Observational constraint on the dark energy scalar field

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Ming-Jian Zhang and Hong Li. Observational constraint on the dark energy scalar field[J]. Chinese Physics C.
Ming-Jian Zhang and Hong Li. Observational constraint on the dark energy scalar field[J]. Chinese Physics C. shu
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Observational constraint on the dark energy scalar field

    Corresponding author: Hong Li, hongli@ihep.ac.cn
  • 1. School of Electronic and Information Engineering, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
  • 2. Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Science, Beijing 100049, China

Abstract: In this paper, we study three scalar fields, quintessence field, phantom field and tachyon field, to explore the source of dark energy via the Gaussian processes method from the background and perturbation growth rate data. The corresponding reconstructions all suggest that the dark energy should be dynamical. Moreover, the quintom field, a combination between quintessence field and phantom field, is powerfully favored by the reconstruction. Their mean values indicate that the potential $ V(\phi) $ in the quintessence field is a double Exponential function, $ V(\phi) $ in phantom field is a double Gaussian function. This reconstruction can provide an important reference on the scalar field study. The two types of data both reveal that the tachyon field is at a disadvantage to describe the cosmic acceleration.

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    I.   INTRODUCTION
    • Multiple experiments, including the type Ia supernova (SNIa) [1, 2], cosmic microwave background (CMB) anisotropies [3], large scale structure [4], and baryon acoustic oscillation (BAO) peaks [5] have consistently approved that our universe was experiencing a process of accelerating expansion. Theoretically, this acceleration needs a new component with repulsive gravity to drive. In numerous theoretical paradigms, the exotic dark energy theory is the most concerned. One essential parameter, the ratio of pressure to energy density, equation of state (EoS) w is designed to understand its nature. For example, EoS of the famous cosmological constant model is $ w = -1 $. In recent analysis [6], this model fits well with the Planck data and other astrophysical data. However, an evolving dark energy are also mildly favoured by many other data, especially in the very recent extended BAO survey [7]. In this paper, we investigate the quintessence field [8-11], phantom field [12], and tachyon field [13-15]. They are all scalar field which can achieve the evolving dark energy. For the quintessence field, it has a positive kinetic energy density with $ -1 \leqslant w \leqslant 1 $. While for the phantom field, it has a negative kinetic energy density with $ w \leqslant -1 $.

      However, numerous observations found that the w was crossing $ -1 $. Regretfully, either the quintessence or phantom scalar field both cannot fulfill this transition. To solve these problems, Feng et al. [16] proposed the quintom model, a combination of quintessence field $ \phi_1 $ and phantom field $ \phi_2 $ in the Lagrangian. When the time derivative of scalar field $ \dot{\phi}_1 > \dot{\phi}_2 $, it leads to $ w \geqslant -1 $; while for $ \dot{\phi}_1 < \dot{\phi}_2 $, we have $ w \leqslant -1 $. To promote the quintom model being a single scalar field, Refs. [17-21] introduce higher derivative operators in the Lagrangian. They found that the models are consistent with the observations.

      We notice that most of the potentials $ V(\phi) $ were built by a parametrization, either in the quintessence, phantom or the quintom model. Their common popular templates include the power-law potential, exponential potential, or trigonometric function potential, and so on. However, a parameterized $ V(\phi) $ template inevitably imposes a prior on the underlying property of cosmic dynamics. In our view, a straightforward manner and template-free study has an advantage to understand the cosmic dynamics.

      In this paper, we focus on a prominent technique, the Gaussian processes (GP) method. Unlike the parametrization constraint, the GP method does not attach to any artificial cosmological template. It is a purely statistical manner. In this process, it presents a requirement for each observational data. That is, they should satisfy a Gaussian distribution. The data sets thus naturally satisfy a multivariate Gaussian distribution. In this process, we need a covariance function $ k(z, \tilde{z}) $ which connects two variables between any two data points. Using the function $ k(z, \tilde{z}) $, information of the variables contained in data can be extrapolated to other redshift which have not been observed. Finally, an involved goal function, such as the potential V in scalar field can be reconstructed via more data. We note that determination of the function $ k(z, \tilde{z}) $ is the primary task in this Gaussian processes. Because this process is independent of any template of the goal function, it has been widely used in many fields, such as the cosmology [22-31]. In our recent work [32], we investigated the dark energy using this method. The EoS w analysis shows that observational data both indicate a dynamical dark energy. However, source of the dynamics is still unknown for us. Therefore, what is the nature of these reconstructed dark energy needs a further understanding. In this paper, we would like to perform relevant analysis via the scalar field. Our goal of the present work is to explore which scalar field may be the dynamical source of dark energy. This test can update our understanding on the cosmic acceleration by presenting a model-independent result. Following our recent work, the data sets we use are still supernova and Hubble parameter, and perturbation data provided by the growth rate of structure. The dynamical models we consider are quintessence, phantom, and tachyon scalar fields.

      This paper is organized as follows: In Section II, we introduce the scalar field and GP approach. And in Section III we introduce the relevant data we use. We present the reconstruction result in Section IV. Finally, in Section V conclusion and discussion are drawn.

    II.   THEORY AND METHOD
    • In this section, we give an introduction about the scalar field and GP approach.

    • A.   Scalar field

    • For a Friedmann-Robertson-Walker universe with flat spatial, we assume it have only dark matter and scalar field. The Friedman equations of this universe are

      $ \begin{aligned}[b] H^2 &= \frac{8\pi G}{3} (\rho_m + \rho_{\phi} ), \\ \frac{\ddot{a}}{a} &= -\frac{4\pi G}{3} (\rho_m + \rho_{\phi} + 3 p_{\phi}) , \end{aligned} $

      (1)

      where the Hubble expansion rate $ H = \dot{a}/a $ is a function of scalar factor $ a(t) $, and the dot denotes derivative with respect to cosmic time t. The parameter $ \rho_{\phi} $ and $ p_{\phi} $ are energy density and pressure of scalar field, respectively. For the dark matter, we assume that its energy density yields $ \rho_m = \rho_{m0} (1+z)^3 $, where $ \rho_{m0} $ is its current energy density. Generally, we introduce the matter energy density parameter $ \Omega_{m0} = \rho_{m0}/\rho_{c0} $, with critical density $ \rho_{c0} = 3H_0^2/ (8 \pi G) $, where $ H_0 $ is the Hubble constant. For the scalar filed, we consider three scenarios in this present paper, namely, the quintessence, phantom and tachyon scalar field.

      For the quintessence scalar field, its energy density and pressure are defined as

      $ \begin{aligned}[b] \rho_{\phi} &= \frac{1}{2}\dot{\phi}^2 + V(\phi) , \\ p_{\phi} &= \frac{1}{2}\dot{\phi}^2 - V(\phi) . \end{aligned} $

      (2)

      For the potential $ V(\phi) $, it is usually parameterized as a function of scalar field $ \phi $. Till now, many models were proposed (see Ref. [33] for a short review), such as the power-law potential $ V(\phi) \propto \phi^{p} $, exponential potential $ V(\phi) \propto e^{-{\rm{\lambda }} \phi} $, inverse power-law potential $ V(\phi) \propto \phi^{-p} $, inverse exponential potential $ V(\phi) \propto e^{{\rm{\lambda }}/ \phi} $, double exponential potential $ V(\phi) \propto V_1 e^{-{\rm{\lambda }}_1 \phi} + V_2 e^{-{\rm{\lambda }}_2 \phi} $, Hilltop potential $ V(\phi) \propto \cos(\phi) $. Meanwhile, some other complex models also can be found in Ref. [34], such as $ e^{{\rm{\lambda }} \phi^2}/\phi^{\alpha} $, $ (\cosh {\rm{\lambda }} \phi -1)^p $, $ \sinh ^{-\alpha} ({\rm{\lambda }} \phi) $, $ [(\phi - B)^{\alpha} + A] e^{-{\rm{\lambda }} \phi} $. By confronting these models with the observational data [35-39], the authors found that some models cannot be discriminated from each other, or some ones are not disfavored by the observational data, such as the inverse power-law potential, inverse exponential potential, even some ones have intrinsic limitation. In this paper, we also have a passion to know what models are suitable for describing the dark energy. Now, our task is to give a solution of the quintessence scalar field. Putting the definition of Eq. (2) to Friedman equations (1), we can solve the quintessence field and its potential as

      $ \begin{aligned}[b] \frac{8\pi G}{3H_{0}^2} \dot{\phi}^{2} &= \frac{1}{3}(1+z) E^{2 \prime} - \Omega_{m0} (1+z)^3 , \\ \frac{8\pi G}{3H_{0}^2} V &= E^2 - \frac{1}{6}(1+z) E^{2 \prime} - \frac{1}{2}\Omega_{m0} (1+z)^3 , \end{aligned}$

      (3)

      where the prime denotes derivative with respect to redshift z; $ E(z) = H(z)/H_0 $ is the dimensionless Hubble parameter. We find that, on the one hand, both the derivative of scalar field $ \dot{\phi}^{2} $ and potential V are in units of $\dfrac{8\pi G}{3H_{0}^2}$. On the other hand, we should note that the function $ \dot{\phi}^{2} $ may be negative when the former term is less than the latter term. If this case happens, it would change into the other model, the phantom scalar field.

      For the phantom scalar field, its energy density and pressure are

      $ \begin{aligned}[b] \rho_{\phi} &= -\frac{1}{2}\dot{\phi}^2 + V(\phi) , \\ p_{\phi} &= -\frac{1}{2}\dot{\phi}^2 - V(\phi) . \end{aligned} $

      (4)

      Performing a similar calculation, we can obtain the phantom field and potential

      $ \begin{aligned}[b] \frac{8\pi G}{3H_{0}^2} \dot{\phi}^{2} &= \Omega_{m0} (1+z)^3 - \frac{1}{3}(1+z) E^{2 \prime} , \\ \frac{8\pi G}{3H_{0}^2} V & = E^2 - \frac{1}{6}(1+z) E^{2 \prime} - \frac{1}{2}\Omega_{m0} (1+z)^3 . \end{aligned} $

      (5)

      Obviously, function $ \dot{\phi}^{2} $ in the Eq. (5) is opposite to the $ \dot{\phi}^{2} $ in Eq. (3). Therefore, the quintessence and phantom field, cannot exist at the same time. Similar to above quintessence scalar field, the cosmologist also modelled a lot of phantom potentials. Caldwell et al. [40] studied this scalar field, and found that $ w<-1 $ would cause a big rip of the universe. Investigation in Ref. [41] considered five models, and showed that they fit well with the observational data, but no one occupies a special position. In order to solve the problem of w crossing $ -1 $ in the near past from $ w > -1 $ to $ w < -1 $, Feng et al. [16] proposed a quintom model with a double exponential potential in the Lagrangian by combining the quintessence field and phantom field

      $ \begin{aligned}[b] {\cal{L}} = &\frac{1}{2} \partial_{\mu} \phi_1 \partial^{\mu} \phi_1 - \frac{1}{2} \partial_{\mu} \phi_2 \partial^{\mu} \phi_2 \\ &- V_0 \left[\exp\left(-\frac{{\rm{\lambda }}}{m_p} \phi_1\right) + \exp\left(-\frac{{\rm{\lambda }}}{m_p} \phi_2\right) \right] , \end{aligned}$

      (6)

      where $ \phi_1 $ and $ \phi_2 $ stand for the quintessence field and phantom field, respectively. We note that it is different from the quintessence field or phantom field. It presents more complicated dynamics. They found that this model also satisfies the observations. The parameters are $ V_0 = 8.38 \times 10^{-126} m_p^4 $, and $ {\rm{\lambda }} = 20 $. This model can realize the transition of w from $ w > -1 $ to $ w < -1 $.

      For the tachyon scalar field, it is a different scalar field from above two scenarios. Its energy density and pressure are

      $ \begin{aligned}[b] \rho_{\phi} & = \frac{V(\phi)}{\sqrt{1-\dot{\phi}^2}} , \\ p_{\phi} & = -V(\phi) \sqrt{1-\dot{\phi}^2} . \end{aligned} $

      (7)

      Combining with the Friedman equations (1), the tachyon field and potential can be solved as

      $ \begin{aligned}[b] \dot{\phi}^{2} &= \frac{(1+z) E^{2 \prime} - 3\Omega_{m0} (1+z)^3}{ 3E^2 - 3\Omega_{m0} (1+z)^3} , \\ \frac{8\pi G}{3H_{0}^2} V &= \sqrt{E^2-\frac{1+z}{3} E^{2 \prime}} \sqrt{E^2 - \Omega_{m0} (1+z)^3} . \end{aligned} $

      (8)

      For this solution, we have several points to note. Firstly, the term $ 1-\dot{\phi}^2 $ in Eq. (7) must be positive. Secondly, we find that the solutions $ \dot{\phi}^{2} $ and V in Eq. (8) are much different from the ones in above two scenarios. For the function $ \dot{\phi}^{2} $, it is immune from the nuisance parameter $\dfrac{8\pi G}{3H_{0}^2}$. For the potential V, it should be non-negative in the square root of Eq. (8). Obviously, any negative values in the Eq. (8) can lead to the failure of tachyon field. In cosmology, several tachyon models were studied. In Ref. [42], the authors numerically investigated the tachyon models with a range of potentials. Comparing with the canonical quintessence models, they have quite similar phenomenology. In Ref. [43], the authors also studied some models, and found that some of them are not strongly disfavoured by observations. For a single tachyon field with an inverse square potential, Guo et al. [44] found that the universe could accelerate only at nearly Planck energy densities. However, the acceleration also can be obtained for multiple tachyon fields at lower-Planck energy densities.

      To obtain the potential $ V(\phi) $, we should solve the field $ \phi $ from the function $ \dot{\phi}^{2} $. Using the relation ${\rm d}t = -\dfrac{1}{(1+z)H} {\rm d}z$, we can transfer the derivative of scalar field $ \dot{\phi}^{2} $ over time t to redshift z, namely,

      $ \left( \frac{{\rm d}\phi}{{\rm d}z} \right)^2 = \frac{\dot{\phi}^{2}}{(1+z)^2 H^2} . $

      (9)

      Here we should be careful for the units of function $\left( \dfrac{{\rm d}\phi}{{\rm d}z} \right)^2$ in different scenarios. In our calculation, we reduce it to a dimensionless quantity. To obtain a dimensionless one, the function $ \dot{\phi}^2 $ in quintessence and phantom fields, and potential $ V(z) $ in these three fields are in units of $\dfrac{8\pi G}{3H_{0}^2}$. Function $\dfrac{{\rm d}\phi}{{\rm d}z}$ is in units of $ H_0 $. Theoretically, the function $\dfrac{{\rm d}\phi}{{\rm d}z}$ can take two signs. Here we consider the positive values. Finally, the scalar field can be obtained by

      $ \phi = \int \frac{{\rm d}\phi}{{\rm d}z} {\rm d}z . $

      (10)

      In our calculation, we take the initial value $ \phi_0 = 0 $. The scalar field $ \phi (z) $ therefore can be obtained over a function of redshift z. After above preparations, the dimensionless potential $ V (z) $ and scalar field $ \phi (z) $ can be reconstructed at the same time. Thus, the potential $ V(\phi) $ can be modelled as a function of scalar field via a model-independent way.

    • B.   Methodology

    • The data we use in this paper are background data from supernova and Hubble parameter; and perturbation data from redshift-space distortions (RSD).

      For the background data, the theoretical distance modulus of supernova in the Friedmann-Robertson-Walker universe is

      $ \mu_{\rm{th}} (z) = 5 {\rm{log}}_{10}d_L(z)+25, $

      (11)

      with the luminosity distance function

      $ d_L(z) = \frac{c}{H_0} (1+z) \int^z_0 \frac{ {\rm{d}} \tilde{z}}{E(\tilde{z})} . $

      (12)

      By introducing a dimensionless comoving luminosity distance

      $ D(z) \equiv \frac{H_0}{c} \frac{d_L (z)}{1+z} , $

      (13)

      we can obtain the relation between Hubble parameter and distance $ D(z) $ via the Eqs. (13) and (12)

      $ E (z) = \frac{ 1}{D'} . $

      (14)

      Therefore, the Hubble parameter data can be used as the derivative of distance function $ D(z) $.

      For the perturbation data, we consider a background universe filled with dark matter and scalar field as the unclustered dark energy. For the dark matter, its density contrast is defined as $ \delta(z) \equiv \dfrac{\delta \rho_m}{\rho_m} (z) $. At scales much smaller than the Hubble radius, evolution of the density contrast should obey a second order differential equation

      $ \ddot{\delta} + 2H \dot{\delta} -4\pi G \rho_m \delta = 0 , $

      (15)

      where $ \rho_m $ is the background matter energy density, $ \delta \rho_m $ represents its first-order perturbation. It is an equation of the matter growth under the assumption of homogeneity and isotropy with zero dark energy perturbations. The density contrast is in the linear regime, i.e., $ \delta \ll 1 $. If the dark energy is clustered, their perturbation would influence the evolution of matter density contrast [45-48]. Thus, it shows that the anisotropic stress of the dark energy fluid has been shown to be an important discriminator between modified gravity and dark energy models. The authors also studied that anisotropic stress affects the weak lensing and galaxy power spectrum.

      According to the relation between scale factor and redshift, we can transfer the derivative of density contrast $ \delta $ over cosmic time t into derivative over redshift z. Hubble parameter in Eq. (15) thus can be expressed as [49, 50]

      $ E^2(z) = 3 \Omega_{m0} \frac{(1+z)^2}{\delta '(z) ^2 } \int_z^{\infty} \frac{\delta}{1+z} (-\delta ') {\rm d} z . $

      (16)

      We find that the Hubble parameter $ E^2(z) $ tends to zero when the redshift in integral $ z \rightarrow \infty $. When the redshift $ z = 0 $, we have the initial condition

      $ 1 = \frac{3 \Omega_{m0}}{\delta '(z = 0) ^2} \int_0^{\infty} \frac{\delta}{1+z} (-\delta ') {\rm d} z . $

      (17)

      Using this initial condition, we consequently rewrite the Hubble parameter in Eq. (16) as

