Cosmological model with decaying vacuum energy from quantum mechanics

We construct the cosmological model to explain the cosmological constant problem. We built the extension of the standard cosmological model $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ 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 powerlike form. This implies that cosmological constant ${\ensuremath{\rho}}_{\mathrm{vac}}=\mathrm{\ensuremath{\Lambda}}(t)={\mathrm{\ensuremath{\Lambda}}}_{\text{bare}}+\frac{\ensuremath{\alpha}}{{t}^{2}}$ where ${\mathrm{\ensuremath{\Lambda}}}_{\text{bare}}$ and $\ensuremath{\alpha}$ are constants. We investigate the dynamics of this model using dynamical system methods due to a link to the $\mathrm{\ensuremath{\Lambda}}(H)$ 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 Friedman-Robertson-Walker universe. Using astronomical data [SNIa, $H(z)$, CMB, BAO] we have estimated the model parameters and compared this model with the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. Our statistical results indicate that the decaying vacuum model is a little worse than the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model. But the decaying vacuum cosmological model explains the small value of the cosmological constant today.

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