⇦Chinese Physics Letters, 2017, Vol. 34, No. 1, Article code 010401 Effect of Phantom Dark Energy on Holographic Thermalization * Xiao-Xiong Zeng(曽晓雄)1,2, Xin-Yun Hu(胡馨匀)3**, Li-Fang Li(李丽仿)4 Affiliations 1School of Material Science and Engineering, Chongqing Jiaotong University, Chongqing 400074 2Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190 3School of Economics and Management, Chongqing Jiaotong University, Chongqing 400074 4State Key Laboratory of Space Weather, Center for Space Science and Applied Research, Chinese Academy of Sciences, Beijing 100190 Received 20 October 2016 *Supported by the National Natural Science Foundation of China under Grant Nos 11405016 and 11365008, and the China Postdoctoral Science Foundation under Grant No 2016M590138.
**Corresponding author. Email: xxzengphysics@163.com Citation Text: Zeng X X, Hu X Y and Li L F 2017 Chin. Phys. Lett. 34 010401 Abstract Holographic thermalization for a black hole surrounded by phantom dark energy is probed. The result shows that the smaller the phantom dark energy parameter is, the easier the is plasma to thermalize as the chemical potential is fixed, the larger the chemical potential is, and the harder the plasma is to thermalize as the dark energy parameter is fixed. The thermalization velocity and thermalization acceleration are presented by fitting the thermalization curves. DOI:10.1088/0256-307X/34/1/010401 PACS:04.70.-s, 04.70.Dy, 04.25.dc © 2017 Chinese Physics Society Article Text In recent years, more and more theoretical physicists have paid attention to the applications of the AdS/CFT correspondence.[1] It can not only check the effectiveness of this correspondence indirectly, but also can provide a method to deal with some problems in strongly coupled systems.[2-7] One of the strongly coupled systems is the quark gluon plasma which is produced in heavy ion colliders such as the RHIC and LHC. The properties of the quark gluon plasma have been investigated extensively and we know that it behaves as an ideal fluid with a very small shear viscosity over the entropy density ratio.[8] However, the process of formation of quark gluon plasma after a heavy ion collision, often referred to as thermalization, has not been well understood until now. Recently some physicists have devoted themselves to addressing this problem from the viewpoint of holography, which is called holographic thermalization. The most prominent property for the thermalization is that it is a non-equilibrium process. From the AdS/CFT correspondence, we know that the initial state before the thermalization is dual to a pure AdS, and the last state after the thermalization is dual to a stationary black hole. Therefore, to describe the thermalization process, one should construct a dynamical background in the bulk, which can be described as black hole formation or black hole merger. There have been some models to study the far-from-equilibrium thermalization behaviors.[9-17] A slightly different and simpler model on this topic was put forward by Balasubramanian et al.,[18,19] where the dynamical background was treated as the gravitational collapse of a shell of dust. They claimed that the thermalization process can be probed by the equal-time two-point correlation functions of local gauge invariant operators, expectation values of Wilson loop operators, and entanglement entropy, which correspond in the gravity side to minimal lengths, areas, and volumes in AdS space, respectively. They found that the thermalization time is closer to the experimental data produced in RHIC and LHC. In addition, they found that the thermalization is a top-down process, in contrast to the predictions of bottom-up thermalization from perturbative approaches,[20] and there is a slight delay in the onset of thermalization. Now such a model has been generalized to the bulk geometry with electrostatic potential,[21-23] high curvature corrections,[24-27] and some other gravity models.