      $ E^2(z) = (1+z)^2 \dfrac{\delta '(z = 0) ^2}{\delta '(z) ^2 } \left[ 1 - \frac{\displaystyle\int_0^{z} \dfrac{\delta}{1+z} (-\delta ') {\rm d} z}{\displaystyle\int_0^{\infty} \dfrac{\delta}{1+z} (-\delta ') {\rm d} z} \right] . $

      (18)

      Observationally, the perturbation $ \delta(z) $ cannot be directly measured by current cosmological surveys, but can be provided by a related observation. It is the growth rate measurement $ f\sigma_8 $ from RSD. Here, the function f is growth rate, which is defined by the derivative of the logarithm of perturbation $ \delta $ with respect to logarithm of the cosmic scale

      $ f \equiv \frac{{\rm d} \, {\rm{ln}} \delta}{{\rm d} \, {\rm{ln}} a} = -(1+z) \frac{{\rm d} \, {\rm{ln}} \delta}{{\rm d} \, z} = -(1+z) \frac{\delta '}{\delta} . $

      (19)

      While the function

      $ \sigma_8 (z) = \sigma_8 (z = 0) \frac{\delta (z)}{\delta (z = 0)} $

      (20)

      is the linear theory root-mean-square mass fluctuation within a sphere of radius $ 8h^{-1} $ Mpc. According to above two definitions, their combination is written as

      $ f \sigma_8 = -\frac{\sigma_8 (z = 0)}{\delta (z = 0)} (1+z) \delta ' , $

      (21)

      which is called the growth rate of structure. Therefore, we can obtain

      $ \delta ' = - \frac{\delta (z = 0)}{\sigma_8 (z = 0)} \frac{f \sigma_8}{1+z} . $

      (22)

      Obviously, we can reconstruct the derivative $ \delta ' $ of the perturbation via the observational RSD data $ f \sigma_8 $. Taking an integral to the two sides of Eq. (22) over redshift, we have

      $ \delta = \delta (z = 0) - \frac{\delta (z = 0)}{\sigma_8 (z = 0)} \int_0^{z} \frac{f \sigma_8}{1+z} {\rm d}z . $

      (23)

      For the constant $ \delta (z = 0) $, we usually consider it as the normalization value $ \delta (z = 0) = 1 $ [32]. For the other constant, we consider it as $ \sigma_8 (z = 0) = 0.8159 $ [6].

      Generally, to obtain the goal function $ g(z) $, the parametrization constraint usually restrict a prior template on it. Different from this method, the GP technique is model-independent, which does not rely on any particular dynamical parametrization. It only needs a probability prior on the goal function $ g(z) $. In our present paper, assuming each observational distance D obeys a Gaussian distribution with mean and variance, the posterior distribution of all observed distance D would obey the joint Gaussian distribution. In this process, we note that covariance function $ k(z, \tilde{z}) $ is a key ingredient. It correlates the distance $ D(z) $ at different points z and $ \tilde{z} $. Commonly, we have several types on the covariance function $ k(z, \tilde{z}) $. Most of them are associated with two hyperparameters $ \sigma_f $ and $ \ell $ which can be determined by the observational data via a marginal likelihood. By training covariance function, we can extend the distance $ D(z) $ to more redshift points. If we want to reconstruct the goal function $ g(z) $, such as w of the dark energy, we should use the relation between the and distance D. Due to this method is model-independent, it has been widely applied to reconstruct the dark energy EoS [22], or to test the concordance model [23, 24].

      In the GP method, many types of the covariance function $ k(z, \tilde{z}) $ are available. In the present paper, we adopt the most commonly used form, squared exponential

      $ k(z, \tilde{z}) = \sigma_f^2 \exp \left[\frac{-|z-\tilde{z}|^2}{2 \ell^2} \right] . $

      (24)

      With the chosen covariance function, the scalar field can be reconstructed. We modify the package GaPP, which is publicly available in Ref. [22]. We also recommend this paper in which more details on the GP method can be found.

      In above work, we introduce the background evolution of scalar field. For a complete study, the perturbation of scalar field has been made extensive research. In the conformal Newtonian gauge, the perturbed metric is

      $ {\rm d}s^2 = a^2(\tau) [(1 + 2\Psi){\rm d}\tau ^2 -(1 -2\Phi){\rm d}x^i {\rm d}x_i] . $

      (25)

      For the quintom field, the perturbation equation has been obtained [51]

      $\begin{aligned}[b] \dot{\delta}_i =& -(1 + w_i) (\theta_i - 3\dot{\Phi}) - 3{\cal{H}}(1-w_i) \delta_i \\ &- 3{\cal{H}} \frac{\dot{w}_i + 3{\cal{H}}(1-w_i^2)}{k^2} \theta_i , \end{aligned}$

      (26)

      $ \dot{\theta}_i = 2{\cal{H}} \theta_i + \frac{k^2}{1 + w_i} \delta_i + k^2 \Psi , $

      (27)

      where the subscript i respectively denotes the quintessence field and phantom field. The function $ \theta_i $ is defined as $ \theta_i = (k^2 / \dot{\phi}_i) \delta \phi_i $. We can refer to the work by Gong-Bo Zhao et al. [52] and Yi-Fu Cai et al. [53]. In Ref. [52], they considered a potential $ V(\phi) = \dfrac{1}{2} m_i^2 \phi_i^2 $. The authors studied the radiation-dominated period and matter-dominated era, respectively. Using the specific scalar factor $ a = A \tau $ and Hubble parameter $ {\cal{H}} = 1/\tau $, they obtained the solution of scalar field $ \phi_1 $ and $ \phi_2 $. However, we note that the potential V, scalar factor a and Hubble parameter $ {\cal{H}} $ in this paper are model-dependent. It is difficult to reconstruct the potential V and scalar field $ \phi $ without an artificial perturbation $ \Psi $ and $ \Phi $. But we also noticed that a model-independent reconstruction of $ f(T) $ gravity has been performed via the Gaussian Processes [54, 55]. In our future work, we also would like to perform a further analysis on the quintom scalar field under the reasonable assumptions and approximations.

