# Anisotropic fractional cosmology: K-essence theory

J. Socorro1<EMAIL_ADDRESS>J. Juan Rosales2<EMAIL_ADDRESS>L. Toledo
Sesma3<EMAIL_ADDRESS>1 Departamento de Física de la División de Ciencias e
Ingenierías de la Universidad de Guanajuato-Campus León,
C.P. 37150, Guanajuato, México.
2 Departamento de Ingeniería Eléctrica, DICIS. Universidad de Guanajuato,
Comunidad de Palo Blanco, C.P. 36885, Salamanca Guanajuato, México.
3 Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria de
Ingeniería Campus Hidalgo, Carretera Pachuca - Actopan Kilómetro 1+500, 42162.
San Agustín Tlaxiaca, Hgo. México

###### Abstract

In the particular configuration of the scalar field K-essence in the Wheeler-
DeWitt quantum equation, for some age in the Bianchi type I anisotropic
cosmological model, a fractional differential equation for the scalar field
arises naturally. The order of the fractional differential equation is
$\beta=\frac{2\alpha}{2\alpha-1}$. This fractional equation belongs to
different intervals, depending on the value of the barotropic parameter; when
$\omega_{X}\in[0,1]$, the order belongs to the interval $1\leq\beta\leq 2$,
and when $\omega_{X}\in[-1,0)$, the order belongs to the interval $0<\beta\leq
1$. In the quantum scheme, we introduce the factor ordering problem in the
variables $(\Omega,\phi)$ and its corresponding momenta
$(\Pi_{\Omega},\Pi_{\phi})$, obtaining a linear fractional differential
equation with variable coefficients in the scalar field equation, then the
solution is found using a fractional power series expansion. The corresponding
quantum solutions are also given. We found the classical solution in the usual
gauge N obtained in the Hamiltonian formalism and without a gauge. In the last
case, the general solution is presented in a transformed time $T(\tau)$,
however in the dust era we found a closed solution in the gauge time $\tau$.

Keywords: Fractional derivative, Fractional Quantum Cosmology; K-essence
formalism; Classical and Quantum exact solutions.

##  I Introduction

Fractional cosmology is a new line of research born approximately twenty years
ago based on fractional calculus (FC). The FC is a non-local natural
generalization to the arbitrary order of derivatives and integrals. Non-local
effects occur in space and time. In the time domain, a non-local description
becomes manifest as a memory effect, and in the space domain, it manifests as
non-homogeneous similarity structures Podlubny ; Uchaikin ; Herman . During
the last decades, FC has been the subject of intense theoretical and applied
research, almost in all areas of the sciences and engineering, from the point
of view of the classical and quantum systems Ca ; Wyss ; Westerlund ; r-s ;
Magin ; Tarasov ; Duarte ; Laskin ; Paulo2020 ; Paulo2021a ; Paulo2023 ,
recently new studies on FC have been made Ortigueira2022a ; Ortigueira2022b ;
Ortigueira2023a ; Ortigueira2023b . This is because, the FC describes more
accurately the complex physical systems and at the same time, investigates
more about simple dynamical systems Rosales1 ; Rosales2 . The general
relativity could not be the exception, in Rami1 ; Rami-2012 ; Rami-2013 ;
Rami2015 ; Rami2016a ; Rami2016b ; Rami2017a ; Rami2017b ; Paulo2021 ;
Paulo2022a ; Paulo2022b  the importance of FC and its potential applications
in cosmology was introduced. In Rami2  the FRW universe was presented in the
context of the variational principle of fractional action. In this new
cosmological formulation, the accelerated expansion of the universe can be
attributed to the fractional dissipative force without the need to introduce
any kind of matter or scalar fields, similar results are obtained in Aspeitia1
; Aspeitia2 . In Rami-2012 , the concept of fractional action cosmology was
applied to massive gravity, where fractional graviton masses are introduced.

Unlike the previously described formalism to obtain fractional cosmology, in
Socorro1  it is mentioned that by quantifying different epochs of the
K-essence theory, a fractional Wheeler-DeWitt equation in the scalar field
component is naturally obtained. Recently, such an equation was solved for
some epochs in the FRW model and communicated in Socorro2 . In this work we
present the continuation of our previous investigation, in this case, we will
analyze the Bianchi type I, which is the anisotropic generalization of the
flat FRW cosmological model. In the quantum scheme, we introduce the factor
ordering problem in the variables $(\Omega,\phi)$ and its corresponding
momenta ($\Pi_{\Omega},\Pi_{\phi}$), obtaining a fractional differential
equation with variable coefficients in the scalar field equation. The solution
is found using a fractional series expansion El-Ajou ; Rida , generalizing our
previous work Socorro2 .

This paper is organized in such a way: in Section 1, we give a brief review of
fractional calculus and the main ideas of the K-essence formalism; in Section
2, we construct the Lagrangian and Hamiltonian densities for the anisotropic
Bianchi type I cosmological model, considering a barotropic perfect fluid for
the scale field in the variable $X$. We found the classical solution in the
usual gauge N obtained in the Hamiltonian formalism, and without a gauge. In
the last case, the general solution is presented in a transformed time
$T(\tau)$, however in the dust era we found a closed solution in the gauge
time $\tau$; in Section 3, the quantization of the model for any era in our
universe is done and present particular scenarios, too. In this section, we
introduce the factor ordering in both variables; finally, in Section 4, the
conclusions are given.

##  II Brief review on fractional calculus and K-essence theory

###  II.1 Brief review on fractional calculus

In the theory of fractional calculus, there are some definitions of fractional
derivatives; Rieman-Liouville, Caputo, Caputo-Fabrizio, Atangana-Baleanu, to
name a few, each with its advantages and disadvantages Ortigueira ; Tarasov1 ;
Sales . In this work, we use the Caputo fractional derivative of order
$\gamma$, defined by using the Riemann-Liouville fractional integral Podlubny

$I^{\gamma}f(t)=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}\frac{f(\tau)}{(t-\tau)^{1-\gamma}}d\tau,\quad\gamma>0,$
(1)

recovering ordinary integral, when $\gamma\to 1$. The Caputo fractional
derivative of order $\gamma\geq 0$ of a function $f(t)$, then, is defined as
the fractional order integral (1) of the integer order derivative

${}^{C}_{0}D^{\gamma}_{t}f(t)=I^{(n-\gamma)}\,_{0}D_{t}^{n}f(t)=\frac{1}{\Gamma(n-\gamma)}\int_{0}^{t}\frac{f^{(n)}(\tau)}{(t-\tau)^{\gamma-n+1}}d\tau,$
(2)

with $n-1<\gamma\leq n\in\mathbb{N}={1,2,...}$, and $\gamma\in\mathbb{R}$ is
the order of the fractional derivative and $f^{(n)}$ are the ordinary integer
derivatives, and $\Gamma(x)=\int_{0}^{\infty}e^{-t}t^{x-1}dt$, is the gamma
function. The Caputo derivative satisfies the following relations

${}^{C}_{0}D^{\gamma}(f(t)+g(t)]$ $\displaystyle=$
${}^{C}_{0}D^{\gamma}f(t)+{{}^{C}_{0}}D^{\gamma}g(t).$ (3)
${}^{C}_{0}D^{\gamma}c$ $\displaystyle=$ $\displaystyle 0,\qquad{\rm
where}\,\,c\,\,{\rm is\,\,a\,\,constant}.$ (4)

The Laplace transform of the function $f(t)$ defined in the ordinary case is
given by

$\mathbb{L}[f(t)]=\int_{0}^{\infty}f(t)e^{-st}dt\equiv F(s),$ (5)

then, the Laplace transform of the Caputo fractional derivative (2) has the
form

$\mathbb{L}[^{C}_{0}D^{\gamma}f(t)]=s^{\gamma}F(s)-\sum_{k=0}^{n-1}s^{\gamma-k-1}f^{(k)}(0),$
(6)

where $f^{(k)}$ is the ordinary derivative. Another definition which will be
used is the Mittag-Leffler function Erdely ; Haubold ; Trifce ,

$\mathbb{E}_{\chi,\sigma}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(n\chi+\sigma)}\qquad(\chi,\,\,\,\sigma>0),$
(7)

for $\sigma=1$, we have one parameter Mittag-Leffler function

$\mathbb{E}_{\chi}(z)=\mathbb{E}_{\chi,1}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(n\chi+1)}\qquad(\chi>0).$
(8)

Another special cases are in Erdely ; Haubold

$\displaystyle\mathbb{E}_{1}(\pm z)$ $\displaystyle=$ $\displaystyle e^{\pm
z},\qquad\mathbb{E}_{2}(z)=\cosh{\sqrt{z}},\qquad\mathbb{E}_{2,1}(-z^{2})=\cos{z},$
$\displaystyle\mathbb{E}_{2,2}(z^{2})$ $\displaystyle=$
$\displaystyle\frac{\sinh{z}}{z},\qquad\mathbb{E}_{2,2}(-z^{2})=\frac{\sin{z}}{z}.$
(9)

Laplace transform (5) of the Mittage-Leffler function is given by the formula

$\int_{0}^{\infty}e^{-st}t^{\chi m+\sigma-1}\mathbb{E}_{\chi,\sigma}^{(m)}(\pm
at^{\chi})dt=\frac{m!\,s^{\chi-\sigma}}{(s^{\chi}\mp a)^{m+1}}.$ (10)

Consequently, the inverse Laplace transform is

$\mathbb{L}^{-1}\Big{[}\frac{m!\,s^{\chi-\sigma}}{(s^{\chi}\mp
a)^{m+1}}\Big{]}=t^{\chi m+\sigma-1}\mathbb{E}_{\chi,\sigma}^{(m)}(\pm
at^{\chi}).$ (11)

This expression will be very useful to obtain analytical solutions of
fractional differential equations using the Laplace transform.

