# Anisotropic chiral cosmology: exact solutions

J. Socorro<EMAIL_ADDRESS>Departamento de Física, DCeI, Universidad
de Guanajuato-Campus León, C.P. 37150, León, Guanajuato, México  S. Pérez-
Payán<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de Ingeniería,
Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana #200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.  Abraham
Espinoza-García<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería, Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana #200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.  Luis Rey
Díaz-Barrón<EMAIL_ADDRESS>Unidad Profesional Interdisciplinaria de
Ingeniería, Campus Guanajuato del Instituto Politécnico Nacional.
Av. Mineral de Valenciana #200, Col. Fraccionamiento Industrial Puerto
Interior, C.P. 36275, Silao de la Victoria, Guanajuaṫo, México.

###### Abstract

In this work we investigate the anisotropic Bianchi type I cosmological model
in the chiral setup, in a twofold manner. Firstly, we consider a quintessence
plus a K-essence like model, where two scalar fields but only one potential
term is considered. Secondly, we look at a model where in addition to the two
scalar fields the two potential terms are taken into account as well as the
standard kinetic energy and the mixed term. Regarding this second model, it is
shown that two possible cases can be studied: a quintom like case and a
quintessence like case. In each of the models we were able to find both
classical and quantum analytical solutions.

##  I Introduction

The incorporation of scalar fields into cosmological models has helped explain
different phenomena of our Universe. For example, single scalar field
cosmological models have been used to describe the inflationary phase and the
late time acceleration of the Universe, also the dark matter component of the
Universe can be accounted as well as unification of the early inflation to the
late acceleration linde1982 ; Linde1983 ; Barrow1993_1 ; Barrow1993_2 ;
Peebles1987 ; Tsujikawa2013 ; Liddle1998 ; Sahni1999 ; Matos2000 ; Urena-
Lopez2016 ; Peebles1998 ; deHaro:2016_1 ; deHaro:2016_2 ; Elizalde:2004mq , to
mention a few.

In recent years multi-scalar field cosmological models have drawn tremendous
attention from the scientific community. In this cosmological scenario two or
more scalar fields are considered to describe adequately the evolution of the
Universe, and generally, the interaction of these scalar fields occur in the
potential, in the kinetic terms or both. The advantages of multi-field
cosmological models (over single field ones) is the introduction of new
degrees of freedom allowing the explanation of several physical phenomena. In
this setup, an inflationary stage can also be achieved Coley:1999mj ;
Copeland:1999cs , even in the case where the fields are non interacting
andrew2007 . Moreover, multi-scalar fields models can also be used to explain
the primordial inflation perturbations analysis Yokoyama:2007dw ; Chiba:2008rp
or the assisted inflation Copeland:1999cs ; andrew1998a . Another appealing
reason to work this models is that when two scalar fields are considered the
crossing of the cosmological constant boundary “-1” can be described, in the
litearature this models are known as quintom models Cai2009 ; Setare2008 ;
Lazkoz2007 ; Leon2018  (single scalar field models do not have this
malleability, since they only describe either the phantom or quintessence
regime). Furthermore, this multi-field models can also tackle the hybrid
inflation of the Universe, which gives an alternative graceful exit in
comparison to the standard inflationary picture chimento ; lindle ; cope ; kim
; omar-epjp2017 ; Wands2008 ; Bond2006 ; Inomata2017 . From a phenomenological
point of view, the most successful models are those that have incorporated
quintessence scalar fields, slow-roll inflation, chiral cosmology connected
with $f(R)$ theories and the nonlinear sigma model OSR ; Liddle1998 ; barrow ;
ferreira ; copeland1 ; copeland2 ; copeland3 ; andrew2007 ; gomez ; capone ;
kolb ; Vagnozzi ; chervon1995 ; Chervon2013 ; Kaiser2014 ; Fomin2017 ;
Chervon2019 ; Paliathanasis2019 ; Paliathanasis2020a ; Paliathanasis2020b ;
Bamba:2012cp ; Dimakis:2020tzc ; Dimakis:2019qfs ; Paliathanasis:2014yfa .

On the other hand, the hypothesis of primordial anisotropy at early stages of
the Universe (even predating inflation), is a tempting proposal that can point
in the right direction regarding the anomalies found in the cosmic microwave
background (CMB) anisotropies on large angular scales. Following these
guidelines, some attempts have been presented in Pereira:2007yy ;
Pitrou:2008gk ; Pereira:2015pga ; Gumrukcuoglu:2007bx , where anisotropic
cosmological models, mostly the Bianchi I model, have been used as a
background space-time in an early anisotropic but homogeneous Universe that
develops isotropization at the onset of inflation, however, the imprints of
such anisotropy would lead to the anomalies found in the thermal maps of the
CMB; and after the inflationary period as a consequence of this
isotropization, the Universe acquires a Friedmann-Lama$\rm\hat{i}$tre-
Robertson-Walker geometry recovering the standard picture of the evolution of
the Universe. Hence, anisotropic cosmological models represent an attractive
arena to test the early stages of the Universe, even if no conclusive evidence
that a primordial anisotropy is needed.

On this regard, multi-field anisotropic cosmological models of inflation have
been explored. In Folomeev:2007uw  the author delves into the study of the
Bianchi type I cosmology considering two interacting scalar fields and a
potential of the form $V(\phi,\chi)\sim\phi^{4}+\chi^{4}$, founding numerical
solutions as well as the asymptotically isotropic Friedmann case. Other
interesting works are presented in omar-epjp2017 ; socorro-libro , where the
potential, with structure $V(\phi,\sigma)\sim e^{\phi+\sigma}$ it is shown to
be a good viable candidate to address the inflationary era in both flat
isotropic and anisotropic space-times. More recently, in sor  the authors
present the case of the anisotropic Bianchi type I cosmology in the multi-
field setup with a potential of the form
$V_{0}e^{-(\lambda_{1}\phi_{1}+\cdots+\lambda_{n}\phi_{n})}$, founding
inflationary exact solutions in a quintessence framework. Additional research
regarding multi-filed anisotropic cosmological models can be found in
Andrianov:2015hba ; GALIAKHMETOV:2014zea ; Abbyazov:2013qqa ; Chervon:2015jji
; Cicciarella:2019ihh ; Herfray:2015fpa ; Leon:2020pfy ; Kaiser:2010ps ;
Kaiser:2013sna ; Chervon2013 ; Beesham:2013rya .

In the present work we present the anisotropic Bianchi type I cosmological
model with two scalar fields in a twofold manner. Following closely the
developments introduced in Socorro:2019vvh ; Socorro:2020nsm , first, we put
forward a simple quintessence plus a K-essence model which arises from
considering the interaction of the two scalar fields but only one term in the
potential. And second, a chiral approach is studied, here in addition to the
two scalar fields we also consider the two potential terms as well as the
standard kinetic energy and the mixed term. For each model, classical and
quantum analytical solutions are found.

This paper is arrange as follows. In section II we introduce the first model,
where the Einstein-Klein-Gordon (EKG) equations are calculated and the
Lagrangian and Hamiltonian approach is implemented in order to find the
corresponding solutions as well as the anisotropic parameters. In section III,
the second model presented, here, after obtaining the Hamiltonian density we
can distinguish two possible scenarios: a quintom like epoch and a
quintessence epoch. For both scenarios the corresponding solutions are
obtained. Section IV is devoted two implement the quantum versions of the
previous two models and the corresponding solutions are obtained. Finally,
section V is left for the final remarks.

##  II First model: quintessence plus K-essence

As mention above, we are going to start by analyzing the quintessence plus
K-essence model. For this purpose we are going to consider the multi-field
Bianchi type I model where the two scalar fields are taken into account but
only one potential term. The Lagrangian density for such a model is

$\rm{\cal
L}=\sqrt{-g}\left(R-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi_{1}\nabla_{\nu}\phi_{1}-\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi_{2}\nabla_{\nu}\phi_{2}+V(\phi_{1})\right)\,,$
(1)

where $\rm R$ is the Ricci scalar, $\rm V(\phi_{1})$ is the corresponding
scalar field potential, and the reduced Planck mass $M_{P}^{2}=1/8\pi G=1$.
The corresponding variations of Eq.(1), with respect to the metric and the
scalar fields give the EKG field equations

$\displaystyle\rm G_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\rm-\frac{1}{2}\left(\nabla_{\alpha}\phi_{1}\nabla_{\beta}\phi_{1}-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}\nabla_{\mu}\phi_{1}\nabla_{\nu}\phi_{1}\right)+\frac{1}{2}g_{\alpha\beta}\,V(\phi_{1})$
(2)
$\displaystyle\rm-\frac{1}{2}\left(\nabla_{\alpha}\phi_{2}\nabla_{\beta}\phi_{2}-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}\nabla_{\mu}\phi_{2}\nabla_{\nu}\phi_{2}\right),$
$\displaystyle\rm\Box\phi_{1}-\frac{\partial V}{\partial\phi_{1}}$
$\displaystyle=$ $\displaystyle\rm
g^{\mu\nu}{\phi_{1}}_{,\mu\nu}-g^{\alpha\beta}\Gamma^{\nu}_{\alpha\beta}\nabla_{\nu}\phi_{1}-\frac{\partial
V}{\partial\phi_{1}}=\rm 0\,,$ (4) $\displaystyle\rm
g^{\mu\nu}{\phi_{2}}_{,\mu\nu}-g^{\alpha\beta}\Gamma^{\nu}_{\alpha\beta}\nabla_{\nu}\phi_{2}=0.$

The line element for the anisotropic cosmological Bianchi type I model in the
Misner’s parametrization is

$\displaystyle\rm ds^{2}$ $\displaystyle=$
$\displaystyle\rm-N^{2}dt^{2}+a_{1}^{2}dx^{2}+a_{2}^{2}dy^{2}+a_{3}^{2}dz^{2},$
(5) $\displaystyle=$
$\displaystyle\rm-N^{2}dt^{2}+e^{2\Omega}\left[e^{2\beta_{+}+2\sqrt{3}\beta_{-}}dx^{2}+e^{2\beta_{+}-2\sqrt{3}\beta_{-}}dy^{2}+e^{-4\beta_{+}}dz^{2}\right],$

where $\rm a_{i}$ ($\rm i=1,2,3$) are the scale factor on directions
$\rm(x,y,z)$, respectively, and N is the lapse function. For convenience, and
in order to carry out the analytical calculations, we consider the following
representation for the line element (5)

$\rm
ds^{2}=-N^{2}dt^{2}+\eta^{2}\left[m_{1}^{2}dx^{2}+m_{2}^{2}dy^{2}+m_{3}^{2}dz^{2}\right],$
(6)

where the relations between both representations (5) and (6) are given by

$\displaystyle\rm\eta$ $\displaystyle=$ $\displaystyle\rm e^{\Omega},$
$\displaystyle\rm m_{1}$ $\displaystyle=$ $\displaystyle\rm
e^{\beta_{+}+\sqrt{3}\beta_{-}},\qquad\frac{\dot{m}_{1}}{m_{1}}=\dot{\beta}_{+}+\sqrt{3}\dot{\beta}_{-},$
$\displaystyle\rm m_{2}$ $\displaystyle=$ $\displaystyle\rm
e^{\beta_{+}-\sqrt{3}\beta_{-}},\qquad\frac{\dot{m}_{2}}{m_{2}}=\dot{\beta}_{+}-\sqrt{3}\dot{\beta}_{-},$
(7) $\displaystyle\rm m_{3}$ $\displaystyle=$ $\displaystyle\rm
e^{-2\beta_{+}},\qquad\qquad\frac{\dot{m}_{3}}{m_{3}}=-2\dot{\beta}_{+},$

where $\eta$ is a function that has information regarding the isotropic
scenario and the $\rm m_{i}$ are dimensionless functions that have information
about the anisotropic behavior of the Universe, such that

$\rm\prod_{i=1}^{3}m_{i}=1,\qquad\prod_{i=1}^{3}a_{i}=\eta^{3},\qquad\rm\sum_{i=1}^{3}\frac{\dot{m}_{i}}{m_{i}}=0,$
(8)

act as constraint equations for the model.

###  II.1 General Solutions to the Field Equations

In this subsection we present the solutions of the field equations for the
anisotropic cosmological model, considering the temporal evolution of the
scale factors with barotropic fluid and standard matter. The solutions
obtained already consider the particular choice of the Misner-like
transformation discussed lines above. Using the metric (6) and a co-moving
fluid, equations (2) take the following form

