Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario

In this work we consider the effects on a quantum cosmology model of dynamical vacuum due to presence of a minimal length introduced by the GUP (generalized uncertainty principle) related to the modified commutation relation $[\hat{x},\hat{p}] := \frac{i\hbar}{ 1 - \beta\hat{P}^2 }$ . We determine the wave function of the Universe $ \psi_{qp}(\xi,t)$, which is solution of the Modified Wheeler-DeWitt equation in the representation of the quase-position space, in the limit where the scale factor of the Universe is small. Thus, physical results can be obtained directly from $ \psi_{qp}(\xi,t)$.


Introduction
The Standard Cosmological Model (SCM) is based on an expanding universe which has a very hot and dense origin. The extrapolation of the SCM for very early times leads to energy scales deep in the Planck regime and, strictly speaking, the expanding universe would have a singular initial state. On the other hand, in order to explain many features of the observed universe today, a period of short but rapid, almost exponential, expansion, which is called inflationary phase, is necessary in the very early universe. So, in the primordial Universe the quantum effects were relevant and a quantum approach is necessary to describe the gravitational effects. A quantum approach to gravity is also necessary in the study of black holes. The features of compact astrophysical objects like neutron stars, and even white dwarfs, may be affected by quantum gravity effects. However, we have no quantum gravity theory which is fully acceptable, although there are many proposal for it. An important aspect of almost all proposals for quantization of the gravity is the prediction of the existence of a minimal length. Thus, an effective description of the effects of quantum gravity or a phenomenological approach can be obtained by way of the introduction of a minimal-length scenario [1,2,3,4]. In addition, the introduction of a minimal length could also solve the problems of the initial singularity (big bang singularity) [5,6,7,8,9,10,11,12,13], of the black hole complete evaporation (catastrophic evaporation of a black hole) [3,14,15,16,17,18], of the cosmological constant [12,19,20,21], of the Chandrasekhar limit [4,18,20], of the instability of the Einstein static universe (emergent universe) [22], of the cosmological constant vanishes in multiverse theories [19], and so on.
We can obtain a minimal-length scenario by modifying the Heisenberg uncertainty principle (HUP) between the position and momentum operators. There are several different proposals of modification of the HUP which introduce a minimal-length scenario. Those modified uncertainty principles are called Generalized Uncertainty Principle (GUP) 1 . GUP's have been derived from different contexts such as string theory [27,28,29,30,31,32], black hole physics [19,33,34,35] and extra dimensions [36].
Although the most common modified or deformed commutation relation (MCR) associated with a GUP (KMM GUP [37]) is quadratic in the momentum operator, effects of a MCR with a linear term in the momentum operator, which leads to a minimal length and a maximum momentum [38], has been studied in contexts of Cosmology models [10,11,39,40,41] but, nevertheless, apparently this GUP leads to non-unitary theories [40] and it is not compatible with current data available [42]. In the most the inflation is described by a scalar field. So, in an inflationary quantum cosmology theory it is necessary to performed the second quantization of that scalar field. Consequently, in a minimallength scenario that scalar field and its conjugate momentum have to obey to modified commutation relations [43,44,45,46]. As it is well known, the ordinary approach of the third quantization 2 of the Wheeler-DeWitt (WDW) equation leads to the vanishing of the cosmological constant. However, it might be possible to obtain a non-zero cosmological constant in minimal-length scenario [19]. Many authors have studied effects of a minimal length in cosmology using a classical approach in which the Poisson brackets are modified according to the correspondence principle [10,22,47,48,49,50,51]. In the vast majority of the GUP's the parameter related to the minimal length (deformation parameter) is positive. Although GUP's with positive deformation parameter prevent black holes evap-orate completely they remove the Chandrasekhar limit 3 , that is, a white dwarf star could turn out arbitrarily large [52,53]. In [18,20] the author shows that a GUP with a negative deformation parameter can restore the Chandrasekhar limit and in spite of allowing a black hole evaporates completely this takes an infinite amount of time. Even though a negative deformation parameter is unusual, it is consistent with a description in which the universe has an underlying crystal lattice-like structure [54]. Mu-In Park proposed that the HUP can be modified to include the cosmology constant term, which is called extend uncertainty principle (EUP) [35,55]. Whereas in a GUP there is a quadratic term in the momentum uncertainty, a EUP has a quadratic term in the position uncertainty. A GUP that has also a quadratic term in the position uncertainty is called GEUP. In [4] the authors use a GEUP to show that a non-zero cosmological constant can restore the Chandrasekhar limit. Effects of a GEUP in cosmology has been studied in [56]. Modified uncertainty principles that induce a maximum length has been employed in order to describe the cosmological particle horizon [57]. This modified uncertainty principle has terms proportional to even powers in the position uncertainty [58]. Cosmological observational data have been used to obtain constraints for GUP parameters [21,42,48,56,59].
Even though the de Sitter Cosmology Model describes the rapidly expansion phase of the Universe, during which vacuum energy dominates, its traditional treatment considers a system without physical content, since there is only single degree of freedom and one constraint [60]. Schutz formalism [61,62], which describes a relativistic fluid interacting with the gravity field, can be used to regard the vacuum as dynamic entity having different degrees of freedom. Besides this process overcomes the difficulties found in de Sitter Model, it allows a natural way of introducing a variable playing the role of time. Then, the Dynamical Vacuum Model is a model of homogeneous and isotropic universe, filled with vacuum fluid whose state equation is P = −ρ, classically treated according to the Schutz canonical formalism. This process rends a linear Hamiltonian in one of the momenta whose associated degree of freedom of the fluid will play the role of time [63].
In general, with the intention of avoiding the problems caused by the Wheeler-DeWitt equation defined in the superspace 4 (the space of all possible three dimensional metrics), the process of quantization occurs into the mini-superspace approach [64], where an infinite number of degrees of freedom of the gravity field is frozen 5 and remaining degrees of freedom are turned operators.
Of course, if the GUP is a fundamental aspect of the nature then not only the position and the momentum have to obey the GUP as well as every others variables which will be quantized [65]. Since a GUP corresponds to a modification of the commutation relation between the operator and its conjugate operator, the Wheeler-DeWitt equation in a minimal-length scenario can be obtained imposing that some or all of those operators (which have come from the quantization of remaining degrees of freedom in the mini-superspace approach) and its conjugate momentum operators satisfy a modified com-mutation relations.
In this work, our primary purpose is to determine the corrections in the wave function of the Universe due to the use of a specific generalized uncertainty principle (GUP) in a quantum cosmology model. Hence, we determine the modified Wheeler-DeWitt Equation up to O(β 2 ) considering a quantum cosmology model of dynamical vacuum in a minimallength scenario induced by the commutation relation proposal by P. Pedram [66,67], which induces a minimal uncertainty in the position, ∆x min = 3 β, and an upper bound on the conjugate momentum, P max = 1 √ β . In above equation β is a parameter related to the minimal length. An important aspect of the GUP associated with this modified commutation relation is that it is not perturbative (in the minimal length), consistent with Doubly Special Relativity (DSR) theories which predicts an upper bound for particle momentum [59,68,69,70] and it is in agreement with several theories proposals of quantum gravity [66]. Then, we find the wave function of the Universe for small scale factors in the formal representation of the "position" space, in fact representation of the scale-factor space 6 . However, we can not obtain from that wave function any physical information since the eigenstates of the "position" operators are not physical states [37]. We overcome this problem for obtaining the wave function in the representation of the quase-position space as a superposition of wave functions in the formal representation of the "position" space [71]. Since it may be questioned whether this procedure is correct, we find the Wheeler-DeWitt equation in the representation of the quase-position space and show that the wave function in the quase-position space previously obtained is solution this equation, as we expected. Last but not least, it should be said that the quantization process occurs into the mini-superspace scenario in which the scale factor is the only degree of freedom. As far as we know, the first applications of a minimal-length scenario to a mini-superspace dynamics can be found in [5,7,8].
In short, in this paper we intend to study a dynamical vacuum, using a description given by the Schutz formalism, in a quantum cosmological scenario where the minimallength proposal is explicitly considered. Our main aim is to verify how GUP affects the solutions for the WDW equation with a dynamical vacuum state, in the mini-superspace, in comparison with the usual approach.
The paper is organized in the following way. In Section 2, we describe the minimallength scenario used, presenting its main results. In Section 3, we describe the ordinary cosmology model, that is, in a scenario without minimal length. We obtain the gravity action and the fluid action according to the Schutz formalism. In Section 4, we determine the modified Wheeler-DeWitt equation in the formal representation of the "position" space and we find its solution for small values of the scale factor. We also discuss about the validity range at which we may consistently work. Then, we find the physically acceptable solutions, that is, the wave function of the Universe in the representation of the quaseposition space. In Section 5, we present our comments and conclusions.