###  II.2 K-essence fractional in the Bianchi I scenario.

One of the fundamental problems of cosmology is to find an explanation
consistent with experiments for the accelerated expansion of the universe.
Many proposals to tackle this task suggest modifying the general relativity
theory. A recent proposal suggests unifying the description of dark matter,
dark energy and inflation, employing a scalar field with a nonstandard kinetic
term, known as K-essence theory. Usually, the action of the K-essence models
Espinoza ; dePutter:2007ny ; Chiba:2009nh ; Bose:2008ew ; Arroja:2010wy ;
Garcia:2012aa  can be written as

$S=\int\,d^{4}x\sqrt{-g}\biggl{(}\frac{1}{2}\mathcal{R}+f(\phi)\mathcal{G}(X)+{\cal
L}_{matter}\biggr{)},$ (12)

$g$ being the determinant of the metric, $\mathcal{R}$ the scalar curvature,
$f(\phi)$ an arbitrary function of the dimensionless scalar field $\phi$,
$X=-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ the canonical
kinetic energy and ${\cal L}_{matter}$ is the corresponding Lagrangian density
of ordinary matter. So, performing the variation of the action (12) with
respect to the metric $g_{\mu\nu}$ and $X$, the field equations are obtained

$\displaystyle
G_{\mu\nu}-f(\phi)\biggl{[}\mathcal{G}_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi+\mathcal{G}g_{\mu\nu}\biggr{]}$
$\displaystyle=$ $\displaystyle T_{\mu\nu},$ (13) $\displaystyle
f(\phi)\biggl{[}\mathcal{G}_{X}\nabla^{\mu}\nabla_{\mu}\phi+\mathcal{G}_{XX}\nabla_{\mu}X\nabla^{\mu}\phi\biggr{]}+\frac{df}{d\phi}\biggl{[}\mathcal{G}-2X\mathcal{G}_{X}\biggr{]}$
$\displaystyle=$ $\displaystyle 0,$ (14)

where, we have assumed that $8\pi G=1$ and a subscript $X$ denotes
differentiation with respect to $X$. K-essence was originally proposed as a
model for inflation; and then, as a model for dark energy, along with
explorations of unifying dark energy and dark matter Bilic:2001cg ;
Bento:2002ps .

Last set of field equations (13) and (14) are the results of considering the
scalar field $X(\phi)$ as part of the matter content, i.e.
$\mathcal{L}_{X,\phi}=f(\phi)\mathcal{G}(X)$, with the corresponding energy-
momentum tensor

$\mathcal{T}_{\mu\nu}^{(\phi)}=f(\phi)\left[\mathcal{G}_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi+\mathcal{G}(X)g_{\mu\nu}\right].$
(15)

Also, considering the energy-momentum tensor of a barotropic perfect fluid,

$T_{\mu\nu}^{(\phi)}=(\rho_{\phi}+P_{\phi})u_{\mu}u_{\nu}+P_{\phi}g_{\mu\nu},$
(16)

with $u_{\mu}$ being the four-velocity satisfying the relation
$u_{\mu}u^{\mu}=-1$, $\rho_{\phi}$ the energy density and $P_{\phi}$ the
pressure of the fluid. To simplify, we are going to consider a comoving
perfect fluid, whose pressure and energy density corresponding to the energy
moment tensor of the field X are

$P_{\phi}(X)=f(\phi)\mathcal{G},\qquad\rho_{\phi}(X)=f(\phi)\left[2X\mathcal{G}_{X}-\mathcal{G}\right],$
(17)

thus the barotropic parameter $\omega_{X}=\frac{P_{\phi}(X)}{\rho_{\phi}(X)}$
for the equivalent fluid is

$\omega_{X}=\frac{\mathcal{G}}{2X\mathcal{G}_{X}-\mathcal{G}}.$ (18)

Notice that the case of a constant barotropic index $\omega_{X}$ (with the
exception $\omega_{X}=0$) can be obtained by the $\mathcal{G}$ function

$\mathcal{G}=X^{\frac{1+\omega_{X}}{2\omega_{X}}}.$ (19)

At this point we can choose

$\mathcal{G}=X^{\alpha},\qquad\alpha=\frac{1+\omega_{X}}{2\omega_{X}}\qquad\to\qquad\omega_{X}=\frac{1}{2\alpha-1}.$
(20)

With this, we can write the states in the evolution of the universe resumed in
the table 1.

$\omega_{X}$ | $\alpha$ | $\mathcal{G}(X)$ | State of evolution
---|---|---|---
1 | 1 | $X$ | Stiff matter
$\frac{1}{3}$ | 2 | $X^{2}$ | Radiation
$\rightarrow 0$ | $\rightarrow\infty$ |  $X^{m}$, $m\rightarrow\infty$ | Dust like
$-1$ | $0$ |  $1$, $f(\phi)=\Lambda=cte$ | Inflation
$-\frac{1}{3}$ | -1 | $\frac{1}{X}$ | Inflation like
$-\frac{2}{3}$ | $-\frac{1}{4}$ | $\frac{1}{\sqrt[4]{X}}$ | Inflation like
Table 1: States of the universe’s evolution according to the barotropic
parameter $\omega_{X}$.

We are interested in the four-dimensional fractional cosmology in the scenario
of k-essence within the anisotropic background, precisely the Bianchi type I,
whose metric has the line element $g_{\alpha\beta}$, which can be read as

$ds^{2}=-N^{2}(t)\,dt^{2}+A^{2}(t)\,dx^{2}+B^{2}(t)\,dy^{2}+C^{2}(t)\,dz^{2},$
(21)

where $N(t)$ is the lapse function, the functions $A(t)$, $B(t)$ and $C(t)$
are the corresponding scale factors in the $(x,y,z)$ directions, respectively.
Moreover, in the Misner’s parametrization, the radii for this anisotropic
background have the explicit form

$A=e^{\Omega+\beta_{+}+\sqrt{3}\beta_{-}},\qquad
B=e^{\Omega+\beta_{+}-\sqrt{3}\beta_{-}},\qquad C=e^{\Omega-2\beta_{+}},$ (22)

where the functions in the radii are dependent on time, $\Omega=\Omega(t)$ and
$\beta_{\pm}=\beta_{\pm}(t)$. In this point, we notice that the line element
(21), in the time $d\tau=Ndt$ reads as

$ds^{2}=-d\tau^{2}+e^{2(\Omega(\tau)+\beta_{+}(\tau)+\sqrt{3}\beta_{-}(\tau))}\,dx^{2}+e^{2(\Omega(\tau)+\beta_{+}(\tau)-\sqrt{3}\beta_{-}(\tau))}\,dy^{2}+e^{2\Omega(\tau)-4\beta_{+}(\tau)}\,dz^{2},$
(23)

and employing the form of the functional ${\cal G}=X^{\alpha}$, and the
following quantities

$\displaystyle\prime=\frac{d}{d\tau}=\frac{d}{Ndt},\qquad
g_{\tau\tau}=g^{\tau\tau}=-1,\qquad{\cal G}_{X}=\alpha
X^{\alpha-1},\qquad{\cal G}_{XX}=\alpha(\alpha-1)X^{\alpha-2},$
$\displaystyle\phi^{,\mu}_{\,;\mu}=g^{\mu\nu}\left(\phi_{,\mu\nu}-\Gamma_{\mu\nu}^{\theta}\phi_{,\theta}\right)=-\left(\phi^{\prime\prime}+3\Omega^{\prime}\phi^{\prime}\right),\qquad
X_{;\mu}\phi^{,\mu}=g^{\mu\nu}\phi_{,\nu}X_{;\mu}=-X^{\prime}\phi^{\prime},$
$\displaystyle
X=\frac{1}{2}\left(\phi^{\prime}\right)^{2},\qquad\left(\phi^{\prime}\right)^{2}=2X,\qquad
X^{\prime}=\phi^{\prime}\phi^{\prime\prime},\qquad\phi^{\prime\prime}=\frac{X^{\prime}}{\phi^{\prime}},$

then the Equation (14) is written as ,

$\alpha
X^{\alpha-1}\left(\phi^{\prime\prime}+3\Omega^{\prime}\phi^{\prime}\right)+\alpha(\alpha-1)X^{\alpha-2}X^{\prime}\phi^{\prime}+(2\alpha-1)X^{\alpha}\frac{d}{d\phi}Lnf=0,$
(24)

which can be transformed into

$\frac{d}{d\tau}\left(LnX+\frac{6\Omega}{2\alpha-1}+Lnf^{\frac{1}{\alpha}}\right)=0,$
(25)

and in turn integrated, resulting

$\int f^{\frac{1}{2\alpha}}(\phi)\,d\phi=\sqrt{2\lambda}\int
e^{-\frac{3\Omega(\tau)}{2\alpha-1}}d\tau,$ (26)

where $\lambda$ is an integration constant and has the same sign as $f(\phi)$.
In the gauge $N=24e^{\frac{3\Omega}{2\alpha-1}}$ the right side is

$\int f^{\frac{1}{2\alpha}}(\phi)\,d\phi=24\sqrt{2\lambda}\,\,(t-t_{i}),$ (27)

where $t_{i}$ is the initial time for the $\alpha$ scenario in the universe.
At this point, we can introduce some structure for the function $f(\phi)$ and
solve the integral.

