system. We also examine the influence of many-body
correlations in the initial state on caustic formation for a Bose gas expanding
from a strictly one-dimensional trap. In addition, we study a similar
arrangement for non-interacting fermions, showing that Fermi surface
discontinuities in the momentum distribution give rise in that case to sharp
peaks in the spatial derivative of the density. We discuss recent experiments
and argue that fringes reported in time of flight images by Chen and co-workers
[Phys. Rev. A 77, 033632 (2008)] are an example of caustic formation.
