Birth of an Isotropic and Homogeneous Universe with a Running Cosmological Constant

1. Introduction

One of the main problems faced by the Standard Model of Cosmology is the origin of spacetime through a big bang singularity. The existence of this singularity in the solutions to Einstein’s equations, when applied to cosmological models, highlights the importance of seeking a quantum gravity theory to eliminate it. Since the early Universe had dimensions much smaller than atomic nuclei, it is valid to suggest that it was governed by the laws of quantum mechanics. The first attempt to find a quantum gravity theory was the quantization of General Relativity, which gave rise to the Wheeler-DeWitt equation [1,2]. The application of this quantum theory to cosmology produced Quantum Cosmology (QC) [3]. Although it is widely accepted that QC is not the fundamental theory for describing the very early universe, many important results have already been obtained from the study of QC models. One of such results is the birth of the universe through a quantum tunneling process. Due to that process, it is possible to show that the universe may originate without the big bang singularity [4,5,6,7,8,9,10,11].

In the literature, there are many works using the quantum tunneling process in different quantum cosmology models [12,13,14,15,16,17,18,19,20,21]. In addition to the investigation of the quantum tunneling process, we may find several other recent works on different aspects of QC. In [22], the authors discuss the time in QC. It is shown that the reparametrization invariance is not guaranteed after one quantizes the model. Reference [23] shows a different quantization process, where the authors use what they call a "super field" consisting of the union of the spacetime wavefunction and the matter fields. The authors of [24] study how a form of dark energy called "phantom energy" affects the wavefunction of the universe which is a solution to the appropriate Wheeler-DeWitt equation. One of the most important results is that the higher the energy content in the model, the greater the probability that the universe will be born with a determined size, avoiding the initial singularity in $a=0$. References [25,26,27] bring a novel approach in which the authors try to use fractional calculus in QC. In [28] the author reviews the DeBroglie-Bohm quantum theory. This is a theory that dispenses with the collapse of the wavefunction and, therefore, is quite appropriate to be applied to QC. In [29] the authors also address the issue of time not being an exactly defined notion in QC and present two solutions with applications focused on QC. In reference [30] the authors employ the fractional Riesz derivative in the Wheeler-DeWitt equation for a closed de Sitter geometry. The authors of [31] study a quantum cosmology model using electromagnetic radiation as a matter content. Reference [32] presents the first study of QC using quantum computers. There, the authors solve a Wheeler-DeWitt equation and show that the use of quantum computers allows one to reach solutions with a high degree of precision. In reference [33] the authors explore a two-dimensional cosmological model of quantum gravity. They do it, initially, by writing the classical sector with the aid of the ADM formalism, and then they solve the Wheeler-Dewitt equation in order to describe the quantum sector. In [34] the authors study approximate solutions to the Wheeler-DeWitt equation and compare the results with cosmological perturbative theories. In the work [35], the author works on the quantization of quantum cosmology models, introducing the concepts of mini-superspace, perfect fluids, and scalar fields, bringing up conceptual problems such as the evolution of time and unitarity, as well as physical interpretations. These are some examples showing that Quantum Cosmology is an active field and grows over time.

Since the seminal work by P. A. M. Dirac in 1937 [36] arguing about the possibility that some of the fundamental constants of nature may vary with the age of the Universe, many works have been done exploring that possibility. Although not mentioned in Dirac’s work, the cosmological constant (CC) is today a very important constant of nature because it is a possible candidate to explain the present accelerated expansion of our Universe [37,38]. Some recent observational evidence shows that the expansion of the universe accelerated faster in the past than it is doing now[39]. This opens up the possibility, among others, for a running CC. In fact, even before this recent observational evidence, several researchers have already considered the possibility of a running CC [40,41,42,43,44,45,46,47]. A promising line of investigation of a running CC is based on the study, using quantum field theory in curved spacetime, of the renormalization group (RG) [45,48,49,50,51,52]. In the RG method, the equation of state for the cosmological constant is exactly given by $p_{\mathrm{\Lambda }}=-{\rho }_{\mathrm{\Lambda }}$ and the CC becomes time dependent. Also, using the RG method it is possible, with the aid of some cosmological observations, to impose phenomenological limits on a certain parameter of the method [51,53]. It certainly improves the predictions of this method.