When we consider the particular mathematical structure for the function
$f(\phi)=p\phi^{m}$ or $f(\phi)=pe^{m\phi}$ with $p$ and $m$ constants, the
classical solutions for the field $\phi$ in quadratures are

$\phi(\tau)=\phi(\tau_{i})+\left\\{\begin{tabular}[]{ll}$\left[\frac{(2\alpha+m)}{2\alpha}{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}\int
e^{-\frac{3\Omega(\tau)}{2\alpha-1}}d\tau\right]^{\frac{2\alpha}{2\alpha+m}},$&\quad$f(\phi)=p\phi^{m},\quad
m\not=-2\alpha$\\\ $Exp\left[{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}\int
e^{-\frac{3\Omega(\tau)}{2\alpha-1}}d\tau\right],$&\quad$f(\phi)=p\phi^{-2\alpha},\quad
m=-2\alpha$\\\
$\frac{2\alpha}{m}\,Ln\left[\frac{m}{2\alpha}{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}\int
e^{-\frac{3\Omega(\tau)}{2\alpha-1}}d\tau\right],$&\quad$f(\phi)=pe^{m\phi},\quad
m\not=0,$\\\ ${p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}\int
e^{-\frac{3\Omega(\tau)}{2\alpha-1}}d\tau$&\quad$f(\phi)=p,\quad
m=0$.\end{tabular}\right.$ (28)

The complete solution to the scalar field $\phi$ depends strongly on the
mathematical structure of the scale factor $\Omega(\tau)$ in the $\alpha$
scenario in our universe. In the gauge $N=24e^{\frac{3\Omega}{2\alpha-1}}$,
these solutions are

$\phi(t)=\phi(t_{i})+\left\\{\begin{tabular}[]{ll}$\left[\frac{12(2\alpha+m)}{\alpha}{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}\left(t-t_{i}\right)\right]^{\frac{2\alpha}{2\alpha+m}},$&\quad$f(\phi)=p\phi^{m},\quad
m\not=-2\alpha$\\\
$Exp\left[24{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}\left(t-t_{i}\right)\right],$&\quad$f(\phi)=p\phi^{-2\alpha},\quad
m=-2\alpha$\\\
$\frac{2\alpha}{m}\,Ln\left[\frac{12m}{\alpha}{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}\left(t-t_{i}\right)\right],$&\quad$f(\phi)=p\,e^{m\phi},\quad
m\not=0,$\\\
$24{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}\left(t-t_{i}\right)$&\quad$f(\phi)=p,\quad
m=0$,\end{tabular}\right.$ (29)

where $t_{i}$ and $\phi(t_{i})$ are the initial time and the scalar field in
this time for the $\alpha$ scenario in the universe. In what follows, we do
the calculations to obtain the scale factor in some cases.

##  III Lagrange and Hamilton formalism

Introducing the line element (21) of the anisotropic Bianchi type I
cosmological model into the Lagrangian (12), we have

$\mathcal{L}_{I}=e^{3\Omega}\left\\{6\frac{\dot{\Omega}^{2}}{N}-6\frac{\dot{\beta}_{+}^{2}}{N}-6\frac{\dot{\beta}_{-}^{2}}{N}-f(\phi)\left(\frac{1}{2}\right)^{\alpha}\left(\dot{\phi}\right)^{2\alpha}N^{-2\alpha+1}\right\\}.$
(30)

Using the standard definition of the momenta
$\Pi_{q^{\mu}}=\frac{\partial\mathcal{L}}{\partial\dot{q}^{\mu}}$, where
$q^{\mu}$ are the coordinate fields $q^{\mu}=(\Omega,\beta_{\pm},\phi)$, we
obtain the momenta associated with each field

$\displaystyle\Pi_{\Omega}$
$\displaystyle=\frac{12}{N}e^{3\Omega}\dot{\Omega},$
$\displaystyle\qquad\dot{\Omega}$
$\displaystyle=e^{-3\Omega}\frac{N\Pi_{\Omega}}{12},$ $\displaystyle\Pi_{\pm}$
$\displaystyle=-\frac{12}{N}e^{3\Omega}\dot{\beta}_{\pm},$
$\displaystyle\qquad\dot{\beta_{\pm}}$
$\displaystyle=-e^{-3\Omega}\frac{N\Pi_{\pm}}{12},,$ $\displaystyle\Pi_{\phi}$
$\displaystyle=-f(\phi)\left(\frac{1}{2}\right)^{\alpha}\frac{2\alpha}{N^{2\alpha-1}}\,e^{3\Omega}{\dot{\phi}}^{2\alpha-1},$
$\displaystyle\qquad\dot{\phi}$
$\displaystyle=-Ne^{-\frac{3\Omega}{2\alpha-1}}\left[\frac{2^{\alpha}}{2\alpha}\frac{\Pi_{\phi}}{f(\phi)}\right]^{\frac{1}{2\alpha-1}},$
(31)

and introducing them into the Lagrangian density, we obtain the canonical
Lagrangian as
$\mathcal{L}_{canonical}=\Pi_{q^{\mu}}\dot{q}^{\mu}-N\mathcal{H}=\Pi_{q^{\mu}}\dot{q}^{\mu}-H$.
When we perform the variation of this canonical Lagrangian with respect to
$N$, $\frac{\delta\mathcal{L}_{canonical}}{\delta N}=0$, we obtain the
constraint $\mathcal{H}=0$. In our model, this is the only constraint
corresponding to the Hamiltonian density, which is weakly zero. So, the
Hamiltonian is

$H=\frac{N}{24}e^{-\frac{3}{2\alpha-1}\Omega}\left\\{e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]-\frac{12(2\alpha-1)}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha
f(\phi)}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}\right\\}.$
(32)

###  III.1 Exact solution in the gauge $N=24e^{\frac{3}{2\alpha-1}\Omega}$.

Using the Hamilton equations for the momenta $\dot{\Pi}_{\mu}=-\frac{\partial
H}{\partial q^{\mu}}$ and coordinates $\dot{q}^{\mu}=\frac{\partial
H}{\partial\Pi_{\mu}}$, we have

$\displaystyle\dot{\Omega}$ $\displaystyle=$ $\displaystyle
2e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{\Omega}$ (33)
$\displaystyle\dot{\beta_{+}}$ $\displaystyle=$
$\displaystyle-2e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{+},$ (34)
$\displaystyle\dot{\beta_{-}}$ $\displaystyle=$
$\displaystyle-2e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\Pi_{-},$ (35)
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle-24\left(\frac{2^{\alpha-1}}{\alpha
f(\phi)}\right)^{\frac{1}{2\alpha-1}}\,\Pi_{\phi}^{\frac{1}{2\alpha-1}},$ (36)
$\displaystyle\dot{\Pi}_{\Omega}$ $\displaystyle=$
$\displaystyle\frac{6(\alpha-1)}{2\alpha-1}e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right],$
(37) $\displaystyle\dot{\Pi}_{\pm}$ $\displaystyle=$ $\displaystyle
0,\qquad\qquad\qquad\Rightarrow\qquad\Pi_{\pm}=p_{\pm}=constant,$ (38)
$\displaystyle\dot{\Pi}_{\phi}$ $\displaystyle=$
$\displaystyle-\frac{12}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\Pi_{\phi}^{\frac{2\alpha}{2\alpha-1}}f^{-\frac{2\alpha}{2\alpha-1}}\frac{\dot{f}}{\dot{\phi}},$
(39)

solving the equation (39) using (36), we have
$\Pi_{\phi}=p_{\phi}f^{\frac{1}{2\alpha}}$, with $p_{\phi}$ an integration
constant. With this result and taking into account the equation (36), we get

$\int
f^{\frac{1}{2\alpha}}\,d\phi=-24\left(\frac{p_{\phi}2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}\left(t-t_{i}\right),$
(40)

that is, similar to (27), previously obtained, when was solved a Klein-Gordon
like equation, directly. Using the Hamiltonian constraint and the solution to
equation (39) found previously, we have

$e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\Pi_{\Omega}^{2}-\Pi_{+}^{2}-\Pi_{-}^{2}\right]=\frac{12(2\alpha-1)}{\alpha}\left(\frac{p_{\phi}^{2\alpha}\,2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}},$
(41)

then, the solution for the momenta becomes

$\Pi_{\Omega}=\eta_{\alpha}\,t+p_{0},$ (42)

where the constant
$\eta_{\alpha}=\frac{72(\alpha-1)}{\alpha}\left(\frac{p_{\phi}^{2\alpha}\,2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}$,
and $p_{0}$ are constants of integration, that introducing in the equation for
$\Omega$, we get the equation for the $\Omega$ function

$\frac{d\Omega}{dt}=2e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left(\eta_{\alpha}t+p_{0}\right),\qquad
e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}=\frac{6(\alpha-1)}{2\alpha-1}\left(\eta_{\alpha}t^{2}+2p_{0}t+p_{1}\right),$
(43)

whose solution becomes

$\Omega=\frac{2\alpha-1}{6(\alpha-1)}\,\ln{\left[\frac{6(\alpha-1)}{2\alpha-1}\left(\eta_{\alpha}t^{2}+2p_{0}t+p_{1}\right)\right]},$
(44)

and the solution for the scalar field is given by the equations (29). The
solutions for the anisotropic function $\beta_{\pm}$ are given by

$\beta_{\pm}(t)=b_{\pm}-\frac{(2\alpha-1)p_{\pm}}{24(\alpha-1)\eta_{\alpha}\sqrt{p_{0}^{2}-\eta_{\alpha}p_{1}}}\,\ln{\left[\frac{\eta_{\alpha}t+p_{0}-\sqrt{p_{0}^{2}-\eta_{\alpha}p_{1}}}{\eta_{\alpha}t+p_{0}+\sqrt{p_{0}^{2}-\eta_{\alpha}p_{1}}}\right]},$
(45)

