Theoretical Study On Generalized F (R) Gravity And Other Modified Gravitational Related Problems

As well-known, our current universe is flat and undergoing a phase of the accelerated ex-pansion, which is supported by the recent observational data sets. In principle, this phenomenon can be explained by either dark energy which constitutes about three-fourths of the whole mat-ter budget of our universe according to the recent WMAP data and Planck data, in which the reason of this phenomenon is due to an exotic component with large negative pressure, or mod-ified theories of gravity. Dark energy and modified theories can be considered to explain the current acceleration of the universe, but they exist essential differences, the former means the discovery of new material composition and the latter means the birth of a new theory of gravity.We discuss some issues in modified theories in this thesis. Firstly, the thermodynamic behavior of field equations for the generalized f(R) gravity with curvature-matter coupling are studied in the two kinds of spacetime, i.e., the both spatially homogenous, isotropic FRW universe and static, spherically symmetric black hole spacetime. The research results show that the field equations of the generalized f(R) gravity with curvature-matter coupling can be cast to the form of the first law of thermodynamics with the so-called the entropy production terms dS and the the generalized second law (GSL) can be given by considering the FRW universe filled only with ordinary matter enclosed by the dynamical apparent horizon with the Hawking temperature. Furthermore, as a concrete example, by utilizing the GSL the constraints on the gravitational model with f1(R)=R+αRl and J2(R)=Rm are also discussed. It is worth noting these results given by us are quite general and can degenerate to the ones in Einstein’s general relativity and pure f(R) gravity with non-coupling and non-minimal coupling as special cases. Comparing with the case of Einstein’s general relativity, the appearance of the entropy production term dS in the first law of thermodynamics demonstrates that the horizon thermodynamics is non-equilibrium one for generalized f(R) gravity with curvature-matter coupling.Secondly, the generalized f(R) gravity with curvature-matter coupling in five-dimensional (5D) spacetime can be established by assuming a hypersurface-orthogonal spacelike Killing vector field of5D spacetime, and it can be reduced to the4D formulism of FRW universe. This theory is quite general and can give the corresponding results to the Einstein gravity, f(R) gravity with both no-coupling and non-minimal coupling in5D spacetime as special cases. Furthermore, in order to get some insight into the effects of this theory on the4D spacetime, by considering a specific type of models with f1(R)=f2R)=αRm and B(Lm)=Lm-ρ, we not only discuss the constraints on the model parameters m, n, but also illustrate the evolutionary trajectories of the scale factor a(t),the deceleration parameter q(t) and the scalar field e(t),Φ(t) in the reduced4D spacetime. The research results show that this type of f(R) gravity models given by us could explain the current accelerated expansion of our universe without introducing dark energy.In the end, a modified f(G) gravity model with coupling between matter and geometry is proposed, which is described by the product of the Lagrange density of the matter and an arbitrary function of the Gauss-Bonnet term.The field equations and the equations of motion corresponding to this model show the non-conservation of the energy-momentum tensor, the presence of an extra-force acting on test particles and the non-geodesic motion. Moreover, the energy conditions and the stability criterion at de Sitter point in the modified f(G) gravity mod-els with curvature-matter coupling are derived, which can degenerate to the well-known energy conditions in general relativity. Furthermore, in order to get some insight on the meaning of these energy conditions, we apply them to the specific models of f(G) gravity and the corre-sponding constraints on the models are given. In addition, the conditions and the candidate for late-time cosmic accelerated expansion in the modified f(G) gravity are studied by means of conditions of power-law expansion and the equation of state of matter less than-1/3.