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We classify singularities in FRW cosmologies, which dynamics can be reduced to the dynamical system of the Newtonian type. This classification is performed in terms of the geometry of a potential function if it has poles. At the sewn singularity, which is of a finite scale factor type, the singularity in the past meets the singularity in the future. We show that such singularities appear in the Starobinsky model in f ( R ^ ) = R ^ + γ R ^ 2 in the Palatini formalism, when dynamics is determined by the corresponding piecewise-smooth dynamical system. As an effect we obtain a degenerate singularity. Analytical calculations are given for the cosmological model with matter and the cosmological constant. The dynamics of model is also studied using dynamical system methods. From the phase portraits we find generic evolutionary scenarios of the evolution of the universe. For this model, the best fit value of Ω γ = 3 γ H 0 2 is equal 9.70 × 10 - 11 . We consider a model in both Jordan and Einstein frames. We show that after transition to the Einstein frame we obtain both the form of the potential of the scalar field and the decaying Lambda term.

We consider FRW cosmology in f ( R ) = R + γ R 2 + δ R 3 modified framework. The Palatini approach reduces its dynamics to the simple generalization of Friedmann equation. Thus we study the dynamics in two-dimensional phase space with some details. After reformulation of the model in the Einstein frame, it reduces to the FRW cosmological model with a homogeneous scalar field and vanishing kinetic energy term. This potential determines the running cosmological constant term as a function of the Ricci scalar. As a result we obtain the emergent dark energy parametrization from the covariant theory. We study also singularities of the model and demonstrate that in the Einstein frame some undesirable singularities disappear.

We study cosmology with running dark energy. The energy density of dark energy is obtained from the quantum process of transition from the false vacuum state to the true vacuum state. We use the Breit–Wigner energy distribution function to model the quantum unstable systems and obtain the energy density of the dark energy parametrization ρ de ( t ) . We also use Krauss and Dent’s idea linking properties of the quantum mechanical decay of unstable states with the properties of the observed Universe. In the cosmological model with this parametrization there is an energy transfer between dark matter and dark energy. The intensity of this process, measured by a parameter α , distinguishes two scenarios. As the Universe starts from the false vacuum state, for the small value of α ( 0 < α < 0.4 ) it goes through an intermediate oscillatory (quantum) regime of the density of dark energy, while for α > 0.4 the density of the dark energy jumps down. In both cases the present value of the density of dark energy is reached. From a statistical analysis we find this model to be in good agreement with the astronomical data and practically indistinguishable from the Λ CDM model.

We derive the Shafieloo, Hazra, Sahni and Starobinsky (SHSS) phenomenological formula for the radioactive-like decay of metastable dark energy directly from the principles of quantum mechanics. To this aim we use the Fock–Krylov theory of quantum unstable states. We obtain deeper insight on the decay process as having three basic phases: the phase of radioactive decay, the next phase of damping oscillations, and finally the phase of power-law decay. We consider the cosmological model with matter and dark energy in the form of decaying metastable dark energy and study its dynamics in the framework of non-conservative cosmology with an interacting term determined by the running cosmological parameter. We study the cosmological implications of metastable dark energy and estimate the characteristic time of ending of the radioactive-like decay epoch to be 2.2 × 10 4 of the present age of the Universe. We also confront the model with astronomical data which show that the model is in good agreement with the observations. Our general conclusion is that we are living in the epoch of the radioactive-like decay of metastable dark energy which is a relict of the quantum age of the Universe.

In this paper, we apply the method of reducing the dynamics of FRW cosmological models with a barotropic form of the equation of state to the dynamical system of the Newtonian type to detect the finite scale factor singularities and the finite-time singularities. In this approach all information concerning the dynamics of the system is contained in a diagram of the potential function V(a) of the scale factor. Singularities of the finite scale factor make themselves manifest by poles of the potential function. In our approach the different types of singularities are represented by critical exponents in the power-law approximation of the potential. The classification can be given in terms of these exponents. We have found that the pole singularity can mimic an inflation epoch. We demonstrate that the cosmological singularities can be investigated in terms of the critical exponents of the potential function of the cosmological dynamical systems. We assume that the general form of the model contains matter and some kind of dark energy which is parameterised by the potential. We distinguish singularities (by an ansatz involving the Lagrangian) of the pole type with the inflation and demonstrate that such a singularity can appear in the past.