$\beta_{\pm}=b_{\pm}-\left(\frac{2\alpha-1}{24(\alpha-1)}\right)\frac{p_{\pm}}{\eta_{\alpha}\sqrt{\lambda_{\alpha}}}\,\ln{\left(\frac{\Sigma_{-}(t)}{\Sigma_{+}(t)}\right)},$
(46)

where

$\Sigma_{\pm}(t)=\eta_{\alpha}t+p_{0}\pm\sqrt{\lambda_{\alpha}},\qquad\text{and}\qquad\lambda_{\alpha}=p_{0}^{2}-\eta_{\alpha}p_{1}>0.$

According to the last expressions, the radii associated with the Bianchi I
have the following behaviour

$\displaystyle A(t)$
$\displaystyle=e^{\Omega+\beta_{+}+\sqrt{3}\,\beta_{-}}=A_{0}\Biggl{[}\frac{6(\alpha-1)}{2\alpha-1}\Biggl{(}\frac{\Sigma_{+}(t)}{\Sigma_{-}(t)}\Biggr{)}^{\frac{p_{+}+\sqrt{3}\,p_{-}}{4\eta_{\alpha}\sqrt{\lambda_{\alpha}}}}(\eta_{\alpha}t^{2}+2p_{0}t+p_{1})\Biggr{]}^{\frac{2\alpha-1}{6(\alpha-1)}},$
(47a) $\displaystyle B(t)$
$\displaystyle=e^{\Omega+\beta_{+}-\sqrt{3}\,\beta_{-}}=B_{0}\Biggl{[}\frac{6(\alpha-1)}{2\alpha-1}\Biggl{(}\frac{\Sigma_{+}(t)}{\Sigma_{-}(t)}\Biggr{)}^{\frac{p_{+}-\sqrt{3}\,p_{-}}{4\eta_{\alpha}\sqrt{\lambda_{\alpha}}}}(\eta_{\alpha}t^{2}+2p_{0}t+p_{1})\Biggr{]}^{\frac{2\alpha-1}{6(\alpha-1)}},$
(47b) $\displaystyle C(t)$
$\displaystyle=e^{\Omega-2\beta_{+}}=C_{0}\Biggl{[}\frac{6(\alpha-1)}{2\alpha-1}\Biggl{(}\frac{\Sigma_{-}(t)}{\Sigma_{+}(t)}\Biggl{)}^{\frac{p_{+}}{2\eta_{\alpha}\sqrt{\lambda_{\alpha}}}}(\eta_{\alpha}t^{2}+2p_{0}t+p_{1})\Biggr{]}^{\frac{2\alpha-1}{6(\alpha-1}},$
(47c)

and the volume of this universe $V(t)=ABC=e^{3\Omega}$ being,

$V(t)=V_{0}\Biggl{[}\frac{6(\alpha-1)}{2\alpha-1}(\eta_{\alpha}t^{2}+2p_{0}t+p_{1})\Biggr{]}^{\frac{2\alpha-1}{2(\alpha-1)}}.$
(48)

Figure 1: Volume of the universe into the radiation and Dust age,
respectively, according to the table 1, we choose $p_{\phi}=2$ and $p_{1}=1$.

Where we have graphed on different time scales in each scenario, in both cases
the volume is increasing, as is shown in figure 1.

###  III.2 Exact solution without gauge N in the time $\tau$

For this case, the Hamilton procedure is not adequate, then we shall use the
Hamilton-Jacobi procedure in order to find the solutions for the remaining
minisuperspace variables, which arises by making the identification
$\frac{\partial S(\Omega,\beta_{\pm},\phi)}{\partial q_{\mu}}=\Pi_{\mu}$ in
the Hamiltonian constraint (32), $H=0$, taking
$S(\Omega,\beta_{\pm},\phi)=S_{\Omega}(\Omega)+S_{+}(\beta_{+})+S_{-}(\beta_{-})+S_{\phi}(\phi)$
which results in

$e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\left(\frac{dS_{\Omega}}{d\Omega}\right)^{2}-\left(\frac{dS_{+}}{d\beta_{+}}\right)^{2}-\left(\frac{dS_{-}}{d\beta_{-}}\right)^{2}\right]-A_{\alpha}\left(\frac{1}{f(\phi)}\right)^{\frac{1}{2\alpha-1}}\,\left(\frac{dS_{\phi}}{d\phi}\right)^{\frac{2\alpha}{2\alpha-1}}=0.$
(49)

Separating this equation, we have

$e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[\left(\frac{dS_{\Omega}}{d\Omega}\right)^{2}-\left(\frac{dS_{+}}{d\beta_{+}}\right)^{2}-\left(\frac{dS_{-}}{d\beta_{-}}\right)^{2}\right]=A_{\alpha}\left(\frac{1}{f(\phi)}\right)^{\frac{1}{2\alpha-1}}\,\left(\frac{dS_{\phi}}{d\phi}\right)^{\frac{2\alpha}{2\alpha-1}}=\ell_{\phi}^{2}.$
(50)

with $\ell_{\phi}$ a separation constant. The solution in the variable $\phi$
is,

$\Pi_{\phi}=\frac{dS_{\phi}}{d\phi}=\left[\frac{\ell_{\phi}^{2}}{A_{\alpha}}\right]^{\frac{2\alpha-1}{2\alpha}}f^{\frac{1}{2\alpha}}(\phi)=p_{\phi}f^{\frac{1}{2\alpha}}(\phi),$
(51)

where $S(\phi)=p_{\phi}\int f^{\frac{1}{2\alpha}}(\phi)\,d\phi$, obtaining
similar results in the Hamilton procedure.

The specific values of the constants are
$p_{\phi}=\left[\frac{\ell_{\phi}^{2}}{A_{\alpha}}\right]^{\frac{2\alpha-1}{2\alpha}}$
and
$A_{\alpha}=\frac{12(2\alpha-1)}{\alpha}\left(\frac{2^{\alpha-1}}{\alpha}\right)^{\frac{1}{2\alpha-1}}$,
in terms of the $\alpha$ parameter.

The other equations are read as

$\displaystyle\left(\frac{dS_{\Omega}}{d\Omega}\right)^{2}$ $\displaystyle=$
$\displaystyle\ell_{+}^{2}+\ell_{-}^{2}+\ell_{\phi}^{2}\,e^{\frac{6(\alpha-1}{2\alpha-1}},$
(52) $\displaystyle\left(\frac{dS_{+}}{d\beta_{+}}\right)^{2}$
$\displaystyle=$ $\displaystyle\ell_{+}^{2},\qquad
S_{+}=s_{+}\pm\ell_{+}\,\beta_{+},$ (53)
$\displaystyle\left(\frac{dS_{-}}{d\beta_{-}}\right)^{2}$ $\displaystyle=$
$\displaystyle\ell_{-}^{2},\qquad S_{-}=s_{-}\pm\ell_{-}\,\beta_{-},$ (54)

where $\ell_{i}^{2}$ are separation constants and $s_{\pm}$ integration
constants. On the other side, recalling the expressions for the momenta we can
obtain solutions for equations (52), (53) and (54) in quadrature; for the
variable $\Omega$ and for $\alpha\not=1$,

###  III.3 Case for $\alpha\not=1$

In this particular case, we have

$d\tau=12\frac{e^{3\Omega}\,d\Omega}{\sqrt{\ell^{2}+\ell_{\phi}^{2}\,e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}}},\qquad\ell^{2}=\ell_{+}^{2}+\ell_{-}^{2},$
(55)

and for the anisotropic variables,

$\Delta\beta_{\pm}=\mp\frac{\ell_{\pm}}{12}\int e^{-3\Omega(\tau)}\,d\tau.$
(56)

For solving equation (55), we employ the transformation in the time variable
$d\tau=e^{\frac{3}{2\alpha-1}\Omega}dT$ and ${\cal
U}=\ell^{2}+\ell_{\phi}^{2}\,e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}$, so,
$d{\cal
U}=\ell_{\phi}^{2}\frac{6(\alpha-1)}{2\alpha-1}e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}d\Omega$,
resulting

$dT=12\frac{e^{\frac{6(\alpha-1)}{2\alpha-1}\Omega}d\Omega}{\sqrt{{\cal
U}}}=\frac{2(2\alpha-1)}{(\alpha-1)\ell_{\phi}^{2}}\frac{d{\cal
U}}{\sqrt{{\cal U}}},$

and the solution is

$T-T_{0}=\frac{4(2\alpha-1)}{(\alpha-1)\ell_{\phi}^{2}}\left(\sqrt{{\cal
U}}-\sqrt{{\cal U}_{0}}\right),$

then for the $\Omega$ variable, we have

$\Omega(T)=Ln\left[\left(\frac{\ell_{\phi}(\alpha-1)}{4(2\alpha-1)}(T-T_{0})+\sqrt{\frac{{\cal
U}_{0}}{\ell_{\phi}^{2}}}\right)^{2}-\left(\frac{\ell}{\ell_{\phi}}\right)^{2}\right]^{\frac{2\alpha-1}{6(\alpha-1)}},$
(57)

and the time transformation becomes

$d\tau=\left[\left(\frac{\ell_{\phi}(\alpha-1)}{4(2\alpha-1)}(T-T_{0})+\sqrt{\frac{{\cal
U}_{0}}{\ell_{\phi}^{2}}}\right)^{2}-\left(\frac{\ell}{\ell_{\phi}}\right)^{2}\right]^{\frac{1}{2(\alpha-1)}}\,dT.$
(58)

To obtain the solutions in the time $T$, for the anisotropic functions
$\beta_{\pm}(T)$, we solve the integral

