Cosmological models with running cosmological term and decaying dark matter

We investigate the dynamics of the generalized $\Lambda$CDM model, which the $\Lambda$ term is running with the cosmological time. On the example of the model $\Lambda \left(t\right)={\Lambda }_{\text{bare}}+\frac{{\alpha }^2}{t^2}$ we show the existence of a mechanism of the modification of the scaling law for energy density of dark matter: ${\rho }_{\text{dm}}\propto a^{-3+\lambda \left(t\right)}$. We use an approach developed by Urbanowski in which properties of unstable vacuum states are analyzed from the point of view of the quantum theory of unstable states. We discuss the evolution of $\Lambda \left(t\right)$ term and pointed out that during the cosmic evolution there is a long phase in which this term is approximately constant. We also present the statistical analysis of both the $\Lambda \left(t\right)$CDM model with dark energy and decaying dark matter and the $\Lambda$CDM standard cosmological model. We use data such as Planck, SNIa, BAO, $H\left(z\right)$ and AP test. While for the former we find the best fit value of the parameter ${\Omega }_{{\alpha }^2,0}$ is negative (energy transfer is from the dark matter to dark energy sector) and the parameter ${\Omega }_{{\alpha }^2,0}$ belongs to the interval $(-0.000040,-0.000383)$ at 2-$\sigma$ level. The decaying dark matter causes to lowering a mass of dark matter particles which are lighter than CDM particles and remain relativistic. The rate of the process of decaying matter is estimated. Our model is consistent with the decaying mechanism producing unstable particles (e.g. sterile neutrinos) for which ${\alpha }^2$ is negative.