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. The $\Lambda(t)$ term
emerges from the covariant theory of the scalar field $\phi$ with the
self-interacting potential $V(\phi)$. On the example of the model
$\Lambda(t)=\Lambda_{\text{bare}}+\frac{\alpha^2}{t^2}$ we show the existence
of a mechanism of the modification of the scaling law for energy density of
dark matter: $\rho_{\text{dm}}\propto a^{-3+\lambda(t)}$. We discuss the
evolution of $\Lambda(t)$ term and pointed out that during the cosmic evolution
there is a long phase in which this term is approximately constant. This effect
justifies Alcaniz and Lima's approach to $\Lambda(H)$ cosmologies. We also
present the statistical analysis of both the $\Lambda(t)$CDM model with dark
energy and decaying dark matter and the $\Lambda$CDM standard cosmological
model. We divide the observational data into two groups: low $z$ data (SNIa,
BAO, $H(z)$ and AP test) and high $z$ data (Planck, WP and lensing). While for
the former we find the best fit value of the parameter $\lambda$ is positive
($\lambda=0.0338$, energy transfer is from the dark energy to dark matter
sector), for the latter we find that $\lambda$ is $-0.0199$ which is an
evidence that the energy transfer is from decaying dark matter. This
disagreement correlates with estimated values of $H_0$ ($67.77$ km/(s Mpc) and
$65.62$ km/(s Mpc) respectively). 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. We show
that in the models of decaying dark matter, the cosmological constant problem
disappears naturally. The model with decaying dark matter possesses one
parameter more but in light of the AIC it is better than the $\Lambda$CDM
standard cosmological model.
