Quantum Creation of a Friedmann-Robertson-Walker Universe: Riesz Fractional Derivative Applied

Abstract

In this work, we apply fractional calculus to study quantum cosmology. Specifically, our Wheeler-DeWitt (WDW) equation includes a Friedman-Robertson-Walker (FRW) geometry, a radiation fluid, a positive cosmological constant ($\mathsf{\Lambda }$), and an ad-hoc potential. We employ the Riesz fractional derivative, which introduces a parameter $\alpha$, where $1<\alpha \le 2$, in the WDW equation. We investigate numerically the tunneling probability for the Universe to emerge using a suitable WKB approximation. Our findings are as follows. When we decrease the value of $\alpha$, the tunneling probability also decreases, suggesting that if fractional features could be considered to ascertain among different early universe scenarios, then the value $\alpha =2$ (meaning strict locality and standard cosmology) would be the most likely. Finally, our results also allow for an interesting discussion between selecting values for $\mathsf{\Lambda }$ (in a non-fractional conventional set-up) versus balancing, e.g., both $\mathsf{\Lambda }$ and $\alpha$ in the fractional framework.

1. Introduction

Since the introduction of the Wheeler-DeWitt equation [1,2], intending to describe the Universe as a quantum mechanical system, quantum cosmology (QC) [3,4,5,6,7] has steadily developed and achieved significant results concerning the quest to understand the origin of the Universe. For seminal contributions and recent reviews, cf. references [8,9].

The framework of QC includes audacious scenarios, namely the spontaneous creation from nothing [10,11,12,13,14]: our time and space emerge through a potential barrier. The universe (we will find ourselves in and observe) tunneling through the potential barrier and appearing to the right of it with a finite size and free from the initial singularity. This concept was broadly embraced, leading to the study of tunneling probabilities (TP) for the birth of the Universe for different cosmological models [15]. Because the regime is of high energy and the time is of the order of Planck time, quantum cosmology has no observable data. For more information on observables in quantum cosmology, their contributions of the theory and future predictions, we suggest reading reference [16].

Given the discussion in the above paragraph it is immensely tempting to explore QC further, trying new tools to better understand the Universe’s origin. One such tool that recently gathered interest is fractional calculus [17], an extension of traditional calculus that allows for the modelling of complex systems and thus may be applied to describe phenomena in classical and quantum cosmology. See, e.g., [18,19] for a broad sample of recently published contributions.

Originating from the works of G. l’Hopital and G. W. Leibniz, and subsequently developed by mathematicians such as Abel, Liouville, and Riemann, an innovative mathematical tool was introduced, allowing derivatives and integrals to assume non-integer orders. This tool enabled the development of fractional calculus and is currently applied in various branches of physics, including field theory, particle physics, and atomic and molecular physics [17]. In addition, fractional calculus has also been used in engineering, biophysics and biomedicine, with encouraging results. For more information see [20,21,22], e.g., and the many references therein.

Depending on the phenomenon to be described, the system of fractional differential equations can be linear [23,24,25,26] or nonlinear [27,28,29]. Especially, there are many important phenomena in nature that can be described by the nonlinear fractional differential equations [27,28,29] or by the nonlinear integer differential equations [30,31,32]. In the present work we will focus on the study of cosmological systems described by linear fractional differential equations.

The application of Fractional calculus to quantum mechanics has allowed a generalization of path integrals and the Schrödinger equation, giving rise to fractional quantum mechanics (FQM) [33]. This new research area has gathered interest, and quite a few publications have appeared in the literature, pointing to potential benefits and new tests of traditional quantum mechanics. There still exist significant open problems to address in FQM. Please, see references [23,24,34,35] for more information. In particular, one of such open problems concerns the tunneling effect. Let us be more concrete. In [36,37], the authors studied quantum tunneling through delta potential barriers and through a rectangular barrier, respectively, using a space fractional Schrödinger equation. Tunneling with fractional time derivatives (Caputo derivative) can be found in [38]. In the first paper, the authors found an analytical expression for the TP as a function of the fractional parameter ($\alpha$): the TP increased as $\alpha$ decreased. In the latter paper, the authors obtained a numerical solution where the TP decreased as $\alpha$ decreased. The scenarios were not so qualitatively different, nevertheless, fractional calculus generated discrepant results regarding the behavior of TP, in terms of $\alpha$.

