Abstract: We present a theory of the density correlations that appear in an atomic Bose-Einstein condensate as a consequence of the emission of correlated pairs of Bogoliubov phonons by a time-dependent atom-atom scattering length. This effect can be considered as a condensed matter analog of the dynamical Casimir effect of quantum field theory. Different regimes as a function of the temporal shape of the modulation are identified and a simple physical picture of the phenomenon is discussed. Analytical expressions for the density correlation function are provided for the most significant limiting cases. This theory is able to explain some unexpected features recently observed in numerical studies of analog Hawking radiation from acoustic black holes.