The cosmological reconstruction scheme for modified F(R) gravity is developed in terms of e-folding (or, redshift). It is demonstrated how any FRW cosmology may emerge from specific F(R) theory. The specific examples of well-known cosmological evolution are reconstructed, including ΛCDM cosmology, deceleration with transition to phantom superacceleration era which may develop singularity or be transient. The application of this scheme to viable F(R) gravities unifying inflation with dark energy era is proposed. The additional reconstruction of such models leads to non-leading gravitational correction mainly relevant at the early/late universe and helping to pass the cosmological bounds (if necessary). It is also shown how cosmological reconstruction scheme may be generalized in the presence of scalar field

Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy–Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied.

In the context of $f(R,T)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. According to restrictions on the background evolution, a specific model within these theories is assumed in order to guarantee the standard continuity equation. Using a completely general procedure, we find the complete set of differential equations for the matter density perturbations. In the case of sub-Hubble modes, the density contrast evolution reduces to a second-order equation. We show that for well-motivated $f(R,T)$ Lagrangians the quasistatic approximation yields to very different results from the ones derived in the frame of the concordance $\Lambda \mathrm{CDM}$ model constraining severely the viability of such theories.

We propose general f(R,T,R_{\mu\nu}T^{\mu\nu}) theory as generalization of covariant Horava-like gravity with dynamical Lorentz symmetry breaking. FRLW cosmological dynamics for several versions of such theory is considered. The reconstruction of the above action is explicitly done, including the numerical reconstruction for the occurrence of \Lambda CDM universe. De Sitter universe solutions in the presence of non-constant fluid are also presented. The problem of matter instability in f(R,T,R_{\mu\nu}T^{\mu\nu}) gravity is discussed.Comment: 12 pages. Analysis extended and minor corrections. Version published in PL

We consider the reconstruction technique in theories with a single or multiple (phantom and/or canonical) scalar fields. With the help of several examples, it is demonstrated explicitly that the universe expansion history, unifying early-time inflation and late-time acceleration, can be realized in scalar-tensor gravity. This is generalized to the theory of a scalar field coupled nonminimally to the curvature and to a Brans-Dicke-like theory. Different examples of unification of inflation with cosmic acceleration, in which de Sitter, phantom, and quintessence type fields play the fundamental role—in different combinations—are worked out. Specifically, the frame dependence and stability properties of de Sitter space scalar field theory are studied. Finally, for two-scalar theories, the late-time acceleration and early-time inflation epochs are successfully reconstructed, in realistic situations in which the more and more stringent observational bounds are satisfied, using the freedom of choice of the scalar field potential, and of the kinetic factor.