$\displaystyle\rm\left(\begin{tabular}[]{c}0\\\ 0\end{tabular}\right)$
$\displaystyle\rm\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{2}}{Nm_{2}}+\frac{\dot{m}_{2}}{Nm_{2}}\frac{\dot{m}_{3}}{Nm_{3}}+\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{3}}{Nm_{3}}+3\left(\frac{\dot{\eta}}{N\eta}\right)^{2}-8\pi
G\rho-\frac{1}{2}\left(\frac{1}{2}\frac{\dot{\phi}_{1}^{2}}{N^{2}}+V(\phi_{1})\right)-\frac{1}{4}\frac{\dot{\phi}_{2}^{2}}{N^{2}}=0,$
(11) $\displaystyle\rm\left(\begin{tabular}[]{c}1\\\ 1\end{tabular}\right)$
$\displaystyle\rm-\frac{\dot{N}}{N^{2}}\left[\frac{\dot{m}_{2}}{Nm_{2}}+\frac{\dot{m}_{3}}{Nm_{3}}+2\frac{\dot{\eta}}{N\eta}\right]+\frac{\ddot{m}_{2}}{N^{2}m_{2}}+\frac{\ddot{m}_{3}}{N^{2}m_{3}}+\frac{\dot{m}_{2}}{Nm_{2}}\frac{\dot{m}_{3}}{Nm_{3}}+2\frac{\ddot{\eta}}{N^{2}\eta}+\left(\frac{\dot{\eta}}{N\eta}\right)^{2}+3\frac{\dot{\eta}}{N\eta}\left[\frac{\dot{m}_{2}}{Nm_{2}}+\frac{\dot{m}_{3}}{Nm_{3}}\right]$
(15)
$\displaystyle\mbox{}\rm+\frac{1}{2}\left(\frac{1}{2}\frac{\dot{\phi}_{1}^{2}}{N^{2}}-V(\phi)\right)+\frac{1}{4}\frac{\dot{\phi}_{2}^{2}}{N^{2}}+8\pi
GP=0,$ $\displaystyle\rm\left(\begin{tabular}[]{c}2\\\ 2\end{tabular}\right)$
$\displaystyle\rm-\frac{\dot{N}}{N^{2}}\left[\frac{\dot{m}_{1}}{Nm_{1}}+\frac{\dot{m}_{3}}{Nm_{3}}+2\frac{\dot{\eta}}{N\eta}\right]+\frac{\ddot{m}_{1}}{N^{2}m_{1}}+\frac{\ddot{m}_{3}}{N^{2}m_{3}}+\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{3}}{Nm_{3}}+2\frac{\ddot{\eta}}{N^{2}\eta}+\left(\frac{\dot{\eta}}{N\eta}\right)^{2}+3\frac{\dot{\eta}}{N\eta}\left[\frac{\dot{m}_{1}}{Nm_{1}}+\frac{\dot{m}_{3}}{Nm_{3}}\right]$
(19)
$\displaystyle\mbox{}\rm+\frac{1}{2}\left(\frac{1}{2}\frac{\dot{\phi}_{1}^{2}}{N^{2}}-V(\phi_{1})\right)+\frac{1}{4}\frac{\dot{\phi}_{2}^{2}}{N^{2}}+8\pi
GP=0,$ $\displaystyle\rm\left(\begin{tabular}[]{c}3\\\ 3\end{tabular}\right)$
$\displaystyle\rm-\frac{\dot{N}}{N^{2}}\left[\frac{\dot{m}_{1}}{Nm_{1}}+\frac{\dot{m}_{2}}{Nm_{3}}+2\frac{\dot{\eta}}{N\eta}\right]+\frac{\ddot{m}_{1}}{N^{2}m_{1}}+\frac{\ddot{m}_{2}}{N^{2}m_{2}}+\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{2}}{Nm_{2}}+2\frac{\ddot{\eta}}{N^{2}\eta}+\left(\frac{\dot{\eta}}{N\eta}\right)^{2}+3\frac{\dot{\eta}}{N\eta}\left[\frac{\dot{m}_{1}}{Nm_{1}}+\frac{\dot{m}_{2}}{Nm_{2}}\right]$
(23)
$\displaystyle\mbox{}\rm+\frac{1}{2}\left(\frac{1}{2}\frac{\dot{\phi}_{1}^{2}}{N^{2}}-V(\phi_{1})\right)+\frac{1}{4}\frac{\dot{\phi}_{2}^{2}}{N^{2}}+8\pi
GP=0,$

here a dot ( $\dot{}$ ) represents a time derivative. The corresponding Klein-
Gordon (KG) equations are given by

$\displaystyle\rm\frac{\dot{N}}{N}\frac{\dot{\phi}_{1}^{2}}{N^{2}}-\frac{\dot{\phi}_{1}\ddot{\phi}_{1}}{N^{2}}-3\frac{\dot{\eta}}{\eta}\frac{\dot{\phi}_{1}^{2}}{N^{2}}-\dot{V}$
$\displaystyle=$ $\displaystyle\rm
0,\qquad\to\qquad\frac{d}{dt}\,Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{1}}\right)=\frac{N^{2}\dot{V}}{{\dot{\phi}_{1}}^{2}},$
$\displaystyle\rm\frac{\dot{N}}{N}\frac{\dot{\phi}_{2}^{2}}{N^{2}}-\frac{\dot{\phi}_{2}\ddot{\phi}_{2}}{N^{2}}-3\frac{\dot{\eta}}{\eta}\frac{\dot{\phi}_{2}^{2}}{N^{2}}$
$\displaystyle=$ $\displaystyle\rm
0,\qquad\to\qquad\frac{d}{dt}\,Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{2}}\right)=0,$
(24)

where from the last equation in (24) it is easy to see that the solution for
the scalar field $\rm\phi_{2}$ (in quadrature form) is given by

$\rm\Delta\phi_{2}=\phi_{20}\int\frac{N}{\eta^{3}}dt,$ (25)

with $\rm\phi_{20}$ an integration constant.

Now, subtracting (15) from the component (19) we obtain

$\rm\frac{\dot{N}}{N^{3}}\left[\frac{\dot{m}_{1}}{m_{1}}-\frac{\dot{m}_{2}}{m_{2}}\right]-\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{3}}{Nm_{3}}+\frac{1}{N^{2}}\left[\frac{\ddot{m}_{2}}{m_{2}}-\frac{\ddot{m}_{1}}{m_{1}}\right]+\frac{\dot{m}_{2}}{Nm_{2}}\frac{\dot{m}_{3}}{Nm_{3}}+3\frac{\dot{\eta}}{N^{2}\eta}\left[\frac{\dot{m}_{2}}{m_{2}}-\frac{\dot{m}_{1}}{m_{1}}\right]=0,$
(26)

noticing that

$\rm\frac{1}{N}\left[\frac{\dot{m}_{2}}{Nm_{2}}-\frac{\dot{m}_{1}}{Nm_{1}}\right]^{\bullet}=\frac{1}{N^{2}}\left[\frac{\ddot{m}_{2}}{m_{2}}-\frac{\ddot{m}_{1}}{m_{1}}\right]-\frac{1}{N^{2}}\left[\left(\frac{\dot{m}_{2}}{m_{2}}\right)^{2}-\left(\frac{\dot{m}_{1}}{m_{1}}\right)^{2}\right]+\frac{\dot{N}}{N^{3}}\left[\frac{\dot{m}_{1}}{m_{1}}-\frac{\dot{m}_{2}}{m_{2}}\right],$
(27)

equation (26) can be rearranged and written as (where ∙ also denotes a time
derivative)

$\rm\frac{1}{N}\left[\frac{\dot{m}_{2}}{Nm_{2}}-\frac{\dot{m}_{1}}{Nm_{1}}\right]^{\bullet}+3\frac{\dot{\eta}}{N\eta}\left[\frac{\dot{m}_{2}}{Nm_{2}}-\frac{\dot{m}_{1}}{Nm_{1}}\right]=0,$
(28)

finally, defining $\rm
R_{21}=\frac{\dot{m}_{2}}{Nm_{2}}-\frac{\dot{m}_{1}}{Nm_{1}}$, the last
equation can be casted as
$\rm\frac{\dot{R}_{21}}{R_{21}}+3\frac{\dot{\eta}}{\eta}=0,$ whose solution is
given by

$\rm R_{21}=\frac{\ell_{21}}{\eta^{3}},$ (29)

where $\rm\ell_{21}$ is an integration constant.

When we perform the same procedure with the other pair of equations, namely,
subtracting (19) from the component (23) one obtains

$\rm\frac{\dot{N}}{N^{3}}\left[\frac{\dot{m}_{2}}{m_{2}}-\frac{\dot{m}_{3}}{m_{3}}\right]-\frac{\dot{m}_{2}}{Nm_{2}}\frac{\dot{m}_{1}}{Nm_{1}}+\frac{1}{N^{2}}\left[\frac{\ddot{m}_{3}}{m_{3}}-\frac{\ddot{m}_{2}}{m_{2}}\right]+\frac{\dot{m}_{1}}{Nm_{1}}\frac{\dot{m}_{3}}{Nm_{3}}+3\frac{\dot{\eta}}{N^{2}\eta}\left[\frac{\dot{m}_{3}}{m_{3}}-\frac{\dot{m}_{2}}{m_{2}}\right]=0,$
(30)

which has the same structure as equation (26). Proceeding in the same manner
as we did above, we define $\rm
R_{32}=\frac{\dot{m}_{3}}{Nm_{3}}-\frac{\dot{m}_{2}}{Nm_{2}}$, obtaining a
differential equation whose solution is analogous to (29),

$\rm R_{32}=\frac{\ell_{32}}{\eta^{3}},$ (31)

where $\rm\ell_{32}$ is an integration constant. And lastly, subtracting (23)
from (15) we get

$\rm R_{13}=\frac{\ell_{13}}{\eta^{3}},$ (32)

$\rm\ell_{13}$ is also a constant that comes from integration, these three
constants satisfy $\rm\ell_{21}+\ell_{32}+\ell_{13}=0$.

From equation (29) we have that

$\rm
2\frac{\dot{m}_{2}}{Nm_{2}}-\frac{\dot{m}_{1}}{Nm_{1}}-\frac{\dot{m}_{2}}{Nm_{2}}=\frac{\ell_{21}}{\eta^{3}},$
(33)

and using the constraints from (8), the last equation reduces to

$\rm
2\frac{\dot{m}_{2}}{Nm_{2}}+\frac{\dot{m}_{3}}{Nm_{3}}=\frac{\ell_{21}}{\eta^{3}},$
(34)

finally, as a last step, utilizing (31) we get

$\rm
3\frac{\dot{m}_{2}}{Nm_{2}}+\frac{\ell_{32}}{\eta^{3}}=\frac{\ell_{21}}{\eta^{3}}.$
(35)

In order to investigate the solution for the last equation we cast it in the
following form

$\rm\frac{\dot{m}_{2}}{Nm_{2}}=\frac{\ell_{21}-\ell_{32}}{3\eta^{3}}=\frac{\ell_{2}}{\eta^{3}},$
(36)

where $\ell_{2}=\frac{\ell_{21}-\ell_{32}}{3}$. The other components can be
obtained in a similar fashion, which read

$\displaystyle\rm\frac{\dot{m}_{3}}{Nm_{3}}=\frac{\ell_{3}}{\eta^{3}},$ (37)
$\displaystyle\rm\frac{\dot{m}_{1}}{Nm_{1}}=\frac{\ell_{1}}{\eta^{3}},$ (38)

the constants being $\rm\ell_{3}=\frac{\ell_{32}-\ell_{13}}{3}$ and
$\rm\ell_{1}=\frac{\ell_{13}-\ell_{21}}{3}$, also, these constants satisfy
$\sum_{j=1}^{3}\ell_{j}=0$. Now that equations (36)-(38) are written in a more
manageable way, obtaining the solutions is straightforward, these are given by

$\rm m_{i}(t)=\delta_{i}Exp\left[\ell_{i}\int\frac{Ndt}{\eta^{3}}\right],$
(39)

where $\rm\Pi_{j=1}^{3}\delta_{j}=1$, setting the gauge $\rm N\to\eta^{3}$,
the solution becomes

$\rm m_{i}(t)\to\alpha_{i}Exp\left[\ell_{i}\Delta t\right].$ (40)

Unfortunately under this approach we could not find analytical solution for
$\eta$, because we need to know the solution for the scalar field $\phi_{1}$
(see equation (11)). To be able to reach a solution we are going to resort to
the Hamiltonian formalism. To this end, we employ equation (1) and the line
element (6), now the Lagrangian density with the scalar field $\rm
V(\phi_{1})=V_{1}e^{-\lambda_{1}\phi_{1}}$ becomes

$\rm{\cal{L}}=\rm\eta^{3}\left(\frac{6}{N}\left(\frac{\dot{\eta}}{\eta}\right)^{2}-\frac{1}{N}\left[\left(\frac{\dot{m}_{1}}{m_{1}}\right)^{2}+\left(\frac{\dot{m}_{2}}{m_{2}}\right)^{2}+\left(\frac{\dot{m}_{3}}{m_{3}}\right)^{2}\right]-\frac{\dot{\phi_{1}}^{2}}{2N}-\frac{\dot{\phi_{2}}^{2}}{2N}+NV_{1}e^{-\lambda_{1}\phi_{1}}\right)\,,$
(41)

where the momenta are

$\begin{split}\rm\Pi_{\eta}&=\rm 12\frac{\eta}{N}\dot{\eta}\\\
\rm\Pi_{\phi_{1}}&=\rm-\frac{\eta^{3}}{N}\dot{\phi}_{1}\\\
\rm\Pi_{\phi_{2}}&=\rm-\frac{\eta^{3}}{N}\dot{\phi}_{2},\\\
\rm\Pi_{1}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{1}}{m_{1}^{2}}\right),\\\
\rm\Pi_{2}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{2}}{m_{2}^{2}}\right),\\\
\rm\Pi_{3}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{3}}{m_{3}^{2}}\right),\end{split}\qquad\begin{split}\rm\dot{\eta}&=\rm\frac{N}{12\eta}\Pi_{\eta},\\\
\rm\dot{\phi}_{1}&=\rm-\frac{N}{\eta^{3}}\Pi_{\phi_{1}}\\\
\rm\dot{\phi}_{2}&=\rm-\frac{N}{\eta^{3}}\Pi_{\phi_{2}},\\\
\rm\dot{m}_{1}&=\rm-\frac{Nm_{1}^{2}\Pi_{1}}{2\eta^{3}},\\\
\rm\dot{m}_{2}&=\rm-\frac{Nm_{2}^{2}\Pi_{2}}{2\eta^{3}},\\\
\rm\dot{m}_{3}&=\rm-\frac{Nm_{3}^{2}\Pi_{3}}{2\eta^{3}},\end{split}$ (42)

leading to the Hamiltonian density, which takes the form

$\rm{\cal
H}=\frac{1}{24\eta}\Pi_{\eta}^{2}-\frac{1}{4\eta^{3}}m_{1}^{2}\Pi_{1}^{2}-\frac{1}{4\eta^{3}}m_{2}^{2}\Pi_{2}^{2}-\frac{1}{4\eta^{3}}m_{3}^{2}\Pi_{3}^{2}-\frac{1}{2\eta^{3}}\Pi_{\phi_{1}}^{2}-\frac{1}{2\eta^{3}}\Pi_{\phi_{2}}^{2}-V_{1}\eta^{3}e^{-\lambda_{1}\phi_{1}}\,.$
(43)

Making the transformation $\rm\Pi_{\eta}=\frac{\partial S}{\partial\eta}$, and
$\rm\Pi_{i}=\frac{\partial S}{\partial m_{i}}$ and choosing $\eta=e^{u}$ and
$\rm m_{i}=e^{u_{i}}$, where $\rm P_{i}=\frac{\partial S}{\partial u_{i}}$ and
$\pi_{u}=\frac{\partial S}{\partial u}$, the Hamiltonian density becomes

$\rm{\cal
H}=\frac{e^{-3u}}{24}\left[\pi_{u}^{2}-6P_{1}^{2}-6P_{2}^{2}-6P_{3}^{2}-12\Pi_{\phi_{1}}^{2}-12\Pi_{\phi_{2}}^{2}-U(u,\phi_{1})\right],$
(44)

where $\rm U(u,\phi_{1})=24V_{1}e^{6u-\lambda_{1}\phi_{1}}$ is the potential
function. In the gauge $\rm N=24e^{3u}$, the Hamilton equations are

$\begin{split}\rm\dot{u}&=\rm 2\pi_{u},\\\ \rm\dot{\pi}_{u}&=\rm
6U,\end{split}\qquad\begin{split}\dot{\phi}_{1}&=\rm-24\Pi_{\phi_{1}},\\\
\dot{\phi}_{2}&=\rm-24\Pi_{\phi_{2}},\end{split}\qquad\begin{split}\rm\dot{\Pi}_{\phi_{1}}&=\rm-\lambda_{1}U,\\\
\rm\dot{\Pi}_{\phi_{2}}&=0,\end{split}\qquad\begin{split}\rm\dot{u}_{i}&=\rm-12P_{i},\\\
\rm\dot{P}_{i}&=0.\end{split}$ (45)

From the Hamilton equations Eq.(45), we can find relations between the scale
factor and the scalar fields, such as

$\displaystyle\rm\dot{\phi}_{1}=\rm-24\Pi_{\phi_{1}}=4\lambda_{1}\pi_{u}+24p_{\phi_{1}}=2\lambda_{1}\dot{u}+24p_{\phi_{1}},$
$\displaystyle\rm\dot{\phi}_{2}=\rm-24\Pi_{\phi_{2}}=24p_{\phi_{2}},$ (46)
$\displaystyle\rm\dot{u}_{i}=\rm-12p_{i},$

where $\rm p_{\phi_{1}}$, $\rm p_{\phi_{2}}$ and $\rm p_{i}$ are integration
constants to be determined by suitable conditions. The solutions of equations
(II.1) read

$\displaystyle\rm\Delta\phi_{1}$ $\displaystyle=$ $\displaystyle\rm
2\lambda_{1}\Delta u+24p_{\phi_{1}}\Delta t,$ (47)
$\displaystyle\rm\Delta\phi_{2}$ $\displaystyle=$ $\displaystyle\rm
24p_{\phi_{2}}\Delta t\,,$ (48) $\displaystyle\rm\Delta u_{i}$
$\displaystyle=$ $\displaystyle\rm 12p_{i}\Delta t,$ (49) $\displaystyle\rm
m_{i}$ $\displaystyle=$ $\displaystyle\rm\beta_{i}e^{-12p_{i}\Delta t},$ (50)

with the constraints between the constants $\rm p_{i}$ that must be fulfil:
$\rm\sum_{i=1}^{3}p_{i}=0$ and
$\rm\sum_{i=1}^{3}p_{i}^{2}=2(p_{2}^{2}+p_{2}p_{3}+p_{3}^{2})$. Equations
(47-50) are expressions similar to the solution found by algebraic
manipulation to (24) for the scalar field $\rm\phi_{2}$ and the Einstein
equation (40) for the $\rm m_{i}$ functions. These expressions are indeed
general relations, since they satisfy the EKG equations Eqs.(11-24).

On the other hand, taking into account the constraint $\rm{\cal H}=0$, we
obtain the temporal dependence for $\rm\pi_{u}(t)$ which allows us to
construct a master equation:

$\rm\frac{d\pi_{u}}{\alpha_{1}\pi_{u}^{2}-\alpha_{2}\pi_{u}-\alpha_{3}}=dt\,,$
(51)

where the parameters $\rm\alpha_{i}$ with $i=1,2,3$, are

$\rm\alpha_{1}=2(3-\lambda_{1}^{2})=2\beta\,,\quad\alpha_{2}=24\lambda_{1}p_{\phi_{1}}\,,\quad\alpha_{3}=72\left[p_{\phi_{1}}^{2}+c^{2}\right]\,,\quad\rm
and\quad c^{2}=p_{\phi_{2}}^{2}+p_{2}^{2}+p_{2}p_{3}+p_{3}^{2}.$ (52)

In the next subsections, we present solutions for three different values of
the parameter $\lambda_{1}$ and also are able to construct the anisotropic
parameters.

###  II.2 Case $\alpha_{1}>0$ and $\lambda_{1}<\sqrt{3}$

For this case, we have that the solution for $\rm\pi_{u}(t)$ is given by

$\rm\pi_{u}=\frac{1}{4\beta}\left[\alpha_{2}-\alpha
Coth\left(\frac{\alpha}{2}t\right)\right]\,,$ (53)

where $\rm\alpha=24\omega_{1}$ with
$\omega_{1}=\sqrt{3p_{\phi_{1}}^{2}+c^{2}\beta}$. The solutions of the set of
variables $\rm(u,u_{i},\phi_{1},\phi_{2})$ and
$\rm(\Pi_{\phi_{1}},\Pi_{\phi_{2}},P_{i})$ are:

$\displaystyle\rm
u=u_{0}+12\frac{\lambda_{1}p_{\phi_{1}}}{\beta}t+\ln{\left[Csch{\left(12\omega_{1}t\right)}\right]}^{1/\beta}\,,$
(54)
$\displaystyle\rm\phi_{1}=p_{\phi_{1_{0}}}+72\frac{p_{\phi_{1}}}{\beta}t-Ln{\left[Sinh{\left(12\omega_{1}t\right)}\right]}^{2\lambda_{1}/\beta}\,,$
(55) $\displaystyle\rm\phi_{2}=p_{\phi_{2_{0}}}+24p_{\phi_{2}}t\,,$ (56)
$\displaystyle\rm u_{i}=-12p_{i}\Delta t\,,$ (57)
$\displaystyle\rm\Pi_{\phi_{1}}=-3\frac{p_{\phi_{1}}}{\beta}+\frac{\lambda_{1}\alpha}{24\beta}coth\left(\frac{\alpha}{2}t\right)\,,$
(58) $\displaystyle\rm\Pi_{\phi_{2}}=-p_{\pi_{2}}\,,$ (59) $\displaystyle\rm
P_{i}=p_{i}\,,$ (60)

here ($\rm u_{0},p_{\phi_{1_{0}}},p_{\phi_{2_{0}}},p_{i}$) are integration
constants. Finally the scale factor $\rm\eta=e^{u}$ and the anisotropic
parameters take the form

$\displaystyle\rm\eta$ $\displaystyle=$
$\displaystyle\rm\eta_{0}Exp\left[12\frac{\lambda_{1}p_{\phi_{1}}}{\beta}t\right]\,Csch^{\frac{1}{\beta}}{\left(12\sqrt{3p_{\phi_{1}}^{2}+p_{\phi_{2}}^{2}\beta}t\right)}\,,$
$\displaystyle\rm m_{i}(t)$ $\displaystyle=$
$\displaystyle\rm\beta_{i}Exp\left[-12p_{i}\Delta t\right],$ (61)

where $\rm\eta_{0}=e^{u_{0}},\sum_{i=1}^{3}p_{i}=0$ and
$\Pi_{i=1}^{3}\beta_{i}=1$.

###  II.3 Case $\alpha_{1}<0$ and $\lambda_{1}>\sqrt{3}$

In this instance its appropriate to take the relation between the momenta

$\rm\Pi_{\phi_{1}}=-\frac{\lambda_{1}}{6}\pi_{u}+p_{\phi_{1}},\qquad\rm
and\qquad\Pi_{\phi_{2}}=-p_{\phi_{2}}=constant,$ (62)

then the constant $\rm-\alpha_{2}=24\lambda_{1}p_{\phi_{1}}$, allowing us to
obtain the temporal dependence for $\rm\pi_{u}(t)$ with which a master
equation can be constructed

$\rm\frac{d\pi_{u}}{-\alpha_{1}\pi_{u}^{2}+\alpha_{2}\pi_{u}-\alpha_{3}}=dt\,,$
(63)

where we have included the minus sign such the constant
$\rm\alpha_{1}=2(\lambda_{1}^{2}-3)=2\beta>0$. Then, defining
$\rm\omega_{1}^{2}=\alpha_{2}^{2}-8\beta\alpha_{3}=576\omega_{2}^{2}$ with
$\omega_{2}^{2}=3\rm p_{\phi_{1}}^{2}-(\lambda_{1}^{2}-3)c^{2}$, we can
rewrite (63) as

$\frac{8\beta\,d\pi_{u}}{\omega_{1}^{2}-\left(4\beta\pi_{u}-24\lambda_{1}\rm
p_{\phi_{1}}\right)^{2}}=\rm dt,$ (64)

where the constraint over the parameters $\rm
p_{\phi_{1}}>c\sqrt{\left(\frac{\lambda_{1}}{\sqrt{3}}\right)^{2}-1}$ must be
satisfied. In order to be able to integrate Eq.(64), as a final step, we
resort to the change of variables $\rm
z=4\beta\pi_{u}-24\lambda_{1}p_{\phi_{1}}$, thus, the solution for the momenta
$\rm\pi_{u}(t)$ becomes

$\rm\pi_{u}=\frac{6\lambda_{1}p_{\phi_{1}}}{\beta}+\frac{6\omega_{2}}{\beta}Tanh\left(12\omega_{2}(t-t_{0})\right).$
(65)

Using the relations from Eq.(45) and after some algebra, the solutions for the
set of variables $\rm(u,\phi_{1},\phi_{2})$ and
$\rm(\Pi_{\phi_{1}},\Pi_{\phi_{2}})$ are:

$\displaystyle\rm
u=u_{0}+\frac{12\lambda_{1}p_{\phi_{1}}}{\beta}(t-t_{0})+\frac{1}{\beta}\,Ln\left[Cosh\left(12\omega_{2}(t-t_{0})\right)\right]\,,$
(66) $\displaystyle\rm u_{i}=-12p_{i}\Delta t\,,$ (67)
$\displaystyle\rm\phi_{1}=\phi_{1_{0}}+72\frac{p_{\phi_{1}}}{\beta}(t-t_{0})+\frac{2\lambda_{1}}{\beta}\,Ln\left[Cosh\left(12\omega_{2}(t-t_{0})\right)\right]\,,$
(68) $\displaystyle\rm\phi_{2}=\phi_{2_{0}}+24p_{\phi_{2}}(t-t_{0}),$ (69)
$\displaystyle\rm\Pi_{\phi_{1}}=-\frac{3p_{\phi_{1}}}{\beta}-\frac{\lambda_{1}\omega_{2}}{\beta}Tanh\left(12\omega_{2}(t-t_{0})\right)\,,$
(70) $\displaystyle\rm\Pi_{\phi_{2}}=-p_{\phi_{2}}\,,$ (71)

where ($\rm u_{0},\phi_{1_{0}},\phi_{2_{0}},p_{i}$) are all integration
constants. Finally the scale factor becomes

$\rm\eta=\eta_{0}\,Exp\left[\frac{12\lambda_{1}p_{\phi_{1}}}{\beta}(t-t_{0})\right]\,\,Cosh^{\frac{1}{\beta}}\left(12\omega_{2}(t-t_{0})\right),$
(72)

with $\rm\eta_{0}=e^{u_{0}}$ and the anisotropic dimensionless function is

$\rm m_{i}(t)=\rm\beta_{i}Exp\left[-12p_{i}\Delta t\right],$ (73)

we can see that Eq.(73) has the same functional form as before (Eq.(II.2)).

###  II.4 Case $\alpha_{1}=0$ and $\lambda_{1}^{2}=3.$

For this case the coefficient $\rm\alpha_{1}=0$ and the master equation to
solve is reduced to

$\rm\int\frac{d\pi_{u}}{\alpha_{2}\pi_{u}-\alpha_{3}}=\int dt\,,$ (74)

thus the solution for $\rm\pi_{u}(t)$ can be obtained relatively easily, which
read

$\rm\pi_{u}(t)=\frac{\alpha_{3}}{\alpha_{2}}+pe^{\alpha_{2}(t-t_{0})}\,,$ (75)

where $\rm p$ is an integration constant. As before, we can use relations from
Eq.(45) and after some manipulation, the solutions for
$\rm(u,\phi_{1},\phi_{2})$ and $\rm(\Pi_{\phi_{1}},\Pi_{\phi_{2}})$ are:

$\displaystyle\rm
u=u_{0}+2\sqrt{3}\frac{p_{\phi_{1}}^{2}+c^{2}}{p_{\phi_{1}}}(t-t_{0})+\frac{\sqrt{3}p}{36p_{\phi_{1}}}e^{24\sqrt{3}p_{\phi_{1}}(t-t_{0})},$
(76) $\displaystyle\rm u_{i}=-12p_{i}\Delta t\,,$ (77)
$\displaystyle\rm\phi_{1}=\phi_{1_{0}}+12\frac{c^{2}-p_{\phi_{1}}^{2}+}{p_{\phi_{1}}}(t-t_{0})+\frac{p}{6}e^{24\sqrt{3}p_{\phi_{1}}(t-t_{0})}\,,$
(78) $\displaystyle\rm\phi_{2}=\phi_{2_{0}}+24p_{\phi_{2}}(t-t_{0})\,,$ (79)
$\displaystyle\rm\Pi_{\phi_{1}}=\frac{1}{2}\frac{p_{\phi_{1}}^{2}-c^{2}+}{p_{\phi_{1}}}-\frac{\sqrt{3}p}{6}e^{24\sqrt{3}p_{\phi_{1}}(t-t_{0})}\,,$
(80) $\displaystyle\rm\Pi_{\phi_{2}}=-p_{\phi_{2}},$ (81)

again ($\rm u_{0},\phi_{1_{0}},\phi_{2_{0}},p_{i}$) are all integration
constants. Finally the scale factor $\rm\eta(t)$ for this case is

$\rm\eta=\eta_{0}Exp\left[2\sqrt{3}\frac{p_{\phi_{1}}^{2}+c^{2}}{p_{\phi_{1}}}(t-t_{0})\right]\,Exp\left[\frac{\sqrt{3}p}{36p_{\phi_{1}}}e^{24\sqrt{3}p_{\phi_{1}}(t-t_{0})}\right]\,,$
(82)

where $\rm\eta_{0}=e^{u_{0}}$, and as before, the anisotropic dimensionless
function $\rm m_{i}(t)$ is the same as in (II.2).

