We use metric formalism in fðRÞ modified gravity to study the dynamics of various systems from the solar system to the cosmological scale. We assume an ansatz for the derivative of action as a function of distance and describe the Pioneer anomaly and the flat rotation curve of the spiral galaxies. Having the asymptotic behavior of action, we propose the action of fðRÞ ¼ ðR þ ÃÞð1 þ lnðR=R c Þ=ðR=R 0 þ 2=ÞÞ where in galactic and solar system scales it can recover our desired form. The vacuum solution of this action also results in a positive late time acceleration for the Universe. We fix the parameters of this model, comparing with the Pioneer anomaly, rotation curve of spiral galaxies, and supernova type Ia gold sample data.

In this paper, we consider Einstein gravity in the presence of a class of nonlinear electrodynamics, called power Maxwell invariant (PMI). We take into account (2 + 1)-dimensional spacetime in Einstein-PMI gravity and obtain its black hole solutions. Then, we regard pure F (R) gravity as well as F (R)-conformally invariant Maxwell theory to obtain exact solutions of the field equations with black hole interpretation. Finally, we investigate the conserved and thermodynamic quantities and discuss about the first law of thermodynamics for the mentioned gravitational models.

We consider the motion of test particles in the spacetime of a black hole in f (R) gravity. The

In this paper, we study thermodynamics and thermodynamic geometry of a black hole surrounded by the perfect fluid in Rastall theory. In particular, we calculate the physical quantity like mass, temperature and heat capacity of the system for two different cases. From the resulting heat capacity, we emphasize stability of the system. Following Weinhold, Ruppiner, Quevedo and HPEM formalism, thermodynamic geometry of this black hole in Rastall gravity is also analyzed. We find that the singular points of the curvature scalar of Ruppeiner and HPEM metrics entirely coincides with zero points of the heat capacity. But there is another divergence of HPEM metric which coincides with the singular points of heat capacity, so we can extract more information of HPEM metric compared with Ruppeiner metric. However, we are unable to find any physical data about the system from the Weinhold and Quevedo formalism.

In this paper, we consider three types (static, static charged, and rotating charged) of black holes in f (R) gravity. We study the thermodynamical behavior, stability conditions, and phase transition of these black holes. It is shown that the number and type of phase transition points are related to different parameters, which shows the dependency of the stability conditions to these parameters. Also, we extend our study to different thermodynamic geometry methods (Ruppeiner, Weinhold, and GTD). Next, we investigate the compatibility of curvature scalar of geothermodynamic methods with phase transition points of the above black holes. In addition, we point out the effect of different values of the spacetime parameters on the stability conditions of mentioned black holes.