  We investigate theoretically the evolution of the two-point density
correlation function of a low-dimensional ultracold Bose gas after release from
a tight transverse confinement. In the course of expansion thermal and quantum
fluctuations present in the trapped systems transform into density
fluctuations. For the case of free ballistic expansion relevant to current
experiments, we present simple analytical relations between the spectrum of
``density ripples'' and the correlation functions of the original confined
systems. We analyze several physical regimes, including weakly and strongly
interacting one-dimensional (1D) Bose gases and two-dimensional (2D) Bose gases
below the Berezinskii-Kosterlitz-Thouless (BKT) transition. For weakly
interacting 1D Bose gases, we obtain an explicit analytical expression for the
spectrum of density ripples which can be used for thermometry. For 2D Bose
gases below the BKT transition, we show that for sufficiently long expansion
times the spectrum of the density ripples has a self-similar shape controlled
only by the exponent of the first-order correlation function. This exponent can
be extracted by analyzing the evolution of the spectrum of density ripples as a
function of the expansion time.
