  We study, within a framework of the classical fields approximation, the
density correlations of a weakly interacting expanding Bose gas for the whole
range of temperatures across the Bose-Einstein condensation threshold. We focus
on elongated quasi-one-dimensional systems where there is a huge discrepancy
between the existing theory and experimental results (A. Perrin et al., Nature
Phys. 8, 195 (2012)). We find that the density correlation function is not
reduced for temperatures below the critical one as it is predicted for the
ideal gas or for a weakly interacting system within the Bogoliubov
approximation. This behavior of the density correlations agrees with the above
mentioned experiment with the elongated system. Although the system was much
larger then studied here we believe that the behavior of the density
correlation function found there is quite generic. Our theoretical studies
indicate also large density fluctuations in the trap in the quasicondensate
regime where only phase fluctuations were expected. We argue that the enhanced
density fluctuations can originate in the presence of interactions in the
system, or more precisely in the existence of spontaneous dark solitons in the
elongated gas at thermal equilibrium.