In the framework of polynomial Palatini cosmology, we investigate a simple cosmological homogeneous and isotropic model with matter in the Einstein frame. We show that in this model during cosmic evolution, early inflation appears and the accelerating phase of the expansion for the late times. In this frame we obtain the Friedmann equation with matter and dark energy in the form of a scalar field with a potential whose form is determined in a covariant way by the Ricci scalar of the FRW metric. The energy density of matter and dark energy are also parameterized through the Ricci scalar. Early inflation is obtained only for an infinitesimally small fraction of energy density of matter. Between the matter and dark energy, there exists an interaction because the dark energy is decaying. For the characterization of inflation we calculate the slow roll parameters and the constant roll parameter in terms of the Ricci scalar. We have found a characteristic behavior of the time dependence of density of dark energy on the cosmic time following the logistic-like curve which interpolates two almost constant value phases. From the required numbers of N-folds we have found a bound on the model parameter.

We consider a cosmology with decaying metastable dark energy and assume that a decay process of this metastable dark energy is a quantum decay process. Such an assumption implies among others that the evolution of the Universe is irreversible and violates the time reversal symmetry. We show that if we replace the cosmological time t appearing in the equation describing the evolution of the Universe by the Hubble cosmological scale time, then we obtain time dependent Λ(t) in the form of the series of even powers of the Hubble parameter H: Λ(t)=Λ(H). Our special attention is focused on radioactive-like exponential form of the decay process of the dark energy and on the consequences of this type decay.

We study the dynamics of cosmological models with a time dependent cosmological term. We consider five classes of models; two with the non-covariant parametrization of the cosmological term Λ : Λ ( H ) CDM cosmologies, Λ ( a ) CDM cosmologies, and three with the covariant parametrization of Λ : Λ ( R ) CDM cosmologies, where R ( t ) is the Ricci scalar, Λ ( ϕ ) -cosmologies with diffusion, Λ ( X ) -cosmologies, where X = 1 2 g α β ∇ α ∇ β ϕ is a kinetic part of the density of the scalar field. We also consider the case of an emergent Λ ( a ) relation obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the study of the dynamics we used dynamical system methods for investigating how an evolutionary scenario can depend on the choice of special initial conditions. We show that the methods of dynamical systems allow one to investigate all admissible solutions of a running Λ cosmology for all initial conditions. We interpret Alcaniz and Lima’s approach as a scaling cosmology. We formulate the idea of an emergent cosmological term derived directly from an approximation of the exact dynamics. We show that some non-covariant parametrization of the cosmological term like Λ ( a ) , Λ ( H ) gives rise to the non-physical behaviour of trajectories in the phase space. This behaviour disappears if the term Λ ( a ) is emergent from the covariant parametrization.

We study the dynamics of cosmological models with a time dependent cosmological term. We consider five classes of models; two with the non-covariant parametrization of the cosmological term [Formula omitted]: [Formula omitted]CDM cosmologies, [Formula omitted]CDM cosmologies, and three with the covariant parametrization of [Formula omitted]: [Formula omitted]CDM cosmologies, where R(t) is the Ricci scalar, [Formula omitted]-cosmologies with diffusion, [Formula omitted]-cosmologies, where [Formula omitted] is a kinetic part of the density of the scalar field. We also consider the case of an emergent [Formula omitted] relation obtained from the behaviour of trajectories in a neighbourhood of an invariant submanifold. In the study of the dynamics we used dynamical system methods for investigating how an evolutionary scenario can depend on the choice of special initial conditions. We show that the methods of dynamical systems allow one to investigate all admissible solutions of a running [Formula omitted] cosmology for all initial conditions. We interpret Alcaniz and Lima's approach as a scaling cosmology. We formulate the idea of an emergent cosmological term derived directly from an approximation of the exact dynamics. We show that some non-covariant parametrization of the cosmological term like [Formula omitted], [Formula omitted] gives rise to the non-physical behaviour of trajectories in the phase space. This behaviour disappears if the term [Formula omitted] is emergent from the covariant parametrization.