Cosmological model with decaying vacuum energy law from principles of
  quantum mechanics
  We construct the cosmological model to explain the cosmological constant
problem. We built the extension of the standard cosmological model $\Lambda$CDM
by consideration of decaying vacuum energy represented by the running
cosmological term. From the principles of quantum mechanics one can find that
in the long term behavior survival probability of unstable states is a
decreasing function of the cosmological time and has the inverse power-like
form. This implies that cosmological constant $\rho_{\text{vac}} = \Lambda(t) =
\Lambda_{\text{bare}} + \frac{\alpha}{t^2}$ where $\Lambda_{\text{bare}}$ and
$\alpha$ are constants. We investigate the dynamics of this model using
dynamical system methods due to a link to the $\Lambda(H)$ cosmologies. We have
found the exact solution for the scale factor as well as the indicators of its
variability like the deceleration parameter and the jerk. From the calculation
of the jerk we obtain a simple test of the decaying vacuum in the FRW universe.
Using astronomical data (SNIa, $H(z)$, CMB, BAO) we have estimated the model
parameters and compared this model with the $\Lambda$CDM model. Our statistical
results indicate that the decaying vacuum model is a little worse than the
$\Lambda$CDM model. But the decaying vacuum cosmological model explains the
small value of the cosmological constant today.
