Integrable higher-dimensional cosmology with separable variables in an Einstein-dilaton-antisymmetric field theory

We consider a $D$-dimensional cosmological model with a dilaton field and two $(D-d-1)$-form field strengths which have nonvanishing fluxes in extra dimensions. Exact solutions for the model with a certain set of couplings are obtained by separation of three variables. Some of the solutions describe accelerating expansion of the $d$-dimensional space. Quantum cosmological aspects of the model are also briefly mentioned.


I. INTRODUCTION
In recent decades, relativistic models with a scalar field in cosmology have much interest, because they are suitable for the inflationary scenario [1] and for tackling the dark energy problems [2,3]. Although numerical solutions or approximate solutions for scale factors are studied in many cases of cosmological models, it would be very interesting to find exact solutions that describe the accelerating universe. In many areas of physics, exact solutions play the most important role in understanding and growing the crude concepts. Recently, many authors have studied integrable models with exponential scalar potentials for cosmology and found various interesting models; for example, some models account for the transient acceleration of the universe [4][5][6][7][8][9][10][11][12][13].
On the other hand, it is known that dilaton gravity arises from a low-energy effective theory of string theory, from certain supergravity theories, and from higher-dimensional theory with extra dimensions. In models based on such theories, scalar fields naturally appears with exponential potentials. In addition, such theories often contain totally antisymmetric tensor fields, of which configuration on the extra space can play an important role in compactification scheme [14][15][16][17][18]. Up to now, some higher-dimensional integrable cosmological models have been studied [19][20][21] and exact analytic solutions are derived in some specific cases with fluxes in extra dimensions [22][23][24][25][26][27][28][29].
In the present paper, we consider analytically soluvable models of D-dimensional cosmology with a scalar dilaton and antisymmetric tensor fields. Integrability does not necessarily mean the existence of analytic solutions. We propose a model in which the equations of motion can be expressed by three separate equations of Liouville-type, and then its cosmological solutions are written in simple elementary functions. We analyze the simple model on the basis of the possibility of temporal accelerating expansion of the (d + 1)-dimensional universe (d + 1 < D).
Here, we illustrate an essential structure of the solvable model we consider in this paper. where σ a = ±1 and the dot denotes the derivative with respect to t, the equations of motion reads the one-dimensional Liouville equation σ aẍa + λ a V a e 2λaxa = 0 .
(1. 2) Then, integrals of motion are (1. 3) We can obtain the following analytic solutions for the equations of motion: • For σ a V a > 0, x a (t) = 1 2λ a ln q 2 a cosh 2 q a |V a |λ a (t − t a ) , E a = σ a q 2 a |V a | 2 (1.4) • For σ a V a < 0, x a (t) = 1 2λ a ln q 2 a sinh 2 q a |V a |λ a (t − t a ) , E a = σ a q 2 a |V a | 2 (1.5) x a (t) = 1 2λ a ln q 2 a sin 2 q a |V a |λ a (t − t a ) , E a = −σ a q 2 a |V a | 2 (1.6) • For V a = 0, we find that x a (t) = q a (t − t a ) and E a = σ a q 2 where q a and t a are integration constants.
Remembering the model is the cosmological one, the Hamiltonian constraint restrict the constants as a σ a 1 2ẋ We present a model which is soluble by using such separation of variables in the next section.
Furthermore, we will see later that the separation of variables is significant for considering the minisuperspace Wheeler-De Witt equation.
The outline of the present paper is the following. In Sec. II, we define our models in which three variables are separable as in the manner mentioned above. The solutions are exhibited in Sec. III and IV, and summarized in Appendix A. Sec. III is devoted to the solutions for the scalar field with the canonical kinetic term, while Sec. IV treats the case of the "phantom" scalar field. The physical scale factor and the physical property of the solutions are discussed in Sec. V. Sec. VI contains the brief description of quantum cosmology of our model through the minisuperspace Wheelr-De Witt equation. We conclude with a discussion in Sec. VII.

II. THE ACTION AND VARIABLES
Let us consider the action of the D-dimensional model where R is the Ricci scalar derived from the metric g M N (M, N = 0, 1, . . . , D − 1), g is the determinant of g M N , and Φ is a real scalar field which has dilaton-like coupling to the two pform field strengths F (l) [p] and F (r) [p] . The constant α represents a scalar (dilaton) self-couplings constant. The constants l and r indicate the couplings between the scalar and the two antisymmetric tensor field strengths. We also use the abbreviation (∇Φ) 2 The action (2.1) is invariant under the following two independent transformations: (2) α ↔ −σ/α and l ↔ r and F (l) If the constant σ is taken to be σ = +1, the kinetic term of the scalar field becomes a canonical one. If we choose σ = −1, the scalar becomes a phantom field [30]. Due to the symmetries, we have only to investigate the cases with 0 < α ≤ 1 to clarify general behaviors of the system. 1 Note also that the choice r = 0 in the action (2.1) reduces the model to that studied by many other authors, such as in Refs. [24,25]. 2 The value of the constant κ will be specified later.
We adopt the following ansätze. We assume the (D − 1)-dimensional space admits the metric of a direct product of a d-dimensional flat Euclidean space and (D−d−1)-dimensional maximally symmetric space. The scale factors and the scalar Φ are considered only to be time-dependent, i.e., are functions of the time coordinate t = x 0 . Therefore we take the metric as follows: Here, we denote the coordinates of the flat space as x i (i = 1, . . . , d) and those of the maximal symmetric extra space as x m (m = d + 1, . . . , D − 1). We use the notation and dΩ 2 [D−d−1] stands for the line element of the extra space whose metric is denoted asg mn . We assume that the Ricci tensor of the extra space is written bỹ where k b is the constant, which has been normalized to be 1 or 0 or −1.
We further consistently assume that the p-form field strengths take "constant" (flux) values in the extra space, thus, and where f is taken as a positive constant (and is possibly quantized as a "magnetic" charge), without loss of generality. Even if we assume non-identical values for two fluxes as classical configurations, the difference in magnitudes of fluxes can be absorbed into the redefinition of the couplings l and r. We now obtain Substituting the anzätze and noting that √ −g ∝ e da+(D−d−1)b+n , we find where the dot indicates the derivative with respect to the time t. Here, if we set as a gauge choice, the reduced cosmological action becomes We now find that the "kinetic" terms, which include the time-derivatives, in the reduced action (2.11) can have a unique quadratic form as follows: the action can be written in three independent variables We write the reduced action as S = L dt with where Note that the coefficients of [dȧ + (D − d − 2)ḃ] 2 in the Lagrangian L is independent of the dilaton coupling α and the kinematical signature σ. Therefore, we take a variable x as throughout this paper. Then, the variable x(t) obeys the equation .
The solution of Eq. (2.16) is: , for k b = +1 , where q 1 and t 1 are constants. respectively.
• (positive E 1 ) where q 1 and t 1 are constants.
The first integral E 1 = − 1 2ẋ 2 + V 1 2 e 2λ 1 x is In the next section, we study the model with σ = +1 and derive exact solutions. The case with σ = −1 is treated in Sec. IV.

III. THE CASE FOR THE CANONICAL KINETIC TERM OF DILATON(σ = 1)
For σ = +1, the reduced cosmological Lagrangian is written as where We permit arbitrary signs of l and r in the present paper. When a negative sign of the coefficient is taken, it yields a "phantom" gauge field. Such a phantom gauge field has been considered once in a cosmological context [31], though the negative value of them may leads to pathological consequences in quantum physics.
The exact solution of the model now can be obtain in each case shown below. In this section, we define integrals of motion as A. Case l > 0 and r > 0 First, in this case, we find that the solutions for y and z of the equations from the reduced Lagrangian L can be written as where q 2 , t 2 , q 3 and t 3 are integration constants. Then, both E 2 and E 3 are positive. Since the Hamiltonian constraint tells defined in Sec. II. The Hamiltonian constraint then reads In this case, the exact solution for y is written as where q 2 and t 2 are constants, and then E 2 > 0. Solutions of other variables are characterized by the following subcategories, according to the signature of E 1 and E 3 .
The solution for x is x −k b (t) and the solution for z is where q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x −k b (t) and the solution for z is where q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solution for z is where q 3 and t 3 are constants. Then the Hamiltonian constraint becomes This case can be regarded as the previous case where the roles of y and z are exchanged mutually. The exact solution for z is written as where q 3 and t 3 are constants, and then E 3 > 0. Solutions of other variables are characterized by the following subcategories, according to the signature of E 1 and E 2 .
The solution for x is x −k b (t) and the solution for y is where q 2 and t 2 are constants. Then, the Hamiltonian constraint becomes The solution for x is x −k b (t) and the solution for y is where q 2 and t 2 are constants. Then, the Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solution for y is where q 2 and t 2 are constants. Then the Hamiltonian constraint becomes D. Case l < 0 and r < 0 In this case, various forms of exact solutions appear, since each integral of motion can take a positive or negative value.
The solution for x is x −k b (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x −k b (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x −k b (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solutions for y and z are where q 2 , t 2 , q 3 and t 3 are constants. Then, the Hamiltonian constraint becomes Now, all the solutions for x, y and z have been shown for σ = +1. In the next section we will examine the case with σ = −1, which corresponds to a phantom dilaton.

