We study flat Friedmann–Robertson–Walker models with a perfect fluid matter source and a scalar field nonminimally coupled to matter having a double exponential potential. It is shown that the scalar field almost always diverges to infinity. Under conditions on the parameter space, we show that the model is able to give an acceptable cosmological history of our Universe, that is, a transient matter era followed by an accelerating future attractor. It is found that only a very weak coupling can lead to viable cosmology. We study in the Einstein frame the cosmological viability of the asymptotic form of a class of $f(R)$ theories predicting acceleration. The role of the coupling constant is briefly discussed.

We propose a new formulation of f(R) gravity, dubbed scalarized f(R) gravity, in which the Legendre transform is included as a dynamical term. This leads to a theory with second-order field equations that describes general relativity with a self-interacting scalar field, without requiring the introduction of conformal frames. We demonstrate that the quadratic version of scalarized f(R) gravity reduces to general relativity with a massive scalar field, and we explore its implications for Friedmann cosmology. Our findings suggest that scalarized f(R) gravity may lead to simplified descriptions of cosmological applications, while the proposed formulation could offer a new perspective on the relationship between f(R) gravity and scalar–tensor theories.

We discuss qualitative features of the conformal relation between certain classes of gravity theories and general relativity, common to different themes such as $f\left(R\right)$, Brans-Dicke-type, and string theories. We focus primarily on the frame relations of the fields involved, slice energy, traceless and Palatini extensions, and selected cosmological applications.

We study the problem of evolution of bulk 5-fluids having an embedded braneworld with a flat, de Sitter, or anti-de Sitter geometry. We introduce new variables to express the Einstein equations as a dynamical system that depends on the equation of state parameter $$\gamma $$γ and exponent $$\lambda $$λ . For linear fluids (i.e., $$\lambda =1$$λ = 1 ), our formulation leads to a partial decoupling of the equations and thus to an exact solution. We find that such a fluid develops a transcritical bifurcation around the value $$\gamma =-1/2$$γ = - 1 / 2 , and study how this behaviour affects to stability of the solutions. For nonlinear fluids, the situation is more diverse. We find an overall attractor at $$\lambda =1/2$$λ = 1 / 2 and draw enough phase portraits to exhibit in detail the overall dynamics. We show that the value $$\lambda =3/2$$λ = 3 / 2 is structurally unstable and typical for other forms of $$\lambda $$λ . Consequently, we observe a noticeable dependence of the qualitative behaviour of the solutions on different ‘polytropic’ forms of the fluid bulk. In addition, we prove the existence of a Dulac function for nonlinear fluids, signifying the impossibility of closed orbits in certain subsets of the phase space. We also provide ample numerical evidence of gravity localizing solutions on the brane which satisfy all energy conditions.