    III.   OBSERVATIONAL DATA
    • In this section, we give an introduction on the related observational data.

      For the supernova data, they are from the joint light-curve analysis (JLA) datasets issued by the SDSS-II and SNLS surveys [56]. For these JLA samples, their redshift have a wide span of $ 0.01 <z <1.3 $. These samples contain a total of 740 SNIa data points. They include three-season data from SDSS-II ($ 0.05 < z <0.4 $), three-year data from SNLS ($ 0.2 < z <1 $), HST data ($ 0.8 < z <1.4 $), and several low-redshift samples ($ z <0.1 $). For these supernova samples, the data are usually presented in tabular form, including distance modulus and errors. For each SNIa, its distance modulus is given by

      $ \mu_{\rm{SN}} = m^{\star}_{B}+\alpha\cdot X_{1}-\beta\cdot {\cal{C}}-M_{B}\;, $

      (28)

      where $ m^{\star}_{B} $ is the observed peak magnitude in rest frame B band. The parameter $ X_{1} $ is the time stretching of light-curve. And parameter $ {\cal{C}} $ describes the supernova color at maximum brightness. The last parameter $ M_{B} $ is absolute B-band magnitude, which is assumed to be related to the host stellar mass ($ M_{\rm{stellar}} $) by a simple step function [56]:

      $ M_{B} = \left\lbrace \begin{array}{l} M^{1}_{B}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; {\rm{for}}\; \; \; M_{\rm{stellar}}<10^{10}M_{\odot},\\ M^{1}_{B}+\Delta_{M}\; \; \; \; \; {\rm{otherwise}}. \\ \end{array} \right. $

      (29)

      Notice that $ \alpha $, $ \beta $, $ M^{1}_{B} $, and $ \Delta_{M} $ are nuisance parameters in our calculation. They should be determined simultaneously with other cosmological parameters.

      To determine the nuisance parameters, the observed data are usually fit in a $ \Lambda $CDM cosmology [27, 56]. In the calculation, the full covariance matrix Cov of the JLA sample is defined by

      $ {\bf{Cov}} = {\bf{D}}_{\rm{stat}}+{\bf{C}}_{\rm{stat}}+{\bf{C}}_{\rm{sys}}\;. $

      (30)

      Here matrix $ {\bf{D}}_{\rm{stat}} $ is the diagonal part of the statistical uncertainty. It can be given by

      $ \begin{aligned}[b] ({\bf{D}}_{\rm{stat}})_{ii} =& \sigma^{2}_{m_{B},i}+\alpha^{2}\sigma^{2}_{X_{1},i}+\beta^{2}\sigma^{2}_{{\cal{C}},i} +2\alpha C_{m_{B}\,X_{1},\,i}-2\beta C_{m_{B}\,{\cal{C}},\,i}\\ &-2\alpha\beta C_{X_{1}\,{\cal{C}},\,i}+\sigma^{2}_{\rm{lens}}+\left(\frac{5\sigma_{z,i}}{z_{i}\ln 10}\right)^{2}+\sigma^{2}_{\rm{coh}}\; . \end{aligned} $

      (31)

      In the first line, $ \sigma_{{m_{B}},i} $ describes the standard errors of the peak magnitude. The $ \sigma_{X_{1},i} $, and $ \sigma_{{{\cal{C}}},i} $ are standard errors of above light-curve parameters $ X_1 $ and $ {\cal{C}} $, respectively. In the second line, the terms $ C_{m_{B}\,X_{1},\,i},\;C_{m_{B}\,{\cal{C}},\,i} $, and $ C_{X_{1}\,{\cal{C}},\,i} $ respectively denote the covariances among observed quantities $ m_{B},\;X_{1},\;{\cal{C}} $ for the i-th SN. In the last line, the term $ \sigma^{2}_{\rm{lens}} $ stands for the variation of magnitudes caused by the gravitational lensing. The second term stands for the uncertainty in cosmological redshift produced by peculiar velocities. The term $ \sigma^{2}_{\rm{coh}} $ is the intrinsic variation in SN magnitude. In Eq. (30), the $ {\bf{C}}_{\rm{stat}} $ and $ {\bf{C}}_{\rm{sys}} $ are respectively the statistical matrices and systematic covariance matrices. They can be given by

      $ {\bf{C}}_{\rm{stat}}+{\bf{C}}_{\rm{sys}} = V_{0}+\alpha^{2}V_{a}+\beta^{2}V_{b} +2\alpha V_{0a}-2\beta V_{0b}-2\alpha \beta V_{ab}\;, $

      (32)

      where $ V_{0} $, $ V_{a} $, $ V_{b} $, $ V_{0a} $, $ V_{0b} $, and $ V_{ab} $ are related matrices which can refer to Ref. [56]. Because of the degeneracy between Hubble constant $ H_{0} $ and parameter $ M_{B} $ in constructing the Hubble diagram, we consider their effects on the scalar field reconstruction in the present paper. According to the investigation in Ref. [27], we respectively have $ (\alpha,\beta,M^{1}_{B},\Delta_{M}) = (0.14\pm0.01,3.10\pm0.09, -19.08\pm 0.02, -0.07\pm0.03) $ in the prior of $ H_{0} = 69.6 \pm 0.7 $ km $ {\rm{s}}^{-1} $ $ {\rm{Mpc}}^{-1} $, and $ (\alpha,\beta,M^{1}_{B},\Delta_{M}) = (0.14\pm0.01,3.11\pm0.09,-19.01\pm0.02, -0.07\pm0.03) $ in the Gaussian prior of $ H_{0} = 73.24 \pm 1.74 $ km $ {\rm{s}}^{-1} $ $ {\rm{Mpc}}^{-1} $. In addition, we should take into account the theoretical initial conditions $ D(z = 0) = 0 $ and $ D'(z = 0) = 1 $ in the related calculation.

