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We introduce the cosmological analysis of the Dirac-Born-Infeld dRGT massive gravity theory which is a new extension of de Rham-Gabadadze-Tolley (dRGT) massive gravity. In this theory, we consider the Dirac-Born-Infeld (DBI) scalar field which is coupled to the graviton field. Moreover, we perform the cosmological background equations, and we demonstrate the self-accelerating background solutions. We show that the theory consists of self-accelerating solutions with an effective cosmological constant. In the following, we exhibit tensor perturbations analyses and achieve the dispersion relation of gravitational waves. We analyze the propagation of gravitational perturbation in the Friedmann-Lema\^itre-Robertson-Walker cosmology in the DBI dRGT massive gravity. Finally, we present the vector and scalar perturbations to show the stability conditions of the theory., Comment: arXiv admin note: text overlap with arXiv:2204.05595

Recently, a Friedmann-Robertson-Walker universe filled with various cosmological fluids has been considered by S.D. Odintsov et al. in [30] from phase space vantage point where various expressions for the Equation-of-State (EoS) parameter were studied. Since these types of EoS parameters are generative of appreciable results in the Hilbert-Einstein model, hence we intend to investigate all the cases in a homogeneous $F(T)$-gravity ($T$ is the torsion) through phase space analysis in precise detail. In short, three viable models of interaction between dark matter and dark energy, including usual-type, power-law type, and oscillating type, are investigated comprehensively. It is indicated that the power-law interaction in the related dynamical systems should be of increasing nature with time to get more critical points. Due to the failure of the oscillating model of ref. [30] in $F(T)$-gravity, four modified models are suggested and examined in both $F(T)$ and Hilbert-Einstein models. As to be seen, the modified models not only are generative of critical points equivalent to that of ref. [30], but also give rise to further critical points covering crucial stages of the evolution of the universe. In the context of these four models, such as the old one, at early times the interactions are negligible and they commence to grow as the cosmic time approaches the late-time in which the unification of early inflation and late acceleration is obtained. Using an indirect method, it is shown that the oscillating models have substantial roles in transitions between eras.

The conditions for which the no boundary proposal may have a classical realization of a continuous change of signature, are investigated for a cosmological model described by FRW metric coupled with a self interacting scalar field, having a noncommutative phase space of dynamical variables. The model is then quantized and a good correspondence is shown between the classical and quantum cosmology indicating that the noncommutativity does not destruct the classical-quantum correspondence. It is also shown that the quantum cosmology supports a signature transition where the bare cosmological constant takes a vast continuous spectrum of negative values. The bounds of bare cosmological constant are limited by the values of noncommutative parameters. Moreover, it turns out that the physical parameters are constrained by the noncommutativity parametres., Comment: 15 pages, 4 figures, Minor revision, references added

In this paper the stationary Klein-Gordon equation is considered for the Coulomb potential in non-commutative space. The energy shift due to noncommutativeity is obtained via the perturbation theory. Furthermore, we show that the degeneracy of the initial spectral line is broken in transition from commutative space to non-commutative space.