The General Property of Tracking and Thawing Models and Their Observational Constraints

Abstract

We study the general property of the evolution of a class of scalar fields with tracking and thawing behaviors. For the tracking solutions, we show explicitly with three different potentials that, independent of initial conditions, there exists a general relation between the equation of state $w_{\varphi }$ and the fractional energy density ${\Omega }_{\varphi }$, so that the scalar field follows the same $w_{\varphi }-{\Omega }_{\varphi }$ trajectory during the evolution. The analytical approximations of the $w_{\varphi }-{\Omega }_{\varphi }$ trajectories are derived even though the analytical expression depends upon the particular form of the potential. For thawing solutions, a universal $w_{\varphi }-{\Omega }_{\varphi }$ relation exists and the relation is independent of both the particular form of the potential and the initial condition of the scalar field. Based on the derived $w_{\varphi }-{\Omega }_{\varphi }$ relation for the thawing models, we derive a tighter upper limit on $w_{\varphi }^{\prime }=d{w}_{\varphi }/dlna$. The observational data is also used to constrain the thawing potential with the help of the universal $w_{\varphi }-{\Omega }_{\varphi }$ relation.

1. Introduction

The observations of type Ia supernova provided the evidence to support the late time cosmic acceleration [1,2,3]. To explain the late time cosmic acceleration, an exotic matter component with negative pressure dubbed as dark energy is usually introduced. The observations suggested that dark energy contributes about 72% to the total energy density in the universe. The simplest candidate for dark energy is the cosmological constant, but the theoretical estimation of the value of the cosmological constant is in discrepancy with the astronomical observations by about 120 orders of magnitude [4], and the cosmological constant as dark energy candidate faces fine tuning and coincidence problems. Based on the concordance $\Lambda$CDM model, the release of Planck first year data on the measurements of the cosmic microwave background anisotropy gave the value of the Hubble constant as $H_0=67.3\pm 1.2$ km/s/Mpc [5,6]. This value is in tension with the local measurement on the Hubble constant $H_0=73.8\pm 2.4$ km/s/Mpc by calibrating the magnitude–redshift relation of 253 type Ia supernovae with over 600 Cepheid variables [7]. With more data and more accurate measurements, the measurement accuracy of the Hubble constant was improved greatly [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27] and the tension became more serious [28]. For example, Planck 2018 data gave the result $H_0=67.4\pm 0.5$ km/s/Mpc [25], the supernovae and $H_0$ for the equation of state (SH0ES) team measured the local Hubble constant as $H_0=73.15\pm 0.97$ km/s/Mpc from Type Ia supernovae (SNe Ia) data by recalibrating the extragalactic distance ladder using a sample of Milky Way Cepheids with the Hubble Space Telescope photometry and Gaia EDR3 parallaxes [13]. By using the tip of the red giant branch method, the local measurements of the Hubble constant gave $H_0=69.8\pm 0.6$ (stat) $\pm 1.6$ (sys)km/s/Mpc [14]. Since the Planck results from the early Universe probe of cosmic microwave background anisotropy depend on the $\Lambda$CDM model and the local measurements from type Ia supernova standard candles suffer the zero-point calibration problem, we are still uncertain about the origin of the tension on the Hubble constant. Gravitational waves as standard sirens provide an independent method of measuring cosmological distance and the observations of gravitational waves and their electromagnetic counterparts may shed light on the origin of the tension on the Hubble constant [29,30,31]. The first gravitational wave event GW170817 from binary neutron star merger and its counterpart GRB 170817A gave $H_0=70.0_{-8.0}^{+12.0}$ km/s/Mpc [32]. Combining the galaxy catalog and 47 gravitational wave events from the third LIGO-Virgo-KAGRA Gravitational-Wave Transient Catalog (GWTC-3), LIGO-Virgo-KAGRA collaborations obtained $H_0={68}_{-6}^{+8}$ km/s/Mpc [33].

