Cosmological implications of the transition from the false vacuum to the
  true vacuum state
  We study the cosmology with the running dark energy. The parametrization of
dark energy with the respect to the redshift is derived from the first
principles of quantum mechanics. Energy density of dark energy is obtained from
the quantum process of transition from the false vacuum state to the true
vacuum state. This is the class of the extended interacting $\Lambda$CDM
models. We consider the energy density of dark energy parametrization
$\rho_\text{de}(t)$, which follows from the Breit-Wigner energy distribution
function which is used to model the quantum unstable systems. The idea that
properties of the process of the quantum mechanical decay of unstable states
can help to understand the properties of the observed universe was formulated
by Krauss and Dent and this idea was used in our considerations. In the
cosmological model with the mentioned parametrization there is an energy
transfer between the dark matter and dark energy. In such a evolutional
scenario the universe is starting from the false vacuum state and going to the
true vacuum state of the present day universe. We find that the intermediate
regime during the passage from false to true vacuum states takes place. The
intensity of the analyzed process is measured by a parameter $\alpha$. For the
small value of $\alpha$ ($0<\alpha <0.4$) this intermediate (quantum) regime is
characterized by an oscillatory behavior of the density of dark energy while
the for $\alpha > 0.4$ the density of the dark energy simply jumps down. In
both cases (independent from the parameter $\alpha$) the today value of density
of dark energy is reached at the value of $0.7$. We estimate the cosmological
parameters for this model with visible and dark matter. This model becomes in
good agreement with the astronomical data and is practically indistinguishable
from $\Lambda$CDM model.
