  We study the fate of algebraic decay of correlations in a harmonically
trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent
experiments on ultracold atoms where power-law correlations have been observed
despite the presence of the external potential. We generalize the spin wave
description of phase fluctuations to the trapped case and obtain an analytical
expression for the one-body density matrix within this approximation. We show
that algebraic decay of the central correlation function persists to lengths of
about 20% of the Thomas--Fermi radius. We establish that the trap-averaged
correlation function decays algebraically with a strictly larger exponent
weakly changing with trap size and find indications that the recently observed
enhanced scaling exponents receive significant contributions from the normal
component of the gas. We discuss radial and angular correlations and propose a
local correlation approximation which captures the correlations very well. Our
analysis goes beyond the usual local density approximation and the developed
summation techniques constitute a powerful tool to investigate correlations in
inhomogeneous systems.