Although more astronomical observations support the late time cosmic acceleration, the question whether the cosmological constant is dark energy remains unanswered. For these reasons, dark energy is usually tackled by a dynamical scalar field called quintessence [34,35,36,37]. As discussed in [38,39,40,41], the value of the Hubble constant $H_0$ can be lowered by quintessence models relative to the cosmological constant. Other possibilities for dark energy such as phantom [42], quintom [43,44,45], tachyon [46,47,48], and k-essence [49] were also considered. Alternatively, the late time cosmic acceleration approach can also be explained by modifying general relativity at the cosmological scale, such as the Dvali–Gabadadze–Porrati model [50], f(R) gravity [51,52,53,54], and dRGT ghost-free massive gravity [55,56]. For a review of dark energy and alternative theories of gravity, please see Refs. [57,58,59,60,61,62,63,64,65].

In general, to discuss the evolution of a scalar field, we need to impose initial conditions, so it is difficult to get the property of the quintessence field from observational data. Therefore, it is worthwhile to seek the general behavior of dynamical dark energy which is independent of initial conditions. Fortunately, there exist general $w_{\varphi }-{\Omega }_{\varphi }$ trajectories for the thawing and tracking solutions [66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88]. For a wide class of quintessence potentials with the tracking solution [37,68], the equation of state parameter $w\left(z\right)$ for the quintessence varies slowly. Motivated by this behavior, Efstathiou approximated $w\left(z\right)$ as $w\left(z\right)=w_0-\alpha ln(1+z)$ in the redshift range $z<4$ [89]. For a wide class of scalar fields, Chevallier–Polarski–Linder (CPL) parametrization with $w\left(a\right)=w_0+w_a(1-a)$ [90,91] was usually used as a model-independent parametrization. Combining the generic $w_{\varphi }-{\Omega }_{\varphi }$ relations and the CPL parametrization, the degeneracy between the two parameters $w_0$ and $w_a$ in the CPL parametrization can be broken by an analytical relation [87,92,93]. In this paper, we study the general property of the evolution for the tracker and thawing fields.

The paper is organized as follows. In Section 2, we discuss the reason for the same $w_{\varphi }-{\Omega }_{\varphi }$ trajectory in the tracker solution. In Section 3, we derive the $w_{\varphi }-{\Omega }_{\varphi }$ trajectories for different potentials with tracking solutions. We derived the thawing solution and obtained the constraint by using observational data for different potentials in Section 4. The conclusion is drawn in Section 5.

2. The Tracking Solution

Using the fractional energy density ${\Omega }_{\varphi }$ and the equation of state parameter $\gamma =1+w_{\varphi }$, the evolution of a scalar field is determined by the following equations:

$$\begin{array}{c}{\Omega }_{\varphi }^{\prime }=3({\gamma }_b-{\gamma }_{\varphi }){\Omega }_{\varphi }(1-{\Omega }_{\varphi }),\end{array}$$

(1)

$$\begin{array}{c}{\gamma }_{\varphi }^{\prime }=(2-{\gamma }_{\varphi })(-3{\gamma }_{\varphi }+\left|\lambda \right|\sqrt{3{\gamma }_{\varphi }{\Omega }_{\varphi }}),\end{array}$$

(2)

$$\begin{array}{c}{\lambda }^{\prime }=-\sqrt{3{\gamma }_{\varphi }{\Omega }_{\varphi }}\lambda \left|\lambda \right|(\Gamma -1),\end{array}$$

(3)

where “′” represents the derivative with respect to $lna$, $w_b=[(1/3)a_{eq}/a]/[1+a_{eq}/a]$ is the equation of state for the background, $\lambda =-V^{-1}dV/d\varphi$ is the roll parameter, and $\Gamma ={\lambda }^{-2}{V}^{-1}{d}^{2}V/d{\varphi }^2$ is the tracker parameter. With these dynamical Equations (1)–(3), we first discuss the behavior of tracking solution. To get the tracking solution, the potential $V\left(\varphi \right)$ must satisfy the condition $\Gamma \ge 1$ [68] so that the absolute value of the roll parameter $\left|\lambda \right|$ either decreases with time or remains a constant. When the tracking solution is reached, ${\gamma }_{\varphi }^{\prime }=0$, the tracker condition [68]:

$${\gamma }_{\varphi }=1+w_{\varphi }=\frac{1}{3}{\lambda }^2{\Omega }_{\varphi }$$

