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.Comment: 24 pages, 15 figures; We pointed out that most stringent limit on parameter \alpha^2 can be obtained if we apply Starobinsky argument and use constraint of Ade et al. (arXiv:1502.01590). Let us note that while the corresponding limit on the parameter \alpha^2 parameter is about twice less than the limit obtained from our estimation, but it is obtained independently of Starobinsky's argumen

Polynomial f(R)f(R) Palatini cosmology -- dynamical system approach

We investigate cosmological dynamics based on $f(R)$ gravity in the Palatini formulation. In this study we use the dynamical system methods. We show that the evolution of the Friedmann equation reduces to the form of the piece-wise smooth dynamical system. This system is is reduced to a 2D dynamical system of the Newtonian type. We demonstrate how the trajectories can be sewn to guarantee $C^0$ extendibility of the metric similarly as `Milne-like' FLRW spacetimes are $C^0$-extendible. We point out that importance of dynamical system of Newtonian type with non-smooth right-hand sides in the context of Palatini cosmology. In this framework we can investigate singularities which appear in the past and future of the cosmic evolution. We consider cosmological systems in both Einstein and Jordan frames. We show that at each frame the topological structures of phase space are different.Comment: RevTeX 4-1, 30 pages, 19 figure

Simple cosmological model with inflation and late times acceleration

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, it appears the early inflation 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 the 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 parametrized through the Ricci scalar. The early inflation is obtained only for an infinitesimally small fraction of energy density of matter. Between the matter and dark energy, there exists interaction because the dark energy is decaying. For 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 behaviour 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 model parameter.Comment: 8 pages, 10 figure

Does the diffusion DM-DE interaction model solve cosmological puzzles?

We study dynamics of cosmological models with diffusion effects modeling dark matter and dark energy interactions. We show the simple model with diffusion between the cosmological constant sector and dark matter, where the canonical scaling law of dark matter $(\rho_{dm,0}a^{-3}(t))$ is modified by an additive $\epsilon(t)=\gamma t a^{-3}(t)$ to the form $\rho_{dm}=\rho_{dm,0}a^{-3}(t)+\epsilon(t)$. We reduced this model to the autonomous dynamical system and investigate it using dynamical system methods. This system possesses a two-dimensional invariant submanifold on which the DM-DE interaction can be analyzed on the phase plane. The state variables are density parameter for matter (dark and visible) and parameter $\delta$ characterizing the rate of growth of energy transfer between the dark sectors. A corresponding dynamical system belongs to a general class of jungle type of cosmologies represented by coupled cosmological models in a Lotka-Volterra framework. We demonstrate that the de Sitter solution is a global attractor for all trajectories in the phase space and there are two repellers: the Einstein-de Sitter universe and the de Sitter universe state dominating by the diffusion effects. We distinguish in the phase space trajectories, which become in good agreement with the data. They should intersect a rectangle with sides of $\Omega_{m,0}\in [0.2724, 0.3624]$, $\delta \in [0.0000, 0.0364]$ at the 95\% CL. Our model could solve some of the puzzles of the $\Lambda$CDM model, such as the coincidence and fine-tuning problems. In the context of the coincidence problem, our model can explain the present ratio of $\rho_{m}$ to $\rho_{de}$, which is equal $0.4576^{+0.1109}_{-0.0831}$ at a 2$\sigma$ confidence level.Comment: 27 pages, 17 figure

Is a pole type singularity an alternative to inflation?

In this paper, we apply a method of reducing the dynamics of FRW cosmological models with the 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 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 mimick 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 the general form of the model contains matter and some kind of dark energy which is parameterized by the potential. We distinguish singularities (by ansatz about the Lagrangian) of the pole type with the inflation and demonstrate that such a singularity can appear in the past.Comment: RevTeX 4.1; 33 pages, 9 figures; ver. 2: new title, new part on the pole type singularities with inflatio

Cosmology with a Decaying Vacuum Energy Parametrization Derived from Quantum Mechanics

