Based on a model of the early Universe with a relativistic Bose-Einstein condensate as the source of inflationary expansion, we calculate the gauge-invariant measure of the cosmological density perturbations caused by the zero-point fluctuations and excitations of the condensate. An approximate conservation law in the gauge-invariant perturbation formulation is used to estimate the magnitude of the perturbations at horizon reentry. The perturbations have a scale-invariant spectrum that depends on the condensate density, the mass, and the coupling constant.