The application of fractional calculus in classical and quantum cosmology has also been of recent interest. The application of fractional calculus in quantum cosmology is usually designated as fractional quantum cosmology (FQC). Let us point out the following references [25,26,39,40]. Notably, FQC usually means taking (spatial) fractional derivatives. An application of interest is the Riesz derivative which affects the kinetic term of the Hamiltonian. Overall, for derivatives with a non-integer (e.g., fractional) order, we can suitably modify the Wheeler-DeWitt equation, which governs the quantum state of the Universe, and discuss fractional dimensions and, most importantly, non-local effects [25,36,37,38]. In fact, the Riesz fractional derivative operates as a non-local operator unless $\alpha =2$. We can also anticipate scale-dependent geometries, which can emerge subtly as mimicked quantum gravity effects.

To be more precise, the Riesz fractional derivative has been considered non-local because its definition involves a convolution between a function and a kernel that considers the function’s values over a certain domain, not just at a single point [17]. In the classical calculus, derivatives measure a function’s local rate of the change at a given point [41]. In contrast, the fractional derivatives, particularly the Riesz fractional derivative, produces a non-local interaction with the function, where the behavior at a point depends on the values of the function at other (even distant) points [17]. Mathematically, the Riesz fractional derivative is often defined as [17],

$$D^{\alpha }f\left(x\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }\widehat{f}\left(k\right){\left|k\right|}^{\alpha }{e}^{ikx}dk.$$

In this definition, there is a Fourier transformation and a power-law factor that introduces the fractional differentiation order ($\alpha$). Its non-local nature arises because the Fourier transform representation incorporates the global information of the function (i.e., the entire function in the frequency domain) rather than just the local information around a point. Consequently, the Riesz derivative responds to changes in the function’s behavior over a wide region, constituting a non-local derivative. This contrasts with the classical derivative, which is strictly local and focuses only on the function’s behavior near the point of interest. If we apply those ideas to a fractional Wheeler-DeWitt equation, we conclude that the behavior of the wave function in a specific region is influenced by the conditions in that region and in other regions, possibly in the entire domain. Thus, a broader influence of quantum mechanical features on semiclassical domains can be substantial. Namely, within a WKB tunneling discussion.

Based in the above discussion, our present work bears a twofold purpose. Firstly, we want contributing to the solution of the problem of seemingly distinct results in references [36,37], regarding tunneling in FQM. In particular, in reference [37] the authors employ a realistic (albeit simplified) approximation for the potential barrier, being rectangular and having a finite width and height, allowing one to estimate TP’s and other effects. As a result of their calculations, they concluded that fractional parameters seem to alter the tunneling dynamics. However, the authors in [37] suggest that more realistic and elaborate potentials should be probed in order to solve the issue: how the TP’s would change when one varies the fractional parameter $\alpha$? That is exactly what we are going to do here. We will investigate a cosmological model endowed with a more realistic and elaborate potential.

Secondly, we are widening the scope of references [15,42] using the fractional Riesz derivative. It is tempting to check if a variation of $\alpha$ can mimic any other alteration we could make in $\mathsf{\Lambda }$, or another parameter in references [15,42], and within which range this may be possible. Moreover, by investigating the fractional Riesz derivative in FQC, we will be studying the TP for the birth of a FRW Universe. As mentioned, above, our cosmological model is endowed with a more realistic and elaborate potential and the TP will be a function of the fractional parameter $\alpha$. Our potential barrier was first introduced in references [15,42] and bears the curvature constant of the Universe $\left(k\right)$, the cosmological constant ($\mathsf{\Lambda }$), that plays the role of dark energy, and a parameter associated with the magnitude of the ad-hoc potential ($\sigma$). Thus, we are interested in comparing our results with the ones obtained in references [15,37,42], contributing to a still scarce study area.

Our paper is structured as follows. In Section 2, we introduce the Wheeler-DeWitt equation and then construct the fractional Schrödinger equation for our model. We solve the WDW equation using the WKB approximation and determine the WKB tunneling probability ($T{P}_{WKB}$). In Section 3, we present the numerical results of $T{P}_{WKB}$, for different selected parameters (namely, $\alpha ,\mathsf{\Lambda },k,\sigma$). We, also, compare the $T{P}_{WKB}$ obtained here, with the ones obtained in references [15,42]. Finally, Section 4 presents our conclusions and discussions about our work. This paper brings new results for the application of fractional calculus in quantum cosmology, thus contributing to the development and comprehension of the area.

2. Fractional WKB Tunneling Probabilites

We want to study how fractional calculus modifies the quantum tunneling studied in references [15,42], where the authors studied the classical and quantum dynamics of the universe with radiation, cosmological constant and an ad hoc potential, for different curvatures.