###  II.5 Anisotropic Parameters

In anisotropic cosmology, the Hubble parameter $\rm H$ is defined in analogy
with the FRW cosmology, that is

$\rm
H=\frac{\dot{a}}{a}=\frac{\dot{\eta}}{\eta}=\frac{1}{3}\left(H_{x}+H_{y}+H_{z}\right),$
(83)

where $\rm H_{x}=\frac{\dot{a}_{1}}{a_{1}}$, $\rm
H_{y}=\frac{\dot{a}_{2}}{a_{2}}$, and $\rm H_{z}=\frac{\dot{a}_{3}}{a_{3}}$.

The scalar expansion $\rm\theta$, the shear scalar $\rm\sigma^{2}$ and the
average anisotropic parameter $\rm\overline{A}_{m}$ are defined as

$\rm\theta=\sum_{i=1}^{3}\frac{\dot{a}_{i}}{a_{i}}=3H,\qquad\sigma^{2}=\frac{1}{2}\left(\sum_{i=1}^{3}H_{i}^{2}-\frac{1}{3}\theta^{2}\right),\qquad\overline{A}_{m}=\rm\frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2},$
(84)

respectively.

Following (Tripathy2012, ), we consider the volume deceleration parameter,

$\rm q(t)=-\frac{v\ddot{v}}{\dot{v}^{2}},$ (85)

where $\rm v=\eta^{3}=a_{1}a_{2}a_{3}$ is the (isotropic) volume function of
the Bianchi type I model, which is in this case given by the exact solution
presented for each case in the $\lambda_{1}$ parameter, giving

$\rm
q(t)=\left\\{\begin{tabular}[]{lr}$\rm-1-\frac{1}{3}\frac{\beta\omega_{1}^{2}}{\left(\lambda_{1}p_{\phi_{1}}Sinh(12\omega_{1}\Delta
t)-\omega_{1}Cosh(12\omega_{1}\Delta t)\right)^{2}}$,&for
$\lambda_{1}<\sqrt{3}$\\\ \\\
$\rm-1-\frac{1}{3}\frac{\beta\omega_{2}^{2}}{\left(\lambda_{1}p_{\phi_{1}}Cosh(12\omega_{1}\Delta
t)+\omega_{2}Sinh(12\omega_{1}\Delta t)\right)^{2}}$,&for
$\lambda_{1}>\sqrt{3}$\\\ \\\
$-1-\frac{4\sqrt{3}pp_{\phi_{1}}^{3}e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}}{\left(pp_{\phi_{1}}e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}+\sqrt{3}\left[p_{\phi_{1}}^{2}+c^{2}\right]\right)^{2}}$,&for
$\lambda_{1}=\sqrt{3}$\end{tabular}\right.$ (86)

these observations indicate that the Universe presents a volume accelerated
expansion in the inflationary epoch. Fig.(1) shows the the temporal evolution
of the deceleration parameter, where $q_{1},q_{2}$ and $q_{3}$ stand for the
solutions $\lambda_{1}<\sqrt{3}$, $\lambda_{1}>\sqrt{3}$,
$\lambda_{1}=\sqrt{3}$, respectively.

Figure 1: Deceleration parameter for the three classical solutions. Here we
have taken $\lambda_{1}=0.5$, $\lambda_{1}^{\prime}=2$, $\rm
p_{\phi_{1}}=0.4$, $\rm p_{\phi_{2}}=0.2$, $p_{2}=p_{3}=0.01$ and p=0.7.

Using the results for the average scale factor $\eta$ and the dimensionless
anisotropic functions $\rm m_{i}$, the average anisotropic parameter is

$\rm\overline{A}_{m}=\left\\{\begin{tabular}[]{lr}$\rm\frac{8}{3}\frac{\beta^{2}(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2})Cosh^{2}(12\omega_{1}\Delta
t)}{\left(\omega_{1}Cosh(12\omega_{1}\Delta
t)-\lambda_{1}p_{\phi_{1}}Sinh(12\omega_{1}\Delta t)\right)^{2}}$,&for
$\lambda_{1}<\sqrt{3}$\\\ \\\
$\rm\frac{8}{3}\frac{\beta^{2}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)Cosh^{2}(12\omega_{2}\Delta
t)}{\left(\omega_{2}\,Sinh(12\omega_{2}\Delta
t)+\lambda_{1}p_{\phi_{1}}Cosh(12\omega_{2}\Delta t)\right)^{2}}$ ,&for
$\lambda_{1}>\sqrt{3}$\\\ \\\
$\frac{96p_{\phi_{1}}^{2}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)}{\left(pp_{\phi_{1}}e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}+\sqrt{3}\left[p_{\phi_{1}}^{2}+c^{2}\right]\right)^{2}}$.&for
$\lambda_{1}=\sqrt{3}$\end{tabular}\right.$ (87)

The other two parameters acquire the form

$\rm\theta=\left\\{\begin{tabular}[]{lr}$\rm\frac{36}{\beta}\left(\lambda_{1}p_{\phi_{1}}-\omega_{1}Ctgh(12\omega_{1}\Delta
t)\right)$,&for $\lambda_{1}<\sqrt{3}$,\\\ \\\
$\rm\frac{36}{\beta}\left[\lambda_{1}p_{\phi_{1}}+\omega_{2}Tanh(12\omega_{2}\Delta
t)\right]$,&for $\lambda_{1}>\sqrt{3}$\\\ \\\ $\rm
6\left(p\,e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}+\sqrt{3}\frac{p_{\phi_{1}}^{2}+c^{2}}{p_{\phi_{1}}}\right)$,&for
$\lambda_{1}=\sqrt{3}$\end{tabular}\right.$ (88)
$\rm\sigma^{2}=\left\\{\begin{tabular}[]{lr}$\rm
648\frac{\frac{8}{9}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)Sinh^{2}(12\omega_{1}\Delta
t)-\left(\omega_{1}Cosh(12\omega_{1}\Delta
t)-\lambda_{1}p_{\phi_{1}}Sinh(12\omega_{1}\Delta
t)\right)^{2}}{\beta^{2}Sinh^{2}(12\omega_{1}\Delta t)}$,&for
$\lambda_{1}<\sqrt{3}$\\\ \\\ $\rm
648\frac{\frac{8}{9}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)Cosh^{2}(12\omega_{1}\Delta
t)-\left(\omega_{2}Sinh(12\omega_{1}\Delta
t)+\lambda_{1}p_{\phi_{1}}Cosh(12\omega_{1}\Delta
t)\right)^{2}}{\beta^{2}\,Cosh^{2}(12\omega_{2}\Delta t)}$,&for
$\lambda_{1}>\sqrt{3}$\\\ \\\ $\rm
18\left\\{32\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)-\left[\frac{p\,p_{\phi_{1}}e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}+\sqrt{3}\left(p_{\phi_{1}}^{2}+c^{2}\right)}{p_{\phi_{1}}}\right]^{2}\right\\}$,&for
$\lambda_{1}=\sqrt{3}$\end{tabular}\right.$ (89)

in Pradhan:2010dm  an references therein, the authors pin down that the red-
shift studies place the limit $\sigma/\theta\leq 0.3$ on the ratio of shear
$\sigma$ to Hubble constant $H$ in the neighborhood of our Galaxy today in
order to have a sufficiently isotropic cosmological model, in this regard we
obtain

$\rm\frac{\sigma^{2}}{\theta^{2}}=\left\\{\begin{tabular}[]{lr}$\rm-\frac{1}{2}+\frac{4}{9}\frac{\beta^{2}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)Sinh^{2}(12\omega_{1}\Delta
t)}{\left(\omega_{1}Cosh(12\omega_{1}\Delta
t)-\lambda_{1}p_{\phi_{1}}\,Sinh(12\omega_{1}\Delta t)\right)^{2}}$,&for
$\lambda_{1}<\sqrt{3}$\\\ \\\
$\rm-\frac{1}{2}+\frac{4}{9}\frac{\beta^{2}(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2})\,Cosh^{2}(12\omega_{2}\Delta)t}{\left(\lambda_{1}p_{\phi_{1}}\,Cosh(12\omega_{2}\Delta
t)+\omega_{2}Sinh(12\omega_{2}\Delta t)\right)^{2}}$,&for
$\lambda_{1}>\sqrt{3}$\\\ \\\
$\rm-\frac{1}{2}+\frac{16p_{\phi_{1}}^{2}\left(\ell_{2}^{2}+\ell_{2}\ell_{3}+\ell_{3}^{2}\right)}{\left[p\,p_{\phi_{1}}e^{24\sqrt{3}p_{\phi_{1}}\Delta
t}+\sqrt{3}\left(p_{\phi_{1}}^{2}+c^{2}\right)\right]^{2}}$,&for
$\lambda_{1}=\sqrt{3}$\end{tabular}\right.$ (90)

from (87) and (90) we can constraint the average anisotropic parameter
$\rm\overline{A}_{m}$ to the following value for both $\lambda_{1}>\sqrt{3}$
and $\lambda_{1}=\sqrt{3}$: $\rm\overline{A}_{m}\leq 3.54$, signaling that the
anisotropic phase still continues.

##  III Second model: chiral anisotropic model

Now we turn our attention to the second model to be considered. In this case
in addition to the two scalar fields we also consider the two potential terms
as well as the standard kinetic energy and the mixed term. The action for such
a Universe is

$\rm{\cal
L}=\sqrt{-g}\left(R-\frac{1}{2}g^{\mu\nu}m^{ab}\nabla_{\mu}\phi_{a}\nabla_{\nu}\phi_{b}+V(\phi_{1},\phi_{2})\right)\,,$
(91)

where $\rm R$ is the Ricci scalar, $\rm V(\phi_{1},\phi_{2})$ is the
corresponding scalar field potential, and $\rm m^{ab}$ is a $2\times 2$
constant matrix and $\rm m^{12}=m^{21}$. The EKG equations are obtained
varying Eq.(91) with respect to the metric and the scalar fields, resulting in

$\rm
G_{\alpha\beta}=\rm-\frac{1}{2}m^{ab}\left(\nabla_{\alpha}\phi_{a}\nabla_{\beta}\phi_{b}-\frac{1}{2}g_{\alpha\beta}g^{\mu\nu}\nabla_{\mu}\phi_{a}\nabla_{\nu}\phi_{b}\right)+\frac{1}{2}g_{\alpha\beta}\,V(\phi_{1},\phi_{2}),$
(92) $\rm m^{ab}\Box\phi_{b}-\frac{\partial V}{\partial\phi_{a}}=\rm
m^{ab}g^{\mu\nu}{\phi_{b}}_{,\mu\nu}-m^{ab}g^{\alpha\beta}\Gamma^{\nu}_{\alpha\beta}\nabla_{\nu}\phi_{b}-\frac{\partial
V}{\partial\phi_{a}}=\rm 0,\,\qquad a,b=1,2.$ (93)

Consequently the Klein-Gordon equations are

$\displaystyle\rm
m^{11}{\phi_{1}^{\prime\prime}}{\phi_{1}^{\prime}}+m^{12}{\phi_{2}^{\prime\prime}}{\phi_{1}^{\prime}}+3\frac{\eta^{\prime}}{\eta}\left(m^{11}{\phi_{1}^{\prime}}^{2}+m^{12}\phi_{1}^{\prime}\phi_{2}^{\prime}\right)+\left(\dot{V}\right)_{\phi_{2}}$
$\displaystyle=$ $\displaystyle\rm 0\,,$ (94) $\displaystyle\rm
m^{22}{\phi_{2}^{\prime\prime}}{\phi_{2}^{\prime}}+m^{12}{\phi_{1}^{\prime\prime}}{\phi_{2}^{\prime}}+3\frac{\eta^{\prime}}{\eta}\left(m^{22}{\phi_{2}^{\prime}}^{2}+m^{12}\phi_{1}^{\prime}\phi_{2}^{\prime}\right)+\left(\dot{V}\right)_{\phi_{1}}$
$\displaystyle=$ $\displaystyle\rm 0,\,$ (95)

here $\prime=d/d\tau$, $\rm d\tau=Ndt$ and
$\left(\rm\dot{V}\right)_{\phi_{i}}$ (with i=1,2) means that the derivative is
calculated maintaining $\phi_{i}$ constant. An equivalent form to write
equations (94) and (95) is

$\displaystyle\rm
m^{11}\dot{\phi}_{1}\frac{d}{dt}Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{1}}\right)+m^{12}\dot{\phi}_{2}\frac{d}{dt}Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{2}}\right)$
$\displaystyle=$
$\displaystyle\rm\frac{N^{2}\left(\dot{V}\right)_{\phi_{2}}}{\dot{\phi}_{1}},$
(96) $\displaystyle\rm
m^{12}\dot{\phi}_{1}\frac{d}{dt}Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{1}}\right)+m^{22}\dot{\phi}_{2}\frac{d}{dt}Ln\left(\frac{N}{\eta^{3}\dot{\phi}_{2}}\right)$
$\displaystyle=$
$\displaystyle\rm\frac{N^{2}\left(\dot{V}\right)_{\phi_{1}}}{\dot{\phi}_{2}}.$
(97)