In this work, we want to study the Universe in its early stages through quantum cosmology. For this purpose, the tunneling probabilities for the emergence of the Universe through a potential barrier are calculated. We do that with the help of the solutions to the Wheeler-DeWitt equation obtained in two different ways: the numerical solution and the WKB approximation. The Universe is described by a Friedmann-Lemaître-Robertson-Walker (FLRW) model with positive curvature of the spatial sections, coupled to a dust perfect fluid and a running cosmological constant, described by the RG method. The latter is introduced to describe the dark energy present in the Universe.

In Section 2, we derive the total Hamiltonian, draw a phase portrait, and solve the Hamilton equations for a FLRW cosmological model, with constant positive spatial sections and coupled to a dust perfect fluid and a running cosmological constant. In Section 3, we quantize the model and solve the resulting Wheeler-DeWitt equation, using a numerical method and the WKB approximation. In Section 4, we compute the WKB ($T{P}_{WKB}$) and the integrated ($T{P}_{int}$) tunneling probabilities for the universe to tunnel through the potential barrier. We investigate how $T{P}_{WKB}$ and $T{P}_{int}$ depend on several parameters of the model. Finally, in Section 5, we summarize the main points and results of the paper. In Appendix A, we compute in detail, with the aid of the Schutz variational formalism, the total Hamiltonian for the dust fluid.

2. The Classical Model

The classical cosmology, based on the FLRW model, has its foundations in Einstein’s General Theory of Relativity (GR) [54], accurately elucidating large-scale phenomena such as black holes, gravitational waves, light deflection, and the precession of Mercury’s orbit. Objectively, in this section, the ADM formalism [55,56] was used to derive the Hamiltonian density of GR, thus making possible to quantize it.

The action integral, written in terms of the matter Lagrangian density (${\mathcal{L}}_m$) and the Ricci scalar (R), is given by,

$$S={\int }_{-\infty }^{+\infty }\left({\displaystyle \frac{R}{2}}+{\mathcal{L}}_m\right)\sqrt{-g}{d}^{4}x.$$

(1)

Where g is the determinant of the metric, ${\mathcal{L}}_m$ is defined as the sum of the contributions from the time-varying CC, ${\mathcal{L}}_{\mathrm{\Lambda }}=p_{\mathrm{\Lambda }}$ (where $p_{\mathrm{\Lambda }}$ is the pressure associated to the time-varying CC), and the perfect fluid ${\mathcal{L}}_{\alpha }=p_{\alpha }$ (where $p_{\alpha }$ is the pressure associated to the fluid). In this work, the following system of units is used: $c=\hslash =8\pi G=1$.

If we introduce the values of R and $\sqrt{-g}$, computed for the FLRW metric with positively curved spatial sections, in Equation (1) we obtain,

$$S=3V_0{\int }_{-\infty }^{+\infty }\left({\displaystyle \frac{\ddot{a}{a}^2}{N}}+{\displaystyle \frac{{\dot{a}}^{2}a}{N}}-{\displaystyle \frac{\dot{N}\dot{a}{a}^2}{N^2}}+Na-{\displaystyle \frac{N{a}^3{\rho }_{\mathrm{\Lambda }}}{3}}+{\displaystyle \frac{N{a}^3\alpha {\rho }_{\alpha }}{3}}\right)dt,$$

(2)

where $a\left(t\right)$ is the scale factor, $N\left(t\right)$ is the lapse function, ${\rho }_{\mathrm{\Lambda }}$ is the time-varying CC energy density (we used that $p_{\mathrm{\Lambda }}=-{\rho }_{\mathrm{\Lambda }}$), ${\rho }_{\alpha }$ is the fluid energy density, $\alpha$ is a constant associated with the fluid Equation (A5), the dot means derivative with respect to time and $V_0$ is defined as,

$$V_0={\int }_{-\infty }^{+\infty }{\displaystyle \frac{r^{2}sin\theta }{{(1-r^2)}^{1/2}}}{d}^{3}x.$$

(3)

