We set up the WheelerDeWitt WDW equation for late gravitational collapse. The fact that the gravitational collapse and the expanding collapsing universe can be described within the realm of the RobertsonWalker metric renders the corresponding WDW equation for collapsing matter a timeless Schrdinger equation. We explore the consequences of such an equation and find the density to be quantized in terms of the Planck density. Apart from that the wave function as a solution of the WDW equation shows that the initial singularity is avoided. We concentrate on different factor orderings in the kinetic term of the equation and show how after splitting off an exponential ansatz new polynomials entering the solution can be constructed. This enables us to conclude that the factor ordering changes the details of the solution and interpretation but overall on a qualitative level the results remain the same. We also probe into the effects of a positive cosmological constant. It offers the possibility of a tunneling scenario at the cosmological horizon.
['Wheeler–DeWitt equation', 'Gravitational collapse', 'Cosmological constant', 'Robertson–Walker metric', 'Black hole', 'White hole', 'General Relativity and Quantum Cosmology']
WheelerDeWitt equation and the late gravitational collapse Effects of factor ordering and the tunneling scenario
{'Wheeler-DeWitt equation and the late gravitational collapse: effects of factor ordering and the tunneling scenario': 'Davide Batic, 1, ∗ M. Nowakowski, 2, † and N. G. Kelkar 3, ‡ 1 Department of Mathematics, Khalifa University of Science and Technology, Main Campus, Abu Dhabi, United Arab Emirates 2 ICTP South American Institute for Fundamental Research, Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 S˜ao Paulo, SP Brazil 3 Departamento de Fisica, Universidad de los Andes, Cra. 1E No. 18A-10, Bogota, Colombia  \n(Dated: September 10, 2024)  \nWe set up the Wheeler-DeWitt (WDW) equation for late gravitational collapse. The fact that the gravitational collapse and the expanding/ collapsing universe can be described within the realm of the Robertson-Walker metric renders the corresponding WDW equation for collapsing matter a timeless Schrodinger equation. We explore the consequences of such an equation and find the density to be quantized in terms of the Planck density. Apart from that, the wave function as a solution of the WDW equation shows that the initial singularity is avoided. We concentrate on different factor orderings in the kinetic term of the equation and show how after splitting off an exponential ansatz, new polynomials entering the solution can be constructed. This enables us to conclude that the factor ordering changes the details of the solution and interpretation, but overall on a qualitative level the results remain the same. We also probe into the effects of a positive cosmological constant. It offers the possibility of a tunneling scenario at the cosmological horizon.  \nPACS numbers:', 'I. INTRODUCTION': "The gravitational collapse is one of the most interesting phenomena which General Relativity has to offer [1-3]. In its extreme case, it gives rise to compact objects like white dwarfs, neutron stars, and black holes [4]. At least in the case of black holes, we expect some effects and fingerprints of quantum gravity, as partly evidenced by the Hawking radiation [5]. Indeed, many suggestions have been put forward in this direction [6, 7]. But one obstacle in the line of research is the choice of the quantum gravity model. Indeed, quantum gravity [8] has already a history close to a hundred years [9, 10] and yet we still cannot decide which one of the many candidates [11] will lead to a consistent quantum theory of gravitation. Different points of view about the nature of spacetime quantization and the lack of experimental data in this region make quantum gravity an active area of research. One of the persistent attempts was to consider General Relativity (GR) as a field theory and quantize it canonically. The prerequisite for such an undertaking is to identify dynamical variables [12] and separate the dynamics from constraints. The result of the canonical quantization is the WDW equation [13] which has been used in quantum gravity over fifty years [14-16] and remains popular until today. The reason for this success is, on the one hand, the fact that canonical quantization has proved otherwise to be a powerful method in quantizing field theories and, on the other hand, that the WDW equation for certain kind of metrics is relatively simple to handle and can be partly interpreted. Regardless of the eventual form that a complete quantum gravity theory takes, it is probable that its connections to canonical quantization and the WDW equation will persist. Although not universally applied (see, for example, [17, 18]), the majority of studies exploring quantum gravity through the WDW equation utilize the Robertson-Walker (RW) metric. This approach is predominantly employed in the field of cosmology [19], where it serves to investigate questions concerning the origins of the universe. Clearly, one expects that quantum effects are important in the early stages of the universe. We emphasize here that the RW metric in GR is also used in gravitational collapse [1] which suggests that one could examine also the WDW equation in the context of a gravitational collapse. Here, we would expect that quantum effects become significant as the collapse progresses towards smaller scale factors. In [26], we have studied such a scenario  \nin its simplest version, i.e., we chose the simplest scheme of the operator factoring and put the cosmological constant to zero. The outcomes were notably promising: the initial singularity was circumvented, density became quantized, and the probability density revealed distinct sharp peaks. These peaks suggest a form of spacetime quantization on a probabilistic basis. Therefore, we revisited the topic to explore the effects of varying factor ordering, incorporate the cosmological constant, and examine a tunneling solution.  \nIn cosmology, the WDW equation is frequently applied in scenarios devoid of matter, a condition that is untenable in the context of collapse. Consequently, we focus on identifying an appropriate Lagrangian, from which the classical Friedmann equations naturally emerge as Euler-Lagrange equations. This Lagrangian is then used to perform the canonical quantization, culminating in the WDW equation with matter. Working under the assumption that all mass is behind the black hole horizon, we investigate the solutions to this equation across various factor ordering schemes. The most basic variant of this model has been examined in [26], yielding the promising outcomes described previously. Although it is not an eigenvalue equation, the requirement for square integrability leads to the quantization of density in terms of the Planck density. Given that factor ordering introduces an ambiguity in selection, investigating these various possibilities holds significant importance. In the initial cases, we successfully solve the corresponding WDW equation using an exponential ansatz. The resultant equation indeed possesses polynomial solutions. This presents a quantization condition analogous to the simplest case, and the physical interpretation remains consistent. These polynomials allow us to construct the complete wave function explicitly. As in the simplest case, the initial singularity is avoided and the scale parameter has preferred values in the probabilistic sense.  \nWith the introduction of a positive cosmological constant, the potential within the WDW equation forms a barrier, setting the stage for an analysis of quantum tunneling. We estimate the tunneling probability, interpreting it as the likelihood of tunneling at the cosmological horizon. Given that in the presence of a positive cosmological constant, the Hamiltonian becomes unbounded, assessing the self-adjointness of this operator is crucial. This assessment is conducted in a dedicated subsection  \nWe draw the reader's attention to the fact that several other papers examined the gravitational collapse in the context of a Schroodinger-like equation. In [27, 28], the divergence from our methodology lies in the Hamiltonian, which distinctly is not the WDW operator. Furthermore, we highlight the notable absence of a time variable in the WDW equation, a prevalent issue in quantum gravity also encountered in the canonical quantization framework [2940]. Contrarily, in studies such as [27, 28], and others like [41-43], Schrodinger-like equations in the form i∂ t Ψ = H Ψ are presented, which starkly contrast with the WDW formulation H Ψ = 0. Thus, these methodologies diverge from ours right from the outset.  \nThe paper is organized as follows. In sections II and III, we list the salient features of [26] where the simplest scheme of factor ordering was used. In section IV, we study a more general factor ordering and compare the outcome with our previous results. Section V is devoted to effects when the cosmological constant Λ is included. We analyze the resulting potential and the self-adjointness of the WDW operator. Next we give a general quantization condition for the density in the presence of this constant. We outline a tunneling scenario connected with Λ. In section VI, we give our conclusions.", 'II. THE WDW EQUATION IN THE GRAVITATIONAL COLLAPSE': "As we mentioned in the introduction, the gravitational aspects of collapsing matter and the salient details of a universe are described by the Friedmann-Robertson-Walker line element with c = 1 [1]  \nds 2 = dt 2 -R 2 ( t ) [ dr 2 1 -kr 2 + r 2 dϑ 2 + r 2 sin 2 ϑdϕ 2 ] . (1)  \nHowever, in cosmology, the curvature parameter k takes on values of ± 1 , 0, rendering r as dimensionless. Meanwhile, the scale factor R carries a dimension of length. In the collapse scenario for cold dust, we have [1]  \nk = 8 π 3 Gρ 0 , (2)  \nwhere G is the Newton's gravitational constant and ρ 0 is the density of a spherically symmetric, isotropic and homogeneous matter distribution. Considering that our main objective is to explore quantum effects in the latter stages of collapse, we note the improbability of densities conforming to the conventional values typically associated with the onset of collapse, as documented in [44]. Our emphasis is, therefore, on the post-black hole formation phase of the collapse. In natural units, where /planckover2pi1 = c = 1, it is obvious that k has the dimension L -2 . The apparent mismatch in dimensions between the cosmological scale factor R and the same factor in the context of gravitational collapse can  \nbe easily remedied by the following rescaling  \na ( t ) = L 0 R ( t ) , ˜ r = r L 0 , L 0 = 1 √ k . (3)  \nAs L 0 carries the dimension of length, the scale factor a now possesses the same dimension. This allows us to recast the line element (1) as  \nds 2 = dt 2 -a 2 ( t ) [ d ˜ r 2 1 -r 2 + ˜ r 2 dϑ 2 + ˜ r 2 sin 2 ϑdϕ 2 ] . (4)  \n˜ This transformation makes possible to draw parallels between gravitational collapse and cosmology when k = 1 in the latter case. Especially, the derivation of the WDW equation without matter will be the same in both cases. For this purpose, we follow the methodology outlined in [46, 47]. We refer for details to [26] and quote here the final result in terms of the action  \nS = 3 π 4 G a 2 ˙ a -3 π 4 G ∫ dt [ a ˙ a 2 -a + Λ 3 a 3 ] -3 π 4 G a 2 ˙ a + S matter , (5)  \n= -3 π 4 G ∫ dt [ a ˙ a 2 -a + Λ 3 a 3 ] + S matter . (6)  \nWe highlight here that constructing a matter Lagrangian that yields the energy-momentum tensor of a perfect fluid is not a straightforward enterprise [48, 49]. However, if we carefully constrain the variation δg µν , we find that a suitable candidate is provided by  \nS matter = ∫ M d 4 x √ -gL matter = -∫ M d 4 x √ -gρ = -2 π 2 ∫ a 3 ρdt. (7)  \nIt results in a total Lagrangian expressed by the equation  \nL = -3 π 4 G ( a ˙ a 2 -a + Λ 3 a 3 ) -2 π 2 a 3 ρ. (8)  \nIndeed, one can show [26] that this Lagrangian leads to the correct Friedmann equation and is suitable for quantization. Taking into account that the Hamiltonian H = π a ˙ a -L has conjugate momentum [46] (we take the opportunity to point out that there is a typographical error in [26] in connection with this equation)  \nπ a = -3 π 2 G a ˙ a, (9)  \nit is straightforward to derive the Hamiltonian  \nH = -G 3 π π 2 a a + 3 π 4 G ( -a + Λ 3 a 3 ) +2 π 2 a 3 ρ. (10)  \nApplying the canonical quantization prescription π a →-id/da , the WDW equation H Ψ( a ) = 0 becomes  \n[ G 3 πa d 2 da 2 + 3 π 4 G ( -a + Λ 3 a 3 ) +2 π 2 a 3 ρ ] Ψ( a ) = 0 . (11)  \nFor later purposes, it is of some importance to note that we have chosen a factor ordering  \nπ 2 a →-d 2 da 2 . (12)  \nThe consequences of different factor ordering choices in the context of gravitational collapse will be further explored in the following section. Finally, we can cast (11) in the form  \n( -d 2 da 2 + V eff ( a ) ) Ψ( a ) = 0 , V eff ( a ) = -9 π 2 4 G 2 ( -a 2 + Λ 3 a 4 ) -6 π 3 G a 4 ρ. (13)  \nAn examination of dimensions reveals that all terms within the round brackets of equation (13) share a consistent dimension of M 2 . This is in contrast to [50] where a similar equation has been independently derived through different means. In the context of cold dust, where ρ = ρ 0 ( a 0 /a ) 3 , the corresponding WDW equation is  \n[ -d 2 da 2 + ( -3 π 2 Λ 4 G 2 a 4 + 9 π 2 4 G 2 a 2 -6 π 3 ρ 0 a 3 0 G a )] Ψ( a ) = 0 . (14)  \nComing back to the original variables R ≡ a = a/L 0 with L 0 defined as in (3), the above equation takes the form  \n˜ Furthermore, if we normalize the radial coordinate r so that ˜ a 0 = 1 [1] and introduce the Planck density ρ Pl = 1 /G 2 as well as the vacuum density associated with the cosmological constant ρ vac = Λ / (8 πG ) then (15) becomes  \nwith  \nThe parameters are defined as follows  \n˜ [ -d 2 d a 2 + ( -3 π 2 Λ L 6 0 4 G 2 ˜ a 4 + 9 π 2 L 4 0 4 G 2 ˜ a 2 -6 π 3 ρ 0 ˜ a 3 0 L 6 0 G ˜ a )] Ψ( ˜ a ) = 0 . (15)  \n( -d 2 d ˜ a 2 + U eff ( ˜ a ) ) Ψ( ˜ a ) = 0 (16)  \nU eff ( ˜ a ) = α ˜ a 4 + β ˜ a ( ˜ a -1) . (17)  \nα = -ρ vac ρ 0 β, β = 81 256 ( ρ Pl ρ 0 ) 2 . (18)", 'III. SOLUTION OF THE WDW EQUATION IN THE CASE p = 0': "In our analysis, we neglect the cosmological constant Λ due to its comparatively negligible contribution to the effective potential. To clarify, the parameter β defined in (18) is extremely large due to the significant difference between the Planck density ( ρ Pl ∼ 5 . 1 × 10 96 kg/m 3 ) and the initial density of the collapsing dust cloud ( ρ 0 ∼ 10 -16 kg/m 3 ). Conversely, the vacuum energy density associated with Λ ( ρ vac ∼ 5 . 9 × 10 -27 kg/m 3 ) is many orders of magnitude smaller. Therefore, the term involving Λ in the effective potential is insignificant compared to the terms involving β , justifying its omission in this context. The simplified effective potential is thus given by  \nU eff ( ˜ a ) = β ˜ a ( ˜ a -1) , (19) which represents a parabola with a minimum at ˜ a = 1 / 2. The equation at hand is not an eigenvalue equation for a shifted harmonic oscillator, but it is clear that there exists a connection. Indeed, by introducing the transformation ̂ a = ˜ a -1 / 2, we can recast the WDW equation into  \n-d 2 Ψ d a 2 + β ̂ a 2 Ψ( ̂ a ) = β 4 Ψ( ̂ a ) . (20)  \n̂ subject to the boundary condition Ψ( ˜ a ) → 0 as ˜ a → + ∞ and with the normalization condition ∫ ∞ 0 | Ψ( ˜ a ) | 2 d ˜ a = 1. These constraints are motivated by the requirement of square integrability and the physical interpretation of the wave function. The condition at infinity ensures that the wave function is normalizable, reflecting the probabilistic nature of quantum mechanics, while the condition at a = 0 corresponds to avoiding the classical singularity, in line with the expectation that quantum gravity effects prevent such singularities. Furthermore, these boundary conditions are analogous to those used in the quantum harmonic oscillator problem, where normalizable solutions must vanish at infinity.  \nIn this reformulation, β emerges as a characteristic parameter, thereby playing a fundamental role in the governing wave equation. If we introduce the dimensionless variable ξ = 4 √ β a , (20) becomes  \n̂ d 2 Ψ dξ 2 = ( ξ 2 -K ) Ψ( ξ ) , K = √ β 4 . (21)  \nThis form is reminiscent of a dimensionless harmonic oscillator [51]. This leads to the following ansatz for the unnormalized wave function  \nΨ( ξ ) = h ( ξ ) e -ξ 2 / 2 , (22)  \nwhich applied to (21) leads to the Hermite differential equation  \nd 2 h dξ 2 -2 ξ dh dξ +( K -1) h ( ξ ) = 0 . (23)  \nSince the requirement for square integrability is satisfied when K = 2 n +1, it leads to  \nβ = 16(2 n +1) 2 , n = 0 , 1 , 2 , · · · . (24)  \nIn other words we obtain a quantization for the density in the form  \nρ 0 ≡ ρ 0 ,n = 9 ρ Pl 64(2 n +1) , n = 0 , 1 , 2 , · · · . (25)  \nFinally, it can be easily checked that the ground state wave function is  \nψ 0 ( ˜ a ) = c 0 e -2 ( ˜ a -1 2 ) 2 , c 0 = √ 2 √ √ π [1 + erf(1)] . (26)  \nFIG. 1: Plot of the probability density | Ψ n | 2 for the ground state n = 0 (dotted line), n = 2 (dashed line) and n = 6 (solid line).  \n<!-- image -->  \nAs n increases, the discrepancy between successive ρ 0 ,n values diminishes, eventually approaching a continuum, a trend supported by the probability density | Ψ n | 2 as depicted in Figure 1. This indicates that at lower n values, the scale parameter exhibits preferred positions in probabilistic terms, while at higher n , it transitions back to a continuous spectrum. Like many phenomena related to quantum effects in the realm of black hole physics, empirical verification remains elusive. However, we expect that quantum gravity theories will naturally avoid the central singularity, an outcome that is in line with various theoretical frameworks, including the Fuzzball paradigm in string theory [20-22], which states that singularities are replaced by complex quantum states (fuzzballs). Although our approach does not explicitly adopt the Fuzzball paradigm, the avoidance of singularities through quantum gravitational effects is a common theme [23-25]. Both our results and the Fuzzball paradigm suggest that a complete theory of quantum gravity will fundamentally change our understanding of black hole interiors.  \nA speculative note concerns the presence of the wave function beyond the value 1: if all matter is enclosed within the black hole horizon, the value 1 denotes the horizon boundary [26]. This can be seen by starting with ρ 0 + M 0 / (4 π/ 3) a 3 s . Using the latter and writing a s = L o ˜ a s , one can infer that ˜ a s = 1 corresponds to a s = 2 GM 0 . Thus, the wave function's extension beyond this value could be associated with phenomena such as Hawking radiation [26]. Alternatively, it may indicate that the black hole is transitioning to a ground state characterized by n = 0, potentially resulting in a black hole remnant.  \nIn conclusion, we end this section by noting a typographical error in equation (43) as cited in [26]. The argument of the Hermite polynomials presented therein should be written as 2 √ 2 n +1( ˜ a -1 / 2).", 'IV. EFFECTS OF DIFFERENT FACTOR ORDERINGS': 'Our focus now shifts to examining how the results of the previous section are influenced by variations in the choice of factor ordering. The ambiguity in factor ordering is a critical aspect to explore, as it is essential to determine whether it leads to qualitative consistency or discrepancy in our findings. The process of quantization inherently involves replacing momenta with differential operators, introducing uncertainties. This necessitates the inclusion of a factor ordering parameter p in the WDW equation, following the prescription outlined in the literature [46, 52, 53]  \nπ 2 a →-a -p [ d da a p d da ] = -( d 2 da 2 + p a d da ) . (27)  \nFor the purpose of achieving congruence with empirical observations, it is essential to meticulously address the intrinsic ambiguity in the WDW equation. This involves discerning the operator ordering factor through the imposition of relevant constraints. A noteworthy candidate for the ordering factor is p = 1, rendering (27) similar to the radial component of the Laplacian operator [54, 55]. Furthermore, [56] derived an exact solution for the WDW equation when subjected to this particular choice. Alternatively, [57], under the premise that the universe wave function is finite, advocated for the adoption of p = -2. Additionally, [58] postulated a boundary condition which stipulates that the ordering parameter be confined within the range 0 ≤ p ≤ 2. It is imperative to emphasize, as pointed out by [56, 59, 60], that the effect of operator ordering becomes prominent only when the universe is relatively small, indicating that the factor ordering issue cannot be resolved purely through the semiclassical limit [61]. Moreover, it is relevant to acknowledge the observations by [56], which cast doubt on the conclusiveness of the arguments presented by [54] in favor of a particular choice for p . Consequently, at this step, we will exercise restraint in committing to a specific value of p . A definitive choice will be contingent upon the solution of the WDW equation satisfying the boundary conditions introduced immediately after equation (20).  \nFor the generalized prescription as illustrated in (27), we can derive the corresponding WDW equation for cold dust from (12). The result is as follows  \n[ d 2 da 2 + p a d da + ( 3 π 2 Λ 4 G 2 a 4 -9 π 2 4 G 2 a 2 + 6 π 3 ρ 0 a 3 0 G a )] Ψ( a ) = 0 . (28)  \nAdhering to the approach adopted in the previous section, we apply the rescaling ˜ a = a/L 0 , in conjunction with the Planck and vacuum densities. This allows us to rewrite the above equation as  \n˜ ˜ ˜ Here, α and β are defined as in (18). For the remaining part of this section, we are particularly interested in the case when Λ = 0, hence, we will set α = 0. In order to proceed with our analysis, we start with the following ansatz  \nd 2 Ψ d a 2 + p a d Ψ d a -[ α ˜ a 4 + β ˜ a ( ˜ a -1) ] Ψ( ˜ a ) = 0 . (29)  \nΨ( a ) = e -√ β 2 ( ˜ a -1 2 ) 2 g ( a ) . (30)  \n˜ a d 2 g d a 2 +( p + √ β ˜ a -2 √ β ˜ a 2 ) dg d a + [ p 2 √ β + ( β 4 -√ β -p √ β ) ˜ a ] g ( ˜ a ) = 0 . (31)  \n˜ ˜ In this context, we focus on the cases where the unknown function g ( ˜ a ) is a polynomial. We can see that the combination of equations (29) and (30) leads to the following differential equation  \n/negationslash  \n˜ ˜ √ β = 4( p +3) , g 1 = -√ β 2 g 0 = -2( p +3) g 0 , p 2 +5 p +4 = 0 . (32)  \n˜ ˜ We initially observe that the constant polynomial solution g ( a ) = g 0 = 0 is attained for √ β = 4( p +1) and p = p 0 = 0. These conditions correspondingly yield β = 16. Encouragingly, as anticipated, this solution indeed reproduces the eigenfunction given by (26) with g 0 = c 0 . Building on this initial result, we now extend our investigation to encompass additional solutions, such as g ( a ) = g 0 + g 1 a . In this case, we find that the latter is indeed a solution, provided that  \nThe latter equation has roots p = -1 and p = -4. The last root must be disregarded in view of the fact that the condition √ β > 0 requires that p > -3. Hence, p = p 1 = -1 and the corresponding eigenfunction is  \nΨ p 1 ( ˜ a ) = g 0 e -4 ( ˜ a -1 2 ) 2 (1 -4 ˜ a ) . (33)  \nAfter the examination of the case of first-degree polynomials, we will now tackle the construction of quadratic polynomials of the form g ( a ) = g 0 + g 1 a + g 2 a 2 . We find that  \nSince p > -5, the only acceptable root is p = p 2 = -5 + √ 5. The corresponding eigenfunction is given as follows  \n˜ ˜ ˜ √ β = 4( p +5) , g 1 = -√ β 2 g 0 = -2( p +5) g 0 , g 2 = √ β 8 ( 4 + √ β ) g 0 = 2( p +5)( p +6) g 0 , ( p +5)( p 2 +10 p +20) = 0 . (34)  \nΨ p 2 ( ˜ a ) = g 0 e -√ 20 ( ˜ a -1 2 ) 2 [ 1 -2 √ 5 ˜ a +2 √ 5(1 + √ 5) ˜ a 2 ] . (35)  \nThe same method applied thus far allows for the construction of additional polynomial solutions of increasingly higher degrees. In what follows, we limit us to summarize the corresponding results in the case of cubic and quartic polynomial solutions  \n1. g ( ˜ a ) = ∑ 3 n =0 g n ˜ a n : we have √ β = 4( p +7) and  \ng 2 = √ β 8 ( 4 + √ β ) g 0 = 2( p +7)( p +8) g 0 , (37)  \ng 1 = -√ β 2 g 0 = -2( p +7) g 0 , (36)  \ng 3 = -β 3 / 2 ( p +2) + 12 pβ +40 β 48( p +2) g 0 = -4( p +7) 2 ( p 2 +12 p +24) 3( p +2) g 0 . (38)  \nThe only root of the equation  \np 5 +32 p 4 +391 p 3 +2268 p 2 +6204 p +6384 = 0 (39)  \nsatisfying the condition p > -7 is p = p 3 = -4. The corresponding eigenfunction reads  \nΨ p 3 ( a ) = g 0 e -6 ( ˜ a -1 2 ) 2 ( 1 -6 ˜ a +24 ˜ a 2 -784 ˜ a 3 ] . (40)  \ng 1 = -√ β 2 g 0 = -2( p +9) g 0 , (41)  \n˜ 2. g ( ˜ a ) = ∑ 4 n =0 g n ˜ a n : we have √ β = 4( p +9) and  \ng 2 = √ β 8 ( 4 + √ β ) g 0 = 2( p +9)( p +10) g 0 , (42)  \ng 3 = -β 3 / 2 ( p +2) + 12 pβ +40 β 48( p +2) g 0 = -16( p +9)( p +10) p +6 g 0 , (43)  \ng 4 = 2( p +9) 2 ( p 4 +29 p 3 +280 p 2 +1032 p +1272) 3( p 2 +5 p +6) g 0 . (44)  \nThe only root of the equation  \np 7 +52 p 6 +1121 p 5 +12950 p 4 +86160 p 3 +328344 p 2 +661776 p +543744 = 0 (45)  \nsatisfying the condition p > -9 is p = p 4 = -2 . 86171. The corresponding eigenfunction is  \nΨ p 4 ( a ) = g 0 e -12 . 276 ( ˜ a -1 2 ) 2 ( 1 -12 . 276 a +87 . 634 a 2 -223 . 392 a 3 +173 . 