The Phase-space analysis of scalar fields with non-minimally derivative coupling

We perform a dynamical analysis for the exponential scalar field with non-minimally derivative coupling. For the quintessence case, the stable fixed points are the same with and without the non-minimally derivative coupling. For the phantom case, the attractor with dark energy domination exists for the minimal coupling only. For the non-minimally derivative coupling without the standard canonical kinetic term, only the de-Sitter attractor exists, and the dark matter solution is unstable.


I. INTRODUCTION
Ever since the discovery of the accelerating expansion of the Universe [1][2][3], theoretical physicists have faced a big challenge to explain this phenomena. A cosmological constant is the simplest way to explain the observed acceleration, but the theoretical prediction of the cosmological constant is at odds with the observed value by 120 orders of magnitude.
Furthermore, a cosmological constant also faces the coincidence problem namely why the energy densities of matter and dark energy nearly equal today. Dynamical fields with scalar field such as quintessence [4][5][6][7][8], phantom [9], tachyon [10][11][12] and k-essence [13] were proposed as dynamical dark energy models. If the accelerating phase is an attractor solution which is independent of initial conditions, then the coincidence problem can be solved. In particular, the dynamical scalar field has an accelerated scaling attractor and the ratio of the energy densities between the scalar field and matter is of order 1. For the quintessence model, exponential potential V (φ) = V 0 exp(−λφ) has scaling attractor solutions [14,15].
For more general scalar fields, scaling attractor solutions were also found in [16][17][18]. However, the attractor solution has Ω m = 0 which is inconsistent with the current observation.
To solve the coincidence problem with the exponential potential, phenomenological interactions between dark energy and dark matter were introduced, but the parameter space was severely constrained [19,20].
More general models for scalar fields with non-minimal coupling to gravity such as ξf (φ)R was also studied extensively. Recently, a universal attractor behaviour for inflaton at strong coupling (ξ ≫ 1) was found for a class of non-minimally coupled scalar field with the potential V (φ) = λ 2 f 2 (φ). However, the combination of the non-minimal coupling term f (φ)R and the Einstein term R can be treated as a special case of the general scalartensor theory F (φ, R). By a conformal transformation, the non-minimal coupling term f (φ)R disappears. If the kinetic term of the scalar field is coupled to curvature, then the model cannot be transformed to scalar-tensor theory by conformal transformation [21]. In four dimensions, Horndeski derived the most general field equations which are at most of second order in the derivatives of both the metric g µν and the scalar field φ and gave the most general Lagrangian which leads to the most general second order equations [22]. In Horndeski theory, the second derivative φ ;µν is coupled to the Einstein tensor by the general If we only consider the non-minimal coupling of the scalar field to the curvature which is quadratic in φ and linear in R, the most general Lagrangian is [21] Due to the divergencies (Rφ ,µ φ) ;µ , (R µν φφ ,µ ) ;µ and (R ,µ φ 2 ) ;µ , only L 1 , L 2 and L 3 are independent. For a massless scalar field, the non-minimally derivative coupling L 1 and L 2 give de-Sitter attractor solution [23,24]. Furthermore, the field equations reduce to the second order equations if κ 2 = −2κ 1 = κ and the non-minimally derivative coupling becomes κG µν φ ,µ φ ,ν [25]. Higgs inflation with λφ 4 potential was then discussed with this non-minimal derivative coupling and it was found that the model does not suffer from dangerous quantum corrections [26]. For a massless scalar field without the canonical kinetic term g µν φ ,µ φ ,ν , the non-minimally derivative coupled scalar field behaves as a dark matter [27,28]. Because of its rich physics, the non-minimally derivative coupling κG µν φ ,µ φ ,ν attracted a lot interests recently [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43]. In this paper, we analyze the dynamical evolution of the scalar field with the non-minimal derivative coupling for an exponential potential.

TIVE COUPLING
The action for the non-minimally derivative coupling scalar field is Where M 2 pl = (8πG) −1 = κ −2 , the coupling constant ω has the dimension of inverse mass, ǫ = 0 corresponds to non-minimally derivative coupling only, ǫ = 1 correspond to the canonical kinetic term, and ǫ = −1 corresponds to phantom case. The energy-momentum tensor for the scalar field is so the energy density and pressure for the scalar field are When the non-minimally derivative coupling is absent, ω = 0, we recover the standard result By using the Friedmann-Robertson-Walker metric, we obtain the cosmological equations from the action (2) and the energy-momentum tensor (3) as, The matter energy density is ρ m ∝ a −3 . For simplicity, we consider the exponential potential In terms of the dimensionless dynamical variables, the cosmological equations (8) and (9) become where x ′ = dx/d ln a, the dimensionless variable z = κ ρ b /3/H for the background matter density satisfies the cosmological constraint ǫx 2 + y 2 + u 2 + z 2 = 1, the auxiliary variables s = −Ḣ/H 2 and t = √ 6φ/Hφ satisfy the following relations, and γ b = 1 + w b . In the above system, if x = u = 0, then the system is not well defined, so we only get those fixed points with which x and u are not zero at the same time. To get the fixed points with x = u = 0, we use the variable v = (ωH) −1 = 3x/u to replace the variable x = uv/3 for ω = 0. When the kinetic energy is negligible,φ = 0, it seems that x should also be zero. Note that u = 0 does not mean the scalar field does not evolve, it just means that the scalar field changes very slowly so thatφ is negligible but not zero. This point can be better understood if we use the dynamical variables x, y and z. The dimensionless energy densities Ω φ = ǫx 2 + y 2 + u 2 = ǫu 2 v 2 /9 + y 2 + u 2 and Ω m = z 2 = 1 − Ω φ . The equation of state parameter w φ of the scalar field is The effective equation of state parameter w ef f of the system is It is obvious that the dynamical equations (12)- (14) consist of an autonomous system. For the case with ǫ = 1 and ω = 0, u = 0 and the system (12)- (14) reduces to the quintessence system [14,15]. For the case with ǫ = −1 and ω = 0, u = 0 and the system (12)- (14) reduces to the phantom system [44]. For the non-minimally derivative coupling case with ǫ = 0, the dynamical analysis was performed in [27]. For the case with ǫ = 1 and ρ b = 0, the dynamical analysis for a power-law potential was dsicussed in [32]. By setting x ′ = y ′ = u ′ = 0 in Eqs.
(12)- (14), we obtain the following critical points: Point C1 with (x c1 , y c1 , u c1 ) = (0, 0, ±1), it exists when ω = 0. This point corresponds to dark matter solution found in [27,28] with the derivative coupled kinetic energy term domination. For this point, we have Ω φ = 1 and w φ = 0 and the scalar field behaves as dark matter even though its potential energy is zero.
Point C3 with (x c3 , y c3 , u c3 ) = (x, 0, 0). The existence condition is ǫ > 0, 0 < ǫx 2 ≤ 1 and γ b = 2. For this point, we have Ω φ = ǫx 2 and w φ = w b = 1, only the canonical kinetic energy of the scalar field contributes to the energy density and the scalar field tracks the stiff matter background.
For this point, we have Ω φ = u 2 and w φ = w b = 0. The non-minimally derivative coupling term makes the only contribution to the energy density of the scalar field and the scalar field tracks the dust background. The scalar field behaves like dark matter [27].
, 0). The existence condition is ǫ > 0, 0 ≤ γ b ≤ 2 and λ 2 > 3ǫγ b . It corresponds to the tracking solution with Ω φ = 3ǫγ b /λ 2 and w φ = w b . Since u c = 0, the contribution from the non-minimally derivative coupling is absent, so the result is the same as the quintessence field.
This point corresponds to matter domination solution with Ω m = 1 and Ω φ = 0.
The fixed points and their existence conditions are summarized in Table I. For the quintessence case with ǫ = 1 and ω = 0, u = 0 and only the critical points C2, C3 and C5-C7 exist, but only the points C2 and C5-C7 were found in [14,15]. For the non-minimally derivative coupling case with ǫ = 0 and ω = 0, only the critical points C1, C4, C7 and C8 present. The dynamical analysis for this case was performed in [27], but only the critical points C4, C7 and C8 were found.
To discuss the stability of the autonomous system we need to expand Eq. (18) around the critical point X c by setting X = X c + U with U the perturbations of the variables considered as a column vector. Thus, for each critical point we expand the equations for the perturbations up to the first order in U as: where the matrix Ξ contains the coefficients of the perturbation equations. If the real parts of the eigenvalues of the matrix Ξ are all negative, then the fixed point is a stable point.
Applying this procedure, we find the eigenvalues of the matrix Ξ and present the stability conditions for the above critical points C1-C8.
For the point C5, the eigenvalues are To keep the real parts of all three eigenvalues negative, we require that 0 < γ b < 2 and λ 2 > 3ǫγ b . The corresponding phase trajectories with different initial conditions for the parameters ǫ = 1, λ = 3 and γ b = 1 are shown in the right panel of Fig. 1.
For the point C7, we use the dynamical variable v instead of x to discuss the dynamical behaviour and the eigenvalues are For the point C8, the dynamical variable v instead of x is used to discuss the dynamical behaviour and it was discussed in [27], this de-Sitter attractor is stable.
The properties of all the critical points are summarized in Tab. I.

III. DISCUSSION AND CONCLUSIONS
For the quintessence case with ǫ = 1 and ω = 0, in addition to the standard stable fixed points C5 and C6, we also find the stable fixed point C3. The fixed points C3 and C5 are tracking solutions, and C6 gives the late time accelerating solution with scalar field domination. For the phantom case with ǫ = −1 and ω = 0, only the stable fixed point C6 exists. For the case with non-minimally derivative coupling only, ǫ = 0 and ω = 0, the dark matter solutions C1 and C4 are unstable, only the de-Sitter attractor exists. For more general case with ω = 0 and ǫ = 0, the stable fixed points C3, C5 and C6 exist only for the quintessence field with ǫ = 1. C3 and C5 are tracking attractors and C6 is accelerating attractor with dark energy totally domination if λ 2 < 2.