The running cosmological constant energy density, ${\rho }_{\mathrm{\Lambda }}$ has the following expression [45,48,49,50,51,52],

$${\rho }_{\mathrm{\Lambda }}={\rho }_{\mathrm{\Lambda }}^0+\nu (H^2-H_0^2),$$

(4)

where ${\rho }_{\mathrm{\Lambda }}^0$ represents the vacuum energy density at present time, H is the Hubble function defined as $H=\dot{a}/a$, $H_0$ is a constant that gives the Hubble function at the present time, and the parameter $\nu$ is a dimensionless constant that emerges from the renormalization process and characterizes the magnitude of quantum effects on the vacuum energy density. It is defined as,

$$\nu ={\displaystyle \frac{\sigma }{12\pi }}{\displaystyle \frac{M^2}{M_P^2}},$$

(5)

where $\sigma$ represents the predominance of bosons ($\sigma =+1$) or fermions ($\sigma =-1$) at high energies, M is a weighted sum of the effective masses of massive virtual particles and $M_P$ is the Planck mass ($M_P\sim 1.22\times {10}^{19}\mathrm{GeV}$). Since $M\ll M_P$, the parameter $\nu$ is typically very small, of the order of ${10}^{-6}$[51,53]. This value reflects the low quantum contribution relative to the total gravitational energy. However, studies suggest that in the early Universe, during a highly anisotropic and dynamic phase, the usual limitations on the sign and magnitude of $\nu$ may not apply. In such conditions, $\nu$ could assume much larger values (both positive and negative). This occurs because spacetime conditions are dominated by intense quantum effects and high-energy interactions that influence vacuum dynamics [52]. This behavior enables exploration of cases where the values of $\nu$ directly impact the isotropization process and the evolution of the metric [52]. The parameter $\nu$ also regulates the temporal variation of the vacuum energy density.

Introducing the value of ${\rho }_{\mathrm{\Lambda }}$ Equation (4) in the action Equation (2), we find,

$$S=V_0{\int }_{-\infty }^{+\infty }\left\{-{\displaystyle \frac{3{\dot{a}}^{2}a}{N}}+3Na+N{a}^3\left[\alpha {\rho }_{\alpha }-{\rho }_{\mathrm{\Lambda }}^0-\nu \left(H^2-H_0^2\right)\right]\right\}dt,$$

(6)

From action (6) one extracts the following Lagrangian density,

$$\mathcal{L}=-{\displaystyle \frac{3{\dot{a}}^{2}a}{N}}+3Na+N{a}^3\left[\alpha {\rho }_{\alpha }-{\rho }_{\mathrm{\Lambda }}^0-\nu \left(H^2-H_0^2\right)\right],$$

(7)

for simplicity, we will consider $V_0=1$.

Now we can compute the canonical momentum,

$$p_a={\displaystyle \frac{\partial \mathcal{L}}{\partial \dot{a}}}=-{\displaystyle \frac{6\dot{a}a}{N}}-2\nu Na\dot{a}=-2\dot{a}a\left({\displaystyle \frac{3}{N}}+\nu N\right)$$

(8)

Introducing $p_a$ Equation (8) into the Hamiltonian density definition, $\mathcal{H}=p_a\dot{a}-\mathcal{L}$ and using the Schutz formalism [57], in order to determine the Hamiltonian of the matter sector (see Appendix A for more details), one obtains,

$$N\mathcal{H}=-{\displaystyle \frac{p_a^2}{4\left(\frac{3}{N}+\nu N\right)a}}-3Na+N{p}_T+N{a}^{3}C,$$

(9)

where $C={\rho }_{\mathrm{\Lambda }}^0-\nu H_0^2$. To avoid factor ordering ambiguities, one must use the following change of variables [58],

$$a={\left({\displaystyle \frac{3x}{2}}\right)}^{2/3},p_a=p_x{a}^{1/2}.$$

(10)

Choosing the gauge $N=1$, the Hamiltonian density becomes,

$$\mathcal{H}=-{\displaystyle \frac{p_x^2}{4\left(3+\nu \right)}}-3{\left({\displaystyle \frac{3x}{2}}\right)}^{2/3}+p_T+{\displaystyle \frac{9}{4}}C{x}^2.$$

(11)

Using the constraint equation $\mathcal{H}=0$ one identifies the potential $V\left(x\right)$,

$$V\left(x\right)=3{\left({\displaystyle \frac{3x}{2}}\right)}^{2/3}-{\displaystyle \frac{9x^2}{4}}C,$$

(12)

which has the shape of a barrier. In Figure 1, we show an example of $V\left(x\right)$ Equation (12), for ${\rho }_{\mathrm{\Lambda }}^0=0.22$, $\nu =0.2$ and $H_0=1$. In this example, the maximum value of $V\left(x\right)$ ($V_{max}$) is 14.1421.

