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This paper studies inflation and isotropization in the quintom model in the Bianchi I, Bianchi III, and Kantowski-Sachs backgrounds. First, we investigate inherent properties and generalize Heusler's proposition. Then by the use of the dynamical system approach, we consider the system in multiplicative and collective modes of potentials. The conclusions of Collins and Hawking and also Burd and Barrow are discussed.

It is well known that in quantum gravity, the very geometry of space and time is subject to continual fluctuation. The mathematical formulation for this old theory is still lacking. This article formulates this more than forty-year-old theory of quantum gravity. On the other hand, recent attention has been paid to fractional gravity. Although this type of gravity leads to brilliant results, we have no deep reason for applying it other than that it works. In this paper, it is demonstrated that quantum gravity equations become fractional when space-time fluctuations are taken into account. Therefore, here a reasonable root and argument for applying fractional gravity are found: ``Fractional quantum gravity is generated by stochastic fluctuations of space-time''. For clarification, Einstein-Hilbert theory along with a scalar field is investigated in deformed and non-deformed minisuperspaces. The results illustrate a transition from decelerated to accelerated expansion (late-time-accelerated expansion).

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.

Recently, a new strategy to the reconstruction of $f(R)$-gravity models based on the Raychaudhuri equation has been suggested by Choudhury et al. In this paper, utilizing this method, the reconstruction of anisotropic Einstein-Maxwell equation in the $1+3$ covariant formalism of $f(R)$-gravity is investigated in four modes: $i.$ Reconstruction from a negative constant deceleration parameter refereeing to an ever-accelerating universe; $ii.$ Reconstruction from a constant jerk parameter $j=1$ which recovers celebrated $\Lambda \text{CDM}$ mode of evolution; $iii.$ Reconstruction from a variable jerk parameter $j=Q(t)$; and $iv.$ Reconstruction from a slowly varying jerk parameter. Furthermore, two suggestions for enhancing the method are proposed., Comment: 19 pages, 8 figures

In this paper, a modified teleparallel gravity action containing a coupling between a scalar field potential and magnetism, in anisotropic and homogeneous backgrounds, is investigated through Noether symmetry approach. The focus of this work is to describe late-time-accelerated expansion. Since finding analytical solutions carrying all conserved currents emerged by Noether symmetry approach, is very difficult, hence regularly in the literature, the authors split the total symmetry into sub-symmetries and then select, usually, some of them to be carried by the solutions. This manner limits the forms of unknown functions obtained. However, in ref. [68], B.N.S. approach was proposed in order to solve such problems but its main motivation was carrying more conserved currents by solutions. In this paper, by eliminating the aforementioned limitation as much as possible, a trick leading to some graceful forms of unknown functions is suggested. Through this fruitful approach, the solutions may carry more conserved currents than usual ways and maybe new forms of symmetries. I named this new approach to be CSSS-trick (Combination of Sub-symmetries through Special Selections). With this approach, it is demonstrated that the unified dark matter potential is deduced by the gauge fields. Utilizing the $\mathfrak{B}\text{-function}$ method, a detailed data analysis of results obtained yielding perfect agreements with recent observational data are performed. And finally, the Wheeler-De Witt (WDW) equation is discussed to demonstrate recovering the Hartle criterion due to the oscillating feature of the wave function of the universe., Comment: 27 pages, 13 figures (26 plots), 1 Table

A technique for the reconstruction of the potential for a scalar field in cosmological models based on induced gravity has recently been developed by Alexander Y. Kamenshchik, Alessandro Tronconi, and Giovanni Venturi \cite{ref1}. In this paper, this extended reconstruction method is utilized to investigate the nature of the scalar potentials of the most extended action of $F(T)$-gravity context. First, a general formalism is formulated and then it is utilized for considering some well-known special cases including `Barotropic Fluid', `Cosmological Constant', and `Modified Chaplygin Gas'. The analysis of the results is carried out by the use of the $\mathfrak{B}\text{-function}$ method which has recently been suggested by the author. As we know, assuming a proportional relation between the scale factor of the $x\text{-direction}$ and the scale factors of $y$ and $z$ directions (i.e. $A=B^{m}$) in dealing with LRS Bianchi-I background is prevalent. Pursuant to observational data, the correct physical range of $m$ is extracted. It is demonstrated that, unlike several papers, $m$ is very close to $1$. Some interesting discussions about the modified and generalized Chaplygin gases are performed. The ranges of the amounts of free parameters of the various types of Chaplygin gases are corrected according to observational data. It is demonstrated that the generalized Chaplygin gas model namely $P=-\sigma_{2}/\rho^{\nu}$ may be developed as $P=-\sigma_{2}/\rho^{\nu(t)}$ (i.e. $\nu=const. \longrightarrow \nu=\nu(t)$). Ultimately, by combining ref. \cite{ref1} and the current paper, a general prescription is recommended for the reconstruction of the potentials of alternative theories of gravity (especially $F(R)$ and $F(T)$), and their relevant analysis.

