We consider a noncommutative (NC) inflationary model with a homogeneous scalar field minimally coupled to gravity. The particular NC inflationary setting herein proposed, produces entirely new consequences as summarized in what follows. We first analyze the free field case and subsequently examine the situation where the scalar field is subjected to a polynomial and exponential potentials. We propose to use a canonical deformation between momenta, in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe, and while the Friedmann equation (Hamiltonian constraint) remains unaffected the Friedmann acceleration equation (and thus the Klein-Gordon equation) is modified by an extra term linear in the NC parameter. This concrete noncommutativity on the momenta allows interesting dynamics that other NC models seem not to allow. Let us be more precise. This extra term behaves as the sole explicit pressure that under the right circumstances implies a period of accelerated expansion of the universe. We find that in the absence of the scalar field potential, and in contrast with the commutative case, in which the scale factor always decelerates, we obtain an inflationary phase for small negative values of the NC parameter. Subsequently, the period of accelerated expansion is smoothly replaced by an appropriate deceleration phase providing an interesting model regarding the graceful exit problem in inflationary models. This last property is present either in the free field case or under the influence of the scalar field potentials considered here. Moreover, in the case of the free scalar field, we show that not only the horizon problem is solved but also there is some resemblance between the evolution equation of the scale factor associated to our model and that for the R 2 (Starobinsky) inflationary model. Therefore, our herein NC model not only can be taken as an appropriate scenario to get a successful kinetic inflation, but also is a convenient setting to obtain inflationary universe possessing the graceful exit when scalar field potentials are present.

By studying the chameleon model during inflation, we investigate whether it can be a successful inflationary model, wherein we employ the common typical potential usually used in the literature. Thus, in the context of the slow-roll approximations, we obtain the number of e-folding for the model to verify the ability of resolving the problems of standard big bang cosmology. Meanwhile, we apply the constraints on the form of the chosen potential and also on the equation of state parameter coupled to the scalar field. However, the results of the present analysis show that there is not much chance of having the chameleonic inflation. Hence, we suggest that if through some mechanism the chameleon model can be reduced to the standard inflationary model, then it may cover the whole era of the universe from the inflation up to the late time.

We consider a noncommutative standard model with a minimal coupling scalar field and a dynamical deformation between the canonical momenta of its scale factor and scalar field, and a chameleon model with a non-minimally coupling scalar field. We indicate that there is a correspondence between these two models, more specific, actually between the noncommutative parameter and the chameleon coupling strength, and also between the matter density of the chameleon model and the noncommutative geometry. In addition, the analogy constrains the type of the matter field in the chameleon model to be nearly a cosmic string-like during the inflation. Thus, the scenario enables the evolution of the universe being described by one single scalar field. That is, the effects of the chameleon and the inflaton can be described by one single scalar field which plays the role of inflaton in the very early universe and then, acts as a chameleon field. Moreover, the proposed correspondence procedure not only sets some constraints on the noncommutative parameter and the chameleon coupling constant, but also nearly specifies functions of the scalar field and its potential.the quintessence models [44]- [49], provides a dynamical alternative to the static cosmological constant. Due to the coupling of the scalar and the matter fields with the gravitational strength, the chameleon field acquires a density-dependent mass and consequently, its property changes depending on the environmental situations. Because of such a dependence, its mass is very light in the cosmological scales, wherein it may play the role of dark energy and causes the cosmic late time acceleration. Although, in the regions of high density, such as on the earth, the chameleon field acquires a large mass that makes its effects being short-ranged and hence, becomes invisible in search for the EPviolation and fifth force in the current experimental and observational tests.However, in Refs. [50,51] by proving two theorems, it has been claimed that the effect of chameleonlike scalar fields is negligible on density perturbations considering the linear scales and also, its influence may not be regarded for the observed cosmic acceleration except as some form of dark energy. Also, in Ref.[52], we considered a coupling between the chameleon field and an unknown matter scalar field, and analyzed a possibility of an influence of the chameleon field on a cosmological acceleration of the universe inflation. In the context of the slow-roll approximations, by employing the potential usually used in this issue in the literature, we evaluated the number of e-folding of the model in order to verify its viability during the inflation. We examined the model for different ranges of the free parameters, however, the results of analysis showed that there is not much chance of having the viable chameleonic universe inflation. That is, as expected for the extreme case, the exponential term (entered due to the conformal factor describing the interaction of the chameleon field with an ambient matter)...

While considering the chameleon scalar field model with the spatially flat FLRW background, we investigate the late-time acceleration phase of the universe, wherein we apply the typical potential usually used in this model. Through setting some constraints on the free parameters of the model, we indicate that the non-minimal coupling between the matter and the scalar field in such a model should be strongly coupled in order to have an accelerated expansion of the universe at the late-time. We also investigate the relative acceleration of the parallel geodesics by obtaining the geodesic deviation equation in the context of chameleon model. Then, through the null deviation vector fields, we obtain the observer area-distance as a measurable quantity to compare the model with other relevant models.