  We analyze density-density correlations of expanding clouds of weakly
interacting two-dimensional Bose gases below and above the
Berezinskii-Kosterlitz-Thouless transition, with particular focus on short-time
expansions. During time-of-flight expansion, phase fluctuations of the trapped
system translate into density fluctuations, in addition to the density
fluctuations that exist in in-situ. We calculate the correlations of these
fluctuations both in real space and in momentum space, and derive analytic
expressions in momentum space. Below the transition, the correlation functions
show an oscillatory behavior, controlled by the scaling exponent of the
quasi-condensed phase, due to constructive interference. We argue that this can
be used to extract the scaling exponent of the quasi-condensate experimentally.
Above the transition, the interference is rapidly suppressed when the atoms
travel an average distance beyond the correlation length. This can be used to
distinguish the two phases qualitatively.