      For the $ H(z) $ data, direct products cannot be obtained from a tailored telescope. But two ways are available to acquire them. The one is called cosmic chronometer, which is the calculation of differential ages of galaxies [57-59]. The other is from the BAO peaks. To be specific, we can deduce it from the galaxy power spectrum [60, 61] or from the Ly$ \alpha $ forest of QSOs [62]. For the latter method, an underlying cosmology increases its model-dependence in the calculation of the sound horizon. In this paper, we therefore, use the 30 cosmic chronometer data points. We compiled them in our recent work [25]. Considering the uncertainty of Hubble constant, the uncertainty of $ E(z) $ can be determined as

      $ \sigma_E^2 = \frac{\sigma_H^2}{H_0^2} + \frac{H^2}{H_0^4} \sigma_{H_0}^2 . $

      (33)

      Using the prior of $ H_0 $, namely, $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $ with 2.4% uncertainty [63] and $ H_0 = 69.6 \pm 0.7 $ km s$ ^{-1} $ Mpc$ ^{-1} $ from the 1% determination [64], uncertainty of the dimensionless Hubble parameter can be calculated. Different from most of previous work, we use them combining with the supernova data, not the usage of $ H(z) $ data alone. That is, the Hubble parameter is used as a derivative of distance D $ D' = \dfrac{1}{E(z)}. $ Meanwhile, we should consider the $ E(z = 0) = 1 $ as an initial condition in our calculation.

      For the RSD data, they can be realized from the galaxy distribution observation. Concretely, they generate from an effect. In the galaxy distribution measurement, the observed distance of the galaxy is different from its true distance in redshift space. This is because velocities in the overdensities deviate from cosmic smooth Hubble flow expansion. We know that cosmic structure growth is correlated with the anisotropy in the clustering of galaxies. From the General Relativity, it can produce an anisotropy. Moreover, smaller deviation from this theory indicates a smaller anisotropic distortion. In virtue of above distinct superiority, the RSD data has a very promising prospect in distinguishing the cosmological models. Because of its sensibility, a similar background evolution yet has a very distinct growth of structure in different cosmological models. Till now, a lot of literatures have used the RSD data to study cosmology. In the present paper, RSD data we utilize are the most recent products from 6dFGS, 6dFGRS, SDSS MGS, SDSS LRG, GAMA, BOSS DR12, WiggleZ, VIPERS, FastSound, BOSS DR14 redshift surveys. Because some data are cosmology-dependent and the covariance for different datasets is unknown [65], we use the compilation from Planck Collaboration [66]. For these data, the BOSS DR12 and WiggleZ have full covariance matrix, including systematic errors.

    IV.   RESULT
    • From above scalar fields and potentials introduction in Section II, we note that their determinations are dependent of matter density parameter $ \Omega_{m0} $, Hubble parameter $ E(z) $ and its derivative $ E'(z) $. For the matter density parameter, we consider a moderate estimation $ \Omega_{m0} = 0.279 \pm 0.025 $ [67]. For the function $ V(\phi) $ that people have been pursuing diligently, a lot of models have been proposed in the past few decades, as introduced in above section. In this paper, we first reconstruct the scalar field $ \phi (z) $ over redshift z and potential $ V(z) $ using the GP method; then we try to fit the function $ V(\phi) $ using their mean values. Due to the model-independence of GP method, we think that it can give a more scientific test on the scalar field. We therefore can expect a better understanding about the dynamics of dark energy.

    • A.   Reconstruction from the JLA and $ H(z) $ data

    • To test the effect of Hubble constant on corresponding reconstructions, we report the results in two subsections.

      $ 1.\;\; H_0=73.24 \pm 1.74 \; {\rm{km}} \;{\rm s} ^{-1} {\rm {Mpc}} ^{-1} $

      In Figs. 1 and 2, we plot the derivative of scalar field $ \dot{\phi}^2 $ and potential V in the quintessence field and tachyon field with $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $. Theoretically, the function $ \dot{\phi}^2 $ should be $ \dot{\phi}^2 \geqslant 0 $. Indeed, the figures show that $ \dot{\phi}^2 $ in these two fields are positive within 68% confidence level at low redshift. However, it turns to negative at high redshift. Moreover, it is difficult to determine the sign of this function within 95% confidence level. For the potential, V in these two fields both increase softly first and then increase sharply at redshift $ z \sim 1.0 $. The initial value is $ V_0 = 0.70 $. In Ref. [42], the authors found that tachyon models considered in their paper present some quite similar phenomenology comparing with the canonical quintessence models. Comparing the reconstructions in these two figures, they are really very similar. Let's return to the function $ \dot{\phi}^2 $ again. The vacillating $ \dot{\phi}^2 $ indicates that the single quintessence field, phantom field or tachyon field are all difficult to be favored by the data. Therefore, we cannot depict the function $ V(\phi) $ using a single field. However, because the function $ \dot{\phi}^2 $ in quintessence field and phantom field are opposite, we can also understand that it keeps switching between the two fields. So, the quintom field proposed by Feng et al. [16] may be a better building on the scalar field. Now we treat the GP reconstruction in Fig. 1 as a quintom field, a combination of quintessence and phantom field. From their mean values, we find that the function $ \dot{\phi}^2 $ changes from $ \dot{\phi}^2 < 0 $ to $ \dot{\phi}^2 > 0 $, which indicates that the scalar field changes from phantom field to quintessence field, with the cosmic evolution. Solving the Eq. (10), we can acquire the scalar field $ \phi (z) $ over the redshift z, as shown in Fig. 3. We find that scalar field $ \phi (z) $ increases with the increasing redshift. Using the mean values of $ \phi (z) $ and $ V (z) $, we eventually obtain the potential $ V (\phi) $ to be a function of field $ \phi $. The potential is also an increasing function. For field $ \phi < 0.23 $, the quintessence field dominates the evolution of the universe. For field $ \phi > 0.23 $, the phantom field plays a dominant role. We treat this transformation as a quintom scalar field. We fit this reconstruction with high R-square $ = 0.9998 $, and find that the quintessence field obeys a 2-order Exponential function, $ V(\phi_q) = 0.7002 e^{0.5978 \phi_q} + 8.449 \times 10^{-6} e^{45.4 \phi_q} $, and phantom field satisfies a 2-order Gaussian function $V(\phi_p) = 1.501 e^{- \left(\frac{\phi_p - 0.2559}{0.03301}\right)^2 } + 0.2606 e^{- \left(\frac{\phi_p - 0.2307}{0.01163}\right)^2 }$, where $ \phi_q $ and $ \phi_p $ are the quintessence field and phantom field, respectively. The fitted potential indicates that each potential should satisfy a double function with high probability. We should reiterate that this fit is performed via their mean values. In the past, many parameterizations, such as the power-law, single exponential, etc. were proposed. With the improvement of observation accuracy, the fitted potential can provide a more important reference.