(4)

is satisfied, the flow parameter $F={\gamma }_{\varphi }/\left({\Omega }_{\varphi }{\lambda }^2\right)=1/3$ and $\beta =-1+(1-w_{\varphi })/\sqrt{12F}=-{\gamma }_{\varphi }/2$. If ${\gamma }_{\varphi }^{\prime \prime }\approx 0$, then ${\gamma }_{\varphi }$ is nearly a constant [68,74]:

$$w_{\varphi }=\frac{w_b(1-{\Omega }_{\varphi })-2(\Gamma -1)}{2\Gamma -1-{\Omega }_{\varphi }}.$$

(5)

It is clear that $w_{\varphi }$ is not a constant in general. To get a nearly constant $w_{\varphi }$, we must require that ${\Omega }_{\varphi }$ is negligibly small and the tracker parameter $\Gamma -1$ is nearly a constant for the tracking solution. The exponential potential has $\Gamma =1$, and the inverse power-law potential $V\left(\varphi \right)=V_0{\varphi }^{-\alpha }$ has $\Gamma =1+1/\alpha$, so they have the tracking behavior. If ${\Omega }_{\varphi }\approx 0$, then:

$$w_{\varphi }\approx w_{\varphi }^{trk}=\frac{w_b-2(\Gamma -1)}{2\Gamma -1},$$

(6)

and the roll parameter $\lambda$ must be large so that the tracker condition can be satisfied. If $\Gamma -1$ is not a constant, then it must be either large or small, so that $w_{\varphi }\approx -1$ or $w_{\varphi }\approx w_b$. For example, the potential $V\left(\varphi \right)=V_{0}exp[-{\left(3\varphi \right)}^{2/3}/2]$ has large $\Gamma -1={\lambda }^2$ and the potential $V\left(\varphi \right)=V_{0}exp(1/\varphi )$ has small $\Gamma -1=2/\sqrt{\lambda }$. In summary, the tracking solution satisfies the two conditions (4) and (6) initially, and the necessary conditions for the tracker behavior to occur are: when the tracker condition (4) is satisfied, the fractional energy density ${\Omega }_{\varphi }$ should be negligible which requires that the initial value of $\left|\lambda \right|$ should be big enough and the roll parameter $\left|\lambda \right|$ either decreases with time or remains a constant.

