A nonlocal gravity model, which does not assume the existence of a new dimensional parameter in the action and includes a function fðh À1 RÞ, with h the d'Alembertian operator, is considered. The model is proven to have de Sitter solutions only if the function f satisfies a certain second-order linear differential equation. The de Sitter solutions corresponding to the simplest case, an exponential function f, are explored, without any restrictions on the parameters. If the value of the Hubble parameter is positive, the de Sitter solution is stable at late times, both for negative and for positive values of the cosmological constant. Also, the stability of the solutions with zero cosmological constant is discussed and sufficient conditions for it are established in this case. New de Sitter solutions are obtained, which correspond to the model with dark matter, and stability is proven in this situation for nonzero values of the cosmological constant.

A nonlocal gravity model is considered which does not assume the existence of a new dimensional parameter in the action and includes a function f (✷ −1 R), with ✷ the d'Alembertian operator. Using a reconstruction procedure for the local scalar-tensor formulation of this model, a function f is obtained for which the model exhibits power-law solutions with the Hubble parameter H = n/t, for any value of the constant n. For generic n-namely except for a few special values which are characterized and also specifically studied-the corresponding function f is a sum of exponential functions. Corresponding power-law solutions are found explicitly. Also the case is solved in all detail of a function f such that the model contains both de Sitter and power-law solutions. *

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We obtain general solutions for some flat Friedmann universes filled with a scalar field in induced gravity models and models including the Hilbert-Einstein curvature term plus a scalar field conformally coupled to gravity. As is well known, these models are connected to minimally coupled models through the combination of a conformal transformation and a transformation of the scalar field. The explicit forms of the self-interaction potentials for six exactly solvable models are presented here. We obtain the general solution for one of the integrable models, namely, the induced gravity model with a power-law potential for the self-interaction of the scalar field. We argue that although being mathematically in a one-to-one correspondence with the solutions in the minimally coupled models, the solutions in the corresponding non-minimally coupled models are physically different. This is because the cosmological evolutions seen by an internal observer connected with the cosmic time can be quite different. The study of a few induced gravity models with particular potentials gives us an explicit example of such a difference.

The possibility to construct inflationary models for the renormalization-group improved potentials corresponding to scalar electrodynamics and to SU (2) and SU (5) models is investigated. In all cases, the tree-level potential, which corresponds to the cosmological constant in the Einstein frame, is seen to be non-suitable for inflation. Rather than adding the Hilbert-Einstein term to the action, quantum corrections to the potential, coming from to the RG-equation, are included. The inflationary scenario is analyzed with unstable de Sitter solutions which correspond to positive values of the coupling function, only. We show that, for the finite SU (2) model and SU (2) gauge model, there are no de Sitter solutions suitable for inflation, unless exit from it occurs according to some weird, non-standard scenarios.Inflation is realized both for scalar electrodynamics and for SU (5) RG-improved potentials, and the corresponding values of the coupling function are seen to be positive. It is shown that, for quite reasonable values of the parameters, the inflationary models obtained both from scalar electrodynamics and from the SU (5) RG-improved potentials, are in good agreement with the most recent observational data coming from the Planck2013 and BICEP2 collaborations.

We investigate the scalar field dynamics of models with non-minimally coupled scalar fields in the presence of the Gauss-Bonnet term and derive the structure of effective potential and conditions for stable de Sitter solutions in general. Specializing to specific couplings, we explore the possibility of realizing the stable de Sitter configurations which may have implications for both the early Universe and late time evolution. * 1 Non-local cosmological models with the Gauss-Bonnet term are actively developed as well [22,23,24,25,26].

A nonlocal gravity model which does not assume the existence of a new dimensional parameter in the action and includes a function f ( −1 R), with the d'Alembertian operator, is studied. By specifying an exponential form for the function f and including a matter sector with a constant equation of state parameter, all available power-law solutions in the Jordan frame are obtained. New power-law solutions in the Einstein frame are also probed. Furthermore, the relationship between power-law solutions in both frames, established through conformal transformation, is substantially clarified. The correspondence between power-law solutions in these two frames is proven to be a very useful tool in order to obtain new solutions in the Einstein frame.

We construct explicit Darboux transformations for a generalized, two-dimensional Dirac equation. Our results contain former findings for the one-dimensional, stationary Dirac equation, as well as for the fully time-dependent case in (1+1) dimensions. We show that our Darboux transformations are applicable to the two-dimensional Dirac equation in cylindrical coordinates and give several examples.