Minimal-Length Scenario
In a quantum approach, a minimal length can be introduced by modifying the HUP in order to raise a non-zero minimal uncertainty in the position 7 . There are many proposals for modification of HUP [14,37,38,50,57,66,67,73].
In this work we concern with the GUP proposed by P. Pedram [66,67], which induces a non-zero minimal uncertainty in the position given by where β is a parameter related to the minimal length. We choose the GUP (1) because it is consistent with several proposals for quantum gravity, such as string theory, loop quantum gravity, and it also introduces a maximal measured momentum 8 , which is in agreement with the DSR. Another important aspect this GUP is that it is not perturbative. Consequently, if β is small then (1) can be expanded until any order in β we wish to do. Since, then related to the GUP (1) we have the modified commutation relation, Although, the representation of the operators: 7 It is no trivial to show that a non-zero minimal uncertainty in position can be interpreted as a minimal length [23,37,72]. 8 It is worth noting that this GUP does not induce a maximum uncertainty in the conjugate momentum.
wherex andp are the ordinary operators of position and momentum satisfying the canonical commutation relation [x,p] := i , is not an exact representation of the algebra (5), it preserves the ordinary form of the position operator. Thus, in this representation of "position" space we have x|P where |x are the state eigenvectors of the position operator. This representation of "position" space is only formal since the eigenvalues of the position operator (6) are not physical states and consequently they do not belong to the Hilbert space. This is because the position operator uncertainty vanishes when it is calculated in any of its eigenstates. But that is physically impossible since ∆x ≥ 3 √ β for all physically allowable state in a minimal-length scenario. However, all information on position can be accessible through the maximal localization states, defined as and In the DGS (Detournay, Gabriel and Spindel) approach [74] the maximal localization states are found to be 9 The representation of quase-position space is obtained by projecting the state vectors onto the maximal localization states, The action of the position and the momentum operators on the quase-position space are given by and with and b := 3π 4 .
Hence, up to O(β 2 ) we have and Finally, one can show that the wave function in the quase-position space is a superposition of the wave functions in the "position" space, given by

The Model
We are going to consider the de Sitter Cosmology Model, which describes the phase of the Universe rapidly expanding, during which the vacuum energy dominates the energy density and gives rise the term corresponding to the cosmological constant Λ.

Gravity Action
The gravity action more general is given by 10 where R is the Ricci curvature scalar, K is the trace of the extrinsic curvature K ij , g is the determinant of the metric g µν , h is the determinant of the induced metric over the three-dimensional spatial hypersurface and ∂M is the bounded of the four-dimensional manifold M. According to the Cosmological Principle, that is, for the homogeneous and isotropic universe the metric is that of FLRW (Friedmann-Lemaître-Robertson-Walker), where N(t) is the lapse function, k = −1, 0, +1 for hyperbolic open, flat open and close universe, respectively, and a(t) is the scale factor of universe.
Using the FLRW metric, Eq. (22), the gravity action becomes Since S g = dtL g we have that the gravity Lagrangian is given by Hence the gravity Hamiltonian, NH g =ȧp a − L g , reads where