The classical dynamics is governed by the Hamilton equations,

$$\left\{\begin{array}{c}\dot{x}={\displaystyle \frac{\partial \mathcal{H}}{\partial p_x}}=-{\displaystyle \frac{p_x}{2(3+\nu )}},\\ \dot{p_x}=-{\displaystyle \frac{\partial \mathcal{H}}{\partial x}}=3{\left({\displaystyle \frac{3x}{2}}\right)}^{-1/3}-{\displaystyle \frac{9C}{2}}x,\hfill \\ \dot{T}={\displaystyle \frac{\partial \mathcal{H}}{\partial p_T}}=1,\hfill \\ \dot{p}T=-{\displaystyle \frac{\partial \mathcal{H}}{\partial T}}=0.\hfill \end{array}\right.$$

(13)

So, now one may impose the constraint equation $\mathcal{H}=0$ leading to an equation for the momentum $p_x$ given by,

$$p_x=\pm {\left[-12\left(3+\nu \right){\left({\displaystyle \frac{3x}{2}}\right)}^{{\displaystyle \frac{2}{3}}}+4\left(3+\nu \right)p_T+9C\left(3+\nu \right)x^2\right]}^{{\displaystyle \frac{1}{2}}}.$$

(14)

We begin studying the dynamics produced by Hamilton equations by constructing the phase portrait of the current model, which is shown in Figure 2. The curves are labeled according to the parameter $p_T$, which takes the values $p_T=0$, 1, 2, 5, 10, 14, $14.145$, 15, 20, 25, …, 200, 250, 300. The phase portrait gives a general idea of all different types of dynamical solutions of the model. As we shall see next, all different types of dynamical solutions can be associated with the four regions identified in the phase portrait Figure 2.

After combining some of the Hamilton equations from Equation (13), one can obtain a second-order differential equation for the classical evolution of the scale factor,

$$\ddot{x}+{\displaystyle \frac{3}{2(3+\nu )}}{\left({\displaystyle \frac{3x}{2}}\right)}^{-1/3}-{\displaystyle \frac{9C}{4(3+\nu )}}x=0.$$

(15)

When solving Equation (15), with the appropriate initial conditions, four different types of classical solutions are found pictured in Figure 3a,d. These solutions are: (i) A contraction solution which has x starting at a high value and decreasing until it reaches $x=0$, which produces a big crunch singularity. This solution can be seen in Figure 3a. For this particular example, the initial conditions were $x_0=100$ and $p_T=-15$. These types of solutions are located in Region II of Figure 2; (ii) An expansion solution where x begins at small values and starts an expansion at an accelerated rate to infinity. This solution can be seen in Figure 3b. For this particular example, the initial conditions were $x_0=2$ and $p_T=15$. These types of solution are located in Region I of Figure 2; (iii) An expansion followed by a contraction solution in which x begins at small values and grows until it reaches a maximum value. Then it begins to shrink until it reaches $x=0$, which produces a big crunch singularity. This solution is shown in Figure 3c. For this particular example, the initial conditions were $x_0=0.1$ and $p_T=2$. These types of solution are located in Region IV of Figure 2; (iv) A bouncing solution, where x starts at a high value, shrinks until it reaches a minimum value and then grows towards infinity. This solution is portrayed in Figure 3d. For this particular example, the initial conditions were $x_0=100$ and $p_T=3$. These types of solution are located in Region III of Figure 2.

