The future of the universe in modified gravitational theories: approaching a finite-time future singularity

We investigate the future evolution of the dark energy universe in modified gravities, including F(R) gravity, and string-inspired scalar Gauss Bonnet and modified Gauss-Bonnet ones, and ideal fluid with an inhomogeneous equation of state (EoS). The modified Friedmann-Robertson-Walker dynamics for all of these theories may be presented in a universal form by using the effective ideal fluid with an inhomogeneous EoS without specifying its explicit form. We construct several examples of a modified gravity which produces accelerating cosmologies ending at the finite-time future singularities of all four known types by applying a reconstruction program. Some scenarios for resolving a finite-time future singularity are presented. Among these scenarios, the most natural one is related to additional modi. cation of the gravitational action in the early universe. In addition, late-time cosmology in the non-minimal Maxwell-Einstein theory is considered. We investigate the forms of non-minimal gravitational coupling which generate finite-time future singularities and the general conditions for this coupling such that the. nite- time future singularities cannot emerge. Furthermore, it is shown that the non-minimal gravitational coupling can remove the finite-time future singularities or make the singularity stronger (or weaker) in modified gravity.