IV. THE CASE FOR A PHANTOM DILATON (σ = −1)
In this section we consider the case with σ = −1. Remembering 0 < α < 1, the reduced action can be written as Similarly to the previous section, we obtain exact solutions for the cases with positive and negative couplings l and r. In this section, we define A. Case l > 0 and r > 0 In this case, the solution for y takes the form where q 2 and t 2 are constants, and then E 2 > 0. Similarly to the several cases in the previous section, several cases are classified below.
The solution for x is x −k b (t) and the solutions for z is given by where q 3 and t 3 are constants. The Hamiltonian constraint becomes The solution for x is x −k b (t) and the solutions for z is where q 3 and t 3 are constants. The Hamiltonian constraint becomes The solution for x is x +−1 (t) and the solutions for z is where q 3 and t 3 are constants. The Hamiltonian constraint becomes In this case, the solutions for y and z are given by where q 2 , t 2 , q 3 and t 3 are constants, and then, E 2 > and E 3 < 0.
The solution for x is x −k b (t). The Hamiltonian constraint reads The solution for x is x +−1 (t). The Hamiltonian constraint reads C. Case l < 0 and r > 0 In this case, all possible signs for E 1 , E 2 , and E 3 can appear.
The solution for x is x −k b (t). The solutions for y and z are . (4.14) The Hamiltonian constraint is The solutions for y and z are The Hamiltonian constraint is The solution for x is x −k b (t). The solutions for y and z are The Hamiltonian constraint is The solution for x is x +−1 (t). The solutions for y and z are The Hamiltonian constraint is The solution for x is x +−1 (t). The solutions for y and z are The Hamiltonian constraint is The solution for x is x +−1 (t). The solutions for y and z are The Hamiltonian constraint is D. Case l < 0 and r < 0 In this case, the solution for z is and then, E 3 < 0.
The solution for x is x −k b (t) and the solution for y is where q 2 and t 2 are constants. The Hamiltonian constraint reads The solution for x is x +−1 (t) and the solution for y is where q 2 and t 2 are constants. The Hamiltonian constraint reads The solution for x is x +−1 (t) and the solution for y is where q 2 and t 2 are constants. The Hamiltonian constraint reads To analyze cosmological behavior closely, we introduce the "physical " (d+1)-dimensional metric and the cosmic time. When we take a representation for D-dimensional metric such as we find that √ −gR is proportional to √ −ḡR + · · ·, whereR is the Ricci scalar of the (d + 1)-dimensional spacetime constructed fromḡ µν . Therefore, the metricḡ µν is considered to define the Einstein frame of the (d + 1)-dimensional spacetime.
In the present study, we should regard the following form for the metric where η is the cosmic time for the (d + 1)-dimensional spacetime and S is the "physical" scale factor of d-dimensional flat space in (d + 1)-dimensional view. Thus, we obtain the relations S(η) = e a(t) They can be written in terms of x, y, and z as follows: where the solutions can be expressed simple functions of η. We first consider these cases.
In all the case examined above, the dilaton field Φ takes a constant value in the final consideration, which corresponds to the extremum of the effective scalar potential coming from the coupling to the constant fluxes, which is proportional to le 2καΦ + re −2κσΦ/α .
The cases with (σ = 1, l < 0, r < 0, k b = −1) and (σ = −1, l < 0, r > 0, k b = −1) yields the exponentially expanding universe. Unfortunately, these are not the most general cosmological solutions of the model. In the next subsection, we will consider asymptotic behaviors of the physical scale factor S(η).

B. asymptotic behaviors of solutions
We have several cases where the asymptotic behavior of S(η), as a function of the cosmic time η can be obtained. If S(t) ∼ e Bt with constant B, S(η) ∼ η 1/d . If S(t) ∼ |t − t 0 | β , S(η) ∼ |η − η 0 | β/(dβ+1) with some constants t 0 and η 0 . The scale of the compact space e b generally shows a similar behavior to S(η).
Thus, some typical cases, in which the solution represents for the expanding universe, can be found as follows: This behavior is found in the cases: This behavior is found in the cases: This behavior is found in the cases: This behavior is found in the cases: This behavior is found in the case: σ = 1, l > 0, r < 0, k b = −1, (E 1 < 0, E 2 > 0, This behavior is found in the cases: This behavior is found in the cases: This behavior is found in the case: σ = 1, l < 0, r > 0, k b = −1, (E 1 < 0, E 2 > 0, This behavior is found in the case: σ = 1, l < 0, r < 0, k b = 0, ±1, (E 1 < 0, E 2 > 0, This behavior is found in the case: σ = 1, l < 0, r < 0, k b = 0, ±1, (E 1 < 0, E 2 > 0, This behavior is found in the case: σ = 1, l > 0, r < 0, k b = −1, t 1 = t 3 .
This behavior is found in the case: σ = 1, l < 0, r > 0, • Expanding and contracting in a finite cosmic time η.
In our model, the eternally expanding universe can be found k b = −1 (σ = 1), similar to the cases with the models studied by many authors [20,21,25].
We here omitted analyses on interesting cases with E 1 > 0 and/or σ = −1, in which complicated evolutions, including a bouncing universe, can be found accordingly to a proper choice of integration constants. We leave the broad study of such cases for future work, and we restrict ourselves to considering a possible accelerating phase in the universe in the rest of this section.