Starting from the FRW metric, using the ADM formalism [43] and the Schutz variational formalism [44,45], the total Hamiltonian of the Universe (H) is written as [15,42],

$$NH=-{\displaystyle \frac{p_a^2}{12}}+p_T-3k{a}^2+\mathsf{\Lambda }{a}^4+V_{ah},$$

(1)

where $p_a$ and $p_T$ are the canonically conjugated momenta to a and T, respectively. Here we are using the natural unit system where $\hslash =c=k_B=8\pi G=1$. The Schutz variational formalism is used to recover the covariance that is lost when using the ADM formalism by rewriting the fluid in terms of potentials, one of which is entropy. In this way, the variable T, associated with the radiation fluid, plays the role of time in this model. For more information see the “Appendix: Hamiltonian for the radiation fluid” in reference [42]. The parameter $\mathsf{\Lambda }$ is the cosmological constant and $V_{ah}$ is the ad hoc potential, first introduced in reference [15]. It is written as,

$$V_{ah}=-{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}}$$

(2)

where $\sigma$ is a dimensionless parameter associated to the magnitude of that potential.

We study the creation of this universe within a quantized model. Specifically, the total Hamiltonian (1) is transformed into an operator using the usual commutation relations. Afterward, we introduce a wave function $\Psi$ that depends on the canonical variables. Imposing that the total Hamiltonian operator annihilates the wave function, we obtain the Wheeler-DeWitt equation,

$$\begin{gathered} \widehat{H}\Psi (a,T)=0, \\ ({\displaystyle \frac{1}{12}}{\displaystyle \frac{{\partial }^2}{\partial a^2}}-i{\displaystyle \frac{\partial }{\partial T}}-3k{a}^2+\mathsf{\Lambda }{a}^4-{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}})\Psi (a,T)=0. \end{gathered}$$

(3)

Equation (3) can be re-cast in the form of a time-dependent Schrödinger equation, with the aid of a new variable $\tau =-T$,

$$({\displaystyle \frac{1}{12}}{\displaystyle \frac{{\partial }^2}{\partial a^2}}-3k{a}^2+\mathsf{\Lambda }{a}^4-{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}})\Psi (a,\tau )=-i{\displaystyle \frac{\partial }{\partial \tau }}\Psi (a,\tau ).$$

(4)

The application of fractional calculus, using the Riesz derivative in the kinetic term of the Hamiltonian operator is given by [25,46],

$${\widehat{H}}_{\alpha }(p,r)=D_{\alpha }{\left|p\right|}^{\alpha }+V\left(r\right),$$

(5)

where $D_{\alpha }$ is a coefficient. For the standard textbook Schrödinger equation we have $i\hslash {\displaystyle \frac{\partial \Psi }{\partial t}}=-{\displaystyle \frac{{\hslash }^2}{2m}}\Delta \Psi +V\Psi$, whereas the fractional QM imports that ${\displaystyle \frac{{\hslash }^2}{2m}}\to D_{\alpha }{(-{\hslash }^2\Delta )}^{{\displaystyle \frac{\alpha }{2}}}$ and with $1<\alpha \le 2$ [23]. The fractional Hamiltonian operator (5) is Hermitian [24,47], that is, it has real eigenvalues that can be related to physical observables. When the eigenvalues associated with the Hamiltonian are not real, one may have complex dimensions. For more information on complex dimension, see reference [48].

Therefore, applying the Riesz derivative in the kinetic term of the Hamiltonian operator (4), we find the equation,

$$(D_{\alpha }[{\displaystyle \frac{{\partial }^2}{\partial a^2}}{]}^{{\displaystyle \frac{\alpha }{2}}}-3k{a}^2+\mathsf{\Lambda }{a}^4-{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}})\Psi (a,\tau )=-i{\displaystyle \frac{\partial }{\partial \tau }}\Psi (a,\tau ),$$

(6)

where we take ${\displaystyle \frac{1}{12}}=D_{\alpha }$, when $\alpha =2$. The Riesz fractional derivative is a spatial, symmetric, nonlocal, linear fractional derivative [17,23,24].

The parameter $\alpha$ is known as the Lévy index and has its domain defined as $\alpha \in (0,2]$ [23,24,33,34]. This parameter allowed the generalization of Brownian motion to Lévy motion, so that Brownian motion became a particular case of Lévy motion when $\alpha =2$. Thus, the path integral over Lévy trajectories allowed the generalization of the Schrödinger equation of quantum mechanics, leading to the fractional Schrödinger equation. However, due to the fractional dimension [33], the parameter $\alpha$ has its domain limited to $(1,2]$ in the application of fractional calculus to quantum mechanics.