Now let us discuss the dependence of the tracking solution on initial conditions. We consider two different initial conditions: (i) If the initial value of the fractional energy density ${\Omega }_{\varphi i}$ is not too small or the initial value of the roll parameter ${\lambda }_i$ is large enough so that ${\lambda }_i^2>3{\gamma }_{\varphi i}/{\Omega }_{\varphi i}$, then from Equation (2), we see that whatever the initial value of $w_{\varphi }$, ${\gamma }_{\varphi }$ will always increase toward 2 until ${\gamma }_{\varphi }^{\prime }=0$. If ${\gamma }_{\varphi }>{\gamma }_b$, then Equation (1) tells us that ${\Omega }_{\varphi }$ will start to decrease because ${\Omega }_{\varphi }^{\prime }<0$. Once ${\Omega }_{\varphi }$ decreases to a small value so that ${\lambda }^2<3{\gamma }_{\varphi }/{\Omega }_{\varphi }$, Equation (2) tells us that ${\gamma }_{\varphi }^{\prime }<0$, then ${\gamma }_{\varphi }$ decreases toward to zero and ${\Omega }_{\varphi }$ starts to increase if ${\gamma }_{\varphi }<{\gamma }_b$. Although $\lambda$ decreases, it can be larger than $3{\gamma }_{\varphi }/{\Omega }_{\varphi }$ if ${\gamma }_{\varphi }\to 0$ because ${\Omega }_{\varphi }$ increases, so ${\gamma }_{\varphi }$ will increase again. When the equation of state parameter ${\gamma }_{\varphi }$ increases away from zero, it will satisfy the relation ${\lambda }^2<3{\gamma }_{\varphi }/{\Omega }_{\varphi }$ and $w_{\varphi }$ will reach the tracker behavior (6), perhaps with several oscillations. However, if the roll parameter decreases too quickly, which can be seen in the potential $V\left(\varphi \right)=V_{0}exp[-{\left(3\varphi \right)}^{2/3}/2]$ with $\Gamma =1+{\lambda }^2$ and ${\lambda }^{\prime }\propto {\lambda }^4$, then there is no tracker solution with a nearly constant $w_{\varphi }$ because $\lambda$ is too small when the tracker condition is satisfied. The reason is as follows: If ${\lambda }_i$ is small, then ${\Omega }_{\varphi i}$ cannot be too small and ${\Omega }_{\varphi }$ may reach 1 very quickly, so there is no tracker solution with nearly constant $w_{\varphi }$. (ii) If the initial value ${\Omega }_{\varphi i}$ or ${\lambda }_i$ is small, and the condition ${\lambda }_i^2<3{\gamma }_{\varphi i}/{\Omega }_{\varphi i}$ is satisfied initially, then independent of the initial value ${\gamma }_{\varphi i}$, Equations (1) and (2) tell us that the equation of state parameter ${\gamma }_{\varphi }$ decreases toward 0 and the fractional energy density ${\Omega }_{\varphi }$ starts to increase if ${\gamma }_{\varphi }<{\gamma }_b$, then the analysis is similar to that in the first case (i). If ${\lambda }_i$ is small, then ${\Omega }_{\varphi }$ always increases although ${\Omega }_{\varphi i}$ is small, and it will reach 1 soon, so there is no tracker solution with nearly constant $w_{\varphi }$ during the matter domination. However, the current values of ${\Omega }_{\varphi }$, $\lambda$, and $w_{\varphi }$ are determined by their initial values. These analyses tell us that there is no tracker solution with nearly constant $w_{\varphi }$ if the initial value of $\lambda$ is small because ${\Omega }_{\varphi }$ reaches 1 very quickly and the scalar field becomes dominant. When ${\Omega }_{\varphi }$ reaches 1, then the evolution of ${\Omega }_{\varphi }$ stops and the dynamical Equations (1)–(3) reduce to the equations for ${\gamma }_{\varphi }$ and $\lambda$. From Equations (2) and (3), we get a relation between $w_{\varphi }$ and $\lambda$. Since ${\Omega }_{\varphi }\approx 1$ and $w_{\varphi }$ approaches $-1$ asymptotically, if $\Gamma$ is a function of $\lambda$, then the relation between $w_{\varphi }$ and $\lambda$ becomes universal which is independent of the initial conditions, so the relation between $w_{\varphi }$ and $w_{\varphi }^{\prime }$ is also universal when $w_{\varphi }\to -1$. For example, there exists the attractor solution ${\Omega }_{\varphi }=1$, $w_{\varphi }=-1$ and $\lambda =0$ for the potential $V\left(\varphi \right)=V_{0}exp[-{\left(3\varphi \right)}^{2/3}/2]$.

Since the tracking conditions (4) and (6) are the initial conditions for the tracking solution, if the same initial value of ${\Omega }_{\varphi }$ is given, then $w_{\varphi }$, ${\Omega }_{\varphi }$, and $\lambda$ will follow the same trajectory, and independent of the initial conditions we will have the same $w_{\varphi }-{\Omega }_{\varphi }$ trajectory for the tracking solution. Although $w_{\varphi }-{\Omega }_{\varphi }$ trajectory is the same, the exact values of $w_{\varphi }$ and ${\Omega }_{\varphi }$ at a moment are still determined by the initial conditions.

3. The Analysis of Different Potentials

For the inverse power-law potential $V\left(\varphi \right)=V_0{\varphi }^{-\alpha }$, the tracker parameter $\Gamma =1+{\alpha }^{-1}$, and the critical point $({\Omega }_{\varphi c},{\gamma }_{\varphi c},{\lambda }_c)=(1,0,0)$ is a stable point. As ${\Omega }_{\varphi }\to 1$, both ${\gamma }_{\varphi }$ and $\lambda$ decrease to zero asymptotically, ${\gamma }_{\varphi }\approx {\lambda }^2/3+2{\lambda }^4/\left(9\alpha \right)$, $\sqrt{{\lambda }^2{\Omega }_{\varphi }}\approx \sqrt{3{\gamma }_{\varphi }}(1-{\gamma }_{\varphi }/\alpha )$ and ${\gamma }_{\varphi }^{\prime }\approx -3(2-{\gamma }_{\varphi }){\gamma }_{\varphi }^2/\alpha$. When the scalar field starts to catch up the background, the $w_{\varphi }-{\Omega }_{\varphi }$ trajectory can be approximated with a linear relation [75,94]:

