The effects of dark energy on the early universe with radiation and Bose-Einstein condensate
G. A. Monerat, F. G. Alvarenga, S. V. B. Gonçalves, R. Fracalossi, G. Oliveira-Neto, C. G. M. Santos, E. V. Corrêa Silva
aa r X i v : . [ g r- q c ] A p r The effects of dark energy on the early universe with radiation andBose-Einstein condensate
G. A. Monerat ∗ , F. G. Alvarenga † , S. V. B. Gon¸calves ‡ , R. Fracalossi § , G. Oliveira-Neto ¶ , C. G. M. Santos ∗∗ , and E. V. Corrˆea Silva †† Departamento de Modelagem Computacional, Instituto Polit´ecnico,Universidade do Estado do Rio de Janeiro,CEP 28.625-570, Nova Friburgo - RJ - Brazil. Departamento de F´ısica, Centro de Ciˆencias Exatas,Universidade Federal do Esp´ırito Santo, CEP 29075-910, Vit´oria, ES, Brazil. Departamento de F´ısica, Instituto de Ciˆencias Exatas, Universidade Federal de Juiz de Fora,CEP 36036-330, Juiz de Fora, Minas Gerais, Brazil. Departamento de F´ısica, Matem´atica e Computa¸c˜ao, Faculdade de Tecnologia,Universidade do Estado do Rio de Janeiro, CEP 27523-000, Resende-RJ, Brazil. (Dated: April 29, 2020)This work analyzes the effects of quantization on a Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) model with positive curvature and material content composed of a Bose-Einsteincondensate, a radioactive fluid and a cosmological constant playing the role of the dark en-ergy of the universe. The quantization of the model was performed using the finite differencemethod in the Crank-Nicolson scheme: solutions of the Wheeler-DeWitt equation are ob-tained, in the form of finite norm wave packets which are well defined in all space, even if the3D-sphere is degenerate. The introduction of the Bose-Einstein condensate and cosmologicalconstant preserves the existence of bounce solutions (for certain choices of parameters andinitial conditions) with exits for inflation (de Sitter solutions). This occurs after the universeemerges from its quantum phase by a tunneling mechanism.
PACS numbers: 98.80.-k, 98.80.Cq, 04.30.-wKeywords: Quantum cosmology, Wheeler-DeWitt equation, Bose-Einstein condensate, Dark energy,Inflation ∗ E-mail: [email protected] † E-mail: fl[email protected] ‡ E-mail: [email protected] § E-mail: [email protected] ¶ E-mail: gilneto@fisica.ufjf.br ∗∗ E-mail: [email protected] †† E-mail: [email protected]
I. INTRODUCTION
Baby universes are interesting laboratories in which the effects of gravitation can be testedin an essentially quantum scenario [1–4]. In fact, the high energy levels associated with the verysmall size of the universe in Planck’s time provide unique characteristics, capable of decisivelyinfluencing cosmic evolution. This means that quantum cosmology is able to provide initialconditions that are essential to justifying the standard cosmological model. As an example, onecan mention inflation, in which the extremely rapid expansion of the universe can be explained byconsidering the quantum effects present in Planck’s time. Furthermore, it is a task of quantumcosmology to solve the most complex of problems in the standard cosmological model: theexistence of an initial singularity. In fact, if one admits that at its beginning the universe hadzero size and infinite energy density, it becomes impossible to physically describe this scenario.However, in the context of quantum cosmology this difficulty can be solved in different ways.One is that of suggesting the birth of the universe ex nihilo via a quantum tunneling process[5, 6]. In this case, the singularity is avoided due to the existence of a potential barrier, fromthe outer side of which the universe has been expanding ever since. The tunneling probabilitydepends on the degrees of freedom of the model under consideration, but it is not possible to takeinto account all the infinite degrees of freedom in the construction of a quantum cosmologicalmodel. Thus, a procedure called quantization in mini-space is used, in which infinite degrees offreedom are frozen and only a very small number of them are quantized. The addition of newdegrees of freedom makes the formulation of the models formidably more complex, however,they should contribute to a more accurate description of the existing processes. New variablescan be obtained by introducing, for example, scalar fields coupled with gravitation. Anotheralternative is to describe the material content that makes up the universe with the introductionof several perfect fluids or even exotic fluids [7–10].An example capable of providing interesting results can be obtained with the addition of aBose-Einsein condensate. In Ref. [11] a scenario was considered in which the material contentof the universe was described by a fluid of radiation and a Bose-Einstein condensate. The Bose-Einstein condensate is well-known and frequently studied in connection with condensed matterphysics. In the context of astrophysics and cosmology, it can be used to explain the origin of darkmatter as well as to describe the evolution of the recent universe [12–16]. When applied to thestudy of the primordial universe, the Bose-Einstein condensate revealed a scale factor behaviorthat depends on the parameter of that fluid. Consequently, classical solutions free of singularitieswere observed. In the quantum case, finite norm wave packets were obtained from the Wheeler-DeWitt equation and the behavior of the scale factor was determined through the many worldsinterpretation . The quantum model is also free of singularities. In the present work, the influenceof the Bose-Einstein condensate is studied in the presence of the cosmological constant, whichplays the role of dark energy in this model. We consider the cosmological constant to be apart of the material content of the universe, responsible for its accelerated expansion, but in itsinitial moments, known as the inflationary period. Due to its important role in our model, wesummarize its historical development as well as its current status in what follows.With a large number of observation instruments arising from great collaborations, such asterrestrial and space telescopes, meteorological balloons, etc, current research in cosmology isat a very advanced stage. The observations obtained so far have produced relevant and alsovery intriguing facts. Among them, we can mention: (i) the rotation curves of galaxies throughdata on the dispersion speed of stars; (ii) galaxies in structures on large cosmological scales;(iii) the accelerated expansion of the universe obtained by the study and observation of SNe Iasupernovae.Several theoretical explanations have been devised to understand and explain these curiousobservations. Among them we find the models that try to explain dark matter, detected byits gravitational effects and responsible for the unusual characteristics of the rotation curves ofgalaxies and for the formation of structures in the universe. Other models try to explain darkenergy, whose effects are only detected on large cosmological scales and would be responsiblefor the accelerated expansion of the universe observed today. Both dark terms, of course, arepart of the material content of the universe, responsible for something like ∼
95% of all thematter-energy that exists. Thus, almost all of this material content is completely unknown tous, with ∼
5% composed of usual baryonic material [17].The cosmological constant has a very curious trajectory in the scenario of Theoretical Physicsin particular. It started with Einstein’s need to adapt his cosmological model, provided by gen-eral relativity, to obtain a static, eternal and immutable universe, which would be in agreementwith the data that existed at the time. The solutions to his equations provided a dynamic uni-verse, a situation not verified by the observations available at the beginning of the 20th century.From the physical point of view, this constant would represent a repulsive force of cosmologicalrange preventing the collapse of the universe caused by gravitational interaction. Notwithstand-ing the discussion about the fact that the introduction of the cosmological constant was an error,the point is that its existence is now important for understanding the nature of dark energy.Despite the observational and theoretical evidence that caused Einstein to reject the cosmo-logical constant and the lack of any direct observational evidence of a non-zero value for theconstant, it simply continued to be used by cosmologists for the most diverse reasons. The con-stant appears in numerous theoretically important solutions in general relativity theory, such asG¨odel’s solution [18, 19], which generates the possibility of traveling to the past through closedcausal loops and the Oszv´ath and Sch¨ucking model [20]. The cosmological constant has alsobeen used to help physicists investigating possible links between general relativity and othertheories [21]. In fact, many researchers have come to consider the cosmological constant as anindispensable element in their descriptions of the primordial universe, in the period called cosmicinflation [22–27].In quantum field theories, vacuum can be defined as the lowest possible energy density,empty of real particles but not of fields, with this empty space filled with particles and antipar-ticles being continuously created and annihilated, without violating the uncertainty principle ofHeisenberg. These fields would have a resulting energy density that would behave and be of thesame nature as the cosmological constant. But this situation gives rise to a big problem: therapid and early expansion of the universe may have been produced from vacuum energy, but itinvolves a very high value for the observed value of the cosmological constant. This discrepancybetween theory (of vacuum energy) and observation (of the cosmological constant) is known asthe cosmological constant problem [28–34].For a long time, the cosmological constant was thought to be exactly zero in our universe.Consequently, it was very surprising in 1998, when observations of type Ia supernovae led tothe discovery that the expansion of the universe is accelerating, implying that Λ is non-zero andpositive. Cosmological observations imply that the vacuum energy density value is [35–38] ρ Λ obs ≈ − GeV . (1)This is a very small number and this smallness in the observed value is the essence of the cosmological constant problem .On the other hand, the theoretical calculations of the quantum field theory produce a veryhigh value for the vacuum energy. We can estimate their value by modeling these fields as asimple collection of independent harmonic oscillators at each point in space and then addingtheir zero point energies. If we apply the quantum field theory up to the Planck scale we obtainthe following estimate [35–38] ρ Λ vac ≈ GeV . (2)Such discrepancy of about 120 orders of magnitude (depending on the dimension used for theenergy density, this number varies slightly) between ρ Λ obs and ρ Λ vac is the central point of the cosmological constant problem. Another issue involving the cosmological constant (considered here as dark energy) has to dowith the fact that its current observed density is of the same order of magnitude as the densityof dark matter Λ (0) vac ∼ Λ (0) obs , (3)implying that we are living in a very peculiar period in the history of the universe that requires avery precise fine-tuning of these quantities. This situation is known as the coincidence problem .[39–42]Consequently, regarding the cosmological constant, we use it quite naturally despite thesetwo problems: the cosmological constant problem and the coincidence problem .Given the difficulties of detecting the particles responsible for dark matter, of discovering thephysical nature of dark energy and of solving the two problems associated to the cosmologicalconstant, research groups around the world consider alternative hypotheses in which, instead ofconsidering the existence of dark matter and dark energy, it is assumed that a theory of gravityworks differently at different scales. Or that the general relativity is not complete and dependson new elements in its action [43–51] . Although some alternative theories reproduce severalastronomical and cosmological observations and even overcome the failures of the dark models,they fail in other aspects in which the dark models are successful.The structure and organization of the article is as follows: in section II we introduce thecharacteristics of the model and we analyze the classic solutions using phase portraits. In sectionIII the model is quantized by the Crank-Nicolson method: we, numerically, solve the Wheeler-DeWitt equation for the model and find wave packets of finite norm. We obtain the quantumtunneling probabilities as functions of the model parameters: the Bose-Einstein parameter ( σ ),the cosmological constant (Λ) and the average energy ( E m ) of the wave packet. Finally, insection IV we present our conclusions. II. THE CLASSICAL MODEL
We consider a spatially isotropic and homogeneous universe with positive spatial section, k = +1, described by the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric as ds = − N ( t ) dt + a ( t ) (cid:18) dr − r + r d Ω (cid:19) , (4)where a ( t ) represents the scale factor of the universe, d Ω is the angular line element of a 2 D sphere, N ( t ) is the lapse function and t is the cosmic time. Here the unit system used will be thenatural one where ~ = 8 πG = c = 1. The matter content that fills the universe is a radiationfluid described by the barotropic equation of state p = ρ , plus a Bose-Einstein condensate,described by the equation [52] p = ωρ + σρ , (5)where the polytropic constant σ describes a repulsive self-interaction when σ >
0, an attractiveself-interaction when σ < ω represents a linear term with − ≤ ω ≤
1, where ω = 1 / ω = 0 is dust, ω = − σ <
0, and ω = − T µν = ( ρ + p ) U µ U ν − pg µν , (6)where U µ is the four-velocity, ρ the energy density, p the pressure of the fluid and g µν representsthe metric tensor. This tensor characterizes an isotropic fluid in comoving coordinates, in whichthe 4-velocity U µ = δ µ is timelike.The classical dynamics of the FLRW is given by the metric (4). As already explained earlier,the matter content is composed of a radiation fluid and an attractive Bose-Einstein condensate( σ < H = − p a − V eff ( a ) + p T , (7)where: a is the scale factor, T is a variable associated to the radiation fluid and p a and p T are, respectively, the momenta canonically conjugated to a and T . We consider the gauge N ( t ) = a ( t ).The term V eff ( a ) in eq. (7) is the effective potential containing the terms related to thecurvature of the positive spatial section, cosmological constant and Bose-Einstein condensate, V eff ( a ) = 3 a − Λ a + σ a ( a + 1) . (8)The potential described by the equation (8) takes the form of one or two potential barriers,depending on the values of the dark energy Λ and the Bose-Einstein condensate polytropicparameter σ . The effective potential is well defined at a = 0, being null at this point. In thelimit a → ∞ , we have V eff ( a ) → −∞ . Fig. 1 shows the effective potential: (a) with one potentialbarrier and (b) with two potential barriers.The classical dynamics is governed by Hamilton’s equations: ˙ a = ∂ H ∂p a = − p a , ˙ p a = − ∂ H ∂a = ∂V eff ( a ) ∂a , ˙ T = ∂ H ∂p T = 1 , ˙ p T = − ∂ H ∂T = 0 , (9)in which the dot indicates conformal time derivative, with dη ≡ a ( t ) dt .The dynamics produced by Hamilton’s equations can be analyzed by the projection on the( a, p a ) plane of the phase portrait of this model. In Fig. 2 we have the phase portraits for thetwo cases of the potential shown in Fig. 1. We observe the existence of fixed points of the centerand of the hyperbolic saddle type.The phase portrait projection in Fig. 2(a) has two fixed points: A ( a = 0 , p a = 0), on theenergy surface P T = 0; and A ( a = 1 . , p a = 0), on the surface P T = 700 . A point A is a center whereas A point is A is a hyperbolic saddle.In particular, A describes an example of the so-called Einstein universe, where gravita-tional attraction is counterbalanced by cosmological repulsion. The red orbit, called a sep-aratrix, because it separates different classes of solutions having the same energy surface (a) (b)Figure 1: Effective potential behavior V eff ( a ) for σ = −
50 and: (a) Λ = 1 .
5; (b) Λ = 0 . a, p a ) plane of the phase diagram of the cosmological model. To illustratewe consider two cases: (a) σ = −
50 and Λ = 1 .
5, and (b) σ = −
50 and Λ = 0 .
01. The orbits hererepresent homogeneous and isotropic universes with cosmological constant, Bose-Einstein condensateand a radiation fluid. P T = 700 . a < . p a > P T < . P T < . p a < P T < . a > . p a <
0) fromsome initial value a and, after reaching a minimum value, (corresponding to p a = 0), expands( p a >
0) towards infinite values of a , tending assymptotically to a solution of De Sitter type.Finally, for P T > . p a > B ( a = 0 , p a = 0), with P T = 0, is a center; B ( a = 1 . , p a = 0), with P T = 704 . B ( a = 5 . , p a = 0), with P T = 163 . B ( a = 11 . , p a = 0), with P T = 241 . B and has the same energy P T = 704 . B , thus having energy P T = 241 . . < P T < . a min = 0) to a maximum ( a max ) value of the scale factor. Another class of bouncingsolutions can be observed in the intermediate region between the two separatrices, representedby red ( P T = 241 . P T = 704 . P T > . p a > P T > . p a < a = 0 (big crunch). The surfaces P T < . a < . P T > . a > . a and p a parameters we find the followingsecond order differential equation, in conformal time, for the classical evolution of the scale0factor of the universe a ( η ), d dη a ( η ) + a ( η ) − a ( η ) + 2 σ a ( η ) ( a ( η ) + 1) − σ a ( η ) ( a ( η ) + 1) = 0 . (10)We will consider initial conditions in different regions of the phase space in Fig. 2(b) andbased on these initial conditions we will solve equation (10) numerically in order to obtain theclassical evolution of the scale factor of the universe in conformal time. For each region ofinterest in Fig. 2(b) we will choose a value for the scale factor a (0) = a situated on a givensurface defined by a specific value of p T ; whereas the initial condition regarding first derivativeof the scale factor will be obtained from the Hamiltonian constraint expression ( H = 0)˙ a (0) = ± p p T − V eff ( a )) , (11)to satisfy Friedmann’s equation for the model. Fig. 3 shows four distinct behaviors (for theparameters σ = − , Λ = 0 .
01) that can be observed in the phase portrait shown in Fig. 2(b),already discussed. Fig. 3 (a) shows universes oscillating between a non-zero minimum value anda maximum value, free of singularities (bouncing solutions). These solutions to the equation (10)are found using as initial conditions a (0) = 6 , ˙ a (0) = 0 . p a (0) = − . p T = 164. Fig. 3 (b) shows the behavior of the scale factor for initial conditions a (0) = 10 , ˙ a (0) = 3 . p a (0) = − . p T = 260. The resultshows a universe without singularities which, upon reaching a state of maximum expansion,starts the contraction until a minimum value of the scale factor is reached: the cosmologicalrepulsion then overcomes the gravitational attraction and the universe expands again towards ade Sitter type configuration. These solutions also feature bouncing solutions. This set of initialconditions was chosen in the phase diagram region between the red and blue curves of Fig.2(b).In its turn, Fig.3(c) was made using the initial conditions a (0) = 10 , ˙ a (0) = 13 . p a (0) = − . p T = 751 , p a < a = 0. In Fig. 3(d) were takenfor initial conditions a (0) = 1 . , ˙ a (0) = − . p a (0) = 39 , p T = 800. The result describes a universe that is born in the singularity at a = 0 and expandsindefinitely to the inflationary phase.1 (a) (b)(c) (d)Figure 3: Classical evolution of the scale factor of the universe in conformal time, for the parameters σ = −
50 and Λ = 0 .
01. The solutions shown in (a) and (b) describe universes free of singularities. Solutions(c) and (d) show universes with singularities. In (c) we see collapsing solutions toward singularities.In (d) we have universes that are born in singularities and continue to expand into a De Sitter typeconfiguration.
III. CANONICAL QUANTIZATION
In order to describe the universe in its primordial phase, it is necessary to quantize the model.This will be done using Dirac’s canonical formalism, which consists of replacing the canonicallyconjugate variables with operators; in particular,ˆ p a → − i ∂∂a , ˆ p T → − i ∂∂T . (12)Quantum effects are dominant in the early universe. They can be described by the so-calledwave function of the universe, Ψ( a, T ), which is explicitly dependent on the degrees of freedom2 a and T .The theory of constrained Hamiltonian systems yields the superhamiltonian constraint which,when quantized and applied to the wave function of the universe, gives rise to an importantquantum gravity equation: the Wheeler-DeWitt equation,ˆ H Ψ( a, T ) = 0 . (13)In the model studied in this paper, the Weeler-DeWitt equation takes the form of a time-dependent Schroedinger equation (cid:18) ∂ ∂a − a + Λ a − σ a ( a + 1) (cid:19) Ψ( a, τ ) = − i ∂∂τ Ψ( a, τ ) , (14)in which the time was reparametrized as τ = − T .The Hamiltonian operator ˆ H must be self-adjoint with respect to the internal product [55](Ψ , Φ) = Z ∞ da Ψ ⋆ ( a, τ )Φ( a, τ ) , (15)in such a way that the structure of Hilbert’s space will be restricted to wave functions thatsatisfy, in general, the following constraint conditions,Ψ(0 , τ ) = Ψ( ∞ , τ ) = 0 or ∂ a Ψ(0 , τ ) = ∂ a Ψ( ∞ , τ ) = 0 , (16)In the present model, we choose the initial normalized wavefunction,Ψ( a,
0) = (cid:18) E m π (cid:19) / a · e − E m a , (17)in which E m is the average radiation energy. It satisfies the Ψ(0 , τ ) = Ψ( ∞ , t ) = 0 boundaryconditions, established by Hartle-Hawking [1].Unlike the case studied by Freitas et al. [11], where there is no dark energy, the presentmodel does not have bound states. As the energy spectrum of this model is not discrete, theGalerkin spectral method used in [11] to solve the Wheeler-DeWitt equation is not suitable inthis case. A. The Finite Difference Method of Crank-Nicolson
The Wheeler-DeWitt equation (14) can be solved by using the finite difference method ofCrank-Nicolson [56], notable for its stability [58, 59]. This method consists of numerically3calculating the values of the Ψ( a, τ ) wave function at points in a grid with discrete values of thevariables a and τ within suitable spatial and time intervals, respectively. The approximation bycentral finite differences in the time variable allows one to write the solution of the equation interms of the Hamiltonian operator ( ˆ H ),Ψ j,n +1 = (cid:18) idt ~ ˆ H (cid:19) − (cid:18) − idt ~ ˆ H (cid:19) Ψ j,n . (18)The j and n indices represent spatial and time indices, respectively, locating points of the grid; dt is the time step and ~ is Planck’s constant. The Hamiltonian operator ˆ H then assumes thethree-diagonal form, ˆ H = V eff ( a ) + ~ mdx − ~ mdx ... − ~ mdx V eff ( a ) + ~ mdx − ~ mdx ... − ~ mdx V eff ( a ) + ~ mdx − ~ mdx ... ... ... − ~ mdx V eff ( a j − ) + ~ mdx . (19) Recently, calculations using this algorithm were applied to FRW models [10, 57]. Thosecalculations involved an effective non-linear potential, in which norm calculations were madeas a function of the number N of points in the grid, at two different time points. The resultshave shown that the norm converges to unity as N grows. The same approach has been used insimpler cosmological models [8, 10, 54].
B. Wave packets and the tunneling mechanism
The Crank-Nicolson method was applied for several values of the parameters of the model. Inall cases, it was possible to obtain a finite and well-defined wave packet for all considered valuesof a , even when the 3D-sphere degenerates ( a = 0). As an example, let us first consider thecase in which the classic effective potential eq. (8) has a single potential barrier with maximumheight V max eff = 700 . | Ψ( a, τ max ) | as a function of the scale factor, calculated at the time τ = τ max , when the packet reaches numerical infinity. Here, the chosen initial condition has anaverage energy E m smaller than the maximum energy of the potential V max eff . As we see in Fig. 4the wave packet, despite having less energy than the top of the barrier, crosses that same barrier,since the return point to the right of the potential barrier in this case is a = 1 . (a) (b)Figure 4: Fig. 4(a) shows the probability density ρ = | Ψ( a, τ max ) | for σ = − , Λ = 1 . , E m = 700, atthe time τ max = 200, when Ψ reaches the numerical infinity at a = 20. We let N = 5000 and dt = 0 . a = 1 . Let us now repeat the study above for the case where the effective potential has two potentialbarriers, as shown in Fig. 1(b). For that, let’s consider σ = −
50 and Λ = 0 .
01. In thiscase the effective potential has its global maximum at a = 1 . V eff = 704 . a = 11 . V eff = 241 . a = 5 . V eff = 163 . E m in the range 241 . < E m < . , the wave function of the universe tunnels through a single potential barrier, during its evolution.The result shows that these wave packets are well defined for all considered values of a , evenwhen the scale factor vanishes ( a → E m in the range 163 . < E m < . a and that they tunnel through the two potential barriers. That indicates the possibility5 (a) (b)Figure 5: Fig. 5(a) shows the probability density ρ = | Ψ( a, τ max ) | for σ = − , Λ = 0 .
01 and E m = 690,at the time τ max = 200, when Ψ reaches the numerical infinity at a = 30. We consider N = 10 ,
000 and dt = 0 .
05. The return point to the right of the potential barrier in this case is a = 1 . of the emergence of the universe to the right of the second potential barrier of smaller height.To exemplify this we consider a wave packet with energy E m = 200 as shown in Fig. 6.For wave packets with average energies E m in the range E m < . a , which tunnel thepotential barrier appearing to the right of the return point. This fact shows that the tunnelingmechanism allows the birth of the universe as a classic system. We built wave packages withenergy E m = 140. The return points on the left and right sides of the potential barrier arerespectively a = 0 . a = 15 . a = 30. In all cases analyzed, the possibility of theuniverse emerging from its primordial phase to the later phase of its evolution by a quantumtunneling mechanism was observed.6 (a) (b)Figure 6: Fig. 6(a) shows the probability density ρ = | Ψ( a, τ max ) | for σ = − , Λ = 0 . , E m = 200,at the time τ max = 200, when Ψ reaches the numerical infinity at a = 30. We consider N = 10 ,
000 and dt = 0 .
05. The return point to the right of the second potential barrier in this case is a = 14 . a = 14 . T P = 1 . · − . TableIII shows T P for other values of the energy E m . C. The tunneling effect at the birth of the universe
The results that appear in Figs. 4, 5, 6 and 7 show that the wave package is able to crossthe potential barriers, indicating that the universe may have appeared by a quantum tunnelingprocess. Next, we will investigate the tunneling probability (
T P ) for a given set of fixed valuesof the model parameters. The expression of
T P is given by
T P = R a max a | Ψ( a, τ max ) | da R a max | Ψ( a, τ max ) | da , (20)where a max represents the chosen numeric infinity. We performed these calculations for differentvalues of the average energy E m (see Fig. 8). These results were compared with the tunnelingprobability in the WKB approach defined in [60] as T P
W KB = 4 (cid:0) θ + θ (cid:1) , with θ = e R a a da √ V eff ( a ) − E ) , (21)where a and a correspond to the return points of the effective potential defined in eq. (8)associated with a given energy E .7 (a) (b)Figure 7: Fig. 7(a) shows the probability density ρ = | Ψ( a, τ max ) | for σ = − , Λ = 0 . , E m = 140,at the time τ max = 200, when Ψ reaches the numerical infinity at a = 30. We consider N = 10 ,
000 and dt = 0 .
05. The return point to the right of the second potential barrier in this case is a = 15 . a = 15 . T P = 1 . · − .Table III shows T P for other values of the energy E m . D. Tunneling probability for the case of one barrier
1. Tunneling probability as a function of E m Fig. 8 compares the probabilities
T P and
T P
W KB as functions of the energy E m , for thepotential in Fig. 1(a). Due to the small value of some T P ’s and
T P
W KB ’s, we plot the logarithmsof those two tunneling probabilities against E m .The result shows that T P and
T P
W KB increase as the average energy E m increases. Forenergies near the top of the potential barrier the probabilities T P coincide with the probabilities
T P
W KB . The behavior of the tunneling probabilities as a function of E m , shown in Fig. 9, isqualitatively the same whatever the value of the cosmological constant Λ: T P and
T P
W KB bothgrow with the average energy E m . In fact, analyzing the tables I and II of the Appendix A, wesee that T P and
T P
W KB , as functions of the average energy E m , increase with the value of Λ.8 PSfrag replacementslog T P
PSfrag replacementslog T P
Figure 8: Tunneling probabilities
T P and
T P
W KB (in log scale) as functions of the average energy E m .Here, σ = − , Λ = 1 . , N = 5000 , dt = 0 . , a max = 20 and τ max = 200. The tunneling probabilitieswere calculated for 147 distinct values of E m , all smaller than the maximum energy of the barrier.Figure 9: Tunneling probabilites T P and
T P wkb (in log scale) as functions of the average energy E m forΛ = 1 .
0. Here, we set the parameters σ = − , N = 5000 , dt = 0 . , a max = 20 and τ max = 200. Thetunneling probabilities were calculated for distinct values of E m , all smaller than the maximum energyvalue of the barrier.
2. Tunneling probability as a function of Λ For cases in which the parameters of the Bose-Einstein condensate and dark energy are,respectively, σ = −
50 and Λ ≥ .
02, we observe the existence of a single potential barrier. Fig.910 shows the behavior of
T P as a function of Λ. Note that
T P grows with Λ. Due to somesmall values of
T P ’s, the log scale is used for plotting that tunneling probability against E m .Using a polynomial interpolation of this data set, we obtain the 11-th degree polynomial T P = 1 . − . + 60 . − . + 303 . − . + 335 . − . + 88 . − . + 4 . − . , (22)shown as the blue curve in Fig. 10.
3. Tunneling probability as a function of σ After studying the behavior of
T P as a function of the parameter σ of the Bose-Einsteincondensate, we observed that T P increases as σ increases. As an example, we plot T P as afunction of σ , by fixing the values of E m and Λ. Fig. 11 shows the behavior of T P as a functionof σ for E m = 680 and Λ = 1 .
5. Due to some small values of
T P , the log scale is used.
E. Tunneling probability for the case of two barriers
1. Tunneling probability as a function of E m For the interval 0 < Λ < .
02, the effective potential V eff ( a ) defined in Eq. (8) has twopotential barriers. Therefore, depending on the value of the average energy E m of the wavepacket, the universe can tunnel through the two barriers to emerge classically to the right of thesecond barrier, as shown in Fig. 12; E max represents the local maximum in the first barrierand E max the local (lower) maximum in the second barrier ( E max > E max ). The in-betweenregion has a local minimum with energy E min .We analyzed wave packets of energies E m < E max . For wave packets with energies E m , suchthat E max < E m < E max , the tunneling probability T P decreases smoothly with E m . Forwave packets with energies E min < E m < E max , there is a sharp drop in T P when comparedto the previous case. For wave packets with energy E m < E min , we find that T P still decreasesas E m decreases, but smoothly. In all the analyzed cases it was possible to observe the existenceof a non-zero T P , thus indicating that it is always possible for the universe to emerge from itsprimordial phase to the later classical phase.0
Figure 10: Tunneling probability
T P (log scale) as a function of the parameter Λ. Here σ = − , N =5000 , dt = 0 . , a max = 20 and τ max = 200. The tunneling probabilities were calculated for 12 distinctvalues of Λ with E m = 680. That value of E m is smaller than V max eff , for all values of Λ considered.Figure 11: Tunneling probability T P (log scale) as a function of the parameter σ . Here N = 5000 , dt =0 . , a max = 20 and τ max = 200. The tunneling probabilities were calculated for 15 distinct values of σ with E m = 680 and Λ = 1 .
5. That value of E m is smaller than V max eff , for all values of σ considered. As an example, we will calculate
T P as a function of the energy E m for wave packets withdifferent values of E m < E max . Let σ = −
50 and Λ = 0 .
01; the potential then reaches itsmaximum value E max = 704 . a = 1 . a = 5 . E min = 163 . a = 11 . E max = 241 . T P as a function of the energy E m , in log scale.1 Figure 12: The case of two potential barriers in the effective potential.Figure 13: Tunneling probability
T P (log scale) versus E m ; we let σ = −
50, Λ = 0 . N = 10 . dt = 0 .
05 and a max = 30. Here V max = 704 . t max = 200. In (a), T P is shown for 241 . ≤ E m < . . ≤ E m < . E m < . T P . F. Emerging into the inflationary phase
One may picture the end of the quantum phase (Planck era) as the conclusion of the tunnelingprocess, when the universe then emerges to the right of the last potential barrier, away from thesingularity a = 0. In all cases analyzed here, it can be seen that after the birth of the universeas a classic system, the scale factor is a >>
1. Therefore, the effective potential described by2(8) takes the asymptotic form V eff ∼ = 3 a − Λ a + σa . (23)This shows that the greater the value assumed by the scale factor of the universe, the smallerthe effect of the Bose-Einstein condensate. Therefore, after the emergence of the universe as aclassic system, its evolution will be determined by the dark energy (Λ > t ) correspond to solutions of the de Sitter type a ( t ) ∼ e q Λ3 t .In what follows we will examine solutions of the second order differential equation (10),choosing as initial conditions a (0) = a , such that a will correspond to the return point for agiven E m , to the right of the potential barrier. The initial value of the first derivative ˙ a (0) willbe provided by eq. (11). We will consider, for example, the case of two potential barriers withΛ = 0 .
01 and σ = −
50, and a wave wave packet with average energy E m = 170 ( E min < E m 51) = 2053 . a , with respectto the conformal time η , against η , in log scale.The behaviors of both the scale factor (Fig. 14) and its derivative (Fig. 15) show the universeleaving its quantum phase and then entering its inflationary phase. IV. CONCLUSIONS A FLRW quantum cosmological model was built in a universe with positive curvature filledwith three components: a radiation perfect fluid, a Bose-Einstein condensate with attractive self-interaction and a positive cosmological constant (dark energy). Its quantization was performedby the method of finite differences in the Crank-Nicolson scheme. Solving the Wheeler-DeWittequation allowed obtaining the wave function of the universe and building wave packets of finitenorm.3 Figure 14: Scale factor of the universe a ( η ) as a function of the conformal time η , for initial conditionstaken right after the Planck era: a (0) = 16, ˙ a (0) = 3 . a ( η ) with respect to the conformal time η , in log scale.The function a ( η ) is shown in Fig. 14. The present model differs from that of [11] solely by the dark energy component, but anew kind