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

Figure 1

The phase portrait of system (33)–(34) with the positive cosmological constant $\mathrm{\Lambda }$ and $\mathrm{\Lambda }(H)=\alpha H^2$. The critical point at the origin represents the de Sitter solution.

Figure 2

The phase portrait of the system in the projective coordinates $(z,u)$ dedicated for the analysis of the behavior of trajectories at the infinity $H^2\to \infty$. There are two critical points: a saddle point representing the Milne universe $H^2\propto a^{-2}$ and an unstable node representing the Einstein-de-Sitter-like universe. It is assumed the parameter $ϵ$ is rather small. The shadowed region is representing the nonphysical domain forbidden for trajectories where $H^2<0$.

Figure 3

The phase portrait of the system in the projective coordinates $(v,w)$ dedicated for the analysis of the behavior of trajectories at the infinity $H^2\to \infty$. There are two critical points: a saddle point representing the Milne universe $H^2\propto a^{-2}$ and an unstable node representing the Einstein-de-Sitter-like universe. It is assumed the parameter $ϵ$ is rather small. The shadowed region is representing the nonphysical domain forbidden for trajectories where $H^2<0$.

Figure 4

Posterior constraints for testing model. Solid lines denote 68% and 95% credible intervals of fully marginalized probabilities, the colors illustrate mean likelihood of the sample. Top left: estimations with the Union2.1 data only. Top right: estimations made using the Union2.1, $h(z)$, BAO, determinations of the Hubble function using the Alcock-Paczynski test data sets. Bottom: estimations made using the Union2.1, $h(z)$, BAO, determinations of Hubble function using the Alcock-Paczynski test and CMB data sets.

Figure 5

Posterior constraints for $\mathrm{\Lambda }\mathrm{CDM}$ model. Solid lines denote 68% and 95% credible intervals of fully marginalized probabilities, the colors illustrate mean likelihood of the sample. Top left: estimations with the Union2.1 data only. Top right: estimations made using the Union2.1, $h(z)$, BAO, determinations of the Hubble function using the Alcock-Paczyński test data sets. Bottom: estimations made using the Union2.1, $h(z)$, BAO, and determinations of Hubble function using the Alcock-Paczynski test and CMB data sets.