E. Capelas de Oliveira and Jayme Vaz Jr, in reference [36], solved the fractional Schrödinger equation for the delta and double delta potential, taking into account that the fractional derivative is a nonlocal operator. M. Hasan and B.P. Mandal, in chapter 4 of referenece [37], studied the tunneling time in space fractional quantum mechanics for a time-independent fractional Schrödinger equation, whose potential barrier was rectangular. Differently from what was proposed by these authors, in their studies, we will obtain the tunneling probability by solving the Wheeler-DeWitt equation, in the form of a time-dependent Schrödinger equation, for a more realistic and elaborate potential barrier. For this, we will use the $WKB$ approximation.

So let us propose a solution for Equation (6) in the following form [42],

$$\Psi (a,\tau ):=\psi \left(a\right)e^{-iE\tau }$$

(7)

where we choose $\psi \left(a\right)=A\left(a\right)e^{i\varphi \left(a\right)}$, with $A\left(a\right)$ being the amplitude, $\varphi \left(a\right)$ the phase, and E is the energy associated to the radiation fluid of the Universe [42].

Substituting Equation (7) into Equation (6), we obtain the following expression,

$$(D_{\alpha }[{\displaystyle \frac{{\partial }^2}{\partial a^2}}{]}^{{\displaystyle \frac{\alpha }{2}}}-3k{a}^2+\mathsf{\Lambda }{a}^4-{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}})\psi \left(a\right)e^{-iE\tau }=-i{\displaystyle \frac{\partial }{\partial \tau }}\psi \left(a\right)e^{-iE\tau },$$

(8)

and therefore,

$${\displaystyle \frac{{\partial }^{\alpha }\psi \left(a\right)}{\partial a^{\alpha }}}+{\displaystyle \frac{1}{D_{\alpha }}}(E-V_{eff}\left(a\right))\psi \left(a\right)=0.$$

(9)

This last equation can be rewritten as,

$${\displaystyle \frac{{\partial }^{\alpha }\psi \left(a\right)}{\partial a^{\alpha }}}+\widehat{\mathcal{K}}{\left(a\right)}^{\alpha }\psi \left(a\right)=0,$$

(10)

where $\widehat{\mathcal{K}}\left(a\right)\equiv ({\displaystyle \frac{1}{D_{\alpha }}}(E-V_{eff}){)}^{{\displaystyle \frac{1}{\alpha }}}$ and $V_{eff}\left(a\right)$ is the effective potential,

$$V_{eff}\left(a\right)\equiv 3k{a}^2-\mathsf{\Lambda }{a}^4+{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}}.$$

(11)

The general solution to Equation (10) is given by,

$$\begin{gathered} \Psi \left(a\right)={\displaystyle \frac{C_1}{{\widehat{\mathcal{K}}\left(a\right)}^{\frac{1}{\alpha }}}}{e}^{\pm i\int \widehat{\mathcal{K}}\left(a\right)da},E>V_{eff}, \\ \Psi \left(a\right)={\displaystyle \frac{C_2}{{\kappa \left(a\right)}^{\frac{1}{\alpha }}}}{e}^{\pm \int \kappa \left(a\right)da},E<V_{eff}, \end{gathered}$$

(12)

where $C_1$ and $C_2$ are constants and $\kappa \left(a\right)\equiv ({\displaystyle \frac{1}{D_{\alpha }}}(V_{eff}-E){)}^{{\displaystyle \frac{1}{\alpha }}}$. When $\alpha =2$, the standard solution in reference [49] is recovered.

The $WKB$ tunneling probability is given by [49],

$$T{P}_{WKB}\equiv {\displaystyle \frac{4}{{(2\theta +\frac{1}{2\theta })}^2}},$$

(13)

where $\theta$ measures the height and thickness of the barrier for a given energy [49]. For our model, we have,

$$\theta =e^{{\int }_{a_L}^{a_R}\kappa \left(a\right)da}.$$

(14)

Here, $a_L$ and $a_R$ are the turning points where the energy line crosses the potential barrier on the left and right sides, respectively. Substituting (14) into (13), we obtain the tunneling probability for a wave function to pass through a high and wide barrier $(\theta \gg 1)$,

$$T{P}_{WKB}\equiv {\displaystyle \frac{4}{(2e^{{\int }_{a_L}^{a_R}\kappa \left(a\right)da}+\frac{1}{2e^{{\int }_{a_L}^{a_R}\kappa \left(a\right)da}}{)}^2}},$$

(15)

where $\kappa \left(a\right)=({\displaystyle \frac{1}{D_{\alpha }}}(3k{a}^2-\mathsf{\Lambda }{a}^4+{\displaystyle \frac{{\sigma }^2{a}^4}{{(a^3+1)}^2}}-E){)}^{{\displaystyle \frac{1}{\alpha }}}$.