C. transient acceleration and the scalar field value moving in a finite range
We cannot tell existence or absence of transient acceleration only judging from asymptotic behavior. Therefore, we should investigate the behavior of S(η) in the intermediate era more closely. To this end, we first observe Thus, for expanding and accelerating physical universe, As already known, the model with k b = −1, i.e., with the hyperbolic internal space, yieds accelerating universe in both cases with no other field content [20,21] and with the single flux and the dilaton field [25]. In our model, therefore, a transient acceleration occurs for a wide range of parameters.
The type of minute behavior of S(η) is diverse in a lot of the solutions. We concentrate ourselves mainly on the case σ = 1, l > 0, r > 0, k b = −1, 0 here, not only because this case yields expanding universe but also because this is only admissible case for quantum field theory in a naive sense. We will, however, add a discussion on a special phantom case in the last of this subsection.
A most remarkable feature of this case in our model is that the value of the dilaton scalar field can be finite throughout evolution of the universe, because of two dilaton couplings to fluxes. Indeed, for the case (σ = 1, l > 0 and r > 0), we find e 2κΦ(t) = r cos 2 θ l sin 2 θ , (5.27) where q, θ , t 2 and t 3 are integration constants. When λ 2 cos θ = λ 3 sin θ, i.e., cos θ = λ 3 / λ 2 2 + λ 2 3 = 1/ √ 1 + α 2 and sin θ = λ 2 / λ 2 2 + λ 2 3 = α/ √ 1 + α 2 , the scalar field Φ behaves as Obviously, this is only the case of finite Φ(t) at t → ∓∞, whose value is given by . Note that for the case with k b = −1, since η → ∞ at t = t 1 , the value of Φ approaches to the constant Φ(t 1 ), while other variables move as If we especially choose t 2 = t 3 , Φ takes a constant value which is the equilibrium point of reduced potential ∝ le 2καΦ + re −2κΦ/α , which is included in the action.

We show the values
versus t for various values of t 2 and t 3 in Figs. 1 and 2. It can be found that the acceleration period becomes earlier when t 2 − t 3 becomes larger positive value. Contrarily, we found that negative t 2 − t 3 leads to a later period of acceleration.  We also found that if the value of α is sufficiently close to unity, there exists an accelerating phase for a wide range of t 2 − t 1 and t 3 − t 1 . We conclude that the solution can give accelerating universes in (d + 1)-dimensional Einstein frame in the model with σ = 1, l > 0, r > 0 and k b = −1 and k b = 0. The result here is qualitatively similar to the result of Ref. [25], where the dilaton coupling to a single flux term is considered.  A similar transient acceleration can be found in a special phantom case, σ = −1, l > 0 and r < 0, and (E 1 < 0, E 2 > 0, E 3 < 0). In this case, we choose the integration constant to satisfy λ 3 sinh θ = λ 2 cosh θ in the solution (A40) and obtain

VI. QUANTUM COSMOLOGY
To study very early universe and especially its initial state, we have to consider quantum nature of cosmology. In our model, we can obtain the minisuperspace Wheeler-De Witt equation by replacingẋ a → −i ∂ ∂xa (where we choose the natural unit = 1) in the Hamiltonian H and regarding the Hamiltonian constraint as HΨ = 0, where Ψ is the wave function of the universe.

