Search for dark energy potentials in quintessence theory

The time evolution of the equation of state $w$ for quintessence models with a scalar field as dark energy is studied up to the third derivative ($d^3w/da^3$) 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 ($V=M^{4+\alpha}/Q^{\alpha}$), the exponential ($V=M^4\exp{(\beta M/Q)}$), the mixed ( $V=M^{4+\gamma}\exp{(\beta M/Q)}/Q^{\gamma}$), the cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types ($V=M^4\exp(-Q^2/\sigma^2)$), 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 form and the evolution of the scalar field ($Q$ and $ \dot{Q} $). Such observations would be the values of $ \Omega_Q, w, dw/da. \cdots $, and $ dw^n/da^n$. From these specific potentials, we can predict the $ n+1 $ and higher derivative of $w$ ; $ dw^{n+1}/da^{n+1}, \cdots$. 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 understood 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 $d^3w/da^3$ are 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 $d^2w/da^2$ in some parameters.


I. INTRODUCTION
The acceleration in the expansion of the universe was discovered by the intensive observations of the cosmology almost ten and several years ago [1,2]. Although the dark energy was introduced to cause the late-time accelerated universe, the physical mechanism and origin have been poorly understood [3,4]. Two theoretical viewpoints have been proposed so far.
One is associated with modification of gravity. The other is associated with matter field theories [5]. From the latter viewpoint, we explore the possibilities of the scalar fields in quintessence models and study how relevant to the dark energy.
In the quintessence models, the scalar fields cause the time evolution of the universe.
Since the scalar field theories involve n independent parameters, we notice 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 this paper, we have carried out the calculations of the third derivative of the equation of state for five scalar potentials to identify the models and to predict the future observations. The parameters in the potentials can be determined by the knowledge of the first, second, and the higher derivatives, with the observable Ω Q and w. The first and second derivatives have been reported in the previous paper [6].
Usually, the variation of the equation of state w for the dark energy is described by [7][8][9][10] w(a) = w 0 + w a (1 − a), 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 parameter space, in this paper, where w a2 = −d 2 w/da 2 and w a3 = −d 3 w/da 3 . One of the new ingredients of this work in comparison with past works is the inclusion of this third derivative for the parameter space.
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; V = M 4 (cos(Q/f ) + 1)) [17,18] and V = M 4 exp(−Q 2 /σ 2 ) [18].
The cosine type is called the pseudo Nambu-Goldstone boson potential [17,18], which is the prototype potential of thawing model. The above mentioned potentials are motivated by particle physics. Investigation of those potentials with the method [6] is another main new ingredient of this work.
The goal of this paper is to explore the dark energy under the quintessence model in a single scalar field by assuming the potential. We have tried to increase the parameter space to examine the features of dark energy, by adding the third derivative. To determine the potential form we must observe the expansion history of the universe. If the parameter number is n for the potential form, it will be necessary for n + 2 independent observations to determine the potential form, Q andQ at some time for the time variation of scalar field.
Such observations would be values of Ω Q , w, dw/da. · · · , and dw n /da n . From these specific potentials, we could predict the n+1 and higher derivative of w ; dw n+1 /da n+1 , · · · . 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 conditions. One of the above mentioned potentials has three parameters, so it is necessary to calculate the fourth derivative of w to estimate the predict values, which is not calculated in this paper. However, the principle would be the same to calculate them. For the dark energy, we consider the scalar field Q(x, t), where the action for this field in the gravitational field is described by where S M is the action of the matter field and G is the gravitational constant, occasionally putting G = 1 [4] . Neglecting the coordinate dependence, the equation for Q(t) becomes where H is the Hubble parameter, overdot is the derivative with time, and V ′ is the derivative with Q. Putting κ = 8π/3, H satisfies the following equation where ρ B , ρ Q and ρ c are the energy density of the background, the scalar field, and the critical density of the universe. The energy density and pressure for the scalar field are written by and respectively. Then the parameter w Q for the equation of state is described by

