The vacuum energy density (Casimir energy) corresponding to a massless scalar quantum field living in different universes (mainly no-boundary ones), in several dimensions, is calculated. Hawking's zeta function regularization procedure supplemented with binomial expansion is shown to be a rigorous and well suited method for performing the analysis. It is compared with other more involved techniques. The principal-part prescription is used to deal with the poles that eventually appear. Results of the analysis are the absence of poles at four dimensions (4D), the total coincidence of the results corresponding to a 3D and a 4D cylinder, and the fact that the vacuum energy density for cylinders is over an order of magnitude smaller than for spheres of the same dimension.