We reconstruct F (R) gravity models with exponential and power-law forms of the scale factor in which bounce cosmology can be realized. We explore the stability of the reconstructed models with analyzing the perturbations from the background solutions. Furthermore, we study an F (R) gravity model with a sum of exponentials form of the scale factor, where the bounce in the early universe as well as the late-time cosmic acceleration can be realized in a unified manner. As a result, we build a second order polynomial type model in terms of R and show that it could be stable. Moreover, when the scale factor is expressed by an exponential form, we derive F (R) gravity models of a polynomial type in case of the non-zero spatial curvature and that of a generic type in that of the zero spatial curvature. In addition, for an exponential form of the scale factor, an F (R) bigravity model realizing the bouncing behavior is reconstructed. It is found that in both the physical and reference metrics the bouncing phenomenon can occur, although in general the contraction and expansion rates are different each other.

We explore cosmology with a bounce in Gauss-Bonnet gravity where the Gauss-Bonnet invariant couples to a dynamical scalar field. In particular, the potential and and Gauss-Bonnet coupling function of the scalar field are reconstructed so that the cosmological bounce can be realized in the case that the scale factor has hyperbolic and exponential forms. Furthermore, we examine the relation between the bounce in the string (Jordan) and Einstein frames by using the conformal transformation between these conformal frames. It is shown that in general, the property of the bounce point in the string frame changes after the frame is moved to the Einstein frame. Moreover, it is found that at the point in the Einstein frame corresponding to the point of the cosmological bounce in the string frame, the second derivative of the scale factor has an extreme value. In addition, it is demonstrated that at the time of the cosmological bounce in the Einstein frame, there is the Gauss-Bonnet coupling function of the scalar field, although it does not exist in the string frame.

We develop an effective Gauss-Bonnet extension of loop quantum cosmology, by introducing holonomy corrections in modified $F(\mathcal{G})$ theories of gravity. Within the context of our formalism, we provide a perturbative expansion in the critical density, a parameter characteristic of loop quantum gravity theories, and we result in having leading order corrections to the classical $F(\mathcal{G})$ theories of gravity. After extensively discussing the formalism, we present a reconstruction method that makes it possible to find the loop quantum cosmology corrected $F(\mathcal{G})$ theory that can realize various cosmological scenarios. We exemplify our theoretical constructions by using bouncing cosmologies, and we investigate which loop quantum cosmology corrected Gauss-Bonnet modified gravities can successfully realize such cosmologies.

We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned.