Accelerating cosmologies in an integrable model with noncommutative minisuperspace variables

We study classical and quantum noncommutative cosmology with a Liouville-type scalar degree of freedom. The noncommutativity is imposed on the minisuperspace variables through a deformation of the Poisson algebra. In this paper, we investigate the effects of noncommutativity of minisuperspace variables on the accelerating behavior of the cosmic scale factor. The probability distribution in noncommutative quantum cosmology is also studied and we propose a novel candidate for interpretation of the probability distribution in terms of noncommutative arguments.


I. INTRODUCTION
Almost two decades ago, the noncommutativity of the spacetime coordinates, such as has been introduced into the study of quantum field theory [1,2]. If the parameters of the noncommutativity θ is taken to be constants, this means the existence of an absolute small scale unit, ∼ |θ|. Indeed, motivation of proposing the noncommutative spacetime was originated from string theory. It is then natural to consider that the noncommutativity may play some key role in scenarios of quantum gravitation theory.
On the other hand, it is known that the expansion rate of our universe is accelerating in the present days [20,21] and also in the very early era of cosmology [22]. Such accelerations can be caused by the dynamics of additional scalar modes in the Einstein gravity. In several models, exact solutions are known and thoroughly investigated [23][24][25][26][27][28][29][30][31][32].
In the present paper, we study classical and quantum noncommutative cosmology with a Liouville-type scalar degree of freedom. The exponential scalar potential naturally appears in string theory. It is known that such a scalar mode also arises from compactification of extra dimensions or pure R p gravity [33]. Specifically, we investigate the effects of noncommutativity in minisuperspace variables on the accelerating behavior of the cosmic scale factor in exact analytical solutions of the model.
We suppose that the interpretation of probability in noncommutative quantum cosmology confronts a subtle problem on variables. In the commutative case, the Wheeler-De Witt equation comes from the Hamiltonian which is represented with the dynamical commutative variables and derivatives by them. Thus, the arguments of the wave function of the universe are not the original noncommutative variables. We should know at least the correspondence between a deformed set of commutative variables and that of noncommutative variables.
By utilizing the exact wave function in the present model, we consider some possibilities in interpretation of the probability distribution in noncommutative quantum cosmology.
One of the possibilities relies on the use of the Wigner function [34]. Since the Wigner function was originally introduced by Wigner in order to treat the quantum statistical physics appropriately, it has been widely applied to the problem in various fields of physics and applied physics [35].
This paper is outlined as follows. In Sec. II, we define the field-theory action of our toy model and describe that equivalent actions can be obtained from higher-dimensional theories and higher-derivative theories. The action for minisuperspace variables of the model are exhibited in Sec. III. In Sec. IV, we review the exact classical commutative cosmological solutions. In Sec. V, we derive the exact classical noncommutative cosmological solution.
We give the deformation of the Poisson algebra here and we find that the calculations can be performed analytically. In Sec. VI, we study the effect of the noncommutativity on the accelerating universe by using the analytical solutions obtained in the previous section. In Sec. VII, the wave function of the universe in our noncommutative model is obtained and connection to the classical solution is described. Section VIII contains a brief description of the Wigner distribution function and deformation in minisuperspace variables. We propose a new interpretation of the distribution function with respect to noncommutative variables.
Finally, Sec. IX is devoted to discussion and outlook.

II. THE SIMPLEST MODELS WITH EXPONENTIAL SCALAR POTENTIALS
Let us consider the action of the D-dimensional model where R is the Ricci scalar derived from the metric g µν (µ, ν = 0, 1, . . . , D − 1), g is the determinant of g µν , and Φ is a real scalar field. The constant α denotes the scalar selfcoupling. We use the abbreviation (∇Φ) 2 ≡ g µν ∂ µ Φ∂ ν Φ. It is known that the exponential potential arises from string theory. In the followings, we exhibit several equivalent models.
A. Higher-dimensional Einstein-antisymmetric field theory with a cosmological constant Let us consider an Einstein-antisymmetric field theory with a cosmological constant in (D + q) dimensions. Its well-known action is We take a representation for the (D + M)-dimensional metric such as where µ, ν = 0, . . . , D − 1 and m, n = D, . . . , D + q − 1. The Ricci tensor of the maximal symmetric extra space, whose metric is denoted asg mn , is assumed to be written as where k b is a constant, which has been normalized to 1, 0, or −1. Further, we assume that the q-form field strength takes a constant value in the extra space; thus, where f is a constant.
Performing the dimensional reduction, we find that the effective D-dimensional action is where we omit the overall constant. Further, defining we obtain the effective action of gravitating scalar field with an exponential potential (2.1).
Then, we identify the coupling parameters in the following three cases: 1 Upon compactification, it is also possible to consider (D − 1)-form field strength. We omit this possibility in this paper only to consider the simplest model (and leave the possibility for future study).
Now, we turn to consider pure R p gravity in D dimensional spacetime [33]. We start with the action (2.11) We can use an auxiliary field χ to obtain classically equivalent action: We can eliminate the χ-dependence in front of the Einstein-Hilbert term R in the action (2.12) by a Weyl transformation. In other words, we consider a Weyl-transformed metric whereR is the Ricci scalar constructed from g µν . To this end, we choose g µν = χ − 2(p−1) D−2g µν in this time. Then, we obtain This action (2.13) is equivalent to the action (2.1) with the coupling constants (2.14) We call this case as Case [RP], for later convenience.

III. ACTION FOR MINISUPERSPACE VARIABLES
Now, we concentrate ourselves on studying cosmological behavior of the model described by the action (2.1). We adopt the following ansätze. The (D − 1)-dimensional space is assumed to be a flat Euclidean space and its scale factors and the scalar Φ are considered to be only time-dependent; i.e., they are functions of the time coordinate t = x 0 . Therefore, we take the metric as follows: Substituting the anzätze and noting that √ −g ∝ e (D−1)a+n , we find where the dot indicates the derivative with respect to time t. Here, if we set as a gauge choice, the reduced cosmological action becomes We now find that the "kinetic" terms, which contain the time derivatives, in the above action can have a form Therefore, we can consider the following reduced Lagrangian of two dynamical variables to analyze the dynamics in minisuperspace: where and The canonical conjugate momenta are and then, the Hamiltonian of the system is found to be The correspondence to the model cases in the previous section is as follows: In the next section, we will give an analytical classical commutative solution for the system described by the Hamiltonian (3.11).

IV. CLASSICAL COMMUTATIVE SOLUTION
In this section, we review the derivation of commutative classical solutions in the system.
As usual dynamical systems, we will work with the Poisson brackets and others are zero. The usual Hamilton's equations for the Hamiltonian (3.11) arė From these equations, we obtain the equations of motion as follows: Note that, because we now consider the cosmological system, the parametric invariance of t → ct (c is a constant) requires if the solution is substituted into the Hamiltonian.
One can easily find that the solution for y(t) is simply given by where P , t 0 and y 0 are constants. The exact solution for x(t), which obeys one-dimensional Liouville equation and the constraint H = 0, is given by The noncommutative classical dynamics of the system will be investigated in the next section.

V. CLASSICAL NONCOMMUTATIVE SYSTEM
At first, we define the minisuperspace variables with noncommutativity. By replacing we obtain the Hamiltonian Here, we consider the Poisson brackets and others are zero. The parameter of the noncommutativity θ is assumed to be a constant.
If we set the noncommutativity parameter θ → 0, the system returns to the commutative system. The Hamilton's equations are now given bẏ The solution for these equations turns out to be [4,6,8,13,14,16,17] • U > 0 which satisfy the Hamiltonian constraint H = 0. It should be noticed that the analytic solutions can be simply classified by the sign of U.
Incidentally, the solutions for commuting variables x and y are given by which satisfy H = 0. Note that x(t) and y(t) depend only on the combination of the parameters θ − ρ. Of course, the variables X(t) and Y (t) constructed from the solutions for x(t) and y(t) with the relation (5.7) are independent of the additional parameter ρ.
In the next section, we study the effect of noncommutativity on the accelerating universe.

VI. EFFECT OF NONCOMMUTATIVITY ON ACCELERATING UNIVERSE
The solutions obtained in the previous section represent the expanding universe in appropriate ranges of parameters. To investigate the evolution of the scale factor, we should regard the following form for the metric: where η is the cosmic time for the D-dimensional spacetime and S is the "physical" scale factor of (D − 1)-dimensional flat space. Thus, we obtain the relations It is difficult to determine the existence or absence of transient acceleration only from analytic methods. Therefore, we should investigate the behavior of S(η) in a numerical plot.
Anyway, analytic solutions are very useful to express numerical values. To this end, we first To summarize, in the model described by the action with an exponential scalar potential (2.1) with V > 0, the noncommutativity reduces the cosmic acceleration if α 2 < D−1 2(D−2) while the noncommutativity enhances the acceleration if α 2 > D−1 2(D−2) .