Taking the metric (5) and pluging it into (91), the Lagrangian density becomes

$\displaystyle\rm{\cal{L}}$ $\displaystyle=$
$\displaystyle\rm\eta^{3}\left(\frac{6}{N}\left(\frac{\dot{\eta}}{\eta}\right)^{2}-\frac{1}{N}\left[\left(\frac{\dot{m}_{1}}{m_{1}}\right)^{2}+\left(\frac{\dot{m}_{2}}{m_{2}}\right)^{2}+\left(\frac{\dot{m}_{3}}{m_{3}}\right)^{2}\right]\right.$
(98)
$\displaystyle\rm\left.-m^{11}\frac{\dot{\phi_{1}}^{2}}{2N}-m^{22}\frac{\dot{\phi_{2}}^{2}}{2N}-m^{12}\frac{\dot{\phi}_{1}\dot{\phi}_{2}}{N}+N\left[V_{1}e^{-\lambda_{1}\phi_{1}}+V_{2}e^{-\lambda_{2}\phi_{2}}\right]\right)\,,$

the momenta are

$\begin{split}\rm\Pi_{\eta}&=\rm 12\frac{\eta}{N}\dot{\eta},\\\
\rm\Pi_{\phi_{1}}&=\rm-\frac{\eta^{3}}{N}\left(m^{11}\dot{\phi}_{1}+m^{12}\dot{\phi}_{2}\right),\\\
\rm\Pi_{\phi_{2}}&=\rm-\frac{\eta^{3}}{N}\left(m^{22}\dot{\phi}_{2}+m^{12}\dot{\phi}_{1}\right),\\\
\rm\Pi_{1}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{1}}{m_{1}^{2}}\right),\\\
\rm\Pi_{2}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{2}}{m_{2}^{2}}\right),\\\
\rm\Pi_{3}&=\rm-\frac{2\eta^{3}}{N}\left(\frac{\dot{m}_{3}}{m_{3}^{2}}\right),\end{split}\qquad\begin{split}\dot{\eta}&=\frac{N}{12\eta}\Pi_{\eta},\\\
\dot{\phi}_{1}&=\rm\frac{N}{\eta^{3}\triangle}\left(-m^{22}\Pi_{\phi_{1}}+m^{12}\Pi_{\phi_{2}}\right),\\\
\dot{\phi}_{2}&=\frac{N}{\eta^{3}\Delta}\left(m^{12}\Pi_{\phi_{1}}-m^{11}\Pi_{\phi_{2}}\right),\\\
\dot{m}_{1}&=-\frac{Nm_{1}^{2}\Pi_{1}}{2\eta^{3}},\\\
\dot{m}_{2}&=-\frac{Nm_{2}^{2}\Pi_{2}}{2\eta^{3}},\\\
\dot{m}_{3}&=-\frac{Nm_{3}^{2}\Pi_{3}}{2\eta^{3}},\end{split}$ (99)

where $\rm\triangle=m^{11}m^{22}-(m^{12})^{2}$. Writing (98) in a canonical
form, i.e. $\mathcal{L}_{can}=\Pi_{q}\dot{q}-N\mathcal{H}$, we can perform the
variation of this canonical Lagrangian with respect to the lapse function $N$,
$\delta\mathcal{L}_{can}/\delta N=0$, resulting in the constraint
$\mathcal{H}=0$, and making the same transformation as in (44), the
Hamiltonian density is

$\rm{\cal
H}=\frac{e^{-3u}}{24}\left[\Pi_{u}^{2}-6P_{1}^{2}-6P_{2}^{2}-6P_{3}^{2}-\frac{12m^{22}}{\triangle}\Pi_{\phi_{1}}^{2}-\frac{12m^{11}}{\triangle}\Pi_{\phi_{2}}^{2}+\frac{24m^{12}}{\triangle}\Pi_{\phi_{1}}\Pi_{\phi_{2}}-24V_{1}e^{-\lambda_{1}\phi_{1}+6u}-24V_{2}e^{-\lambda_{2}\phi_{2}+6u}\right]\,.$
(100)

Proposing the following canonical transformation on the variables
$\rm(\eta,\phi_{1},\phi_{2},u_{i})\leftrightarrow(\xi_{1},\xi_{2},\xi_{3},u_{i})$

$\begin{split}\rm\xi_{1}&=-6u+\lambda_{1}\phi_{1},\\\
\xi_{2}&=-6u+\lambda_{2}\phi_{2},\\\
\xi_{3}&=-4u+\frac{\lambda_{1}}{6}\phi_{1}+\frac{\lambda_{2}}{6}\phi_{2},\\\
\rm u_{i}&=\rm
u_{i},\end{split}\quad\longleftrightarrow\quad\begin{split}u&=\rm\frac{\xi_{1}+\xi_{2}-6\xi_{3}}{12},\\\
\rm\phi_{1}&=\rm\frac{3\xi_{1}+\xi_{2}-6\xi_{3}}{2\lambda_{1}},\\\
\rm\phi_{2}&=\rm\frac{\xi_{1}+3\xi_{2}-6\xi_{3}}{2\lambda_{2}},\end{split}$
(101)

and setting the gauge $\rm N=24e^{3u}$, allows us to find a new set of
conjugate momenta $\rm(\pi_{1},\pi_{2},\pi_{3})$

$\displaystyle\rm\Pi_{u}$ $\displaystyle=\rm-6\pi_{1}-6\pi_{2}-4\pi_{3},$
$\displaystyle\rm\Pi_{\phi_{1}}$
$\displaystyle=\rm\lambda_{1}\pi_{1}+\frac{\lambda_{1}}{6}\pi_{3},$ (102)
$\displaystyle\rm\Pi_{\phi_{2}}$
$\displaystyle=\rm\lambda_{2}\pi_{2}+\frac{\lambda_{2}}{6}\pi_{3},$

which finally leads us to the Hamiltonian density

$\displaystyle\rm{\cal H}$ $\displaystyle=\rm
12\left(3-\frac{\lambda_{1}^{2}m^{22}}{\triangle}\right)\pi_{1}^{2}+12\left(3-\frac{\lambda_{2}^{2}\\*m^{11}}{\triangle}\right)\pi_{2}^{2}+\left(16+\frac{-\lambda_{1}^{2}m^{22}+2\lambda_{1}\lambda_{2}m^{12}-\lambda_{2}^{2}m^{11}}{3\triangle}\right)\pi_{3}^{2}$
$\displaystyle\rm+12\left[\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{1}^{2}m^{22}}{3\triangle}\right)\pi_{1}+\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{2}^{2}m^{11}}{3\triangle}\right)\pi_{2}\right]\pi_{3}$
$\displaystyle\rm-6P_{1}^{2}-6P_{2}^{2}-6P_{3}^{2}+24\left(3+\frac{\lambda_{1}\lambda_{2}m^{12}}{\triangle}\right)\pi_{1}\pi_{2}-24\left(V_{1}e^{-\xi_{1}}+V_{2}e^{-\xi_{2}}\right),$
(103)

the parameter $\rm\triangle$ is the same that was defined after equations
(99). The form that the Hamiltonian density (103) acquires after applying the
transformation (101) into (100) will, in the end, allows us to obtain the
solutions for this model. First, let’s compute Hamilton’s equations, which
read

$\displaystyle\rm\dot{\xi}_{1}$ $\displaystyle=$ $\displaystyle\rm
24\left(3-\frac{\lambda_{1}^{2}m^{22}}{\triangle}\right)\pi_{1}+24\left(3+\frac{\lambda_{1}\lambda_{2}m_{12}}{\triangle}\right)\pi_{2}+12\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{1}^{2}m^{22}}{3\triangle}\right)\pi_{3},$
$\displaystyle\rm\dot{\xi}_{2}$ $\displaystyle=$ $\displaystyle\rm
24\left(3-\frac{\lambda_{2}^{2}\\*m^{11}}{\triangle}\right)\pi_{2}+24\left(3+\frac{\lambda_{1}\lambda_{2}m_{12}}{\triangle}\right)\pi_{1}+12\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{2}^{2}m^{11}}{3\triangle}\right)\pi_{3},$
$\displaystyle\rm\dot{\xi}_{3}$ $\displaystyle=$ $\displaystyle\rm
12\left[\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{1}^{2}m^{22}}{3\triangle}\right)\pi_{1}+\left(4+\frac{\lambda_{1}\lambda_{2}m^{12}-\lambda_{2}^{2}m^{11}}{3\triangle}\right)\pi_{2}\right]+2\left(16+\frac{-\lambda_{1}^{2}m^{22}+2\lambda_{1}\lambda_{2}m^{12}-\lambda_{2}^{2}m^{11}}{3\triangle}\right)\pi_{3},$
$\displaystyle\rm\dot{\pi}_{1}$ $\displaystyle=$
$\displaystyle\rm-24V_{1}e^{-\xi_{1}},$ (104) $\displaystyle\rm\dot{\pi}_{2}$
$\displaystyle=$ $\displaystyle\rm-24V_{2}e^{-\xi_{2}},$
$\displaystyle\rm\dot{\pi}_{3}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\rm\dot{P}_{i}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\rm\dot{u}_{i}$ $\displaystyle=$ $\displaystyle-12P_{i},$

from this last set of equations is straightforward to see that
$\rm\pi_{3}=p_{3}$ and $\rm P_{i}=n_{i}$ are constants and the solutions to
$\rm u_{i}=u_{i_{0}}-12n_{i}\Delta t$. Taking the time derivative of the first
equation in (104), we obtain

$\rm\rm\ddot{\xi}_{1}=\rm-576V_{1}\left(3-\frac{\lambda_{1}^{2}m^{22}}{\triangle}\right)e^{-\xi_{1}}-576V_{2}\left(3+\frac{\lambda_{1}\lambda_{2}m^{12}}{\triangle}\right)e^{-\xi_{2}}.$
(105)

The main purpose of introducing the transformation (101) was to be able to
separate the set of equations arising from the Hamiltonian density (103). To
reach to a solution for our problem we set to zero the coefficient that is
multiplying the mixed momenta term in (103), which sets the following
constraint on the matrix element $\rm m^{12}$

$\rm
m^{12}=\frac{\lambda_{1}\lambda_{2}}{6}\left(1\pm\sqrt{1+36\frac{m^{11}m^{22}}{\lambda_{1}^{2}\lambda_{2}^{2}}}\right),$
(106)

which implies that the second term in the square root of (106) is a real
number, say $\ell=\rm 36(m^{11}m^{22}/\lambda_{1}^{2}\lambda_{2}^{2})$
$\in\mathbb{R}^{+}$, giving the same weight to the matrix elements $\rm
m^{11}$ and $\rm m^{22}$, whose values are $\rm
m^{11}=\frac{1}{6}\sqrt{\ell}\lambda_{1}^{2}$ and $\rm
m^{22}=\frac{1}{6}\sqrt{\ell}\lambda_{2}^{2}$. We are going to distinguish two
possible scenarios for $\rm m^{12}$ as: $\rm
m^{12}_{+}=\frac{1}{6}\lambda_{1}\lambda_{2}\left(1+\sqrt{1+\ell}\right)>0$
and $\rm
m^{12}_{-}=-\frac{1}{6}\lambda_{1}\lambda_{2}\left(\sqrt{1+\ell}-1\right)<0$.
This two choices of $\rm m^{12}$ enables us to have a quintom like case and
quintessence like case, respectively. With these two possible values for the
matrix element $\rm m^{12}$ we can see that
$\triangle_{+}=-\frac{1}{18}\lambda_{1}^{2}\lambda_{2}^{2}\left(1+\sqrt{1+\ell}\right)<0$
for $\rm m^{12}_{+}$ and
$\triangle_{-}=\frac{1}{18}\lambda_{1}^{2}\lambda_{2}^{2}\left(\sqrt{1+\ell}-1\right)>0$
for $\rm m^{12}_{-}$.

###  III.1 Quintom like case

We begin by analyzing the quintom like case, for which the matrix element $\rm
m^{12}_{-}=-\frac{1}{6}(\sqrt{1+\ell}-1)\lambda_{1}\lambda_{2}$, the
Hamiltonian density is rewritten as,

$\rm{\cal
H}=-\frac{\pi_{1}^{2}}{\mu_{{}_{\ell}}}-\frac{\pi_{2}^{2}}{\mu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\pi_{1}+\pi_{2}\right)\pi_{3}+\left(16-\frac{1}{18c_{{}_{\ell}}}\right)\pi_{3}^{2}-6\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}},$
(107)

also we have define the parameters
$\rm\mu_{{}_{\ell}}=\sqrt{\ell}/36\left(1+\sqrt{1+\ell}-\sqrt{\ell}\right)$
and $\rm
c_{{}_{\ell}}=\sqrt{\ell}/{36\left[\left(1+\sqrt{1+\ell}\right)+\sqrt{\ell}\right]}$.
Thus, Hamilton equations for the new simplified coordinate $\rm\xi_{i}$ are

$\displaystyle\rm\dot{\xi}_{1}$ $\displaystyle=$
$\displaystyle\rm-\frac{2\pi_{1}}{\mu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\pi_{3},$
$\displaystyle\rm\dot{\xi}_{2}$ $\displaystyle=$
$\displaystyle\rm-\frac{2\pi_{2}}{\mu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\pi_{3},$
(108) $\displaystyle\rm\dot{\xi}_{3}$ $\displaystyle=$
$\displaystyle\rm+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\pi_{1}+\pi_{2}\right)+2\left(16-\frac{1}{18c_{{}_{\ell}}}\right)\pi_{3},$

the equations for $\rm\dot{\pi}_{i}$ remain the same as in (104). Taking the
derivative of the first equation of (III.1) yields

$\rm\ddot{\xi}_{1}=\frac{48V_{1}}{\mu_{{}_{\ell}}}e^{-\xi_{1}},$ (109)

which has a solution of the form

$\rm
e^{-\xi_{1}}=\frac{\mu_{{}_{\ell}}r_{1}^{2}}{24V_{1}}\,Sech^{2}\left(r_{1}t-q_{1}\right).$
(110)

From (III.1) we can see that $\dot{\xi}_{2}$ has the same functional structure
as $\dot{\xi}_{1}$, therefore its solution will be of the same form as (110),
so we have

$\rm
e^{-\xi_{2}}=\frac{\mu_{{}_{\ell}}r_{2}^{2}}{24V_{2}}\,Sech^{2}\left(r_{2}t-q_{2}\right),$
(111)

where $\rm r_{i}$ and $\rm q_{i}$ (with $\rm i=1,2$) are integration
constants, both at (110) and (111). Reinserting these solutions into Hamilton
equations for the momenta, we obtain

$\displaystyle\rm\pi_{1}$ $\displaystyle=$
$\displaystyle\rm\alpha_{1}-\mu_{{}_{\ell}}\,r_{1}\,Tanh\left(r_{1}t-q_{1}\right),$
(112) $\displaystyle\rm\pi_{2}$ $\displaystyle=$
$\displaystyle\rm\alpha_{2}-\mu_{{}_{\ell}}\,r_{2}\,Tanh\left(r_{2}t-q_{2}\right).$
(113)

With (112) and (113), it can be easily check that the Hamiltonian is
identically null when

$\rm\alpha_{1}=\alpha_{2}=\frac{72\mu_{{}_{\ell}}-1}{6}p_{3},\qquad
p_{3}^{2}=\frac{\mu_{\ell}(r_{1}^{2}+r_{2}^{2})+6n^{2}}{4(72\mu_{{}_{\ell}}+1)},\qquad
n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2},$ (114)

where $\rm n^{2}$ belongs to the contribution on the anisotropic functions.
Now we are in position write the solutions for the $\rm\xi_{i}$ coordinates,
which read

