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Recent astronomical observations indicate that our Universe is undergoing a period of an accelerated expansion. While there are many cosmological models, which explain this phenomenon, the main question remains which is the best one in the light of available data. We consider ten cosmological models of the accelerating Universe and select the best one using the Bayesian model comparison method. We demonstrate that the ΛCDM model is most favored by the Bayesian statistical analysis of the SNIa, CMB, BAO and H(z) data.

In the paper we investigate observational constraints on coupling to gravity constant parameter ξ using distant supernovae SNIa data, baryon oscillation peak (BOP), the cosmic microwave background radiation (CMBR) shift parameter, and H(z) data set. We estimate the value of this parameter to constrain the extended quintessence models with nonminimally coupled to gravity phantom scalar field. The combined analysis of observational data favors a value of ξ which lies in close neighborhood of the conformal coupling. While our estimations are model dependent they give rise to a indirect bound on the Equivalence Principle.PACS numbers: 98.80. Es, 98.80.Cq, 95.36.+x At the present, scalar fields play the crucial role in modern cosmology. In the inflationary scenario they generate an exponential rate of evolution of the universe as well as a density fluctuations due to vacuum energy.Observations of distant supernovae support the cosmological constant term. But two problems emerge in this context. Namely, the fine tuning and cosmic coincidence problems. The lack of some fundamental mechanism which sets the cosmological constant almost zero is called the cosmological constant problem. The second problem called "cosmic conundrum" is the question why the energy densities of both dark energy and dark matter are nearly equal at the present epoch. One of the solutions to this problem offers the idea of quintessence [1,2] which is a version of the time varying cosmological constant conception.All these models base on the assumption that there is a minimal coupling of scalar field to gravity (ξ = 0). This a priori assumption requires some justification. There are many theoretical arguments suggesting that the non-minimal coupling (NC for short) should be considered. The nonzero ξ comes from quantum corrections [3], renormalization of classical theory that shifts it to one with nonzero ξ [4], in relativity the value of ξ = 1/6 (conformal coupling) is distinguished. Only in this case the Einstein equivalence principle is not violated (for details see [5] The main goal of this paper is a statistical estimation of the coupling parameter from the astronomical observations. For this aim we consider the spatially flat FRW model where the source of gravity is a noninteracting mixture of dust matter and non-minimally coupled phantom scalar field.We treat ξ as a free parameter in the model and we are looking for constraints on it's value from observational cosmology. For simplicity it is assumed a simple and natural form of the potential function of the scalar field. For the vanishing coupling constant (minimal coupling) this potential corresponds to chaotic inflation [9]. This paradigm of inflation can be extended by including NC [6]. Also the scalar field non-minimally coupled to gravity are simple, non-exotic models of phantoms give rise to superacceleration [10]. Tsujikawa and Gumjudpai [11] also investigated constraints on the NC parameter from the CMB observations. Recently Jankiewicz and Kephart [12] applied the method of WKB a...

We study new FRW type cosmological models of modified gravity treated on the background of Palatini approach. These models are generalization of Einstein gravity by the presence of a scalar field non-minimally coupled to the curvature. The models employ Starobinsky's term in the Lagrangian and dust matter. Therefore, as a by-product, an exhausted cosmological analysis of general relativity amended by quadratic term is presented. We investigate dynamics of our models, confront them with the currently available astrophysical data as well as against ΛCDM model. We have used the dynamical system methods in order to investigate dynamics of the models. It reveals the presence of a final sudden singularity. Fitting free parameters we have demonstrated by statistical analysis that this class of models is in a very good agreement with the data (including CMB measurements) as well as with the standard ΛCDM model predictions. One has to use statefinder diagnostic in order to discriminate among them. Therefore Bayesian methods of model selection have been employed in order to indicate preferred model. Only in the light of CMB data the concordance model remains invincible.

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