$\displaystyle\int e^{-3\Omega(\tau)}d\tau$ $\displaystyle=$
$\displaystyle\int
e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega(T)}dT=\int\frac{dT}{\left(\frac{\ell_{\phi}(\alpha-1)}{4(2\alpha-1)}(T-T_{0})+\sqrt{\frac{{\cal
U}_{0}}{\ell_{\phi}^{2}}}\right)^{2}-\left(\frac{\ell}{\ell_{\phi}}\right)^{2}}$
(59) $\displaystyle=$
$\displaystyle\frac{4(2\alpha-1)}{\ell(\alpha-1)}Ln\left[\frac{\frac{\ell_{\phi}^{2}(\alpha-1)}{4(2\alpha-1))}(T-T_{0})+\sqrt{\mu_{0}}-\ell}{\frac{\ell_{\phi}^{2}(\alpha-1)}{4(2\alpha-1))}(T-T_{0})+\sqrt{\mu_{0}}+\ell}\right],$

then, the anisotropic functions (56) become

$\beta_{\pm}(T)=\beta_{\pm}(T_{i})\mp\frac{\ell_{\pm}}{3}\frac{(2\alpha-1)}{\ell(\alpha-1)}Ln\left[\frac{\frac{\ell_{\phi}^{2}(\alpha-1)}{4(2\alpha-1))}(T-T_{0})+\sqrt{\mu_{0}}-\ell}{\frac{\ell_{\phi}^{2}(\alpha-1)}{4(2\alpha-1))}(T-T_{0})+\sqrt{\mu_{0}}+\ell}\right].$
(60)

and the scalar field (28), takes the form

$\phi(T)=\phi(T_{i})+\left\\{\begin{tabular}[]{ll}$\left[\frac{(2\alpha+m)}{2\alpha}{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}(T-T_{i})\right]^{\frac{2\alpha}{2\alpha+m}},$&\quad$f(\phi)=p\phi^{m},\quad
m\not=-2\alpha$,\\\
$Exp\left[{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}(T-T_{i})\right],$&\quad$f(\phi)=p\phi^{-2\alpha},\quad
m=-2\alpha$,\\\
$\frac{2\alpha}{m}\,Ln\left[\frac{m}{2\alpha}{{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}(T-T_{i})\right],$&\quad$f(\phi)=pe^{m\phi},\quad
m\not=0,$\\\
${{p^{-\frac{1}{2\alpha}}\sqrt{2\lambda}}}(T-T_{i}),$&\quad$f(\phi)=p,\quad
m=0,$\end{tabular}\right.$ (61)

On the other side, the only state when the time $\tau=T$, corresponds to the
scenario, $\alpha\to\infty$, which is calculated below.

####  III.3.1 Dust scenario, $\alpha\to\infty$

For this particular case, we have

$\Omega(\tau)=Ln\left[\left(\frac{\ell_{\phi}\,}{8}(\tau-\tau_{0})+\sqrt{\frac{{\cal
U}_{0}}{\ell_{\phi}^{2}}}\right)^{2}-\left(\frac{\ell}{\ell_{\phi}}\right)^{2}\right]^{\frac{1}{3}},$
(62)

then the volume function becomes

$V(\tau)=V_{0}\left\\{\left(\frac{\ell_{\phi}\,}{8}(\tau-\tau_{0})+\sqrt{\frac{{\cal
U}_{0}}{\ell_{\phi}^{2}}}\right)^{2}-\left(\frac{\ell}{\ell_{\phi}}\right)^{2}\right\\},$
(63)

and the anisotropic functions are

$\beta_{\pm}(\tau)=\beta_{\pm}(\tau_{0})\mp\frac{\ell_{\pm}}{3}\frac{2}{\ell}Ln\left[\frac{\frac{\ell_{\phi}^{2}}{8}(\tau-\tau_{0})+\sqrt{\mu_{0}}-\ell}{\frac{\ell_{\phi}^{2}}{8}(\tau-\tau_{0})+\sqrt{\mu_{0}}+\ell}\right].$
(64)

We can see that the scalar field constant $\ell_{\phi}$ is huge, the
anisotropic function goes to constant, and the anisotropic model can be
isotropic one. We rewrite the corresponding solutions in the scalar field
(28), for this scenario

$\phi(\tau)=\phi(\tau_{0})+\left\\{\begin{tabular}[]{ll}$\left[{{\sqrt{2\lambda}}}(\tau-\tau_{0})\right],$&\quad$f(\phi)=p\phi^{m},\quad
m\not=-2\alpha,$\\\
$Exp\left[{{\sqrt{2\lambda}}}(\tau-\tau_{0})\right],$&\quad$f(\phi)=,\quad
m=-2\alpha,$\\\
$\,Ln\left[{{\sqrt{2\lambda}}}(\tau-\tau_{i})\right],$&\quad$f(\phi)=pe^{2\alpha\phi},$\\\
${{\sqrt{2\lambda}}}(\tau-\tau_{0}),$&\quad$f(\phi)=p,\quad
m=0.$\end{tabular}\right.$ (65)

##  IV Quantum Regime

The WDW equation for these models is obtained by making the usual substitution
$\Pi_{q^{\mu}}=-i\hbar\partial_{q^{\mu}}$ in (32), and promoting the classical
Hamiltonian density in the differential operator applied to the wave function
$\Psi(\Omega,\beta_{\pm},\phi)$, $\hat{\cal H}\Psi=0$; we have

$\hbar^{2}e^{-\frac{6(\alpha-1)}{2\alpha-1}\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{+}^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{-}^{2}}\right]-\frac{12(2\alpha-1)}{\alpha}\,\left(\frac{2^{\alpha-1}}{\alpha{{f(\phi)}}}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\frac{2\alpha}{2\alpha-1}}\frac{\partial^{\frac{2\alpha}{2\alpha-1}}}{\partial\phi^{\frac{2\alpha}{2\alpha-1}}}\Psi=0.$
(66)

This fractional differential equation of degree
$\beta=\frac{2\alpha}{2\alpha-1}$, belongs to different intervals, depending
on the value of the barotropic parameter Socorro2 . We can write this equation
in terms of the $\beta$ parameter, we have

$\hbar^{2}e^{-3(2-\beta)\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{+}^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{-}^{2}}\right]-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha{{f(\phi)}}}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\frac{\partial^{\beta}}{\partial\phi^{\beta}}\Psi=0.$
(67)

For simplicity, the factor $e^{-3(2-\beta)\Omega}$ may be the factor ordered
with $\hat{\Pi}_{\Omega}$ and $f^{-\frac{1}{2\alpha-1}}(\phi)$ may be the
factor ordered with $\frac{\partial^{\beta}}{\partial\phi^{\beta}}$ in many
ways, we employ it might be called a semi-general factor ordering, which, in
this case, would order the terms $e^{-3(2-\beta)\Omega}\hat{\Pi}^{2}_{\Omega}$
as
$-e^{-(3(2-\beta)-Q)\Omega}\,\partial_{\Omega}e^{-Q\Omega}\partial_{\Omega}=-e^{-3(2-\beta)\Omega}\,\partial^{2}_{\Omega}+Q\,e^{-3(2-\beta)\Omega}\partial_{\Omega},$
where $Q$ is any real constant that measures the ambiguity in the factor
ordering in the variables $\Omega$ and its corresponding momenta. For the
other factor ordering, we make the following calculation which, in this case,
would order the terms
$\frac{g(\phi)}{f^{\frac{1}{2\alpha-1}}(\phi)}\frac{\partial^{\beta}}{\partial\phi^{\beta}}$,
where in the particular case we choose $g(\phi)=\phi^{s}$, similarly to
$f(\phi)$ in the classical case, that is

$f^{-\frac{1}{2\alpha-1}}(\phi)\,\phi^{-s}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}\phi^{s}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}=f^{-\frac{1}{2\alpha-1}}(\phi)\frac{\partial^{\beta}}{\partial\phi^{\beta}}+f^{-\frac{1}{2\alpha-1}}(\phi)\,\phi^{-s}\left[\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}\phi^{s}\right]\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}},$
(68)

where the Caputo fractional derivative of
$\left[\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}\phi^{s}\right]$
becomes Podlubny ,

${}_{0}^{c}D_{x}^{\beta/2}\phi^{s}=\frac{\Gamma(s+1)}{\Gamma(s-\beta/2+1)}\phi^{s-\beta/2}.$
(69)

Thus, the equation (68) is rewritten as

$\displaystyle
f^{-\frac{1}{2\alpha-1}}(\phi)\,\phi^{-s}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}\phi^{s}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}$
$\displaystyle=$ $\displaystyle
f^{-\frac{1}{2\alpha-1}}(\phi)\frac{\partial^{\beta}}{\partial\phi^{\beta}}+f^{-\frac{1}{2\alpha-1}}(\phi)\,\phi^{-s}\frac{\Gamma(s+1)}{\Gamma(s+1-\beta/2)}\phi^{s-\beta/2}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}$
(70) $\displaystyle=$ $\displaystyle
f^{-\frac{1}{2\alpha-1}}(\phi)\frac{\partial^{\beta}}{\partial\phi^{\beta}}+f^{-\frac{1}{2\alpha-1}}(\phi)\,\frac{\Gamma(s+1)}{\Gamma(s+1-\beta/2)}\phi^{-\beta/2}\frac{\partial^{\beta/2}}{\partial\phi^{\beta/2}}$

