The effect of curvature in thawing models

We study the evolution of spatial curvature for thawing class of dark energy models. We examine the evolution of the equation of state parameter, $w_\phi$, as a function of the scale factor $a$, for the case in which the scalar field $\phi$ evolve in nearly flat scalar potential. We show that all such models provide the corresponding approximate analytical expressions for $w_\phi(\Omega_\phi,\Omega_k)$ and $w_\phi(a)$. We present observational constraints on these models.


I. INTRODUCTION
About a decade ago, current measurements of redshift and luminosity-distance relations of Type Ia Supernovae (SNe) [1] indicate that the expansion of the universe presents an accelerated phase [2,3]. In fact, the astronomical measurements showed that Type Ia SNe at a redshift of z ∼ 0.5 were systematically fainted which could be attributed to an acceleration of the universe caused by a non-zero vacuum energy density. As this shows a result, that the pressure and the energy density of the universe should violate the strong energy condition, ρ φ + 3 p φ > 0, where ρ φ and p φ are energy density and pressure of some exotic, unknown and unclustered matter component, dubbed dark energy [4] (see also Refs. [5,6] for recent reviews). A direct consequence of this, is that the pressure must be negative.
The first step toward understanding the property of dark energy is to make clear whether it is a simple non-vanishing cosmological constant or its genesis comes from other sources which dynamically change in time. It is possible to distinguish between these two possibilities by taking into account the evolution of the equation of state parameter defined by In what concern to the dynamical dark energy (or quintessence) its physics is described by a scalar field, φ, (quintessence scalar field), with canonical momentum [21]. One of the main characteristic of the quintessence field is when it rolls the self interacting potential curve. It will provide a negative pressure if the potential curve is quite flat. In this way, the quintessence scalar field evolves slowly enough to drive the present cosmic acceleration.
Since the evolution of the quintessence scalar field my be described by the change of the equation of state parameter w φ , so that we could distinguish two possible situations: the case in which dω φ /dφ < 0 and dω φ /dφ > 0. The former case is referred as the freezing and the later the thawing scenarios, respectively [22](see also Ref. [23] for details). While the observational data up to now are not discriminating in the sense that we could not distinguish between a freezing or a thawing phases by the variation of the equation of state parameter, it is expect that will be able to do so with the next decade high-precision astronomical observations.
On the other hand, in what concern to the curvature of the universe, today we do not know precisely the geometry of the universe, since we do not know the exact amount of matter present in the Universe. Various tests of cosmological models, including spacetime geometry, galaxy peculiar velocities, structure formation and very early universe descriptions (related to the Guths inflationary universe model [24]) support a flat universe scenario.
However, by using the seven-year Wilkinson Microwave Anisotropy Probe (WMAP) data combined with measurements of Type Ia SNe and Baryon Acoustic Oscillations (BAO) in the galaxy distribution, it was reported that the value for the curvature density parameter, −0.0068 (68% CL) represents a preferred model, which is slightly closed [25,26]. In this paper we would like to study some of the consequences that this slightly curvature may have on the evolution of the universe, together with the the situation in which the thawing cosmological evolution for the quintessence scalar field is invoked. The outline of the paper goes as follow, in section II we present the model to be study. Section III, deals with the fundamental field equations which allow then and the dynamical system. Finally, in section IV we conclude with our finding.