$${\Omega }_{\varphi }=\frac{1-2w_i+4w_i^2}{(1-w_i^2)w_i}(w_{\varphi }-w_i),$$

(7)

where $w_i=w_{\varphi }^{trk}$. Therefore:

$$\sqrt{{\lambda }^2{\Omega }_{\varphi }}-\sqrt{3{\gamma }_{\varphi }}\approx -\frac{{\gamma }_b-{\gamma }_i}{{\gamma }_i}\frac{{\gamma }_{\varphi }\sqrt{3{\gamma }_{\varphi }}}{2-{\gamma }_{\varphi }}\left(1-\frac{{\gamma }_{\varphi }}{{\gamma }_i}\right).$$

(8)

Substitute the above relation (8) into Equations (1) and (2), we derive the approximate relation:

$$\begin{array}{c}\hfill {\Omega }_{\varphi }=\left[1-\frac{(1-w_i^2)w_i}{1-2w_i+4w_i^2}\frac{1}{{\gamma }_i-{\gamma }_{\varphi }}{\left(\frac{{\gamma }_{\varphi }}{{\gamma }_i}\right)}^{1/2+{\gamma }_i{\gamma }_b/4({\gamma }_b-{\gamma }_i)}\right.\\ \hfill {\left.exp\left[\frac{{\gamma }_i{\gamma }_b}{{\gamma }_i-{\gamma }_b}\left(\frac{1}{{\gamma }_{\varphi }}-\frac{1}{{\gamma }_i}\right)\right]\right]}^{-1}.\end{array}$$

(9)

The ${\Omega }_{\varphi }-w_{\varphi }$ trajectories for the inverse power-law potential along with the analytical approximation (9) are shown in Figure 1. It is clear that the analytical result (9) approximates the ${\Omega }_{\varphi }-w_{\varphi }$ trajectory well, especially for smaller $\alpha$ as shown in Figure 1, and the inverse power-law potential $V\left(\varphi \right)=V_0{\varphi }^{-2}$ is excluded by observational data.

For the potential $V\left(\varphi \right)=V_{0}exp(1/\varphi )$, $\Gamma =1+2/\sqrt{\lambda }$. The tracking solution starts with ${\gamma }_{\varphi }^{trk}={\gamma }_b/(1+4/\sqrt{\lambda })\approx {\gamma }_b$ because $\lambda$ is large due to the tracker condition. As ${\gamma }_{\varphi }\to 0$ and ${\Omega }_{\varphi }\to 1$, we have $\lambda \sqrt{{\Omega }_{\varphi }}\approx \sqrt{3{\gamma }_{\varphi }}[1-2{\left(3{\gamma }_{\varphi }\right)}^{3/4}/3]$ and ${\gamma }_{\varphi }^{\prime }\approx -4{\left(3{\gamma }_{\varphi }\right)}^{7/4}/3$. The $w_{\varphi }-{\Omega }_{\varphi }$ relation for the tracking solution can be approximated with:

$${\Omega }_{\varphi }={\left[1+3{\gamma }_{\varphi }^{1/4}{(1-{\gamma }_{\varphi })}^{-15/2}exp[-{\left(3{\gamma }_{\varphi }\right)}^{1/4}-{\left(3{\gamma }_{\varphi }\right)}^{-3/4}]\right]}^{-1}.$$

(10)

The ${\Omega }_{\varphi }-w_{\varphi }$ trajectory and the analytical approximation (10) are shown in Figure 1. As seen from Figure 1, the analytical relation (10) approximates the trajectory well and the potential $V\left(\varphi \right)=V_{0}exp(1/\varphi )$ is consistent with observational data.

