Constraints on the Phase Plane of the Dark Energy Equation of State

Classification of dark energy models in the plane of w and w', where w is the dark energy equation of state and w' its time-derivative in units of the Hubble time, has been studied in the literature. We take the current SN Ia, CMB and BAO data, invoke a widely used parametrization of the dark energy equation of state, and obtain the constraints on the w -w' plane. We find that dark energy models including the cosmological constant, phantom, non-phantom barotropic fluids, and monotonic up-rolling quintessence are ruled out at the 68.3% confidence level based on the current observational data. Down-rolling quintessence, including the thawing and the freezing models, is consistent with the current observations. All the above-mentioned models are still consistent with the data at the 95.4% confidence level.


I. INTRODUCTION
Compelling evidence from different types of observation shows that the expansion of the universe is accelerating at late times (see [1] for a review). Within the framework of general relativity, this indicates that there should exist an energy source with significant negative pressure, termed dark energy, to drive this acceleration. The nature of dark energy is generally regarded as one of the most tantalizing problems in cosmology. Many dark energy models have been proposed and studied (see [1,2], and references therein). While the cosmological constant remains the simplest realization of dark energy, current observations do not rule out the possibility of time-evolving dark energy [1]- [3].
In the pursuit of revealing the nature of dark energy, cosmological observations serve to constrain the behavior of dark energy. Theoretical studies, on the other hand, should determine whether dark energy models can be distinguished by their observational consequences.
The ratio of pressure to energy density for dark energy, the equation of state w = p/ρ, is  [4]- [8]. It is found that different dark energy models are bounded in different sectors in the w-w ′ plane.
In this paper, on the one hand, we gather the bounds for various dark energy models in the w-w ′ plane. On the other hand, we obtain the constraints on the w-w ′ plane in the redshift region 0 < z < 1, by adopting a widely used parametrization [3,9,10], , based on the current observational data. The data set we use includes the recently compiled "Constitution set" of Type Ia supernovae (SN Ia) data [11]- [17], the cosmic microwave background (CMB) measurement from the five-year Wilkinson Microwave Anisotropy Probe (WMAP) [18], and the baryon acoustic oscillation (BAO) measurement from the Sloan Digital Sky Survey (SDSS) [19] and the 2dF Galaxy Redshift Survey (2dFGRS) [20]. We then compare the dark energy models with the constraints on the w-w ′ plane for 0 < z < 1. The work close to ours is that of Barger et al. [21], in which they used the earlier data set and examined the dark energy models in the w 0 -w a plane only at the redshift z = 1 .

Quintessence
The quintessence model [22]- [24], which invokes a time-varying scalar field, generally allows its energy density and equation of state to evolve with time, and has w > −1. The equation of motion of the quintessence field isφ + 3Hφ + V ,φ = 0, where H =ȧ/a is the Hubble expansion rate, and V ,φ = dV /dφ. In terms of w and w ′ , the equation of motion can be written as [25] where the minus sign corresponds toφ > 0 and the plus sign to the opposite, Ω φ (a) is the dimensionless energy density of the quintessence field, and x q = (1 + w)/(1 − w). For the down-rolling quintessence field (V < 0), the left-hand side of Eq. (1) is positive, and the bound of w and w ′ can be obtained as w ′ > −3(1 − w)(1 + w) [5,6]. The up-rolling quintessence field (V > 0) takes the other side, w ′ < −3(1 − w)(1 + w). The bound of the tracker quintessence [26] is obtained in [5,6]. However, strong acceleration today, with w −0.7, requires the breakdown of tracking [7]. The bound should only apply to the high redshift [7], z ≫ 1, which is not the region of interest in this paper. A conjectured limit of quintessence has been proposed in [7] as V /(−V ,φ ) < M P , where M P is the Plank mass. However, the physical origin of this limit is not clear [7]. We therefore do not impose this constraint on the quintessence model. Caldwell and Linder identified two categories of quintessence models, "thawing" and "freezing", based on their dynamical behavior [4]. For the thawing models, the equation of state is w ≈ −1 at early times, but grows less negative with time as w ′ > 0. The bounds of the thawing models are (1 + w) < w ′ < 3(1 + w). For the freezing models, initially the equation of state is w > −1 with w ′ < 0, but the field is frozen at late times where w → −1 and w ′ → 0. The bounds of the freezing models are 3w(1 + w) < w ′ < 0.2w(1 + w). Note that the upper bound for the freezing models is only valid for z < 1.

Phantom
The phantom model has negative kinetic energy and the equation of state w < −1 [27].
The equation of motion of the phantom field isφ + 3Hφ − V ,φ = 0. In terms of w and w ′ , the equation of motion can be written as [28] ± where the plus sign corresponds toφ > 0 and the minus sign to the opposite, Ω φ (a) is the dimensionless energy density of the phantom field, and x p = −(1 + w)/(1 − w). For the up-rolling phantom field (V > 0), the left-hand side of Eq. (2) is positive, and the bound of w and w ′ can be obtained as w ′ < −3(1 − w)(1 + w). The down-rolling phantom field (V < 0) takes the other side w ′ > −3(1 − w)(1 + w). Note that Eq. (2) and the bounds are different from those obtained in [6].

