Linear Newtonian perturbation theory from the Schr\"odinger-Poisson equations

We obtain solutions to the coupled Schr\"odinger-Poisson equations. The solutions describe the evolution of cold dark matter density perturbations in an otherwise homogeneous expanding Friedmann universe. We discuss the relationships between descriptions of cold dark matter in terms of a pressureless fluid, in terms of a wavefunction, of a classical scalar field, and a quantum scalar field. We identify the regimes where the various descriptions coincide and where they differ.