Assuming this factor ordering for the Wheeler-DeWitt equation, we get

$\displaystyle
e^{-3(2-\beta)\Omega}\left[-\frac{\partial^{2}\Psi}{\partial\Omega^{2}}+Q\frac{\partial\Psi}{\partial\Omega}+\frac{\partial^{2}\Psi}{\partial\beta_{+}^{2}}+\frac{\partial^{2}\Psi}{\partial\beta_{-}^{2}}\right]$
$\displaystyle-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha{{f(\phi)}}}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\frac{\partial^{\beta}}{\partial\phi^{\beta}}\Psi-\frac{24}{\beta}\,\left(\frac{2^{\alpha-1}}{\alpha{{f(\phi)}}}\right)^{\frac{1}{2\alpha-1}}\,\hbar^{\beta}\phi^{-s}\frac{\Gamma(s+1)}{\Gamma(s+1-\mu)}\frac{\partial^{\frac{\beta}{2}}}{\partial\phi^{\frac{\beta}{2}}}\Psi=0.$
(71)

Using the ansatz for the wave function
$\Psi(\Omega,\beta_{+},\beta_{-},\phi)={\cal A}(\Omega){\cal
B_{+}}(\beta_{+}){\cal B_{-}}(\beta_{-}){\cal C}(\phi)$, we obtain the
following differential equations on the corresponding variables

$\displaystyle\frac{d^{2}{\cal A}}{d\Omega^{2}}-Q\frac{d{\cal
A}}{d\Omega}-\left[\pm\left(\frac{\rho}{\hbar}\right)^{2}\,e^{3(2-\beta)\Omega}+\rho_{1}^{2}\right]{\cal
A}$ $\displaystyle=$ $\displaystyle 0,$ (72) $\displaystyle\frac{d^{2}{\cal
B_{+}}}{d\beta_{+}^{2}}-\rho_{2}^{2}{\cal B_{+}}$ $\displaystyle=$
$\displaystyle 0,$ (73) $\displaystyle\frac{d^{2}{\cal
B_{-}}}{d\beta_{-}^{2}}-\rho_{3}^{2}{\cal B_{-}}$ $\displaystyle=$
$\displaystyle 0,\qquad\rho_{3}^{2}=\rho_{2}^{2}-\rho_{1}^{2}$ (74)
$\displaystyle\phi^{\gamma}\frac{d^{2\gamma}{\cal
C}}{d\phi^{2\gamma}}+\frac{\Gamma(s+1)}{\Gamma(s+1-\gamma)}\frac{d^{\gamma}{\cal
C}}{d\phi^{\gamma}}\pm\left(\frac{p_{\phi}\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\gamma\rho^{2}}{12\hbar^{2\gamma}}{\cal
C}$ $\displaystyle=$ $\displaystyle 0,\quad\beta=2\gamma,\quad 0<\gamma\leq
1.$ (75)

We can see that the fractional differential equation (75) has variable
coefficients, so to solve it we can use the fractional power series El-Ajou ;
Rida , also $f(\phi)=p_{\phi}\phi^{m}$, with $p_{\phi}$ a constant and
choosing $\frac{m}{2\alpha-1}=-\gamma$, as particular case, which implies that
the parameter $m=-\alpha$ in sense that $\gamma$ parameter is in accordance
with its original definition (see equation (75) and definition of the $\beta$
parameter, or the equation in the text after equation (76)).

Following the book of Polyanin polyanin  (page 179.10), we find the solution
for the first equation, considering different values in the factor ordering
parameter, (we take the corresponding sign minus in the constant $\rho^{2}$)

$\mathcal{A}(\Omega)=e^{\frac{Q\Omega}{2}}\biggl{[}C_{1}K_{\nu}\biggl{(}\frac{\rho}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\biggr{)}+C_{2}I_{\nu}\biggl{(}\frac{\rho}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\biggr{)}\biggr{]},$
(76)

where $K_{\nu}(z)$ and $I_{\nu}(z)$ are the modified Bessel functions, and
order $\nu=\frac{\sqrt{Q^{2}+4\rho_{1}^{2}}}{6(1-\gamma)}$ with
$\gamma=\frac{\beta}{2}=\frac{\alpha}{2\alpha-1}$ the new order in the
fractional derivative. However, for physical conditions we will only take the
modified Bessel $K_{\nu}$.

The corresponding quantum solution for equations (73) and (74) are

$\displaystyle\mathcal{B}_{+}=a_{0}e^{\rho_{2}\beta_{+}}+a_{1}e^{-\rho_{2}\beta_{+}},$
(77)
$\displaystyle\mathcal{B}_{-}=b_{0}e^{\rho_{3}\beta_{-}}+b_{1}e^{-\rho_{3}\beta_{-}}.$
(78)

with $\rho^{2}$ and $\rho_{i}^{2}$ are separation constants.

The solution of the equation (75) with positive sign and
$f(\phi)=p_{\phi}=constant$, with zero factor ordering, may be obtained by
applying direct and inverse Laplace transform Rosales2 ; Socorro2 , providing

${\cal C}_{+}(\phi,\gamma)=\mathbb{E}_{2\gamma}\left(-z^{2}\right),\qquad
z={\left(\frac{p_{\phi}\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2(2\alpha-)}}}\frac{\sqrt{\gamma}\rho}{2\sqrt{3}\hbar^{\gamma}}\phi^{\gamma},\qquad
0<\gamma\leq 1,$ (79)

where $\mathbb{E}_{2\gamma}$ is the Mittag-Leffler function (7), then, the
probability density of the wave function for this particular case becomes

$|\Psi|^{2}=\psi_{0}^{2}e^{Q\Omega\pm 2\rho_{2}\beta_{+}\pm
2\rho_{3}\beta_{-}}\,K_{\nu}^{2}\left[\frac{\rho}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\right]\,\mathbb{E}^{2}_{2\gamma}(-z^{2}),$
(80)

and its corresponding plot for two values in the ordering parameter Q, as is
shown in figure 2.

Figure 2: Probability density (80) of the universe dominated by Dust era, in
this stage of the universe shows that the probability density has a decay,
both in $\Omega$ and $\phi$. Both are plotting with $\rho=0.5$,
$\gamma=\frac{1}{2}$, $\rho_{1}=0.1$; to the left figure we take $Q=6$,
$\nu=2.00111$, and right figure $Q=2$, $\nu=0.669992$.

Table II shows the differential equations obtained from (75), depending on the
values of $\omega_{X}$, $\alpha$ and $\gamma$

$\omega_{X}$ | $\alpha$ | $\gamma$ | Fractionary equation
---|---|---|---
1 | 1 | 1 | $\phi\frac{d^{2}{\cal C}}{d\phi^{2}}+\frac{\Gamma(s+1)}{\Gamma(s)}\frac{d{\cal C}}{d\phi}\pm p_{\phi}\frac{\mu^{2}}{24\hbar^{2}}{\cal C}=0$
$\frac{1}{3}$ | 2 | $\frac{2}{3}$ | $\phi^{\frac{2}{3}}\frac{d^{\frac{4}{3}}{\cal C}}{d\phi^{\frac{4}{3}}}+\frac{\Gamma(s+1)}{\Gamma(s+\frac{1}{3})}\frac{d^{\frac{2}{3}}{\cal C}}{d\phi^{\frac{2}{3}}}\pm(p_{\phi})^{\frac{1}{3}}\frac{\mu^{2}}{18\hbar^{\frac{4}{3}}}{\cal C}=0$
$\rightarrow 0$ | $\rightarrow\infty$ | $\frac{1}{2}$ | $\phi^{\frac{1}{2}}\frac{d{\cal C}}{d\phi}+\frac{\Gamma(s+1)}{\Gamma(s+\frac{1}{2})}\frac{d^{\frac{1}{2}}{\cal C}}{d\phi^{\frac{1}{2}}}\pm\par\frac{\mu^{2}}{24\hbar}{\cal C}=0$
$-1$ | 0 | 0 | $without\,\,equation$
$-\frac{1}{3}$ | -1 | $\frac{1}{3}$ | $\phi^{\frac{1}{3}}\frac{d^{\frac{2}{3}}{\cal C}}{d\phi^{\frac{2}{3}}}+\frac{\Gamma(s+1)}{\Gamma(s+\frac{2}{3})}\frac{d^{\frac{1}{3}}{\cal C}}{d\phi^{\frac{1}{3}}}\pm\left(\frac{p_{\phi}\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\mu^{2}}{36\hbar^{\frac{2}{3}}}{\cal C}=0$
$-\frac{2}{3}$ | $-\frac{1}{4}$ | $\frac{1}{6}$ | $\phi^{\frac{1}{6}}\frac{d^{\frac{1}{3}}{\cal C}}{d\phi^{\frac{1}{3}}}+\frac{\Gamma(s+1)}{\Gamma(s+\frac{5}{6})}\frac{d^{\frac{1}{6}}{\cal C}}{d\phi^{\frac{1}{6}}}\pm\left(\frac{p_{\phi}\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\mu^{2}}{72\hbar^{\frac{1}{3}}}{\cal C}=0$
Table 2: Fractionary equation in the field $\phi$ according to the barotropic
parameter $\omega_{X}$.