      Figure 1.  (color online) GP reconstruction in the quintessence field for JLA and $ H(z) $ data with Hubble constant $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      Figure 2.  (color online) GP reconstruction in the tachyon field for JLA and $ H(z) $ data with Hubble constant $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      Figure 3.  (color online) The field $ \phi (z) $ and potential $ V(\phi) $ from their mean values for JLA and $ H(z) $ data with Hubble constant $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      $ 2.\;\; H_0=69.6 \pm 0.7 \; {\rm {km}} \;{\rm s} ^{-1} {\rm {Mpc}} ^{-1} $

      In this prior, the JLA and $ H(z) $ data present a slightly different reconstruction on these scalar fields.

      In Fig. 4, we plot the reconstructions in quintessence field. Firstly, compared with the reconstructions in Fig. 1, the function $ \dot{\phi}^2 $ also increases first and then decreases with the increasing redshift. Secondly, we find that mean values of the function $ \dot{\phi}^2 >0 $, which means that the quintessence scalar field is favored to a certain degree. This situation is different from above reconstruction. However, what we should pay special attention to is that it still cannot prevent the derivative $ \dot{\phi}^2 < 0 $ at higher redshift within 68% confidence level. That is, considering the uncertainties of $ \dot{\phi}^2 $, the quintom field is still a favourite model. This result is consistent with the reconstruction in $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $. Last, the reconstruction of $ V(z) $ shows that the data present an increasing potential. Especially for redshift $ z \gtrsim 1 $, it increases sharply. At redshift $ z = 0 $, we have a model-independent estimation $ V_0 = 0.71 $, which is similar as above reconstruction.

      Figure 4.  (color online) GP reconstruction in the quintessence field for JLA and $ H(z) $ data with Hubble constant $ H_0 = 69.6 \pm 0.7 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      In Fig. 5, we plot the reconstructions in tachyon field. We find that the data present a similar reconstruction as the quintessence field. That is, mean values of the function $ \dot{\phi}^2 $ are also positive. A slight difference is that the potential $ V(z) $ decreases first and then increases. Within 68% confidence level, the function $ \dot{\phi}^2 < 0 $ is still supported by the data. So, the tachyon field cannot be convincingly favored by the data.

      Figure 5.  (color online) GP reconstruction in the tachyon field for JLA and $ H(z) $ data with Hubble constant $ H_0 = 69.6 \pm 0.7 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      In Fig. 6, we plot the scalar field $ \phi (z) $ and potential $ V(\phi) $ using their mean values. This figure shows that the field $ \phi (z) $ also increases in the two models. But in the middle redshift, they evolve slightly differently. For the initial value of potential, the data present a same $ V_0 = 0.71 $ as in the quintessence field. For the potential $ V(\phi) $, they are same as each other at field $ \phi <0.10 $. However, in the middle region, they present a significant difference. Therefore, the potential $ V(\phi) $ in these two fields may reflect different models.

      Figure 6.  (color online) The field $ \phi (z) $ and potential $ V(\phi) $ from their mean values for JLA and $ H(z) $ data with Hubble constant $ H_0 = 69.6 \pm 0.7 $ km s$ ^{-1} $ Mpc$ ^{-1} $.

      Now, we fit the function $ V(\phi) $ in different fields, and present the list in Table 1. We find that $ V(\phi) $ in the quintessence field and tachyon field are really different. For the quintessence field, the mean values favor a double Exponential function, $ V(\phi) = 0.7177 e^{-0.341 \phi} + 3.102 \times 10^{-4} e^{27.23 \phi} $. While for the tachyon field, it needs a more complex double Gaussian function.

      JLA+$ H(z) $: $ (H_0=73.24 \pm 1.74 ) $ quintom
      Exponential: $ V(\phi_q) = 0.7002 e^{0.5978 \phi_q} + 8.449 \times 10^{-6} e^{45.4 \phi_q} $
      Gaussian : $V(\phi_p) = 1.501 e^{- \left(\frac{\phi_p - 0.2559}{0.03301}\right)^2 } + 0.2606 e^{- \left(\frac{\phi_p - 0.2307}{0.01163}\right)^2 }$
      JLA+$ H(z) $: $ (H_0=69.6 \pm 0.7) $ quintessence
      Exponential: $ V(\phi) = 0.7177 e^{-0.341 \phi} + 3.102 \times 10^{-4} e^{27.23 \phi} $
      JLA+$ H(z) $: $ (H_0=69.6 \pm 0.7) $ tachyon
      Gaussian : $V(\phi) = 1.192 \times 10^{3} e^{- \left(\frac{\phi - 0.6406}{0.1324}\right)^2 } + 0.7157 e^{- \left(\frac{\phi - 0.0302}{0.3793}\right)^2 }$
      RSD: quintom
      Exponential: $ V(\phi_q) = 0.4981 e^{3.328 \phi_q} + 6.073 \times 10^{-7} e^{39.42 \phi_q} $
      Gaussian : $V(\phi_p) = 1.951 e^{- \left(\frac{\phi_p - 0.4315}{0.1615}\right)^2 } + 0.4538 e^{- \left(\frac{\phi_p - 0.3107}{0.06524}\right)^2 }$

      Table 1.  Function $ V(\phi) $ obtained by their mean values in different scalar fields for different observational data.

    • B.   Reconstruction from the RSD data

    • In Fig. 7, reconstruction in quintessence field is plotted. We find that these results are similar as the reconstruction for JLA and $ H(z) $ data. Firstly, function $ \dot{\phi}^2 $ from the background data and RSD data both present a transformation from positive to negative, namely, a direct transformation from quintessence field to phantom field. However, difference from the background data is that the $ \dot{\phi}^2 $ from the RSD data changes more dramatically. Moreover, $ \dot{\phi}^2 $ at high redshift from the RSD data are completely negative within 68% confidence level. It indicates that the RSD data are more potential to support the quintom model. Secondly, for the potential V, it is also an increasing function, but increases slower than that from the background data. The initial value of potential is $ V_0 = 0.50 $, which is also different from $ V_0 = 0.70 $ by the background data. Therefore, we think that the RSD data may present a different scalar field model.