Barotropic fluids
Barotropic fluids are those for which the pressure is an explicit function of the energy density, p = f (ρ) (see [8] and references therein). The expression for w ′ can be written as [5,8] The sound speed for a barotropic fluid is given by c 2 s = dp/dρ. To ensure stability, we must have c 2 s ≥ 0, which gives the bound w ′ ≤ 3w(1 + w) for non-phantom (w > −1) barotropic fluids [5,8]. For causality, we further require c 2 s ≤ 1 [29], which gives the bound The classification of the above-mentioned dark energy models in the w-w ′ plane is shown in Fig. 1. Note that all of the bounds are valid at late times for 0 < z < 1.

A. Observational data
We use the combined data set from three types of observations including the SN Ia observation, the CMB measurement, and the BAO measurement. We assume that the universe is flat in this paper.
We use the Constitution set of SN Ia data compiled by Hicken et al. [11]- [17], which provides the information of the luminosity distance and the redshift. The SN Ia samples lie in the redshift region 0 < z < 1.55. The luminosity distance-redshift relation is given by We use the CMB shift parameter measured by the five-year WMAP observation [18], where H 0 is the Hubble constant and Ω m is the dimensionless matter density at present. We use the BAO measurement from the joint analysis of the SDSS and 2dFGRS data [19,20], and D A (z) is the angular diameter distance, To obtain the constraints on the w-w ′ plane, we invoke a broadly used form of parametrization of the equation of state [3,9,10], The constraint of w 0 , w a and Ω m is obtained by fitting the three parameters to this combined data set. The estimate of the parameters are found to be w 0 = −0.89 +0.12 −0.14 , w a = −0.18 +0.71 −0.74 , Ω m = 0.25 +0.03 −0.02 . The two-dimensional constraint of w 0 -w a is obtained and shown in Fig. 2.

B. Results of the constraints on the w-w ′ plane
We reconstruct the the w-w ′ plane via Eq. (8) and at late times for 0 < z < 1. At each redshift, the two-dimensional constraint is obtained by converting the points on the boundaries of the confidence regions in the w 0 -w a plane to the corresponding points in the w-w ′ plane, following Eq. (8) and Eq. (9), for the 68.3% and the 95.4% confidence regions respectively. 1 Since the transformation between (w, w ′ ) and (w 0 , w a ) is linear, each point inside a confidence region in the w 0 -w a plane gives a distinct point inside the corresponding confidence region in the the w-w ′ plane.
In the w-w ′ plane, we find that the cosmological constant, the phantom models, the up-rolling quintessence models, and the non-phantom barotropic fluids lie outside the 68.3% confidence region in the redshift regions 0 < z < 1, 0.18 < z < 0.22, 0.4 < z < 1 and 0.7 < z < 1, respectively. This shows that the four models are ruled out at the 68.3% confidence level. On the contrary, the down-rolling quintessence models, including the thawing and the freezing models, overlap with the 68.3% confidence region for 0 < z < 1. All of the models overlap with the 95.4% confidence region for 0 < z < 1. Samples of the constraints on the w-w ′ plane at redshifts z = 0, 0.2, 0.7 and 1, together with the models, are shown in Fig. 3.

IV. TEST OF THE METHOD
In Sec. III B, we invoked a criterion that a model is ruled out at the 68. today. The trajectories of the models in the w-w ′ plane are shown in Fig. 4 in the redshift region 0 < z < 1.
In the test, we realize each fiducial model by simulating the SN Ia, the BAO and the CMB data assuming current data quality. 1000 sets of simulated data are generated and fitted with the three parameters w 0 , w a and Ω m . The values of w 0 and w a are converted to w and

V. CONCLUSION AND DISCUSSION
Applying the bounds for various dark energy models in the w-w ′ plane for redshift 0 < z < 1, we find that several models including the cosmological constant, phantom, nonphantom barotropic fluids, and monotonic up-rolling quintessence are ruled out at the 68.3% confidence level based on the current observational data. On the other hand, down-rolling quintessence, including the thawing and the freezing models, is consistent with the current observations. All the models are still consistent with the data at the 95.4% confidence level.
Using the same SN Ia data set, Shafieloo et al. [30] and Huang et al. [31] also found the cosmological constant inconsistent with the data at the 68.3% confidence level. Barger et al. [21] found the non-phantom barotropic fluids excluded at the 95.4% confidence level based on the earlier data set. We notice that there was a time the observations favored w(z = 0) ≤ −1 [13] but now the observations favor w(z = 0) ≥ −1. However, the conclusions are drawn at the 68.3% confidence level at most. It is hoped that the next-generation observations will constrain the dark energy equation of state an order of magnitude better [1,3]. We shall be curves. The symbols "T", "F", "B", and "P" denote the "thawing", "freezing", "non-phantom barotropic", and "phantom" models, respectively. The quintessence models correspond to the region for w > −1 . The cosmological constant corresponds to the point (−1, 0). The bold solid curve is both the lower bound for the non-phantom barotropic models and the bound that separates the down-rolling and up-rolling quintessence models (down-rolling takes the upper side).
The dotted curve is the bound that separates the down-rolling and up-rolling phantom models (up-rolling takes the lower side).