For the potential $V\left(\varphi \right)=V_0{sinh}^{-\beta }\left(\alpha \varphi \right)$, the tracker parameter $\Gamma =1+1/\beta -{\alpha }^2\beta /{\lambda }^2$. By setting ${\Omega }_{\varphi }^{\prime }={\gamma }_{\varphi }^{\prime }={\lambda }^{\prime }=0$, we obtain the following critical points. Point C1 with $({\Omega }_{\varphi C1},{\gamma }_{\varphi C1},{\lambda }_{C1})=(1,2,\alpha \beta )$: it exists for all parameters, the point is a saddle point when $\lambda >\sqrt{6}$ and unstable when $\lambda <\sqrt{6}$. Point C2 with $({\Omega }_{\varphi C2},{\gamma }_{\varphi C2},{\lambda }_{C2})=(0,2,\lambda )$: it exists for all parameters and the point is an unstable point. Point C3 with $({\Omega }_{\varphi C3},{\gamma }_{\varphi C3},{\lambda }_{C3})=(0,0,\lambda )$: it exists for all parameters and the point is a saddle point for $-1<w<1$. Point C4 with $({\Omega }_{\varphi C4},{\gamma }_{\varphi C4},{\lambda }_{C4})=(1,0,\lambda )$: it exists for all parameters and the point is a stable point only when $\lambda =0$. Point C5 with $({\Omega }_{\varphi C5},{\gamma }_{\varphi C5},{\lambda }_{C5})=(3{\gamma }_b/{\lambda }^2,{\gamma }_b,\alpha \beta )$: the existence condition is ${\lambda }^2>3(1+w_{\varphi })$, the point is a stable point for $3(1+w_{\varphi })<{\lambda }^2<24{(1+w_{\varphi })}^2/(7+9w_{\varphi })$ and saddle point when ${\lambda }^2>24{(1+w_{\varphi })}^2/(7+9w_{\varphi })$. Point C6 with $({\Omega }_{\varphi C6},{\gamma }_{\varphi C6},{\lambda }_{C6})=(1,{\alpha }^2{\beta }^2/3,\alpha \beta )$: the existence condition is ${\lambda }^2<6$, the point is a stable point for ${\lambda }^2<3(1+w_{\varphi })$ and saddle point when $3(1+w_{\varphi })<{\lambda }^2<6$. If we choose $\alpha =0.1$ and $\beta =2$, the $w_{\varphi }-{\Omega }_{\varphi }$ relation for the tracking solution can be approximated with:

$${\Omega }_{\varphi }=1-28\times 3^{-3/2}{\gamma }_{\varphi }^{17/4}.$$

(11)

The ${\Omega }_{\varphi }-w_{\varphi }$ trajectory and the analytical approximation (11) are shown in Figure 1. As seen from Figure 1, the analytical relation (11) approximates the trajectory well and the model $V\left(\varphi \right)=V_0{sinh}^{-2}\left(0.1\varphi \right)$ is excluded by observational data.

Although we derived analytical approximations for the ${\Omega }_{\varphi }-w_{\varphi }$ trajectories, it seems to be different for different potentials, and the tracking behavior depends on the particular form of the potential, so it is difficult to reconstruct the general property of the tracker potential from observational data. However, the situation is different for the thawing solution.

4. The Thawing Solution

For the thawing solution, $w_{\varphi }$ starts with the initial value $-1$ and starts to increase. From Equation (3), it is easy to see that $\lambda$ will keep to be a constant when $w_{\varphi }=-1$. Taking the approximation ${\gamma }_{\varphi }\ll 1$ and $\lambda \approx {\lambda }_i$, we get [88]:

$$\begin{array}{c}\hfill \frac{d{\gamma }_{\varphi }}{d{\Omega }_{\varphi }}=\frac{-6{\gamma }_{\varphi }+2\lambda \sqrt{3{\gamma }_{\varphi }{\Omega }_{\varphi }}}{3{\gamma }_b{\Omega }_{\varphi }(1-{\Omega }_{\varphi })},\\ \hfill {\gamma }_{\varphi }=\frac{{\lambda }_i^2}{3}{\left(1+\frac{1}{2}{\gamma }_b\right)}^{-2}{\Omega }_{\varphi }{(1-{\Omega }_{\varphi })}^{2/{\gamma }_b}\\ \hfill {}_2{F}_1^2\left(\frac{1}{{\gamma }_b}+\frac{1}{2},\frac{1}{{\gamma }_b}+1,\frac{1}{{\gamma }_b}+\frac{3}{2};{\Omega }_{\varphi }\right),\end{array}$$