Our Hamiltonian becomes
noticing that the definition of λ 2 and λ 3 is different in each case. Owing to the separation of variables, the wave function is expressed by superposition of the multiplicative form, Let us first consider the case σ = 1, l > 0 and r > 0. Then, the normalizable wave function can be written in the form where A(q, θ) is the amplitude and the eigenfunctions F ν and G ν are defined as [13,32] F ν (z) = 1 2 cos(νπ/2) and c 1 and c 2 are constants. In this expression, we adopted the wave normalization found in Refs. [33], so that where The Gaussian wave packet is often considered [13,[34][35][36][37] in a semiclassical analysis of quantum cosmology. There is another possibility that the amplitude A is independent of θ, which is naturally motivated from the form of (6.3) because the integral region, or moduli space, of θ is apparently finite. To grasp succinctly what occurs by taking this assumption, we consider a simple calculation. We show |ψ q (ξ 1 , ξ 2 )| 2 with q = 4 in Fig. 5, where ψ q (ξ 1 , ξ 2 ) ≡ 2π 0 dθ 2(2) iq cos θ Γ(−iq cos θ) K iq cos θ (e ξ 1 ) 2(2) iq sin θ Γ(−iq sin θ) K iq sin θ (e ξ 2 ) . (6.6) Many peaks of the function are located in the region (ξ 1 < 0, ξ 2 < 0) and considerably high peaks are found at ξ 1 ∼ ξ 2 . Because both the eigenfunction of ξ 1 and that of ξ 2 have the incoming wave and reflected wave from the potential wall, the interference of the four waves generates complicated wave pattern. Nevertheless, we find that a chain of peaks appears in the line ξ 1 ∼ ξ 2 , which seems plausible from symmetry.
From this fact, we expect that, if the amplitude A(q) is assumed to have sufficiently narrow width, peaks of the probability density |Ψ| 2 appear at the discrete positions where λ 2 y ∼ λ 3 z, i.e., Φ ∼ 0 for α ∼ 1. An interpretation from this shape of the partial wave function is accepted that the initial state of the universe possesses a small but finite e a , whose possible value is somewhat discretized, with the stationary value of Φ.
Next let us consider the phantom case σ = −1, r < 0, and k b = 1. Then, the normalizable wave function can be written in the form where A ′ is the amplitude and we rearrange the integration constants.
We consider again the case A ′ = A ′ (q) be a Gaussian with a narrow width. As in the previous case, we expect the high peaks at λ 1 x ∼ λ 3 z 0. Then, the initial state of the universe is equipped with some finite e a and some b ∝ Φ of a finite value.
The interpretation we presented here is very qualitative, regrettably. To obtain more quantitative results, we should study the wave function carefully, by taking account of a normalization measure, which can be dependent somewhat on θ, and detailed calculation of superposition. This study will be done in future work.

VII. SUMMARY AND DISCUSSION
In this paper, a class of analytical cosmological solutions is considered in an integrable higher-dimensional model with a scalar field and an antisymmetric tensor field. The scalar field is either canonical (σ = +1) or phantom one (σ = −1).
In Sec. V, we have looked for solutions for the accelerating universe. We have found that the expanding universe with transient acceleration is obtained in the case with hyperbolic or flat internal space with positive energy density of dilaton and antisymmetric fields (σ = 1, l > 0 and r > 0), except for other special cases.
One of the special feature of our model is that the value of Φ can be finite both at the beginning, S ∼ 0 and at the far future of the universe, η → ∞. The coupling between the dilaton field and additional gauge or matter fields can trigger off some cosmological timedependent phenomina, though the present form fields unfortunately have difficulty in being interpreted in the (d + 1)-dimensional universe. The study on inclusion of matter fields is significant in any cases, even apart from pursuing exact solutions.
We have argued that quantum cosmology of some specific cases briefly. Throughout this paper, we have found that separable variables have made the analyses simple both in classical and quantum cosmological behavior of the scale factor. However, further deeper investigation is required especially for cosmology with initial bouncing behavior, on which has not been touched in the present study. We should also consider various aspects of wave functions of the universe in our model in more precise manner. We left these subjects to study for future work.

Appendix A: Summary of solutions
Here, we summarize the solutions and show the expressions for two scale factors a and b and the scalar field Φ. They are found, by using x, y and z, as follows: (A3) We will exhibit them, categorizing the sign of σ, the signs of l and r, and the values for k b , in this order. In the following expressions, t 1 , t 2 , t 3 , q and θ are integration constants.
Therefore, the possible cases are listed below: In each case, the case r = 0 can be considered as the limit r → 0 of the solution for a finite r. This is because if we set |r| = e −2q ′ λ 3 t 0 , we found that with noting that the time-reversal invariance of the equation of motion and the multiplicative constant factor is irrelevant for the case. 4 Thus, we would not exhibit explicit form of solutions in the case r = 0 in this paper. The case l = 0 can be analyzed similarly (and we do not repeat the analysis). By using the l ↔ r symmetry, the case of arbitrary dilaton coupling α can be covered by considering the limit r → 0 properly as indicated above.