[28-42] In this Letter, we intend to study holographic thermalization in the bulk surrounded by phantom dark energy. The phantom dark energy is the scalar field with a negative kinetic term with an equation of state parameter $\omega < -1$. As is already known, the presence of cosmological phantom fields continues to receive support from both collected observational data[43] and theoretical models.[44] All these programs have pointed out an accelerated expansion of the universe, dominated by an exotic fluid of negative pressure. Furthermore, there is evidence suggesting that the exotic fluid could be of phantom nature.[45,46] Since then, an interest in phantom fields has grown and resulted in many phantom black hole solutions.[47,48] In recent years, many issues pertaining to phantom black holes, such as thermodynamic stability[49] and light paths[50,51] have been dealt with. In this work, we intend to study the effect of the phantom dark energy parameter on the holographic thermalization. The solution of the phantom Reissner–Nordstrom AdS black brane is $$ ds^{2}=-F(r)dt^{2}+F^{-1}(r)dr^{2}+r^{2}dx_i^{2},~~ \tag {1} $$ in which $i=1$ and 2, and $$ F(r)=-\frac{2M}{r}+\eta \frac{q^2}{r^2}+\frac{r^2}{L^2},~~ \tag {2} $$ where $L$ is the radius of the AdS, $\eta$ is the phantom dark energy parameter. For $\eta=1$, the solution represents the Reissner–Nordstrom-AdS black hole. The chemical potential at infinity is $$ u=\frac{q}{r_{\rm h}},~~ \tag {3} $$ where $r_{\rm h}$ is the event horizon determined by $F(r_{\rm h})=0$. According to the definition of the surface gravity, we can also obtain the Hawking temperature emitting from the black brane, $$ T=\frac{\kappa}{2\pi}=\frac{1}{4\pi}\Big(\frac{2M}{r^2_{\rm h}}-\eta \frac{2q^2}{r_{\rm h}^3}+\frac{2r_{\rm h}}{L^2}\Big),~~ \tag {4} $$ which can be viewed in the framework of AdS/CFT as the equilibrium temperature of the dual field theory living on the boundary. With the Eddington–Finkelstein coordinate $dv=dt+\frac{dr}{F(r)}$, Eq. (1) changes into $$ ds^2=\frac{1}{z^2}[-H(z) d{v}^2-2 dzdv+dx_i^2],~~ \tag {5} $$ where $z=\frac{L^2}{r}$, and $$ H(z)=1-2 M z^3+\eta q^2 z^4,~~ \tag {6} $$ in which we have set $L=1$. Treating the mass parameter in Eq. (6) as an arbitrary function of $v$, Eq. (5) can be regarded as a gravitational collapse solution surrounded by phantom dark energy.[18,19,21,22] As can be seen, such a metric is sourced by the null dust with the energy momentum tensor flux and gauge flux[21,22] $$\begin{align} 8\pi G T_{\mu\nu}^{\rm matter}=\,&z^{2}[\dot{M}(v)-2zq(v)\dot{q}(v)]\delta_{\mu v}\delta_{\nu v},\\ 8\pi G J^{\mu}_{\rm matter}=\,&z^4 \dot{q}(v)\delta^{\mu z},~~ \tag {7} \end{align} $$ where the dot stands for derivative with respect to coordinate $v$; $M(v)$ and $q(v)$ are the mass and charge of a collapsing black brane, respectively. To describe the thermalization more comprehensively, we should have an initial state and an equilibrium state, which is dual to a pure AdS and a stationary black brane, respectively. Recent investigations show that as the mass $M(v)$ and charge $q(v)$ are written as the smooth functions[18,19,21,22] $$\begin{align} M(v)=\,&\frac{M}{2}\Big( 1+\tanh \frac{v}{v_0}\Big),~~ \tag {8} \end{align} $$ $$\begin{align} q(v)=\,&\frac{q}{2}\Big( 1+\tanh \frac{v}{v_0}\Big),~~ \tag {9} \end{align} $$ one can construct such a gravity model, where $v_0$ represents a finite shell thickness. In the limit $v\rightarrow -\infty$, the mass vanishes and the background in Eq. (5) corresponds to a pure AdS space while in the limit $v\rightarrow \infty$, the mass is a constant and the background corresponds to a phantom Reissner–Nordstrom-AdS black brane. Thus for different values of the time $v$, the background in Eq. (5) stands for different stages of gravitational collapse, which represents different stages of the thermalization process in the dual conformal field theory. As the gravity model describing the thermalization is constructed, we will choose proper observables to probe it. In this work, we use the Wilson