A new approach, "$\mathfrak{B}\text{-Function}$ Method", to the analysis of the reconstruction methods, phase space, and exact solutions of the alternative theories of gravity is suggested. The $\mathfrak{B}\text{-function}$ method which is constructed by suggesting eight new model-independent physical theorems, extracts the physically admissible ranges of the evolution of the parameters (i.e. the variables and constant parameters of action and those which emerged by differential equations as the constants of integration) and helps one for quick data analysis. The most significant feature and beauty of the $\mathfrak{B}\text{-function}$ method are that all conditions leading to physically admissible domains, can be expressed only by one new function, $\mathfrak{B}\text{-Function}$, which is the Lie derivative of $\ln(\mathcal{F}(x))$ along a vector field, $\mathbf{X}=x+c$, on a singleton $Q=\{x\}$.

As usual, we observe an unknown coupling function, i.e. $F(\varphi)$, with a function of torsion and also curvature, i.e. $f(T)$ and $f(R)$, generally depending on a scalar field. In $f(R)$ case, it comes from quantum correlations and other reasons. Now, what if beside this term in $f(T)$-gravity context, we enhance the action through another term which depends on both scalar field and its derivatives? In this paper, we have added such an unprecedented term in the generic common action of $f(T)$-gravity such that in this new term, an unknown function of torsion has coupled with an unknown function of both scalar field and its derivatives. We have explained why we can append such term, in the introduction in details. By the use of Noether symmetry approach, we have considered its behavior and effect. We have shown that it does not produce an anomaly, but rather it works successfully and numerical analysis of the exact solutions of field equations coincide with all most important observational data particularly late-time-accelerated expansion. So, this new term may be added to the gravitational actions of $f(T)$-gravity.

Unlike Noether symmetry, a metric independent general conserved current exits for non-minimally coupled scalar-tensor theory of gravity, if the trace of the energy momentum tensor vanishes. Thus, in the context of cosmology, a symmetry exists both in the early vacuum and radiation dominated era. For slow roll, symmetry is sacrificed, but at the end of early inflation, such a symmetry leads to a Friedmann-like radiation era. Late time cosmic acceleration in the matter dominated era is realized in the absence of symmetry, in view of the same decayed and red-shifted scalar field. Thus, unification of early inflation with late time cosmic acceleration with a single scalar field, may be realized., Comment: 9 pages, 7 figures

In this paper, we present the Noether symmetries of locally rotationally symmetric Bianchi type I (LRS BI), an anisotropic model, in the context of the teleparallel gravity. We study a certain modified teleparallel theory based on the action that, in particular, contains a coupling between the scalar field and field strength (magnetism part). We derive the symmetry generators and show that, by means of cyclic variables approach, we can not obtain a suitable solution for field equations. Hence by the use of B.N.S. approach, we solve the equations which carry Noether currents as well. By data analysis of the obtained results, we show compatible results with observational data at the last half the age of universe which is accelerating., Comment: arXiv admin note: text overlap with arXiv:1610.08099

In this paper, we have presented the Noether symmetries of flat FRW spacetime in the context of a new action in Teleparallel Gravity which we construct it based on $f(R)$ version. This modified action contains a coupling between scalar field potential and magnetism. Also, we introduce an innovative approach (B.N.S. Approach) for exact solutions which carry more conserved currents than Noether approach. By data analysis of the exact solutions, obtained from Noether approach, late time acceleration and phase crossing are realized, and some deep connections with observational data such as the age of the universe, the present amounts of the scale factor as well as the state and deceleration parameters are observed. In B.N.S. approach, we have considered dark energy dominated era.