3. Results

With the Fractional $WKB$ Tunneling Probability, Equation (15), we are now able to compute the tunneling probabilities for the creation of the universe: an application of a simplified version of fractional calculus, in other words, the Riesz derivative in QC. In order to help us computing the $T{P}_{WKB}$ Equation (15), we will use the Maple 18 Symbolic Software.

Equation (15) has 5 (five) parameters: $\left(1\right)$ the curvature k, $\left(2\right)$ the radiation energy E, $\left(3\right)$ the cosmological constant $\mathsf{\Lambda }$, $\left(4\right)$ the ad hoc potential parameter $\sigma$ and $\left(5\right)$ the FC $\alpha$. $D_{\alpha }$ is a constant. We will investigate how the $T{P}_{WKB}$ depends on each of these five parameters. In order to do that, we will fix the value of four parameters and compute the $T{P}_{WKB}$ for several different the values of the remaining one. In this way, we will determine how the $T{P}_{WKB}$ depends on this fixed parameter. Then, we will repeat the same procedure for the other four parameters. For a better visualization of our results, we will show several graphs of the $T{P}_{WKB}$. Let us be more concrete. To compare our results with the ones obtained in reference [42], we will use the same values for the four parameters k, E, $\mathsf{\Lambda }$ and $\sigma$, which the authors used in reference [42]. All our results for the cases where $\alpha \ne 2$ will be exhibited in Section 3.2. Before that, in Section 3.1, we will comment on the constant $D_{\alpha }$.

3.1. Tunneling Probability When $\alpha =2$

Here, we call attention to the constant $D_{\alpha }$ present in the $T{P}_{WKB}$ Equation (15). This constant is dependent on the parameter $\alpha$ and its definition involves some arbitrariness. An important condition that quantity must satisfy is that, when $\alpha =2$ the result must be the constant of the standard model, without the fractional calculus. For fractional quantum mechanics, the constant $D_{\alpha }={\displaystyle \frac{1}{2m}}$ is recovered when $\alpha =2$ [33,34].

In the present paper, we will choose one of the simplest possible values for $D_{\alpha }$, such that it has the correct limit when $\alpha =2$. In this way, we define $D_{\alpha }={\left({\displaystyle \frac{1}{12}}\right)}^{{\displaystyle \frac{\alpha }{2}}}$. When we substitute $\alpha =2$ in Equation (15), we obtain the same equation and results for the $T{P}_{WKB}$ as in the model in reference [42]. In a future work, we want to explore in details this arbitrariness, which may lead to some quantitatively different results. In order to give some idea about these quantitatively different results, we show in Appendix A some results for $T{P}_{WKB}$ obtained with a different choice for $D_{\alpha }$.

3.2. Tunneling Probability When $\alpha \ne 2$

We now investigate quantitatively how the fractional calculus features, namely the presence of the Riesz derivative, modify the quantum tunneling probabilities. To do that, we will study how the $T{P}_{WKB}$ Equation (15) depends on the parameters: k, E, $\mathsf{\Lambda }$, $\sigma$ and $\alpha$. We will employ the same values used for these parameters in reference [42], in order to compare our results with theirs.

3.2.1. $T{P}_{WKB}$ as a Function of E

Concerning the variation of the energy E, we will fix $\mathsf{\Lambda }=1.5$ and $\sigma =-50$ like in [42]. All values for the energy E are below $V_{eff}=691.51$, which is the maximum value of the barrier. We computed numerically Equation (15) for different values of E, $\alpha$ ($1<\alpha <2$) and k. We found that when E and $\alpha$ increase, the tunneling probabilities increase too. When we vary only the geometry $\left(k\right)$ keeping the other parameters fixed, we obtain that $T{P}_{WKB}$ is larger for the hyperbolic spatial sections ($k=-1$). An example of these results can be found in Figure 1.

3.2.2. $T{P}_{WKB}$ as a Function of $\mathsf{\Lambda }$

The following study is for the cosmological constant $\mathsf{\Lambda }$, which plays the role of dark energy in this model. We compute numerically Equation (15), for different values of $\mathsf{\Lambda }$, $\alpha$ ($1<\alpha <2$) and $k=0, \pm 1$, but with $\sigma =-50$ and $E=690$. We can see in Figure 2, as an example of our general result, that $T{P}_{WKB}$ increases when $\alpha$ and $\mathsf{\Lambda }$ increase and is larger for $k=-1$.

3.2.3. $T{P}_{WKB}$ as a Function of $\sigma$

Let us now vary the parameter $\sigma$, where $\sigma$ is a dimensionless parameter associated with the magnitude of the ad hoc potential [15]. Following the same procedure, fixing the parameters $E=680$ and $\mathsf{\Lambda }=1.5$, we vary and take several values of $\alpha$ and $\sigma$, for different curvatures of the space-like sections $\left(k\right)$. As an example of our results, we can see in Figure 3, that $T{P}_{WKB}$ increases when $\sigma$ and $\alpha$ increase. The biggest value of $T{P}_{WKB}$ is for $k=-1$ geometry, if compared to $k=0,1$ geometries.

3.3. Analysis and Implication of the Riesz Derivative in the Calculation of the Tunneling Probability

The Riesz derivative is an operator often used in fractional calculus and acts only on the kinetic term of the Hamiltonian operator (6). Thus, the effective potential (11) is identical to the effective potential of the model in reference [42]. In Section 3.2.1, we noticed that the $T{P}_{WKB}$ decreases when we decrease the value of the parameter $\alpha$, even when the energy E remains the same as the value used in reference [42]. In the non-fractional model, in order to decrease the TP, while keeping the energy fixed, we need to modify the potential barrier to become e.g., wider and higher. On the other hand, in the model with the Riesz derivative, this can go quite different. Let us be more specific.

In Figure 4, Figure 5 and Figure 6, we draw the potential barriers for three different cases: (i) the fractional case with $\alpha =1.9397961$ (blue line), (ii) the non-fractional case with $\mathsf{\Lambda }=0.7$ (red line), and (iii) the non-fractional case with the same values of $\mathsf{\Lambda },\sigma ,k$ as in case $\left(i\right)$ (gold dot line). This last case (iii), is the one studied in reference [42]. In each Figure, the three cases are models with the same values of k. The horizontal black line, representing the constant energy $E=200$, illustrates how the energy line intersects the barriers. We can see that the potential barriers in cases (i) and (iii) coincide. However, they generate different $T{P}_{WKB}$’s for the same energy, as per informed from Figure 1, Figure 2 and Figure 3. It means that, the introduction of the Riesz derivative modifies in a significant way the $T{P}_{WKB}$’s associated with a given QC model.

On the other hand, we find that the results of $T{P}_{WKB}$ for the non-fractional model with $\mathsf{\Lambda }=0.7$ closely mimic the results found for the fractional model with $\mathsf{\Lambda }=1.5$ and $\alpha =1.9397961$, as we can see from the Figure 7, Figure 8 and Figure 9. In each Figure, the two cases are models with the same values of k. The potential barriers of the two models can be seen in Figure 4, Figure 5 and Figure 6.

This new and interesting result suggests that modifying the parameter $\alpha$ in the fractional model may be comparable to modifying the potential barrier in the non-fractional model, i.e., its height and width. In Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, we show some results for $\theta$ Equation (14) and $T{P}_{WKB}$ Equation (15) for different values of E, regarding the fractional model with $\mathsf{\Lambda }=1.5$ and $\alpha =1.9397961$, and the non-fractional model with $\mathsf{\Lambda }=0.7$.

• For hyperbolic space-like sections

Table 1. Comparative table of $\theta$ for different energies in the fractional and non-fractional models for $k=-1$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
−1	10	$1.002778169\times {10}^{88}$	$1.303841382\times {10}^{88}$
	170	$3.873356742\times {10}^{58}$	$1.577277123\times {10}^{60}$
	340	$3.066824521\times {10}^{34}$	$1.353799484\times {10}^{36}$
	500	$7.888358713\times {10}^{16}$	$5.091334256\times {10}^{17}$
	660	463.1719831	400.5848243

Table 1. Comparative table of $\theta$ for different energies in the fractional and non-fractional models for $k=-1$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
−1	10	$1.002778169\times {10}^{88}$	$1.303841382\times {10}^{88}$
	170	$3.873356742\times {10}^{58}$	$1.577277123\times {10}^{60}$
	340	$3.066824521\times {10}^{34}$	$1.353799484\times {10}^{36}$
	500	$7.888358713\times {10}^{16}$	$5.091334256\times {10}^{17}$
	660	463.1719831	400.5848243