B. Time variation of w Q
It is assumed that the current value of w Q is slightly different from a negative unity by ∆(> 0) By using Eq. (8),Q 2 is written asQ which becomes, using the density parameter Ω Q = ρ Q /ρ c , Combining Eqs. (10) and (11), V is given by From Eqs. (11) and (12),Q is expresseḋ Since ρ c is given by the observation through the Hubble parameter H,Q is determined by Ω Q and ∆, which also determine the value of V . If we adopt the form and parameters of each potential, the value of V could be used to estimate the value of Q. Actually, the evolution of H in Eq. (4) depends on the background densities which include radiation density. The effect of radiation density can be ignored in the near past (z ≤ 10 3 ) and so is not considered in this work.

C. First derivative of w Q
To investigate the variation of w Q , we calculate dw Q /da, using Eqs. (4), (6) and (7), If the first derivative is observed, V ′ V M pl is specified by, where M pl is the Planck mass. To investigate further, we must consider each potential form.
D. Second derivative of w Q From Eq. (14), the second derivative of w Q is given by After the calculation in the paper [6], d 2 w/da 2 becomes In the limit ∆ → 0, the signature of d 2 w Q /da 2 is positive under the condition V ′ /V = 0.
From this equation, we estimate d 2 w Q /da 2 for each potential in the following. If the first E. Third derivative of w Q From Eq. (16), the third derivative of w Q is given by The detailed derivation of this equation is described in the Appendix.
In the limit ∆ → 0 and under the condition V ′ V = 0, the signature of In the next section, we investigate the potential forms. Although potential parts such as V ′ /V, V ′′ /V, andV ′′′ /V are varying, the coefficients do not change in Eq. (19). Thus it is convenient to define the following notations, By using these notations, Eqs. (19) and (20) become and respectively.

III. FREEZING MODEL
In the freezing model, w Q will approach to −1. Then the first derivative of w Q is expected not positive ( dw Q /da ≤ 0). If it is necessary, we adopt the current scale factor is a = 1.
In the following, we investigate the power inverse potential V = M 4 (M/Q) α ; (α > 0), the exponential potential V = M 4 exp(βM/Q); (β > 0), and the mixed type potential The parameters of the potential are M and α. From Eq. (12), Q is given by Then Q 0 , Ω Q , ∆, and α determine the parameter M, which means that parameters determining the accelerating behavior are Q 0 , Ω Q , ∆, and α. The problem is how to estimate Q 0 and α.

First derivative
From Eq. (26), Q is derived as then Q/M pl is given by If dw Q /da is observed, Q 0 /α will be determined by the observed values Ω Q , ∆, and dw Q /da.
Since the following relations are derived we substitute them into Eq. (17) and obtain If dw Q /da is observed, (Q/M pl )/α will be determined by Eq. (28). If d 2 w Q /da 2 is observed, one could estimate the value of α from the above equation as

Third derivative
Since the following equations are derived we substitute them into Eq. (22), using Eq. (21), and obtain Because we get Q, α through the observations of dw Q /da, d 2 w Q /da 2 , we could predict the third derivative for this potential.
This potential has also two independent parameters of β and M. From Eq. (12), the potential relates to the observables which is written by

First derivative
Since the first derivative of the potential is Using Eqs. (26), (5), and (13), the first derivative of w Q becomes We get βM Q 2 M pl from the observables In the following Eq. (40), we could estimate M pl /Q by the observables. After then we could estimate βM/Q by the observables through this equation.

Second derivative
Since the second derivative of the potential is it is derived The second derivative of w Q is obtained by Eq. (17) Then we find out The value of β is estimated through Eq. (34). As the two parameters of β and M are specified, it becomes possible to predict the third deivative of w.

Third derivative
Since the third derivative of the potential is Then the third derivative of w is given through Eqs. (22) and (21) by This is the predictive value for this potential.
There are three parameters ζ, M, and γ for this potential. If we use the relation of Eq.
(12) for the potential with the observables, the parameter M is expressed by where there are three unspecified γ, ζ, and Q parameters.