loop to probe the thermalization. According to the AdS/CFT correspondence, the expectation value of the Wilson loop can be expressed approximated as[19,52] $$ \langle W(C)\rangle \approx e^{-\frac{\tilde{A}_{\rm ren}({\it \Sigma})}{2\pi\alpha'}},~~ \tag {10} $$ where $C$ is the closed contour, ${\it \Sigma}$ is the minimal bulk surface ending on $C$ with $A_{\rm ren}$ being its renormalized minimal area surface, and $\alpha'$ is the Regge slope parameter. Here we are focusing solely on the rectangular space-like Wilson loop. In this case, the enclosed rectangle can always be chosen to be centered at the coordinate origin and lying on the $x_1$–$x_2$ plane with the assumption that the corresponding bulk surface is invariant along the $x_2$ direction. For notational simplicity, $x_1$ and $x_2$ will be replaced by $x$ and $y$, respectively. For the Vaidya-like AdS black branes, the area of the minimal area surface can be written as $$ \tilde{A}=\int_{\frac{l}{2}}^{\frac{l}{2}}dx \frac{\sqrt{1-2z'(x)v'(x)-H(v,z) v'(x)^2}}{z(x)^2},~~ \tag {11} $$ where $l$ is the boundary septation along $x$ direction. From Eq. (11), we can obtain the motion equations of $z(x)$ and $v(x)$ as follows: $$\begin{align} 0=\,&4-4v'(x)^2H(v,z)-8v'(x)z'(x)\\ &-2z(x)v''(x)+z(x)v'(x)^2\partial_zH(v,z),~~ \tag {12} \end{align} $$ $$\begin{align} 0=\,&v'(x)z'(x)\partial_zH(v,z)+\frac{1}{2}v'(x)^2\partial_vH(v,z)\\ &+v''(x)H(v,z)+z''(x).~~ \tag {13} \end{align} $$ To solve the motion equations, we use the following boundary conditions $$ z(0)=z_{\star},~v(0)=v_{\star},~v'(0)=z'(0)=0.~~ \tag {14} $$ In addition, the area is divergent due to its contribution near the AdS boundary, to eliminate it, we should impose a cutoff near the boundary $$ z\Big(\frac{l}{2}\Big)=z_0,~v\Big(\frac{l}{2}\Big)=t_0,~~ \tag {15} $$ where $z_0$ is the IR radial cut-off, and $t_0$ is the time that the minimal area surface approaches to the boundary, which is called the thermalization time usually. As the divergent part of $\tilde{A}$ is subtracted, we can obtain the renormalized minimal area surface, $$ \tilde{A}_{\rm ren}=2\int_0^{\frac{l}{2}}dx \frac{z^2_{\star}}{z(x)^4}-\frac{2}{\tilde{z}_0}.~~ \tag {16} $$ where $\frac{2}{\tilde{z}_0}$ is the contribution of the minimal area surface in the AdS boundary.[18,19] According to the AdS/CFT correspondence, we know that the electromagnetic field in the bulk is dual to the chemical potential in the dual quantum field theory, therefore we will use the electromagnetic field defined in Eq. (3) to explore the effect of the chemical potential on the thermalization process in the AdS boundary. As Refs. [21,53], we will employ the relation $$ \frac{u}{T}=\frac{2q (1+\eta q^2)}{p (3-\eta q^2)},~~ \tag {17} $$ to check the effect of the chemical potential on the thermalization time. During the numerics, we will set $v_0=z_0=0.01$ and $r_{\rm h}=1$. For the phantom dark energy parameter, we know $\eta \leq -1$, thus we will choose $\eta=-1$, $-$2 and $-3$ in our numerical result. We will also study the case $\eta=1$, and in this case the solution stands for the Reissner–Nordstrom-AdS black brane. From Eq. (17), we know that for $\eta=1$, the chemical potential raises from $0\rightarrow \infty$ provided $q$ changes from $(0,\sqrt{3})$. However, for $\eta \leq -1$, the chemical potential is not monotonous as the charge increases, as shown in Fig. 1. For $\eta=-1$, $-$2 and $-3$, the chemical potential is monotone increasing for $0 < q < 0.49$, $0 < q < 0.35$ and $0 < q < 0.28$, respectively, and monotone decreasing for the other values. To investigate conveniently, we will choose $q=0.5$, 0.7 and 0.9 in this study. With this choice, from Fig. 1 we know that for the negative phantom dark energy parameters, as the charge increases, the chemical potential decreases, which is different from the case $\eta=1$ where the chemical potential increases.
Table 1. Change of $u/T$ to the charge $q$. The red, green and blue lines correspond to $q=-1$, $-$2 and $-$3, respectively.