Fluid Action
In this Model the Universe is fully completed by a perfect fluid. Now, we are going to employ the Schutz's formalism [61,62] for the dynamics description this fluid interacting with the gravitational field [63]. For the fluid whose the state equation is where ρ is the density and α depends on the type of fluid, the pression is given by where µ is the specific enthalpy and S is the entropy. In the Schutz's formalism the four-velocity is defined by means of six potentials: Since ǫ and η are associated with rotation movement, they vanish. From the normalization condition, we get The fluid action is given by Using Eq. (31) into Eq. (28) and because the model is homogeneous and isotropic, we have Thus, the fluid Lagrangian is The conjugate canonical momenta to φ and S are and Therefore, the fluid Hamiltonian, NH F =φp φ +Ṡp S − L F , is given by From Eqs. (36) and (25), it follows that the total Hamiltonian, H = H g + H F , is Now, performing the canonical transformation (re-parametrization) and the total Hamiltonian becomes Note that p T is linear in Eq. (41). When the quantization occurs, this allows us to consider t = −T as playing role of time and to obtain an equation like Schroedinger.

The modified Wheeler-DeWitt Equation
The quantization process takes place in the framework of mini-superspace and according with the Wheeler-DeWitt quantization scheme, and such that H −→Ĥ and thus satisfying the Wheeler-DeWitt equation, In order to obtain a minimal-length scenario we demand that with the representation of the operators as follow: where thex andp operators satisfy the canonical commutation relation, Using the representation (46) and (47), the modified Wheeler-DeWitt equation turns out where we have made T = −t. Now, projecting Eq. (50) onto the formal representation of the "position" space, we have where we have used Eqs. (8) and (9).
Assuming that the solution of the above equation can been written as ψ(x, t) = ϕ(x)τ (t), we found the stationary solutions and the time independent equation Lastly, considering a flat universe (k = 0), without any other matter content, fully completed by a dynamical vacuum with state equation and Λ = 0 11 , Eq. (53) up to O (β 2 ) becomes 12

Solution of the modified Wheeler-DeWitt equation
It is no possible to solve Eq. (55) analytically since we do not know all initial or boundary conditions 13 , that is, ϕ (N ) (x 0 ), for N = 0, 1, ..., 5. A way to get around this problem is considering x small. This is well reasonable since we are interested in solutions describing the Universe in its quantum regime, that is, in the initial phases of the Universe, when the scale factor was small 14 , and as it is known quantum effects are significant only for small values of the scale factor.
Initially, we take into account Eq. (55) in the limit β = 0, which we named ordinary Wheeler-DeWitt equation, 11 Note that if Λ > 0 we can always redefine ω′ := ω + 2Λ. 12 From now on, for the sake of simplicity we are going to omit the subscript ω. 13 In Ref. [2] the author has solved a modified Wheeler-DeWitt equation of forth order d 4 dx 4 using the Sommerfeld polynomial method. However, the solution holds dependent on four parameters that must be determined by the initial conditions, which are not known.
14 Later on, we will discuss more carefully the issue if x can rigorously describe the scale factor.
It is easy to see that in the limit for small x and Now, imposing the condition 15 that ϕ 0 (0) = 0 we get B = 0, and the solution is given by Remembering that and retaining the first three significant terms of ϕ 0 (x) we get It is appropriate we rewrite the solution (63) absorbing C 1 in a normalization constant, which we are going to omit without affecting the results, We can obtain an approximate solution for Eq. (55), in the range of x small, using Eq. (64) into O (β) and O (β 2 ) terms. Thus, and With the approximation ϕ ≈ x into Eqs. (65) and (66), Eq. (55) becomes Subsequently, we are going to find the approximate solution of the above equation in two different ways.

Solution: first method
Since x = 0 is an ordinary point of Eq. (67) we can find a solution in power series of x. Then, replacing ϕ(x) = ∞ n=0 a n x n into Eq. (67) and retaining only significant terms, we have The parameters A, B and C are determined by requiring that (68) satisfies Eq. (67) up to order chosen in x. Then, substituting (68) into (67) we have A = 1, B = 38 43 and C = 3 7 . Therefore, the solution reads as

Solution: second method
The same previous solution can be obtained by taking the approximation following Integrating twice the above equation we get where α 1 and α 2 are integration constants. After that, using (64) into (71) we obtain The constants α 1 and α 2 can be determined by demanding that ϕ(x) −→ ϕ 0 (x) when β −→ 0. In the similar way, the parameters A ′ , B ′ and C ′ are found requiring that the solution (72) satisfies Eq. (67). Consequently, we again have the solution (69), as we expected.