      Figure 7.  (color online) GP reconstruction in the quintessence field for RSD data.

      In Fig. 8, we plot the reconstruction in tachyon field. For the mean values of function $ \dot{\phi}^2 $, they also transfer from positive to negative, which is similar as Fig. 2. Considering their uncertainties, it cannot ensure the function $ \dot{\phi}^2 > 0 $ at high redshift within 68% confidence level. Therefore, we think that the tachyon field cannot be convincingly favored by the data. This determination is same as the results for JLA and $ H(z) $ data. For the potential V, it is also an increasing function. But considering their errors, the potential V at high redshift is negative, which is invalid from the definition in Eq. (8). In short, we think that the tachyon field is at a disadvantage to describe the cosmic evolution.

      Figure 8.  (color online) GP reconstruction in the tachyon field for RSD data.

      In Fig. 9, we plot the field $ \phi (z) $ and function $ V(\phi) $ using their mean values. As pointed out above, the function $ \dot{\phi}^2 $ in Fig. 7 cannot fulfill $ \dot{\phi}^2 > 0 $ at all redshift. It transfers from positive to negative at redshift $ z \sim 0.90 $. Scalar fields also naturally change from quintessence field to phantom field, as shown in this figure. Similar as above reconstruction in the background data, the field $ \phi (z) $ is also a monotone increasing function. To describe the reconstructed scalar field, we understand it as a quintom field, which changes from phantom field to quintessence field with the cosmic evolution. The picture shows that it is also a complicated model similar as Fig. 3. From the Table 1, we find that $ V(\phi_q) $ should satisfy a double Exponential function, and $ V(\phi_p) $ a double Gaussian function, which is fully consistent with the reconstruction from background data.

      Figure 9.  (color online) The field $ \phi (z) $ and potential $ V(\phi) $ from their mean values for RSD data.

    V.   CONCLUSION AND DISCUSSION
    • In this paper, a test with model-independence on the scalar field is put into operation to explore the source of dark energy, using the Gaussian processes approach. We respectively use the combination of supernova data and $ H(z) $ data, and growth rate data.

      Although we have investigated the dark energy using GP method in our previous work [32], we should emphasize that it still has important physical significance. Comparing with our previous study, this paper shows clearly which scalar field is the candidate of dark energy. From the reconstructing result, we find that it is not the quintessence field or the phantom field. It's probably another substance, the quintom field. Moreover, the fit of potential $ V(\phi) $ is so far beyond our imagination. It can provide an important reference on the scalar field study.

      In past several years, the scalar field has been studied via many parameterizations. Focus of attention was which template is the best dynamical description of dark energy. Work in this paper presents a template-free analysis. We not only reconfirm that the dark energy should be dynamical, but also reconstruct the potential over the scalar field.

      From the background data, we find that they do not favor a single quintessence field or phantom field or tachyon field in the prior of $ H_0 = 73.24 \pm 1.74 $ km s$ ^{-1} $ Mpc$ ^{-1} $. Their mean values indicate that they favor a quintom field, which is a transformation between phantom field and quintessence field. The fitted potential $ V(\phi) $ in quintessence field is a double Exponential function, $ V(\phi) $ in phantom field is a double Gaussian function, as shown in Table 1. We also test the effect from Hubble constant, and find that $ H_0 $ has a marked influence on the reconstruction. When considering their uncertainties, the reconstructions also favor the quintom field.

      Our study also solves another puzzle. In previous work [42], it was found that the tachyon models present quite analogous phenomenology to canonical quintessence models. We find that they are really similar at low redshft (or small $ \phi $). However, they also present an unnegligible difference at middle redshift, as shown in Fig. 6. Moreover, the background data and RSD data both reveal that the tachyon field is at a disadvantage to describe the cosmic acceleration.

      From the RSD data, they also prefer to a quintom field. This determination is identical with above analysis from the background data. Moreover, the RSD data are more potential to support the quintom model within 68% confidence level. Their mean values show that the potential $ V(\phi) $ is fully consistent with the reconstruction from background data.

      Argument about the dynamics of dark energy has been going on around whether it evolved or not. Our analysis highly reveals that the dark energy should be dynamical, regardless of it is from background data, i.e., supernova and $ H(z) $ data or perturbation data, i.e., RSD data. Moreover, the corresponding reconstructions favor the quintom field. In recent work by Zhao et al. [7], they investigated the Kullback–Leibler divergence using the latest data. The data include CMB temperature and polarization anisotropy spectra, supernova, BAO from the clustering of galaxies and from the Lyman-$ \alpha $ forest, Hubble constant and $ H(z) $. The study reveals that the dynamical dark energy can moderate the Hubble constant tension. Moreover, it is preferred at a 3.5$ \sigma $ C.L. In addition, the forthcoming dark energy survey DESI++ would be capable of providing a decisive Bayesian evidence. In our future work, we also would like to invest more observational data on the study of scalar field, to make a clearer analysis on cosmic dynamics.

      Another point we should emphasize is the importance of Hubble constant. We note that it has a notable influence on the determination of dark energy dynamics. The tension in $ H_0 $ has aroused great concern. Some Refs. [68, 69] think that it may be a signature of new physics. Till now, its measurement window has been opened from traditional Cepheids, Tip of the Red Giant Branch, SNIa, Surface Brightness Fluctuations, Masers, and Gravitational Lens Time Delays, to fashionable Gravitational-wave [70]. The detection of GW170817 has detected the merger of a binary neutron-star system with a strong signal. The identification of its host galaxy has obtained a completely independent and consistent determination with existing measurements [71]. Moreover, it also can be measured with neutron star black hole mergers from advanced LIGO and Virgo [72]. The future multi-messenger astronomy will enable the $ H_0 $ and cosmic dynamics to be constrained to high precision.

    ACKNOWLEDGMENTS
    • We thank the anonymous referee whose suggestions greatly helped us improve this paper. M.-J. Zhang would like to thank Shulei Ni, De-Liang Wu, Hua Zhai and Xiao Wang for valuable discussion.

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