From de Sitter to de Sitter: decaying vacuum models as a possible solution to the main cosmological problems

Decaying vacuum cosmological models evolving smoothly between two extreme (very early and late time) de Sitter phases are capable to solve or at least to alleviate some cosmological puzzles, among them: (i) the singularity, (ii) horizon, (iii) graceful-exit from inflation, and (iv) the baryogenesis problem. Our basic aim here is to discuss how the coincidence problem based on a large class of running vacuum cosmologies evolving from de Sitter to de Sitter can also be mollified. It is also argued that even the cosmological constant problem become less severe provided that the characteristic scales of the two limiting de Sitter manifolds are predicted from first principles.