One of the greatest challenges in cosmology today is to determine the nature of dark energy, the sourse of the observed present acceleration of the Universe. Besides the vacuum energy, various dark energy models have been suggested. The Friedmann -Robertson -Walker (FRW) spacetime plays an important role in modern cosmology. In particular, the most popular models of dark energy work in the FRW spacetime. In this work, a new class of integrable FRW cosmological models is presented. These models induced by the well-known Painlevé equations. Some nonintegrable FRW models are also considered. These last models are constructed with the help of Pinney, Schrödinger and hypergeometric equations. Scalar field description and two-dimensional generalizations of some cosmological models are presented. Finally some integrable and nonintegrable F (R) and F (G) gravity models are constructed.

In this article, we investigate the modified F (T ) gravity, which is non-minimally coupled with the Dirac (fermion) field in Friedmann-Robertson-Walker space-time. Point-like Lagrangian is derived and modified Friedmann equations and Dirac equations for the fermion field are obtained by using the Lagrange multiplier. The Noether symmetry method related to differential equations is a useful tool for studying conserved quantities. In addition, this method is very useful for determining the unknown functions that exist in the point-like Lagrangian. Using this method, the form of the coupling between gravity and matter, the self-consistent potential, the symmetry generators, the form of F (T ) gravity and the first integral (Noether charge) or a conserved quantity for this model are determined. Cosmological solutions that have a power-law form and describe the late time accelerated expansion of the Universe are obtained.

In this paper, we obtained a class of oscillatory, cyclic and knot type solutions from the non-linear Friedmann equations. This is performed by choosing specific forms of energy density and pressure of matter. All the expressions written here are in dimensionless form. We show that evolutionary path taken by the spatial coordinates in the model follow various knots, specifically trefoil and eight-knots. We provide several examples and plot relevant cosmological parameters in figures. Our cyclic models can be interpreted as a periodic cosmological model, such that early and late time acceleration are unified under the same mechanism. Finally we have presented some examples of knot universes for the Bianchi -I spacetime.

In this paper, we have investigated trace-anomaly (TA) driven inflation in Generalized Teleparallel gravity (namely f (T )). A quasi de Sitter (dS) scenario for inflation is proposed like in the case of R 2 gravity. It has been found that in f (T ) gravity, the dS solution for inflation is unstable due to the conformal anomaly (CA). As an illustrative example, we have analyzed trace anomaly driven inflation in a T 2 + f (T )-model. Furthermore, a model with a non zero cosmological constant is studied. We have shown that dS solution always remains unstable also in the presence of cosmological constant, but for Anti dS (AdS) case the situation dramatically changes.

The existing analysis reports a reconstruction scheme of the newly proposed gravity say f(Q) gravity through the scale factor of the form a(t)=a0tn=11+z by describing the power-law cosmology. The reconstructed f(Q) gravity models disclosed how this modified gravity model is capable to replicate dissimilar epochs of the cosmological history. Also, the reconstructed f(Q) gravity models are castoff to develop the expressions for density and pressure and the equation of state parameter. We reconstruct two cases of interacting fluid scenario ghost and pilgrim dark energy with pressureless dark matter. The physical behavior of the models is talked over the evolution of the Universe is accelerated. Moreover, the well-known cosmological planes i.e., (ωD−ωD′) and (r − s) constructed for our models, also include a comparison of our findings of these dynamical parameters with observational constraints. It is also quite interesting to mention here that the results of the equation of state parameter, (ωD−ωD′) and (r − s)-planes coincide with the modern observational data.

In this paper, we review the so-called Myrzakulov Gravity models (MG-N, with N=I,II,…,VIII) and derive their respective metric-affine generalizations (MAMG-N), discussing also their particular sub-cases. The field equations of the theories are obtained by regarding the metric tensor and the general affine connection as independent variables. We then focus on the case in which the function characterizing the aforementioned metric-affine models is linear and consider a Friedmann-Lemaître–Robertson–Walker background to study cosmological aspects and applications. Historical motivation for this research is thoroughly reviewed and specific physical motivations are provided for the aforementioned family of alternative theories of gravity.

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