  In the present mainstream cosmology, matter and spacetime emerged from a
singularity and evolved through four distinct periods: early inflation,
radiation, dark matter and late-time inflation (driven by dark energy). During
the radiation and dark matter dominated stages, the universe is decelerating
while the early and late-time inflations are accelerating stages. A possible
connection between the accelerating periods remains unknown, and, even more
intriguing, the best dark energy candidate powering the present accelerating
stage (Lambda-vacuum) is plagued with the cosmological constant and coincidence
puzzles. Here we propose an alternative solution for such problems based on a
large class of time-dependent vacuum energy density models in the form of power
series of the Hubble rate, Lambda=Lambda(H). The proposed class of
Lambda(H)-decaying vacuum model provides: i) a new mechanism for inflation
(different from the usual inflaton models), (ii) a natural mechanism for a
graceful exit, which is universal for the whole class of models; iii) the
currently accelerated expansion of the universe, iv) a mild dynamical dark
energy at present; and v) a final de Sitter stage. Remarkably, the late-time
cosmic expansion history of our class of models is very close to the
concordance LambdaCDM model, but above all it furnishes the necessary smooth
link between the initial and final de Sitter stages through the radiation- and
matter-dominated epochs.