903 a 4 ] . (46)  \n˜ ˜ ˜ ˜ ˜ All constructed solutions are regular at ˜ a = 0 and they satisfy the boundary condition Ψ( ˜ a ) → 0 as ˜ a → + ∞ with the normalization condition ∫ ∞ 0 | Ψ( ˜ a ) | 2 d ˜ a = 1. For plots of the probability densities corresponding to the eigenfunctions, as specified in equations (26), (33), (35), (40) and (45) across different values of the ordering parameter p , we refer to Figure 2 and 3. From the analysis of the previously obtained polynomial solutions, a pattern emerges, suggesting the following quantization condition  \n√ β = 4(2 n +1+ p ) 2 , n = 0 , 1 , 2 , · · · . (47)  \nFIG. 2: Plots of the probability density | Ψ p i | 2 for p 0 = 0 (dotted line), p 1 = -1 (dashed line), p 2 = -5 + √ 5 (solid line).  \n<!-- image -->  \nFIG. 3: Plots of the probability density | Ψ p i | 2 for p 0 = -4 (long dashed line) and p 4 = --2 . 86171 (solid line).  \n<!-- image -->  \nHere, n represents the degree of the polynomial solution. By using equation (18), we can reformulate equation (47) as  \nρ 0 = 9 ρ Pl 64(2 n +1+ p ) , n = 0 , 1 , 2 , · · · . (48)  \nTo ensure consistency, we observe that the above result correctly reproduces equation (25) when p = 0. Qualitatively, we also confirm the behavior of the probability density: the probabilistic values of the scale factor exhibit more peaks for the preferred values if we go beyond the ground state. Finally, we observe that the quantization condition (48) constrains the possible values that the ordering factor may take on. More precisely, we need to require that p is not a negative odd number, that is p = -(2 n +1) with n ∈ N ∪{ 0 } .  \n/negationslash  \nIt is important to note that the conclusions reached in equations (33), (35), (40), and (46) are subject to the specific choice of operator ambiguities of a polynomial kind, as expressed in equation (27). If different functions, such as non-polynomial forms, were used for the factor ordering, the resulting WDW equation and its solutions would be significantly different. The redefinition would no longer be guaranteed to be a polynomial in a , leading to potentially different physical interpretations.', 'V. EFFECTS OF THE COSMOLOGICAL CONSTANT': 'Incorporating the cosmological constant into our analysis highlights three key areas worth investigating. Firstly, from a mathematical perspective, the importance of examining the self-adjointness of the Hamiltonian arises. The introduction of the cosmological constant renders the potential unbounded from below, making this examination crucial. Secondly, the issue of quantization persists, especially under conditions where Λ is maintained as non-zero. Thirdly, the configuration of the potential now indicates the possibility of a tunneling solution, facilitated by the  \npresence of the cosmological horizon. This scenario mirrors the classical understanding described in [62], where the effective potential in the geodesic equation for a Schwarzschild-deSitter metric exhibits a local maximum, suggesting a similar tunneling phenomenon.', 'A. Analysis of the potential': "Transitioning to the de Sitter scenario, which is distinguished by a positive cosmological constant (Λ > 0), implies a non-zero vacuum energy density ( ρ vac > 0). To provide a grasp of the magnitudes of the coefficients introduced in equation (18), let us consider a few illustrative densities. The Planck density, ρ Pl , is notably large at 5 . 1 · 10 96 Kg/m 3 . In stark contrast, the vacuum energy density, ρ vac , is significantly smaller, at 5 . 9 · 10 -27 Kg/m 3 . For a typical cloud core, we anticipate an initial density, ρ 0 , approximately 10 -16 Kg/m 3 [44]. With the densities defined, we estimate an order of magnitudes using  \nρ vac ρ 0 /lessmuch 1 /lessmuch ρ Pl ρ 0 , (49)  \nwhere ρ vac /ρ 0 ∼ 10 -11 and ρ Pl /ρ 0 ∼ 10 112 . Consequently, | α | ∼ 10 214 and β ∼ 10 224 emerge. This implies that the terms with α in the effective potential become influential only at large values of ˜ a . This can be easily seen if we rewrite (17) as follows  \nThen, Figure 4 clearly shows that the effective potential exhibits a minimum between 0 and 1 followed by a high maximum for large values of ˜ a . A straightforward application of perturbation theory indicates that the smallest  \nV eff ( ˜ a ) = U eff ( ˜ a ) β = -/epsilon1 ˜ a 4 + ˜ a ( ˜ a -1) , /epsilon1 = ρ vac ρ 0 . (50)  \nFIG. 4: Graphical representation of the effective potential (50) for /epsilon1 = 10 -11 . The potential exhibits a classical forbidden region between 1 and 3 . 16 · 10 5 . The maximum occurs at 2 . 24 · 10 5 .  \n<!-- image -->  \npositive intersection ˜ a -occurs at  \nwhile the minimum is located at  \n˜ a -= 1 + /epsilon1 + O ( /epsilon1 2 ) , (51)  \n˜ a m = 1 2 + /epsilon1 4 + O ( /epsilon1 2 ) . (52)  \nFor both ˜ a -and ˜ a m the first-order corrections prove to be extraordinarily small, approximately of the order of 10 -11 . However, for later purposes we also need precise formulae for the positive intersections of the effective potential with  \nthe ˜ a -axis and the exact locations of the minimum and maximum in the potential. As a first step in this direction we rewrite the effective potential as follows  \nNote that the cubic polynomial in equation (53) is already in reduced form. By employing Vieta's theorem, if ˜ a i (with i = 1 , 2 , 3) denotes the roots of the reduced cubic, we deduce ˜ a 1 + ˜ a 2 + ˜ a 3 = 0 and ˜ a 1 ˜ a 2 ˜ a 3 = -2 q . Given that q > 0, if the cubic has three real roots, there can only be either one or two negative roots. The number of real solutions for our cubic equation is determined by the sign of the discriminant  \nV eff ( ˜ a ) = -/epsilon1 ˜ a ( ˜ a 3 +3 p ˜ a +2 q ) , p = -1 3 /epsilon1 , q = 1 2 /epsilon1 . (53)  \nD = q 2 + p 3 = 1 27 /epsilon1 3 ( 27 4 /epsilon1 -1 ) . (54)  \nSince /epsilon1 /lessmuch 1, we are clearly in the regime /epsilon1 < 4 / 27 and we immediately conclude that D < 0. This implies that the cubic in (53) has three real roots. We must make two important observations at this stage. Firstly, in the current regime, aside from ˜ a = 0, the effective potential reveals two positive roots, henceforth referred to as a ± . Subsequently, we will rename ˜ a 1 as ˜ a -. It is noteworthy that the potential displays both a minimum and maximum value, which we will calculate shortly. In analogy with the standard Schrodinger equation, we observe a classically forbidden region ˜ a -< ˜ a < ˜ a + and two classically allowed regions, specifically when 0 < ˜ a < ˜ a -and ˜ a > ˜ a + . As detailed in [63], we define  \nr = 1 √ 3 /epsilon1 , cos φ = 3 2 √ 3 /epsilon1. (55)  \nRecalling that /epsilon1 /lessmuch 1 allows to conclude that φ lies within the range (0 , π/ 2). With this observation, the two positive roots can be expressed as  \n˜ a ± = 2 √ 3 /epsilon1 cos ( π ∓ φ 3 ) , φ = cos -1 ( 3 2 √ 3 /epsilon1 ) . (56)  \n˜ ˜ a + = 1 √ /epsilon1 + O (1) , (57)  \nSince ( π -φ ) / 3 and ( π + φ ) / 3 fall within the ranges π/ 6 < ( π -φ ) / 3 < π/ 3 and π/ 3 < ( π + φ ) / 3 < π/ 2, respectively, we infer that ˜ a + > ˜ a -. Taking into account that a + admits the perturbative expansion  \nthe separation between these turning points, denoted as ∆ a ± = a + -a -, can be calculated for the case of /epsilon1 /lessmuch 1 as  \nIn order to compute the maximum and minimum displayed in Figure 4 for the case /epsilon1 /lessmuch 1, we impose that d V eff /d ˜ a = 0. This condition gives rise to the following reduced cubic  \nThe corresponding discriminant  \n˜ ˜ ˜ ∆ ˜ a ± = 1 √ /epsilon1 + O (1) . (58)  \n˜ a 3 +3 p ˜ a +2 q = 0 , p = -1 6 /epsilon1 , q = 1 8 /epsilon1 . (59)  \nD = 1 216 /epsilon1 3 ( 27 8 /epsilon1 -1 ) (60)  \nis clearly negative since /epsilon1 /lessmuch 1. As a consequence, the cubic above admits three real roots. Following [63], we define  \n̂ r = 1 √ 6 /epsilon1 , cos ̂ φ = 3 4 √ 6 /epsilon1, (61)  \nby means of which the two positive roots can be expressed as  \n˜ a M = 2 √ 6 /epsilon1 cos ( π -̂ φ 3 ) , ˜ a m = 2 √ 6 /epsilon1 cos ( π + ̂ φ 3 ) , ̂ φ = cos -1 ( 3 4 √ 6 /epsilon1 ) . (62)  \nA consistency check shows that the perturbation expansion for ˜ a m given by (62) coincides with (52) while the corresponding expansion for the maximum is given by  \nIt is gratifying to observe that ˜ a M → + ∞ as /epsilon1 → 0 which correctly signals that the effective potential has a global minimum when /epsilon1 = 0. Taking into account that /epsilon1 /lessmuch 1, the effective potential at these points is given by  \nIt follows that the height of the barrier is  \n˜ a M = 1 √ 2 /epsilon1 + O (1) . (63)  \nV eff ( ˜ a m ) = -1 4 -/epsilon1 16 + O ( /epsilon1 2 ) , V eff ( ˜ a M ) = 1 4 /epsilon1 + O ( 1 √ /epsilon1 ) . (64)  \n∆ V eff = V eff ( ˜ a M ) -V eff ( ˜ a m ) = 1 4 /epsilon1 + O ( 1 √ /epsilon1 ) , (65)  \nBy means of (50) we can switch back to the effective potential U eff and find that  \n∆ U eff ∼ β 4 /epsilon1 = 81 ρ 2 Pl 1024 ρ 0 ρ vac ∼ 7 . 