First derivative
Since the following relations are derived, we substitute them into Eq. (??) From the observables, including ∆, Ω Q , and dw Q /da, we could estimate where we put Since the following equations are obtained we substitute them into Eq. (17) and derive for M 2 From the observables, including dw 2 Q /da 2 , we could estimate Y . If we make the square of X Then we could estimate γ M pl Q 2 + 2ζ from X and Y .

Third derivative
There is still unspecified parameter, which is different from the potentials with two parameters. Checking the third derivative of w Q in Eq. (19), there is still unknown term V , which must be investigated. The third derivative of the potential is then V ′′′ /V is given by If we use the third power of X, where we have used XY = X 3 − γ 2 M pl Q 3 + 4ζ 2 Q M pl . If dw 3 Q /da 3 is observed, Z could be estimated from Eq. (19). So it is possible to specify three parameters γ, ζ, and Q/M pl from the observables X, Y, and Z.
From Eq. (51), we put 2ζ into Eq. (48) and obtain and put it into Eq. (54). Then we get Because Q/M pl is derived from the above equation as γ is estimated from Eq. (54), ζ is derived through Eq. (51), and M is estimated by Eq.
(45), respectively. For this potential, three parameters are specified through the observations dw Q /da, d 2 w Q /da 2 , and d 3 w Q /da 3 . However, it is necessary to calculate the fourth derivative of the potential to predict d 4 w Q /da 4 .

IV. THAWING MODEL
The definition of the thawing model is that equation of state is w = −1 at early times and then it increases from −1, so it is expected Two parameters are M and f , where f (> 0) is the energy scale of spontaneous symmetry break down. The potential is related to the observation by Eq. (12) as

First derivative
Since the first derivative of the potential is where we put X = V ′ /V . From Eq. (14), the first derivative of w Q becomes If dw Q /da is observed, X is estimated from If dw Q /da ≥ 0, then X < 0. It means

Second derivative
Since the second derivative of the potential is The second derivative of w Q is derived by Eq. (17) as If d 2 w Q /da 2 is observed, it becomes possible to estimate Y as From Eqs. (57) and (58), f is estimated by which is equivalent, from X = − 1 f tan −1 (Q/2f ), to From Eq. (56), M is also determined. Then it becomes possible to predict the third derivative.

Third derivative
Since the following relations are derived the third derivative of w Q is given through Eq. (22) by This is the predictable value for this potential.
Two parameters are M and σ.

First Derivative of w Q
Since the first derivative of the potential is From Eq. (??), If dw Q /da is observed, 2Q σ 2 could be estimated.

Second derivative
Since the second derivative of the potential is Because 2Q σ 2 can be derived when dw Q /da is observed, 2Q , the second derivative is given by If d 2 w Q /da 2 is observed, σ is specified by The value Q and M can be also specified by Eq. (68) and Eq. (12), respectively.

Third derivative
Since the third derivative of the potential is If the parameters are specified when dw Q /da and dw 2 Q /da 2 are observed, it will become possible to predict the third derivative which is given by

