Cosmology with decaying cosmological constant -- exact solutions and model testing

We study dynamics of $\Lambda(t)$ cosmological models which are a natural generalization of the standard cosmological model (the $\Lambda$CDM model). We consider a class of models: the ones with a prescribed form of $\Lambda(t)=\Lambda_{\text{bare}}+\frac{\alpha^2}{t^2}$. This type of a $\Lambda(t)$ parametrization is motivated by different cosmological approaches. We interpret the model with running Lambda ($\Lambda(t)$) as a special model of an interacting cosmology with the interaction term $-d\Lambda(t)/dt$ in which energy transfer is between dark matter and dark energy sectors. For the $\Lambda(t)$ cosmology with a prescribed form of $\Lambda(t)$ we have found the exact solution in the form of Bessel functions. Our model shows that fractional density of dark energy $\Omega_e$ is constant and close to zero during the early evolution of the universe. We have also constrained the model parameters for this class of models using the astronomical data such as SNIa data, BAO, CMB, measurements of $H(z)$ and the Alcock-Paczy{\'n}ski test. In this context we formulate a simple criterion of variability of $\Lambda$ with respect to $t$ in terms of variability of the jerk or sign of estimator $(1-\Omega_{\text{m},0}-\Omega_{\Lambda,0})$. The case study of our model enable us to find an upper limit $\alpha^2 < 0.012$ ($2\sigma$ C.L.) describing the variation from the cosmological constant while the LCDM model seems to be consistent with various data.