We critically discuss current research on black hole (BH) solutions in f ( R ) gravity and shed light on its geometrical and physical significance. We also investigate the meaning, existence or lack thereof of Birkhoff’s theorem (BT) in this kind of modified gravity. We then focus on the analysis and search for non-trivial (i.e. hairy) asymptotically flat (AF) BH solutions in static and spherically symmetric (SSS) spacetimes in vacuum having the property that the Ricci scalar does not vanish identically in the domain of outer communication. To do so, we provide and enforce regularity conditions at the horizon in order to prevent the presence of singular solutions there. Specifically, we consider several classes of f ( R ) models like those proposed recently for explaining the accelerated expansion in the Universe and which have been thoroughly tested in several physical scenarios. Finally, we report analytical and numerical evidence about the absence of geometric hair in AFSSSBH solutions in those f ( R ) models. First, we submit the models to the available no-hair theorems (NHTs), and in the cases where the theorems apply, the absence of hair is demonstrated analytically. In the cases where the theorems do not apply, we resort to a numerical analysis due to the complexity of the non-linear differential equations. With that aim, a code to solve the equations numerically was built and tested using well-known exact solutions. In a future investigation we plan to analyze the problem of hair in de Sitter and anti-de Sitter backgrounds.

We consider metric f(R) theories of gravity without mapping them to their scalar-tensor counterpart, but using the Ricci scalar itself as an ``extra'' degree of freedom. This approach avoids then the introduction of a scalar-field potential that might be ill defined (not single valued). In order to explicitly show the usefulness of this method, we focus on static and spherically symmetric spacetimes and deal with the recent controversy about the existence of extended relativistic objects in certain class of f(R) models.

We present a new parametrization for the equation of state (EoS) ${\omega }_X=P_X/{\rho }_X$, which can reproduce a $f(R)$-like evolution with a precision between [0.5%–0.8%] over the numerical solutions. Also, our proposal can render a variety of popular $f(R)$ models that are considered as viable candidates for the cosmic late time acceleration. By using observational data from baryonic acoustic oscillations, supernovae and cosmic chronometers we investigate the constraints on the new EoS parameters. This proposal set an EoS formulation which can be used in an efficient way and makes a good candidate to be implemented in a variety of surveys in order to test the $f(R)$ generic behavior.

We derive a class of non-static inhomogeneous dust solutions in f ( R ) gravity described by the Lemaître–Tolman–Bondi (LTB) metric. The field equations are fully integrated for all parameter subcases and compared with analogous subcases of LTB dust solutions of GR. Since the solutions do not admit regular symmetry centres, we have two possibilities: (i) a spherical dust cloud with angle deficit acting as the source of a vacuum Schwarzschild-like solution associated with a global monopole, or (ii) fully regular dust wormholes without angle deficit, whose rest frames are homeomorphic to the Schwarzschild–Kruskal manifold or to a 3d torus. The compatibility between the LTB metric and generic f ( R ) ansatzes furnishes an ‘inverse procedure’ to generate LTB solutions whose sources are found from the f ( R ) geometry. While the resulting fluids may have an elusive physical interpretation, they can be used as exact non-perturbative toy models in theoretical and cosmological applications of f ( R ) theories.

We focus on the analysis of three inequivalent equations of state of geometric dark energy in $f(R)$ cosmology that have been considered in the past and discuss their differences, advantages and drawbacks.

A general formulation to compute habitable zones around binary stars is presented. A habitable zone in this context must satisfy two separate conditions: a radiative one and one of dynamical stability. For the case of single stars, the usual concept of circumstellar habitable zone is based on the radiative condition only, as the dynamical stability condition is taken for granted (assuming minimal perturbation from other planets). For the radiative condition, we extend the simple formulation of the circumstellar habitable zone for single stars, to the case of eccentric stellar binary systems, where two sources of luminosity at different orbital phases contribute to the irradiance of their planetary circumstellar and circumbinary regions. Our approach considers binaries with eccentric orbits and guarantees that orbits in the computed habitable zone remain within it at all orbital phases. For the dynamical stability condition, we use the approach of invariant loops developed by Pichardo et al. to find regions of stable, non-intersecting orbits, which is a robust method to find stable regions in binary stars, as it is based in the existence of integrals of motion. We apply the combined criteria to calculate habitable zones for 64 binary stars in the solar neighbourhood with known orbital parameters, including some with discovered planets. Formulae and interpolating tables are provided, so the reader can compute the boundaries of the habitable zones for an arbitrary binary system, using the stellar flux limits they prefer. Together with the formulae provided for stable zones, these allow the computation of both regions of stability and habitability around any binary stellar system. We found 56 per cent of the cases we consider can satisfy both restrictions, this is a very important constriction to binary systems. Nevertheless, we conclude that these systems where a dynamical and radiative safe zone exists, must be considered strong candidates in the search for habitable planets.