Table 2. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=-1$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
−1	10	$9.944667312\times {10}^{-177}$	$5.882344812\times {10}^{-177}$
	170	$6.665381380\times {10}^{-118}$	$4.019610680\times {10}^{-121}$
	340	$1.063217573\times {10}^{-69}$	$5.456212944\times {10}^{-73}$
	500	$1.607040038\times {10}^{-34}$	$3.857773981\times {10}^{-36}$
	660	0.000004661379104	0.000006231744824

Table 2. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=-1$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
−1	10	$9.944667312\times {10}^{-177}$	$5.882344812\times {10}^{-177}$
	170	$6.665381380\times {10}^{-118}$	$4.019610680\times {10}^{-121}$
	340	$1.063217573\times {10}^{-69}$	$5.456212944\times {10}^{-73}$
	500	$1.607040038\times {10}^{-34}$	$3.857773981\times {10}^{-36}$
	660	0.000004661379104	0.000006231744824

• For flat space-like sections

Table 3. Comparative table of the parameter $\theta$ for different energies in the fractional and non-fractional models for $k=0$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
0	10	$1.793016944\times {10}^{92}$	$1.146483507\times {10}^{91}$
	170	$1.379498224\times {10}^{61}$	$1.674829609\times {10}^{62}$
	340	$5.585366235\times {10}^{35}$	$2.076589094\times {10}^{37}$
	500	$3.790604212\times {10}^{17}$	$2.537403708\times {10}^{18}$
	660	1191.527234	1069.113378

Table 3. Comparative table of the parameter $\theta$ for different energies in the fractional and non-fractional models for $k=0$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
0	10	$1.793016944\times {10}^{92}$	$1.146483507\times {10}^{91}$
	170	$1.379498224\times {10}^{61}$	$1.674829609\times {10}^{62}$
	340	$5.585366235\times {10}^{35}$	$2.076589094\times {10}^{37}$
	500	$3.790604212\times {10}^{17}$	$2.537403708\times {10}^{18}$
	660	1191.527234	1069.113378

Table 4. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=0$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
0	10	$3.110507212\times {10}^{-185}$	$7.607892668\times {10}^{-183}$
	170	$5.254818360\times {10}^{-123}$	$3.564993479\times {10}^{-125}$
	340	$3.205506704\times {10}^{-72}$	$2.318989928\times {10}^{-75}$
	500	$6.959581392\times {10}^{-36}$	$1.553176674\times {10}^{-37}$
	660	$6.959581392\times {10}^{-7}$	$8.748876416\times {10}^{-7}$

Table 4. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=0$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
0	10	$3.110507212\times {10}^{-185}$	$7.607892668\times {10}^{-183}$
	170	$5.254818360\times {10}^{-123}$	$3.564993479\times {10}^{-125}$
	340	$3.205506704\times {10}^{-72}$	$2.318989928\times {10}^{-75}$
	500	$6.959581392\times {10}^{-36}$	$1.553176674\times {10}^{-37}$
	660	$6.959581392\times {10}^{-7}$	$8.748876416\times {10}^{-7}$

• For spherical space-like sections

Table 5. Comparative table of the parameter $\theta$ for different energies in the fractional and non-fractional models for $k=1$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
1	10	$1.577640978\times {10}^{97}$	$1.907005973\times {10}^{94}$
	170	$1.313683533\times {10}^{64}$	$2.830562481\times {10}^{64}$
	340	$1.380589165\times {10}^{37}$	$3.946000113\times {10}^{38}$
	500	$1.992360778\times {10}^{18}$	$1.370930167\times {10}^{19}$
	660	3160.340886	2945.792223

Table 5. Comparative table of the parameter $\theta$ for different energies in the fractional and non-fractional models for $k=1$.

k	E	${\mathit{\theta }}_{\mathit{non}-\mathit{fractional}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{\theta }}_{\mathit{fractional}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
1	10	$1.577640978\times {10}^{97}$	$1.907005973\times {10}^{94}$
	170	$1.313683533\times {10}^{64}$	$2.830562481\times {10}^{64}$
	340	$1.380589165\times {10}^{37}$	$3.946000113\times {10}^{38}$
	500	$1.992360778\times {10}^{18}$	$1.370930167\times {10}^{19}$
	660	3160.340886	2945.792223

Table 6. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=1$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
1	10	$4.017756788\times {10}^{-195}$	$2.749766985\times {10}^{-189}$
	170	$5.794533632\times {10}^{-129}$	$1.248114729\times {10}^{-129}$
	340	$5.246516932\times {10}^{-75}$	$6.422229384\times {10}^{-78}$
	500	$2.519208036\times {10}^{-37}$	$5.320706876\times {10}^{-39}$
	660	$1.001226000\times {10}^{-7}$	$1.152380106\times {10}^{-7}$