Validity range
In order that the solution (69) works consistently the first disregarded term in ϕ 0 (x) must be much smaller than the last kept terms in O(β) and O (β 2 ), that is, x ≪ β ω 1 8 . Since β is much small, that drastically reduces the validity range for our solution. We can improve this if terms of higher powers in x are taken account in order O (β 0 ) part of the solution. Therefore we can consider √ xJ 1/6 (K 0 x 3 ) as the part in O (β 0 ) of ϕ(x), that is, . It is clear that terms of order such that x 6n−2 < β 2 ω 1−n lie far outside the validity range and they must throw away. The same reasoning applies in the case of terms in O (β 2 ) when compared to terms in O(β). This way, with the purpose of increasing the validity range of our solution we take account the two first significant terms in O(β). In conclusion, in light of the forgoing the solution turns out 16 In dealing with expansions of small parameters (in this case β) it is necessary to take care to consistently work when x goes to an extremely small value. It is easy to see that terms in β 0 are like ω N x 1+6N , terms in β are like βω N x 5+6N and terms in β 2 are like β 2 ω N x 3+6N . Therefore, the terms in βωx 5 and βω 2 x 11 are negligible when x < √ β and the term βω 2 x 11 is negligible when √ β < x < β 1/8 . Note that the term in β 0 x always will be greater than the term in β 2 ωx 3 for x small.
Keep in mind that in fact solution (73) is ϕ ω (x), that is, eigenfunctions whose associated eigenvalues are values of the cosmology constant.

Physically acceptable solutions
As we have already said, we can not obtain directly from ϕ(x) physical results because thex operator eigenstates do not belong to the Hibert space. Nevertheless, the projections of the state vectors |ϕ on the maximal localization states, that is, the wave functions in the representation of the quase-position space do. The wave functions in the quaseposition space can easily be obtained from (73) by using Eq. (20). Therefore, replacing (73) into (20) we have 17 ϕ qp (ξ) = ξJ 1/6 K 0 ξ 3 + β − 16 + 12b 2 ωξ 5 + 4864 385 Hence, above quase-position wave function represents the probability amplitude for the Universe being maximally localized around the position ξ.
The reader may be questioning if the correct is to determine the solution of the modified Wheeler-DeWitt equation in the representation of the quase-position space, ψ ml ξ | −p 2 a + 24ωâ 4 |ψ(t) = 0.

Conclusion
In this work, we saw that the study of the primordial Universe requires a quantum approach, in which the gravitation effects must be taken in consideration. It follows the need to implement a minimal-length scenario, which is carried out by using a GUP. Since we chose the GUP (1) we can expand the representation of the momentum until O(β 2 ) and thus to obtain a modified WDW equation up to O(β 2 ), too.
We found the modified WDW equation in the formal representation of "position" space, Eq. (55), and its solution ϕ(x), Eq.(73), because it is simpler than in the representation of quase-position space. However, we can not obtain directly from ϕ(x) physical results. Consequently we obtained the wave function the Universe in the representation of quaseposition space, ϕ qp (ξ), as a superposition of two wave functions of the Universe in the formal representation of "position" space, ϕ qp (ξ) = 1 √ 2 [ϕ(ξ + x min ) + ϕ(ξ − x min )]. With the aim of insuring our result, we found the modified WDW equation in the representation of quase-position space, Eq.(76), and we checked that ϕ qp (ξ) is actually its solution.
The ignorance of the initial or boundary conditions, that is, of the derivatives of the wave function does not allow us to find an exact solution for modified WDW equation, what forced us to seek a solution for small values of the scale factor of the Universe.
In next works, we are going to study the effects on the scale factor evolution, applications of the obtained results and comparisons with others models and GUP's.