$\displaystyle\rm\xi_{1}$
$\displaystyle=\rm\beta_{1}+Ln\left[Cosh^{2}\left(r_{1}t-q_{1}\right)\right],$
(115) $\displaystyle\xi_{2}$
$\displaystyle=\rm\beta_{2}+Ln\left[Cosh^{2}\left(r_{2}t-q_{2}\right)\right],$
(116) $\displaystyle\rm\xi_{3}$
$\displaystyle=\rm\beta_{3}+p_{3}\left[16\left(1+72\mu_{{}_{\ell}}\right)-8\frac{\mu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t-\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\mu_{{}_{\ell}}\,Ln\,\left[Cosh\left(r_{1}t-q_{1}\right)\,Cosh\left(r_{2}t-q_{2}\right)\right],$
(117)

here the $\rm\beta_{i}$, with $\rm i=1,2,3$, terms are constants coming from
integration. Applying the inverse canonical transformation we obtain the
solutions in the original variables $\rm(\eta,\phi_{1},\phi_{2})$ as

$\begin{split}\rm\eta&=\rm\eta_{0}+\frac{1}{12}Ln\left[Cosh^{2}\left(r_{1}t-q_{1}\right)Cosh^{2}\left(r_{2}t-q_{2}\right)\right]-\frac{1}{2}p_{3}\left[16\left(1+72\mu_{{}_{\ell}}\right)-8\frac{\mu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t\\\
&\rm+\frac{1}{2}\mu_{\ell}\,\left(48-\frac{1}{3c_{{}_{\ell}}}\right)Ln\,\left[Cosh\left(r_{1}t-q_{1}\right)\,Cosh\left(r_{2}t-q_{2}\right)\right],\\\
\rm\phi_{1}&=\rm\phi_{10}+\frac{1}{2\lambda_{1}}Ln\left[Cosh^{6}\left(r_{1}t-q_{1}\right)Cosh^{2}\left(r_{2}t-q_{2}\right)\right]-\frac{3}{\lambda_{1}}p_{3}\left[16\left(1+72\mu_{{}_{\ell}}\right)-8\frac{\mu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t\\\
&\phantom{{}={}}\rm+\frac{3}{\lambda_{1}}\mu_{\ell}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)Ln\,\left[Cosh\left(r_{1}t-q_{1}\right)\,Cosh\left(r_{2}t-q_{2}\right)\right],\\\
\rm\phi_{2}&=\rm\phi_{20}+\frac{1}{2\lambda_{2}}Ln\left[Cosh^{2}\left(r_{1}t-q_{1}\right)Cosh^{6}\left(r_{2}t-q_{2}\right)\right]-\frac{3}{\lambda_{2}}p_{3}\left[16\left(1+72\mu_{{}_{\ell}}\right)-8\frac{\mu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t\\\
&\phantom{{}={}}+\frac{3}{\lambda_{2}}\mu_{\ell}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\rm
Ln\,\left[Cosh\left(r_{1}t-q_{1}\right)\,Cosh\left(r_{2}t-q_{2}\right)\right],\end{split}$
(118)

where $\eta_{0},\phi_{10}$ and $\phi_{20}$ are given in terms of the
$\beta_{i}$ constants as

$\eta_{0}=\frac{\beta_{1}+\beta_{2}-6\beta_{3}}{12},\quad\phi_{10}=\frac{3\beta_{1}+\beta_{2}-6\beta_{3}}{2\lambda_{1}},\quad\phi_{20}=\frac{\beta_{1}+3\beta_{2}-6\beta_{3}}{2\lambda_{2}}.$
(119)

###  III.2 Quintessence like case

Now we turn our attention to the quintessence like case, for which the matrix
element $\rm
m^{12}_{+}=\frac{1}{6}\left(1+\sqrt{1+\ell}\right)\lambda_{1}\lambda_{2}$,
then the Hamiltonian density describing this quintessence model is rewritten
as

$\rm{\cal
H}=\frac{\pi_{1}^{2}}{\nu_{{}_{\ell}}}+\frac{\pi_{2}^{2}}{\nu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\pi_{1}+\pi_{2}\right)\pi_{3}+\left(16-\frac{1}{18c_{{}_{\ell}}}\right)\pi_{3}^{2}-6\left(P_{1}^{2}+P_{2}^{2}+P_{3}^{2}\right)-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}},$
(120)

here we define the parameter
$\rm\nu_{{}_{\ell}}=\sqrt{\ell}/{36\left(\sqrt{1+\ell}+\sqrt{\ell}-1\right)}$
and $\rm
c_{{}_{\ell}}=\sqrt{\ell}/{36\left(+\sqrt{\ell}+1-\sqrt{1+\ell}\right)}$.

From (120) we can calculate Hamilton equations for the phase space spanned by
$\rm(\xi_{i},\pi_{i})$, given by

$\displaystyle\rm\dot{\xi}_{1}$ $\displaystyle=$
$\displaystyle\rm\frac{2\pi_{1}}{\nu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\pi_{3},$
$\displaystyle\rm\dot{\xi}_{2}$ $\displaystyle=$
$\displaystyle\rm\frac{2\pi_{2}}{\nu_{{}_{\ell}}}+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\pi_{3},$
(121) $\displaystyle\rm\dot{\xi}_{3}$ $\displaystyle=$
$\displaystyle\rm\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\pi_{1}+\pi_{2}\right)+2\left(16-\frac{1}{18c_{{}_{\ell}}}\right)\pi_{3},$
$\displaystyle\rm P_{i}$ $\displaystyle=$ $\displaystyle\rm n_{i}=constant,$

as in the quintom case $\rm\dot{\pi}_{i}$ remain the same as in (104).
Proceeding in a similar way as in the previous case, we take the derivative of
the first equation in (121), obtaining

$\rm\ddot{\xi}_{1}=-\frac{48V_{1}}{\nu_{{}_{\ell}}}e^{-\xi_{1}},$ (122)

which the corresponding solution is

$\rm
e^{-\xi_{1}}=\frac{\nu_{{}_{\ell}}r_{1}^{2}}{24V_{1}}\,Csch^{2}\left(r_{1}t-q_{1}\right).$
(123)

Also in this quintessence like setting, the $\dot{\xi}_{2}$ functional form is
the same as $\dot{\xi}_{1}$, indicating that the solution is of the same type
as (123), that is

$\rm
e^{-\xi_{2}}=\frac{\nu_{{}_{\ell}}r_{2}^{2}}{24V_{2}}\,Csch^{2}\left(r_{2}t-q_{2}\right),$
(124)

in (123) and (124) the $\rm r_{i}$ and $\rm q_{i}$ (with $\rm i=1,2$) are
constants coming from integration. With (123) and (124) at hand, we can
reinsert them into Hamilton equations for the momenta, giving

$\displaystyle\rm\pi_{1}$ $\displaystyle=$ $\displaystyle\rm-
a_{1}+\nu_{{}_{\ell}}\,r_{1}\,Coth\left(r_{1}t-q_{1}\right),$ (125)
$\displaystyle\rm\pi_{2}$ $\displaystyle=$ $\displaystyle\rm-
a_{2}+\nu_{{}_{\ell}}\,r_{2}\,Coth\left(r_{2}t-q_{2}\right),$ (126)

where it can be easily verify that with (125) and (126) at hand the
Hamiltonian is identically zero when

$\rm a_{1}=a_{2}=\frac{72\nu_{{}_{\ell}}+1}{6}p_{3},\qquad
p_{3}^{2}=\frac{\nu_{{}_{\ell}}(r_{1}^{2}+r_{2}^{2})+6n^{2}}{4(72\nu_{{}_{\ell}}-1)},\qquad
n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}.$ (127)

So, the solutions for the $\rm\xi_{i}$ coordinates become

$\displaystyle\rm\xi_{1}$
$\displaystyle=\rm\beta_{1}+Ln\left[Sinh^{2}\left(r_{1}t-q_{1}\right)\right],$
(128) $\displaystyle\xi_{2}$
$\displaystyle=\rm\beta_{2}+Ln\left[Sinh^{2}\left(r_{2}t-q_{2}\right)\right],$
(129) $\displaystyle\rm\xi_{3}$
$\displaystyle=\rm\beta_{3}-p_{3}\left[16\left(72\nu_{{}_{\ell}}-1\right)-8\frac{\nu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t+\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\,\nu_{{}_{\ell}}Ln\,\left[Sinh\left(r_{1}t-q_{1}\right)\,Sinh\left(r_{2}t-q_{2}\right)\right],$
(130)

where $\rm\beta_{i}$ are integration constants. After applying the inverse
canonical transformation we get the solutions in terms of the original
variables $\rm(\Omega,\phi_{1},\phi_{2})$ as

$\begin{split}\rm\eta&=\rm\eta_{0}+\frac{1}{12}Ln\left[Sinh^{2}\left(r_{1}t-q_{1}\right)Sinh^{2}\left(r_{2}t-q_{2}\right)\right]+\frac{1}{2}p_{3}\left[16\left(72\nu_{\ell}-1\right)-8\frac{\nu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t\\\
&\rm-\frac{1}{2}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\,\nu_{\ell}Ln\,\left[Sinh\left(r_{1}t-q_{1}\right)\,Sinh\left(r_{2}t-q_{2}\right)\right],\\\
\rm\phi_{1}&=\rm\phi_{10}+\frac{1}{2\lambda_{1}}\biggl{[}Ln\left[Sinh^{6}\left(r_{1}t-q_{1}\right)Sinh^{2}\left(r_{2}t-q_{2}\right)\right]-6\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\nu_{\ell}\times\\\
&\phantom{{}={}}\rm
Ln\,\left[Sinh\left(r_{1}t-q_{1}\right)\,Sinh\left(r_{2}t-q_{2}\right)\right]\biggr{]}\rm+\frac{3}{\lambda_{1}}p_{3}\left[16\left(72\nu_{\ell}-1\right)-8\frac{\nu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t,\\\
\rm\phi_{2}&=\rm\phi_{20}+\frac{1}{2\lambda_{2}}\biggl{[}Ln\left[Sinh^{2}\left(r_{1}t-q_{1}\right)Sinh^{6}\left(r_{2}t-q_{2}\right)\right]-6\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\nu_{\ell}\times\\\
&\phantom{{}={}}\rm
Ln\,\left[Sinh\left(r_{1}t-q_{1}\right)\,Sinh\left(r_{2}t-q_{2}\right)\right]\biggr{]}\rm+\frac{3}{\lambda_{2}}p_{3}\left[16\left(72\nu_{\ell}-1\right)-8\frac{\nu_{{}_{\ell}}}{c_{{}_{\ell}}}\right]\Delta
t,\end{split}$ (131)

where $\eta_{0},\phi_{10}$ and $\phi_{20}$ are given in terms of the
$\beta_{i}$ constants as

$\displaystyle\eta_{0}=\frac{\beta_{1}+\beta_{2}-6\beta_{3}}{12},\quad\phi_{10}=\frac{3\beta_{1}+\beta_{2}-6\beta_{3}}{2\lambda_{1}},\quad\phi_{20}=\frac{\beta_{1}+3\beta_{2}-6\beta_{3}}{2\lambda_{2}}.$
(132)

It is clear that the standard quintessence model with two scalar fields cannot
be reproduce under this approach, because when we set $\rm m^{12}=0$, this
imply that parameter $\ell$ is equal to zero, then, the matrix elements $\rm
m^{11}=m^{22}$ are zero too, this was the challenge to resolve.

##  IV Quantum Approach

On the Wheeler-DeWitt (WDW) equation there are a lot of papers dealing with
different problems, for example in Gibbons & Gishchuk, (1989), they asked the
question of what a typical wave function for the universe is. In Ref. Zhi,
(1987) there appears an excellent summary of a paper on quantum cosmology
where the problem of how the universe emerged from big bang singularity can no
longer be neglected in the GUT epoch. On the other hand, the best candidates
for quantum solutions become those that have a damping behavior with respect
to the scale factor, since these allow to obtain good classical solutions when
using the WKB approximation for any scenario in the evolution of our universe
Hartle & Hawking, (1983); Hawking, (1984).

In this section we present the quantum version of the classical anisotropic
cosmological models studied above along with its solutions. Since we already
have the classical Hamiltonian density, the quantum counterpart can be
obtained making the usual replacement
$\rm\Pi_{q^{\mu}}=-i\hbar\partial_{q^{\mu}}$. First we modified the classical
Hamiltonian density (44) in order to consider the factor ordering problem
between the function $\rm e^{-3\Omega}$ and its moment $\rm\pi_{u}$,
introducing the linear term as $\rm e^{-3u}\pi_{u}^{2}\to
e^{-3u}\left[\pi_{u}^{2}+Qi\hbar\pi_{u}\right]$ where Q is a real number that
measure the ambiguity in the factor ordering.