of solution shows up. In [11] only bounded solutions are obtained. (The scale factorvaries in a finite range, at the classical and quantum levels; at the classical level, some arebouncing solutions; at the quantum level, they are free of singularities.) When dark energyis introduced, however, asimptotically De Sitter soluions are also obtained — i.e., solutionsleading to an inflationary phase — in addition to the bounded solutions. All classicals solutionsare represented in the phase portraits in Figs. 2 and 3.4We studied the possibility of the birth of the universe through a quantum tunneling process.The present model is very interesting because, depending on the values of Λ (cosmologicalconstant) and σ (polytropic constant), one may have a potential with one or two barriers. Wecomputed the tunneling probability ( T P ) for both cases, for different values of the radiationenergy ( E m ) of the wave packet. We noticed, for both cases, that T P increases when E m increases. We also computed, for the case of a single barrier, T P as a function of Λ and σ .We observed that T P increases as either both parameters increase. Finally, we computed theclassical universe evolution using as initial conditions the value of scale factor just to the rightof the rightmost barrier of the potential, and of its first derivative, given by the Hamiltonianconstraint. 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Appendix A: Tunneling Probabilities E m T P a a T P WKB 700 4 . . − . . − 690 4 . . − . . − 680 3 . . − . . − 670 3 . . − . . − 660 3 . . − . . − 650 3 . . − . . − 640 3 . . − . . − 630 3 . . − . . − 620 3 . . − . . − 610 3 . . − . . − 600 3 . . − . . − 590 3 . . − . . − 580 3 . . − . . − 570 3 . . − . . − 560 3 . . − . . − 550 3 . . − . . − 540 2 . . − . . − 530 2 . . − . . − 520 2 . . − . . − 510 2 . . − . . − 500 2 . . − . . − 490 2 . . − . . − 480 2 . . − . . − 470 2 . . − . . − 460 2 . . − . . − 450 2 . . − . . − 440 2 . . − . . − 430 2 . . − . . − 420 1 . . − . . − 410 1 . . − . . − 400 1 . . − . . − 390 1 . . − . . − 380 1 . . − . . − 370 1 . . − . . − 360 1 . . − . . − 350 1 . . − . . − E m T P a a T P WKB 340 1 . . − . . − 330 1 . . − . . − 320 1 . . − . . − 310 9 . . − . . − 300 9 . . − . . − 290 8 . . − . . − 280 7 . . − . . − 270 6 . . − . . − 260 5 . . − . . − 250 5 . . − . . − 240 4 . . − . . − 230 3 . . − . . − 220 3 . . − . . − 210 2 . . − . . − 200 2 . . − . . − 190 1 . . − . . − 180 1 . . − . . − 170 9 . . − . . − 160 6 . . − . . − 150 4 . . − . . − 140 3 . . − . . − 130 1 . . − . . − 120 1 . . − . . − 110 5 . . − . . − 100 2 . . − . . − 90 8 . . − . . − 80 2 . . − . . − 70 7 . . − . . − 60 1 . . − . . − 50 5 . . − . . − 40 2 . . − . . − 30 8 . . − . . − 20 2 . . − . . − 10 4 . . − . . − . . − . . − Table I: Tunneling probabilities for the parameters σ = − 50, Λ = 1 . N = 5 . 000 (spatial discretization), dt = 0 . 05 and a max = 20. Here V max = 700 . τ max = 200. E m T P a a T P WKB 700 4 . . − . . − 690 4 . . − . . − 680 3 . . − . . − 670 3 . . − . . − 660 3 . . − . . − 650 3 . . − . . − 640 3 . . − . . − 630 3 . . − . . − 620 3 . . − . . − 610 3 . . − . . − 600 3 . . − . . − 590 3 . . − . . − 580 3 . . − . . − 570 3 . . − . . − 560 3 . . − . . − 550 3 . . − . . − 540 2 . . − . . − 530 2 . . − . . − 520 2 . . − . . − 510 2 . . − . . − 500 2 . . − . . − 490 2 . . − . . − 480 2 . . − . . − 470 2 . . − . . − 460 2 . . − . . − 450 2 . . − . . − 440 2 . , . − . . − 430 2 . . − . . − 420 1 . . − . . − 410 1 . . − . . − 400 1 . . − . . − 390 1 . . − . . − 380 1 . . − . . − 370 1 . . − . . − 360 1 . . − . . − 350 1 . . − . . − E m T P a a T P WKB 340 1 . . − . . − 330 1 . . − . . − 320 1 . . − . . − 310 9 . . − . . − 300 9 . . − . . − 290 8 . . − . . − 280 7 . . − . . − 270 6 . . − . . − 260 5 . . − . . − 250 5 . . − . . − 240 4 . . − . . − 230 3 . . − . . − 220 3 . . − . . − 210 2 . . − . . − 200 2 . . − . . − 190 1 . . − . . − 180 1 . . − . . − 170 9 . . − . . − 160 6 . . − . . − 150 4 . . − . . − 140 3 . . − . . − 130 1 . . − . . − 120 1 . . − . . − 110 5 . . − . . − 100 2 . . − . . − 90 8 . . − . . − 80 2 . . − . . − 70 6 . . − . . − 60 1 . . − . . − 50 5 . . − . . − 40 2 . . − . . − 30 8 . . − . . − 20 2 . . − . . − 10 4 . . − . . − . . − . . − Table II: Tunneling probabilities for the parameters σ = − 50, Λ = 1 . N = 5 . 000 (spatial discretization), dt = 0 . 05 and a max = 20. In the table V max = 700 . τ max = 200. E m T P a a 700 3 . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . , . − . . − . . − . . − . . − . . − . . − . . − E m T P a a 280 6 . . − . . − . . − E max . . − . . − . . − . . − . . − . . − . . − . . − . . − E min . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − Table III: Tunneling probabilities for the parameters σ = − 50, Λ = 0 . N = 10 . dt = 0 . 05 and a max = 30. Here V max = 704 . τ max = 200, E min = 163 . E max = 241 ..