###  IV.1 Solution to FDE associated with the different state evolutions

We write the fractional differential equation (75) as follows,

$\phi^{\gamma}\frac{d^{2\gamma}{\mathcal{C}}}{d\phi^{2\gamma}}+A\frac{d^{\gamma}\mathcal{C}}{d\phi^{\gamma}}+B\,\mathcal{C}=0,\quad
0<\gamma\leq 1,$ (81)

where we have made the simplifications
$A=\frac{\Gamma(s+1)}{\Gamma(s+1-\gamma)}$ and
$B_{(\alpha,\gamma)}=\pm\left(\frac{p_{\phi}\alpha}{2^{\alpha-1}}\right)^{\frac{1}{2\alpha-1}}\frac{\gamma\mu^{2}}{12\hbar^{2\gamma}}$.
The last linear fractional differential equation (81), will be solved using
the fractional power series El-Ajou ; Rida

$\mathcal{C}=\sum^{\infty}_{n=0}a_{n}\,\phi^{n\gamma}.$ (82)

Then, the fractional derivatives are

$\displaystyle\frac{d^{\gamma}\mathcal{C}}{d^{\gamma}\phi}$
$\displaystyle=\sum_{n=1}^{\infty}a_{n}\,\frac{\Gamma[n\gamma+1]}{\Gamma[(n-1)\gamma+1]}\,\phi^{(n-1)\gamma},$
(83a) $\displaystyle\frac{d^{2\gamma}\mathcal{C}}{d^{2\gamma}\phi}$
$\displaystyle=\sum_{n=2}^{\infty}a_{n}\,\frac{\Gamma[n\gamma+1]\Gamma[(n-1)\gamma+1]}{\Gamma[(n-1)\gamma+1]\Gamma[(n-2)\gamma+1]}\,\phi^{(n-2)\gamma}.$
(83b)

Replacing the expressions (83) into (81), we get

$\sum_{n=2}^{\infty}a_{n}\,\frac{\Gamma[n\gamma+1]}{\Gamma[(n-2)\gamma+1]}\,\phi^{(n-1)\gamma}+A\sum_{n=1}^{\infty}a_{n}\,\frac{\Gamma[n\gamma+1]}{\Gamma[(n-1)\gamma+1]}\,\phi^{(n-1)\gamma}+B_{(\alpha,\gamma)}\sum^{\infty}_{n=0}a_{n}\,\phi^{n\gamma}=0.$
(84)

Now, taking $\ell=n-1$ into the first and second terms, and $n=\ell$ into the
third term of (84), we have

$\sum_{\ell=1}^{\infty}a_{\ell+1}\,\frac{\Gamma[(\ell+1)\gamma+1]}{\Gamma[(\ell-1)\gamma+1]}\,\phi^{\ell\gamma}+A\sum_{\ell=0}^{\infty}a_{\ell+1}\,\frac{\Gamma[(\ell+1)\gamma+1]}{\Gamma[\ell\gamma+1]}\,\phi^{\ell\gamma}+B_{(\alpha,\gamma)}\sum^{\infty}_{\ell=0}a_{\ell}\,\phi^{\ell\gamma}=0.$
(85)

Shifting one place in the second and third summations, we have

$\displaystyle\sum_{\ell=1}^{\infty}a_{\ell+1}\,\frac{\Gamma[(\ell+1)\gamma+1]}{\Gamma[(\ell-1)\gamma+1]}\,\phi^{\ell\gamma}+$
$\displaystyle
A\,\frac{\Gamma[\gamma+1]}{\Gamma[1]}a_{1}+A\sum_{\ell=1}^{\infty}a_{\ell+1}\,\frac{\Gamma[(\ell+1)\gamma+1]}{\Gamma[\ell\gamma+1]}\,\phi^{\ell\gamma}$
$\displaystyle+B_{(\alpha,\gamma)}a_{0}+B_{(\alpha,\gamma)}\sum^{\infty}_{\ell=1}a_{\ell}\,\phi^{\ell\gamma}=0.$
(86)

From last expression (86), we get $(s\not=0)$

$a_{1}=-\frac{B_{(\alpha,\gamma)}}{A\Gamma(\gamma+1)}=-\frac{\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}}{\Gamma[s+1]\,\Gamma[\gamma+1]}a_{0},$
(87)

and the recurrence relationships between the parameters $a_{\ell}$, is

$a_{\ell+1}=-\frac{\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}\,\Gamma[\ell\gamma+1]\,\Gamma[(\ell-1)\gamma+1]\,}{\Gamma[(\ell+1)\gamma+1]\biggl{\\{}\Gamma[s+1-\gamma]\,\Gamma[\ell\gamma+1]+\Gamma[s+1]\,\Gamma[(\ell-1)\gamma+1]\biggr{\\}}}\,a_{\ell},\qquad\forall\,\ell\geq
1.$ (88)

Some terms of this relation are

$\displaystyle a_{1}$
$\displaystyle=-\frac{\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}}{\Gamma[s+1]\,\Gamma[\gamma+1]}a_{0},$
$\displaystyle a_{2}$
$\displaystyle=\frac{\left(\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}\right)^{2}}{\Gamma[s+1]\Gamma[2\gamma+1]\left(\Gamma[s+1]+\Gamma[s+1-\gamma]\Gamma[\gamma+1]\right)}a_{0},$
$\displaystyle a_{3}$
$\displaystyle=-\frac{\Gamma[\gamma+1]}{\Gamma[s+1]\Gamma[3\gamma+1]\left(\Gamma[s+1-\gamma]\Gamma[2\gamma+1]+\Gamma[s+1]\Gamma[\gamma+1]\right)}\times$
$\displaystyle\qquad\qquad\Biggl{(}\frac{\left(\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}\right)^{3}}{\left(\Gamma[s+1]+\Gamma[s+1-\gamma]\Gamma[\gamma+1]\right)}\Biggr{)}a_{0},$
(89) $\displaystyle a_{4}$
$\displaystyle=\frac{\Gamma[\gamma+1]\Gamma[2\gamma+1]}{\Gamma[s+1]\Gamma[4\gamma+1]\left(\Gamma[s+1-\gamma]\Gamma[3\gamma+1]+\Gamma[s+1]\Gamma[2\gamma+1]\right)}\times$
$\displaystyle\qquad\Biggl{(}\frac{(\Gamma[s+1-\gamma]\,B_{(\alpha,\gamma)})^{4}}{\left(\Gamma[s+1\gamma]\Gamma[2\gamma+1]+\Gamma[s+1]\Gamma[\gamma+1]\right)\left(\Gamma[s+1]+\Gamma[s+1-\gamma]\Gamma[\gamma+1]\right)}\Biggr{)}a_{0},$
$\displaystyle\vdots$

Then, the solution of the fractional equation (81) has the form

$\displaystyle\mathcal{C}_{s,\alpha,\gamma}$ $\displaystyle=$ $\displaystyle
a_{0}\left[1-\frac{\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}}{\Gamma[s+1]\,\Gamma[\gamma+1]}\phi^{\gamma}\right.$
$\displaystyle\left.+\frac{\left(\Gamma[s+1-\gamma]B_{(\alpha,\gamma)}\right)^{2}}{\Gamma[s+1]\Gamma[2\gamma+1]\left(\Gamma[s+1]+\Gamma[s+1-\gamma]\Gamma[\gamma+1]\right)}\phi^{2\gamma}+\ldots\right].$

For the Dust-like scenario (see table 2), $\alpha\to\infty$ and
$\gamma=\frac{1}{2}$, then $B_{(\infty,\frac{1}{2})}=\frac{\mu^{2}}{24\hbar}$.
The solution associated with this fractional differential equation is given in
fractional series form by,

$\displaystyle\mathcal{C}_{s,\to\infty,\frac{1}{2}}$ $\displaystyle=$
$\displaystyle
a_{0}\left[1-\frac{2\Gamma[s+\frac{1}{2}]B_{(\infty,\frac{1}{2})}}{\Gamma[s+1]\sqrt{\pi}}\,\phi^{\frac{1}{2}}+\frac{2\left(\Gamma[s+\frac{1}{2}]B_{(\infty,\frac{1}{2})}\right)^{2}}{\Gamma[s+1](2\Gamma[s+1]+\Gamma[s+\frac{1}{2}]\sqrt{\pi})}\,\phi\right.$
$\displaystyle\left.-\frac{2\left(2\Gamma[s+\frac{1}{2}]B_{(\infty,\frac{1}{2})}\right)^{3}}{3\sqrt{\pi}\Gamma[s+1](2\Gamma[s+\frac{1}{2}]+\Gamma[s+1]\sqrt{\pi})(2\Gamma[s+1]+\Gamma[s+\frac{1}{2}]\sqrt{\pi})}\,\phi^{\frac{3}{2}}+\cdots\right],$

where we employed $\Gamma[\frac{3}{2}]=\frac{\sqrt{\pi}}{2}$ and
$\Gamma[\frac{5}{2}]=\frac{3\sqrt{\pi}}{4}$.

For the radiation stage, $\alpha=2$ and $\gamma=\frac{2}{3}$, then
$B_{(2,\frac{2}{3})}=\frac{\sqrt[3]{p_{\phi}}\mu^{2}}{18\hbar^{\frac{4}{3}}}$,
the solution for the fractional differential equation is

$\displaystyle\mathcal{C}_{s,2,\frac{2}{3}}$ $\displaystyle=$ $\displaystyle
a_{0}\left[1-\frac{3\Gamma[s+\frac{1}{3}]B_{(2,\frac{2}{3})}}{2\Gamma[s+1]\Gamma[\frac{2}{3}]}\,\phi^{\frac{2}{3}}+\frac{\left(3\Gamma[s+\frac{1}{3}]B_{(2,\frac{2}{3})}\right)^{2}}{4\Gamma[s+1]\Gamma[\frac{4}{3}]\left(3\Gamma[s+1]+2\Gamma[s+\frac{1}{3}]\Gamma[\frac{2}{3}]\right)}\,\phi^{\frac{4}{3}}\right.$
$\displaystyle\left.-\frac{\frac{2}{3}\Gamma[\frac{2}{3}]\left(3\Gamma[s+\frac{1}{3}]B_{(2,\frac{2}{3})}\right)^{3}}{2\Gamma[s+1]\left(2\Gamma[s+\frac{1}{3}]\Gamma[\frac{4}{3}]+\Gamma[s+1]\Gamma[\frac{2}{3}]\right)\left(3\Gamma[s+1]+2\Gamma[s+\frac{1}{3}]\Gamma[\frac{2}{3}]\right)}\,\phi^{\frac{8}{3}}+\cdots\right],$

with $\Gamma[\frac{2}{3}]=1.35412$ and $\Gamma[\frac{4}{3}]=0.89298$.

