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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 $Λ$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 $ρ_{\text{vac}} = Λ(t) = Λ_{\text{bare}} + \fracα{t^2}$ where $Λ_{\text{bare}}$ and $α$ are constants. We investigate the dynamics of this model using dynamical system methods due to a link to the $Λ(H)$ 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 $Λ$CDM model. Our statistical results indicate that the decaying vacuum model is a little worse than the $Λ$CDM model. But the decaying vacuum cosmological model explains the small value of the cosmological constant today.