(12)

where ${}_2{F}_1(a,b,c,x)$ is the hypergeometric function. Note that this approximation is invalid when ${\gamma }_{\varphi }\sim 1$. If we take ${\gamma }_b=1$, then we recover the familiar $w_{\varphi }-{\Omega }_{\varphi }$ relation [81,92]:

$$\begin{array}{cc}\hfill {\gamma }_{\varphi }=& \frac{{\lambda }_i^2}{3}{\left[\frac{1}{\sqrt{{\Omega }_{\varphi }}}-\left(\frac{1}{{\Omega }_{\varphi }}-1\right)\left({tanh}^{-1}\sqrt{{\Omega }_{\varphi }}\right)\right]}^2\hfill \\ \hfill =& (1+w_0){\left[\frac{1}{\sqrt{{\Omega }_{\varphi }}}-\left(\frac{1}{{\Omega }_{\varphi }}-1\right){tanh}^{-1}\left(\sqrt{{\Omega }_{\varphi }}\right)\right]}^2\hfill \\ \hfill & \times {\left[\frac{1}{\sqrt{{\Omega }_{\varphi 0}}}-({\Omega }_{\varphi 0}^{-1}-1){tanh}^{-1}\sqrt{{\Omega }_{\varphi 0}}\right]}^{-2}.\hfill \end{array}$$

(13)

Note that the last equality also holds for thawing phantom models [78,92]. For thawing phantom models with the potentials $V\left(\varphi \right)\sim {\varphi }^6$, $V\left(\varphi \right)\sim {\varphi }^2$, $V\left(\varphi \right)\sim {\varphi }^{-2}$, and $V\left(\varphi \right)\sim exp(-\lambda \varphi )$, it was explicitly shown that the analytical result (13) gives the behavior of $w\left(a\right)$ [78]. Therefore, we can use $w\left(a\right)$ given by Equation (13) to approximate thawing scalar fields including quintessence and phantom. If ${\gamma }_{\varphi }$ is near zero, the scalar field is not much different from the cosmological constant, then we can approximate the dynamics of the thawing field by the SSLCPL model [87,92] with $w\left(a\right)=w_0+w_a(1-a)$ and:

$$\begin{array}{c}\hfill w_a=6(1+w_0)\frac{({\Omega }_{\varphi 0}^{-1}-1)[\sqrt{{\Omega }_{\varphi 0}}-{tanh}^{-1}\left(\sqrt{{\Omega }_{\varphi 0}}\right)]}{{\Omega }_{\varphi 0}^{-1/2}-({\Omega }_{\varphi 0}^{-1}-1){tanh}^{-1}\left(\sqrt{{\Omega }_{\varphi 0}}\right)}.\end{array}$$

(14)

Although we discuss quintessence field only, the SSLCPL model can also approximate the thawing phantom fields.

As ${\Omega }_{\varphi }\to 0$ and $w_{\varphi }\to -1$, ${\gamma }_{\varphi }\to {\lambda }_0^2{\Omega }_{\varphi }/3{(1+{\gamma }_b/2)}^2$, we get the flow parameter $F=1/3{(1+{\gamma }_b/2)}^2$ and $\beta ={\gamma }_b/2$, the results are the same as those found in [84]. If $w_{\varphi }$ starts to increase during the matter domination, ${\gamma }_b=1$ and $F=4/27$. Therefore, $4/27\le F\le 1/3$ for the thawing solution, a tighter upper limit $w_{\varphi }^{\prime }\le 3(1-w_{\varphi }^2)/2$ is then obtained. As discussed in [88], the approximation (13) works better for smaller ${\lambda }_i$ and the $w_{\varphi }-{\Omega }_{\varphi }$ relation (13) approximates the trajectory for the power-law potential better than that for the pseudo-Nambu Goldstone boson potential.