###  IV.1 Quantum Anisotropic Quintessence-K-essence Model

The quantum version for the first cosmological model we employ the modified
Hamiltonian density,

$\rm{\cal
H}=\pi_{u}^{2}+Qi\hbar\pi_{u}-12\Pi_{\phi_{1}}^{2}-12\Pi_{\phi_{2}}^{2}-6P_{1}^{2}-6P_{2}^{2}-6P_{3}^{2}-24V_{1}e^{6u-\lambda_{1}\phi_{1}}\,,$
(133)

in order to obtain the WDW equation, we implement the following change of
variables
$\rm(u,\phi_{1},\phi_{2},u_{i})\leftrightarrow(\xi_{1},\xi_{2},\xi_{3})$

$\begin{split}\rm\xi_{1}&=\rm 6u-\lambda_{1}\phi_{1},\\\ \xi_{2}&=\rm u,\\\
\xi_{3}&=\phi_{2},\end{split}\qquad\longleftrightarrow\qquad\begin{split}\rm
u&=\rm\xi_{2},\\\ \phi_{1}&=\frac{-\xi_{1}+6\xi_{2}}{\lambda_{1}},\\\
\phi_{2}&=\xi_{3},\\\ \rm u_{i}&=\rm u_{i},\end{split}$ (134)

where $\rm u_{i}$ are the conjugate coordinate to momenta $\rm P_{i}$, and
also, obtaining a new set of conjugate momenta (in the same manner as (42)),
of the variables $\rm(\xi_{1},\xi_{2},\xi_{3})$, namely
$\rm(\pi_{1},\pi_{2},\pi_{3})$, which read

$\rm\pi_{u}=6\pi_{1}+\pi_{2},\qquad\Pi_{\phi_{1}}=-\lambda_{1}\pi_{1},\qquad\Pi_{\phi_{2}}=\pi_{3},$
(135)

which in turn transforms the Hamiltonian density (133) as

$\rm{\cal H}=\rm
12\left(3-\lambda_{1}^{2}\right)\pi_{1}^{2}+\pi_{2}^{2}+12\pi_{1}\pi_{2}-12\pi_{3}^{2}+i\hbar
Q(6\pi_{1}+\pi_{2})-6P_{1}^{2}-6P_{2}^{2}-6P_{3}^{2}-24V_{1}e^{\xi_{1}}.$
(136)

Introducing the replacement $\rm\pi_{q^{\mu}}=-i\hbar\partial_{q^{\mu}}$, the
WDW equation becomes

$\begin{split}\rm{\cal
H}\Psi=-12\hbar^{2}\left(3-\lambda_{1}^{2}\right)\frac{\partial^{2}\Psi}{\partial\xi_{1}^{2}}-\hbar^{2}\frac{\partial^{2}\Psi}{\partial\xi_{2}^{2}}-12\hbar^{2}\frac{\partial^{2}\Psi}{\partial\xi_{1}\partial\xi_{2}}+12\hbar^{2}\frac{\partial^{2}\Psi}{\partial\xi_{3}^{2}}+Q\hbar^{2}\left(6\frac{\partial\Psi}{\partial\xi_{1}}+\frac{\partial\Psi}{\xi_{2}}\right)\\\
+6\hbar^{2}\left(\frac{\partial^{2}\Psi}{\partial
u_{1}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{2}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{3}^{3}}\right)-24V_{1}e^{\xi_{1}}\Psi=0,\end{split}$ (137)

due that the scalar potential does not depend on the coordinates
$\rm(\xi_{2},\xi_{3},u_{i})$, we propose the following ansatz for the wave
function
$\rm\Psi(\xi_{1},\xi_{2},\xi_{3},u_{i})=e^{-(a_{2}\xi_{2}+a_{3}\xi_{3}+a_{4}u_{1}+a_{5}u_{2}+a_{6}u_{3})/\hbar}G(\xi_{1})$
where the $\rm a_{i}$ are arbitrary constants. Introducing the mentioned
ansatz in (137) we have that

$\rm-12\hbar^{2}\left(3-\lambda_{1}^{2}\right)\frac{1}{G}\frac{d^{2}G}{d\xi_{1}^{2}}+6\hbar\left(2a_{2}+\hbar
Q\right)\frac{1}{G}\frac{dG}{d\xi_{1}}-a_{2}(a_{2}+\hbar
Q)+12a_{3}^{2}+6a_{0}^{2}-24V_{1}e^{\xi_{1}}=0,$

identifying $a_{0}^{2}=a_{4}^{2}+a_{5}^{2}+a_{6}^{2}$, where we also divided
the whole equation by the ansatz; this in turn leads us to the following
differential equation

$\rm\frac{d^{2}G}{d\xi_{1}^{2}}-\frac{2a_{2}+\hbar
Q}{2\hbar(3-\lambda_{1}^{2})}\frac{dG}{d\xi_{1}}+\frac{1}{12\hbar^{2}(3-\lambda_{1}^{2})}\left[24V_{1}e^{\xi_{1}}+\eta\right]G=0,$
(138)

here $\eta=a_{2}(a_{2}+\hbar\rm Q)-12a_{3}^{2}-6a_{0}^{2}$. The last equation
can be casted as $\rm y^{\prime\prime}+ay^{\prime}+\left(be^{\kappa
x}+c\right)y=0$ (and whose solutions will depend on the value of
$\lambda_{1}$) polyanin , where

$\rm
y=Exp\left({-\frac{ax}{2}}\right)Z_{\nu}\left(\frac{2\sqrt{b}}{\kappa}e^{\frac{\kappa
x}{2}}\right),$ (139)

here $\rm Z_{\nu}$ is the Bessel function and $\nu=\sqrt{a^{2}-4c}/\kappa$
being the order. The corresponding relations between the coefficients of (138)
and $\rm a,b,c$ and $\kappa$ are

$\rm a=\rm\left\\{\begin{tabular}[]{lr}$\rm\frac{2a_{2}+\hbar
Q}{2\hbar(\lambda_{1}^{2}-3)},$&when $\lambda_{1}^{2}>3$\\\ \\\
$\rm-\frac{2a_{2}+\hbar Q}{2\hbar(3-\lambda_{1}^{2})},$&when
$\lambda_{1}^{2}<3$\end{tabular}\right.$ (140) $\rm
b=\rm\left\\{\begin{tabular}[]{ll}$\rm-\frac{2V_{0}}{\hbar^{2}(\lambda^{2}-3)},$&when
\,\,$\lambda_{1}^{2}>3$\\\ \\\
$\rm\frac{2V_{0}}{\hbar^{2}(3-\lambda^{2})},$&when
\,\,$\lambda_{1}^{2}<3$\end{tabular}\right.$ (141) $\rm
c=\rm\left\\{\begin{tabular}[]{ll}$\rm-\frac{\eta}{12\hbar^{2}\left(\lambda_{1}^{2}-3\right)}$,&when
\,\,$\lambda_{1}^{2}>3$\\\ \\\
$\rm\frac{\eta}{12\hbar^{2}\left(3-\lambda_{1}^{2}\right)}$,&when
\,\,$\lambda_{1}^{2}<3$\end{tabular}\right.$ (142) $\kappa=1,$ (143)

according to the constant b, the solution to the function G becomes

$\displaystyle\rm G(\xi_{1})=\rm Exp\left(-\frac{2a_{2}+\hbar
Q}{4\hbar(\lambda_{1}^{2}-3)}\xi_{1}\right)\,\,\,K_{\nu_{1}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{\lambda^{2}-3}}\,\,e^{\frac{\xi_{1}}{2}}\right),\qquad\lambda_{1}^{2}>3$
(144) $\displaystyle\rm G(\xi_{1})=\rm Exp\left(\frac{2a_{2}+\hbar
Q}{4\hbar(3-\lambda_{1}^{2})\xi_{1}}\right)\,\,\,J_{\nu_{2}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{3-\lambda_{1}^{2}}}\,\,e^{\frac{\xi_{1}}{2}}\right),\qquad\lambda_{1}^{2}<3$
(145)

and the wavefunction takes the form

$\displaystyle\rm\Psi_{\nu_{1}}=\rm Exp\left(-\frac{2a_{2}+\hbar
Q}{4\hbar(\lambda_{1}^{2}-3)}\xi_{1}-\frac{a_{2}\xi_{2}+a_{3}\xi_{3}}{\hbar}-\frac{a_{4}u_{1}+a_{4}u_{2}+a_{6}u_{3}}{\hbar}\right)\,\,\,K_{\nu_{1}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{\lambda_{1}^{2}-3}}\,\,e^{\frac{\xi_{1}}{2}}\right),\qquad\lambda_{1}^{2}>3$
(146) $\displaystyle\rm\Psi_{\nu_{2}}=\rm Exp\left(\frac{2a_{2}+\hbar
Q}{4\hbar(3-\lambda_{1}^{2})}\xi_{1}-\frac{a_{2}\xi_{2}+a_{3}\xi_{3}}{\hbar}-\frac{a_{4}u_{1}+a_{4}u_{2}+a_{6}u_{3}}{\hbar}\right)\,\,\,J_{\nu_{2}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{3-\lambda^{2}}}\,\,e^{\frac{\xi_{1}}{2}}\right),\qquad\lambda_{1}^{2}<3.$
(147)

where $\rm\nu_{1}=\sqrt{\left(-\frac{2a_{2}+\hbar
Q}{4\hbar(\lambda_{1}^{2}-3)}\right)^{2}+\frac{4\eta}{12\hbar^{2}(\lambda_{1}^{2}-3)}}$
and $\rm\nu_{2}=\sqrt{\left(\frac{2a_{2}+\hbar
Q}{4\hbar(3-\lambda_{1}^{2})}\right)^{2}-\frac{4\eta}{12\hbar^{2}(3-\lambda_{1}^{2})}}$
are the corresponding orders of the wave function. Applying the inverse
transformation on the variables $\rm\xi_{i}$, we can write the wave function
in terms of the original variables
$\rm(A=e^{\Omega},\phi_{i},m_{i}=e^{u_{i}})$, which read

$\displaystyle\rm\Psi_{\nu_{1}}=\rm
m_{1}^{-\frac{a_{4}}{\hbar}}m_{2}^{-\frac{a_{5}}{\hbar}}m_{3}^{-\frac{a_{6}}{\hbar}}A^{-\alpha_{1}}\,Exp\left(\frac{2a_{2}+\hbar
Q}{4\hbar(\lambda_{1}^{2}-3)}\lambda_{1}\phi_{1}-\frac{a_{3}}{\hbar}\phi_{2}\right)\,\,\,K_{\nu_{1}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{\lambda_{1}^{2}-3}}\,\,A^{3}e^{\frac{\lambda_{1}}{2}\phi_{1}}\right),\qquad\lambda_{1}^{2}>3$
(148) $\displaystyle\rm\Psi_{\nu_{2}}=\rm
m_{1}^{-\frac{a_{4}}{\hbar}}m_{2}^{-\frac{a_{5}}{\hbar}}m_{3}^{-\frac{a_{6}}{\hbar}}A^{-\alpha_{2}}\,Exp\left(-\frac{2a_{2}+\hbar
Q}{4\hbar(\lambda_{1}^{2}-3)}\lambda_{1}\phi_{1}-\frac{a_{3}}{\hbar}\phi_{2}\right)\,\,\,J_{\nu_{2}}\left(\frac{2}{\hbar}\sqrt{\frac{2V_{0}}{3-\lambda^{2}}}\,\,A^{3}e^{\frac{\lambda_{1}}{2}\phi_{1}}\right),\qquad\lambda_{1}^{2}<3.$
(149)

with $\alpha_{1}=\frac{1}{\hbar}\left(a_{2}+\frac{3}{2}\frac{2a_{2}+\hbar
Q}{\lambda_{1}^{2}-3}\right)$ and
$\alpha_{2}=\frac{1}{\hbar}\left(a_{2}-\frac{3}{2}\frac{2a_{2}+\hbar
Q}{3-\lambda_{1}^{2}}\right)$.

Figure 2: Behavior of the probability density for $\lambda_{1}<\sqrt{3}$, for
$\rm Q=1$, $\lambda_{1}$, $a_{2}=0.6$, $a_{3}=1$ and $a_{4}=a_{5}=a_{6}=0.3$.

In Fig(2) we can see the behavior of the probability density of the wave
function for the solution $\lambda_{1}<\sqrt{3}$. It is observed that the
evolution of the wave function with respect of the scale factor is damped,
which is a good characteristic and this kind of behavior also have been
reported in sor ; Socorro:2020nsm ; Socorro:2018amv . In comparison with
isotropic model Socorro:2020nsm , we can see that the anisotropies shrink the
probability density of the wave function.

Figure 3: Behavior of the probability density for $\lambda_{1}>\sqrt{3}$. For
both figures $\lambda_{1}=6$, $a_{2}=2$, $a_{3}=1$, $a_{4}=a_{5}=a_{6}=0.3$
whereas for the left figure $\rm Q=-2$ and for the right figure $\rm Q=-10$.

In Fig.(3) we can observed the evolution of the wave function for the solution
$\lambda_{1}>\sqrt{3}$. In this particular case the values of $\rm Q$ act as a
retarder (for negative values) for the wave function and compresses the length
over the axis were the scalar field evolves (this should also retard the
inflation epoch), but still having the damped behavior. Contrasting this
results with those of the isotropic treatment Socorro:2020nsm , we can see
that the anisotropies shrink the the probability density of the wave function
along the evolution of the scalar field.

Finally, for the particular case of $\lambda_{1}=\sqrt{3}$ the quantum
solution for the function $\rm G(\xi_{1})$ becomes

$\rm G(\xi_{1})=G_{0}Exp\left[\frac{\eta}{6\hbar(2a_{2}+\hbar
Q)}\xi_{1}\right]\,Exp\left(\frac{4V_{1}}{\hbar(2a_{2}+\hbar
Q)}e^{\xi_{1}}\right),$

and the wave function is

$\rm\Psi(A,\phi_{i},m_{i})=\Psi_{0}m_{1}^{-\frac{a_{4}}{\hbar}}m_{2}^{-\frac{a_{5}}{\hbar}}m_{3}^{-\frac{a_{6}}{\hbar}}\,A^{r}\,Exp\left(-\frac{a_{3}}{\hbar}\phi_{2}-\frac{\lambda_{1}}{6\hbar(2a_{2}+\hbar
Q)}\phi_{1}\right)\,Exp\left[\frac{4V_{1}}{\hbar(2a_{2}+\hbar
Q)}A^{6}e^{-\lambda_{1}\phi_{1}})\right].$ (150)

where the constant $\rm r=-\frac{a_{2}}{\hbar}+\frac{\eta}{\hbar(2a_{2}+\hbar
Q)}$.