VII. WAVE FUNCTION OF THE UNIVERSE
In a commutative model, we can obtain the minisuperspace Wheeler-De Witt equation by replacing π x → −i ∂ ∂x and π y → −i ∂ ∂y in the Hamiltonian H and regarding the Hamiltonian constraint as HΨ(x, y) = 0, where Ψ(x, y) is the wave function of the universe [37].
Deformation of the Wheeler-De Witt equation in our noncommtative case with the Hamiltonian H θ can be performed by These operators satisfy the relations and other commutators vanish.
The corresponding Wheeler-De Witt equation is fonud to be 1 2 3) The solution of the equation can be written as [3,31] Ψ and In the above expressions, A ν is the overall amplitude for ν and c 1 , c 2 and c 3 are normalization factors. J ν (z) and K ν (z) represent the Bessel function and the modified Bessel function of the second kind, respectively.
In the noncommutative case, we would like to consider Ψ(X, Y ) instead of Ψ(x, y), i.e., the expression in the main noncommutative variables. If we take the parameter choice ρ = −θ, we find that Y equals y. In our model, moreover, the relationΠ Y Ψ ν = νΨ ν holds.
Therefore, we can regard X ∼ x − θP approximately at this parameter choice, where P is the most probable value for the peaked wave packet.
First, we consider the case with U > 0 (7.5). We set the factors as [31] c 1 = ν/λ tanh(νπ/(2λ)) , c 2 = 0 . Next, we consider the case with U < 0 (7.7). We set the factors as  In each case, one finds that the peak of the distribution function reproduces a classical trajectory well, even with the simplest rectangular form of amplitude adopted here.
The interpretation of the noncommutative variables, especially X in the present case may not be accepted in general noncommutative dynamics, since the 'expectation value' P for ν corresponds to a unique constant in our classical model.
In the next section, we propose another candidate of probability distribution by using the Wigner function and examine its validity by analysis with our soluble model.

VIII. WIGNER FUNCTION AND DEFORMATION
Generally speaking, the Wigner function [34,35] is defined, in terms of a wave function φ(q), by The Wigner function has beautiful properties, such as whereφ(p) is the Fourier transform of φ(q).
The application of the Wigner function has been considered already in quantum cosmologies with deformed phase spaces with slightly different motivations from ours [41,42]. Our present aim is to construct the probability distribution whose arguments are noncommutative variables, say, X and Y , from the Wigner function defined by the wave function of commutative variables x and y.
We start with the Wigner distribution function constructed from the wave function Ψ(x, y) (7.4), where we leave the variable x untouched: Therefore, the Fourier transform between y and ν is easily obtained as a simple form: Now, we come to an idea that one may interpret X = x − θp y as in the relation (5.7) with ρ = −θ. To this end, we define and integrate out the orthogonal variableX. Namely, we propose a (Fourier transformed) probability distributioñ dXW (x(X,X), ν, p y (X,X)) , (8.6) where x(X,X) = 1 1 + θ 2 (X + θX) , p y (X,X) = To compare the new probability distribution with that considered in the last section, we consider the Fourier transform of ρ(X, Y ): These expressions show the fact that̺ W (X, ν) can be very close to̺(X, ν) if the amplitude A ν has the compact form of which central value is located at ν ∼ P . One can also find that ̺ W (X, ν)=̺(X, ν) exactly in the commutative case with θ = 0.  between FIG. 4 and FIG. 5. Similarly, the case with U < 0 is shown in FIG. 6 and FIG. 7. Any remarkable difference cannot be found also in this case.
Consequently, it is safely to say that the newly defined distribution function̺ W (X, Y ) is valid for describing the present model. Note that our present model and analysis may be very specific; we have only studied quantum cosmologies in terms of wave packets, which have rather semiclassical properties. The study in terms of generic wave functions will be expected in future work. Nonetheless, the verification of the validity through the exact analytic solutions in the present work is the first essential step to study the noncommutative quantum cosmology with noncommutative variables.

IX. DISCUSSION AND OUTLOOK
In this paper, a noncommutative deformation of the minisuperspace variables is studied by means of an integrable model. Its analytical solutions are obtained in classical and quantum cosmology.
It has been already known that the D-dimensional model with an exponential scalar potential V 2 e 2αφ , which is related to higher-dimensional/higher-derivative theories, gives rise to an accelerating universe. We find that the noncommutativity suppresses acceleration if the coupling α is small whereas it enhances acceleration if the coupling α is large. The critical value is given by α 2 = D−1 2(D−2) in our model. We have managed to interpret the probability distribution in noncommutative quantum cosmology. We first showed that the peak of the wave packet reproduces the classical trajectory by using exact solutions with an interpretation of the noncommutative variables in the present model. Next, we proposed a new probability distribution in noncommutative quantum cosmology constructed from the Wigner function. Its validity in the present solvable model is confirmed numerically.
In future study, we will investigate general noncommutative cosmology by using the