V. CONCLUSIONS AND DISCUSSION
If more details of the accelerating universe is observed, it will be important to find out the time variation of the equation of state to understand the so-called dark energy. Many potentials are proposed to explain the acceleration in the context of quintessence with a single scalar field. It is necessary to distinguish which type of the potential will be the theory to explain the expansion. To differentiate the potential, it is necessary to specify the parameters of the potential. In this paper we have studied the method to find out the potential and calculated the third derivative of the equation of state for several potentials for this purpose.
At present, backward observation of large-scale structure of the universe has been undertaken to estimate w Q at the age (1 + z) [10]. Actually, the values of w(a = a 0 ), dw/da [7][8][9] and d 2 w/da 2 [6] have been pursued to be determined from the observation: Since observations of the third and higher derivatives of w could be expected in the future we have calculated the third derivative of w for general potential V and applied to five typical potentials. Three are the freezing model; the inverse power type (V = M 4+α /Q α ), the exponential type (V = M 4 exp (βM/Q)), and the mixed type (V = M 4 /Q γ exp(ζQ 2 /M 2 pl ), and two are the thawing model; the PNGB type (V = M 4 (cos(Q/f ) + 1)), and the Gaussian type (V = M 4 exp (Q 2 /σ 2 )). The four of them have two parameters and one has three parameters to identify the form.
The common points of these potentials are that it is necessary to observe the n derivatives of w Q (dw Q /da, d 2 w Q /da 2 , · · · , d n w Q /da n ) to specify the n parameters of the potential. It becomes possible to predict the n + 1 derivative of w Q (d n+1 w Q /da n+1 ) from the specific potential.
For example about the inverse power-law potential (V = M 4+α /Q α ), the observed first and second derivatives (dw Q /da, d 2 w Q /da 2 ) with H, w Q , and Ω Q could determine the two parameters of the potential M and α. The point is that the third derivative (d 3 w Q /da 3 ) is described by the current values of parameters Ω Q , w Q , and its time derivatives, dw Q /da, and d 2 w Q /da 2 . If it is predicted value, it could be understood that the dark energy would be described by the quintessence with a single scalar field of this potential. At least it will satisfy the necessary condition. It seems to be difficult to define the sufficient condition for the model of the dark energy. However, in principle, the higher derivatives d n w Q /da n (n ≥ 3) could be predicted from the specific potentials.@ The evolution of forward and/or backward time variation could be analyzed at some fixed time point. If the potential is known, the evolution will be estimated from values Q andQ at this point, because the equation for scalar field is the second derivative equation as in Eq. (4).
About the derivative of w Q , if w Q (z i ) at redshit z i is observed, the derivative of w Q is given by dw Q /da ≃ (w Q (z i ) − w Q (z i+1 ))/(a(z i ) − a(z i+1 )), for a(z i ) > a(z i+1 ). The n-th derivative of w Q could be derived through the observation w Q (z i ), where i takes 1, 2, · · · , n+ 1, respectively. The differences of w Q (z i ) could give the higher derivative of w Q (z i ). If we get the form of the potential, we could predict any higher derivative of w Q by the observables through the method developed in the paper [6] and this paper.@ If ∆ < 0, we must consider fully different models such as phantoms [19], quintom [20] or k-essence [21]. There are other models which are proposed to explain dark energy such as chameleon field [22], tachyon field [23], dilaton field [24], holographic dark energy [25], modified gravity theory [26], and so on. These models should be considered to parameterize the characteristic features in relation with the high derivatives of the accelerated expansion velocity and observable quantities.
Here, we describe the calculation to derive the third derivative of w Q . It should be noted that the third derivative is denoted by the superscript by (3) and the differences should be noted by parentheses ( ), curly brackets { } , and brackets [ ].
From Eq. (16), the third derivative is given as where the dot symbol · means the derivative with time.
The derivative of the first term in the above equation becomes then Since the second derivatives are described in the paper [6], the third derivatives of p Q , and ρ Q are shown here as, Hereafter we calculate each term of Eq. (A3) consequently. Beforehand we show the necessary peaces as [6]ä Next we calculate the elements appeared in this section as Hereafter we calculate each term in the bracket [ ] of Eq. (A3) separately.
At first, the second term becomes ASince the left curly bracket { } part is calculated in the paper [6], the right parenthesis ( ) part is calculated as Then the second term within bracket [ ] of Eq. (A3) becomes Next we consider the first term; Two elements of the above equation are following, then we need to calculate the following two parts The second part (ii) becomes The right hand part of the first term becomes At last we calculate the first term as We concentrate the first and fourth terms and sort out as Then p To combine the first and second term, we must multiply the first term by × Subsequently we concentrate the second and third terms, and sort out as B The necessary elements are displayed; Then Eq. (A23) becomes So we must multiply the whole equation by and arrange as Then the first Equation of p Q becomes as To put together the above whole calculations, the first term of Eq. (A3) becomes Here we arrange the above equation by adjusting the factor Then we put togather the above equation by Eq. (A10) times by − and facotorize by