Table 6. Comparative table of tunneling probabilities for different energies in the fractional and non-fractional models for $k=1$.

k	E	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{non}-\mathit{fractional}}}$  $(\mathsf{\Lambda }=0.7)$	${\mathit{TP}}_{{\mathit{WKB}}_{\mathit{fractional}}}$ $(\mathsf{\Lambda }=1.5)$ and $\mathit{\alpha }=1.9397961$
1	10	$4.017756788\times {10}^{-195}$	$2.749766985\times {10}^{-189}$
	170	$5.794533632\times {10}^{-129}$	$1.248114729\times {10}^{-129}$
	340	$5.246516932\times {10}^{-75}$	$6.422229384\times {10}^{-78}$
	500	$2.519208036\times {10}^{-37}$	$5.320706876\times {10}^{-39}$
	660	$1.001226000\times {10}^{-7}$	$1.152380106\times {10}^{-7}$

Thus, analyzing the results qualitatively through the Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 and quantitatively through the Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6, we see that the shapes of the barriers and the results for $\theta$ and $T{P}_{WKB}$, of the fractional model with $\mathsf{\Lambda }=1.5$ and $\alpha =1.9397961$ and the non-fractional model with $\mathsf{\Lambda }=0.7$, are close. This convergence between the fractional and non-fractional models is lost for small $\mathsf{\Lambda }$, on the order of ${10}^{-2}$, since the behavior of the effective potential for $k=1$ is modified.

4. Conclusions and Discussions

In this work, we investigated whether a concrete tool of fractional calculus may influence the tunneling probability computed using the $WKB$ approximation. For this purpose, a $FRW$ model was employed, containing a radiation fluid, a positive cosmological constant, and an ad-hoc potential. Moreover, we used the Riesz fractional derivative in the Hamiltonian operator, which was applied to the kinetic term. In this manner, we made it possible to investigate the behavior of the tunneling probability as a function of the new fractional parameter $\alpha$.

Our results show that when $\alpha =2$, we obtain the same results found in the study carried out in reference [42]. When we allow $\alpha$ to vary, we observe that when $\alpha$ decreases, within its domain $1<\alpha \le 2$, the tunneling probability decreases. Thus, the Universe is more likely to be created for higher values of $\alpha$, namely $\alpha =2$.

These results allowed us to conclude that although our potential barrier is the same as that established in reference [42], applying fractional calculus in the kinetic term of the Hamiltonian operator modifies the tunneling probabilities. Upon investigating this feature in more detail, we noticed that the decrease of the parameter $\alpha$ seems to produce an effect similar to a modification on the potential barrier of the non-fractional model in reference [42]. More precisely, in order to achieve that we have to make the potential barrier broader and higher by modifying its parameters $\mathsf{\Lambda },\sigma ,k$. To substantiate this observation, we presented an example with a fractional model where $\mathsf{\Lambda }=1.5$ and $\alpha =1.9397961$ and a non-fractional model with $\mathsf{\Lambda }=0.7$. We showed that the tunneling probabilities computed for both models are very close. This was checked for other ranges, as presented in the previous section. Please compare Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 and Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

In addition to the qualitative analysis of the potential barriers in the fractional and non-fractional models, through the Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9, we performed a quantitative analysis through $\theta$, which informs about the height and length of the barriers. The results, found in Table 1, Table 2 and Table 3, confirm the similarity of the barriers and the values of the tunneling probabilities between the models.

Finally, our results also allow for an interesting discussion between selecting values for $\mathsf{\Lambda }$ (in a non-fractional conventional set-up) versus balancing, e.g., both $\mathsf{\Lambda }$ and $\alpha$ in the fractional framework. Concretely, as pointed out above, the $T{P}_{WKB}$ in the non-fractional model if, e.g., $\mathsf{\Lambda }=0.7$, is very close to the $T{P}_{WKB}$ computed if in the fractional model we choose, e.g., $\mathsf{\Lambda }=1.5,\alpha =1.9397961$. This suggests that if fractional derivatives were present in a model with a certain $\mathsf{\Lambda }$, the universe described by this model could be born with almost the same probability of another universe with a smaller value of $\mathsf{\Lambda }$, provided that in the second model there were only usual derivatives.

These new results also contribute and shed additional light on the use of fractional calculus in quantum cosmology. More precisely, on the calculation of tunneling probabilities by using a more realistic and elaborate potential than those studied in references [37] and [36]. Lastly, the application of fractional calculus in quantum cosmology, specially in the calculation of tunneling probabilities, is relatively new and requires further investigation.