3. Canonical Quantization

The next step is the quantization of the present model. We shall do that following Dirac’s formalism [59,60], which consists of replacing the canonical variables x and T and their canonically conjugated momenta with its respective operators. After that, one must introduce the wavefunction of the Universe, which is a function of the operators associated with the canonical variables, written as $\mathrm{\Psi }(x,T)$. The quantum effects are prevailing in the early Universe and the wavefunction of the Universe, is responsible to describe those quantum effects. One initiates the quantization process by replacing the canonical momenta $p_x$ and $p_T$ with their respective operators:

$$p_x⟶-i{\displaystyle \frac{\partial }{\partial x}},p_T⟶-i{\displaystyle \frac{\partial }{\partial T}}.$$

(16)

Next, we must impose the constraint equation $\widehat{H}\mathrm{\Psi }(x,T)=0$, where $\widehat{H}$ is the Hamiltonian operator. This equation gives rise to the Wheeler-DeWitt equation [1,2]. Using the Hamiltonian from Equation (9) with gauge choice $N=1$ and doing $\tau =-T$, one obtains the Wheeler-DeWitt equation for this model:

$$\left({\displaystyle \frac{1}{4(3+v)}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}-3{\left({\displaystyle \frac{3x}{2}}\right)}^{2/3}+{\displaystyle \frac{9x^2}{4}}C\right)\mathrm{\Psi }(x,t)=-i{\displaystyle \frac{\partial }{\partial \tau }}\mathrm{\Psi }(x,t).$$

(17)

3.1. WKB Tunneling Probability

Assuming that one may write the solution to equation (17) in the form $\mathrm{\Psi }(x,\tau )=\psi \left(x\right)e^{-iE\tau }$ and using it in Equation (17), one finds,

$${\displaystyle \frac{{\partial }^2}{\partial x^2}}\psi \left(x\right)+4(3+\nu )[E-V\left(x\right)]\psi \left(x\right)=0,$$

(18)

with $V\left(x\right)$ being given by Equation (12). Now, suppose that the WKB approximation is valid for the present model. Once obtained, the WKB solution will be used to calculate the tunneling probabilities through the potential barrier $V\left(x\right)$. Following the steps to obtain the WKB solution, one begins writing the solution $\psi \left(x\right)$ as $\psi \left(x\right)=\mathcal{A}{e}^{i\phi \left(x\right)}$. Using this ansatz in equation (18) and assuming that $d^2\mathcal{A}\left(x\right)/d{x}^2$ is negligible compared to the other terms, the solution may be written as [61],

$$\psi \left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{A}{\sqrt{K\left(x\right)}}}exp\left(i{\int }_x^{x_l}K\left(x\right)dx\right)+{\displaystyle \frac{B}{\sqrt{K\left(x\right)}}}exp\left(-i{\int }_x^{x_l}K\left(x\right)dx\right)\hfill & 0\le x\le x_l,\hfill \\ {\displaystyle \frac{C}{\sqrt{\kappa \left(x\right)}}}exp\left(-{\int }_{x_l}^x\kappa \left(x\right)dx\right)+{\displaystyle \frac{D}{\sqrt{\kappa \left(x\right)}}}exp\left({\int }_{x_l}^x\kappa \left(x\right)dx\right)\hfill & x_l\le x\le x_r,\hfill \\ {\displaystyle \frac{F}{\sqrt{K\left(x\right)}}}exp\left(i{\int }_{x_r}^{x}K\left(x\right)dx\right)+{\displaystyle \frac{G}{\sqrt{K\left(x\right)}}}exp\left(-i{\int }_{x_r}^{x}K\left(x\right)dx\right)\hfill & x_r\le x<\infty ,\hfill \end{array}\right.$$

(19)

where $A,B,C,D,F$ and G are constants, $x_l$ is the value of x which the energy E intercepts the potential $V\left(a\right)$ from the left side, and $x_r$ is the value of x where the energy E intercepts the potential $V\left(a\right)$ from the right side. Also, the $K\left(x\right)$ and $\kappa \left(x\right)$ terms are,

$$\left\{\begin{array}{cc}K\left(x\right)=\sqrt{4(3+\nu )(E-V(x\left)\right)}\hfill & E>V\left(x\right),\hfill \\ \kappa \left(x\right)=\sqrt{4(3+\nu )\left(V\right(x)-E)}\hfill & E<V\left(x\right).\hfill \end{array}\right.$$

(20)