With the approximation (14), we use the 740 spectroscopically confirmed type Ia supernovae data from three years of the SDSS-II supernovae survey and SNLS3 supernovae data [95], the combination of Planck 2013 results on the measurements of the cosmic microwave background power spectra with the nine-year WMAP polarization low-multipole likelihood data [5,6,96], the 8 baryon acoustic oscillation data from the 6 dFGS survey, the baryon oscillation spectroscopic survey, the SDSS survey and the WiggleZ dark energy survey [97,98,99,100,101], and the 21 Hubble parameter $H\left(z\right)$ data [102,103,104,105] to constrain $w_{\varphi 0}$ and ${\Omega }_{\varphi 0}$, and the results are shown in Figure 2. As discussed above, the approximation (14) holds for both quintessence and phantom thawing fields, so the $2\sigma$ constraint $-1.23<w_{\varphi 0}<-0.85$ applies for both quintessence and phantom thawing fields. From Figure 2, we also see that the $2\sigma$ bound on the roll parameter is $|{\lambda }_i|=1.013$ for the quintessence thawing field and $|{\lambda }_i|=1.199$ for the phantom thawing field. From the marginalized 95% contours of $w_{\varphi 0}$ and ${\Omega }_{\varphi 0}$ obtained in Figure 2, we reconstruct the thawing potential $V\left(\varphi \right)$ by using the following relations:

$${\left(\frac{d\varphi }{dlna}\right)}^2=3m_{pl}^2{\Omega }_{\varphi }\left(a\right)|1+w\left(a\right)|,$$

(15)

$$V\left(a\right)=\frac{1}{2}{\rho }_{cr0}(1-w\left(a\right)){\Omega }_{\varphi }\left(a\right){(H\left(a\right)/H_0)}^2,$$

(16)

where $m_{pl}={\left(8\pi G\right)}^{-1/2}$ and the current critical density ${\rho }_{cr0}=3m_{pl}^2{H}_0^2$. The reconstructed potential is shown in Figure 3. From Figure 3, we see that the power-law potential $V\left(\varphi \right)=V_0{\varphi }^{0.8}$ is consistent with the observational data. The power-law potential $V\left(\varphi \right)=V_0{\varphi }^{0.8}$ as a inflationary model is also consistent with the measurements on the cosmic microwave background anisotropy. We also find that the potential $V\left(\varphi \right)=V_0{\varphi }^{0.7}$ is consistent with the $2\sigma$ upper bound and the potential $V\left(\varphi \right)=V_0{\varphi }^{1.1}$ is consistent with the $2\sigma$ lower bound.

5. Conclusions

To guarantee an almost constant $w_{\varphi }$ when the tracking solution begins, the fractional energy density ${\Omega }_{\varphi }$ should be small and the roll parameter $\lambda$ must be large enough so that the tracker condition can be satisfied, and $\left|\lambda \right|$ does not increase with time. Therefore, a nearly constant tracker parameter with $\Gamma >1$ and large initial value of the roll parameter $\lambda$ are the necessary conditions for tracking solutions. Although the current value of ${\Omega }_{\varphi }$ and $w_{\varphi }$ depend on the initial conditions for the tracking solutions, the $w_{\varphi }-{\Omega }_{\varphi }$ trajectory is independent of the initial conditions. We derived the analytical approximations of the $w_{\varphi }-{\Omega }_{\varphi }$ trajectory for the potentials $V\left(\varphi \right)\sim {\varphi }^{-\alpha }$ and $V\left(\varphi \right)\sim exp(1/\varphi )$. Unfortunately, the analytical approximations are different for different potentials.

For the thawing solutions, $w_{\varphi }\approx -1$ initially, the roll parameter changes very slowly and it can be approximated as a constant, a general $w_{\varphi }-{\Omega }_{\varphi }$ relation (12) is then obtained. Based on the asymptotical behavior of the $w_{\varphi }-{\Omega }_{\varphi }$ relation, we derive the tighter upper bound $w_{\varphi }^{\prime }\le 3(1-w_{\varphi }^2)/2$. In general, ${\Omega }_{\varphi 0}$ and $w_{\varphi 0}$ depend on the initial conditions, so it will be difficult to constrain general scalar fields by using the observational data. However, the $w_{\varphi }-{\Omega }_{\varphi }$ trajectories for both the tracking and thawing solutions are independent of the initial conditions, so it will be easier to constrain the tracker and thawing potentials. We apply the observational data to reconstruct the thawing potential and we find that the power-law potential $V\left(\varphi \right)\sim {\varphi }^{0.8}$ is consistent with the observational data. We also find that the potential $V\left(\varphi \right)=V_0{\varphi }^{0.7}$ is consistent with the $2\sigma$ upper bound and the potential $V\left(\varphi \right)=V_0{\varphi }^{1.1}$ is consistent with the $2\sigma$ lower bound.