We are going to present the graphical behaviour of the wave function for the
Dust-like case, in which the solution is constrained to the variables $\Omega$
and $\phi$. This means we are going to shrink the directions $\beta_{+}$ and
$\beta_{-}$. So, the wave function takes the form

$\displaystyle\Psi(\Omega,\beta_{+},\beta_{-},\phi)$
$\displaystyle=e^{\frac{Q\Omega}{2}}\biggl{[}C_{1}K_{\nu}\biggl{(}\frac{\rho}{3\hbar(1-\gamma)}e^{3(1-\gamma)\Omega}\biggr{)}\biggr{]}\times$
$\displaystyle\Biggl{[}1-a_{1}\phi^{\frac{1}{2}}+a_{2}\phi-
a_{3}\phi^{\frac{3}{2}}+a_{4}\phi^{2}-a_{5}\phi^{\frac{5}{2}}+a_{6}\phi^{3}-a_{7}\phi^{\frac{7}{2}}+a_{8}\phi^{4}+\ldots\Biggr{]}.$
(93)

By restraining ourselves to the values of
$\nu=\frac{\sqrt{Q^{2}+4\rho_{1}^{2}}}{6(1-\gamma))}$, $Q=10,6,2,1.6$,
$\rho=\frac{1}{2}$, $\gamma=\frac{1}{2}$, $s=1$, $\rho_{1}=0.1$ and $C_{1}=1$,
we see that the wave function ((93)) can be rewritten as

$\displaystyle\Psi^{2}(\Omega,\phi)$ $\displaystyle=$ $\displaystyle
e^{Q\Omega}\,K^{2}_{\nu}\left(\frac{2}{3}e^{\frac{3\Omega}{2}}\right)\left[1-\frac{1}{24}\phi^{1/2}+\frac{\sqrt{\pi}}{576(4+\pi)}\phi-\frac{2\sqrt{\pi}}{41472(4+\pi)}\phi^{3/2}\right.$
(94)
$\displaystyle\left.+\frac{\pi^{\frac{3}{2}}}{663552(8+3\pi)(4+\pi)}\phi^{2}-\frac{\pi^{\frac{3}{2}}}{69672960(8+3\pi)(4+\pi)}\phi^{\frac{5}{2}}\right.$
$\displaystyle\left.+\frac{\pi^{\frac{5}{2}}}{334430208(8+3\pi)(4+\pi)(32+15\pi)}\phi^{3}\right.$
$\displaystyle\left.-\frac{\pi^{\frac{5}{2}}}{45649723392(8+3\pi)(4+\pi)(32+15\pi)}\phi^{\frac{7}{2}}\right.$
$\displaystyle\left.+\frac{5\pi^{\frac{7}{2}}}{208684449792(8+3\pi)(4+\pi)(32+15\pi)(192+105\pi)}\phi^{4}\right]^{2}\,,$

where we have taken the cut-off order in $\phi^{4}$ and the a’s parameter are
read from (89). In the following, we present some plots of the probability
density of the wave function, including the factor ordering parameter Q and
particular values in the parameter $\rho$, $\rho_{1}$ and the particular value
to the ordering parameter s=1. In all of them we observe that for any value of
Q, the probability density decays with respect to the scale factor, but has a
different evolution in the scalar field. For small Q’s, the quantum universe
has considerable existence in the evolution with respect to the scale factor
and then decays. On the other hand, for large Q’s, this interval is small.
What we can say about the evolution of the scalar field is that at small Q’s,
the scalar field appears faster than for large Q’s, which enters late, but has
existed forever (see figure 3).

Figure 3: Probability density (94) of the universe dominated by Dust era, in
this stage of the universe shows that the probability density has a decay in
$\Omega$ and exhibits considerable growth for certain values of $\phi$. The
plots have the parameters $\rho=0.5$, $\gamma=\frac{1}{2}$, $\rho_{1}=0.1$.
The Q and $\nu$ parameters taken the values, $Q=10,6,2,1.6$,
$\nu=3.334$,2.00111,0.0669992,0.537489, respectively,

##  V Conclusions

Unlike the previous work Socorro2 , in the present paper we employed a
barotropic equation with perfect fluid for energy-momentum tensor in the
K-essence scalar field into the Lagrangian and Hamiltonian formalism,
obtaining the momentum of a scalar field with fractional numbers, while the
momentum of the scale factor appears in the usual way. We obtained the
classical solutions for different scenarios in the universe, employing
different times $(t,T(\tau),\tau)$. In the quantum scheme, we include the
factor ordering problem, and we find a fractional differential equation for
the scalar field with variable coefficients, which was solved using the
fractional series expansion. With this in mind, we visualize two alternatives
in our analysis; the first one is within the traditional expectation over the
behaviour of the probability density, that the best candidates for quantum
solutions are those that have a damping behaviour with respect to the scale
factor, appearing in all scenarios under our study, without saying anything
about the scalar field. The other scenario is when we keep the scale factor,
and we consider the values of the scalar field as significant in the quantum
regime, appearing in various scenarios in the behaviour of the universe;
mainly in those where the universe has a huge behaviour, for example, in the
actual epoch, where the scalar field appears as background.

In other words, the interpretation of the probability density of the
unnormalized wave function is given, when we demand that $\Psi$ does not
diverge when the scale factor A (or $\Omega$) goes to infinity, and the scalar
field is arbitrary. However, the evolution with the scalar field is important
in this class of theory and others, as it appears in some stages of the
evolution of our Universe.

In the reference Micolta-2023 , the gravitational action integral is altered
by hand, leading to a modified Friedmann-type equation. They employed the
dynamical system approach in order to find the balance points providing a
range for the order of the fractional derivatives in their investigation of
the cosmological universe, and they do mention that it can be confirmed that
the solutions isotropize at a late time. In our approach, this occurs when the
$\ell_{\phi}$ is huge (see equation (64)); due to that, the anisotropic
parameters become constants. On the other hand, as we use the volume of the
universe $V=ABC=e^{3\Omega(\tau)}$ on any time scale, it depends only on the
$\Omega$ function and not on the anisotropic parameters $\beta_{\pm}$. Not so,
the scale factors $A$, $B$ and $C$, which do depend on these. It remains to
study whether the fractional derivatives alter the gravitational part and how
the universe’s singularity can be avoided because, from our approach, this
part needs to be revised since the gravitational part is not altered from the
point of view of the equations of motion. It is modified in the scalar field
part.

It would be interesting to extend the Bohm-type semiclassical formalism in
this context, which we will do in future work. Much work has been made in this
direction paulo2012 ; Priscila2013 ; 1511.08382 ; omar2017 ; 1710.08666 ;
2012.10814 ; 2103.00802 , where the quantum potential emerges as the imaginary
part in the Bohm formalism, appearing as a constraint equation. In this sense,
in the reference Jalalzadeh-2023  appears an approach based on the
semiclassical limit of fractionary quantum cosmology using the Riesz
derivative. It would be interesting to continue under our focus, where the
corresponding Friedmann equation and the Hubble parameter depend on the Levy’s
fractional parameter, which is associated with the concept of Lévy path in the
corresponding quantum cosmology.

We briefly illustrate the main results of this work.

  1. 1.

Using the K-essence formalism in a general way, applied to anisotropic Bianchi
type I cosmological model, we found the Hamiltonian density in the scalar
field momenta raised to power with non-integers, which produces in the quantum
scheme a fractional differential equation in a natural way. We include the
factor ordering problem in both variables $(\Omega,\phi)$ and its momenta
$(\Pi_{\Omega},\Pi_{\phi})$, with the order $\beta=\frac{2\alpha}{2\alpha-1}$,
where $\alpha\in(-1,\infty)$, and it was solved in a general way, we include
two particular scenarios of our Universe.

  2. 2.

We found the solution in the classical scheme employing two gauges,
$N=24e^{3\Omega}$, for two forms of the function $f(\phi)$ in the time $t$;
however, when we let the Lagrange multiplier N, we need to employ a
transformed time $T(\tau)$ for solving the classical equation, and only in the
dust era, we recover the gauge time $\tau$.

  3. 3.

In the quantum regime, when we include the factor ordering problem, the
fractional differential equation in the scalar field appears with variable
coefficients, and it was necessary to use the fractional series expansion to
solve it in a general way.

  4. 4.

In one of our analysis presented on the probability density, we consider the
values of the scalar field as significant in the quantum regime, appearing in
various scenarios in the behaviour of the universe; mainly in those where the
universe has a huge behaviour, for example, in the actual epoch, where the
scalar field appears as a background, the quantum regime appears with big
values, but it presents a moderate development in other scenarios with
different ordering parameters Q and s.

###### Acknowledgements.

J.S. was partially supported by PROMEP grants UGTO-CA-3. Authors were
partially supported SNI-CONACyT. J. Rosales is supported by PROMEP grants
UGTO-CA-20 nonlinear photonics and the Department of Electrical Engineering.
This work is part of the collaboration within the Instituto Avanzado de
Cosmología and Red PROMEP: Gravitation and Mathematical Physics under project
Quantum aspects of gravity in cosmological models, phenomenology and geometry
of space-time. L.T.S. is supported by Secretaria de Investigación y Posgrado
del Instituto Politécnico Nacional, grant SIP20211444. Many calculations were
done by Symbolic Program REDUCE 3.8..

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