Using matrix notation, one can set forth the connection between coefficients $A,B,C,D,F$ and G [61],

$$\left({\displaystyle \genfrac{}{}{0pt}{}{A}{B}}\right)=\left(\begin{array}{cc}2\theta +{\displaystyle \frac{1}{2\theta }}& i\left(2\theta -{\displaystyle \frac{1}{2\theta }}\right)\\ -i\left(2\theta -{\displaystyle \frac{1}{2\theta }}\right)& 2\theta +{\displaystyle \frac{1}{2\theta }}\end{array}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{F}{G}}\right).$$

(21)

The parameter $\theta$ gives the barrier height and length. It is given by,

$$\theta =exp\left({\int }_{x_l}^{x_r}\kappa \left(x\right)dx\right).$$

(22)

Having the WKB solution, one is now able to compute the transmission coefficient, i.e., the tunneling probability. If one considers that there is no wave coming from the right ($G=0$), the tunneling probability, being denoted by $T{P}_{WKB}$, will be [61],

$$T{P}_{WKB}={\displaystyle \frac{{\left|F\right|}^2}{{\left|A\right|}^2}}={\displaystyle \frac{4}{{\left(2\theta +\frac{1}{2\theta }\right)}^2}}.$$

(23)

For the present model, $\theta$ will be,

$$\theta =exp\left({\int }_{x_l}^{x_r}\sqrt{4(3+\nu )\left(3{\left({\displaystyle \frac{3x}{2}}\right)}^{2/3}-{\displaystyle \frac{9x^2}{4}}C-E\right)}dx\right).$$

(24)

3.2. Integrated Tunneling Probability

The operator $\widehat{H}$ Equation (17) is self-adjoint [62] with respect to the internal product,

$$(\mathrm{\Psi },\Phi )={\int }_0^{\infty }dx{\mathrm{\Psi }}^{*}(x,\tau )\Phi (x,\tau ),$$

(25)

if the wavefunctions satisfy one of the next boundary conditions, $\mathrm{\Psi }(0,\tau )=0$ or ${\mathrm{\Psi }}^{\prime }(0,\tau )=0$, where the prime ′ means the partial derivative with respect to x. In the present paper, we shall restrict our attention to wave functions satisfying $\mathrm{\Psi }(0,\tau )=0$. We also demand that $\mathrm{\Psi }(x,\tau )\to 0$ when $x\to \infty$. Along with the WKB solution, we also solve the Wheeler-DeWitt equation (17) numerically. In order to do that, we employ a finite difference procedure based on the Crank-Nicolson method [63]. We tested the validity of our numerical solutions by computing the norm of the wavefunction for different times, and as a result, we found that it was always preserved. That test is commonly performed in order to evaluate the validity of numerical solutions to quantum mechanical systems [13,14,15,17]. The next step in order to solve the Wheeler-DeWitt equation Equation (17), is furnishing an initial wavefunction. That initial wavefunction fixes an energy for the dust. One may understand the relationship between the initial wavefunction energy and the dust energy because the time variable is related to the degree of freedom of the dust. Hence, the energies of the stationary states are associated with the energies of the dust fluid.

Among the several possibilities, as a matter of simplicity, we have chosen the following initial wavefunction,

$$\mathrm{\Psi }(x,0)={\displaystyle \frac{8\sqrt[4]{2}{E}^{3/4}x{e}^{-4x^{2}E}}{\sqrt[4]{\pi }}}$$

(26)

where E represents the mean kinetic energy associated to the dust energy and $\mathrm{\Psi }(x,0)$ is very concentrated in a region close to $x=0$. The initial wavefunction Equation (26) must satisfy the following condition, in order to be normalized, ${\int }_0^{\infty }{\left|\mathrm{\Psi }(x,0)\right|}^{2}dx=1$. Since we want to calculate the tunneling probability, we must compute the outgoing wavefunction. In other words, we must compute the part of the wavefunction which tunnels the potential barrier. The outgoing wavefunction, propagates to infinity in the positive x direction, as time goes to infinity. Once we are performing a numerical calculation, we must specify a maximum value, in the x direction. Let us call that number $x_{max}$. The behavior of all wavefunctions, we computed, and their time evolution show that they are well defined in the whole space.