II. THE MODEL
The Friedmann equation in which curvature is taken into account becomes given by where the Hubble parameter H =ȧ/a, with a dot representing a derivative with respect to the cosmological time, a is the scale factor, and the curvature parameter k = 0, +1, and −1 represents flat, closed and open spatial section, respectively. Here, we use units for which 8πG = 1. The total energy density ρ is given by ρ = ρ φ + ρ m , where ρ φ and ρ m are the energy density of dark energy and dark matter, respectively. We will assume that these two components are conserve separately, satisfying the continuity equationṡ where w φ is the equation of state parameter introduced in the introduction.
We assume that the dark energy is modelled by a minimally-coupled scalar field φ, where the pressure and density of the scalar field are given by and respectively. Here, V (φ) represents the effective potential associated to the scalar field.
In term of the scalar field, Eq.(3) can be written as Equation (6) indicates that the field rolls down the hill in the scalar potential, V (φ), but its motion is damped by a term proportional to H.
Following, a similar technique developed in Ref. [27], Eqs. (1) and (6) can be expressed in terms of new variables x, y, λ, and Ω k , defined by where K ≡ −ka −2 and a prime denote the derivative with respect to ln a, and The density parameter Ω φ is expressed in terms of the variables x 2 and y 2 in such a way that while, the equation of state parameter is given by Eqs. (1) and (6) can be written in terms of the new variables Eqs.(9)- (13), so that we get where In the thawing model we have w φ ∼ −1 and thus the γ parameter satisfies γ = 1 + w φ ≪ 1.
We obtain At this point we would like to stress two assumptions that we are considering: the first is γ ≪ 1, which corresponds to ω φ ∼ −1, as discussed previously. The second assumption we make is that the scalar field begins with an initial value in a potential which is nearly flat. In this way, following [27], we assume that λ is approximately constant, so that where λ 0 is a small constant evaluated at φ = φ 0 , the initial value of the scalar field which corresponds to when it stars to roll down the potential.
Let us first to consider the evolution of the system using initial values for the curvature where we have expanded and maintained the lowest order terms in Ω k and γ. Taking the boundary value γ = 0 at Ω φ = 0 (see Ref. [27]). The resulting solution is where Note that in the limit of a flat universe, i.e., Ω k → 0, we recover the expression given in Ref. [27].
In the same way we can derive an approximate solution from Eqs. (21) and (22) under the same approximations (γ ≪ 1 and Ω k ≪ 1) from which we get From Eqs. (26) and (29) we get where the function G(Ω φ ) is given by We can use equation (21) to solve for Ω φ as a function of a and thus determine w φ (a).
Taking the limit γ ≪ 1 and Ω k ≪ 1 in equation (21) gives the following solution where Ω φ0 and Ω k0 are the present values of Ω φ , Ω k , respectively, and we take a In Fig.(4) we show the dependence of the parameter w φ as a function of the scale factor a, for different values of the curvature parameter Ω k with w 0 = −0.95 and Ω φ0 = 0.7. Note that w φ (a) is not sensible to the value of Ω k = 0 (see Ref. [27]).
We should mention that if we look for numerical solution to our set of dynamical Eqs., in which a scalar potential, such that V (φ) ∼ φ 2 , φ −2 , exp[−φ], etc, is used, we observe that there is no much changes when them are compared with that shown in Ref. [27], where Ω k = 0 was taken into account. Having an approximated expression for w φ (a) we can use it to perform a Bayesian analysis using SNIa observations, BAO distances and CMB shift parameter. In this work, we use the Supernova Cosmology Project Union sample [28], having 307 SN distributed over the range 0.015 < z < 1.551. We fit the (theoretical) distance modulus µ(z) th defined by to the observational ones µ(z) obs . Here H 0 = 100hkm s −1 Mpc −1 is the Hubble constant and the luminosity distance is defined by d L (z) = (1 + z)r(z) where and µ 0 = 42.38 − 5 log 10 h. Sinn(x) = sin x, x, sinh x for Ω k < 0, Ω k = 0, and Ω k > 0 respectively. The second major input for parameter determination comes from the baryon acoustic oscillations (BAO) detected by Eisenstein et al. [29]. In our work, we add the following term to the χ 2 of the model: where A is a distance parameter defined by and A BAO = 0.469, σ A = 0.017, and z BAO = 0.35. The CMB shift parameter R is given by [30] R(z * ) = Ω m H 2 0 r(z * ).
Here the redshift z * (the decoupling epoch of photons) is obtained by using the fitting function [31] z where the functions g 1 and g 2 are given as The WMAP-7 year CMB data alone yields R(z * ) = 1.726 ± 0.018 [32]. Defining the corre- one can deduce constraints on Ω φ0 , ω 0 and Ω k0 . A joint analysis using SN+BAO+CMB leads to the best fit values showed in Fig.5, where we see the cross section of the χ 2 function in terms of the parameters Ω φ0 , ω 0 and Ω k0 . The two horizontal lines indicate the 90% and 99% confidence range for each parameter.
The analysis shows that considering thawing quintessence with an explicit curvature term is consistent with observations. This is exactly the conclusion of [27] for the flat case in quintessence. However, as was demonstrated in [33], relaxing the slow-roll assumption, the equation of state parameter for different thawing potentials looks appreciably different.
In the following, we consider both quintessence and Tachyon field models, and two scalar field potentials; V = φ and V = φ −2 . In figure ?? conditions λ ini ≃ 1 (assuming that the potential is not flat) and γ ini ≃ 0 (the equation of state parameter can vary from its freezing state (w = −1) until today.)

IV. CONCLUSIONS
In the present work we have studied the thawing dark energy scenarios in which the effect of curvature was taking into account. We have plotted numerically trajectories in the (γ, Ω φ ), (Ω k , Ω φ ) and (Ω k , γ) for a potential nearly flat.
We have shown that all such models converge to a common behavior and we have find the corresponding approximate analytical expressions for γ(Ω φ ) given by Eq.(30) and for w φ (a) in the cases when γ ≪ 1 and Ω k ≪ 1. Here, we noted that an analitical solution for w φ (a) is not very perceptible to the value of Ω k = 0. A Bayesian analysis using SNIa data was performed to constraint the best fit parameters using our analytic function, w φ (a). This analysis shows that current data does not rule out the model. In this way, the motivation is to see whether one can distinguish thawing dark energy models from Ω k = 0 models using this method.