Within the quantum mechanical treatment of the decay problem one finds that at late times $t$ the survival probability of an unstable state cannot have the form of an exponentially decreasing function of time $t$ but it has an inverse power-like form. This is a general property of unstable states following from basic principles of quantum theory. The consequence of this property is that in the case of false vacuum states the cosmological constant becomes dependent on time: $\Lambda - \Lambda_{\text{bare}}\equiv \Lambda(t) -\Lambda_{\text{bare}} \sim 1/t^{2}$. We construct the cosmological model with decaying vacuum energy density and matter for solving the cosmological constant problem and the coincidence problem. We show the equivalence of the proposed decaying false vacuum cosmology with the $\Lambda(t)$ cosmologies (the $\Lambda(t)$CDM models). The cosmological implications of the model of decaying vacuum energy (dark energy) are discussed. We constrain the parameters of the model with decaying vacuum using astronomical data. For this aim we use the observation of distant supernovae of type Ia, measurements of $H(z)$, BAO, CMB and others. The model analyzed is in good agreement with observation data and explain a small value of the cosmological constant today.Comment: 9 pages, 4 figures. Talk given at Seventh International Workshop DICE 2014: Spacetime -- Matter -- Quantum Mechanics ... news on missing links, Castiglioncello (Tuscany, Italy), September 15--19, 201

Do sewn singularities falsify the Palatini cosmology?

We investigate further (cf. arXiv:1512.01199, JCAP01 (2016) 040) Starobinsky cosmological model $R+\gamma R^2$ in the Palatini formalism with Chaplygin gas and baryonic matter as a source. For this aim we use dynamical system theory. The dynamics is reduced to the 2D sewn dynamical system of a Newtonian type (a piecewise-smooth dynamical system). We classify all evolutional paths in the model as well as trajectories in the phase space. We demonstrate the presence of a degenerate freeze singularity (glued freeze type singularities) for the positive $\gamma$. In this case it is a generic feature of early evolution of the universe. We point out that a degenerate type III of singularity can be considered as an endogenous model of inflation between the matter dominating epoch and the dark energy phase. We also investigate cosmological models with negative $\gamma$. It is demonstrated that $\gamma$ equal zero is a bifurcation parameter and dynamics qualitatively changes in comparison to positive $\gamma$. Instead of the big bang the sudden bounce singularity of a finite scale factor appears and there is a generic class of bouncing solutions sewn along the line $a=a_{\text{sing}}$. And we argue that the presence of sudden singularities in an evolutional scenario of the Universe falsifies the negative $\gamma$ in the Palatini cosmology. Only very small values of $\Omega_{\gamma}$ parameter are admissible if we requires that agreements physics with the $\Lambda$CDM model. From the statistical analysis of astronomical observations, we deduce that the case of negative values of $\Omega_\gamma$ can be rejected even if it may fit better to the data.Comment: 26 pages, 15 figures, v3: change of title, more discussion on singularitie

Inflationary cosmology with Chaplygin gas in Palatini formalism

We present a simple generalisation of the $\Lambda$CDM model which on the one hand reaches very good agreement with the present day experimental data and provides an internal inflationary mechanism on the other hand. It is based on Palatini modified gravity with quadratic Starobinsky term and generalized Chaplygin gas as a matter source providing, besides a current accelerated expansion, the epoch of endogenous inflation driven by type III freeze singularity. It follows from our statistical analysis that astronomical data favors negative value of the parameter coupling quadratic term into Einstein-Hilbert Lagrangian and as a consequence the bounce instead of initial Big-Bang singularity is preferred.Comment: 19 pages, 6 figures, typos corrected; version accepted by JCA

Dynamics of the diffusive DM-DE interaction--dynamical system approach

We discuss dynamics of a model of an energy transfer between dark energy (DE) and dark matter (DM). The energy transfer is determined by a non-conservation law resulting from a diffusion of dark matter in an environment of dark energy. The relativistic invariance defines the diffusion in a unique way. The system can contain baryonic matter and radiation which do not interact with the dark sector. We treat the Friedman equation and the conservation laws as a closed dynamical system. The dynamics of the model is examined using the dynamical systems methods for demonstration how solutions depend on initial conditions. We also fit the model parameters using astronomical observation: SNIa, $H(z)$, BAO and Alcock-Paczynski test. We show that the model with diffuse DM-DE is consistent with the data.Comment: 25 pages, 11 figures; v.2: corrected formulas, concept of early energy falsified; accepted in JCA