Figure 4 gives an example of the probability density as a function of the scale factor x at the moment ${\tau }_{max}=10$, when $\mathrm{\Psi }$ reaches the numerical infinity, at $x_{max}=100$. In this example, we draw the potential written in Eq.(12), which is a barrier, with values ${\rho }_{\mathrm{\Lambda }}^0=0.22$, $\nu =0.2$ and $H_0=1$. In this example, the maximum potential value is $V_{max}=14.1421$ located at $x=12.5353$. The fluid energy is $E=14.1$, which is smaller than $V_{max}$. Then, for this example, the tunneling process will occur. For that fluid energy E, the wavefunction will reach the potential barrier at $x_l=11.6072$ and, after tunneling, it will leave the potential barrier at $x_r=12.6396$.

In order to obtain the integrated tunneling probability, denoted by $T{P}_{int}$, we use the following definition [13,14],

$${\mathit{TP}}_{\mathit{int}}={\displaystyle \frac{{\int }_{x_r}^{\infty }{\left|\mathrm{\Psi }(x,{\tau }_{max})\right|}^{2}dx}{{\int }_0^{\infty }{\left|\mathrm{\Psi }(x,{\tau }_{max})\right|}^{2}dx}}.$$

(27)

Here, $\mathrm{\Psi }(x,{\tau }_{max})$ is the wavefunction of the universe calculated at the time $\tau ={\tau }_{max}$, which is the moment $\mathrm{\Psi }$ reaches the numerical infinity $x_{max}$. $T{P}_{int}$ can be understood as the odds for the Universe to be found at the right side of the potential barrier.

4. Results

In this section we will compute the tunneling probabilities $T{P}_{int}$ and $T{P}_{WKB}$ for the potential $V\left(x\right)$ given in Equation (12) and how they change due to variations of energy E and the parameters ${\rho }_{\mathrm{\Lambda }}^0$, $\nu$ and $H_0$. Aiming to investigate the behavior of probabilities in the face of changes in parameters and energy, we fix them as well as the energy, except for the quantity under study. This strategy is applied to each one of the parameters and energy. The x values where the energy E intercepts the potential, $x_l$ and $x_r$, are also relevant. They must be determined to calculate the probabilities.

4.1. The Phenomenological Parameter $\nu$

To study how the tunneling probabilities $T{P}_{int}$ and $T{P}_{WKB}$ are affected by changes in the $\nu$ parameter, we fixed the values of parameters ${\rho }_{\mathrm{\Lambda }}^0$ and $H_0$ and the energy E. We chose ${\rho }_{\mathrm{\Lambda }}^0=0.22$, $H_0=1.0$ and $E=4$ (this value is smaller than $V_{max}$). The range of values used to examine the behavior of $T{P}_{int}$ and $T{P}_{WKB}$ begin at $\nu =-0.02$ and go all the way up to $\nu =0.021$ in steps of $0.001$. Since the Universe under study is very primitive, $\nu$ may have negative or positive values. One can see the results of the calculations in Table 1 and Figure 5, which shows a comparison between $T{P}_{int}$ and $T{P}_{WKB}$. As one can see, the change in $T{P}_{int}$ is much more subtle than the change in $T{P}_{WKB}$. We have further analyzed each of the tunneling probabilities individually. It is possible to see in Figure 6 that both decrease when the parameter $\nu$ grows. Table 1 shows the parameter $\nu$ values, the tunneling probabilities and the x values where the energy intercepts the potential ($E=V\left(x\right)$) from the left side ($x_l$) and from the right side ($x_r$).

4.2. The Cosmological Constant Energy Density ${\rho }_{\mathrm{\Lambda }}^0$

To examine how the tunneling probabilities $T{P}_{int}$ and $T{P}_{WKB}$ are affected by changes in the ${\rho }_{\mathrm{\Lambda }}^0$ parameter, we fixed values for parameters $\nu$ and $H_0$ and the energy E. Here, we chose $\nu =0.2$, $H_0=1.0$ and $E=10$ (this value is smaller than $V_{max}$). The range of values used to examine the behavior of $T{P}_{int}$ and $T{P}_{WKB}$ begin at ${\rho }_{\mathrm{\Lambda }}^0=0.22$ and go all the way up to ${\rho }_{\mathrm{\Lambda }}^0=0.23$ in steps of $0.0005$. One can see the results of the calculations in Table 2 and Figure 7, which shows a comparison between $T{P}_{int}$ and $T{P}_{WKB}$. Table 2 shows the parameter ${\rho }_{\mathrm{\Lambda }}^0$ values, the tunneling probabilities and the x values where the energy intercepts the potential ($E=V\left(x\right)$) from the left side ($x_l$) and from the right side ($x_r$). Figure 7 shows that $T{P}_{int}$ and $T{P}_{WKB}$ increase as one increases the parameter ${\rho }_{\mathrm{\Lambda }}^0$.

