We demonstrate that there appear finite-time future singularities in f(T) gravity with T being the torsion scalar. We reconstruct a model of f(T) gravity with realizing the finite-time future singularities. In addition, it is explicitly shown that a power-law-type correction term T$\beta${} ($\beta${}>1) such as a T2 term can remove the finite-time future singularities in f(T) gravity. Moreover, we study f(T) models with realizing inflation in the early universe, the $\Lambda${}CDM model, little rip cosmology and pseudo-rip cosmology. It is demonstrated that the disintegration of bound structures for little rip and pseudo-rip cosmologies occurs in the same way as in gravity with corresponding dark energy fluid. We also discuss that the time-dependent matter instability in the star collapse can occur in f(T) gravity. Furthermore, we explore thermodynamics in f(T) gravity and illustrate that the second law of thermodynamics can be satisfied around the finite-time future singularities for the universe with the temperature inside the horizon being the same as that of the apparent horizon.

In this work, we analyse static spherically symmetric solutions in the framework of mimetic gravity, an extension of general relativity where the conformal degree of freedom of gravity is isolated in a covariant fashion. Here we extend previous works by considering, in addition, a potential for the mimetic field. An appropriate choice of such a potential allows for the reconstruction of a number of interesting cosmological and astrophysical scenarios. We explicitly show how to reconstruct such a potential for a general static spherically symmetric space-time. A number of applications and scenarios are then explored, among which are traversable wormholes. Finally, we analytically reconstruct potentials, which leads to solutions to the equations of motion featuring polynomial corrections to the Schwarzschild space-time. Accurate choices for such corrections could provide an explanation for the inferred flat rotation curves of spiral galaxies within the mimetic gravity framework, without the need for particle dark matter.

We study inflation induced by (power-low) scalar curvature corrections to General Relativity. The class of inflationary scalar potentials $V($\sigma${})$\sim${}\text{exp}[n$\sigma${}]$, $n$ general parameter, is investigated in the Einstein frame, and the corresponding actions in the Jordan frame are derived. We found the conditions for which these potentials are able to reproduce viable inflation according to the last cosmological data and lead to large scalar curvature corrections that emerge only at a mass scale larger than the Planck mass. The cosmological constant may appear or be set equal to zero in the Jordan frame action without changing the behavior of the model during inflation. Moreover, polynomial corrections to General Relativity are analyzed in detail. When de Sitter space-time emerges as an exact solution of the models, it is necessary to use perturbative equations in the Jordan framework to study their dynamics during the inflation. In this case, we demonstrate that the Ricci scalar decreases after a correct amount of inflation, making the models consistent with the observable evolution of the Universe.