The time evolution of the equation of state
w for quintessence scenario with a scalar field as dark energy is studied up to
the third derivative (d3w/da3) with respect to the scale factor a, in order to
predict the future observations and specify the scalar potential parameters
with the observables. The third derivative of w for general potential V is
derived and applied to several types of potentials. They are the inverse
power-law (V = M4 + α/Qα), the exponential ？, the mixed , the cosine ？and the Gaussian types , which are prototypical potentials for the freezing and
thawing models. If the parameter number for a potential form is n, it is
necessary to find at least for n + 2 independent observations to identify the
potential for0m and the evolution of the scalar field (Q and

Abstract:
We were interested, along this work, in the phenomena of the quintessence and the inflation due to the F-harmonic maps, in other words, in the functions of the scalar field such as the exponential and trigo-harmonic maps. We showed that some F-harmonic map such as the trigonometric functions instead of the scalar field in the lagrangian, allow, in the absence of term of potential, reproduce the inflation. However, there are other F-harmonic maps such as exponential maps which can’t produce the inflation; the pressure and the density of this exponential harmonic field being both of the same sign. On the other hand, these exponential harmonic fields redraw well the phenomenon of the quintessence when the variation of these fields remains weak. The problem of coincidence, however remains.

Abstract:
A model is presented where the quintessence parameter, w, is related to a time-varying gravitational constant. Assuming a present value of w = -0.98 , we predict a current variation of ？/G = -0.06H_{0}, a value within current observational bounds. H_{0} is Hubble’s parameter, G is Newton’s constant and ？ is the derivative of G with respect to time. Thus, G has a cosmic origin, is decreasing with respect to cosmological time, and is proportional to H_{0}, as originally proposed by the Dirac-Jordan hypothesis, albeit at a much slower rate. Within our model, we can explain the cosmological constant fine-tuning problem, the discrepancy between the present very weak value of the cosmological constant, and the much greater vacuum energy found in earlier epochs (we assume a connection exists). To formalize and solidify our model, we give two distinct parametrizations of G with respect to “a”, the cosmic scale parameter. We treat G^{-1} as an order parameter, which vanishes at high energies; at low temperatures, it reaches a saturation value, a value we are close to today. Our first parametrization for G^{-1} is motivated by a charging capacitor; the second treats G^{-1}(a) by analogy to a magnetic response, i.e., as a Langevin function. Both parametrizations, even though very distinct, give a remarkably similar tracking behavior for w(a) , but not of the conventional form, w(a) = w_{0} + w_{a}(1-a) , which can be thought of as only holding over a limited range in “a”. Interestingly, both parametrizations indicate the onset of G formation at a temperature of approximately 7×10^{21} degrees Kelvin, in contrast to the ΛCDM model where G is taken as a constant all the way back to the Planck temperature, 1.42×10^{32} degrees Kelvin. At the temperature of formation, we find that G has increased to roughly 4×10^{20} times its current value. For most of cosmic evolution, however, our variable G model gives results similar to the predictions of the ΛCDM model, except in the very early universe, as we shall demonstrate. In fact, in the limit where w approaches -1, the expression, ？/G , vanishes, and we are left with the concordance model. Within our framework, the emergence of dark energy over matter at a scale of a ≈ 0.5 is that point where

Abstract:
We show some characteristics of three exact solutions to Einstein's gravity equation minimally coupled to a Quintessence field. Besides eternal inflation, several other interesting inflationary processes, such as transitory inflation, are obtained in these solutions. Singularity is avoided in some special cases.

Abstract:
For deeper understanding the process of baryonic matter evolution in the expanding Universe it is necessary to know the physical property of concrete field that represents the background of substrate type of dark energy. Beside, it is necessary to explore in details the influence of such field on the continuous medium of baryonic matter. These statements were realized for the quintessence field that describes by two gravitating scalar fields. They give own contributions at the total pressure and at the total mass density of baryonic matter. It allowed show that evolution of baryonic matter’s density perturbations obeys the equation of forced oscillations and admits the resonance case, when amplitude of baryonic matter’s density perturbations gets the strong short-time splash. This splash interprets as a new macroscopic mechanism of the initial matter density perturbations appearance.

Abstract:
we study the cosmic evolution in recent times within the induced gravity theory. taking into account the experimental constraints on the parameter of the theory, we study the quintessence dynamics. we obtain the possible models and discuss the physical consequences in cosmology and particle physics.

Abstract:
In this
paper, I suggest a possible explanation for the accelerating expansion of the
universe. This model does not require any dark energy or quintessence. Rather,
the idea is to suggest a different view on the origin of general relativity.
Since it is very difficult to say something in general, I will mainly restrict
myself to the case of very low curvature. The question about the underlying
reasons for the acceleration is also closely related to the question whether
the universe is a finite or infinite. It is part of the purpose of this paper
to argue that a phase of accelerating expansion may be very well compatible
with the idea of a closed universe.

Abstract:
The lack of Birkhoff
theorem in finite-range gravitation reveals nonzero acceleration of the test
body inside the massive spherical shell, as well as breakdown of screening
inside the charged conductor gives rise to acceleration of the test charge. An
application of this effect to the motion of galaxies in Local Group allows to
constraint quintessence parameter in some massive gravitational theories.

Abstract:
This paper reviews the development of f(R) gravity theory and Phantom and Quintessence fields. Specifically, we present a new general action of f(R) gravity and Phantom and Quintessence fields coupled to scalar curvature. Then, we deduce Euler-Lagrange Equations of different fields, matter tensor and effective matter tensor. Additionally, this paper obtains the general pressure, density and speed sound of the new general field action, and investigates different cosmological evolutions with inflation. Further, this paper investigates a general f(R) gravity theory with a general matter action and obtains the different field equations, general matter tensor and effective matter tensor. Besides, this paper obtains the effective Strong Energy Condition (SEC) and effective Null Energy Condition (NEC). Then, we prove that when f(R) approaches to R, the effective SEC and the effective NEC approach to the usual SEC and the usual NEC, respectively. Finally, this paper presents a general action of f(R) gravity, Quintessence and Phantom fields and their applications.

Abstract:
we discuss an effective cosmology a la brans-dicke with two interacting scalar fields: a non-minimally coupling massive inflaton higgs-like scalar field φ interacting with a minimally coupling massless scalar field χ. several features are observed and discussed in some details.