4.3. Energy E

To study how the tunneling probabilities $T{P}_{int}$ and $T{P}_{WKB}$ are affected by changes in the fluid energy E, we fixed values for the potential barrier parameters. Here, we chose ${\rho }_{\mathrm{\Lambda }}^0=0.22$, $\nu =0.2$ and $H_0=1.0$. The maximum value of this potential is $V_{max}=14.1421$. The energy values used to examine the behavior of $T{P}_{int}$ and $T{P}_{WKB}$ begin at $E=0.5$ and go all the way up to $E=14$ in steps of $0.5$. We also include the values $E=14.05,14.07$ and $14.09$. One can see the results of the calculations in Table 3 and Figure 8, which shows a comparison between $T{P}_{int}$ and $T{P}_{WKB}$. Table 3 shows the energy values, the tunneling probabilities and the x values where the energy intercepts the potential ($E=V\left(x\right)$) from the left side ($x_l$) and from the right side ($x_r$). Figure 8 shows that $T{P}_{int}$ and $T{P}_{WKB}$ grow as one increases the energy.

4.4. $H_0$

To examine how the tunneling probabilities $T{P}_{int}$ and $T{P}_{WKB}$ are affected by changes in the $H_0$ parameter, we fixed values for parameters $\nu$ and $\mathrm{\Lambda }$ and determined a fixed energy value. Here, we chose $\nu =0.2$, ${\rho }_{\mathrm{\Lambda }}^0=0.22$ and $E=5$ (this value is smaller than $V_{max}$). The range of values used to examine the behavior of $T{P}_{int}$ and $T{P}_{WKB}$ begin at $H_0=1.0$ and go all the way up to $H_0=1.01$ in steps of $0.0005$. One can see the results of the calculations in Table 4 and Figure 9, which shows a comparison between $T{P}_{int}$ and $T{P}_{WKB}$. Table 4 shows the parameter $H_0$ values, the tunneling probabilities and the x values where the energy intercepts the potential ($E=V\left(x\right)$) from the left side ($x_l$) and from the right side ($x_r$). Figure 9 shows that $T{P}_{int}$ and $T{P}_{WKB}$ decrease as one increases the parameter $H_0$.

5. Conclusions

In the present work, the probability for the birth of a homogeneous and isotropic FLRW Universe with positively curved spatial sections ($k=+1$) was studied. The matter content of the model is composed of a dust perfect fluid. The potential barrier of the model originates from the geometry of space-time, the fluid energy density and the presence of running cosmological constant, which can be interpreted as the zero-point energy of the quantum vacuum in Quantum Field Theory (QFT) [64]. The Hamiltonian was found using the ADM formalism and the model was quantized using the Dirac formalism.

We explicitly calculated the tunneling probabilities $T{P}_{WKB}$ and $T{P}_{int}$ for the birth of the Universe as a function of the phenomenological parameter ($\nu$), the dust energy E, the cosmological constant energy density (${\rho }_{\mathrm{\Lambda }}^0$) and the Hubble parameter ($H_0$).

It was observed that both tunneling probabilities, $T{P}_{WKB}$ and $T{P}_{int}$, decrease as one increases $\nu$. It was also noted that $T{P}_{WKB}$ and $T{P}_{int}$ grow as E increases, indicating that the Universe is more likely to be born with higher values of the dust energy E. The same is observed for the ${\rho }_{\mathrm{\Lambda }}^0$ parameter, that is, $T{P}_{WKB}$ and $T{P}_{int}$ are bigger for higher values of ${\rho }_{\mathrm{\Lambda }}^0$. Finally, the tunneling probabilities decrease as one increases the value of $H_0$.

So, the best conditions for the Universe to be born, in the present model, would be having the higher possible values for E and $\mathrm{\Lambda }$ and the lowest possible values for $\nu$ and $H_0$.