9 · 10 223 . (66)", 'B. Spectral considerations on the Hamiltonian': "When dealing with differential operators, especially in quantum mechanics, we often want to ensure that the operator is self-adjoint. This is crucial because self-adjoint operators have real eigenvalues, which correspond to physically observable quantities. However, not all differential operators possess this property. Their self-adjointness can depend on the behavior of solutions to the associated differential equation at certain endpoints of an interval. Our variable a varies in the domain (0 , ∞ ) and not on the whole real line. Therefore, the spectral properties of our problem are connected to the non self-adjointness of the radial momentum operator in standard quantum mechanics [64].  \nGiven a Sturm-Liouville type operator  \nLy = -d 2 y dx 2 + U ( x ) y, 0 ≤ x < + ∞ , (67)  \nwhere U is the potential, the behavior of solutions at the endpoints can be classified into two main categories: the limit point case (LP) and the limit circle case (LC). We say that (67) is in the LP case, if there is only one linearly independent solution to the differential equation that is square-integrable as x approaches the boundary. In this scenario, the operator naturally exhibits self-adjointness across a broad spectrum of boundary conditions. On the other hand, (67) is in the LC case, whenever there are two linearly independent solutions to the differential equation and both are square-integrable as x → + ∞ . In this scenario, since self-adjointness is not guaranteed, one must choose appropriate boundary conditions. More precisely, there can be a whole family of possible self-adjoint extensions, each corresponding to a different boundary condition. The distinction between the LP and LC cases is crucial. In quantum mechanics, a self-adjoint Hamiltonian ensures that the energy eigenvalues are real and that the time evolution of a system is unitary. If an operator is in the LC case, and we do not handle it correctly, we might end up with a non-self-adjoint Hamiltonian, leading to non-physical results. In our present problem, we observe that due to the fact that the effective potential (17) is unbounded from below, the Hamiltonian operator  \nH = -d 2 d a 2 + U eff ( ˜ a ) , U eff ( ˜ a ) = α ˜ a 4 + β ˜ a ( ˜ a -1) (68)  \n˜ with α and β defined as in (18) is not essentially self-adjoint. Nevertheless, we will soon show that the Schrodinger equation (16) always admits a square integrable solution on the interval [0 , ∞ ). Moreover, since U eff ( ˜ a ) ≥ 0 on the interval [ ˜ a -, ˜ a + ], the first corollary after Theorem 3 . 3 in [65] predicts that such a solution has at most one zero on the aforementioned interval. In order to convince ourselves that the Hamiltonian H defined in (68) is in the limit circle case, we need to study the behavior of the solution in the regimes ˜ a → ∞ and ˜ a → 0. Firstly, we observe that our (68) is a special case of (4.1) and (4.2) in [65] with κ = 0. Moreover, since the effective potential is asymptotically  \ndivergent, we are in case c) with f ( ˜ a ) = √ -U eff ( ˜ a ) (see page 84 in [65]). In order to apply Theorem 4 . 6 therein, we need to check if the effective potential satisfies the conditions  \n˜ ˜ where ˜ a 0 in the above integrals can be chosen arbitrarily large. Here, a prime denotes differentiation with respect to ˜ a . Let us pick ˜ a 0 /greatermuch 1 so that we only need to consider the asymptotic behavior of the integrands above. A routine calculation shows that  \n∫ ∞ ˜ a 0 | U ' eff ( ˜ a ) | 2 | U eff ( a ) | 5 / 2 d ˜ a < ∞ , ∫ ∞ ˜ a 0 | U '' eff ( ˜ a ) | 2 | U eff ( a ) | 3 / 2 d ˜ a < ∞ , (69)  \n| U ' eff ( ˜ a ) | 2 | U eff ( a ) | 5 / 2 = 16 √ β/epsilon1 a 4 + O ( 1 a 5 ) , | U '' eff ( ˜ a ) | 2 | U eff ( a ) | 3 / 2 = 12 √ β/epsilon1 a 4 + O ( 1 a 6 ) (70)  \n˜ Ψ ± , ∞ ( ˜ a ) = 1 4 √ -U eff ( ˜ a ) exp ( ± i ∫ ˜ a ˜ a 0 √ -U eff ( s ) ds ) [1 + O (1)] . (71)  \n˜ ˜ ˜ ˜ ˜ ˜ and we immediately realize that both conditions appearing in (69) are fulfilled. Then, Theorem 4 . 6 implies that (16) admits a pair of solutions with the following asymptotic behavior as a →∞  \nIf we take into account that ˜ a 0 /greatermuch 1, we can further approximate the above formula as follows  \nΨ ± , ∞ ( ˜ a ) = 1 a e ± i 3 √ β/epsilon1 ˜ a 3 [1 + O (1)] , (72)  \nd 2 Ψ 0 d a 2 + β ˜ a Ψ( ˜ a ) = 0 . (73)  \n˜ which is reminiscent of (5.13) in [52]. Note that the second factor in (72) oscillates and is bounded. Hence, both solutions have integrable squares on the interval [ ˜ a 0 , ∞ ) even though the original Hamiltonian is not essentially selfadjoint. In order to investigate the solutions in a neighborhood of ˜ a = 0, we first observe that in the limit of ˜ a → 0 (16) becomes  \n˜ Its solution can be expressed in terms of Airy functions as follows  \n˜ ˜ ˜ Note that both Airy functions are finite at ˜ a = 0 and square integrable on the interval [0 , c ] with c /lessmuch 1. Hence, according to theorem D in [66], we conclude that the differential operator (68) is in the limit circle case. It is also worth emphasizing that potentials unbounded from below are not merely theoretical artifacts. They have practical relevance in high-energy physics, particularly in the context of field localization on the 3-brane within braneworld models featuring warped extra dimensions [67]. As highlighted in [68], these types of potentials can be approximated with a significant degree of accuracy in real-world mesoscopic systems.  \nΨ 0 ( a ) = c 1 Ai( -3 √ β a ) + c 2 Bi( -3 √ β a ) . (74)  \nThe fact that equation (68) falls within the limit circle case suggests that the operator is not self-adjoint unless additional boundary conditions are introduced or a suitable domain of definition for the operator is specified.  \nHowever, for those magnitudes of the parameter /epsilon1 associated with various black hole scenarios (refer to Table I), it can be demonstrated that our Hamiltonian acquires Parity-Time (PT) symmetry in the manner described by [69]. It is inherently time-invariant, as time does not feature in the WDW equation. To verify the parity symmetry across the /epsilon1 ranges outlined in Table I, introducing a change of variable ˜ a = x + ˜ a m , accompanied by a vertical shift by V eff ( ˜ a M ), proves useful. Finally, with the help of (52), (63), and (64), we can rewrite equation (16) as  \n-dψ dx 2 + β [ -/epsilon1x 4 + x 2 -1 4 /epsilon1 ] ψ ( x ) = 0 . (75)  \nIt is straightforward to verify that the operator above maintains its invariance under the transformation x → -x . With an appropriately defined inner product, the eigenfunctions of a PT-symmetric Hamiltonian possess positive norms and demonstrate unitary time evolution [70]. These characteristics are essential for the consistency of quantum theories. Furthermore, [70] demonstrated that all eigenvalues of a PT-symmetric Hamiltonian with an inverted quartic potential are real and positive. This property is corroborated by our findings in the following subsection, where we establish a quantization condition for the density ρ 0 , expressed in relation to the Planck density.  \nLast but not least, an intriguing observation from equation (72) hints at a form of tunneling. Despite the probability density | Ψ ± , ∞ | 2 ∼ ˜ a -2 decaying as ˜ a →∞ , it indicates a non-negligible probability in the region ( ˜ a + , ∞ ). To delve deeper into this phenomenon, let us first observe that in the /epsilon1 → 0 limit, equation (57) which depicts the most significant intersection of the effective potential with the a -axis, simplifies to  \nwhere we employed (50) and k is defined via (2). This result implies that the cosmological horizon is adjusted by a factor of √ k . Consequently, ˜ a + can be understood as the cosmological horizon in our context. To understand the behavior of (76) in the Λ → 0 limit, we focus on its dependence on Λ. First of all, this equation represents the cosmological horizon, where k is related to the initial density ρ 0 of the collapsing dust cloud. As Λ → 0, the term √ 3 k Λ diverges. This implies that the cosmological horizon a + moves to infinity. Physically, this is consistent with the classical picture where, in the absence of a cosmological constant, there is no cosmological horizon, and the potential barrier introduced by Λ vanishes.  \n˜ ˜ a + = 1 √ /epsilon1 = √ 3 k Λ , (76)", 'C. An approximated quantization condition': 'An approximated quantization condition can be derived by studying the differential equation (16) in a neighborhood of the minimum in the effective potential. To this purpose, we can add and subtract the positive term -U eff ( ˜ a m ) to the effective potential and recast the WDW equation in the equivalent form  \n˜ Through this process, we have effectively shifted the potential upward by -U eff ( ˜ a m ). This newly introduced term now serves as a positive spectral parameter. Taking into account that ˜ a m satisfies the equation dU eff /d ˜ a = 0 and performing a Taylor expansion around a = a m , it is not difficult to verify that  \n-d 2 Ψ d a 2 +[ U eff ( ˜ a ) -U eff ( ˜ a m )] Ψ( ˜ a ) = -U eff ( ˜ a m )Ψ( ˜ a ) . (77)  \n˜ ˜ U eff ( ˜ a ) -U eff ( ˜ a m ) = β [ -/epsilon1 ( ˜ a -˜ a m ) 4 -4 /epsilon1 ˜ a m ( ˜ a -˜ a m ) 3 +(1 -6 /epsilon1 ˜ a 2 m )( ˜ a -˜ a m ) 2 ] . (78)  \nU eff ( ̂ a ) = β [ (1 -6 /epsilon1 ˜ a 2 m ) ̂ a 2 -4 /epsilon1 ˜ a m ̂ a 3 -/epsilon1 ̂ a 4 ] . (80)  \n̂ ˜ ˜ -d 2 Ψ d ̂ a 2 + U eff ( ̂ a )Ψ( ̂ a ) = -U eff ( ˜ a m )Ψ( ̂ a ) , ̂ a ≥ -˜ a m (79)  \nFinally, by means of the coordinate change a = a -a m we can rewrite (77) as follows  \nwith  \nIn a neighborhood of ̂ a = 0, (79) becomes the equation of a harmonic oscillator, namely  \n-d 2 Ψ o d a 2 + K ̂ a 2 Ψ o ( ̂ a ) = E Ψ o ( ̂ a ) , K = β (1 -6 /epsilon1 ˜ a 2 m ) , E = -U eff ( ˜ a m ) . (81)  \nK = β [ 1 -3 2 /epsilon1 + O ( /epsilon1 2 ) ] , (82)  \n̂ Since K is positive as it can be seen from the following expansion in the small parameter /epsilon1  \nwe can proceed as in [45]. Namely, we introduce the new variable ξ = 4 √ K a and cast (81) into the form  \n̂ d 2 Ψ o dξ 2 +( λ -ξ 2 )Ψ o ( ξ ) = 0 , λ = E √ K . (83)  \nThen, the quantization condition reads [45]  \n2 n +1 = -U eff ( ̂ a m ) √ β (1 -6 /epsilon1 ˜ a 2 m ) , n = 0 , 1 , 2 , · · · . (84)  \nExpanding the r.h.s. of the above expression in the small parameter /epsilon1 yields  \n2 n +1 = √ β 4 [ 1 + /epsilon1 + O ( /epsilon1 2 ) ] , n = 0 , 1 , 2 , · · · . (85)  \nAs a consistency check for the above procedure, we observe that in the case /epsilon1 = 0 the above formula correctly reproduces the quantization condition obtained in [26]. Finally, with the help of (18) we can rewrite (85) as  \n16(2 n +1) 2 = 81 256 ( ρ Pl ρ 0 ) 2 [ 1 + 2 ρ vac ρ 0 + O ( ρ 2 vac ρ 2 0 )] , n = 0 , 1 , 2 , · · · . (86)  \nThe approximated quantization condition derived in this section highlights the discrete nature of the density ρ 0 in the presence of a positive cosmological constant Λ. By expanding the effective potential around its minimum and drawing an analogy to a quantum harmonic oscillator, we obtain (86). This condition shows that ρ 0 is quantized, with levels influenced by the Planck density and the vacuum energy density, reflecting the impact of quantum gravitational effects.', 'D. WKB analysis and tunneling probability': "Let us first rewrite the Schrodinger equation (16) as follows  \n-µ 2 d 2 Ψ d a 2 + V ( ˜ a )Ψ( ˜ a ) = 0 , V ( ˜ a ) = -˜ a + ˜ a 2 -/epsilon1 ˜ a 4 , µ 2 = 1 β , ˜ a ≥ 0 . (87)  \n˜ We recall that this is a one-dimensional Schrodinger equation for a particle with zero total energy and half the unit mass. Note that according to our previous estimates µ ∼ 10 -112 and /epsilon1 ∼ 10 -11 . This observation suggests that we can construct a WKB solution in terms of the small parameter µ . Let us introduce the local momentum p 2 ( ˜ a ) = -V ( ˜ a ) which in turn allows defining a local de Broglie wavelength  \nIn view of the notation introduced above, it is convenient to rewrite (87) as follows  \nλ ( ˜ a ) = µ p ( ˜ a ) . (88)  \n-µ 2 d 2 Ψ d a 2 = p 2 ( ˜ a )Ψ( ˜ a ) , ˜ a ≥ 0 . (89)  \n˜ The potential admits three physical turning points at ˜ a = 0 and ˜ a ± which are given by (51) and (57), respectively. Again, drawing an analogy to the standard Schrodinger equation, we can say that there are two classically allowed regions where V ( ˜ a ) < 0 and p 2 ( ˜ a ) > 0. The first region denoted by the symbol I occurs for ˜ a ∈ (0 , ˜ a -) while the second region called III extends over the interval ( ˜ a + , ∞ ). In the classically forbidden region, we have V ( ˜ a ) > 0 and thus p 2 ( ˜ a ) < 0. We use the symbol II to denote such region, which is limited to the interval ( ˜ a -, ˜ a + ). If we write the wave function as  \nwhere ρ is the probability density, it is straightforward to verify that  \nΨ( ˜ a ) = √ ρ ( ˜ a ) e i µ S ( ˜ a ) , (90)  \nΨ ∗ ( ˜ a ) d Ψ d a = 1 2 dρ d a + i µ ρ ( ˜ a ) d S d a (91)  \n˜ ˜ ˜ and therefore, the only non-zero current component is the one along the a -axis and is given by the formula  \n˜ ˜ We recall that in classical physics a fluid with density ρ ( x ) moving with velocity v ( x ) has a current density ρv = 2 ρp which contrasted with (92) leads to the conclusion  \n˜ J ˜ a = 2 µ /Ifractur ( Ψ ∗ ( ˜ a ) d Ψ d a ) = 2 ρ ( ˜ a ) d S d a . (92)  \np ( x ) ∼ d S d ˜ a . (93)  \nObserve that the equality does not exactly hold in the above formula in general unless we are in the WKB regime. In our WKB approximation scheme, it is more convenient to introduce the ansatz  \nΨ( a ) = e i µ S ( ˜ a ) , (94)  \n˜ where /Rfractur ( S ) and /Ifractur ( S ) represent the phase and magnitude of the wave function. By replacing (94) into (89), we obtain the following nonlinear differential equation for S  \nIf we take µ to be the small parameter in a systematic expansion of S of the form  \n( dS d ˜ a ) 2 -iµ d 2 S d ˜ a 2 = p 2 ( ˜ a ) . (95)  \nS ( a ) = S 0 ( a ) + µS 1 ( a ) + O ( µ 2 ) , (96)  \n˜ ˜ ˜ replacing (96) into (95) and neglecting terms of order µ 2 and higher lead to the equation  \nThe above equation is satisfied provided that  \n[ ( dS 0 d ˜ a ) 2 -p 2 ( ˜ a ) ] + µ ( 2 dS 0 d ˜ a dS 1 d ˜ a -i d 2 S 0 d ˜ a 2 ) + O ( µ 2 ) = 0 . (97)  \n( dS 0 d a ) 2 = p 2 ( ˜ a ) , (98)  \n˜ dS 1 d a = i 2 d 2 S 0 /d ˜ a 2 dS 0 /d ˜ a . (99)  \n˜ Equation (98) can be immediately integrated with solutions  \nS 0 ( ˜ a ) = ± ∫ ˜ a ˜ a 0 p ( s ) ds, (100)  \ndS 1 d ˜ a = i 2 d d ˜ a ln p ( ˜ a ) . (101)  \nwhere ˜ a 0 is an initial point to be accordingly adjusted before performing the integral. Moreover, with the help of (98) equation (99) can be rewritten as  \nThis is readily solved to give  \nLet us now reconstruct the wave function to this order of approximation by means of (100) and (102). We find that  \niS 1 ( ˜ a ) = -1 2 ln p ( ˜ a ) + C ' . (102)  \nΨ( ˜ a ) = A √ p ( a ) e ± i µ ∫ ˜ a ˜ a 0 p ( s ) ds . (103)  \n˜ This is the basic solution in the WKB approximation. We do not attempt to normalize this wave function because the region of validity of this approximation is still unclear and will be discussed in this section. Let us recall that the observables for the basic solution are", '1. the probability density given by': "ρ ( ˜ a ) = | Ψ( ˜ a ) | 2 = | A | 2 p ( a ) , (104)  \n˜ where we used (103). Clearly, ρ is higher when p is small in those regions.  \n- 2. The probability current according to (92) coupled with (93) and (104) simplifies to  \nJ ˜ a = 2 | A | 2 . (105)  \nClearly, the current is constant. This is expected because a position dependent current for the zero energy eigenstate is not possible as it would violate the current conservation equation ∂ ˜ a J ˜ a + ∂ t ρ = 0, given that ρ is time-independent. For a discussion of the probability current in the context of the WDW equation and its implications for quantum cosmology, we refer to the seminal papers [13, 71].  \nUsing (103), we can finally write the general solutions that apply to classically allowed and classically forbidden regions. In the allowed regions I and III, we have -V ( ˜ a ) > 0 with p 2 ( ˜ a ) = -V ( ˜ a ) and hence, the general solution is a superposition of two basic solutions with waves propagating in opposite direction, namely  \n˜ ˜ Ψ III ( ˜ a ) = A III 4 √ -V ( a ) e i µ ∫ ˜ a ˜ a 0 ,III √ -V ( s ) ds + B III 4 √ -V ( a ) e -i µ ∫ ˜ a ˜ a 0 ,III √ -V ( s ) ds , . (107)  \nΨ I ( ˜ a ) = A I 4 √ -V ( a ) e i µ ∫ ˜ a ˜ a 0 ,I √ -V ( s ) ds + B I 4 √ -V ( a ) e -i µ ∫ ˜ a ˜ a 0 ,I √ -V ( s ) ds , (106)  \n˜ ˜ Note that the waves with coefficients A I and A III move to the right while the second waves with coefficients B I and B III move to the left. On the forbidden region II the potential is positive and p 2 ( ˜ a ) = -V ( ˜ a ), so the general solution reads  \n˜ ˜ Since in the expressions (106)-(108) the potential vanishes at the turning point, we need to discuss the validity region of our WKB approximation. To this purpose, it is convenient to go back to the expansion (97). We must have that the terms of order µ in the differential equation are much smaller in magnitude than the O (1) terms. At each of these orders, we have two terms that are set equal to each other by the differential equations (98) and (99). Hence, it suffices to check that one of the O ( µ ) terms is much smaller than one of the O (1) terms. For example, we must have | µS ' 0 S ' 1 | /lessmuch | S ' 0 | 2 where the prime denotes differentiation with respect to ˜ a . Since | S ' 0 | = | p | , we obtain | µS ' 1 | /lessmuch | p | . By means of (101), we note that | S ' 1 | = | p ' / 2 p | and therefore, | µp ' /p | /lessmuch | p | . With the help of (88) we get, | λp ' | /lessmuch | p | which says that the changes in the local momentum over a distance equal to the de Broglie wavelength are small compared to the momentum. Alternatively, we can rewrite the latter condition as  \nΨ II ( ˜ a ) = A II 4 √ V ( a ) e 1 µ ∫ ˜ a ˜ a 0 ,II √ V ( s ) ds + B II 4 √ V ( a ) e -1 µ ∫ ˜ a ˜ a 0 ,II √ V ( s ) ds . (108)  \nthat is the local de Broglie wavelength must vary slowly. In our case, we find with the help of (88) that  \n∣ ∣ ∣ ∣ dλ d ˜ a ∣ ∣ ∣ ∣ /lessmuch 1 , (109)  \nSince the left-hand side of the above expression can be expressed in terms of the potential as follows  \n∣ ∣ ∣ ∣ d d ˜ a ( 1 p )∣ ∣ ∣ ∣ /lessmuch 1 µ = √ β ∼ 10 112 . (110)  \nwe conclude that our WKB approximation will hold at all those point on the positive a -axis such that  \n∣ ∣ ∣ ∣ d d ˜ a ( 1 p )∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ p ' p 2 ∣ ∣ ∣ ∣ ∣ = ∣ ∣ ∣ ∣ ∣ V ' V 3 / 2 ∣ ∣ ∣ ∣ ∣ (111)  \n∣ ∣ As can be seen in Figure 5, the condition (112) is violated only when the object lingers in a region extremely close to the turning points of the potential. To understand how close we can go to the point ˜ a = 0, we can go back to (112) from which we find that the solution Ψ I given by (106) will be valid provided that ˜ a /greatermuch 10 -75 and | ˜ a -˜ a -| /greatermuch 10 -75 . Similarly, it can be checked that ψ II and Ψ III holds whenever | ˜ a -˜ a -| /greatermuch 10 -75 and | ˜ a -˜ a + | /greatermuch 10 -77 . Let us now briefly explore a global upper bound for the tunneling probability. We will begin by noting that as ˜ a approaches infinity, the wave function does not vanish asymptotically, as it can be evinced from the asymptotic behavior described  \n˜ ∣ ∣ ∣ ∣ V ' V 3 / 2 ∣ ∣ ∣ ∣ /lessmuch 1 µ . (112)  \nFIG. 5: Plot of | V ' /V 3 / 2 | for /epsilon1 = 10 -11 and V defined as in (87). In view of the condition (112), the WKB approximation holds exceptionally well everywhere modulo some tiny intervals centered at the turning points of the potential.  \n<!-- image -->  \nby (72). This observation signals that in the present case the conclusion stated in [72], according to which the wave function should vanish within the barrier, cannot be applied. Consequently, we can calculate the tunneling probability using the following formula  \nT WKB = exp ( -2 µ ∫ ˜ a + ˜ a -√ V ( ˜ a ) d ˜ a ) . (113)  \n∫ ˜ a + ˜ a -√ V ( ˜ a ) d ˜ a ≤ √ V ( ˜ a M )( ˜ a + -˜ a -) . (114)  \nIf we observe that the function √ V ( ˜ a ) shares its maximum location, denoted as ˜ a M , with V ( ˜ a ) within the interval [ ˜ a -, ˜ a + ], we can establish an upper bound for it by using √ V ( ˜ a M ). Consequently, the integral in (113) can be bounded as follows  \nAt this point, a comment is in order. Firstly, the right-hand side in the above expression depends on the parameter /epsilon1 . Secondly, as indicated in Table I, the values of /epsilon1 associated with astrophysically relevant black holes are tiny. This observation leads us to the reasonable conclusion that it is justifiable to expand the combination √ V ( ˜ a M )( ˜ a + -˜ a -) with respect to the small parameter /epsilon1 . More precisely, by means of (56), (63) and (64) we find that  \nTABLE I: Numerical values of the parameter /epsilon1 defined in (50) in the case of different black hole scenarios. Here, M /circledot = 1 . 989 · 10 30 Kg denotes the solar mass.  \n·  \nGiven the very small magnitude of /epsilon1 , we can bound the tunneling probability from above by  \n√ V ( ˜ a M )( ˜ a + -˜ a -) = 1 2 /epsilon1 [ 1 -( 3 2 + √ 2 ) √ /epsilon1 + O ( /epsilon1 ) ] . (115)  \nT WKB ≈ exp ( -√ β /epsilon1 ) = exp ( -9 ρ Pl 8 ρ vac ) ∼ exp ( -10 123 ) . (116)  \nWe acknowledge here the extremely small value of the tunneling probability. We attribute it to the fact that the tunneling seems to be macroscopic. Indeed, a positive cosmological constant causes the effective potential of the radial geodesic equation of motion to develop a local maximum far away from the center [62]. The same happens in the WDW equation.  \nThe second notable feature of this upper bound is that it does not show any dependence on the physical parameters of the object which underwent the gravitational collapse. Moreover, in the realm of black hole to white hole transitions, various models and theories have been proposed to understand the underlying mechanisms. The present approach, which rests on the application of the WDW equation in the context of gravitational collapse in a universe with a positive cosmological constant, predicts a very small tunneling probability for a black hole to emerge as a white hole, on the order of exp( -10 123 ).", 'E. Dwell time or residence time in the barrier': "This section has a twofold purpose. In spite of the fact that the WDW equation has no time, we would like to show that it is possible to introduce non-classical time concepts based on quantum mechanics. Secondly, the estimate of the so-called tunneling time will reinforce our conclusion about the macroscopic nature of the tunneling.  \nApart from the more commonly evaluated barrier penetration or the transmission probability discussed above, there exists a lot of interest in the time spent by the tunneling particle in the allowed and forbidden regions in tunneling. Indeed, starting with the pioneering work of Wigner on the phase time delay [74], there exist several definitions such as the Larmor, traversal, residence or dwell time, and even complex times, which are often derived within the framework of semi-classical approximations (see [75] for a review of these concepts). Among the various definitions, the dwell time seems to have found a straightforward interpretation in terms of physically measurable quantities such as the lifetimes of decaying nuclei [76, 77]. In the context of the present work, an estimate of the dwell time in the barrier region may allow us to comment on the possibility of the transition from the black hole to a white hole.  \nIn order to briefly introduce the concept, let us consider an arbitrary barrier V ( x ) in one-dimension confined to an interval ( x 1 , x 2 ). The dwell time in the barrier is given as  \nτ D = ∫ x 2 x 1 | Ψ( x ) | 2 dx j . (117)  \nHere Ψ( x ) is the time independent solution of the Schrodinger equation in the given region and j the current density. In what follows, we shall evaluate the dwell time in a semiclassical approximation for the wave function in the present work. However, one can always define a dwell time as above in a given region of space, be it with an exact or an approximate wave function. Considering the analogy between the standard time independent Schrodinger equation in one-dimension and equation (87), namely,  \n-µ 2 L 2 0 d 2 Ψ da 2 + V ( a/L 0 )Ψ( a/L 0 ) = 0 , (118)  \nwe can write the 'average dwell time' in the forbidden region (under the barrier) as [77]  \nτ II D = L 0 2 µ ∫ ˜ a + ˜ a -d ˜ a √ V (˜ a ) exp [ -2 ∫ ˜ a ˜ a -da ' √ V ( a ' ) ] , (119)  \nwhere we have substituted for ˜ a = a/L 0 in (87). Note that the normalization of the wave function cancels with the same factor appearing in the expression for the current density, and hence, the dwell time can be evaluated simply with the knowledge of the potential barrier and the classical turning points.  \nIn order to get an estimate of the order of magnitude of the dwell time in the barrier region, let us consider the barrier to be rectangular, with the height given by its maximum value. We would therefore obtain a lower limit on the dwell time with the substitution, V (˜ a ) → V (˜ a M ). Thus, the lower bound on the dwell time in region II is given by  \nτ II D = L 0 4 µV (˜ a M ) ( 1 -e -2 √ V (˜ a M )(˜ a + -˜ a -) ) . (120)  \nWith L 0 = 1 / √ k = 3 / (8 πGρ 0 ) = 4 . 23 × 10 12 s and using the values of the barrier from Figure 4, τ II D has a lower bound of approximately 10 113 seconds or 10 96 billion years.  \nEquation (117) is, in principle, the definition of an average dwell time or the average amount of time spent in the region ( x 1 , x 2 ) regardless of how the particle escaped (by reflection or transmission). The transmission and reflection dwell times for the particular cases when the particle is bound in the barrier and later either got transmitted or reflected is written by replacing the current density j by a transmitted and reflected flux, j T = j T and j R = j R respectively. Here, T and R are the transmission and reflection coefficients and R + T = 1 due to conservation of probability. One would then obtain [78]  \n1 τ D = T τ D + R τ D = 1 τ D,T + 1 τ D,R . (121)  \nThe lower bound on τ II D,T would thus be extremely large.  \nThis result stands in stark contrast to the models discussed in [7]. For instance, the spherical shells model mentioned in [73] and further analyzed in [79, 80] suggests that the exterior region of a black hole may undergo a transition to a white hole, with the timescales of this transition being relatively short. Furthermore, the outer horizon undergoes three distinct stages before matter starts to emerge again: the stages of being a black hole, transitioning, and then becoming a white hole. The phase where it acts as a black hole persists until the shift towards the white hole phase starts. This shift can happen almost instantly or take a longer duration, leading to an intermediate state termed by [73] as the 'gray horizon'-a blend of the black and white hole phases. Once this transition ends, the system stabilizes into a white hole state. To be precise, the duration for which the horizon remains a black hole is the period during which distant observers perceive it as such. However, it is worth mentioning that in models where time is symmetrical, a black hole horizon that exists for an extended period suggests a similarly prolonged existence for the white hole horizon. This prolonged white hole phase could be problematic because of recognized instabilities in white hole configurations, as pointed out by [81, 82].  \nThe authors in [83] constructed a metric representing the black hole tunneling into a white hole. This was achieved by employing classical equations outside the quantum region, estimating the onset of quantum gravitational phenomena, and considering indirect evidence of quantum gravity's effects. It is important to note, as highlighted by the authors, that this approach is not derived from first principles. A comprehensive theory of quantum gravity would be required for such a derivation. One of the key aspects of this model is that the time it takes for this transition to occur is influenced by the Schwarzschild mass of the black hole, more precisely, the larger the Schwarzschild mass, the longer it might take for the black hole to transition into a white hole. On the other hand, if we use a theory of quantum gravity such as the one represented by the WDW equation, and we compare the model in [83] with the predictions emerging from our approach, there is a stark contrast. The WDW equation suggests that the likelihood of a black hole transitioning into a white hole is exceedingly rare, almost to the point of being negligible.  \nIn [84], the study focused on quantum tunneling from black holes, exploring the theoretical possibility of particles tunneling into their time-reversed counterparts, i.e. white holes. The dynamics of black-to-white hole bounces from matter collapse was further analyzed by [85].  \nWe conclude this section by mentioning that [79, 80, 86, 87] introduced the concept of 'oscillating' black holes, which periodically transition between black and white hole states. Such periodic behavior, if validated, could introduce novel astrophysical phenomena and challenge our current understanding of black holes. However, the WDW equation, when applied to gravitational collapse, casts doubt on these oscillating solutions because it suggests that the probability of black to white hole transition is extremely low. This indicates that the likelihood of observing oscillating black holes might be minimal. On the other hand, in light of the implications emerging from the WDW equation, the idea of oscillating black holes, while intriguing, requires further rigorous investigation and empirical validation.", 'VI. CONCLUSIONS': 'The Friedmann-Robertson-Walker metric is applicable to expanding or contracting universes as well as to collapsing matter. This fact makes the application of the WDW equation in the late gravitational collapse an appealing tool to study quantum effects in a black hole after all matter has entered the horizon. Indeed, many features of the WDW equation in collapse resemble the application of WDW in cosmology. In particular, the equation becomes a timeless Schrodinger-like equation of the form H Ψ = 0. Assuring the correct Lagrangian with matter suitable for quantization, we have taken up the task to examine the above-mentioned quantum effects, paying attention to the ambiguity of factor ordering. In the simplest case where the kinetic term becomes a double derivative with respect to the scale factor, the solution of the wave function is related to the one dimensional harmonic oscillator. Although the WDW equation is not inherently an eigenvalue problem, it results in a quantization of the density in terms of the Planck density. We reach the classical limit in the case of a large quantum number n . The probability density displays certain preferred values, which makes the spacetime look quantized in probabilistic sense. The central singularity is avoided.  \nThe question arises whether these nice properties change if we introduce more complicated factor orderings. We solve the corresponding WDW equations just by the standard ansatz of an exponential multiplied by polynomials. We can construct these polynomials explicitly and at the same time obtain a quantization condition which is similar to the original one in the simplest case. The probability distribution has again preferred values, even though the latter are distributed in a different manner as compared to the simplest case. The classical central singularity does not appear.  \nA tunneling scenario emerges when we switch on a positive cosmological constant. The barrier which gives rise to the tunneling is also encountered in classical gravity [62]. We could give an estimate of the tunneling probability, which turns out to be very small. This is not surprising, as we talk here about a macroscopic tunneling.  \nIt is satisfying to see these features coming out of the WDW equation applied to collapse. It might take a long time till we settle down the full theory of quantum gravity, and even longer till we can confirm it by observations. But we have certain expectations what such a theory would predict. For instance, we would indeed expect the central singularity is absent in the quantum version. Secondly, we might also suspect that the spacetime will appear if not fully discretized, at least taking probabilistically preferred values. Thirdly, as in the standard quantum mechanics, one would expect some quantities to be quantized.  \nLast but not least, we have focused on the quantum gravitational effects during the collapse process, as described by the WDW equation. While Hawking radiation is a significant effect, its impact on the immediate dynamics of collapse is negligible within the WDW formalism. A comprehensive treatment of Hawking radiation would require a different approach beyond the scope of this work.', 'Acknowledgments': "N.G.K. thanks the Faculty of Science, Universidad de Los Andes, Colombia, for financial support through Grant No. INV-2023-162-2841. We would like to thank Prof. R. Baier for useful discussions on the self-adjointness of the WDW operator.  \n- [1] S. Weinberg, Gravitation and Cosmology: Principles and Applications of The General Theory of Relativity , John Wiley & Sons (1972).\n- [2] P. S. 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