Scalar-tensor theories of gravitation attract again a great interest since the discovery of the Chameleon mechanism and of the Galileon models. The former allows reconciling the presence of a scalar field with the constraints from Solar System experiments. The latter leads to inflationary models that do not need ad hoc potentials. Further generalizations lead to a tensor-scalar theory, dubbed the “Fab Four,” with only first and second order derivatives of the fields in the equations of motion that self-tune to a vanishing cosmological constant. This model needs to be confronted with experimental data in order to constrain its large parameter space. We present some results regarding a subset of this theory named “John,” which corresponds to a nonminimal derivative coupling between the scalar field and the Einstein tensor in the action. We show that this coupling gives rise to an inflationary model with very unnatural initial conditions. Thus, we include the term named “George,” namely, a nonminimal, but nonderivative, coupling between the scalar field and Ricci scalar. We find a more natural inflationary model, and, by performing a post-Newtonian analysis, we derive the set of equations that constrain the parameter space with data from experiments in the Solar System.

We propose a stellar model under the [Formula: see text] gravity following Mazur–Mottola’s conjecture[Formula: see text] known as gravastar which is generally believed as a viable alternative to black hole. The gravastar consists of three regions, viz., (I) interior region, (II) intermediate shell region, and (III) exterior region. The pressure within the interior core region is assumed to be equal to the constant negative matter-energy density which provides a constant repulsive force over the thin shell region. The shell is assumed to be made up of fluid of ultrarelativistic plasma and following the Zel’dovich’s conjecture of stiff fluid3 it is also assumed that the pressure which is directly proportional to the matter-energy density according to Zel’dovich’s conjecture, does cancel the repulsive force exerted by the interior region. The exterior region is completely vacuum and it can be described by the Schwarzschild solution. Under all these specifications, we find out a set of exact and singularity-free solutions of the gravastar presenting several physically valid features within the framework of alternative gravity, namely [Formula: see text] gravity,4 where the part of the gravitational Lagrangian in the corresponding action is taken as an arbitrary function of torsion scalar [Formula: see text] and the trace of the energy–momentum tensor [Formula: see text].

The aim of this paper is to study some examples of exponentially harmonic maps. We study such maps firstly on flat euclidean and Minkowski spaces and secondly on Friedmann-Lema ître universes. We also consider some new models of exponentially harmonic maps which are coupled with gravity which happen to be based on a generalization of the lagrangian for bosonic strings coupled with dilatonic field.

We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic maps. We showed that some F-harmonic map such as the trigonometric functions instead of the scalar field in the lagrangian, allow, in the absence of term of potential, reproduce the inflation. However, there are other F-harmonic maps such as exponential maps which can’t produce the inflation; the pressure and the density of this exponential harmonic field being both of the same sign. On the other hand, these exponential harmonic fields redraw well the phenomenon of the quintessence when the variation of these fields remains weak. The problem of coincidence, however remains