###  IV.2 Quantum Anisotropic Quintom Case

For the second cosmological model, the quintom like case, the quantum version
of this model is obtained applying, again, the recipe
$\rm\Pi_{q^{\mu}}=-i\hbar\partial_{q^{\mu}}$ to the Hamiltonian density (107),
hence

$\begin{split}\rm\biggl{[}\frac{\hbar^{2}}{\mu_{{}_{\ell}}}\frac{\partial^{2}}{\partial\xi_{1}^{2}}+\frac{\hbar^{2}}{\mu_{{}_{\ell}}}\frac{\partial^{2}}{\partial\xi_{2}^{2}}-\hbar^{2}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\frac{\partial^{2}}{\partial\xi_{3}\partial\xi_{1}}+\frac{\partial^{2}}{\partial\xi_{3}\partial\xi_{2}}\right)-\hbar^{2}\left(16-\frac{1}{18c_{{}_{\ell}}}\right)\frac{\partial^{2}}{\partial\xi_{3}^{2}}\\\
+6\hbar^{2}\left(\frac{\partial^{2}\Psi}{\partial
u_{1}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{2}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{3}^{2}}\right)-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}}\biggr{]}\Psi=0,\end{split}$
(151)

because the scalar potential does not depend on the coordinate $\rm\xi_{3}$,
we propose the following ansatz for the wave function
$\rm\Psi(\xi_{1},\xi_{2},\xi_{3})=e^{(a_{3}\xi_{3}+a_{4}u_{1}+a_{4}u_{2}+a_{6}u_{3})/\hbar}\,{\cal
A}(\xi_{1}){\cal B}(\xi_{2})$ where $\rm a_{i}$ (with $\rm i=3,4,5,6$) are an
arbitrary constants. Substituting and dividing by the ansatz in (151), we
obtain

$\rm\frac{\hbar^{2}}{\mu_{{}_{\ell}}{\cal A}}\frac{d^{2}{\cal
A}}{d\xi_{1}^{2}}+\frac{\hbar^{2}}{\mu_{{}_{\ell}}{\cal B}}\frac{d^{2}{\cal
B}}{d\xi_{2}^{2}}-a_{3}\hbar\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\frac{1}{{\cal
A}}\frac{d{\cal A}}{d\xi_{1}}+\frac{1}{{\cal B}}\frac{d{\cal
B}}{d\xi_{2}}\right)-a_{3}^{2}\left(16-\frac{1}{18c_{{}_{\ell}}}\right)+6a_{0}^{2}-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}}=0,$
(152)

with $\rm a_{0}^{2}=a_{4}^{2}+a_{5}^{2}+a_{6}^{2}$, where we can separate the
equations as

$\displaystyle\rm\frac{d^{2}{\cal
A}}{d\xi_{1}^{2}}-\frac{a_{3}\mu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\frac{d{\cal
A}}{d\xi_{1}}-\frac{\mu_{{}_{\ell}}}{\hbar^{2}}\left(\frac{a_{3}^{2}}{2}\left(16-\frac{1}{18c_{{}_{\ell}}}\right)-3a_{0}^{2}-\alpha^{2}+24V_{1}e^{-\xi_{1}}\right){\cal
A}=0,$ (153) $\displaystyle\rm\frac{d^{2}{\cal
B}}{d\xi_{2}^{2}}-\frac{a_{3}\mu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\frac{d{\cal
B}}{d\xi_{2}}-\frac{\mu_{{}_{\ell}}}{\hbar^{2}}\left(\frac{a_{3}^{2}}{2}\left(16-\frac{1}{18c_{{}_{\ell}}}\right)-3a_{0}^{2}+\alpha^{2}+24V_{2}e^{-\xi_{2}}\right){\cal
B}=0,$ (154)

with $\alpha^{2}$ being the separation constant. The corresponding solutions
of (153) and (154) have the following form polyanin

$\rm
Y(x)=Exp\left({-\frac{ax}{2}}\right)Z_{\nu}\left(\frac{2\sqrt{b}}{\lambda}e^{\frac{\lambda
x}{2}}\right),$ (155)

here $\rm Z_{\nu}$ are the generic Bessel function with order
$\rm\nu=\sqrt{a^{2}-4c}/\lambda$. If $\sqrt{b}$ is real, $\rm Z_{\nu}$ are the
ordinary Bessel function, otherwise the solution will be given by the modified
Bessel function. Making the following identifications

$\displaystyle\rm\lambda$ $\displaystyle=$ $\displaystyle-1,$ (156)
$\displaystyle\rm a$ $\displaystyle=$
$\displaystyle-\frac{a_{3}\mu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right),$
(157) $\displaystyle\rm b_{1,2}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{{}_{\ell}}}{\hbar^{2}}24V_{1,2},$ (158)
$\displaystyle\rm c_{\mp}$ $\displaystyle=$
$\displaystyle-\frac{\mu_{{}_{\ell}}}{\hbar^{2}}\left(a_{3}^{2}\left(8-\frac{1}{36c_{{}_{\ell}}}\right)-3a_{0}^{2}\mp\alpha^{2}\right),$
(159) $\displaystyle\rm\nu_{\mp}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\rm a^{2}}{\mu_{\ell}}+4\rm c_{\mp}},$ (160)

we can check that the value for $\sqrt{b}$ is imaginary, which as already
mentioned, gives a solution in terms of the modified Bessel function $\rm
Z_{\nu}=K_{\nu}$ whose order lies in the reals. Thus, the wave function is

$\begin{split}\rm\Psi_{\nu_{\pm}}=Exp\left[\left(\frac{\mu_{{}_{\ell}}}{2\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)(\xi_{1}+\xi_{2})+\frac{\xi_{3}}{\hbar}+\frac{a_{4}u_{1}+a_{5}u_{2}+a_{6}u_{3}}{\hbar}\right)a_{3}\right]\rm
K_{\nu_{-}}\left(\frac{4}{\hbar}\sqrt{6V_{1}\mu_{{}_{\ell}}}e^{-\frac{\xi_{1}}{2}}\right)\times\\\
K_{\nu_{+}}\left(\frac{4}{\hbar}\sqrt{6V_{2}\mu_{{}_{\ell}}}e^{-\frac{\xi_{2}}{2}}\right).\end{split}$
(161)

###  IV.3 Quantum Anisotropic Quintessence Case

Lastly, we are going to consider the quantum version of the anisotropic
quintessence like case. As in the previous two subsections, what we want is to
obtain an equation of the form $\rm{\cal H}\Psi(\xi_{i})=0$, to achieve this
we introduce the standard prescription
$\Pi_{q}^{\mu}=-i\hbar\partial_{q^{\mu}}$ in (120), obtaining

$\displaystyle\rm\left[-\frac{\hbar^{2}}{\nu_{{}_{\ell}}}\frac{\partial^{2}}{\partial\xi_{1}^{2}}-\frac{\hbar^{2}}{\nu_{{}_{\ell}}}\frac{\partial^{2}}{\partial\xi_{2}^{2}}-\hbar^{2}\left(24+\frac{1}{3\nu_{{}_{\ell}}}\right)\left(\frac{\partial^{2}}{\partial\xi_{3}\partial\xi_{1}}+\frac{\partial^{2}}{\partial\xi_{3}\partial\xi_{2}}\right)-\hbar^{2}\left(12+\frac{1}{18\nu_{{}_{\ell}}}\right)\frac{\partial^{2}}{\partial\xi_{3}^{2}}\right.$
$\displaystyle\left.\rm+6\hbar^{2}\left(\frac{\partial^{2}\Psi}{\partial
u_{1}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{2}^{2}}+\frac{\partial^{2}\Psi}{\partial
u_{3}^{2}}\right)-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}}\right]\Psi=0,$ (162)

we can see that the scalar potential does not depend on the coordinates
$\rm\xi_{3},u_{i}$, consequently we propose the following ansatz for the wave
function
$\rm\Psi(\xi_{1},\xi_{2},\xi_{3})=e^{(b_{3}\xi_{3}+b_{4}u_{1}+b_{5}u_{2}+b_{6}u_{3})/\hbar}\,{\cal
A}(\xi_{1}){\cal B}(\xi_{2})$ where $\rm b_{i}$ (with $\rm i=3,4,5,6$) are an
arbitrary constant. Applying and dividing by the ansatz in (162) we get

$\rm-\frac{\hbar^{2}}{\nu_{{}_{\ell}}{\cal A}}\frac{d^{2}{\cal
A}}{d\xi_{1}^{2}}-\frac{\hbar^{2}}{\nu_{{}_{\ell}}{\cal B}}\frac{d^{2}{\cal
B}}{d\xi_{2}^{2}}-b_{3}\hbar\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\left(\frac{1}{{\cal
A}}\frac{d{\cal A}}{d\xi_{1}}+\frac{1}{{\cal B}}\frac{d{\cal
B}}{d\xi_{2}}\right)-b_{3}^{2}\left(16-\frac{1}{18c_{{}_{\ell}}}\right)+6b_{0}^{2}-24V_{1}e^{-\xi_{1}}-24V_{2}e^{-\xi_{2}}=0,$
(163)

with $\rm b_{0}^{2}=b_{4}^{2}+b_{5}^{2}+b_{6}^{2}$, separating the equations
we have that

$\displaystyle\rm\frac{d^{2}{\cal
A}}{d\xi_{1}^{2}}+\frac{b_{3}\nu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\frac{d{\cal
A}}{d\xi_{1}}+\frac{\nu_{{}_{\ell}}}{\hbar^{2}}\left(b_{3}^{2}\left(8-\frac{1}{36c_{{}_{\ell}}}\right)-3b_{0}^{2}-\alpha^{2}+24V_{1}e^{-\xi_{1}}\right){\cal
A}=0,$ (164) $\displaystyle\rm\frac{d^{2}{\cal
B}}{d\xi_{2}^{2}}+\frac{b_{3}\nu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)\frac{d{\cal
B}}{d\xi_{2}}+\frac{\mu_{{}_{\ell}}}{\hbar^{2}}\left(b_{3}^{2}\left(8-\frac{1}{36c_{{}_{\ell}}}\right)-3b_{0}^{2}+\alpha^{2}+24V_{2}e^{-\xi_{2}}\right){\cal
B}=0,$ (165)

where $\alpha^{2}$ is the separation constant. These last two equations are
similar to the quantum quintom like case (153) and (154). Proceeding in a
similar fashion as the previous subsection (IV.2), we make the following
identifications

$\displaystyle\rm\lambda$ $\displaystyle=$ $\displaystyle-1,$ (166)
$\displaystyle\rm a$ $\displaystyle=$
$\displaystyle\frac{b_{3}\nu_{{}_{\ell}}}{\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right),$
(167) $\displaystyle\rm b_{1,2}$ $\displaystyle=$
$\displaystyle\frac{\nu_{{}_{\ell}}}{\hbar^{2}}24V_{1,2},$ (168)
$\displaystyle\rm c_{\mp}$ $\displaystyle=$
$\displaystyle\frac{\nu_{{}_{\ell}}}{\hbar^{2}}\left(b_{3}^{2}\left(8-\frac{1}{36c_{{}_{\ell}}}\right)-3a_{0}^{2}\mp\alpha^{2}\right),$
(169)

and conclude that the solutions are given by the ordinary Bessel function
$J_{\nu}$ with order $\rm\nu_{\mp}=\sqrt{(\rm a^{2}/\nu_{\ell})+4\rm
c_{\mp}}$. Thus, the wave function becomes

$\begin{split}\rm\Psi_{\nu_{\pm}}=Exp\left[\left(\frac{\nu_{{}_{\ell}}}{2\hbar}\left(48-\frac{1}{3c_{{}_{\ell}}}\right)(-\xi_{1}-\xi_{2})+\frac{\xi_{3}}{\hbar}\right)b_{3}+\frac{b_{4}u_{1}+b_{5}u_{2}+b_{6}u_{3}}{\hbar}\right]\rm
J_{\nu_{-}}\left(\frac{4}{\hbar}\sqrt{6V_{1}\nu_{{}_{\ell}}}e^{-\frac{\xi_{1}}{2}}\right)\times\\\
J_{\nu_{+}}\left(\frac{4}{\hbar}\sqrt{6V_{2}\nu_{{}_{\ell}}}e^{-\frac{\xi_{2}}{2}}\right).\end{split}$
(171)

##  V Final Remarks

In this work we have studied the anisotropic Bianchi type model in the chiral
cosmology setup in a twofold way. In the first model we consider two scalars
fields but only one potential term. In the second one, additionally to the two
scalar fields, we also consider both terms in the potential as well as the
standard kinetic energy and the mixed term. For both models we did a classical
and quantum treatment, obtaining exact analytical solutions for both
scenarios.

In the first model, which can be thought as a quintessence plus K-essence
model, our findings show that the volume of the Universe grows in a
accelerated manner for each of the three exact solutions that were found. This
feature can be seen from Fig.(1), where solutions for $\lambda_{1}<\sqrt{3}$
and $\lambda_{1}=\sqrt{3}$ have a similar behavior whereas the solution for
$\lambda>\sqrt{3}$ has a more faster evolution. After a certain amount of
time, the three solutions stabilized at the same value of -1. Also, because of
the ratio of the shear to scalar expansion bound: $\sigma/\theta\leq 0.3$
Pradhan:2010dm , we were able to constrain the value of the anisotropic
parameter $\rm\overline{A}_{m}$ for two of the solutions, however the
anisotropy continue, because for the cases $\lambda_{1}>\sqrt{3}$ and
$\lambda_{1}=\sqrt{3}$ the anisotropic parameter becomes
$\rm\overline{A}_{m}\leq 3.54$. In the quantum regime we were also able to
find exact solutions. For the particular case of $\lambda_{1}<\sqrt{3}$ we
found that wave function has a damped behavior as the scale factor evolves, as
can be seen in Fig.(2), this distinctive mark have also been reported in sor ;
Socorro:2020nsm ; Socorro:2018amv . In contrast with the isotropic treatment
Socorro:2020nsm , we found that the anisotropies shrink the probability
density of the wave function. For the solution $\lambda_{1}>\sqrt{3}$, it is
found that the damped behavior still exist, but the parameter $\rm Q$ acts as
a retarder (for negative values) for the wave function and the length over the
axis were the field evolves is compressed as shown in Fig.(3), signaling that
the inflation epoch should also be retarded in time. In this case the
anisotropies shrink the probability density along the evolution of the scalar
field. Finally, equation (150) depicts the quantum solution for the case
$\lambda_{1}=\sqrt{3}$.

For the second model under study, we consider both potential terms, in
addition to the two scalars fields. In this setup two possible avenues were
distinguished: a quintom one and a quintessence one. Classical exact solutions
for the former are given by equations (118), while the solutions for the
latter are given by equations (131). For the quantum counterpart exact
solutions were also obtained. The quantum solutions for both the anisotropic
quintom case and the anisotropic quintessence case are given in terms of an
exponential functions (that has the anisotropic information) multiplied by the
modified Bessel function $K_{\pm\nu}$ and the ordinary Bessel function
$J_{\pm\nu}$, as depicted in Eq.(161) and Eq.(171), respectively.

###### Acknowledgements.

This work was partially supported by PROMEP grants UGTO-CA-3. J.S. were
partially supported SNI-CONACYT. 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. Many calculations where done
by Symbolic Program REDUCE 3.8.

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