$q=0.5$ $q=0.7$ $q=0.9$
$v_{\star}=-0.888$ $\eta=1$ 0.989522 0.999862 1.01398
$\eta=-1$ 0.969131 0.9597 0.947474
$\eta=-2$ 0.959202 0.94115 0.918376
$\eta=-3$ 0.949734 0.923542 0.891606
$v_{\star}=-0.555$ $\eta=1$ 1.34905 1.37703 1.41573
$\eta=-1$ 1.29452 1.26975 1.23855
$\eta=-2$ 1.26889 1.22264 1.16637
$\eta=-3$ 1.24432 1.17899 1.10331
$v_{\star}=-0.333$ $\eta=1$ 1.52816 1.56579 1.61914
$\eta=-1$ 1.45507 1.42259 1.38149
$\eta=-2$ 1.42132 1.36125 1.29001
$\eta=-3$ 1.38917 1.30584 1.21261
Since different initial times $v_{\star}$ correspond to different stages of the thermalization, we study whether the phantom dark energy coefficient and chemical potential have the same effect on the thermalization for different $v_{\star}$. The result is listed in Table 1. From Table 1, we know that as the phantom dark energy parameter decreases, the thermalization time increases. As the charge increases, the thermalization time increases for the positive dark energy parameter while it decreases for the negative parameters. With the numeric result of $z(x)$, we can further obtain the renormalized minimal area surface defined in Eq. (16). The result is shown in Figs. 2 and 3. We are interested in the $l$ independent quantity $\overline{\delta A}\equiv A_{\rm ren}/l$, $\overline{\delta A_{\rm q}}\equiv A_{\rm q}/l$, where $A_{\rm q}$ is the renormalized minimal area surface at the equilibrium state. To observe the thermalization conveniently, we plot the quantity $\overline{\delta A}-\overline{\delta A_{\rm q}}$. In this case, the thermalized state is labeled by the null point of $\overline{\delta A}-\overline{\delta A_{\rm q}}$. In each picture, the vertical axis indicates the renormalized minimal area surface while the horizontal axis indicates the thermalization time $t_0$.
cpl-34-1-010401-fig1.png
Fig. 1. Change of $u/T$ to the charge $q$. The red, green and blue lines correspond to $\eta=-1$, $-$2 and $-$3, respectively.
cpl-34-1-010401-fig2.png
Fig. 2. Thermalization of the renormalized minimal area surface for different phantom dark energy parameters with the same chemical potential at the same boundary separation $l=2.2$. The purple, red, green, and blue lines correspond to $\eta=1$, $-1$, $-2$ and $-3$, respectively.
From Fig. 2, we know that as the phantom dark energy parameter decreases, the thermalization time increases. This phenomenon is more obvious as the charge raises. Especially for the case $q=0.9$, we find that the difference of the thermalization curve for different phantom dark energy parameters is most obvious. The effect of the chemical potential on the thermalization is plotted in Fig. 3. In Fig. 3(a), we know that as the chemical potential increases, the thermalization time enhances, which is consistent with that obtained in Refs. [21–23]. In Figs. 3(b)–3(d), we find that as the charge increases, the thermalization time decreases. Note that for the case $\eta=-1$, $-2$ and $-3$, the chemical potential is monotonously decreasing as the charge increases. Thus we can also obtain that as the chemical potential increases, the thermalization time rises. Namely the chemical always delays the thermalization, though this seems different for the negative and positive parameters as the charge increases.
cpl-34-1-010401-fig3.png
Fig. 3. Thermalization of the renormalized minimal area surface for different chemical potentials with the same phantom dark energy parameter at the same boundary separation $l=2.2$. The purple, red, and green lines correspond to 0.5, 0.7 and 0.9, respectively.
cpl-34-1-010401-fig4.png
Fig. 4. Comparison of the thermalization process between the numerical curves and fitting functions.
