In this work we have investigated the dynamics of a recent modification to the general theory of relativity, the energy-momentum squared gravity model $f(R,{\mathbf{T}}^{\mathbf{2}})$, where $R$ represents the scalar curvature and ${\mathbf{T}}^{\mathbf{2}}$ the square of the energy-momentum tensor. By using dynamical system analysis for various types of gravity functions $f(R,{\mathbf{T}}^{\mathbf{2}})$, we have studied the structure of the phase space and the physical implications of the energy–momentum squared coupling. In the first case of functional where $f(R,{\mathbf{T}}^{\mathbf{2}})=f_0{R}^n({\mathbf{T}}^{\mathbf{2}}{)}^m$, with $f_0$ constant, we have shown that the phase space structure has a reduced complexity, with a high sensitivity to the values of the $m$ and $n$ parameters. Depending on the values of the $m$ and $n$ parameters, the model exhibits various cosmological epochs, corresponding to matter eras, solutions associated to an accelerated expansion, or decelerated periods. The second model studied corresponds to the $f(R,{\mathbf{T}}^{\mathbf{2}})=\alpha R^n+\beta {({\mathbf{T}}^{\mathbf{2}})}^m$ form with $\alpha$, $\beta$ constant parameters. In this case a richer phase space structure is obtained which can recover different cosmological scenarios, associated to matter eras, de–Sitter solutions, and dark energy epochs. Hence, this model represent an interesting cosmological model which can explain the current evolution of the Universe and the emergence of the accelerated expansion as a geometrical consequence.

In this paper, we analyze Vaidya spacetime with an energy dependent metric in Galileon gravity's rainbow. This will be done using the rainbow functions which are motivated from the results obtained in loop quantum gravity approach and noncommutative geometry. We will investigate the Gravitational collapse in this Galileon gravity's rainbow. We will discuss the behavior of singularities formed from the gravitational collapse in this rainbow deformed Galileon gravity

In this paper, we intend to evaluate and analyze the statefinder parameters in $f(T)$ cosmology. Friedmann equation in $f(T)$ model is taken, and the statefinder parameters ${r,s}$ are calculated. We consider a model of $f(T)$ which contains a constant, linear and non-linear form of torsion. We plot $r$ and $s$ in order to characterize this model in the ${r,s}$ plane. We found that our model $f(T)=2C_1 \sqrt{-T} +\alpha T+C_2,$ predicts the decay of dark energy in the far future while its special case namely teleparallel gravity predicts that dark energy will overcome over all the energy content of the Universe

In this paper, we will analyze the energy dependent deformation of massive gravity using the formalism of massive gravity's rainbow. So, we will use the Vainshtein mechanism and the dRGT mechanism for the energy dependent massive gravity, and thus analyze a ghost free theory of massive gravity's rainbow. We study the energy dependence of a time-dependent geometry, by analyzing the radiating Vaidya solution in this theory of massive gravity's rainbow. The energy dependent deformation of this Vaidya metric will be performed using suitable rainbow functions.Comment: 13 pages, 8 figure