Cosmology with a Decaying Vacuum Energy Parametrization Derived from
  Quantum Mechanics
  Within the quantum mechanical treatment of the decay problem one finds that
at late times $t$ the survival probability of an unstable state cannot have the
form of an exponentially decreasing function of time $t$ but it has an inverse
power-like form. This is a general property of unstable states following from
basic principles of quantum theory. The consequence of this property is that in
the case of false vacuum states the cosmological constant becomes dependent on
time: $\Lambda - \Lambda_{\text{bare}}\equiv \Lambda(t) -\Lambda_{\text{bare}}
\sim 1/t^{2}$. We construct the cosmological model with decaying vacuum energy
density and matter for solving the cosmological constant problem and the
coincidence problem. We show the equivalence of the proposed decaying false
vacuum cosmology with the $\Lambda(t)$ cosmologies (the $\Lambda(t)$CDM
models). The cosmological implications of the model of decaying vacuum energy
(dark energy) are discussed. We constrain the parameters of the model with
decaying vacuum using astronomical data. For this aim we use the observation of
distant supernovae of type Ia, measurements of $H(z)$, BAO, CMB and others. The
model analyzed is in good agreement with observation data and explain a small
value of the cosmological constant today.