The thermalization curves can be fitted as an analytical function with respect to the thermalization time.[25] For the case $q=0.7$, the fitting functions of the thermalization curves for $\eta=1$, $-1$, $-2$ and $-3$ can be expressed as $$\begin{align} &-0.532952+0.0142505t_0+0.276771t_0^2\\ &+0.111791t_0^3-0.260812t_0^4+0.151048t_0^5\\ &-0.0328631t_0^6-0.427766+0.0156803t_0\\ &+0.00149207t_0^2+0.442568t_0^3-0.52845 t_0^4\\ &+0.346392t_0^5-0.0937172t_0^6-0.380358\\ &+0.0222836t_0-0.18342 t_0^2+0.76406t_0^3\\ &-0.913827t_0^4+0.638737 t_0^5-0.179835t_0^6\\ &-0.335312+0.0126767t_0-0.244621t_0^2\\ &+0.73413 t_0^3-0.859683t_0^4+0.690742t_0^5\\ &-0.219645 t_0^6.~~ \tag {18} \end{align} $$ Figure 4 is the comparison of the numerical curves and fitting function curves. It is obvious that at the order of $t_0^6$, the thermalization curves can be described well by the fitting functions. With the analytical functions of the thermalization curves, we can obtain the thermalization velocity, and thermalization acceleration conveniently. From the thermalization velocity curves in Fig. 5, we find that the thermalization process is similar for different phantom dark energy parameters, namely at the middle stage of the thermalization process there is a phase transition point which divides the thermalization into an acceleration phase and a deceleration phase. This result is reasonable because the thermalization cannot always accelerate to approach an equilibrium state. The phase transition points can also be read off from the null point of the acceleration curves, which are plotted in Fig. 6. For the negative phantom dark energy parameter, as the parameter decreases, the phase transit decreases. For all the phantom dark energy parameters, the initial acceleration of renormalized minimal area surface decreases as the state parameter decreases. Correspondingly, the initial thermalization velocity of minimal area surface decreases gradually. In addition, from Figs. 5 and 6, we find that the maximum acceleration and maximum velocity increase as the phantom dark energy parameters decrease.
cpl-34-1-010401-fig5.png
Fig. 5. Thermalization velocity of the renormalized minimal area surface at $q=0.7$. The red, green, orange, and brown lines correspond to $\eta=1$, $-1$, $-2$ and $-3$, respectively.
cpl-34-1-010401-fig6.png
Fig. 6. Thermalization acceleration of the renormalized minimal area surface at $q=0.7$. The red, green, orange, and brown lines correspond to $\eta=1$, $-1$, $-2$ and $-3$, respectively.
In summary, we have constructed a gravitational collapse model which is interpolated between a pure AdS space and a phantom Reissner–Nordstrom AdS black brane, and probed the gravitational collapse behavior with the minimal area surface. According to the language of AdS/CFT correspondence, this is dual to probe the thermalization behavior of the quark gluon plasma by the expectation value of the Wilson loop in the dual conformal field theory. We find that for a fixed chemical potential, the smaller the phantom dark energy parameter is, the easier the plasma thermalizes. For a fixed dark energy parameter, the larger the chemical potential is, the harder the plasma is to thermalize. For the effect of chemical potential on the thermalization time, there are two points that should be stressed. (1) From Fig. 3, we know that as the charge rises, the thermalization time enhances for the case $\eta=1$, while it decreases for the case $\eta=-1$, $-2$ and $-3$. However, in the dual conformal field theory, the quantity which is meaningful physically is the chemical potential but not the charge. With the consideration for $q>0.5$, the chemical potential rises for $\eta=1$, while it decreases for $\eta=-1$, $-2$ and $-3$ as the charge grows. Thus in the parameter range chosen, we can obtain that the chemical potential delays the thermalization for all the phantom dark energy parameters. (2) From Fig. 1, we know that the chemical potential is not a monotone function with respect to the charge. For $0 < q < 0.49$, $0 < q < 0.35$ and $0 < q < 0.28$, the chemical potential is monotonously increasing as the charge rises for $\eta=-1$, $-2$ and $-3$, respectively. Thus if we choose the smaller charge, e.g., $q=0.2$, the chemical potential promotes the thermalization, which is different from the previous result that the chemical potential delays the thermalization. Thus strictly speaking, the effect of the chemical potential on the thermalization time depends on the dark energy parameters. We also obtain the fitting functions of the thermalization curves, and with the functions, we further investigate the thermalization velocity and thermalization acceleration. From the thermalization velocity curves, we know that in the middle stage of the thermalization there is a phase transition point which separates the thermalization into an acceleration phase and a deceleration phase. The phase transition point is found to be shifted for different dark energy parameters. In particular, for $\eta=-1$, $-2$ and $-3$, the phase transit decreases as the phantom dark energy parameter decreases. Both the maximum acceleration and maximum velocity are found to be increased as the dark energy parameters decrease. 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