In the context of $f(R,T)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. According to restrictions on the background evolution, a specific model within these theories is assumed in order to guarantee the standard continuity equation. Using a completely general procedure, we find the complete set of differential equations for the matter density perturbations. In the case of sub-Hubble modes, the density contrast evolution reduces to a second-order equation. We show that for well-motivated $f(R,T)$ Lagrangians the quasistatic approximation yields to very different results from the ones derived in the frame of the concordance $\Lambda \mathrm{CDM}$ model constraining severely the viability of such theories.

We develop a number of novel “black-bounce” spacetimes. These are specific regular black holes where the area radius always remains nonzero, thereby leading to a “throat” that is either timelike (corresponding to a traversable wormhole), spacelike (corresponding to a “bounce” into a future universe), or null (corresponding to a “one-way wormhole”). We first perform a general analysis of the regularity conditions for such a spacetime and then consider a number of specific examples. The examples are constructed using a mass function similar to that of Fan-Wang and fall into several particular cases, such as the original Simpson-Visser model, a Bardeen-type model, and other generalizations thereof. We analyze the regularity, energy conditions, and causal structure of these models. The main results are several new geometries, more complex than before, with two or more horizons, with the possibility of an extremal case. We derive a general theorem regarding static spacetime regularity and another general theorem regarding (non)satisfaction of the classical energy conditions.

We investigate the cosmological reconstruction in an anisotropic universe for both the homogeneous and inhomogeneous content of the universe. Special attention is given to three interesting cases: Bianchi type I, Bianchi type III, and Kantowski-Sachs models. The de Sitter, power-law, and general exponential solutions are assumed for the scale factor in each spatial direction and the corresponding cosmological models are reconstructed. Moreover, for the general exponential solutions—from which the de Sitter and power-law solutions may be obtained—we obtain models which reproduce the early Universe (assuming inflation) and the late-time accelerated expanding Universe. The models obtained for the late-time Universe are consistent with a known result in the literature where a power-law type correction in $T$ is added to a power-law type of $f(T)$ for guaranteeing the avoidance of the big rip and the big freeze.

We obtain a class of regular black hole solutions in four-dimensional $f(R)$ gravity, $R$ being the curvature scalar, coupled to a nonlinear electromagnetic source. The metric formalism is used and static spherically symmetric spacetimes are assumed. The resulting $f(R)$ and nonlinear electrodynamics functions are characterized by a one-parameter family of solutions which are generalizations of known regular black holes in general relativity coupled to nonlinear electrodynamics. The related regular black holes of general relativity are recovered when the free parameter vanishes, in which case one has $f(R)\propto R$. We analyze the regularity of the solutions and also show that there are particular solutions that violate only the strong energy condition.