_{1}

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
). Such observations would be the values of ΩQ, w, dw/da, ???, and dwn/dan. From these specific potentials, we can predict the n + 1 and higher derivative of w; dwn + 1/dan + 1, ???. Since four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of w for them to estimate the predict values. If they are tested observationally, it will be under- stood whether the dark energy could be described by the scalar field with this potential. At least it will satisfy the necessary conditions. Numerical analysis for d3w/da3 is made under some specified parameters in the investigated potentials, except the mixed one. It becomes possible to distinguish the potentials by the accurate observing dw/da and d^{2}w/da^{2} in some parameters.

There are mainly two theoretical viewpoints to explain the accelerated universe. One is related to modification of gravity and the other is associated with vacuum energy and/or matter field theories. Taking the latter viewpoint, we investigate the scalar fields in quintessence scenario how relevant it to the dark energy.

In this scenario, the potential of the scalar field has n independent parameters, so we recognize that in principle n time derivatives of the equation of state with observable Ω_{Q} and w are enough to specify the scalar potentials and to predict the higher derivatives. In the paper [

Usually, the variation of the equation of state w for the dark energy is described by [

where a, w_{0}, and w_{a} are the scale factor (a = 1 at current), the current value of w(a) and the first derivative of w(a) by w_{a} = −dw/da, respectively.

We have extended the parameters pace, in the paper [

where

Recent Planck and other observations for w(z) are shown in ^{2}w/da^{2} are estimated in

We follow the single scalar field formalism of Steinhardt et al. (1999) [^{4+}^{α}/Q^{α} (inverse power law) [

We study other two potentials for so-called thawing model, in which the field is nearly constant at first and then starts to evolve slowly down the potential;

Because four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of w for them to estimate the predict values. If they are the predicted one, it will be understood that the dark energy could be described by the scalar field with this potential. At least it will satisfy the necessary

z = 0.1 | z = 0.3 | z = 0.5 | dw/da | d^{2}w/da^{2} | sign | |
---|---|---|---|---|---|---|

w(z) | −0.90 | −0.91 | −0.85 | 0.45 | 5.42 | a) |

w(z) | −0.90 | −0.91 | −0.90 | 0.17 | 1.39 | b) |

w(z) | −0.90 | −0.93 | −0.95 | 0.23 | 0.16 | c) |

w(z) | −0.90 | −0.93 | −0.958 | 0.18 | −0.48 | d) |

conditions. Numerical analysis are made for d^{3}w/da^{3} under some specified parameters in the investigated potentials except mixed one which has three parameters [

For the dark energy, we consider a scalar field Q(x,t), where the action for this field in the gravitational field is described in [

Neglecting the coordinate dependence, the equations for Q(t) becomes

where H is the Hubble parameter, over-dot is the derivative with time, and V' is the derivative with Q. The equation of state w_{Q} due to the scalar field is described by

We put w_{Q} = −1 + ∆ for the later convenience (0 < ∆ < 0.2).

The detailed calculations of the second, and third derivatives of w_{Q} for potentials are displayed in the paper [

In ^{4+}^{α}/Q^{α} with ∆ = 0.1 by the red solid curve in the dw_{Q}/da and d^{2}w_{Q}/da^{2} coordinates. The signature of α will change beyond the parabolic curve. We assume α > 0, so that the upper part of the red curve is forbidden for this potential and the freezing type potentials as well. The green (inner) dotted curve is the case of

The interesting point is that the forbidden regions for the freezing type potentials are allowed region for the thawing type potentials and the reverse is also true. It is possible to distinguish the potentials among each type due to the different predicted values of d^{3}w_{Q}/da^{3} [_{Q}/da, d^{2}w_{Q}/da^{2}, d^{3}w_{Q}/da^{3} and other parameters such as ∆.

At present, backward observations, such as Planck, baryon acoustic oscillation, Supernova Ia, Hubble constant, weak lensing, and red shift distortion, have been undertaken to estimate w_{Q} at the age (1 + z) as in _{0}), dw/da and d^{2}w/da^{2} have been estimated which are presented in _{Q}/da and d^{2}w_{Q}/da^{2} plane in

The adopted values from observation show −0.5 < d^{2}w_{Q}/da^{2}< 6 within the region 0.1 < dw_{Q}/da < 0.5. Although there is a lot of uncertainty, at the moment, it seems to be preferable for the thawing model against the freezing model under the comparison with the numerical results and the observations [

About observations in

Usually matter density increases as_{Q} and p_{m} are pressure for scalar field and matter. There seems to be no such features that w increases with z in the observations in

If w < −1 which means ∆ < 0 is correct in

Tetsuya Hara, (2016) Quest for Potentials in the Quintessence Scenario. Journal of Applied Mathematics and Physics,04,211-214. doi: 10.4236/jamp.2016.42027